Volume 3
Issue 1

Meta-Level Abduction
Katsumi Inoue

1
7

Abduction and Beyond in Logic Programmingwith Application to Morality
Luís Moniz Pereira and Ari Saptawijaya

37

A Dynamic Approach to Peirce’s Interrogative
Construal of Abductive Logic
Minghui Ma and Ahti-Veikko Pietarinen

73

May 2016

Editorial Preface
Lorenzo Magnani

Contents

Course of Action Recommendations and Their
Place in Developmental Abduction
Donna E. West
123
Abduction: from the Ignorance Problem tothe Ignorance Virtue
Tommaso Bertolotti, Selene Arfiniand Lorenzo Magnani

151

Journal of Logics
and their Applications
Issue 1

May 2016

Special Issue
The IfColog Journal of Logics and their applications

105

The IfCoLog

Volume 3

Abduction and Beyond. Methodologicaland Computational Aspects of Creativity
in Natural Sciences
Andrés Rivadulla

ISSN PRINT 2055-3706
ISSN ONLINE 2055-3714

Frontiers of Abduction
Guest Editor
Lorenzo Magnani

Volume 3
Issue 1

Meta-Level Abduction
Katsumi Inoue

1
7

Abduction and Beyond in Logic Programmingwith Application to Morality
Luís Moniz Pereira and Ari Saptawijaya

37

A Dynamic Approach to Peirce’s Interrogative
Construal of Abductive Logic
Minghui Ma and Ahti-Veikko Pietarinen

73

May 2016

Editorial Preface
Lorenzo Magnani

Contents

Course of Action Recommendations and Their
Place in Developmental Abduction
Donna E. West
123
Abduction: from the Ignorance Problem tothe Ignorance Virtue
Tommaso Bertolotti, Selene Arfiniand Lorenzo Magnani

151

Journal of Logics
and their Applications
Issue 1

May 2016

Special Issue
The IfColog Journal of Logics and their applications

105

The IfCoLog

Volume 3

Abduction and Beyond. Methodologicaland Computational Aspects of Creativity
in Natural Sciences
Andrés Rivadulla

ISSN PRINT 2055-3706
ISSN ONLINE 2055-3714

Frontiers of Abduction
Guest Editor
Lorenzo Magnani

IFCoLog Journal of Logic and its
Applications
Volume 3, Number 1
May 2016

Disclaimer
Statements of fact and opinion in the articles in IfCoLog Journal of Logics and their Applications
are those of the respective authors and contributors and not of the IfCoLog Journal of Logics andtheir Applications or of College Publications. Neither College Publications nor the IfCoLog Journalof Logics and their Applications make any representation, express or implied, in respect of theaccuracy of the material in this journal and cannot accept any legal responsibility or liability forany errors or omissions that may be made. The reader should make his/her own evaluation as tothe appropriateness or otherwise of any experimental technique described.

c

Individual authors and College Publications 2016
All rights reserved.

ISBN 978-1-84890-214-5
ISSN (E) 2055-3714
ISSN (P) 2055 3706

College Publications
Scientific Director: Dov Gabbay
Managing Director: Jane Spurr

http://www.collegepublications.co.uk

Printed by Lightning Source, Milton Keynes, UK
All rights reserved. No part of this publication may be reproduced, stored in a retrieval system ortransmitted in any form, or by any means, electronic, mechanical, photocopying, recording or otherwise
without prior permission, in writing, from the publisher.

Editorial Board

Editors-in-Chief
Dov M. Gabbay and Jörg Siekmann
Marcello D’Agostino
Natasha Alechina
Sandra Alves
Arnon Avron
Jan Broersen
Martin Caminada
Balder ten Cate
Agata Ciabttoni
Robin Cooper
Luis Farinas del Cerro
Esther David
Didier Dubois
PM Dung
Amy FeltyDavid Fernandez DuqueJan van Eijck

Melvin Fitting
Michael Gabbay
Murdoch Gabbay
Thomas F. Gordon
Wesley H. Holliday
Sara Kalvala
Shalom Lappin
Beishui LiaoDavid Makinson
George Metcalfe
Claudia Nalon
Valeria de Paiva
Jeff ParisDavid Pearce
Brigitte Pientka
Elaine Pimentel

iii

Henri Prade
David Pym
Ruy de Queiroz
Ram Ramanujam
Chrtian Retoré
Ulrike Sattler
Jörg Siekmann
Jane Spurr
Kaile Su
Leon van der Torre
Yde Venema
Rineke Verbrugge
Heinrich Wansing
Jef Wijsen
John Woods
Michael Wooldridge

iv

Scope and Submissions

This journal considers submission in all areas of pure and applied logic, including:pure logical systems
proof theory
constructive logic
categorical logic
modal and temporal logic
model theory
recursion theory
type theory
nominal theory
nonclassical logics
nonmonotonic logic
numerical and uncertainty reasoning
logic and AI
foundations of logic programming
belief revisionsystems of knowledge and belief
logics and semantics of programming
specification and verification
agent theory
databases

dynamic logic
quantum logic
algebraic logic
logic and cognition
probabilistic logic
logic and networks
neuro-logical systems
complexity
argumentation theory
logic and computation
logic and language
logic engineeringknowledge-based systems
automated reasoning
knowledge representation
logic in hardware and VLSI
natural language
concurrent computation
planning

This journal will also consider papers on the application of logic in other subject areas:
philosophy, cognitive science, physics etc. provided they have some formal content.
Submissions should be sent to Jane Spurr (jane.spurr@kcl.ac.uk) as a pdf file, preferablycompiled in LATEX using the IFCoLog class file.

v

vi

Contents

Editorial Preface . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Lorenzo Magnani

1

ARTICLES
Meta-Level Abduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Katsumi Inoue

7

Abduction and Beyond in Logic Programming with Application to Morality 37
Luı́s Moniz Pereira and Ari Saptawijaya
A Dynamic Approach to Peirce’s Interrogative Construal of Abductive Logic 73
Minghui Ma and Ahti-Veikko Pietarinen
Abduction and Beyond. Methodological and Computational Aspects ofCreativity in Natural Sciences . . . . . . . . . . . . . . . . . . . . . . . . . . . 105
Andrés Rivadulla
Course of Action Recommendations and Their Place in Developmental
Abduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 123
Donna E. West
Abduction: from the Ignorance Problem to the Ignorance Virtue . . . . . . 153
Tommaso Bertolotti, Selene Arfini and Lorenzo Magnani

vii

viii

Editorial Preface
Lorenzo Magnani
University of Pavia, Italylmagnani@unipv.it

This special issue of the IfColog Journal of Logics and their Applications “Frontiers of Abduction” is based on a selection of papers concerning abduction that aresituated at the crossroad of logic, epistemology, and cognitive science.
The status of abduction – that Hintikka considered the fundamental problem
of contemporary epistemology [7] – is very controversial. When dealing with abductive reasoning, misinterpretations and equivocations are common. What are thedifferences between abduction and induction? What are the differences between abduction and the well-known hypothetico-deductive method? What did Peirce mean
when he considered abduction a kind of inference? Does abduction involve only thegeneration of hypotheses or their evaluation too? Are the criteria for the best explanation in abductive reasoning epistemic, or pragmatic, or both? How many kinds of
abduction are there? The papers presented in this issue aim at increasing knowledge– logical, cognitive, epistemological – about creative and expert inferences.
Analyzing various reasoning and modeling practices provides a way of specifying
the nature of some abductive reasoning processes. Following [12; 14] we have to
distinguish between sentential, model-based, and manipulative abduction. In thepast decades many attempts have been made to model abduction by developing someformal tools in order to illustrate its computational properties and the relationships
with the different forms of deductive reasoning (see, for example, [4]). Some ofthese formal models of abductive reasoning are based on the theory of the epistemicstate of an agent [3], where the epistemic state of an individual is modeled as aconsistent set of beliefs that can change by expansion and contraction (belief revisionframework). These formal tools, often related to the tradition of nonmonotonic
logic, logic programming, semantic tableaux, dynamic and adaptive logic, etc., areusually devoted to illustrating sentential abduction and present some limitations[13]. It may be said that the traditional logical accounts of abduction certainly
illustrate much of what is important in abductive reasoning, especially the objectiveof selecting a set of hypotheses (diagnoses, causes) that are able to dispense good
(preferred) explanations of data (observations), but fail in accounting for many casesof explanations occurring in science or in everyday reasoning. The logical papers
Vol. 3 No. 3 2016
IFCoLog Journal of Logic and its Applications

Lorenzo Magnani
presented in this special issue aim at overcoming these limitations. Moreover, if
we want to provide a suitable framework for analyzing the most interesting casesof conceptual changes in science we cannot limit ourselves to the sentential viewof abduction but we have to consider a broader inferential one, encompassing bothsentential and model-based sides of creative abduction.1
Many kinds of abductions involving analogies, diagrams, thought experimenting, visual imagery, etc. (for example in scientific discovery processes), can be called
model-based. I believe that research in cognitive science, especially cognitive psychology, artificial intelligence, and computational philosophy, have established thatheuristic procedures are abductive and reasoned. Among the various kinds of modelbased and abductive reasoning, analogy has received particular attention from thepoint of view of computational models designed to simulate aspects of human analogical thinking: for example, Thagard et al. have developed ARCS (Analog Retrieval
by Constraint Satisfaction; [24]) and ACME (Analogical Constraint Mapping Engine; [8]), computational programs that are built on the basis of a multiconstraint
theory. Thagard and Croft have also shown that analogical reasoning is present insome forms of model-based questioning relevant to technological innovation [23]. In
the last three decades many authors have demonstrated how the practices of analogical modeling, visual modeling, and thought experimenting have played generativeroles in concept formation in science. They also developed several accounts of how
these model-based reasoning tools function in conceptual innovation and change.
For example, Craig, Nersessian, and Catrambone [5] argued that some commonsense and epistemological solutions to target problems can be generated basing on
perceptual simulations of analogous but superficially dissimilar source problems.
The concept of manipulative abduction [17] is devoted to capturing the role ofaction and of external representations in many interesting situations: action provides otherwise unavailable information that enables the agent to solve problems by
starting and performing a suitable abductive process of generation or selection of hypotheses [15], which occurs in a highly structured cognitive and semiotic framework[16].
Finally, the important GW and AKM schemas of abduction are clearly illustrated
in [6]: indeed the classical schematic representation of abduction is expressed bywhat [6] call AKM-schema, which is contrasted to their own (GW-schema): A
stands for Aliseda ([1; 2]), K for Kowalski ([10]), Kuipers ([11]), and Kakas et al.
([9]), and M stands for Magnani ([14]) and Meheus ([22]). On this issue cf. [18;19], in which the recent so-called EC-model of abduction is illustrated as well.
In [20] some model-based aspects concerning the abductive role of diagrams in mathematicalthinking is addressed. An integration of the computational account of abduction with some ideas
developed inside the so–called dynamical approach is illustrated in [21].
12Editorial Preface
The contributions to this special issue are written by interdisciplinary researchers
in logic, epistemology, and cognitive science. They aim at increasing knowledge
about abductive reasoning by illustrating some of the most recent results andachievements. Three papers are centered on fundamental logical aspects: the statusof logical models of abduction in the so called meta-abduction (Katsumi Inoue),the interplay between logic programming and abduction, and the application of
abductive logic to morality (Luís Moniz Pereira and Ari Saptawijaya), Peirce’s interrogative construal of abductive logic in a “dynamic perspective” (Minghui Ma
and Ahti-Veikko Pietarinen). The remaining three papers illustrate some recent issues concerning the status of abduction with respect to logico-cognitive problems:from the methodological and computational aspects of abductive creativity in natural sciences (Rivadulla), to the role of recommendations as imperative propositionsin the active operation of abductive reasoning (West), to the very recent research on
the intertwining between abduction and ignorance (Bertolotti, Arfini and Magnani).
Several papers concerning abduction, practical reasoning, and scientific discovery deriving from the presentations given at the Conferences MBR (Model-Based
Reasoning), promoted since 1998 by the editor of the present special issue, arecontained in the following books: Model-Based Reasoning in Scientific Discovery,edited by L. Magnani, N.J. Nersessian, and P. Thagard (Kluwer Academic/Plenum
Publishers, New York, 1999; Chinese edition, China Science and Technology Press,
Beijing, 2000), which was based on the papers presented at the first “model-based
reasoning” international conference, held at the University of Pavia, Pavia, Italy
in December 1998. Other two volumes were based on the papers presented at thesecond “model-based reasoning” international conference, held at the same place in
May 2001: Model-Based Reasoning. Scientific Discovery, Technological Innovation,
Values, edited by L. Magnani and N.J. Nersessian (Kluwer Academic/Plenum Publishers, New York, 2002) and Logical and Computational Aspects of Model-Based
Reasoning, edited by L. Magnani, N.J. Nersessian, and C. Pizzi (Kluwer Academic,
Dordrecht, 2002). Another volume, Model-Based Reasoning in Science and Engineering, edited by L. Magnani (College Publications, London, 2006), was based on
the papers presented at the third “model-based reasoning” international conference,
held at the same place in December 2004. The volume Model-Based Reasoning in
Science and Medicine, edited by L. Magnani and L. Ping (Springer, Heidelberg/Berlin 2006), was based on the papers presented at the fourth “model-based reasoning”
conference, held at Sun Yat-sen University, Guangzhou, P. R. China. The volume
Model-Based Reasoning in Science and Technology. Abduction, Logic, and Computational Discovery, edited by L. Magnani, W. Carnielli and C. Pizzi (Springer, Heidelberg/Berlin 2010), was based on the papers presented at the fifth “model-based
reasoning” conference, held at the University of Campinas, Campinas, Brazil, in3

Lorenzo Magnani
December 2009. Finally, the volume Model-Based Reasoning in Science and Technology. Theoretical and Cognitive Issues, edited by L. Magnani, (Springer, Heidelberg/Berlin 2013), was based on the papers presented at the sixth “model-based
reasoning” conference, held at Fondazione Mediaterraneo, Sestri Levante, Italy, June
2015.
Some interesting papers related to problem of abduction can be found in previous Special Issues of Journals: in Philosophica: Abduction and Scientific Discovery,
61(1), 1998, and Analogy and Mental Modeling in Scientific Discovery, 61(2) 1998; inFoundations of Science: Model-Based Reasoning in Science: Learning and Discovery,5(2) 2000, all edited by L. Magnani, N.J. Nersessian, and P. Thagard; in Foundationsof Science: Abductive Reasoning in Science, 9, 2004, and Model-Based Reasoning:
Visual, Analogical, Simulative, 10, 2005; in Mind and Society: Scientific Discovery:
Model-Based Reasoning, 5(3), 2002, and Commonsense and Scientific Reasoning,
4(2), 2001, all edited by L. Magnani and N.J. Nersessian; in the Special Issue ofthe Logic Journal of the IGPL: Abduction, Practical Reasoning, and Creative Inferences in Science, 14(1) (2006); in the two Special Issues of Foundations of Science:
Tracking Irrational Sets: Science, Technology, Ethics, and Model-Based Reasoningin Science and Engineering, 13(1) and 13(2) (2008), all edited by L. Magnani. Other
technical logical papers presented at MBR09_BRAZIL have been published in a special issue of the Logic Journal of the IGPL: Formal Representations in Model-Based
Reasoning and Abduction, 20(2) (2012), edited by L. Magnani, W. Carnielli, and C.
Pizzi. Finally, technical logical papers presented at MBR12_ITALY have been published in a special issue of the Logic Journal of the IGPL: Formal Representations
in Model-Based Reasoning and Abduction, 21(6) (2013), edited by L. Magnani.
Lorenzo Magnani
University of Pavia
Pavia, Italy, January 2016

References
[1]
[2]
[3]

A. Aliseda. Seeking Explanations: Abduction in Logic, Philosophy of Science andArtificial Intelligence. PhD thesis, Amsterdam: Institute for Logic, Language andComputation, 1997.
A. Aliseda-Llera. Abductive Reasoning: Logical Investigations into Discovery and Explanation. Springer, Berlin, 2006.
C. Boutilier and V. Becher. Abduction as belief revision. Artificial Intelligence, 77:43–
94, 1995.
4

Editorial Preface[4]

T. Bylander, D. Allemang, M. C. Tanner, and J. R. Josephson. The computational
complexity of abduction. Artificial Intelligence, 49:25–60, 1991.
[5] D.L. Craig, N.J. Nersessian, and R. Catrambone. Perceptual simulation in analogicalproblem solving. In L. Magnani and N.J. Nersessian, editors, Model-Based Reasoning:
Science, Technology, Values, pages 167– 189, New York, 2002. Kluwer Academic/Plenum Publishers.
[6] D.M. Gabbay and J. Woods. The Reach of Abduction. North-Holland, Amsterdam,
2005. Volume 2 of A Practical Logic of Cognitive Systems.
[7] J. Hintikka. What is abduction? the fundamental problem of contemporary epistemology. Transactions of the Charles S. Peirce Society, 34:503–533, 1998.
[8] K. J. Holyoak and P. Thagard. Analogical mapping by constraint satisfaction. Cognitive
Science, 13:295–355, 1989.
[9] A Kakas, R. A. Kowalski, and F. Toni. Abductive logic programming. Journal of Logic
and Computation, 2(6):719–770, 1993.
[10] R. A. Kowalski. Logic for Problem Solving. Elsevier, New York, 1979.
[11] T. A. F. Kuipers. Abduction aiming at empirical progress of even truth approximation
leading to a challenge for computational modelling. Foundations of Science, 4:307–323,
1999.
[12] L. Magnani. Abductive reasoning: philosophical and educational perspectives inmedicine. In D.A. Evans and V.L. Patel, editors, Advanced Models of Cognition for
Medical Training and Practice , pages 21–41, Berlin, 1992. Springer.
[13] L. Magnani. Limitations of recent formal models of abductive reasoning. In A. Kakasand F. Toni, editors, Abductive Reasoning. KKR-2, 17th International Joint Conferenceon Artificial Intelligence (IJCAI 2001), pages 34–40, Berlin, 2001. Springer.
[14] L. Magnani. Abduction, Reason, and Science. Processes of Discovery and Explanation.
Kluwer Academic/Plenum Publishers, New York, 2001.
[15] L. Magnani. Epistemic mediators and model-based discovery in science. In L. Magnani
and N. J. Nersessian, editors, Model-Based Reasoning: Science, Technology, Values,
pages 305–329, New York, 2002. Kluwer Academic/Plenum Publishers.
[16] L. Magnani. Abduction and cognition in human and logical agents. In S. Artemov,
H. Barringer, A. Garcez, L. Lamb, and J. Woods, editors, We Will Show Them: Essaysin Honour of Dov Gabbay, pages 225–258, London, 2005. King’s College Publications.
[17] L. Magnani. Abductive Cognition. The Epistemological and Eco-Cognitive Dimensionsof Hypothetical Reasoning. Springer, Heidelberg/Berlin, 2009.
[18] L. Magnani. Is abduction ignorance-preserving? Conventions, models, and fictions inscience. Logic Journal of the IGPL, 21(6):882–914, 2013.
[19] L. Magnani. The eco-cognitive model of abduction. ’Aπαγωγ ή now: Naturalizing the
logic of abduction. Journal of Applied Logic, 13:285–315, 2015.
[20] L. Magnani and R. Dossena. Morphodynamical abduction: causation by attractors
dynamics of explanatory hypotheses in science. Foundations of Science, 10:107–132,
2005.

5

Lorenzo Magnani[21] L. Magnani and M. Piazza. Perceiving the infinite and the infinitesimal world: unveilingand optical diagrams and the construction of mathematical concepts. Foundations of
Science, 10:7–23, 2005.
[22] J. Meheus, L. Verhoeven, M. Van Dyck, and D. Provijn. Ampliative adaptive logics
and the foundation of logic-based approaches to abduction. In L. Magnani, N. J.
Nersessian, and C. Pizzi, editors, Logical and Computational Aspects of Model-Based
Reasoning, pages 39–71. Kluwer Academic Publishers, Dordrecht, 2002.
[23] P. Thagard and D. Croft. Scientific discovery and technological innovation: ulcers,
dinosaur extinction, and the programming language java. In N.J. Nersessian L. Magnani
and P. Thagard, editors, Model-Based Reasoning in Scientific Discovery, pages 125–137,
New York, 1999. Kluwer Academic/Plenum Publishers.
[24] P. Thagard, K. J. Holyoak, P. Nelson, and G. Gochfeld. Analog retrieval by constraint
satisfaction. Artificial Intelligence, 46:259–310, 1990.

6

M ETA -L EVEL A BDUCTION
K ATSUMI I NOUE
National Institute of Informatics, JapanTokyo Institute of Technology, Japaninoue@nii.ac.jp

Abstract
Meta-level abduction (MLA) has been proposed as a method to abduce missing
laws in completing proofs and explaining observations at the meta-level. Based ona simple logic of causality, Inoue et al. (2010) firstly proposed meta-level abduction
to discover physically unobserved causality in terms of hidden rules to explain given
empirical rules with respect to skills for music playing. Meta-level abduction has also
been applied to completion of biological networks containing both positive and negative causal effects (Inoue et al., 2013). In this paper, we define a general framework
for meta-level abduction together with a logical system for it, and analyze its potential
power in various patterns of abductive reasoning. We will see that meta-level abduction
can realize second-order existential abduction by Schurz (2008). Moreover, meta-level
abduction can be coordinated with selective, creative and other types of abductions.

1

Abduction and Meta-Reasoning

1.1 Abductive Reasoning
Abduction is one of the three fundamental modes of reasoning characterized by Peirce [58],the others being deduction and induction. Abduction amounts to concluding the minor
premise from the major premise and the conclusion, and its original inferential style isgiven as follows.
The (surprising) fact, ψ, is observed;
But if ϕ were true, ψ would be a matter of course;
Hence, there is reason to suspect that ϕ is true.
This corresponds to the rule of the form, called the fallacy of affirming the consequent:
ψ

ϕ→ψ
.
ϕ

(1)

Vol. 3 No. 3 2016
IFCoLog Journal of Logic and its Applications

I NOUEϕ is called an explanans for an explanandum ψ.
In the field of Artificial Intelligence (AI), logic of abduction has often been studied
from the point of view of automated reasoning [56, 29] and its applications to diagnosis andrecognition [35]. There, it is often assumed in (1) that both ϕ and ψ are propositional or
first-order formulas and the arrow → is interpreted as an implication connective in such alogic. The major premise (ϕ → ψ) is either given in a prior (or background) knowledge Σ,i.e., (ϕ → ψ) ∈ Σ, or is inferred from it, i.e., Σ |= (ϕ → ψ). In either case, the inference
rule (1) can be expressed in the meta-theoretical relation:Σ ∧ ϕ |= ψ.

(2)Under these assumptions,1 abductive proof procedures have been designed in AI, often
based on deduction with the resolution principle [67], e.g., [63, 14, 62, 11, 28, 74, 36,65]. From the theorem-proving viewpoint, abduction augments sufficient conditions that are
missing in the premises, i.e., background knowledge Σ, to enable a derivation, i.e., proof,of the given observation ψ. This inference fills the gap in a proof of the observation fromthe premises. An inferred sufficient condition ϕ is called a hypothesis or an explanation.
Note that a hypothesis can be any formula, e.g., a conjunction (set) of atoms, literals orrules. Moreover, a proof system or a consequence relation |= can be based on any classicalor non-classical logic.
Abduction plays essential roles in knowledge discovery in science and technology [45].
Then, abduction has been used since 2000s in the field of inductive logic programming(ILP), which originally focused on induction of logic programs [51]. The use of backgroundknowledge in scientific applications has directed an attention of ILP to theory completion[49] rather than classical learning tasks such as concept learning and classification. There,
abduction is mainly used to complete proofs of observations from incomplete backgroundknowledge, while induction refers to generalization of the abduced cases.
In any case, abductive inference by way of (2) is meta-theoretical, that is, logic of
abduction is given at the meta-level.

1.2 Meta-Reasoning
On the other hand, meta-reasoning has been intensively investigated in AI or logic programming from a distinct viewpoint from abduction. Meta-reasoning has been used for themeta-level control of reasoning, thereby enabling us the introspective monitoring of the reasoning process at the object level, e.g., [59, 8, 68]. Meta-reasoning has also been applied tometa-programming in logic programming, e.g., [7, 24, 38, 23, 13]. Bowen and Kowalski’s
Usually, the consistency relation that Σ ∧ ϕ is consistent is also required. Additionally, ϕ is often to be
as minimal as possible by virtue of Occam’s Razor.
18

M ETA -L EVEL A BDUCTION
meta-interpreter [7] and its “Vanilla” variant [24] have been most widely used to implement
meta-logic in logic programming, which are given as:
solve(true).
solve(A ∧ B) ← solve(A) ∧ solve(B).
solve(¬A) ← not solve(A).

solve(A) ← rule(A ← B) ∧ solve(B).

Here, variables are capitalized in logic programming, e.g., A, B and C,2 and not represents
negation as failure [10]. The predicates solve and rule are meta-predicates, and all constructs with them are atoms. An atom of the form solve(A) reflects that A is proved fromthe program, and rule(A ← B) denotes that (A ← B) is a rule in the program.3 Note thata fact of the form A can be expressed as rule(A ← true) in a meta-program. In general,
any inference rule can be expressed as such a meta-rule, e.g.,solve(A ← B) ← solve(A ← C) ∧ solve(C ← B).
solve(¬A) ← solve(B ← A) ∧ solve(¬B).

Note that those meta-level axioms have been used for deduction in meta-programming or
deduction control [59].

1.3 Meta-Level Abduction
Now, a question arises. What happens if abduction is applied for such meta-level axioms?
For example, suppose that the meta-rule representing Modus Ponens (MP)solve(A) ← rule(A ← B) ∧ solve(B)as well as the meta-level expression of a minor premise q
solve(q),2Those free variables are assumed to be universally quantified at the front of each rule. Conventionally,
the connective ← is used in logic programming instead of → as material implication, that is, (A ← B) and(B → A) is logically equivalent.
3
The connective symbol ← in the atom rule(A ← B) is used here for notational convenience. Instead, an
atom of the form rule(A ← B) should be expressed as rule(A, B) in logic programming. The meta-predicatessolve and rule are originally represented as demo and clause, respectively. The meta-predicate demo takes
another argument T to represent a theory in the meta-interpreter in [7] and demo(T, A) represents that A isprovable in T , while the Vanilla meta-interpreter [24] omits the theory argument. The meta-predicate clausehas two arguments as clause(A, B), which represents the rule (A ← B).

9

I NOUEare contained in a background theory Σ. Given the meta-level expression of a consequent p
as an observation
solve(p),we can abduce by (2)rule(p ← q).

In this example, (p ← q) is a rule, hence law abduction [45] is realized, yet this is an
ordinary abduction in AI, called fact abduction in philosophy [71, 27], since it only abduces
the atom with the meta-predicate rule. Inoue et al. have called this inference meta-level
abduction (MLA) [32, 31], since abduction is performed at the meta-level rather than at
the object level. It is not called “meta-abduction” because meta-abduction would mean
abduction about abduction, cf., [4]. Meta-level abduction is abduction at the meta-level andthus is abduction to augment what the meta-theory is intended to infer, but could be also
applied for meta-abduction if the meta-level axioms are given for abduction.
It is notable that most previous abductive methods in the AI literature deal with fact
abduction. Yet, it is possible to implement law abduction by providing a predetermined setof candidate rules (called abducible rules). Previous abductive methods of abduction thus
select proper combinations of abducible facts or abducible rules. In this sense, abduction in
AI is generally selective [45, 46]. In contrast, meta-level abduction does not need any such
abducible rules in advance.
Meta-level abduction has been applied to discover physical skills in terms of hidden
rules to explain given empirical rules in the domain of cello playing in [32]. By representing
rule structures of the problem in a form of causal networks, meta-level abduction infers
missing links and unknown nodes from incomplete networks to complete paths from stimuli
to effects. Then, applicability of meta-level abduction to deal with networks expressing both
positive and negative causal effects has been examined in [31]. Such negative causal effects
are known in biology as inhibitory effects, which are essential in gene regulatory, signaling
and metabolic networks. Such biological applications of meta-level abduction to networks
with inhibition have been reported in treatment of hypertension [41], completion of p53
signaling networks [31] and completion of glucose repression networks [77]. Meta-level
abduction has also been extended to realize analogical abduction by adding meta-axioms
for analogy, which has been applied to cello exercises to obtain analogical explanations forskill acquisition [20].

1.4 Related Research
Meta-reasoning has been intensively applied to logic programming [38, 23, 13], and it ispossible to implement abductive procedures using meta-programming to perform abduction [74, 17, 36, 2]. Such abductive meta-interpreters should be distinguished from ab10

M ETA -L EVEL A BDUCTION
duction at the meta-level, and law abduction had not been considered in the literature ofmeta-reasoning before MLA was proposed in [32]. For example, Kowalski discussed theimportance of both abduction and meta-reasoning in [38], but did not mention their combination there. Christiansen [9] computes both abduction and induction in a unified system ofmeta-programming, and uses separate forms of reasoning in their actual computation followed by the system. In [9], the demo predicate is used to generate parts of programs thatare necessary to derive the goal based on techniques of constraint logic programming.
Muggleton et al. [52, 53] propose meta-interpretive learning (MIL) for law abduction.
Similar to MLA, MIL uses an abductive meta-interpreter for generating rules from the prestored rule schemas. However, the use of meta-interpreters in MIL is different from that in
MLA; MLA applies abduction for meta-level axioms, but MIL uses abduction for objectlevel theories with higher-order abducible rules. Predicate invention [50] is an important
method in ILP to introduce new objects and relations, and both MLA and MIL have implemented predicate invention by abduction but in different ways. MLA in [32] introduces
new objects or propositions as new terms of meta-predicates, but MIL in [52] introduces
new symbols which represent relations.
Some attempts in AI and ILP have contributed to abduction and induction in causal
theories, which are formally represented as sets of individual rules between causes and theireffects. The event calculus [39] is a meta-theory for reasoning about time and action in theframework of logic programming. Abductive event calculus [15] is an abductive extensionof the event calculus, and can be regarded as a kind of meta-level abduction. Abductive
event calculus has been extended for applications to planning, e.g., [72], but has never beenused for abducing causal theories.
Bharathan and Josephson [4] use a specific structure of abduction that is tailored forbelief revision, then meta-abductive reasoning is performed over the records of an agent’s
reasoning trace to make requisite changes to beliefs. Such a meta-abduction reasons about
reasoning states and steps as a controller of abduction, whereas our MLA performs abduction about observations using a meta-theory as a background knowledge. Pereira and Pinto[60] investigate side-effects of interest called inspection points in choosing abductive solutions. Inspection points can be embedded as meta-predicates on top of existing abductive
systems. Similar to [4], this is a work on controlling abduction at the meta-level.
Gabbay and Woods [21] show how to make an abductive logic based on some base logic
and the labeled deduction system. In this sense, the logic of abduction is discussed at themeta-level, thereby formalizing abduction in a flexible manner upon any base logic. MLA
can also be based on any base logic and can be applied to meta-level axioms of such a logic.
There are several attempts to propose meta-level inference rules (or structural rules)
that represent rationality postulates for abductive/explanatory relations, thereby enabling a
logical characterization of abduction at the meta-level [18, 48, 42, 3, 6]. Compared withthese works, MLA does not offer a logic for abduction itself, since MLA just applies ab11

I NOUE
duction to a meta-theory in any base logic using any abductive computation for that logic.
It would thus be possible to write such structural rules for abduction as meta-level axioms
and then to perform MLA upon them. Such a possibility will be further discussed later in
Section 3.2.
In the rest of this paper, we analyze MLA by giving a general framework that subsumes
specific MLA systems in [32, 31]. Based on this new MLA framework, we reveal thepotential inference power of MLA and give a general perspective of MLA towards creative
abduction.

2 Patterns of Abduction
Our goal is to see the potential power of meta-level abduction from the point of view offorms of explanans and explanundums. To this end, this section reviews the patterns of
abduction investigated by Hirata [26], Magnani [45, 46], Schurz [71] and Hoffmann [27],which are then utilized in a mixed way in later sections.
Hirata [26] classifies abduction into 5 types in the context of logic programming: ruleselecting, rule-finding, rule-generating, theory-selecting, and theory-generating abductions.
This classification is based on works on abduction in AI and abductive and inductive logic
programming, and focuses mainly on how hypotheses are obtained with or without background theories.
Magnani [45, 46] gives a wider perspective on different epistemological types of abduction, and in particular shows two distinct notions of abduction, i.e., selective and creative
abductions. The task in selective abduction is to abduce hypotheses by selecting them froman encyclopedia of pre-stored entities in the language of hypotheses, while a discovery of
new hypotheses is in order in creative abduction.
Schurz [71] then further classified patterns of abduction broadly into 4 kinds: factual,law, theoretical-model, and second-order existential abductions, which are further divided
into 8 patterns in Table 1. Both theoretical-model abduction and 2nd-order existential
abduction are kinds of creative abduction discussed by Magnani. According to Schurtz,
theoretical-model abduction introduces a new theoretical model that consists of facts, laws
and concepts without using any new facts and concepts, that is, the organization and combination of its components are new, while the components themselves are already known.
Classification among 2nd-order existential abduction is based on the situations in which abduction is performed, and all patterns in it generate new laws with new concepts.4 Schurz

4

In [71], “Hypothetical cause abduction” is further divided into “Speculative abduction”, “Strict common
cause abduction”, “Statistical factor analysis”, and “Abduction to reality”. Schurz also specified in his classification how each abduction is driven, which is omitted in Table 1.

12

M ETA -L EVEL A BDUCTION
Kind of abduction
Factual abduction— Observable-fact abduction— 1st-order existential abduction— Unobservable-fact abduction
Law abduction
Theoretical-model abduction
2nd-order existential abduction— Micro-part abduction— Analogical abduction— Hypothetical cause abduction

Explanandum
Single facts
Single facts
Single facts
Single facts
Empirical laws
General empirical phenomena
General empirical phenomena
General empirical phenomena
General empirical phenomena
General empirical phenomena

Explanans
New facts
Factual reasons
Facts with new unknown individuals
Unobservable facts
New laws
New theoretical models
New laws with new concepts
Microscopic compositions
New laws with analogical concepts
Hidden (unobservable) causes

Table 1: Patterns of abduction by Schurz (part)
Explanans
Facts
Types (or concepts)
Laws
Theoretical models
Representations

Exists in our mind
Selective fact abduction
Selective type abduction
Selective law abduction
Selective model abduction

Exists in our culture / Historically new
P-creative/H-creative fact abduction
P-creative/H-creative type abduction
P-creative/H-creative law abduction
P-creative/H-creative model abduction

Selective meta-diagrammatic abduction

P-creative/H-creative meta-diagrammatic abduction

Table 2: Hoffmann’s taxonomy of kinds of abduction
considered both factual and law abductions are selective abduction, but the distinction between factual and law abductions are mainly with regard to the form of explanations.5
Hoffmann [27] then argued that factual and law abductions can be creative too, so this
syntactic distinction should be orthogonal to that of selective and creative abductions. Hoffmann’s fact abduction is similar to Schurz’s factual abduction, but type abduction is distinguished from it to denote abduction of new concepts. Law abduction by Hoffmann can also
include new concepts. Table 2 is the summary of patterns of abduction that are modified
by Hoffmann.6 Hoffmann’s meta-diagrammatic abduction partially correspond to Schurz’s
2nd-order existential abduction, but is mostly creative, since completely new theoreticalmodels are required by changing or developing new representation systems.

2.1 Existing Abductive Systems
Before we go deep into the characterization of meta-level abduction (MLA), here we connect past works on abductive inference systems with patterns of abduction.
5
Roughly speaking, Hirata’s rule-selecting and rule-finding abductions in [26] correspond to Schurz’s factual abduction and analogical abduction, respectively. On the other hand, rule-generating and theory-selecting
abductions in [26] correspond to law abduction. Hirata’s theory-generating abduction is more like (explanatory)
induction in ILP, in particular is equal to Shapiro’s model inference system [73].
6
For creative abduction, “P” stands for “psychological”, and “H” stands for “historical” [27].

13

I NOUEAs mentioned in Section 1.3, most abductive procedures which are developed in AI andlogic programming deal with fact abduction only. In particular, those resolution-based abductive procedures, e.g., [63, 14, 62, 28, 74, 36], are of this kind. First-order existential abduction by Schurz has been realized in such procedures as in [63, 14, 28, 65] by computing
hypotheses with existentially quantified variables. Other types of abductive computation
are also mostly considered for fact abduction or selective abduction [35]. Console et al.
[11] have shown an object-level characterization of meta-level expression (2) in terms of
deduction and completion. This work uses deduction for fact abduction.
Some works on law abduction had existed in AI before MLA was introduced in [32].
In most works, a set of candidate rule patterns called abducible rules must be pre-stored inadvance, hence they are all instances of selective abduction. The framework of abductive
systems with such abducible rules was firstly considered by Poole [61], which associates aunique name with each ground instance of an abducible rule schema, and those rule names
are abduced by fact abduction. This is feasible when we know exact patterns of rules asstrong biases, and offers a convenient method to reduce selective law abduction to fact
abduction. However, it is impossible to prepare all patterns of rules in advance in order toabduce missing rules.
Inductive Logic Programming (ILP) [51] is a subfield of machine learning in AI whichuses logic programming as a uniform representation for (training) examples/observations,background knowledge and hypotheses. As its name indicates, ILP aims at developmentof theory and practice of induction, but it has been recognized that explanatory induction
can be viewed as abduction [19], in particular as law abduction. An early work by Shapiro
called Model Inference System [73] builds a theory from scratch within a given language,
and can thus be viewed as theoretical-model abduction. Predicate invention [50] had been
intensively investigated in the initial stage of ILP research and has recently been revisited[51, 53], since it should play an important role in discovery. Predicate invention itself
corresponds to type abduction by Hoffmann, and ILP with predicate invention partiallyrealizes creative abduction, since it introduces new concepts and new laws.
Fact abduction has been used to implement law abduction in several ILP systems. Including [69] and other works in [19], most such previous systems use abduction to computeinductive hypotheses cleverly, and such integration is useful in theory refinement. CFinduction [30] can induce explanatory rules as hypotheses, hence realizes law abduction in
first-order clausal theories. CF-induction can directly induce first-order full clausal theories
at the object level based on inverse entailment, viz., Σ ∧ ¬ψ |= ¬ϕ for (2), but predicate
invention should be realized by inverse resolution [50] and the search space for hypothesis enumeration is huge in general. TAL [12] translates an ILP problem into an equivalentabductive problem, then an abductive procedure is used to abduce a hypothesis for an observation from the background theory and a pre-stored inductive bias called a top theory.
TAL uses the naming method to abduce rules as in [61]. Meta-interpretive learning (MIL)14

M ETA -L EVEL A BDUCTION[52, 53] is a further refined ILP method to generate rules from the pre-stored rule schemas,
and hence realizes law abduction in a selective way. Using MIL, predicate invention as wellas learning of recursive programs can be efficiently implemented by way of abduction withrespect to a meta-interpreter of logic programming.
MLA was introduced in [32] as a method to discover unknown relations from incomplete knowledge bases. The main objective to use MLA in [32] is to provide knowledge
representation and a reasoning method for law abduction. MLA in [32] and its nonmonotonic extension [31] have been implemented in SOLAR [54, 55], an automated deduction
system for consequence finding [40, 28, 47] in first-order logic. Meta-level abduction by
SOLAR is powerful enough to infer missing rules, missing facts, and unknown causes involving predicate invention [50] in the form of existentially quantified hypotheses.

2.2 Lifting to the Meta-Level
Based on the summary of abductive computation in the last subsection, we should choose
an appropriate procedure for a pattern of abduction. Most methods are for selective abduction, and abductive procedures developed in AI and logic programming are for fact
abduction. Law abduction has been implemented in ILP, and predicate invention introduces
new concepts. Some ILP works can be considered to realize creative abduction. In thetheoretical-model abduction by Schurz [71], multiple laws and facts are newly combined toform a hypothesis to explain multiple observations. Hence, from the computational viewpoint, we do not have to distinguish theoretical-model abduction from law abduction. In
Section 3, we will see that MLA can realize 2nd-order existential abduction by Schurz [71]at the object level, although MLA in [32] itself can be classified as fact abduction (in thesense of Hoffmann [27]) at the meta-level, which we refer to as fact meta-level abduction
hereafter.
The last observation allows us to expect that meta-level abduction can lift the type of
abduction to one level higher. After we verify that fact meta-level abduction can actuallyrealize creative law abduction at the object level in Section 3, we will consider law metalevel abduction and its potential usage in Section 4. Then, we will conclude that MLA canbe coordinated with selective, creative and other types of abductions by [45].

3Fact Meta-Level Abduction

Apart from the specific form of MLA in [32, 31], we here give a general definition of MLA.
We suppose a language L that includes both a meta-level language LM and an objectlevel language LO as well as their amalgamation [7]. The language LM contains higherorder (meta-)predicates and predicate/function/constant variables, while the language LO
contains predicates, functions, constants and variables at the object level. It is important15

I NOUE
here that the domain of the language L is assumed to contain all possible individuals, butsome of them are not practically stored in the domain in advance. That is, new individuals
and new relations can be introduced in the language when they become necessary. We thensuppose a meta-theory T in L, which is composed of elements in LM with constructs fromLO in their arguments. We also assume a consequence relation |= over L.

Definition 3.1. An MLA framework is defined as a triple (T, A, G), where T is a metatheory in L, A is a set of formulas in L called abducibles and G is a formula in L called an
observation or a goal. An abductive task is to find an explanation E of G such that (i) E isa conjunction of abducibles from A, and (ii) T ∧ E |= G.7

Definition 3.2. A theory for fact MLA is an MLA framework (T, A, G), in which A is a setof facts from L.

As case studies of fact MLA, Section 3.1 will review an MLA framework introducedin [32] for causal networks, and Section 3.2 will provide a new logical characterization of
causal networks and abduction on them. Section 3.3 will revisit an extension of MLA for
nonmonotonic causal networks introduced in [31]. All these MLA frameworks are analyzed
to see their abductive power with regard to patterns of abduction.

3.1 Simple Logic of Causality
In [32], a background theory is assumed to be represented in a network structure called a
causal network. A causal network is a directed hyper-graph, which consists of a set of nodes
and a set of (directed) (hyper-)arcs (or links). Each node in a causal network represents someevent, fact or proposition. A direct causal link corresponds to a directed arc, and a causal
chain is represented by the reachability through direct causal links between two nodes.8
A first-order language can be associated to express causal networks as follows. A node isassociated with a proposition or a (ground) atom in the object-level language. A link isassociated with a (ground) rule in the object-level language. Namely, when there is a directcausal link from a node s to a node g, it is expressed as a rule of the form (g ← s). In
this case, we define that linked(g, s) is true as in (3) at the meta-level, where linked is ameta-predicate.9
gi
si
linked(g, s)(3)
7

As in Footnote 1, the consistency of T ∧ E as well as the minimality of E are usually required.
The interpretation of a “cause” here just represents the connectivity, which may refer to any dependency
such as a physical, chemical, conceptual, epidemiological, social, structural, or statistical dependency [57].
A “direct cause” here simply represents the adjacent connectivity in a causal graph. In general, treatment of
causality requires tackling many important issues that have been discussed in such as [25, 76, 57], but we do
not intend to argue what it should be in this paper. Instead, we here use the notion of causality as a means ofconnecting observations and abductive explanations.
9
The meta-predicate linked is expressed as connected in [32].
816M ETA -L EVEL A BDUCTION
The statement that “g is jointly caused together directly by s and t”, written as (g ← s ∧t) at the object level, is expressed in a disjunction of the form (4) at the meta-level (seeEquation (7) for why the disjunction is used here instead of conjunction).
gi

si

AND
)
e

iP
P
PP i
t

linked(g, s) ∨ linked(g, t)(4)

In the above expression, (i) each atom at the object level is represented as a term at the metalevel, and (ii) each rule, i.e., a causal link, at the object level is represented as a (disjunctive)
fact at the meta-level. For object-level propositions g and s, caused(g, s) is defined true
if there is a causal chain from s to g, where caused is another meta-predicate.10 The
caused/2 relation is recursively defined with the linked/2 facts as follows.
Definition 3.3. A theory of Simple Logic for Causality (SLC) is represented as a metatheory TSLC that consists of a set of meta-facts of the form (3) or (4) mentioning objectlevel direct causal links, together with the two meta-rules:11
caused(X, Y ) ← linked(X, Y ).

caused(X, Y ) ← linked(X, Z) ∧ caused(Z, Y ),

(5)
(6)

A theory for SLC-based MLA is an MLA framework (TSLC , ASLC , G), in which ASLC isa set of disjunctive facts with the meta-predicate linked and G is a disjunctive fact with themeta-predicate caused.
Law abduction can be realized in an SLC-based MLA as follows. Suppose an observation (or a goal) G represented as a causal chain caused(g, s), we want to know why (orhow) G can be caused. G corresponds to an empirical rule to be explained if it is an observation or represents a virtual goal to be achieved if it is a goal. An abductive task is then tofind hidden rules that establish a connection from the source s to the target g by filling thegaps in causal networks. For this, an explanation consisting of missing causes (links) andmissing facts (nodes) for the causal network are abduced in an SLC-based MLA.
For example, suppose the observation O = caused(g, s) ∧ caused(h, s), that is, themultiple causal chains between two goal facts g, h and the source fact s. Examples of

The meta-level expression caused(g, s) can be related to the input-output pair (s, g) in [43] or the production rule (s ⇒ g) in [5].
11
The notation for meta-theories here is the same as that for logic programming.
1017

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minimal explanations of O containing two intermediate nodes are as follows.
H1 :

H2 :

gi

Xi
i
P

hi

S
o

PP i
s

)

i
Y

hi
gi Xi

si


S i
Y

∃X∃Y (linked(g, X) ∧ linked(h, Y )∧ linked(X, s) ∧ linked(Y, s))∃X∃Y (linked(g, X) ∧ linked(X, Y )∧ linked(h, Y ) ∧ linked(Y, s))

H1 and H2 represent different connectivities, and we may want to enumerate different typesof network structures that are missing in the original causal network. Moreover, these hypotheses contain existentially quantified variables, where X and Y are newly invented here.
Those new terms can be regarded as either some existing nodes or new unknown nodes.
Since new formulas can be produced at the object level, predicate invention [50] is partiallyrealized here.12
Connecting the meta-level and the object-level representation, meta-literals in SLC havebeen translated to first-order logic formulas at the object level in [32]: An atom linked(s, t)
at the meta-level corresponds to a rule (s ← t) at the object level. Then, a disjunction
linked(g, s) ∨ linked(g, t) (4) corresponds tolinked(g, s) ∨ linked(g, t) ⇔ (g ← s) ∨ (g ← t) ≡ (g ← s ∧ t).

(7)

Based on this translation, the soundness and completeness of law abduction in SLC-based
MLA can be derived. Given a meta-theory TSLC in SLC, let λ(TSLC ) be the object-level
theory obtained by replacing every linked(t1 , t2 ) (t1 and t2 are terms) appearing in TSLC \{(5), (6)} as (disjunctive) facts with the formula (t1 ← t2 ). Note that replacement by λ in
TSLC does not apply to the meta-axioms (5) and (6) as these are subtracted, hence λ(TSLC )does not contain any meta-predicate.
Proposition 3.0.1. [31, Theorem 1] Suppose an SLC-based MLA (TSLC , ASLC , G). Let
G be a meta-formula of the form G = (caused(g, s1 ) ∨ · · · ∨ caused(g, sn )). Let E be an
explanation of G constructed from ASLC . Then,
TSLC ∧ E |= G

iff

λ(TSLC ) ∧ λ(E) |= (g ← s1 ∧ · · · ∧ sn ).

Besides the use in law abduction, SLC-based MLA can also be applied to fact abduction [32]. Abduction of facts at the object level can be formalized as query answering at
12One could argue that this form of predicate invention can be regarded as a realization of object invention
rather than relation invention. Since a concept can be represented as an object, type or relation, it can be
regarded as a concept invention, yet we use predicate invention as a conventional term in ILP.

18

M ETA -L EVEL A BDUCTION
the meta-level. Given a goal of the form caused(g, X), abduction of causes is computed byanswer substitutions to the variable X. To perform selective fact abduction, an abducible literal a at the object level is associated with the fact caused(a, a) at the meta-level. Finally,
law abduction and fact abduction can be combined in the form of conditional query answering [33], which extracts answers in a query with additional abduced conditions. Thus,
SLC-based MLA enables us to abduce both rules and facts [32].
In summary, introduction of new facts, new rules along with new concepts by SLCbased MLA indicates that MLA is able to abduce new theoretical models within SLC. In
other words, SLC-based MLA realizes 2nd-order existential abduction at the object-levelby performing abduction in SLC.

3.2 General Logic of Causality
Here we consider a generalization of SLC called the general logic of causality (GLC). In
the correspondence between SLC-based MLA and object-level abduction, each direct causeof the form linked(g, s) is translated to (g ← s) in SLC. The material implication ← has
the property of transitivity, which corresponds to transitivity of causality via (5) and (6) ina causal network. Another device in SLC is that a conjunctive causal effect (4) is translated
to a disjunction of the form (7), which also depends on the classical interpretation of ←.

Under GLC, we should not consider any particular interpretation of the meta-predicate
linked so that its semantics is left open. This is generally desirable, since causality cannot
be interpreted as material implication. Instead, the meaning of causality at the object levelis now abstracted away. Once a set of linked atoms are given as a causal network, themeaning of the chained causality predicate caused should be determined by the postulates
that are written in inference rules.
In GLC, we will write a conjunctive cause (4) as a direct conjunctive cause of the form
linked(g, s∧t) by allowing any conjunction in an argument of meta-facts, thereby avoiding
a disjunctive fact of the form (7) used in SLC. More generally, a causal network can be given
as a set of atoms of the form linked(F, G), where F and G are Boolean formulas. Similarly,
we can allow atoms of the form caused(F, G) with Boolean formulas F and G in GLC.
To propose the postulates for causal networks in GLC, let us denote the meta-atoms
linked(g, s) and caused(g, s) as (s 7→ g) and (s  g), respectively, by simply reversing
the order of two arguments. Let  be a standard Tarski consequence relation such thatα  β iff α logically entails β. Then, the logic of GLC is then expressed in the following
inference rules.
19

I NOUE

α 7→ ψ
αψ
αβ
βψ
αψ
αβ
βψ
αψ
αϕ
αψ
αϕ∧ψ
αβ
α∧β ψ
αψ(Reflection)(Weakening)(Strengthening)
(And)
(Cut)

(8)
(9)
(10)
(11)
(12)

The rule of Reflection (8) corresponds to the meta-axiom (5) in SLC, and is the means toconnect the direct causal links 7→ with the causal relation , while all other causal statements
are inferred by other inference rules. Note that we do not include the inference rule:αψα 7→ ψ

and in particular (α 7→ α) is not satisfied unless it is explicitly given as a self-loop in the input network. This is because the consequence relation is not generally considered as a causal
relation and is in fact unnecessary for our purpose. However, a consequence relation is useful for weakening and strengthening a causal statement. The rule of (Right) Weakening (9)often appears in other consequence relations, and the rule of (Left) Strengthening (10) is
also called Monotonicity [37]. The Cut rule (12) originated in Gentzen systems and is also
an abbreviation of Cumulative Transitivity [37]. Note that Cut and Strengthening imply thetransitivity:13
αβ
βψ
αψ

(Transitivity)(13)

If necessary, we can introduce more inference rules. For example, if a nondeterministic
effect is given in the form of linked(g ∨ h, s)14 or a disjunctive cause is put inside a link
13

There are pros and cons for the claim that causation is transitive. On the negative side, see [25] for someexamples. Moreover, the monotonicity by Strengthening apparently fails to hold in nonmonotonic logics, which
implies that GLC is not always applicable. We do not argue the appropriateness of these properties here, but canmerely think that the causality notion in SLC/GLC is nothing but the reachability in a causal network defined
via the relation linked. Note that the problem of nonmonotonicity is partly handled in Section 3.3.
14
The meta-atom linked(g ∨ h, s) in GLC can be expressed as the disjunction linked(g, s) ∨ linked(h, s)
in SLC [32].

20M ETA -L EVEL A BDUCTION
like linked(g, s ∨ t), we need an inference rule to handle disjunctions:
(Or)

αψ
βψ
α∨β ψ

(14)

Using this inference system, we can deduce all rational causal statements. Here, causality in
GLC is viewed as a deductive consequence relation, rather than an abductive consequence
relation in such as [18, 42],15 and its logical concept is somewhat similar to logics in [43,6],16 although these works are related to each other. The next proposition shows that GLC
can derive all causal statements that are derived by SLC. Since GLC allows more generalstatements like caused(p ∧ q, r ∨ s), GLC is stronger (and is thus more general) than SLC.
Proposition 3.0.2. Suppose a theory T for SLC, and let T 0 be a theory for GLC obtained
from T \ {(5), (6)} by replacing every disjunctive fact of the form (linked(p, q1 ) ∨ · · · ∨linked(p, qn )) with a meta-fact of the form (q1 ∧ · · · ∧ qn 7→ p). If G = (caused(g, s1 ) ∨· · · ∨ caused(g, sn )) is derived from T in SLC, then G0 = (s1 ∧ · · · ∧ sn  g) is derived
from T 0 in GLC.
Proof. In GLC, Reflection (8) corresponds to the meta-axiom (5) in SLC. Moreover, themeta-axiom (6) in SLC can be obtained by Transitivity (13) (which is a consequence ofStrengthening (10) and Cut (12)) and Reflection. Hence, the two meta-axioms in SLC also
hold in GLC, and all inferences with these axioms in SLC can be obtained in GLC.
Next we consider inference involving disjunctive facts in SLC. This involves the caseinference for each disjunct in a disjunction, and we can show that the corresponding inference is obtained in GLC. Although a complete proof needs a structural induction, we
here show an example in the case of 2 disjuncts. Suppose a conjunctive direct link represented by linked(q, p1 ) ∨ linked(q, p2 ) in SLC. From caused(p1 , s1 ) and caused(p2 , s2 ),G = caused(q, s1 ) ∨ caused(q, s2 ) is derived in SLC, meaning that s1 and s2 jointly cause
q. The corresponding inference can be obtained in GLC as follows.
s1  p1s 2  p2
(Strengthening)s1 ∧ s2  p1 s1 ∧ s2  p2
p ∧ p2 7→ q (Reflection)(And) 1s1 ∧ s2  p1 ∧ p2
p1 ∧ p2  q(*)
s1 ∧ s2  q
The last inference (*) is by Transitivity, which is obtained by Strengthening and Cut. This
proves that G0 = (s1 ∧ s2  q) holds.

15
One would claim that an abductive consequence relation should not be the same as that for deduction, sothat some restriction is put in the condition of a rule to work with it under an abductive context [18, 48].
16
In input/output logics [43], the motivation and rationale of inference rules (9–12,14) are explained fromthe philosophy of norms and deontic logic [44]. Although norms and causal statements are quite different innature, Bochman adopted a similar set of rules for his logic of causality [5] and a causal logic of abduction [6].

21

I NOUELet  be a consequence relation such that α  β iff β  α.17 Writing down themeta-rules (8–12,14) in a logic program at the meta-level, we can define MLA on GLC.
Definition 3.4. A theory of General Logic of Causality (GLC) is represented as a metatheory TGLC that consists of a set of facts with the meta-predicate linked mentioning
object-level causal links, together with the meta-rules:caused(X, Y ) ← linked(X, Y ).

(15)

caused(X, Y ) ← caused(X, Z) ∧ (Z  Y ).

(17)caused(X, Y ) ← (X  Z) ∧ caused(Z, Y ).

(16)

caused(X ∧ Y, Z) ← caused(X, Z) ∧ caused(Y, Z).

(18)caused(X, Y ∨ Z) ← caused(X, Y ) ∧ caused(X, Z).

(20)

caused(X, Y ) ← caused(X, Y ∧ Z) ∧ caused(Z, Y ).

(19)

A theory for GLC-based MLA is an MLA framework (TGLC , AGLC , G), in which AGLC isa set of facts with the meta-predicate linked and G is a fact with the meta-predicate caused.
In the same way as SLC-based MLA, law abduction with predicate invention is realizedin GLC-based MLA. A conjunctive cause can be abduced in the form of linked(g, s ∧ t) in
GLC rather than a disjunction of linked literals in SLC. Compared with SLC, GLC is more
complete but generally needs more inference steps for an underlying abductive procedure,
since there are more possible rule instances to be examined in each step so that the possible
search space for GLC is larger than that for SLC.
As mentioned earlier, GLC does not provide a specific correspondence between themeta-level and object-level formulas. Instead, the correctness of MLA is given with respect
to the inference system.
Proposition 3.0.3. Suppose a GLC-based MLA (TGLC , AGLC , G). Let E be a conjunctionof atoms from AGLC . Then, E is an explanation of G = caused(g, s) if and only if (s  g)
is derived from TGLC ∧ E using the inference rules (8–12,14).

Proof. Given a causal network N , we can show that G = caused(g, s) is derived from
N and the rules (15–20) if and only if (s  g) is derived from N using the inference
rules (8–12,14). Let T interchangeably denote both a logic program defined with N andthe rules (15–20) and the inference system with N and the rules (8–12,14). The proposition
then easily follows by replacing T with TGLC ∧ E.
17

If α and β are conjunctions of propositional atoms, α  β is equivalent to α ⊆ β, where a conjunction is
identified with the set of its conjuncts.

22

M ETA -L EVEL A BDUCTION
In summary, GLC offers a logic for causal networks in a general setting. Unlike SLC,
GLC does not need a translation of meta-level formulas into object-level formulas. Like
SLC, GLC-based MLA realizes 2nd-order existential abduction at the object level.

3.3 Causality Logic with Inhibition
In GLC-based MLA, the meaning of causes was not explicitly argued, yet links in a causal
network have been of one kind, i.e., linked(g, s), representing that g directly depends ons somehow. Depending on applications, GLC can give a specific semantics to the relation
linked; but once its semantics is determined, it is applied to all causal statements. However, this is not desirable in many real-world situations, since there are multiple types of
causalities in general. At least, we can easily consider two types of causalities, i.e., positive
and negative causal effects. With this regard, we can understand that each arc of the form
linked(g, s) only represents positive effects.
In [31], applicability of MLA is thus extended to deal with networks containing both
positive and negative causal effects. Such networks are seen in many domains like physics,
ecology, economics and social science. In particular in biological domains, inhibition effects negatively in gene regulatory, signaling and metabolic pathways. Two types of directcausal relations, triggered and inhibited, are thus introduced as meta-predicates in [31].
Note that we still do not give any specific meaning to positive and negative causality but
just require that they should be opposed notions. The notion of causal chains is also divided
into two types: the positive one (promoted) and the negative one (suppressed), respectively
corresponding to chained relations of triggered and inhibited.
Definition 3.5. A theory of Logic of Causality with Inhibition (LCI) is represented asa meta-theory TLCI that consists of a set of (disjunctive) facts with the meta-predicatestriggered and inhibited mentioning object-level causal relations, together with the metarules:
promoted(X, Y ) ← triggered(X, Y ).

(21)

promoted(X, Y ) ← inhibited(X, Z) ∧ suppressed(Z, Y ).

(23)

suppressed(X, Y ) ← inhibited(X, Z) ∧ promoted(Z, Y ).

(25)

promoted(X, Y ) ← triggered(X, Z) ∧ promoted(Z, Y ).

(22)

suppressed(X, Y ) ← inhibited(X, Y ).

(24)

suppressed(X, Y ) ← triggered(X, Z) ∧ suppressed(Z, Y ).

(26)

A theory for LCI-based MLA is an MLA framework (TLCI , ALCI , G), in which ALCI isa set of (disjunctive) facts with the meta-predicates triggered and inhibited and G is a(disjunctive) fact with the meta-predicate promoted and suppressed.
23

I NOUE
LCI-based MLA reduces to SLC-based MLA when there is no inhibitor in a causal
network. In this case, the first two axioms (21) and (22) for promoted have the samestructure as (5) and (6). In the presence of inhibitors, we assume that a node X can bepromoted if an adjacent inhibitor for X is blocked (23). On the other hand, a negative
causal chain to X can be established if negative influence is propagated to X either directly
by an adjacent inhibitor of an active (24) or promoted (25) item or indirectly by a trigger ofsuppressed (26). In this way, we can establish the connection between X and Y by mixing
both positive and negative links. By (21–26), all possible paths from a source to a target,
which is either positive or negative, can be obtained by meta-level abduction.
In this formalization of LCI-based MLA, both promoted(g, s) and suppressed(g, s)
can be explained at the same time. This is not surprising because we can give any virtual
goal and abduction can achieve it either positively or negatively. Although different sets of
abducibles lead to different consequences, incompatible conclusions can be derived fromthe same set of abducibles. To detect such a potential inconsistency of a combination of
abducibles, the following integrity constraint can be placed at the meta-level.
← promoted(X, Y ) ∧ suppressed(X, Y ).

(27)

Incorporation of negative effects in causal networks introduces non-linear dynamics into
the system. In the logical setting, this phenomenon is related to nonmonotonic reasoning.
For example, biological networks are modeled to prefer inhibition to activation in [31]: atrigger of g works if there is no inhibitor for g, but if an inhibitor is added then the trigger
stops working. MLA has been combined with default reasoning in [31] to implement thisnonmonotonicity in the framework of abduction with default assumptions [33]. Alternatively, this nonmonotonic version can be expressed by replacing the three rules (21,22,26)
containing triggers with normal defaults in default logic [66]:triggered(X, Y ) : promoted(X, Y ),promoted(X, Y )triggered(X, Z) ∧ promoted(Z, Y ) : promoted(X, Y ),promoted(X, Y )triggered(X, Z) ∧ suppressed(Z, Y ) : suppressed(X, Y ).
suppressed(X, Y )Applications to p53 signal networks [64] are presented in [31] as case studies, in which
meta-level abduction reproduces theories explaining how tumor suppressors work [75].
Analysis of such abstract signaling networks, although simple, tackles a fundamental inference problem in computer-aided scientific research including Systems Biology, such that,given an incomplete causal network, the goal is to infer possible connections and functions of network entities in order to reach the target entities from the sources.18 Network
18

Several logic-based models of biological systems and reasoning on them can be found in [16].

24M ETA -L EVEL A BDUCTION

UV

p53
Mdm2
A

X

B
C

cancerFigure 1: Completion of the p53 network [75, 31]
completion in signaling networks is recognized particularly important, since it is hard toobserve activity levels and quantities of proteins in living organisms [1]. This feature of
non-observability also exists in completing causal networks with respect to physical skills
for cello playing [32], in which internal causation in human bodies cannot be observed directly. Hence abduction in these domains is inevitably unobservable fact/law abduction.
Moreover, it is micro-part abduction or hypothetical cause abduction defined by Schurz[71], both of which belong to 2nd-order existential abduction.
Inoue et al. [31] show how the nonmonotonic version of LCI-based MLA works for thefollowing biological scenario made by [75].
Ultraviolet light (UV) causes stress, which influences the growth of tumors andinduces the upregulation of the p53 protein. p53 leads to protect the target cellfrom cancer by binding its transactivator domain to the promoters of the gene.
But the activity of p53 in the cell is influenced by its interactions with otherproteins. The mediator complex p53/Mdm2, which is formed by binding thetransactivator domain of p53 to Mdm2, inhibits p53 from tumor suppression.
This scenario can be simply represented at the meta-level by the facts:
triggered(cancer, uv),

triggered(p53, uv),

inhibited(cancer, a),triggered(a, p53),

inhibited(a, b),(28)

triggered(b, p53 ∧ mdm2),

where “a” is the inhibitory domain of p53 and “b” is the p53/Mdm2 complex. In [31], theatom of the form triggered(X, Y ∧ Z) is replaced with the disjunction (triggered(X, Y ) ∨25

I NOUE
triggered(X, Z)) like (4) in SLC.19 Alternatively, we can introduce meta-axioms to deal
with a conjunction in an argument of promoted/2 and suppressed/2 like (16–19) in GLC.20
The scenario is depicted in the solid lines of the causal network in Figure 1. Note that effectof p53 in the target cell is nonmonotonic. If only UV exists, cancer may appear. But thenp53 is activated by UV, which suppresses cancer. Then, if Mdm2 is introduced and is boundwith p53, cancer appears again. That is, inhibiting an inhibition (double inhibitions) leads
to promotion.
In the setting of [75], a biologist considers some another tumor suppressor gene X,
whose function is not well known but whose mutants are highly susceptible to cancer. In
an experiment, exposure of the cell to high level UV does not lead to cancer, under theconditions that the initial concentration of Mdm2 is high and that a high level of gene
expression of the X protein is also observed. Hence we are given an empirical rule
∃S(suppressed(cancer, S)),

which should be achieved by MLA with the abducibles {triggered(_, _), inhibited(_, _)}and by finding a source S of this suppression. The goal thus cannot be given as a simple
input-output pair, but should contain an unknown source S, since it is hard to identify which
conditions suppress cancer.
Then LCI-based MLA produces the following hypothesis among others:
triggered(x, uv) ∧ ∃Y (triggered(Y, p53 ∧ x) ∧ inhibited(b, Y )).

(29)

This hypothesis is depicted in dashed lines in Figure 1. The variable Y in (29) represents
a new complex synthesized from X and p53. Hence, MLA generates a new mutant Y (“C”
in Figure 1) by combining X and p53, which then inhibits the existing complex b. The
hypothesis (29) indicates that X influences p53 protein stability. UV causes stress then induces high expression of X, which then binds to p53, so p53 is stabilized and formation of
Mdm2-p53 complex is prevented. Hence, p53 (“A”) can be functional as a tumor suppressor. This solution inverts the activation effect of double inhibitions again, that is, the triple
inhibitions finally result in the suppression. The hypothesis (29) is actually suggested in[75], but an important thing here is that all abduced links in such a hypothesis are newly
generated by MLA and that the new node Y is automatically invented during inference.21
19
In [31], the correspondence between meta-level atoms and object-level formulas are given as follows:
A trigger triggered(g, s) is interpreted as (g ← s) ∧ (¬g ← ¬s) at the object level, and an inhibitor
inhibited(g, s) is interpreted as (¬g ← s) ∧ (g ← ¬s) at the object level. It turns out that the implication ←in this correspondence cannot be equivalent to material implication in propositional logic.
20
This extension of GLC to deal with both positive and negative causal effects has not yet been completed,
and its nonmonotonic version with default logic is left open too.
21
In contrast, Tran and Baral [75] consider an action theory for this p53 network, and prepare all possible
ground candidate nodes and links as abducibles in advance. Hence, [75] performs selective law abduction, in
which no new concept is automatically generated.

26

M ETA -L EVEL A BDUCTION
Those obtained hypotheses are important in the sense that the activation/inhibition mechanism of p53 is linked to some proteins that might not have been found out yet. MLA is thus
crucial for this discovery task, and inferred hypotheses can suggest to scientists necessary
experiments with gene knockout mice as minimally as possible.
In summary, LGI-based MLA can realize 2nd-order existential abduction, can abduce
unobservable facts, new rules and new concepts, and can invent new objects and relations
at the object level. Hence, LGI-based MLA is creative abduction.

4 Law/Theoretical-Model Meta-Level Abduction
Definition 4.1. A theory for law MLA is an MLA framework (T, A, G), in which A is a setof rules from the language L.

The difference between fact MLA and law MLA lies only in the form of their explanations. As we have already mentioned in Section 2, selective law abduction can be simulated
by fact abduction by the naming method. Selective law MLA is hence reduced to fact MLA
in Section 3. On the other hand, theoretical-model abduction and law abduction are not necessarily distinguished from the point of view of computation, as discussed in Section 2.1.
Hence, we consider creative law MLA here.
Law MLA could be called meta-level induction, since abduction of rules, i.e., explanatory induction is performed at the meta-level. Theoretically, any ILP system that is capableof law abduction can be used for law MLA. But, what can be abduced by law MLA? The
main motivation of law MLA is to abduce meta-level axioms. Such meta-theories are foundfor deduction, nonmonotonic reasoning [37], abduction [18, 48, 42, 3, 6], causal theories[43, 5], deontic logic [44], etc. If we abduce laws for deduction by law MLA, it is abduction of deductive systems. If abduced laws are for abduction, it is abduction of abductive
systems. Therefore, by law MLA, we can devise a part of deductive/abductive system.
One way to construct meta-rules for a domain by law MLA is to learn them from examples and a part of background meta-theory in the spirit of meta-level induction. For
example, let T be a theory consisting of an incomplete meta-theory from TSLC and themeta-expression of direct causal links:
T = (6) ∧ linked(a, b) ∧ linked(b, c).

Given G = caused(a, c) as the observation, CF-induction [30] and some other ILP systems
can construct the meta-rule (5) as a hypothesis to complete the meta-axioms of TSLC . We
can see that this abductive computation is nothing but an ordinary hypothesis-finding in ILP,
but a meta-rule is abduced with respect to a meta-theory.
One might think that we can use meta-meta-rules to abduce meta-rules by fact MLA
in the same way that meta-rules are used to abduce object-level rules by fact MLA. In this27

I NOUEcase, a meta-meta-interpreter is necessary, but often the same meta-interpreter is reused
as a meta-meta-interpreter in logic programming. Now let us see the work by Bundy andSterling [8], in which a meta-theory of algebra is proposed to derive new control informationby proving theorems and a meta-meta-level language is developed to control the search forproving theorems by expressing proof plans. Here is an example of meta-meta-rules in [8]:prove-by-induction(Conjecture) ←structural-induction(Conjecture, Scheme) ∧prove-base-case(Conjecture, Scheme) ∧prove-step-case(Conjecture, Scheme).

Using this meta-meta-axiom and given a target equation eq1 and a partial inductive proof
including the base case proof, abducing prove-step-case(eq1, list-recursion) by fact MLA
does not involve law abduction. Although this hypothesis tells us that the scheme of list
recursion can be used for a proof plan, the meta-meta-atom alone does not provide a metatheory on how to prove the step case but should be defined by another meta-meta-axiom.
That is to say, we must distinguish meta-meta-level abduction from meta-level abductionof meta-theories, as in the case where meta-level abduction was distinguished from metaabduction in Section 1.3.

5

Representation Meta-Level Abduction

Hoffmann’s representation abduction or meta-diagrammatic abduction [27] requires representational changes for a theory. As its name indicates, meta-diagrammatic abduction itself
is a kind of meta-level abduction that abduces a new domain theory. This implies that theunderlying assumption of the language L of MLA considered in Section 3 such that the domain of L should contain all possible individuals might not be adequate. Hence, it is hard
to realize representation abduction even at the object level, regardless of with or without
MLA.22 Consideration of representation MLA is thus done by a thought experiment.
A representation change often appears in scientific discovery as a paradigm shift whena totally new theory emerges. For example, non-Euclidean geometry changed geometricalpostulates of Euclidean geometry, and classical physics was replaced by quantum physics
with revolutionary ideas. In the course of such a discovery, a new representation system isabduced, and a new axiom set is introduced into the new framework. Hence abducing anessential change in a domain is representation abduction at the object level. On the other
hand, for such invented non-Euclidean geometries, Hilbert studied axiomatic systems of22

Hoffmann [27] includes selective meta-diagrammatic abduction in his classification (see Table 2), which
could be computed by MLA. However, it is not clear how a collection of representations is stored in advance.

28

M ETA -L EVEL A BDUCTION
both Euclidian and non-Euclidian geometries in his “Foundations of Geometry” [22]. As
Hilbert later invoked the term metamathematics, his work on axiomatic systems exactly
describes a meta-theory of those variants of geometries. Therefore, consideration of theseaxioms is exactly representation MLA.
In general, extraction of a hidden logic from a domain or inventing a new logic fromobservations in a new situation invokes representation MLA. This inference would also benecessary for AI programs or agents to automatically learn logics from observations. For
this, Sakama and Inoue [70] consider the question “Can machines learn logics?”:
There are two machines A and B. The machine A is capable of deductive reasoning: it has a set L of axioms and inference rules. Given a set S of formulas
as an input, the machine A produces (a subset of) the logical consequences
T h(S) as an output. On the other hand, the machine B has no axiomatic system for deduction, while it is equipped with a machine learning algorithm C.
Given input-output pairs (S1 , T h(S1 )), . . . , (Si , T h(Si )), . . . of A as an input
to B, the problem is whether one can develop an algorithm C which successfully reproduces the axiomatic system L for deduction.
Meta-level one-step deduction rules including MP are learned in [70]. Although this problem supposes learning of deduction, the scenario can be applied to learning abduction andother non-standard logics. The input examples to representation MLA is a set of observed input-output pairs in the form of pairs of input formulas and their consequencesin the target logic. These input examples can be either given in such an abstract way as({p ∨ q, p → r, q → r)}, {p ∨ q, r}) or can come from a specific domain like Euclidian geometry in the form of pairs of domain axioms and their consequences. Moreover,they are not necessarily given from a teacher, but can be implicitly hidden in log files ofdynamic systems or in dialogues with unknown agents, so that the machine should identify
those input-output relations automatically. Then the output of representation MLA is a setof meta-theoretical inference rules for the domain or inference patterns of the agents. To
solve this problem, simple machine learning from observable samples would not work, but
abduction of the underlying logic is necessary.23 Once a logic is learned in this way, it must
be refined or revised through a continuous, cyclic process between observations and abduction [19, 45] on meta-theoretical relations. This process would thus introduce a dynamics
of incremental perfection of theories.
A similar mechanism is necessary for one to acquire or learn complex items like skills
of music playing, cultural knowledge, science and advanced technologies, in which one
23

There is an attempt to learn the underlying rules of dynamics from traces of state transition [34], which
is applied to learn cellular automata and gene regulatory networks. This idea could be extended to learn theunderlying logic in more general settings.

29

I NOUE
tries to explain them or understand what causes them with the intellectual faculties rather
than by surface learning through experience.

6

Conclusion

The method of meta-level abduction proposed in [32] has been analyzed from the viewpoint of patterns of abduction. By generalizing the framework of MLA in [32], several
patterns of MLA are considered in this paper. Fact MLA is strong enough to realize 2ndorder existential abduction at the meta-level. Law MLA can produce meta-axioms in logics.
Representation MLA can abduce logics themselves.
To realize law abduction by fact MLA, we have mainly focused on the logic of causality,i.e., SLC and GLC, in the form of causal networks. Other meta-theories like control ofinference [59, 8] can be used for meta-level abduction too. For such meta-programs, MLA
will abduce a control atom like [60], but unlike SLC/GLC-based MLA this abduction is notlaw abduction any more. Hence, the expectation in Section 2.2 that MLA can lift the typeof abduction to one level higher is not always true, but depends on the meta-theory. SLC
and GLC deal with meta-theories for causal networks, and hence are useful for establishingand reconstructing causal relations between items. That is why fact abduction on thesemeta-theories can lead to law abduction at the object level.
There are several open problems on MLA. Technically, extension of GLC to deal with
inhibition like LCI is a future work. Inhibition effects introduce negative effects, but there
are other types of negations: an inhibition usually prevents causes, and a constraint stipulates that some causal statement never happens. We should distinguish those several meanings of negations in a logical way. For example, the statement that “α is not cause by β”
can be expressed as ¬caused(α, β). Inference of negation could be implemented by either
negation as failure or explicit negation in logic programming. Instead of using default logic
in Section 3.3, we could express default reasoning in logic programs with negation.
So, why don’t we use MLA more in various domains? In [31], it has been argued:
Problem solving with meta-level abduction consists of three steps: (I) designof meta-level axioms; (II) representation of domain knowledge at the metalevel; and (III) restriction of the search space to treat large knowledge. This
work supposes an incomplete network for Step (II), and hence representationof a problem is rather tractable. On the other hand, we have made great effort
on Step (I), by gradually extending the positive causal networks to alternating
axioms and then to nonmonotonic formalization. This formalization process
was based on trial and error by running SOLAR many times.
Law MLA can thus contribute to Step (I) above. We did not discuss about Step (III) in thispaper, and it is left open for both fact MLA and law MLA. This last topic is important not30

M ETA -L EVEL A BDUCTION
only for efficient computation of hypotheses but for one to select appropriate hypotheses
easily. Discovering useful methods for constraint generation is left open, and another useof meta-level abduction such as control of reasoning will be useful too. Finally, refining
theories through continuous uses of meta-theoretical abduction should be more investigated.

Acknowledgements
The author would like to thank reviewers for valuable comments on this paper, and is alsoindebted to Gauvain Bourgne, Andrew Cropper, Christine Froidevaux, Koichi Furukawa,
Stephen Muggleton, Hidetomo Nabeshima, Adrien Rougny, Chiaki Sakama and Yoshitaka
Yamamoto for fruitful discussions on the topic of this paper.

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Received January 2015

36

Abduction and Beyondin Logic Programmingwith Application to Morality
Luís Moniz Pereira
NOVA Laboratory for Computer Science and Informatics
Departamento de Informática, Faculdade de Ciências e Tecnologia
Universidade Nova de Lisboa, Portugal
lmp@fct.unl.pt
Ari Saptawijaya
Faculty of Computer Science at Universitas Indonesia, Depok, Indonesia
saptawijaya@cs.ui.ac.id

Abstract
In this paper we emphasize two different aspects of abduction in Logic Programming (LP): (1) the engineering of LP abduction systems, and (2) application of LP abduction, complemented with other non-monotonic features, tomodel morality issues. For the LP engineering part, we present an implemented
tabled abduction technique in order to reuse priorly obtained (and tabled) abductive solutions, from one abductive context to another. Aiming at the interplay between LP abduction and other LP non-monotonic reasoning, this tabledabduction technique is combined with our own-developed LP updating mechanism – the latter also employs tabling mechanisms, notably incremental tablingof XSB Prolog. The first contribution of this paper is therefore a survey of
our tabled abduction and updating techniques, plus further development of ourpreliminary approach to combine these two techniques.
The second contribution of the paper pertains to the application part. We
formulate a LP-based counterfactual reasoning, based on Pearl’s structural theory, via the aforementioned unified approach of our LP abduction and updating.
We thank the referees for their constructive comments and suggestions. Both authors acknowledge
the support from FCT/MEC NOVA LINCS PEst UID/CEC/04516/2013. Ari Saptawijaya acknowledges the support from FCT/MEC grant SFRH/BD/72795/2010. We thank Terrance Swiftand David Warren for their expert advice in dealing with implementation issues in XSB, andEmmanuelle-Anna Dietz for fruitful discussions on counterfactuals.
Vol. 3 No. 3 2016
IFCoLog Journal of Logic and its Applications

Pereira and Saptawijaya
The formulation of counterfactuals allows us to subsequently demonstrate its
applications to model moral permissibility, according to the Doctrines of Double
and Triple Effect, and to provide its justification. The applications are shown
through classic moral examples from the literature, and tested in our prototype,
Qualm, an implementation reifying the presented unified approach.

1

Introduction

Abduction has been well studied in Logic Programming (LP) [11, 26, 25, 13, 17, 4],establishing theoretical results that support the implementation of LP abduction
systems. LP abduction has been applied in a variety of areas, such as in diagnosis[18], planning [15], scheduling [28], reasoning of rational agents and decision making[31, 42], knowledge assimilation [27], natural language understanding [5], security
protocols verification [1], and systems biology [50]. Such applications demonstrate
the potential of LP abduction and motivates further advancements of LP abduction
systems to prepare them for even more challenging applications.
In this paper we report two results of our recent research in LP abduction concerning: (1) the engineering of LP abduction systems, and (2) application of LP
abduction, complemented with other non-monotonic reasoning, to modeling a number of issues in morality.
Engineering of LP abduction systems We address two challenges in engineering a LP abduction system. For one, in abduction, finding some best explanations(i.e. adequate abductive solutions) to an observed evidence, or finding assumptions
that can justify a goal, can be costly. It is often the case that abductive solutions
found within one context are also relevant in a different context, and can be reusedwith little cost. In LP, absent of abduction, goal solution reuse is commonly addressed by employing a tabling mechanism [62]. Therefore, tabling appears to beconceptually suitable for abduction, so as to reuse abductive solutions. In practice, abductive solutions reuse is not immediately amenable to tabling, because such
solutions go together with an abductive context.
We conceptualize in [57] a tabled abduction technique, to benefit from LP tablingmechanisms in contextual abduction, i.e., to reuse priorly obtained (and tabled)
abductive solutions, from one abductive context to another. The technique is underpinned by the theory of Abdual [4], for computing abduction in LP over WellFounded Semantics [66]. The tabled abduction technique is implemented in XSBProlog [64], resulting in a LP abduction system Tabdual. It concretely realizes theabstract theory of Abdual, while also taking care of pragmatic issues, due to thepresence of tabled abduction, to foster the practicality of Tabdual.
38

Abduction and Beyond in LP with Application to Morality
Another challenge we address is the interplay between LP abduction and other
LP non-monotonic reasoning. It is apparent that when it comes to applications,notably in reasoning of rational agents and decision making, abduction needs tobe enriched with other features. For instance, such applications are susceptible
to knowledge updates and changes, due to incomplete information of the world or
the evolution of the agents themselves. In this case, abduction systems need to becomplemented with LP updating feature.
Aiming at that goal, we propound yet another use of tabling mechanisms, notably the so-called incremental tabling, for LP updating. Incremental tabling, alsoavailable in XSB Prolog, is an advanced tabling feature that ensures the consistencyof answers in a table with all dynamic information on which the table depends. It
does so by incrementally maintaining the table, rather than by recomputing answers in the table from scratch to keep it updated. This incremental table maintenance is achieved by automatically propagating assertion or retraction of information
bottom-up; such propagation is typically triggered by the invocation of a given goal.
In [56, 55], we conceptualize a technique with incremental tabling that permits areconciliation of high-level top-down deliberative reasoning about a goal, with autonomous low-level bottom-up world reactivity to ongoing updates. The technique,dubbed Evolp/r, is theoretically based on Dynamic Logic Programs [3] and its
subsequent development, Evolving Logic Programs (Evolp) [2].
In [58] we preliminarily bring LP abduction and LP updating into a unified
approach based on the aforementioned two implementation techniques: Tabdual
and Evolp/r, respectively. It aims at the use of joint tabling of both features, so as
to benefit from reusing the abductive solutions obtained in one context for another,while also allowing top-down goal-driven incremental tabling of updates by upwards
propagation. The approach in [58] borrows the concept of hypothesis generation[42]. While this concept permits generating only abductive explanations relevant
for the problem at hand, the preliminary unified approach [58] limits the benefit oftabled abduction. In this paper we further develop this unified approach by leaving
out the hypotheses generation concept in order to uphold the idea and the benefitof tabled abduction. The unified approach is implemented and applied to formulate
counterfactual reasoning, which in turn is used to address morality issues, as we
detail next.
Applications of LP abduction to Morality Computational morality (alsoknown as machine ethics, machine morality, artificial morality and computational
ethics) is a burgeoning field of inquiry that emerges from the need of imbuing autonomous agents with the capacity of moral decision-making. It has particularly
attracted interest from the artificial intelligence community and has brought to39

Pereira and Saptawijaya
gether perspectives from various fields, amongst them: philosophy, cognitive science,
neuroscience and primatology. The overall result of this interdisciplinary researchis therefore not only important for equipping agents with the capacity of making
moral judgments, but also for helping us better understand morality, through thecreation and testing of computational models of ethical theories.
The potential of LP to computational morality has been reported in [30, 45,21]. In [59] we summarize our work emphasizing the application of LP abduction
and other LP-based reasoning for modeling the dynamics of knowledge and moral
cognition of an individual agent, where we employ LP abduction jointly with othernon-monotonic reasoning. The reader is also referred to [46] which summarizes:(1) our study with other co-authors on collective morals and the emergence, in a
population, of evolutionarily stable moral norms, of fair and just cooperation, inthe presence of cognitive aspects, such as intention recognition, commitment, andmutual tolerance through apology; and (2) our view on bridging individual andcollective morality, as they are necessarily intertwined.
In the second part of the paper we report the continuation of our prior work formodeling the dynamics of knowledge and moral cognition of an individual agent,
now benefiting from the integration of Tabdual and Evolp/r discussed earlier. It
is important to note that the application we show in this paper neither aims at defending any considered moral principles nor resolving the dilemmas appearing in theexamples, as even philosophers naturally are split over opinions on them. Instead, its
purpose is to show that our unified approach of LP abduction and updating (with its
implementation) are capable and appropriate for expressing viewpoints on morality
issues discussed herein, in accordance with the results and the theory argued in theliterature.
We start by applying LP abduction and updating to model counterfactual reasoning, as people typically reason about what they should or should not have done
when they examine decisions in moral situations [36, 34, 52, 38]. It is therefore
natural for them to engage counterfactual thoughts in such settings.
Counterfactuals capture the process of reasoning about a past event that did notoccur, namely what would have happened had this event occurred; or, vice-versa,
to reason about an event that did occur but what if it had not. An example, takenfrom [8]: Lightning hits a forest and a devastating forest fire breaks out. The forest
was dry after a long hot summer and many acres were destroyed. One may think ofa counterfactual about it, e.g., “if only there had not been lightning, then the forest
fire would not have occurred”.
Counterfactuals have been widely studied in philosophy [32, 9, 20], psychology[35, 51, 36, 8, 14, 38], as well as from the computational viewpoint [19, 41, 6, 40, 67].
Research in LP to model counterfactual reasoning is nevertheless still limited. In
40Abduction and Beyond in LP with Application to Morality[41], counterfactuals, based on Lewis’s counterfactuals [32], are evaluated using contradiction removal semantics of LP. The semantics defines the most similar worlds
by removing contradictions from the associated program, obtaining the so-called
maximal non-contradictory submodels of the program. In [6], Probabilistic LP language P-log with Stable Model Semantics is employed to encode Pearl’s probabilistic
causal model [40] of counterfactual reasoning. In [67], Pearl’s approach is encodedusing a different Probabilistic LP, viz., CP-logic. None of these LP-based approaches
involves abduction in modeling counterfactuals.
In this paper we specifically adopt Pearl’s approach [40], abstaining from probabilities, but resorting to LP abduction and updating. LP abduction is employedfor providing background conditions from observations made or evidences given,
whereas defeasible LP rules allow adjusting the current model via hypothetical updates of intervention. Therefore, our first application is to formulate an implemented
evaluation procedure for counterfactuals via LP abduction and updating. The implementation is based on our unified approach of Tabdual and Evolp/r, andnamed Qualm (its ongoing development is available from https://github.com/
merah-putih/qualm). Qualm is appropriately used for applications concerned withthe reasoning of agents, in particular those involving counterfactuals when probabilities are not known or needed. We specifically look into its challenging applications
in moral reasoning (for our related work on probabilistic moral reasoning, cf. [21]).
We apply counterfactuals (by resorting to its LP formulation), together with a
combination of LP abduction and updating, to model morality issues. Counterfactuals are engaged to examine moral permissibility of an action by distinguishingwhether an effect of an action is a cause for achieving a morally dilemmatic goal or
merely a side-effect of that action. The distinction is essential for establishing moral
permissibility from the viewpoints of the Doctrines of Double Effect [37] and of
Triple Effect [29], as scrutinized herein through several classic moral examples fromthe literature. We also apply counterfactuals to address a form of moral justification.
Through examples, we show how compound counterfactuals may provide moral justification for what was done due to lack of the current knowledge. We also touchupon Scanlon’s contractualism [60] as a reference for employing counterfactuals toprovide a justified, but defeasible, exception to permissibility of actions.
The paper is organized as follows. Section 2 reviews necessary background on
abduction in LP. We discuss the LP engineering part by presenting the concept oftabled abduction, LP updating with incremental tabling, and the interplay between
them in Section 3. The application part is subsequently elaborated in the next
two sections: the formulation of counterfactuals with LP abduction and updating
is detailed in Section 4, whereas Section 5 presents applications of counterfactuals
to model morality issues. The reader may skip the technical LP engineering part41Pereira and Saptawijaya(Section 3) without loss of understanding of the application part. We conclude in
Section 6 with some remarks on related issues that may trigger future work.

2

Abduction in Logic Programming

We start by recapping basic notation in LP and review how abduction is expressedand computed in LP.
A literal is either an atom B or its default negation not B, named positive andnegative literals, respectively. They are negation complements to each other. The
atoms true and false are true and false, respectively, in every interpretation. A logic
program is a set of rules of the form H ← B, naturally read as “H if B”, where its
head H is an atom and its (finite) body B is a sequence of literals. When B is empty(equal to true), the rule is called a fact and simply written H. A rule in the formof a denial, i.e., with false as head, is an integrity constraint (IC).
In LP, an abductive hypothesis (abducible) is a 2-valued positive literal Ab or
its negation complement Ab∗ (denotes not Ab), whose truth value is not initiallyassumed, and it does not appear in the head of a rule. An abductive logic programis one allowing abducibles in the body of rules. An observation O is a set of literals,
analogous to a query in LP; we use the usual LP notation ?- to denote a query.
Abduction in LP can be accomplished by a top-down query-oriented procedurefor finding a query solution, i.e., a consistent abductive solution, by need. The
solution’s abducibles are leaves in its procedural query-rooted call-graph, where thegraph is recursively generated by the procedure calls from literals in bodies of rules
to heads of rules, and thence to the literals in a rule’s body.
Example 1. A library in a city is close with two possible explanations: it is closein the weekend, or when it is not weekend, there are no librarians working. Thesedays, librarians are often absent from their work because they participate in a strike.
On the other hand, a museum in that city is only close when there is a special visit
by important guests. This example can be expressed by a logic program below(weekend, strike, and special_visit are abducible atoms):
close_library ← weekend.
close_library ← weekend∗ , absent.
absent ← strike.
close_museum ← special_visit.

Consider the query ?- close_library. It induces the call-graph with close_libraryas its root and, through procedure calls, it ends with two leaves: one leaf containing
abducible weekend as an abductive solution, and the other containing abducibles[weekend∗ , strike] as another solution.
The correctness of this top-down computation requires the underlying semantics
to be relevant, as it avoids computing a whole model (to warrant its existence) in
42

Abduction and Beyond in LP with Application to Morality
finding an answer to a query. Instead, it suffices to use only the rules relevant to thequery – those in its procedural call-graph – to find its truth value. In Example 1,
rule close_museum ← special_visit is not relevant to answer the aforementioned
query, and thus is not involved in the call-graph.
The 3-valued Well-Founded Semantics or WFS [66] enjoys this relevancy property [12], i.e., it permits finding only relevant abducibles and their truth value via
the aforementioned top-down query-oriented procedure. Those abducibles not mentioned in the solution are indifferent to the query, e.g. abducible special_visit isindifferent to query ?- close_library. Due to its relevancy property, the approaches
in this paper are based on WFS. Moreover, the XSB Prolog system [64], wherein all
the prototypes of our research are implemented, computes the WFS.

3

Tabling in Contextual Abduction and LP Updating

The progress of LP has been promoting new techniques for engineering LP-based
systems, most notably tabling mechanisms [62]. Tabling affords solutions reuse,rather than recomputing them, by keeping in tables subgoals and their answers
obtained by query evaluation. It is supported, to a different extent, by a number of
Prolog systems.
In abduction, finding explanations to an observed evidence, or finding assumptions that can justify a goal, can be costly. We discuss in Section 3.1, that abduction
may benefit from tabling mechanisms by reusing priorly obtained abductive solutions from one abductive context to another. We subsequently discuss a unified
approach of LP abduction and tabling via their joint tabling, in Section 3.2.

3.1

Tabling in Contextual Abduction

Motivation and Idea Example 2 illustrates the motivation and the idea of tabledabduction.
Example 2. A plane may be hit by lightning, when it flies in a stormy area and at a
low altitude. A case where it flies into a stormy area is when it enters cumulonimbus
cloud (a dense towering cloud associated with thunderstorms). Being hit by lightningin such a stormy area may crash the plane. This example is expressed in the program
below (cumulonimbus and low_altitude are abducible atoms):
storm ← cumulonimbus.
hit_lightning ← storm, low_altitude.
crash ← hit_lightning, storm.

Suppose three queries are invoked, asking for individual explanations of storm,
hit_lightning, and crash, in that order. The first query, ?- storm, is satisfied
43Pereira and Saptawijaya
simply by having [cumulonimbus] as the abductive solution for storm, and tablingit.
Executing the second query, ?- hit_lightning, amounts to satisfying the two
subgoals in its body, i.e., abducing low_altitude followed by invoking storm. Since
storm has previously been invoked, we can benefit from reusing its solution, insteadof recomputing, given that the solution was tabled. That is, query ?- hit_lightning
can be solved by extending the current ongoing abductive context [low_altitude] of
subgoal storm with the already tabled abductive solution [cumulonimbus] of storm,
yielding the abductive solution [cumulonimbus, low_altitude].
The final query ?- crash can be solved similarly. Invoking the first subgoal
hit_lightning results in the priorly registered abductive solution: [cumulonimbus,
low_altitude], which becomes the current abductive context of the second subgoal storm. Since [cumulonimbus, low_altitude] subsumes the previously obtained(and tabled) abductive solution [cumulonimbus] of storm, we can then safely take[cumulonimbus, low_altitude] as the abductive solution to query ?- crash.
Example 2 shows how a tabled abductive solution [cumulonimbus], the abductive solution of the first query ?- storm, can be reused from one abductive context
of storm (viz., [low_altitude] in the second query, ?- hit_lightning) to anothercontext (viz., [cumulonimbus, low_altitude] in the third query, ?- crash). In practice a repeatedly called rule, like storm, may have a body comprising many subgoals,
causing potentially expensive recomputation of its abductive solutions, if they havenot been tabled.
Contextual Abduction with Tabling Tabled abduction with its prototype
Tabdual consists of a program transformation that provides self-sufficient programtransforms, which can be directly run to enact abduction by means of Tabdual’s
library of reserved predicates. We describe below how the idea in Example 2 is conceptually realized by contextual abduction with tabling in Tabdual. The reader isreferred to [54] for a more formal and technical treatment of the transformation.
Example 2 indicates two key ingredients of tabled abduction in Tabdual:
1. Abductive contexts, which relay the ongoing abductive solution from one subgoal to subsequent subgoals in the body of a rule, as well as from the head tothe body of a rule, via input and output contexts.
2. Tabled predicates, which table the abductive solutions for predicates definedin the program, such that they can be reused from one abductive context toanother.
44

Abduction and Beyond in LP with Application to Morality
Example 3 shows how these two ingredients figure in the program transform of
Example 2.
Example 3. Consider rule storm ← cumulonimbus of Example 2. A new predicate
stormab is introduced to table the abductive solution of storm,1 defined as:stormab ([cumulonimbus]).
It is a simple tabled fact, as the rule of storm is defined solely by the abducible
cumulonimbus, without any other literals in its body.
We defined similar tabled predicates hit_lightningab and crashab for the othertwo rules (of hit_lightning and crash, respectively):
hit_lightningab (E) ← storm([low_altitude], E).

crashab (E2 ) ← hit_lightning([ ], E1 ), storm(E1 , E2 ).

(1)
(2)

Predicate hit_lightningab (E) is a predicate that tables, in E, the abductive
solution of hit_lightning. Its definition follows from the original definition of
hit_lightning. Two extra (and new) parameters, that serve as input and output
contexts, are added to the subgoal storm. Notice how low_altitude, the abducible
appearing in the body of the original rule of hit_lightning, becomes the input abductive context of storm. This subgoal storm requires a rule of storm(I, O), definedin rule (3) below, that reuses the solution tabled in stormab to produce the output
abductive context O from its input context I.
Predicate crashab tables the abductive solution for crash in its argument E2 . The
rule expresses that the tabled abductive solution E2 of crashab is obtained by relaying the ongoing abductive solution stored in context E1 from subgoal hit_lightning
to subgoal storm in the body, given the empty input abductive context (i.e., [ ]) of
hit_lightning (because there is no abducible by itself in the body of the original
rule of crash).
Now that we have tabled predicates, the remaining rules to define are those thatreuse the tabled solutions. That is, we define the rule of storm(I, O) that reuses
the solution tabled in stormab , and produce the output abductive context O fromits input context I (the other rules hit_lightning(I, O) and crash(I, O) are definedsimilarly):
storm(I, O) ← stormab (E), produce_context(O, I, E).

(3)This rule expresses that the output abductive solution O of storm is obtained fromthe solution entry E of stormab and the given input context I of storm, via the1

In XSB Prolog, this tabling effect is achieved by declaring predicate stormab as a tabled
predicate.

45Pereira and Saptawijaya
Tabdual system predicate produce_context(O, I, E). This system predicate concerns itself with: whether E is already contained in I and, if not, whether there
are any abducibles from E, consistent with I, that can be added to produce O.
For example, produce_context in rule (1) adds the tabled solution cumulonimbus
to the input abductive context [low_altitude] of storm, thus producing the output[cumulonimbus, low_altitude]. On the other hand, in rule (2), the tabled solution cumulonimbus of storm is already contained in the input abductive context
E1 = [cumulonimbus, low_altitude], yielding the latter as the output E2 = E1 .
Note that if E is inconsistent with I then the specific entry E cannot be reusedwith I, produce_context fails and another entry E is sought. For instance, suppose storm is invoked with the input abductive context [cumulonimbus∗ ]. This
goal storm([cumulonimbus∗ ], O) will fail, as produce_context ends up with an inconsistent solution containing cumulonimbus and cumulonimbus∗ . In summary,produce_context should guarantee that it produces a consistent output context O
from I and E that encompasses both.
Abduction under Negative Goals We have seen in Example 3 abduction performed under positive goals via abductive contexts and tabling. A common approach
for abducing under negative goals is by first computing all abductive solutions of its
corresponding positive goal, and then having to negate their disjunction. Tabdual
avoids such computation of all abductive solutions by employing the dual transformation [4] in contextual abduction.
The dual transformation is based on the idea of making negative goals ‘positive’
literals. It thus permits obtaining one abductive solution at a time, just as if we
treat abduction under positive goals. The dual transformation defines for each atom
A and its set of rules R in a program P , a set of dual rules whose head not_A is true
if and only if A is false by R in the employed semantics of P . Note that, instead of
having a negative goal not A as the rules’ head, we use its corresponding ‘positive’
literal, not_A. Example 4 illustrates the main idea of how the dual transformation
is employed in Tabdual.
Example 4. Recall Example 1 and the two rules of close_library:close_library ← weekend.
close_library ← weekend∗ , absent.
The dual transformation will create a set of dual rules for close_library. Thesedual rules falsify close_library with respect to its two rules. That is, the rule ofnot_close_library is defined by falsifying both the first rule and the second rule,denoted below by predicate close_library ∗1 and close_library ∗2 , respectively:
46

Abduction and Beyond in LP with Application to Morality
not_close_library(C0 , C2 ) ← close_library ∗1 (C0 , C1 ), close_library ∗2 (C1 , C2 ).
Note the input and output abductive context parameters that figure in the rule:
C0 and C2 , in the head, and similarly in each subgoal of the rule’s body, whereintermediate context C1 relays the ongoing abductive solution from close_library ∗1
to close_library ∗2 .
Next, we define how the first and the second rules of close_library are falsified,i.e., rules for close_library ∗1 and close_library ∗2 . These rules are naturally definedby falsifying the body of close_library’s first rule and second rule, respectively. In
case of close_library ∗1 : the first rule of close_library is falsified only by abducing
the negation of weekend, viz., weekend∗ . Therefore, we have:close_library ∗1 (I, O) ← weekend∗ (I, O).
Notice that weekend∗ is abduced by invoking the subgoal weekend∗ (I, O). This
subgoal is defined as follows (other abducibles, both positive and negative ones,
transform similarly):
weekend∗ (I, O) ← insert_abducible(weekend∗ , I, O).

(4)where insert_abducible(A, I, O) is a Tabdual system predicate that inserts abducible weekend∗ into input context I, resulting in output context O. As in thepredicate produce_context, it also maintains the consistency of the context, failing
if inserting A results in an inconsistent one.
In case of close_library ∗2 : the second rule of close_library is falsified by alternatively failing one subgoal in its body at a time, i.e. by negating weekend∗ or,alternatively by negating absent.
close_library ∗2 (I, O) ← weekend(I, O).
close_library ∗2 (I, O) ← not_absent(I, O).
where weekend(I, O) is similarly defined as in the above rule of weekend∗ (I, O),
and not_absent(I, O) is defined by the dual transformation for absent.
Consider now query ?- not_close_library([ ], O). Provided the dual rules for
close_library and absent, we obtain a single solution [weekend∗ , strike∗ ]. Note that
insert_abducible rules out [weekend∗ , weekend] as a solution, due to its inconsistency.
Abduction with Integrity Constraints

47

Pereira and Saptawijaya
Example 5. Recall Example 1 plus new information: librarians may also be freefrom working in case the library is being renovated, expressed as:
absent ← renov.
Query ?- close_library([ ], T ) now returns an additional solution, viz. [weekend∗ ,renov]. Let us further suppose that the municipal authority permits any renovations
to take place only on weekends, expressed as an IC below:
f alse ← weekend∗ , renov.
The queries should now respect the IC, too. That is, it should be conjoined with
not_false to ensure that all integrity constraints are satisfied:?- close_library([ ], T ), not_f alse(T, O).
where the dual transformation provides the definition for not_f alse. Note that theabductive solution for close_library is further constrained by passing it to the subsequent subgoal not_false for confirmation, via the intermediate context T . Using
the Tabdual approach, we can easily check that [weekend∗ , renov] is now ruled out
from the solutions.
Tabdual also takes care of abduction over non-ground programs and programswith loops. Abduction over programs with loops sometimes occur in real applications, e.g., in psychiatric diagnosis, cf. [18] that deals with non-stratified negation:“The patient has an Adjustment Disorder if he does not have Alzheimer’s Dementia,
and has Alzheimer’s Dementia if the patient does not have an Adjustment Disorder”,
which corresponds to rules adjustment_disorder ← not alzheimer_dementia andalzheimer_dementia ← not adjustment_disorder. Tabdual also benefits fromother XSB’s features to foster its practicality, e.g., constructing dual rules by need
only. The reader is referred to [54] for details of implementation aspects.

3.2

Combining LP Abduction and Updating

In addition to abduction, an agent may learn new knowledge from the external world
or update itself internally on its own decision in order to pursue its present goal. It
is therefore natural to accommodate abduction systems with knowledge updating.
In Section 3.2.1 we summarize an implemented approach for LP updating, called
Evolp/r, that employs incremental tabling [53], an advanced tabling feature of XSB
Prolog that lends itself to dynamic environments and evolving systems. Having thetwo tabling-based approaches for different purposes, viz., Tabdual and Evolp/r,
we combine them in a unified approach, as discussed in Section 3.2.2.
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Abduction and Beyond in LP with Application to Morality
3.2.1

Evolp/r for LP Updating

Evolp/r is theoretically underpinned by Evolving Logic Programs (Evolp) [2];the latter is itself based on Dynamic Logic Program [3]. Though its syntax andsemantics are based on Evolp, they differ in several respects, which characterize
Evolp/r, as per below. The reader is referred to [55] for a more detailed theoreticalbasis of Evolp/r.
Restriction to Fluent Updates While Evolp allows full-blown rule updates,
Evolp/r restricts updates to state-dependent literals only (commonly known as
fluents), i.e., literals whose truth value may change from one state to another. Syntactically, every fluent F is accompanied by its fluent complement ∼F . Program
updates are enacted by asserting F , whereas its retraction is realized by assertingits complement ∼F at a later state. It thus admits non-monotonicity of a fluent, as
the latter update supervenes the former.
Though updates in Evolp/r are restricted to fluents only, it nevertheless still
permits rule updates by introducing a rule name fluent that uniquely identifies therule for which it is introduced. Such a rule name fluent is placed in the body of a rule
to turn the rule on and off, cf. [48]; this being achieved by asserting or retractingthat specific fluent. For instance, while Evolp allows asserting rule absent ← renov
in Example 5, Evolp/r does this by introducing a unique rule name fluent, sayrule(absent←renov) , in the body of that rule (this is internally done by Evolp/r via
a program transformation):absent ← rule(absent←renov) , renov.

(5)

and asserting the rule name fluent rule(absent←renov) at state T , making it true at T
and thus turning the rule on. On the other hand, asserting ∼rule(absent←renov) at a
subsequent state T 0 > T , making the rule name fluent false later at T 0 , and hence
turning the rule off from then on.
Incremental Tabling of Fluents The first approach of Evolp/r [56] preliminarily exploits the combination of two tabling features in XSB Prolog: (1) incremental tabling [53], which ensures the consistency of answers in tables with all dynamic
facts and rules upon which the tables depend; and (2) answer subsumption [63],which allows tables to retain only answers that subsume others wrt. some orderrelation.
Incremental tabling of fluents is employed in Evolp/r to automatically maintainthe consistency of program states due to assertion and retraction of fluent literals,
whether obtained as updated facts (wrt. extensional fluent) or concluded by rules49Pereira and Saptawijaya(wrt. to intensional fluent). On the other hand, answer subsumption of fluent literals
allows to address the frame problem, by automatically keeping track of only theirlatest assertion or retraction wrt. the state of a given query. Tabling both a fluent
and its complement provides a convenient formulation of counterfactual reasoning(cf. Section 3.5 of [47]).
The combined use of incremental tabling and answer subsumption is realized
in the incrementally tabled predicate f luent(F, HtF , Qt) for fluent literal F , where
HtF and Qt are the states (timestamps) when it holds true (holds-time) and when
it is queried (query-time), resp. Invoking f luent(F, HtF , Qt) thus, either looks foran entry in its table, if one exists; otherwise, it invokes dynamic definitions of fluent
F , and returns the latest time HtF fluent F is true wrt. a given query-time Qt.
In order to return only the latest time HtF when F is true (wrt. Qt), f luent/3
is tabled using answer subsumption on its second argument. Predicate f luent/3
is then employed in Evolp/r system predicate holds(F, Qt) to query whether F
holds true at query-time Qt. It does so by looking for the entry for F in the table off luent(F, HtF , Qt), obtaining the latest time HtF ≤ Qt, and guarantees that F is
not supervened by its complement ∼F , i.e., Ht∼F ≤ HtF , where Ht∼F is obtainedby invoking f luent(∼F, Ht∼F , Qt)).
While answer subsumption is shown useful in this approach to avoid recursing
through the frame axiom by allowing direct access to the latest time when a fluent
is true, it requires f luent/3 to have query time Qt as its argument. Consequently,
it may hinder the reusing of tabled answers of f luent/3 by similar goals which differ
only in their query-time. Ideally, the state of a fluent literal in time depends solely
on the changes made to the world, and not on whether that world is being queried.
The above issue is addressed in the second approach [55], where the use of incremental tabling in Evolp/r is fostered further, while leaving out the problematicuse of answer subsumption. The main idea, not captured in the first approach, isthe perspective that knowledge updates (either self or world wrought changes) occur
whether or not they are queried: the former take place independently of the latter,i.e., when a fluent is true at Ht, its truth lingers on independently of Qt. Consequently, from the standpoint of the tabled f luent predicate definition, Qt no longer
becomes its argument: we now have incremental tabled predicate f luent(F, HtF ).
Invoking f luent(F, HtF ) thus amounts to look for an entry in its table, if one exists;
otherwise, it invokes dynamic definitions of fluent F , and returns a holds-time HtF(a state when fluent F is true). Since answer subsumption is left out, more than
one instance of holds-time HtF of fluent F may be tabled.
The implementation details of Evolp/r are beyond the scope of this paper, but
can be found in [55]. We only illustrate, in Example 6, how the state information,viz., holds-time, figures in a rule transform.
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Abduction and Beyond in LP with Application to Morality
Example 6. Recall rule (5): absent ← rule(absent←renov) , renov. For the purposeof this example, assume that renov is not an abducible, but instead a fluent. Its
rule transform is shown below (abstracted away from implementation details):
absent(T ) ← f luent(rule(absent←renov) , T1 ), f luent(renov, T2 ), latest([T1 , T2 ], T ).
where the reserved predicate latest determines a holds-time T of fluent absent in thehead by which inertial fluent in its body (i.e., fluents rule(absent←renov) and renov)holds the latest.
The main characteristics of the second approach are summarized below.
• Though more than one instance of holds-time HtF of fluent F may be tabled,
this approach still permits avoiding top-down recursion or bottom-up iteration
through the frame axiom. This is taken care of by the underlying tablingmechanism: look-up a collection of states a fluent literal is true in the table,
pick-up the most recent one, and ensure that its complement, with a later
state, does not exist in the table. More precisely, querying whether fluent
F holds true at query-time Qt amounts to looking for entries for fluent F in
the table, by invoking f luent(F, HtF ), in order to obtain the highest state
HtF ≤ Qt, and to guarantee that F is not supervened by its complement ∼F ,by invoking f luent(∼F, Ht∼F ) and obtaining the highest state Ht∼F such that
Ht∼F ≤ HtF .
• Propagation of fluent updates is controlled by initially keeping them pending
in the database. On the initiative of an actual (top-goal) query, just those
updates up to the actual query time are activated, via their incremental assertions, which in turn, automatically trigger system-level incremental bottom-up
recomputation of other tabled fluent literals (viz., updating the state information Ht of these fluents). This updating technique thus allows high-level topdown deliberative reasoning about a query to be reconciled with autonomous
low-level (via tabling) bottom-up world reactivity to ongoing updates.
• For facilitating the propagation of fluent complements, Evolp/r borrows thedual transformation from Tabdual. That is, the transformation derives rules
for the dual negation fluent literal ∼F from their corresponding positive rules offluent F . It thus helps propagate this dual negation fluent literal incrementally,in order to establish whether the fluent or rather its complement is true at somestate.
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Pereira and Saptawijaya
3.2.2

Combining Tabdual and Evolp/r

We have presented in the previous sections approaches for LP abduction (Tabdual)
and updating (Evolp/r). Both approaches enjoy the benefit of tabling features,
albeit their different usage. We now recap a unified approach to integrate Tabdual
and Evolp/r for keeping the benefit of tabling in each individual approach.
Our initial attempt, as reported in [58], employs the concept of hypotheses generation from [42], where the notion of expectation is employed to express preconditions
for enabling the assumption of an abducible. An abducible A can be assumed, orconsidered, if there is an expectation for it, and no expectation to the contrary:consider(A) ← expect(A), not expect_not(A), A.
Note that this concept consequently requires every rule with abducibles in its bodyto be preprocessed, by substituting abducible A with consider(A). The need and therule definitions for expect(A) and expect_not(A) depend on the encoded problem.
While this concept generates only abductive explanations relevant for the problem at hand, the approach in [58] limits the benefit of tabled abduction. That is,
instead of associating tabled abduction to a particular atom (e.g., stormab in Example 3) for a subsequent reuse in another context, a single (and generic) tabled
predicate considerab is introduced for tabling only this single abducible A being
considered. Though expect(A) and expect_not(A) may have rules stating conditions for expecting a or otherwise, it is unnatural that such rules would have otherabducibles as a condition for abducing A, as consider/1 concerns conditions for one
abducible only, according to its intended semantics and use. That is, a consider/1
call will not depend on another consider/1.
Such a generic tabling by consider_ab is indeed unnecessary and not intended
by tabled abduction. In this section we remedy the joint tabling approach of [58] inorder to uphold the idea and the benefit of tabled abduction. As in [58], the joint
tabling approach in combining Tabdual and Evolp/r naturally depends on twopieces of information carried from each of these techniques, viz. abductive contexts
and the state when a fluent holds true, respectively. They keep the same functionin the integration as in their respective individual approach. Example 7 shows how
both entries figure in a remedied rule transform.
Example 7. Recall rule hit_lightning ← storm, low_altitude in Example 2. Its
rule transform is shown below (abstracted away from its rule name and other implementation details):
hit_lightningab (E2 , T ) ← low_altitude([ ], E1 , T1 ), storm(E1 , E2 , T2 ),latest([T1 , T2 ], T ).
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(6)

Abduction and Beyond in LP with Application to Morality
where E2 is an abductive solution tabled by predicate hit_lightningab , obtained byrelaying the ongoing abductive solution in context E1 from subgoal low_altitude
to subgoal storm in the body, given the empty input abductive context (i.e., [ ]) of
low_altitude. As in Example 6, the reserved predicate latest determines the state
T of hit_lightning from low_altitude and storm that latest holds.
There are two points to note in the above remedied transformation. First, theabducible low_altitude in the body is now called explicitly (rather than immediately
placed as the input abductive context of storm, cf. rule (1) in Example 3). This is toanticipate possible updates on this abducible. Such an abducible update may takeplace when one wants to commit to a preferred explanation and fix it in the program
as a fact, a case that we show in Section 4 to fix an abduced background context
of a counterfactual. Having an abducible as an explicit subgoal in the body thus
facilitates bottom-up propagation of its updates (induced by incremental tablingin Evolp/r), due to the apparent dependency between the tabled predicate in thehead (e.g., hit_lightningab ), and the abducible goal in the body (e.g., low_altitude).
Second, by prioritizing abducible goals in the body (to occur before any other nonabducible goals), Tabdual’s benefit, of reusing tabled abductive solutions from one
context to another, is still obtained. In Example 7, calling abducible low_altitudewith the empty input abductive context – solved using its definition similar to rule(4) – provides storm with an actual input abductive context E1 = [low_altitude].
It thus achieves the same effect as simply having [low_altitude] as the input context
of storm (cf. rule (1) of Example 3). Using a similar transform as in rule (6), therule transform for crashab is given below:
crashab (E1 , T ) ← hit_lightning([ ], E1 , T1 ), storm(E1 , E2 , T2 ), latest([T1 , T2 ], T ).
The tabled solution in hit_lightningab (rule (6)) can be reused via the definition
hit_lightning below (this definition is similar to rule (3) in Section 3.1, just adding
the state information T ):hit_lightning(I, O, T ) ← hit_lightningab (E, T ), produce_context(O, I, E).
We may observe that, like in Example 3 of Tabdual, the subgoal storm in theabove rule of crashab is called with a different actual input abductive context (viz.,
E1 = [cumulonimbus, low_altitude]) from that of the same subgoal in rule (6).
Finally, the different purposes of the dual program transformation, employedboth in Tabdual and Evolp/r, are now consolidated within this unified approach:• Following the name convention for dual predicates in Tabdual, the dual negation ∼F of fluent literal F is now renamed to not_F in the unified approach.
53Pereira and Saptawijaya• As in the positive rule transform, the abductive context and state information(viz., holds-time) jointly figure in the parameters of dual predicates. Recall
Example 4. Considering close_library as a fluent, the dual transformation of
the unified approach results in rules below (abstracted away from the two rulename fluents of close_library and other implementation details):
not_close_library(C0 , C2 , Ht) ← close_library ∗1 (C0 , C1 , Ht1 ),close_library ∗2 (C1 , C2 , Ht2 ),latest([Ht1 , Ht2 ], Ht)(7)close_library ∗1 (I, O, Htwe∗ ) ← weekend∗ (I, O, Htwe∗ ).
close_library ∗2 (I, O, Htwe ) ← weekend(I, O, Htwe ).
close_library ∗2 (I, O, Htna ) ← not_absent(I, O, Htna ).

From the abduction perspective, it helps to efficiently deal with downwards byneed abduction under negated goals, e.g., abduction via close_library ∗i due toinvoking the goal not_close_library([ ], O, T ). The state when an abducible isabduced, e.g., Htwe∗ of weekend∗ in the rule of close_library ∗1 , is determinedin rule (7) by Ht (viz., by the reserved predicate latest): it can be a concrete
timestamp Ht2 or the actual query-time Qt (if Ht2 is not concrete yet, whichis the case when Ht2 itself refers to another abduction state).
From the updating perspective, it helps to incrementally propagate upwards
the dual negation complement of a fluent. For instance, updating the programwith weekend∗ at Htwe∗ will incrementally propagate Htwe∗ to determine Htof fluent not_close_library, in this case via close_library ∗1 .

4

Counterfactuals with LP Abduction and Updating

In this section we propose an approach that makes use of LP abduction and updating for counterfactual reasoning [47]. It is based on Pearl’s approach [40] to evaluate
counterfactuals, and adapts it to LP. In Pearl’s approach, counterfactuals are evaluated based on a probabilistic causal model and a calculus of intervention. Its mainidea is to infer background circumstances that are conditional on current evidences,and subsequently to make a minimal required intervention in the current causal
model, so as to comply with the antecedent condition of the counterfactual. The
modified model serves as the basis for computing the counterfactual consequent’s
probability.
Our LP-based approach abstains from the use of probability, and concentrates
on pure non-probabilistic counterfactual reasoning in LP, by resorting to LP abduction and updating, to determine the logical validity of counterfactuals under54

Abduction and Beyond in LP with Application to Morality
Well-Founded Semantics (WFS). Nevertheless, the approach is also adaptable toother semantics, e.g., Weak Completion Semantics [24] is employed in [44].
In the sequel, the Well-Founded Model of program P is denoted by W F M (P ).
The logical consequence relation |= is defined such that for formula F , P |= F iff F
is true in W F M (P ).

4.1

Causation and Intervention in LP

Two important ingredients in Pearl’s approach of counterfactuals are causal modeland intervention. Given that the inferential arrow in a LP rule representing causal
direction, causation can be captured by LP abduction. It expresses a causal source
by providing an explanation to a given observation. With respect to Pearl’s causal
model: (1) An observation O corresponds to Pearl’s definition for evidence. That
is, O has rules concluding it in program P , and hence does not belong to the set of
abducibles A; (2) Pearl’s model comprises a set of background variables (also known
as exogenous variables), whose values are conditional on case-considered observed
evidences and are not causally explained in the model. In terms of LP abduction,
they correspond to a set of abducibles E ⊆ A that provide abductive explanations
to observation O. Indeed, these abducibles have no preceding causal explanatory
mechanism, as they have no rules concluding them in the program.
Whereas abduction permits obtaining explanations to observations, the evaluation of counterfactual “if P re had been true, then Conc would have been true”,
following Pearl’s approach, requires the so-called intervention. This is achievedby explicitly imposing the desired truth value of P re, and subsequently checking
whether the predicted truth value of Conc consistently follows from this intervention. As described in Pearl’s approach, such an intervention establishes a required
adjustment, so as to ensure that the counterfactual’s antecedent be met. It permits
the value of the antecedent to differ from its actual one, whilst maintaining theconsistency of the modified model.
LP updating complements LP abduction in establishing an apposite adjustment
to the causal model by hypothetical updates of causal intervention on the program,affecting defeasible rules. Moreover, LP updating facilitates fixing the initial abduced background context of the counterfactual being evaluated in the program, by
updating the program with the preferred explanation to the current observations.
Next, we detail the roles of LP abduction and updating in our procedure for
evaluating the validity of counterfactuals.
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Pereira and Saptawijaya

4.2

Evaluating Counterfactuals in LP

The procedure to evaluate counterfactuals in LP essentially takes the three-stepprocess of Pearl’s approach as its reference. That is, each step in the LP approach
captures the same idea of its corresponding step in Pearl’s. The three-step procedure
is detailed as follows.
Let program P encode the modeled situation on which counterfactuals are evaluated. Consider a counterfactual “if P re had been true, then Conc would have beentrue”, where P re and Conc are finite conjunctions of literals.
1. Abduction: Perform abduction to explain past circumstances in the presenceof evidence (i.e., factual observation).2 More to the point, compute an explanation E ⊆ A to the observation O. The selected explanation is fixed (via
updating) as the abduced background context in which the counterfactual isevaluated (“all other things being equal”), to obtain program P ∪ E.

2. Action: For each literal L in conjunction P re, introduce a pair of reserved
meta-predicates make(B) and make_not(B), where B is the atom in L. These
two meta-predicates are introduced for the purpose of establishing causal intervention: they are used to express hypothetical alternative events to be
imposed. This step comprises two stages:(a) Transformation:• Add rule B ← make(B) to program P ∪ E.
• Add not make_not(B) to the body of each rule in P whose headis B. If there is no such rule, add rule B ← not make_not(B) to
program P ∪ E.

Let (P ∪ E)τ be the resulting transform.

(b) Intervention: Adjust the resulting transform (P ∪E)τ to comply with theantecedent of the counterfactual. The antecedent of the counterfactual is
imposed with an intervention on the program, by updating the programwith the hypothetical fluents corresponding to the required intervention:
update program (P ∪ E)τ with literal make(B) or make_not(B), for
L = B or L = not B, resp. Assuming that P re is consistent, make(B)and make_not(B) will not be imposed at the same time.
Let (P ∪ E)τ,ι be the program obtained after these hypothetical updates
of intervention.
2

Factual observations are those explicitly given, and not presupposed from the considered counterfactual. Implicit presuppositions, either those appearing in counterfactual or indicative conditionals, cannot immediately be regarded as observations.

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Abduction and Beyond in LP with Application to Morality
3. Prediction: Verify whether the consequent of the counterfactual deductively
follows: (P ∪ E)τ,ι |= Conc. Additionally, ensure that all ICs (if any) are alsosatisfied.
The counterfactual “if P re had been true, then Conc would have been true” isthus valid given an observation O iff O is explained by E ⊆ A, (P ∪ E)τ,ι |= Conc,and all ICs (if any) are satisfied in W F M ((P ∪ E)τ,ι ).
We illustrate in Example 8 how this three-step procedure is actually realized using a combination of LP abduction and updating to evaluate the validity of counterfactuals. The reader is referred to [47] for its formal procedure and implementation
details. The procedure has been implemented in a prototype, Qualm, on top of ourunified approach of LP abduction and updating (cf. Section 3.2.2), in XSB Prolog.
Example 8. Recall the example in Section 1: Lightning hits a forest and a devastating forest fire breaks out. The forest was dry after a long hot summer and many
acres were destroyed. Let us consider more abductive causes for the forest fire: storm(which implies lightning hitting the ground) or barbecue. Note that dry leaves areimportant for forest fire in both cases. This example is expressed in the program
below, where the set of abducibles is A = {storm, barbecue, storm∗ , barbecue∗ }:f ire ← barbecue, dry_leaves.
f ire ← barbecue∗ , lightning, dry_leaves.
lightning ← storm.
dry_leaves.
The explicit use of barbecue∗ in the second rule of f ire is intended so as to havemutual exclusive explanations.
Consider counterfactual “if only there had not been lightning, then the forest firewould not have occurred”, where the factual observations is O = {lightning, f ire}.
Note that the observations assure us that both the antecedent and the consequent
literals of the counterfactual were factually false, ensuring that the latter is relevant.
1. Abduction: Abduce consistent explanations E ⊆ A to the above factual
observations O, viz., E1 = {storm, barbecue∗ } and E2 = {storm, barbecue}.
Say E1 is preferred for consideration. The abduced background context for thecounterfactual is fixed by updating the program with E1 , obtaining P ∪ E1 .
2. Action: The transformation results in program (P ∪ E1 )τ :f ire ← barbecue, dry_leaves.
f ire ← barbecue∗ , lightning, dry_leaves.
lightning ← storm, not make_not(lightning).
dry_leaves.
lightning ← make(lightning).

Program (P ∪ E1 )τ is updated with make_not(lightning) as the requiredcausal intervention, viz., “if there had not been lightning”.
57Pereira and Saptawijaya
3. Prediction: Verify if the conclusion “the forest fire would not have occurred” holds in WFS. Indeed, (P ∪ E1 )τ,ι |= not f ire. That is, not f ire
holds with respect to the intervened modified program for explanation E1 ={storm, barbecue∗ } and the intervention make_not(lightning). Thus, thecounterfactual is valid.
Example 9. Continuing Example 8, if E2 is instead preferred to update P , thecounterfactual is no longer valid. In this case, (P ∪ E1 )τ = (P ∪ E2 )τ , and therequired causal intervention is also the same: make_not(lightning). But we now
have (P ∪ E2 )τ,ι 6|= not f . Indeed, this conforms with our understanding that theforest fire would still have occurred but due to an alternative cause, viz., a barbecue.
Skeptical and credulous counterfactual evaluations could ergo be defined, i.e.,by evaluating the proferred counterfactual for each abduced background context.
Given that step 2 can be accomplished by a one-time transformation, such skeptical and credulous counterfactual evaluations require only executing step 3 for eachbackground context fixed in step 1.

5

Applications to Computational Morality

Counterfactual theories are very suggestive of a conceptual relationship to a form of
debugging, namely in view of correcting moral blame, since people ascribe abnormal
antecedents an increased causal power, and are also more likely to generate counterfactuals concerning abnormal antecedents. Two distinct processes can be identified
when people engage in counterfactual thinking. For one, its frequent spontaneous
triggers encompass bad outcomes and “close calls” (some harm that was close to happening). Second, such thinking comprises a process of finding antecedents which,if mutated, would prevent the bad outcome from arising. When people employ
counterfactual thinking, they are especially prone to change abnormal antecedents,
as opposed to normal ones. Following a bad outcome, people are likely to conceive of the counterfactual “if only [some abnormal thing] had not occurred, thenthe outcome would not have happened”. See [51] for a review. Such a conceptionof counterfactuals in morality is discussed below by examining the issues of moral
permissibility (Section 5.1) and its justification (Section 5.2).

5.1

Moral Permissibility

In this section we look into the application of counterfactuals for examining viewpoints on moral permissibility, exemplified by classic moral dilemmas from the literature on the Doctrines of Double Effect (DDE) [37] and of Triple Effect (DTE)58

Abduction and Beyond in LP with Application to Morality[29]. Examining moral permissibility with counterfactuals according to these twowell-known doctrines does not require formalizing them, but instead detailing, according to our approach, their materialization in concrete moral dilemmas. Classic
dilemmas regarding these doctrines are presented, so the results of counterfactual
evaluation are readily comparable to those intended in the literature. Moreover,
our applications focus on showing the process of moral reasoning through counterfactuals, rather than representing formally the notions of obligation, prohibition,and permissibility in a formal logic (say, as deontic logic operators). Such distinct
research direction has been discussed elsewhere, e.g., [49, 7].
DDE is often invoked to explain the permissibility of an action that causes aharm by distinguishing whether this harm is a mere side-effect of bringing about agood result, or rather a means to bringing about the same good end [37]. In [22],DDE has been utilized to explain the consistency of judgments, shared by subjects
from demographically diverse populations, on a series of moral dilemmas.
Counterfactuals may provide a general way to examine DDE in dilemmas, e.g.,
the classic trolley problem [16], by distinguishing between a cause and a side-effectas a result of performing an action to achieve a goal. This distinction between causes
and side-effects may explain the permissibility of an action in accordance with DDE.
That is, if some morally wrong effect E happens to be a cause for a goal G that one
wants to achieve by performing an action A, and not a mere side-effect of A, thenperforming A is impermissible. This is expressed by the counterfactual form below,
in a setting where action A is performed to achieve goal G:
If not E had been true, then not G would have been true.
The evaluation of this counterfactual form identifies permissibility of action A fromits effect E, by identifying whether the latter is a necessary cause for goal G or amere side-effect of action A. That is, if the counterfactual proves valid, then E is instrumental as a cause of G, and not a mere side-effect of action A. Since E is morally
wrong, achieving G that way, by means of A, is impermissible; otherwise, not. Notethat the evaluation of counterfactuals in this application is considered from the perspective of agents who perform the action, rather than from others’ (e.g., observers).
Moreover, our emphasis on causation in this application focuses on agents’ deliberate actions, rather than on causation and counterfactuals in general. See [9, 40, 23]
for a more general and broad discussion on causation and counterfactuals.
Examples 10 and 11 apply this counterfactual form in two off-the-shelf militarycases from [61]: terror bombing vs. tactical bombing. The former refers to bombinga civilian target during a war, thus killing civilians, in order to terrorize the enemy,
and thereby get them to end the war. The latter case is attributed to bombing a
military target, which will effectively end the war, but with the foreseen consequenceof killing the same number of civilians nearby. According to DDE, terror bombing
59Pereira and Saptawijaya
fails permissibility due to a deliberate element of killing civilians to achieve the goalof ending the war, whereas tactical bombing is accepted as permissible.
Example 10. Terror bombing is expressed by the program below, where abducibles
are Ateb = {terror_bombing, terror_bombing ∗ }:end_war ← terrorize_enemy.
terrorize_enemy ← kill_civilian.
kill_civilian ← bomb_civilian.
bomb_civilian ← terror_bombing.
We consider end_war as the goal and the counterfactual “if civilians had not been
killed, then the war would not have ended”, provided the factual observation O ={kill_civilian, end_war}. It has a single explanation Eteb = {terror_bombing}.
Given the transform kill_civilian ← bomb_civilian, not make_not(kill_civilian)
and the intervention make_not(kill_civilian), the counterfactual is valid, because
not end_war holds in WFS with respect to the intervened modified program andthe background context Eteb , i.e., (P ∪ Eteb )τ,ι |= not end_war. That means themorally wrong kill_civilian is instrumental in achieving the goal end_war: it is a
cause for end_war by performing terror_bombing and not merely its side-effect.
Hence terror_bombing is DDE morally impermissible.
Example 11. Tactical bombing with the same goal end_war is modeled by the
program below, where Atab = {tactical_bombing, tactical_bombing ∗ }:end_war ← bomb_military.
bomb_military ← tactical_bombing.
kill_civilian ← tactical_bombing.
The counterfactual is the same and now we have Etab = {tactical_bombing} as the
only explanation to the same observation O = {kill_civilian, end_war}. By imposing the intervention make_not(kill_civilian) on the rule transform kill_civilian ←tactical_bombing, not make_not(kill_civilian), one can verify that the counterfactual is not valid, because end_war holds in WFS with respect to the intervened modified program and the background context Etab . Therefore, the morally
wrong kill_civilian is just a side-effect in achieving the goal end_war. Hencetactical_bombing is DDE morally permissible.
This application of counterfactuals can be extended to distinguish moral permissibility according to DDE vs. DTE. DTE [29] refines DDE particularly on the notion
about harming someone as an intended means. That is, DTE distinguishes further
between doing an action in order that an effect occurs and doing it because that effect will occur. The latter is a new category of action, which is not accounted for in
DDE. Though DTE also classifies the former as impermissible, it is more tolerant to
the latter (the third effect), i.e., it treats as permissible those actions performed justbecause instrumental harm will occur. Kamm [29] proposed DTE to accommodate
a variant of the trolley problem, viz., the Loop Case [65]: A trolley is headed toward60Abduction and Beyond in LP with Application to Morality
five people walking on the track, and they will not be able to get off the track in time.
The trolley can be redirected onto a side track, which loops back towards the five. A
fat man sits on this looping side track, whose body will by itself stop the trolley. Is it
morally permissible to divert the trolley to the looping side track, thereby hitting the
man and killing him, but saving the five? This case strikes most moral philosophersthat diverting the trolley is permissible [39]. Referring to a psychology study [22],56% of its respondents judged that diverting the trolley in this case is also permissible. To this end, DTE may provide the justification, that it is permissible because
it will hit the man, and not in order to intentionally hit him [29]. Nonetheless, DDE
views diverting the trolley in the Loop case as impermissible.
We use counterfactuals to capture the distinct views of DDE and DTE in the
Loop case.
Example 12. We model the Loop case with the program below, where save, divert,
hit, tst, mst stand for save the five, divert the trolley, man hit by the trolley, train
on the side track and man on the side track, respectively, with save as the goal andabducibles Aloop = {divert, divert∗ }:save ← hit.
hit ← tst, mst.
tst ← divert.
mst.
DDE views diverting the trolley impermissible, because this action redirects thetrolley onto the side track, thereby hitting the man. Consequently, it prevents thetrolley from hitting the five. To come up with the impermissibility of this action,
it is required to show the validity of the counterfactual “if the man had not been
hit by the trolley, the five people would not have been saved”. Given the factual
observation O = {hit, save}, its only explanation is Eloop = {divert}. Note thatrule hit ← tst, mst transforms into hit ← tst, mst, not make_not(hit), and the required intervention is make_not(hit). The counterfactual is therefore valid, because
not save holds in WFS of the intervened modified program, given the backgroundcontext Eloop . This means hit, as a consequence of action divert, is instrumental asa cause of goal save. Therefore, divert is DDE morally impermissible.
DTE considers diverting the trolley as permissible, since the man is alreadyon the side track, without any deliberate action performed in order to place him
there. In Example 12, we have the fact mst ready, without abducing any ancillaryaction. The validity of the counterfactual “if the man had not been on the side
track, then he would not have been hit by the trolley”, which can easily be verified,ensures that the unfortunate event of the man being hit by the trolley is indeed
the consequence of the man being on the side track. The lack of deliberate action(exemplified here by pushing the man – push for short) in order to place him on theside track, and whether the absence of this action still causes the unfortunate event(the third effect) is captured by the counterfactual “if the man had not been pushed,
61Pereira and Saptawijaya
then he would not have been hit by the trolley”. This counterfactual is not valid,because the factual observation O = {push, hit} has no explanation E ⊆ Aloop , i.e.,
push 6∈ Aloop , and no fact push exists either. This means that even without thishypothetical but unexplained deliberate action of pushing, the man would still havebeen hit by the trolley (just because he is already on the side track). Though hit
is a consequence of divert and instrumental in achieving save, no deliberate action
is required to cause mst, in order for hit to occur. Hence divert is DTE morally
permissible.
Example 13. Consider a variant of the Loop case, viz., the Loop-Push Case (cf.
Extra Push Case in [29]). Differently from the Loop case, now the looping side trackis initially empty, and besides the diverting action, an ancillary action of pushing a
fat man in order to place him on the side track is additionally performed. This caseis modeled by the program below, where Alp = {divert, push, divert∗ , push∗ }:save ← hit. hit ← tst, mst. tst ← divert. mst ← push.
Recall the counterfactuals in the discussion of DDE and DTE of the Loop case:• “If the man had not been hit by the trolley, the five people would not have beensaved.” The observation O = {hit, save} provides an extended explanation
Elp1 = {divert, push}, i.e., the pushing action needs to be abduced for having
the man on the side track, so the trolley can be stopped by hitting him. The
same intervention make_not(hit) is applied to the same transform, resulting
in a valid counterfactual, because not save holds in WFS of the intervened
modified program with the background context Elp1 .
• “If the man had not been pushed, then he would not have been hit by thetrolley.” The factual observation is O = {push, hit}, explained by Elp2 ={divert, push}. As there is no rule for push, rule push ← not make_not(push)
is introduced, and the intervention make_not(push) renders not hit to hold in
WFS of the intervened modified program, given Ep2 as its background context.
Thus, whereas this counterfactual is not valid in DTE of the Loop case, it isvalid in the Loop-Push case.
From the validity of these two counterfactuals it can be inferred that, given thediverting action, the ancillary action of pushing the man onto the side track causes
him to be hit by the trolley, which in turn causes the five to be saved. In the LoopPush, DTE agrees with DDE that such a deliberate action (pushing) performed inorder to bring about harm (the man hit by the trolley), even for the purpose of agood or greater end (to save the five), is likewise impermissible.
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Abduction and Beyond in LP with Application to Morality

5.2

Moral Justification

Counterfactuals may as well be suitable to address moral justification, via ‘compound
counterfactuals’: Had I known what I know today, then if I were to have doneotherwise, something preferred would have followed. Such counterfactuals, typicallyimagining alternatives with worse effect – the so-called downward counterfactuals[35], may provide moral justification for what was done due to a lack in the current
knowledge. This is accomplished by evaluating what would have followed if theintent would have been otherwise, other things (including present knowledge) being
equal. It may justify that what would have followed is no morally better than theactual ensued consequence.
Example 14. Consider a scenario developed from the Loop case of the trolley
problem, which happens on a particularly foggy day. Due to the low visibility, theagent saw only part of the looping side track, so the side track appeared to theagent rather as a straight non-looping one. The agent was faced with a situation
whether it was permissible for him to divert the trolley. The knowledge base of theagent with respect to this scenario is shown in a simplified program below. Notethat divert(X) is an abducible atom.
run_sidetrack(X)
hit(X, Y )save_f rom(X)save_f rom(X)sidetrack(straight)
sidetrack(loop)
f oggy.

←
←
←
←
←
←

divert(X).
run_sidetrack(X), on_sidetrack(Y ).
sidetrack(straight), run_sidetrack(X).
sidetrack(loop), hit(X, Y ), heavy_enough(Y ).
f oggy.
not f oggy.
on_sidetrack(man).
heavy_enough(man).

Taking save_f rom(trolley) as the goal, the agent performed counterfactual reasoning “if the man had not been hit by the trolley, the five people would not havebeen saved”. Given the abduced background context divert(trolley), one can verify
that the counterfactual is not valid. That is, the man being hit by the trolley is justa side-effect of achieving the goal, and thus divert(trolley) is morally permissible
according to DDE. Indeed, this case resembles the original trolley problem (alsocommonly known as the Bystander case).
At some later time point, the fog has subsided, and by then it was clear to
the agent that the sidetrack was looping to the main track. This is achieved by
updating the program with not f oggy, rendering sidetrack(loop) true. There are
two standpoints on how the agent can justify his action divert(trolley). For one, itcan employ the aforementioned form of compound counterfactual “Had I known that63

Pereira and Saptawijaya
the sidetrack is looping, then if I had not diverted the trolley, the five would havebeen saved” as a form of self-justification. Given the present knowledge that the side
track is looping, the inner counterfactual is not valid, meaning that to save the fivepeople, diverting the trolley (with the consequence of the man being hit) is required.
Moreover, the counterfactual “if the man had not been hit by the trolley, the fivepeople would not have been saved”, in the abduced context divert(trolley), is valid,meaning that this action is DDE impermissible (cf. Example 12). Therefore, theagent can justify that what would have followed (given its present knowledge, i.e.,
sidetrack(loop)) is no morally better than the actual one, when there was lack of thatknowledge: its decision divert(trolley) at that time was instead DDE permissible.
Qualm can evaluate such compound counterfactuals, thanks to its implemented
incremental tabling of fluents (Section 3.2.1). Because fluents and their state information are tabled, events in the past subjected to hypothetical updates of intervention can readily be accessed (in contrast to a destructive database approach, cf.
[31]). Indeed, these hypothetical updates take place without requiring any undoing of other fluent updates, from the state those past events occurred in up to thecurrent one; more recent updates are kept in tables and readily provide the current
knowledge. The discussion of this technique can specifically be found in Section 3.5
of [47].
A different standpoint from where to justify its action is by resorting to Scanlon’s contractualism [60]. Scanlon argues that moral permissibility can be addressed
through the so-called deliberative employment of moral judgments. That is, thequestion of the permissibility of actions is answered by identifying the justified but
defeasible argumentative considerations, and their exceptions. It is based on a viewthat moral dilemmas typically share the same structure: (1) They concern generalprinciples that in some cases admit exceptions; (2) They raise questions about whenthose exceptions apply. With this structure, an action is determined impermissible
through deliberative employment when there is no countervailing consideration thatwould justify an exception to the applied general principle. In this vein, for theexample we are currently discussing, the DTE serves as the exception to justify thepermissibility of the action divert(trolley) when the side track was known to belooping, as shown through counterfactual reasoning in Example 12.
We extend now Example 14 to further illustrate how moral permissibility of actions is justified through defeasible argumentative considerations according to Scanlon’s contractualism.
Example 15. As the trolley approached, the agent realized that the man was notheavy enough to stop it. This information is acknowledged by the agent throughupdating its knowledge base with not heavy_enough(man). But there was a heavy
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Abduction and Beyond in LP with Application to Morality
cart on the bridge over the looping sidetrack that the agent could push to place iton the side track, and thereby stop the trolley. This scenario is expressed with rules
below (push(X) is an abducible atom), in addition to the program of Example 14:
on_sidetrack(X) ← on_bridge(X), push(X).
on_sidetrack(Y ) ← push(X), inside(Y, X).
on_bridge(cart).

(8)
(9)

heavy_enough(cart).

The second rule of on_sidetrack is an extra knowledge of the agent, that if an object
Y is inside the pushed object X, then Y will be on the side track, too.
The goal save_f rom(trolley) now succeeds with [divert(trolley), push(cart)] as
its abductive solution. But the agent subsequently learned that a fat man, who
was heavy enough, was inside the cart: the agent updates its knowledge base withinside(f at_man, cart) and heavy_enough(f at_man). As a consequence, this man
was also on the side track and hit by the trolley, which can be verified by query?- hit(trolley, f at_man).
In this scenario, a deliberate action of pushing was involved that consequently
placed the fat man on the side track (as verified by ?- on_sidetrack(f at_man))and the man being hit by the trolley is instrumental to save the five people fromthe track (as verified by the counterfactual “if the fat man had not been hit by
the trolley, the five people would not have been saved”). Nevertheless, the agent
may justify the permissibility of its action by arguing that its action is admitted by
DTE. In this case, the fat man being hit by the trolley is just a side-effect of theagent’s action push(cart) in order to save the five people. Indeed, this justification
can be shown through reasoning on the counterfactual “if the cart had not been
pushed, then the fat man would not have been hit by the trolley”, which is valid
given the abduced background context push(cart). Furthermore, the observation
hit(trolley, f at_man) cannot be explained by push(f at_man) given the absence ofthe fact on_bridge(f at_man), i.e., the hypothetical action push(f at_man) is not
the causal source for the fat man being hit by the trolley.
All moral examples presented in this paper have been successfully tested in
Qualm. Examples 14 and 15 together actually form one single program, where
Qualm features (derived from its constituents, Tabdual and Evolp/r) are exercised. For instance, in Example 14, after updating the program with not f oggy, reinvoking the goal save_f rom(trolley) reuses the abductive solution divert(trolley)
tabled from the previous invocation of run_sidetrack(trolley). Moreover, thistabled solution is involved in (as the context of) the deliberative reasoning forthe goal on_sidetrack(man) when hit(trolley, man) is called. It thus provides a65Pereira and Saptawijaya
computational model of collaborative interaction between deliberative and reactive
reasoning in a form of the dual-process model [10, 33], when viewing tabling as aform of low-level reactive behavior. Another feature used, from Evolp/r, is thatrules (8) and (9) are first switched off in Example 14, via the rule name fluent mechanism (cf. Section 3.2.1), as they are not applicable in the considered scenario. Only
later, in Example 15, are they switched on again to allow abducing additional action
push(cart).

6

Summary and Concluding Remarks

Abduction has recently been on the back burner in LP. One of our research contributions is to address challenges in engineering LP abduction systems, to revive
abduction in LP. We present an implemented tabled abduction technique, Tabdual, to benefit from LP tabling mechanisms in contextual abduction, so that priorly
obtained (and tabled) abductive solutions can be reused from one abductive context
to another.
Aiming at facilitating the interplay between LP abduction and other LP nonmonotonic reasoning, we have also combined this tabled abduction technique with
our own-developed LP updating Evolp/r; the latter employs an incremental tabling
feature of XSB Prolog, and permits a reconciliation of high-level top-down deliberative reasoning about a goal, with autonomous low-level bottom-up world reactivity
to ongoing updates.
The other contribution is that of applying our unified approach of LP abduction
and LP updating to formulate counterfactuals reasoning based on Pearl’s structural
theory, but omitting formal probability, given the lack of pure non-probabilistic
counterfactual reasoning in LP. In our LP-based counterfactual approach, abduction
specifically plays an important role, to explain past circumstances in the presence ofevidence (factual observation), and fix (via LP updating) considered explanations as
the abduced background context in which the counterfactual is to be evaluated. Our
implemented approach and its prototype Qualm is based on the Well-Founded Semantics, but adaptable to other semantics as well, e.g., Weak Completion Semantics[24] is employed in [44], where manual updating is used.
We also apply counterfactuals to model morality issues. Counterfactuals are
employed to examine moral reasoning about permissibility by resorting to our LP
approach, to distinguish between causes and side-effects as a result of agents’ actions to achieve a goal. Counterfactuals are also used to support justifying moral
permissibility of agents’ actions, either by a compound counterfactual formulation
or by arguing an appropriate moral principle as an applicable exception, following
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Abduction and Beyond in LP with Application to Morality
Scanlon’s contractualism.
Side-effects in abduction have been investigated in [42, 43] through the concept
of inspection points; the latter are construed in a procedure by ‘meta-abducing’a specific abducible abduced(A) whose function is only that of checking that its
corresponding abducible A is indeed already adopted elsewhere. Therefore, theconsequence of the action that triggers this ‘meta-abducing’ is merely a side-effect.
Indeed, inspection points may be employed to distinguish a cause from a mereside-effect, and thus may provide an alternative or supplement to counterfactuals
employed for the above same purpose of examining moral permissibility.
The moral examples in the present paper do not exploit the use of ICs. The use of
ICs within abduction, but without counterfactuals, for morality has been explored inour related work [45]. They are particularly useful to rule out impermissible actions.
One of the difficulties in using an IC to express impermissibility is that it requires therepresentation to be crafted in sufficient detail in order for the IC to be applicable.
While we use counterfactuals to examine permissibility (so we are not bound to havea subtle problem representation), ICs can be used for other purposes, e.g., to choose
among mutually exclusive abducibles. On the other hand, ICs should be treatedcarefully in counterfactuals, because an intervention may render ICs unsatisfiable,
and hence their body’s support may need to be abductively revised in order to
re-impose satisfaction.
While moral examples in this paper are based on those from the literature witheither their conceptual or empirical results readily available, a more systematic evaluation of counterfactual reasoning in computational morality may be considered as
future work. For instance, an empirical study with human subjects, say in collaboration with cognitive scientists, may be conducted to provide insights on relevant
counterfactuals in examining moral permissibility of considered cases, as well as
in examining argumentative processes of moral reasoning in justifying permissibility
via counterfactuals. The study will then be able to guide the form of counterfactuals
to represent and to reason about using the approach fostered in this paper.

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72

A Dynamic Approach to Peirce’s
Interrogative Construal of Abductive
Logic
Minghui Ma∗
Institute for Logic and Intelligence, Southwest University, Chongqing
mmh.thu@gmail.com
Ahti-Veikko Pietarinen†
Center for Epistemology and Cognition, Xiamen University, Xiamen
Peirce Research Centre, University of Helsinki, Helsinki
ahti-veikko.pietarinen@helsinki.fi

Abstract
Peirce’s novel, post-1903 interrogative construal of abductive logic is studiedin terms of a dynamic approach. The information flow in abduction is interpreted by a dynamic mechanism over ranges. An abductive logic is proposedthat analyzes conjecture-making in scientific discovery. It explains how the updates on models representing the inquirers’ information proceed when they are
faced with new surprising facts. The formal semantics for this logic is givenby neighborhood models. We apply our abductive logic to an analysis of someexamples of scientific discovery.

We thank the two anonymous reviewers of this journal for the abductive feedback that has contributed so much to the improvement of the present paper. We also thank the audience of the 15th
Congress of Logic, Methodology and Philosophy of Science held in Helsinki, August 2015, in whicha part of the present paper was presented in a special symposium on Pragmati(ci)st Philosophy of
Science: Old and New organized by Chiara Ambrosio.
∗
This work is supported by China National Fund of Humanities and Social Sciences (grant no.
14ZDB016).
†
This work is supported by the Academy of Finland (project 1270335) and the Estonian Research Council (project PUT 267) (Diagrammatic Mind: Logical and Cognitive Aspects of Iconicity,
Principle Investigator A.-V. Pietarinen).
Vol. 3 No. 3 2016
IFCoLog Journal of Logic and its Applications

Ma and Pietarinen

1

Introduction

Peirce discovered abductive—which he also variously termed hypothetical, retroductive, adductive or presumptive—reasoning in his critics on logic he was developingin the late 19th-century. Initially, Peirce conceived deductive logic as the logic ofmathematics, and the inductive and abductive logic as the logic of science. Later, hetook these three types of reasoning to represent different stages of scientific inquiry
rather than different types of reasoning.
According to Peirce, “the art of making explanatory hypotheses is the supremebranch of logic” (Peirce to J. Royce, June 30, 1913).1 What is it that such an art
harbours? In July 16, 1905, Peirce drafted a letter which he intended to send to
Victoria Welby: “This kind of reasoning by which theories are formed under theinspiration of the facts I call Abduction . . . to warrant my use of the word ‘abduction’ to denote the process of conjecturally setting up a theory suggested by thestudy of surprising facts to explain their being surprising. It is very remarkablethat few logicians have ever attempted to define the circumstances under which an
explanatory hypothesis is called for”.2
In this paper, we develop upon Peirce’s unpublished remarks concerning his late,post-1903 development of abduction. His novel point is that one can characterize
abduction as “reasoning from surprise to inquiry”. Compared with the rule of Modus
Tollens, he formulated abduction as follows: given a (surprising) fact C, if A implies
C, then it is to be inquired whether A plausibly holds. The important discovery isthat the conclusion is presented in a kind of interrogative mood. But the interrogative mood does not merely mean that a question is raised. In fact, it means that thepossible conjecture A becomes the subject of inquiry: the purpose is to determine
whether that A is indeed plausible or not. Peirce termed such mood “the investigandmood”. Hence abduction can be viewed as the dynamic process toward a plausible
conjecture and, ultimately, toward a limited set of the most plausible conjectures.
We reconstruct Peirce’s interrogative construal of abductive logic in terms ofa dynamic approach, interpreting the information flow in abduction by a dynamic
mechanism over epistemic ranges. The first, preliminary analysis is based on propositional logic. Reasoning is a dynamic process from premisses to the conclusion. Theinterrogative mood shows up as an uncertainty in the final range of truth-value configurations. Although this analysis is far too simple to capture the full meaning of1
From the Papers of Josiah Royce deposited at the Harvard University Archives, published in[28, pp. 765–766].
2This letter was never sent and it has not been published before. It is preserved among theoriginal correspondence [24], but it was omitted from the publication of the Peirce–Welby correspondence [13]. A complete transcription of this letter is to appear in [28, pp. 842–845].

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A Dynamic Approach to Peirce’s Abductive Logic
the special mood that is required, in drafting plausible conjectures we neverthelessuncover the dynamic perspective on reasoning, a necessary preliminary step towards
a more full-blown logic of abduction.
The second step for achieving the analysis of Peirce’s interrogative construalof abduction is to introduce a logic of making conjectures. Here conjectures arepropositions which are obtained by the agent by guessing from a surprising fact.
Some of them may become plausible after a dynamic process of inquiring. We
introduce a special interpretation to the modal operator 2A which can mean, forexample, that ‘A is conjectured’, or alternatively, and historically and conceptuallyperhaps even more accurately, that ‘it is hoped that A’.3
The models for our logic are neighborhood models. The advantage of using suchmodels is that one can easily express that a proposition is a conjecture in the epistemic or mental range of the current state. If we were to use standard models ofmodal logic, we should admit principles of logical omniscience. However, the logic ofneighbourhood models does not assume logical omniscience, with the sole exceptionof closure under logical equivalents and, in some cases, closure under valid implications (when monotonicity and non-emptiness is assumed). Second, using neighbourhood models can represent conjectures in an insightful manner. The semantic
value of a proposition is supposed to be a set of possible worlds. In neighbourhoodmodels, to be in a neighbourhood of the current world is to be a conjecture (proposition) at the current world. Third, each standard (Hintikka-Kripke) model is in fact
equivalent to a neighbourhood model, but the converse does not hold. This meansthat ne! ighbourhood models are more general than standard structures. In order
to interpret the dynamic process of forming a plausible conjecture, we introduce anupdating mechanism that changes the neighborhood model into one in which weachieve a plausible conjecture.
The third step we take is to introduce dynamic operators into the logic of conjecture. It is here that we achieve the full dynamic logic of abduction. We introduce
the dynamic operator on the modal one, [!F ]2A, which means that A preserves its
status as a conjecture after the surprising fact F has occurred. In other words, theconjecture, or the hope that one levies on the hypothesis, survives the experience
encountering a series of further surprising facts. The dynamic operator is interpreted
by updating a neighborhood model by such new facts F . Of course, a scientist ora research team expects to encounter a vast number of such surprising facts in thecourse of inquiry. It is the hallmark of frontier sciences that one is on the relentless
3‘To hope’ may even be slightly weaker than ‘to conjecture’. Certainly it is weaker than belief.
Peirce did not take abduction to result in anything stronger than conjecturing: “processes of thoughtcapable of producing no conclusion more definite than a conjecture . . . I now call Abduction” (MS293, 1906-7).

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Ma and Pietarinen
lookout for new sources of information to discover new and surprising facts. In looking for warrants for their hypotheses, scientists seldom rest their case on evidence
already in their possession.
The final step of our work is to test the dynamic logic of abduction by two
examples, one from Dewey and the other from Peirce.

2

Peirce’s Abductive Logic in Perspective

In his early works from the 1860s, Peirce used syllogism to prove that there are only
three fundamental types of reasoning: deduction, abduction and induction. These
correspond to the following three figures of syllogism:
Figure I.
Deduction
S is M
M is P
S is P

Figure II.
Abduction
S is M
P is M
S is P

Figure III.
Induction
M is S
M is P
S is P

One of Peirce’s examples of abduction (termed that early time the hypothetic
argument) under Figure II was: Light is polarizable. Ether waves are polarizable.
Therefore, light is ether waves (W 1:427, 1866). The form of the hypothetic argument
is the only one that enables us, Peirce explains, to “see the why of things” (W 1:428).
Later in the late 1870s and 1880s [21, 22], he described abduction and induction
as inversions of a deductive syllogism in the following sense. We say that the major
premise of a syllogism in Figure I is Rule, its minor premise Case, and its conclusion
Result. Then abduction is the inference from a result and a rule to a case, andinduction is the inference from a case and a result to a rule. Table 1 below presents
three examples of these three types of inference.
Later still, towards the turn of the century Peirce arrived at the view that thesethree types of inference are in fact three stages in scientific inquiry [30]. The firststage is abduction, by which one or more hypotheses or conjectures that may explainsome surprising fact is set forth. The second stage is deduction, by which thenecessary consequences of the hypothesis are traced. The third stage is induction,
which puts the consequences to test and generalizes its conclusion. Every example ofscientific inquiry is for Peirce bound to follow the pattern of abduction–deduction–induction. It was moreover shown in [30] that even deduction itself breaks intosuch patterns: abduction, for instance, appears as a ‘moment’, or the first phase, of
deductive reasoning, which is concerned with definition and logical analysis.4
4

The reason for what might strike one as a surprising assertion of deduction also involving

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A Dynamic Approach to Peirce’s Abductive Logic

Deduction
Rule:
Case:
Result:Abduction
Rule:
Result:Case:
Induction
Case:
Result:Rule:

All the beans in this bag are white.
These beans are from this bag.
These beans are white.
All the beans in this bag are white.
These beans are white.
These beans are from this bag.
These beans are from this bag.
These beans are white.
All the beans in this bag are white.

Table 1: Figures of Syllogism
Of the three methods, Peirce took deduction to be the most secure and the least
fertile, while abduction is, in contrast, the most fertile and the least secure. Since
abduction is not secure, it is that mode of reasoning that is best suited to deal withuncertainty and guess, prevalent in everyday scientific practice. But it therefore also
requires some compelling reasons for its validity and justification in order for us toregard it as a trustworthy mode of reasoning in the sciences [30].
The central question of Peirce’s late formulation of abductive logic is to interpret
it as a logic of scientific discovery. This in view, Peirce came to generalize thesyllogistic description of abduction. The following logical form has come to be thestandard—and even termed the “classical” one in [31]—description:
The surprising fact, C, is observed.
But if A were true, C would be a matter of course.
Hence, there is reason to suspect that A is true.
The truth is that Peirce happened to deliver this schema relatively casually, during
the seventh of his 1903 Harvard Lectures (CP 5.189). This widely popular scheme
nevertheless became, quite unwarrantably, the form routinely referred to in thesubsequent literature. It by no means was his best or the final formulation of what
the logical form of abduction ought to look like. It is telling that after 1903, Peirce
abduction is that deduction is not only demonstrative but is importantly reliant on theorematic
types of reasoning. And it is such theorematic reasoning that comes to the fore in the tasks oflogical analysis and defining concepts, both of which Peirce took to pertain to deduction in its firstphase, followed by demonstration.

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hardly ever refers to it. On the contrary, he goes on to propose new logical forms
for abduction, more often in fact termed retroduction than abduction.
The matter has consequently caused some serious confusions in the literature.
The 1903 schema indeed encounters several problems. Any interpretation of abductive reasoning as the logic of scientific discovery which would exclusively be based on
the analysis and application of that 1903 schema would be insufficient or misleadingat least for the following reasons:(1) Recognition of what counts as ‘surprising’ requires a separate explanation. It
involves a special kind of scientific intelligence and experience by which toobserve, recognize and identify such facts.
(2) The conditional major premiss is not in an indicative but in a subjunctive
mood, making it much harder to be analyzed in semantical and logical precision.
(3) The A’s and C’s are not singular propositions but conceivable states of things(the A’s) and complex facts (the C’s). For instance the design of controlled
experiments is an example of the tasks that aim at spelling out the possible
dependent relationships between A’s and C’s.
(4) The schema is an idealized sketch that masks much of the complexity involvedin actual processes of scientific discovery, including the role of backgroundtheories and research agendas, and the connected questions concerning the
economy of research. Peirce attached a great deal of importance to such issuesof economy. He for instance remarked that “the whole question of what one
out of a number of possible hypotheses ought to be entertained becomes purely
a question of economy” (CP 6.528).
(5) The schema might indicate, misleadingly, that the essence of abductive reasoning merely deflates into that of the inference to the best explanation (IBE).
As to the last point, the difference in brief is that abduction allows generating more
hypotheses than the IBE account does. Abduction is the method for theory building,while IBE compares explanatory values of hypotheses. IBE has no account of whata scientific guess may be, although the latter is basically tantamount to Peirce’s
conception of abduction. Reasons for keeping the two clearly separate are also
found in how abduction copes with scientific uncertainties and ignorance that falls
outside of standardized probabilistic methods.
This last, deflationist assimilation might even suggest some a reduction of abduction to Bayesian, confirmational modes of reasoning. That would be an outright78

A Dynamic Approach to Peirce’s Abductive Logic
wrong direction to look for the essence of abductive reasoning. It would strip abduction of its most important features that enabled Peirce to delineate it as the only
“originary” (CP 5.171) mode of reasoning: reasoning that not only is vaguely classified as creative in suggesting new ideas but also and more importantly, originates
by coping with fundamental uncertainties in real discovery. A research project that
proposes to cope with such uncertainties and structural ignorance in some novel andunexpected ways may stand a chance of increasing the plausibility, but not thoseof probability, credibility, or degrees of beliefs in hypotheses, thus increasing thepossibility of progress in not eliminating the lines of research that the probabilistic
assessment schemas would have done at the outset.5
Before moving on to the main logical form for abduction, we note that Peircealso gave the following description in one of the late unpublished manuscripts:
In the inquiry, all the possible significant circumstances of the surprising phenomenon are mustered and pondered, until a conjecture furnishes some possible
Explanation of it, by which I mean a syllogism exhibiting the surprising fact asnecessarily following from the circumstances of its occurrence together with thetruth of the conjecture as premisses. (MS 843, p. 41, 1908; cf. CP 6.469–470)

The explaining syllogism here is as follows:
If A were true, C would be observable.
A is true.
Therefore, C is observable.
Here the major premiss is in a subjunctive mood, and the reasoning is by Modus
Ponens on the subjunctive and the minor premiss. We take this kind of reasoning
quite unsatisfactory, however, as that applies the MP to a subjunctive conditional.
A highly important logical form of abduction is found in the July 16, 1905
unpublished and unsent letter which Peirce drafted to Victoria Welby. In this form,
the major premiss is in an indicative and the conclusion in an interrogative mood.
Peirce explains the interrogative mood as follows:[The] “interrogative mood” does not mean the mere idle entertainment of anidea. It means that it will be wise to go to some expense, dependent upon theadvantage that would accrue from knowing that Any/Some S is M, provided
that expense would render it safe to act on that assumption supposing it to
be true. This is the kind of reasoning called reasoning from consequent toantecedent. For it is related to the Modus Tollens thus:
5

See [5, 9, 14, 20, 27, 30, 36] for related criticisms on those that have conflated IBE and abductionin the past, and [29] for a criticism of Bayesian approach from the strategic perspective of gametheory. Few genuinely productive research projects have ex ante priors to work with in the earlystages of inquiry.

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Ma and Pietarinen
ModusTollens
If A is true, C is not trueis
But C is true
is notTherefore, A is not true

Abduction
If A is true, C is not trueis
But C is not trueisTherefore, Is A not true?

Instead of “interrogatory”, the mood of the conclusion might more accurately
be called “investigand”, and be expressed as follows:“It is to be inquired whether A is not true.”
The reasoning might be called “Reasoning from Surprise to Inquiry” (Peirce to
Welby, July 16, 1905, added emphases).

The term ‘investigand’ is highly suggestive here. We can relate it to a multiplicity
of issues that constitute the initial course of scientific thought, among them thefollowing six: (i) Noticing the surprising phenomenon, (ii) searching for pertinentcircumstances, (iii) asking a question, (iv) forming a conjecture, (v) remarking thatthe conjecture at least appears to explain the surprising phenomenon, (vi) adopting
the conjecture as an imaginable, comprehensible and plausible one, (v) paying close
attention to economic factors and cost-benefit analysis in regard to the potential
testing of the hypothesis, and (vi) a recommendation for a reasonably safe matterof course to act on those assumptions that suppose certain situations that scientifichypotheses propose. All these phases and aspects of inquiry are constitutive of thefirst, abductive stage of inquiry. Its heart is that of forming the conjecture.6
The forming of conjectures has often been described, and occasionally even by
Peirce himself, as an act or a flash of insight, or having an instinct for guessing
right. There are reasons to take conjecture-forming to be a genuinely logical process, however, and the present paper is a contribution to those reasons. Glossing
abduction as an act or a flash of insight, or an instinctive form of behaviour, both
fail to capture the crucial logical characteristics involved in that mode of reasoning,
however, as has been pointed out in [3].
In the 1905 letter, Peirce’s remarkable move is to take abduction to be a Modus
Tollens from the premises “If A is true, C is not true” and “But C is not true”
to the interrogative, “Is A not true?”. He describes the process illustratively as
6

A reviewer points out that these aspects of inquiry that we take to pertain to the abductive
stage of reasoning are somehow similar to, although definitely more detailed than, what was proposed in [34]. The two papers are similar in that they attempt at a dynamic characterization of
abduction in the overall framework of epistemic logic. Where the two approaches differ is epistemological: we interpret the essential modality as “conjecturing” or “hoping that”, which is even
weaker than belief.

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A Dynamic Approach to Peirce’s Abductive Logic“Reasoning from Surprise to Inquiry”. In this reasoning, the mood of the conclusion
is a peculiar combination of interrogative and imperative moods; a mood which we
may not find in natural languages. Unlike in the 1903 schema, the new schema also
is a non-committal to the suspectibility of A’s truth. Rather, the commitment is togo ahead with the investigation: “it is to be inquired whether A is not true”. Peirce
termed this the “investigand” mood. This is a clear development of his 1903 view,according to which abduction “commits us to nothing. It merely causes a hypothesis
to be set down upon our docket of cases to be tried” (CP 5.603, 1903), which he
now realizes as too poor an account. The 1905 schema is equally a development of
his suggestion a few years earlier that the abductive conclusion is a ‘Maybe A’. This
he now also finds too weak and indecisive, even though a ‘May-be’ it might comejust slightly above the pure possibility in its strength. Abductive conclusions must
become committals to action, that something is to be done, while suggesting somereal pointers towards what is it that one ought to do. What the 1905 abductive
schema does, then, is to embed the key qualities of the economy of research into its
logical form. Questions of economy are not extra-logical or supererogatory questions;they are part and parcel of the logical form of abductive reasoning.7
We note that it is here in the development that takes place in 1905 that abduction, now conceived as a reasoning process that ranges from the preliminary feelingsof surprise to the initiation of large-scale scientific endeavours, comes strikingly close
to what Hintikka’s interrogative model of inquiry has purported to accomplish [14].
The details of those connections are yet to be spelled out.
What is also important in Peirce’s draft letter is that the major premiss is nolonger presented in the subjunctive mood. While subjunctive moods are surely thehallmarks both of scientific precepts as well as Peirce’s philosophy of pragmaticism,
our task is simplified as the models for such logical reasoning on indicative conditionals are easier to formulate. These 1905 remarks on the logical form of abduction
are our guidelines in our study of the logical form of abduction in the present paper.
Niiniluoto [18] observed that the branch of applied mathematics that studies inverse problems deals with abductive types of inference. This proposal needs serious
qualifications. Niiniluoto takes abduction to be reasoning from effects to causes.
This is at the same time a limiting view of abduction and a too coarse-grained one.
It is limitative, as it implies that some (e.g., well-posed, continuous and parametricmodel) inverse problems, which in reality are largely the matter of deductive inferences, would become instances of abductive inference. For example, in well-posed7

For example, Peirce recommends relying on early guesses, those that are “gravid with young
truth” (EP 2:472), the notion which he had gleaned from his studies on the logic of the history ofscientific discoveries. Why this works is that it expedites the inquiry that sooner or later would
have! discovered those conjectures.

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Ma and Pietarinen
inverse problems, where the relevant parameters or properties of models are known
so that the solution depends continuously on the available data, the predominantmode of reasoning is deductive rather than abductive one.
Niiniluoto’s proposal is too coarse, on the other hand, as conceiving abduction
as an inverse reasoning from effects to causes masks the details of what is goingon in scientific reasoning and even may misleadingly suggest that abduction has
such explanatory virtues as confirmatory degrees of beliefs in the likelihood of the
hypotheses.
Peirce’s interrogative construal, however, suggests both a broader and a more
specific view of abduction. It is broader, as it is fitted also for situations in whichthose strict cause-effect relationships that Niiniluoto requires may be unobtainable. Those situations typically concern problem contexts that are severely understructured. There are endless examples of such contexts in the sciences. In the areaof inverse problems, for example, such contexts give rise to ill-posed problems, suchas those where the time-reversibility fails when the converse of a continuous mapping is a discontinuous one and so that analog samples no longer work, or where themodels are non-parametric, where there is too much noise, or where there is only
some anecdotal data at hand, and so on. These are very common situations in contemporary science and fieldwork. Yet the investigation must go on. What would ascientist do? Examples such as these cry out for abduction in its interrogative or investigand mood, where the scientists are “guessing at the unknown unknowns” [26].
For example, when confidence regions or choosing good statistical parameters that
do not ‘under-smooth’ are at issue, a statistician typically makes a guess. But suchguesses are not blind guesses. They are guided by a relatively precise understandingof the kind of uncertainty and ignorance involved in the situation. Examples of suchtypes of abductive reasoning in contemporary sciences are numerous, and have been
explored in [27].

3

A Preliminary Analysis of Peirce’s Abduction

In order to avoid abduction from deteriorating into deductive, over-simplified or
static schemas, which would appear more like black boxes from evidence, effects or
conclusions to hypotheses, we develop here a first, preliminary analysis of what the
dynamic processes mean in reasoning in general.
Logical reasoning can be viewed as a dynamic process of information flow frompremises to the conclusion. The notion of information and its content in logic has
been studied in [4]. We first explain the information flow in a deductive reasoning,
and then extend this perspective to Peirce’s abductive reasoning. This, we empha82

A Dynamic Approach to Peirce’s Abductive Logic
sise, is a preliminary analysis following a straightforward interpretation of Peirce’s
abduction using classical truth-value configurations of propositions. Although it
quickly turns out that this first analysis is much too simple to capture the meaningof scientific inquiry or the making of plausible conjectures of the abductive kind, we
can learn the basic idea of logical dynamics of information flow from these remarks.
Consider the Modus Tollens (MT for short). Assume that you know that C istrue if A is true, and you learn that C is not true. Then logic tells you that A is nottrue. Formally, we write
A → C, ¬C ` ¬A.

What is the information flow in this piece of deductive reasoning? Intuitively, theinformation provided by the premises is a range of states which tells you the possibilities given by the truth values of propositions. Let T be the truth value true, andF be the truth value false. The rule of MT involves two propositions, and hence
there are only four possibilities regarding their truth values. This is represented inTable 2.
TT
FT

TF
FF

Table 2: Range I
We call each possibility a state. Every state consists of a truth value of A and atruth value of C. The range of states represents the basic information of MT, that
is, all the truth-value configurations for propositions that occur in MT. Identifying
the range of states is not a part of the MT reasoning. When the reasoning is started,
the information concerning the range of states will be changed as the premises arebeing fed in.
The major premise of the MT is the proposition A → C. It tells us that C is true
if A is true, that is, the state where A is true but C is false does not occur. This is
the logical information contained in the implication. Now we start the reasoning in
MT. Input A → C. The range I in Table 2 is changed into the range II in Table 3.
TT
FT

—
FF

Table 3: Range II
The state TF is removed from the range I by the new information A → C. Then
we input the minor premise ¬C which means that ¬C is true. The information
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Ma and Pietarinen
involved in ¬C is that the opposite truth value is obtained if we give a truth value
for C. Hence C is false. And hence the states TT and FT cannot survive since they
allow only T for C. Finally, we obtain the final range of states in Table 4.
—
—

—
FF

Table 4: Range III
Let us call this elimination process the dynamic process of the range of information. It is seen to provide a rough characterization of what deductive reasoning
means:• A pattern of propositional reasoning P is deductive, if and only if there is no
uncertainty at the final stage of the dynamic process of the range of information
on P .
Note that the uncertainty concerns not the truth values per se but whether we arein the position to determine the conclusions’s truth value or not. Naturally, there
is no state at the final stage of the dynamic process in which the conclusion in P
would come out as false. In other words, deductive reasoning is maximally secure: itapproaches certainty to the fullest degree when compared with other, non-deductive
kinds of reasoning. But of course the conclusion in deductive reasoning can neverbe any more certain than any of its premises.
As in all reasoning, information flows in deductive reasoning. Information as arange provides an appropriate basis for such dynamics. Every pattern of deductive
reasoning in propositional logic can be interpreted as a dynamic process exhibitedin the range of information.
Let us now consider Peirce’s abduction that has an interrogative conclusion as
well as a negated minor premise. One direct formalization of it is to use classicalpropositional logic with a question mark in the conclusion. We can have the following
forms to be contrasted with MT:
A → C, C ` ?¬A.

A → ¬C, ¬C ` ?¬A.

(1)
(2)

Given the classical reading of negation, these two patterns are essentially the same.
Peirce’s 1905 formulation of abduction took both to correspond to MT. The key
ingredient is the question mark ?, which represents the interrogative mood in the84A Dynamic Approach to Peirce’s Abductive Logic
conclusion. Recall that the conclusion ?¬A is to indicate that ‘it is to be inquired
whether A is not true’.8
With respect to (1), using our analysis of information flow as a MT we have thefollowing dynamic process of changes in the range of information:
TT
FT

TF
FF

=⇒

A→C

TT
FT

—
FF

=⇒
C

TT
FT

—
—

At the final stage, we arrive at the range of two states TT and FT in which
we are uncertain whether a definite truth value of A can be established. In Peirce’s
terms, the reasoning now is less secure than the previous one. The uncertaintyof this kind is what is contained in the conclusion of abductive reasoning in theinterrogative mood (?¬A).
For Peirce, however, less secure reasoning is in a certain quite concrete sense
superior to the more secure, deductive type of reasoning: the former is more valuable
in its productiveness. Since our hypotheses, models, simulations, scenarios andultimately theories are bound to be fallible, the value of abduction lies in its power
to open up new, future lines of investigation. The trade-off is that when the security
decreases, the amount of “uberty” (Peirce’s term for those useful and valuable typesof productiveness that we find thriving in uncertainty) increases.
Likewise, for (2) we have the following dynamic process:
TT
FT

TF
FF

=⇒

A→¬C

—
FT

TF
FF

¬C

=⇒

—
—

TF
FF

The final truth values of A measure the uncertainty that remains in the final
stage of the reasoning process.
Peirce’s abduction as the MT with an interrogative conclusion can be viewed
as reasoning in which the range of information changes along with the input of thepremises, achieving a certain measure of uncertainty about the conclusion in the
final stage. It is not difficult to see that every abductive reasoning pattern given inclassical propositional logic can be analyzed in terms of such ranges of information
as shown above.
8

In fact, the negation symbol in Peirce’s conclusion ?¬A can be removed, since the uniform
pattern for (1) and (2) is A → C, C ` ?A. What is significant is the meaning of the question mark.

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Ma and Pietarinen
Moreover, we can generalize the MT-abduction in the following way:
A1 → C, . . . , An → C, C ` ?¬A1 & . . . & ?¬An

A1 → ¬C, . . . , An → ¬C, ¬C ` ?¬A1 & . . . & ?¬An .

(3)
(4)

In (3) and (4), we have n possible reasons A1 . . . An . The conclusion is the conjunction of interrogative assertions concerning these reasons. Naturally, we could equallywell consider disjunctive cases that inquire “whether some of these possibilities are
not the case?”. Obviously, our analysis on the dynamics of abductive reasoning canbe applied to all these patterns.
Up to now we have achieved an interpretation of the interrogative mood in the
conclusion ?¬A of abduction; namely, it is uncertainty about the establishment of
the truth value of A. However, this obviously is not a sufficient interpretation ofwhat ?¬A means in Peirce’s novel 1905 schema of abduction. It suffices to recall
that the starting point of abductive stage of inquiry is the observed surprising fact,
and the end of abduction is a formulation of a conjecture submitted to investigation.
Intuitively, ?¬A means that it is to be inquired whether ¬A is a plausible conjecture
or not. Uncertainty of course cannot do the whole job of living inquiry. It only
nurtures it by increasing the degrees of freedom for the lines of research scientificprojects can undertake. The inquiry comes to its full blossom only after satisfactoryinterrogative conclusions have been made. But then the obvious question that arises
is: what is the model for selecting plausible reasons for a surprising fact among somany other reasons? Even further, is there, at least in an ideal case, a unique, mostplausible such reason? To address these questions, we need to seek and find answers
to ?¬A1 , . . . , ?¬An just as the schemas (3) and (4) invite us to do.
Given a surprising fact F , there may be many, or even an endless number of,potential propositions that could count as possible reasons that could explain how F
ceases to be surprising. In order to form plausible conjectures, on the other hand, weneed to state which propositions are within the range of such plausible ones. Even
more specifically, we might need to find a way to state which ones among the full
spectrum of plausible conjectures are more plausible than others, or even further,which among the more plausible ones may be counted among the most plausible such
hypotheses. In the following sections, we will use neighborhood models and their
updates in order to achieve a better logical grasp of how inquirers may find plausible conjectures. The questions concerning establishment of hierarchies or orderings
between a number of plausible conjectures is left for a further occasion.
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A Dynamic Approach to Peirce’s Abductive Logic

4

A Logic of Conjecture

In this section, we introduce a logic of conjecture which is interpreted in neighborhood models. This logic describes the relationship between the occurrence of asurprising fact and the investigand, namely the reason to initiate inquiry that could
ascertain the plausibility of the conjecture. After a surprising fact F occurs to anagent, some possible propositions that might support F occur to the agent which
we call conjectures. It is the surprising facts that force the agents to produce suchcandidates for possible reasons to account for those facts.
To develop a logic of conjecture we need a formal language to express that “theagent conjectures that” A is true. A modal epistemic language that is widely usedin (dynamic) epistemic logic will come in handy.
The desired language LEL consists of a denumerable set V of propositional variables, propositional connectives ¬ (negation), ∨ (disjunction) and the modal attitudeoperator 2. The set of all formulas is defined inductively as follows:
A ::= p | ¬A | (A ∨ A) | 2A,
where p ∈ V. Other propositional connectives ∧ (conjunction) and → (implication)
can be defined as usual. We read the formula 2A as that the agent conjectures that
A is true, while its dual 3A, that the agent does not conjecture that A is false, may
be read as that the agent reckons the conjecture A possible. (That is, “it is possible,
for all that the agent can conjecture, that A”.)99

Instead of conjecturing, the modal attitude involved in Peirce’s abductive reasoning could
well be be taken to be that of hope, in the qualified sense of the scientific hope on our or future
generations’ powers to conjecture or guess right. Thus we could also read the modal operator as“the agent hopes that A would be the case”. Evidence for ‘hope’ as the central attitude involved inabduction comes from Peirce’s letter to F. A. Woods (MS L 477, 1913): “I have always, since earlyin the sixties, recognized three different types of reasoning, viz.: First, Deduction which depends on
our confidence in our ability to analyze the meanings of the signs in or by which we think; second,Induction, which depends upon our confidence that a run of one kind of experience will not bechanged or cease without some indication before it ceases; and third, Retroduction, or Hypothetic
Inference, which depends on our hope, sooner or later, to guess at the conditions under which a given
kind of phenomena will present itself” (added emphasis). Hope, we believe, is an all-important andneglected modal attitude, and even more important than belief, in scientific reasoning and discovery.
It does not seem to have been studied before in philosophy of science or in modal logics. In logic
it would give rise to a new class of ‘spestic’ logics. In philosophy of science, it would replace thedoxastic approaches that are either too far removed from the actual practice of scientists or else
in a conceptual deadlock when founded on the orthodox Bayesian reasoning. To have ‘hope’ inthe truth of propositions comes moreover closer in spirit to Peirce’s limit notion of truth and his
logic of inquiry than belief does. Far from meaning ‘faith’, hope is an expression of a genuine (i.e.,
non-sham and non-fake) scientific attitude which meets fundamental uncertainty concerning future
data, and where beliefs or degrees of belief in the hypothesis would come out as much too strong.

87Ma and Pietarinen
Remark 1. One may wonder about the meaning of the question mark in our notation ?¬A in Section 3 that formalizes the interrogative conclusion. As commented
above, the interrogative conclusion is not merely a question that asks weather A istrue or not. The meaning of ?¬A cannot be identified with the meaning of a purequestion as they occur in natural language. There are indeed many erotetic logics
of question in the literature. For example, [32] take asking a question as drawing
an equivalence relation in the epistemic Hintikka-Kripke model. But we cannot see
how such interpretations could be applied to abduction at all. A similar comment
applies to the suitability of inquisitive semantics [7] for abductive problems, as while
perhaps mildly pragmatistic in its motivation, the models of inquisitive logic fail toreach the generality of our neighbourhood models. In fact, we intend to interpreted?¬A as a determination to find out whether the conjecture A is plausible or not. It
is a request for finding new data, not unlike those questions that can be interpreted
as requests put upon the sources of information to bring it about whether somethingis the case. The proposition that A is a conjecture is expressed by the formula 2A.
Hence ?¬A means that 2A is either plausible or not. The presupposition of such a‘question’ is of course readily established. In the abduction A → C, C ` ?¬A, theproposition C is the surprising fact, and the premiss A → C suggests A to acquire
the status of a conjecture concerning novel findings. But it is the plausibility of A
that really is in question in our framework.
The models for the language LEL that we use are neighborhood models. They
are used in non-normal modal logics (cf. [6]) to represent the information range andalso do so in our analysis of the dynamic process of abduction. We recall some basic
definitions which can be found in [17].
Definition 2. A neighborhood frame is a pair F = (W, τ ), where W is a non-empty
set and τ : W → P(P(W )) is a function from W to the power set of the power
set of W . For each w ∈ W , a set X ∈ τ (w) is called a neighborhood of w. A
neighborhood model is a structure M = (W, τ, V ), where (W, τ ) is a neighborhood
frame and V : V → P(W ) is a valuation.
Given a neighborhood model M = (W, τ, V ) and w ∈ W , we may think of points
in W as the possible worlds or mental states of the inquirer or the group of scientists.
The set τ (w) represents the set of all possible propositions subject to the scientificattitudes of the agent, namely conjectures that the agent or the team may makeat the current world w. Using such neighborhood models one can easily updatethem by deleting propositions which are not plausible. This is the advantage of ourapproach that we take, as compared to the standard Kripke models commonly usedin epistemic logic.
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A Dynamic Approach to Peirce’s Abductive Logic
Definition 3. The truth of a formula A at a world w in a model M (notation:
M, w |= A) can be defined recursively as follows:
M, w |= p iff w ∈ V (p).
M, w |= ¬A iff M, w 6|= A.
M, w |= A ∨ B iff M, w |= A or M, w |= B.
M, w |= 2A iff JAKM ∈ τ (w), where JAKM = {u ∈ W | M, u |= A}.

We say that a formula A is valid, if JAKM = W for every neighborhood modelM = (W, τ, V ).

We say that a neighborhood frame F = (W, τ ) is monotone when for all w ∈ W ,if X ∈ τ (w) and X ⊆ Y ⊆ W , then Y ∈ τ (w). Naturally the same applies to themodels, too. Over monotone neighborhood models, we rewrite the semantic clausefor conjecturing as:
M, w |= 2A iff there exists X ∈ τ (w) such that X ⊆ JAKM .

The attitude logic considered here is the set of all formulas valid over all neighborhood models, which is finitely axiomatizable as the modal logic E in [6] is. Thatis, it is the smallest set of modal formulas containing the classical propositional logic
which is closed under the following rules:(MP): from A and A → B infer B.
(E): from A ↔ B infer 2A ↔ 2B.

Moreover, the modal logic M which is the extension of E that adds the monotonicity
rule (M): from A → B infer 2A → 2B, is sound and complete with respect to theclass of all monotone neighborhood models (cf. [6]).
Next, we analyze the updates of a neighborhood model and apply that analysis
to the dynamic process of abduction. Given a neighborhood model M = (W, τ, V )and w ∈ W , the neighborhoods of w are in the set τ (w). Consider now a surprising
fact F appearing to the agent. The τ (w) will be changed when the agent asks which
propositions in τ (w) are plausible conjectures for F . Note that the denotation of F
in M is its truth set JF KM . The update can be described as follows:
If a neighborhood X ∈ τ (w) supports F , then keep F in the updated
model. Otherwise, delete X from τ (w) in the updated model.

We need to explain what supporting means. Formally, a neighborhood X ∈ τ (w)supports F if and only if X ⊆ JF KM . Then we can obtain the following formal
definition of the update of a neighborhood model (cf. [17]):89

Ma and Pietarinen
Definition 4. Given a neighborhood model M = (W, τ, V ) and a formula F , theupdated model MF = (JF KM , τ F , V F ) of M by F is defined as follows:τ F (w) = {X ⊆ JF KM | X ∈ τ (w)}, for every w ∈ JF KM .
V F (p) = V (p) ∩ JF KM , for every p ∈ V.

Note that the updated model is monotone if M is monotone: suppose X ∈ τ F (w)and X ⊆ Y ⊆ JF KM . Then X ∈ τ (w). Hence Y ∈ τ (w) by the monotonicity of τ .
Therefore, Y ∈ τ F (w).

To support a surprising fact F is to have at least one conceivable situation,
namely the truth set generated by that fact. To have an updated neighborhood
τ F (w) is exactly to have the range of propositions which support F .

5

Two Examples

We present two examples from the classical literature that illustrate the dynamics
of the logic of abduction.
Example 5. First, we analyze the following example in the context of commonplacereasoning given by Peirce’s student John Dewey [8]:Projecting nearly horizontally from the upper deck of the ferryboat on which
I daily cross the river, is a long white pole, bearing a gilded ball at its tip. It
suggested a flagpole when I first saw it; its color, shape, and gilded ball agreed
with this idea, and these reasons seemed to justify me in this belief. But soon
difficulties presented themselves. The pole was nearly horizontal, an unusual
position for a flagpole; in the next place, there was no pulley, ring, or cordby which to attach a flag; finally, there were elsewhere two vertical staffs fromwhich flags were occasionally flown. It seemed probable that the pole was notthere for flag flying.
I then tried to imagine all possible purposes of such a pole, and to consider
for which of these it was best suited: (a) Possible it was an ornament. But as
all the ferryboats and even the tugboats carried like poles, this hypothesis was
rejected. (b) Possible it was the terminal of a wireless telegraph. But the sameconsiderations made this improbable. Besides, the more natural place for sucha terminal would be the highest part of the boat, on top of the pilot house. (c)Its purpose might be to point out the direction in which the boat is moving.
In support of this conclusion, I discovered that the pole was lower than thepilot house, so that the steersman could easily see it. Moreover, the tip wasenough higher than the base, so that, from the pilots’s position, it must appear
to project far out in front of the boat. Moreover, the pilot being near the front90

A Dynamic Approach to Peirce’s Abductive Logicof the boat, he would need some such guide as to its direction. Tugboats would
also need poles for such a purpose. This hypothesis was so much more probablethan the others that I accepted it. I formed the conclusion that the pole was set
up for the purpose of showing the pilot the direction in which the boat pointed,
to enable him to steer correctly. ([8, pp. 70–71])

Dewey’s example relates to familiar, common sense observations. However, one isusing abduction in ordinary life much in the same sense as in complex scientificinquiry. At the beginning, the surprising fact is the following proposition:
F : There is a long white pole bearing a gilded ball at its tip and near to the upper
deck of the ferryboat.
Then the following four investigands are considered to be possible, consistent hypotheses:
A1 : Is it not that the long white pole is a flagpole?
A2 : Is it not an ornament?
A3 : Is it not the terminal of a wireless telegraph?
A4 : Is it not the pole for pointing out the direction in which the boat is moving?
All these hypotheses support F at some stage. It is the proposition F that makes
the agent to suggest possible reasons A1 to A4 . The surprising fact eliminates otherunrelated propositions about the fact. For example, the fact that the pole was madeby a certain company is irrelevant to the situation at the current state that theagent wants to know the function of the pole. Thus we achieve a stage which canbe represented as the following neighborhoods of the current state:
A1
A3

A2
A4

Some of these neighborhoods are dropped when some more surprising facts areobserved. Consider the following facts appearing in the example:
C1 : There are similar flagpoles on other places.
C2 : Every boat has a pole like this in the same position.
C3 : Similar boats carried similar poles, and a better place for the terminal of awireless telegraph would be in a higher position.
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Ma and Pietarinen
C4 : The pole was lower than the pilot house.
When the fact C1 is suggested, A1 does not support it. Hence A1 is deleted fromthe neighborhoods. Similarly after C2 and C3 are suggested, A2 and A3 are deleted.
Finally, A4 supports facts like C4 , and it is thus kept in the neighborhoods andidentified as the only remaining explanation for the function of the pole.
Dewey’s example conveys well his insight that the investigand has the status ofa working hypothesis: a preliminary, incomplete conjecture whose plausibility canchange as the inquiry goes on.10
Example 6. Our second example is the famous Kepler case, namely Kepler’s discovery of the Martian orbit being elliptical. This was Peirce’s favourite argument
for the reality of abductive reasoning in the history of science.
For example, at a certain stage of Kepler’s eternal exemplar of scientific reasoning, he found that the observed longitudes of Mars, which he had long triedin vain to get fitted with an orbit, were (within the possible limits of error of
the observations) such as they would be if Mars moved in an ellipse. The facts
were thus, in so far, a likeness of those of motion in an elliptic orbit. Kepler
did not conclude from this that the orbit really was an ellipse; but it did incline
him to that idea so much as to decide him to undertake to ascertain whether
virtual predictions about the latitudes and parallaxes based on this hypothesis
would be verified or not. This probational adoption of the hypothesis was an
Abduction. (CP 2.96, c.1902)

After Peirce, the details of Kepler’s five-year process of discovery have been discussed
abundantly by the historians and philosophers of science (see e.g. [15, 16, 35]).
Here we relate those steps that lead to Kepler’s discovery to the dynamic process
of abduction.
At the beginning of Kepler’s quest, the surprising fact is the following proposition:
F : The planet Mars has non-uniform speed.
Years of painstaking investigation had Kepler consider the following four propositions at the various stages of his inquiry as those plausible hypotheses consistent
with the fact F :A1 : Is not the orbit of Mars circular in the geometric sense? (This is Kepler’sVicarious Hypothesis, the goal of which was to develop a new mathematicalmodel by which to calculate Mars’s longitudinal positions.)10

We thank Sami Paavola for this remark.

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A Dynamic Approach to Peirce’s Abductive Logic
A2 : Is not the orbit of Mars circular and epicyclic in the physical sense? (This is
Kepler’s Area Method, the goal of which was to replace the geometric equantpoint (A1 ) by two further ideas: the area principle and the epicycle theory
that describe the real physical forces.)
A3 : Is not the orbit of Mars oval shaped?
A4 : Is not the orbit of Mars an ellipse?
All these hypotheses support F at some stage. A1 supports F in its geometric structure. A2 supports F as regards to the behavior of physical forces. A3 supports thefact F , since the oval-shaped curve is an approximation of the circle in the quadrants
as well as the observed errors in the octants. The fact is being supported by A4 ,namely the elliptical orbit, too, as elliptical shapes are mathematical approximations
to oval-shaped orbits.
More facts were then observed along Kepler’s investigation:
C1 : The equant point makes the angular velocity of Mars around it uniform. Therefore, the planet’s speed along its path is also non-uniform, though according
to the Vicarious Hypothesis, the equant point is not a physically real point inspace.
C2 : There is no acceptable model of Mars to be derived from the area method, as
the calculated longitudes do not fully agree with A1 .
C3 : The calculated longitudes do not fully agree with A3 with an auxiliary ellipsethat approximates oval-shaped orbit.
C4 : Observations and implications of the physical theory can be saved only if theorbit of Mars is an ellipse.
Since no purely geometric description of the celestial movement was satisfactoryfor Kepler, A1 fails to support C1 . A2 fails to support C2 due to the absence of
any observed actual physical fact to imply the hypothesis. (It is at this point of
encountering the observed errors in the longitudinal measurements that Kepler infact began to doubt the circular orbit assumption, in other words his finding of thefamous 8-minute discrepancy in the longitudinal measurement in the octants, inmodern terms the eccentricity in the orbital motion of planets.) Finally, A3 fails to
support C3 , as calculating the longitudes with an auxiliary ellipse also resulted insimilar reversed errors in the octants.
We are therefore left with A4 , the elliptical orbit hypothesis. The fact C4 , itneeds to be remarked, is supported by A4 in the sense that—unlike is the case with93

Ma and Pietarinen
A1 —A4 does not require any mathematical and geometrical account that has no
known relation to reality (known to Kepler at the time of his inquiry, that is). So it
is A4 that is kept in the neighborhoods and taken as the sole remaining explanation
for the surprising fact observed concerning the non-uniform movement of Mars.
In this example, Peirce assigns abductive conclusions the status of probationary
hypotheses much like Dewey’s working hypotheses later on were intended to capture.
Those hypotheses are invitations to further inquiry, which is what the investigand
mood aims to convey.

6

A Logic of Abduction

In Section 4 we defined the logic of making conjectures, in which no dynamic mechanism is introduced. In this section, we present the logic of abduction that we desire,
which is the dynamic extension of the logic of conjecture. This logic was presented
in [17] in the setting of monotone neighborhood models. Here we represent it in theclass of all neighborhood models, as that is what the abductive reasoning modes
really call for.
The language of the dynamic conjecture logic LDC is obtained from LEL byadding dynamic operators of the form [!F ] where F is a formula. The set of LDC formulas is defined recursively as follows:
A ::= p | ¬A | (A ∨ A) | 2A | [!A]A,
where p ∈ V. Define h!AiB := ¬[!A]¬B. Intuitively, [!F ]A means that the hypothesis
A becomes true after the occurrence of the surprising fact F .11
In this way we can read the formula [!F ]2A as follows: Agent’s conjecturing or
hoping for A to turn out to be a plausible one is survived in the updated neighborhood of the current state. In other words, it is not only the proposition but theattitude the agent or the team of inquirers have towards the status of the proposition A that survives in the face of some surprising fact F . And when that happens,
F ceases to be surprising and serves as evidence by virtue of which the abductive
hypothesis acquires or increases its plausibility.
11

Or, alternatively but not equivalently, the hypothesis remains unfalsified. The two are notequivalent, since while the first affirms that A is the case, the second, Popper-style update only
states that the hypothesis has not been proved to be false. We thank the reviewer for raising
the point that the first interpretation may be seen as the natural one, where A becomes the caseafter the occurrence of the surprising fact, whereas the second deals with whether the hypothesis(conjecture) A has been discarded or not, something that has to do with whether A is found in theagent’s neighbourhood.

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A Dynamic Approach to Peirce’s Abductive Logic
Definition 7. Let M = (W, τ, V ) be a neighborhood model. The truth of an LDC formula A (notation: M, w |= A) is defined recursively as the semantics for LS ,adding the following semantic clause:
M, w |= [!F ]A iff M, w |= F implies MF , w |= A.
The logic DC is defined as the set of all LDC -formulas valid over the class of allneighborhood models. The logic DCM is the set of all LDC -formulas valid over theclass of all monotone neighborhood models.
These two logics are finitely axiomatizable, by adding the reduction axioms (R1 –
R5 ) in Table 5 to E and M respectively.
(R1 )
(R2 )
(R3 )
(R4 )
(R5 )

[!F ]p ↔ (F → p)[!F ]¬A ↔ (F → ¬[!F ]A)[!F ](A ∨ B) ↔ [!F ]A ∨ [!F ]B[!F ]2A ↔ (F → 2h!F iA)[!F ][!G]A ↔ [!(F ∧ [!F ]G)]B

Table 5: Reduction Axioms
Theorem 8. The logic DC (DCM) is sound and complete with respect to the classof all (monotone) neighborhood models.
Proof. Here we sketch the proof (cf. [17]). Assume that A 6∈ DC. By the reduction
axioms R1 to R5 , A is equivalent to a formula A∗ ∈ LEL , i.e., without occurrences ofdynamic operators. Then A∗ 6∈ E, for otherwise A would be a theorem of DC since
DC is an extension of E. By the completeness of E, there exists a neighborhoodmodel M and a state w such that M, w 6|= A∗ . Then M, w 6|= A. Hence thecompleteness is obtained. For the soundness, it suffices to verify the validity of those
reduction axioms. Here we recall only the validity of the reduction axiom [!F ]2A ↔
(F → 2h!F iA). Assume M, w |= [!F ]2A and M, w |= F . Hence MF , w |= 2A.
Then JAKMF ∈ τ F (w). Hence JAKMF ⊆ JF KM and JAKMF ∈ τ (w). It is easy tocheck that JAKMF = JF ∧ [!F ]AKM = Jh!F iAKM ∈ τ (w). Hence M, w |= 2h!F iA.
Conversely, assume M, w |= F → 2h!F iA and M, w |= F . Then M, w |= 2h!F iA.
Hence Jh!F iAKM ∈ τ (w). Then JAKMF ∈ τ (w). Since JAKMF ⊆ JF KM , we have
JAKMF ∈ τ F (w). Hence MF , w |= 2A. Therefore M, w |= [!F ]2A. The case for the
logic DCM is quite similar.
95

Ma and Pietarinen
In the updated model MF = (JF KM , τ F , V F ) of a given neighborhood model M
by a surprising fact F , the support relation is involved: a neighborhood X ∈ τ (w)survives in τ F (w) if and only if X supports F . This fact allows us to introduce a
general logic for support. We introduce a binary operator ≺ for the support relation.
We read A ≺ B as that the proposition A supports B in the sense of turning B intoa plausible conjecture.
The language LS of the support logic S consists of a denumerable set V of propositional variables, propositional connectives ¬ and ∨, and the binary supporting
operator ≺. The set LS of all formulas is defined recursively as follows:
A ::= p | ¬A | (A ∨ A) | (A ≺ A) | UA,
where p ∈ V. Other propositional connectives (>, ⊥, ∧, → and ↔) are defined as
usual. Define 2A := A ≺ > and 3A := ¬2¬A. The unary operator U is the global
modality. We read UA as that A holds everywhere.
The semantics for LS will be given in arbitrary neighborhood models (and so it
is not restricted to monotone ones).
Definition 9. Given a neighborhood model M = (W, τ, V ) and w ∈ W , the truthof a formula A at a state w in M (notation: M, w |= A) is defined recursively as
follows:
M, w |= p iff w ∈ V (p)M, w |= ¬A iff M, w 6|= A
M, w |= A ∨ B iff M, w |= A or M, w |= B
M, w |= A ≺ B iff JAKM ∈ τ (w) and JAKM ⊆ JBKM
M, w |= UA iff JAKM = W ,where JAKM = {u ∈ W | M, u |= A} and JBKM = {v ∈ W | M, v |= B} are thetruth sets of A and B in M. We say a formula A is valid, if M, w |= A for every
neighborhood model M and state w in M.
It follows from the semantics directly that the formula(A ≺ B) ↔ 2A ∧ U(A → B)
is in S. Moreover, by the definition, we haveM, w |= 2A iff JAKM ∈ τ (w).

Apparently, this coincides with the semantics of 2 as given in Definition 3.
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A Dynamic Approach to Peirce’s Abductive Logic
The support logic S is defined as the set of all valid formulas in our semantics.12
Now the task is to explore the dynamic extension of S. The language LDS of the
dynamic extension is obtained from LS by adding dynamic operators of the form[!F ]A. We use DS to denote the dynamic logic of support. For getting a complete
axiomatization of DS, it remains to work out the reduction axioms for ≺ and U.
Theorem 10. The following reduction axioms are valid:(R6 ) [!F ]UA ↔ (F → U[!F ]A).

(R7 ) [!F ](A ≺ B) ↔ (F → 2h!F iA ∧ [!F ]U(A → B)).

Proof. For (R6 ), assume M, w |= [!F ]U and M, w |= F . Then MF , w |= UA. Let v
be any state in M. Assume M, v |= F . Then v ∈ JF KM . Since MF , w |= UA, wehave MF , v |= A. Hence M, v |= [!F ]A. Then M, w |= U[!F ]A. Conversely, assume
M, w |= F → U[!F ]A and M, w |= F . Then M, w |= U[!F ]A. Let v be any state in
MF . Then M, v |= F and M, v |= [!F ]A. So MF , v |= A. Hence MF , w |= UA.
For (R7 ), since (A ≺ B) ↔ 2A ∧ U(A → B) ∈ S, by reduction axioms (R5 ) and(R6 ), we get (R7 ).
The reduction axioms reveal how the relation of supporting is affected by updates
as they reduce dynamic formulas to static formulas.
Theorem 11. The dynamic logic of support DS is axiomatized by all theorems of
S and reduction axioms (R1 )–(R5 ) in Table 5 and (R6 ) in Theorem 10.
Proof. The proof is quite similar to the proof of Theorem 8.

7

The Two Examples Revisited

It remains to be checked how DC serves as the logic of abduction in some concrete
cases. To do so, we consider the two examples given in Section 5.
Example 5* . Let M = (W, τ, V ) be a neighborhood model and w ∈ W satisfying
the following conditions:• τ (w) = {V (Ai ) | V (Ai ) 6= ∅ & 1 ≤ i ≤ 4};

12

The logic S may be axiomatized by adding, first, the formula (A ≺ B) ↔ 2A ∧ U(A → B) into
S and, second, adding additional axioms for the universal modality U to the logic E. The global
modality is introduced in order to add the reduction axiom (R7 ) in Theorem 10. The classicalmodal logic with universal modality is studied systematically in [10]. How to use global modalities
in neighborhood models is currently unknown. This problem will be addressed in another work.
Here we concentrate on the logic S defined in the semantic way.

97

Ma and Pietarinen• V (Ci ) 6= ∅ for 1 ≤ i ≤ 4.
During the abductive process of investigation, the agent who observed the ferryboat’s
device made a series of conjectures A1 , . . . , A4 . Then M, w |= 2Ai for 1 ≤ i ≤ 4.
This is verified by the fact that V (Ai ) ∈ τ (w). When the first surprising fact C1
occurs, we have that M, w 6|= A1 ≺ C1 . This is determined by the agent’s knowledge
on the relationship between the surprising fact C1 and the conjecture A1 . Hence themodel M is updated by C1 and we get the new model MC1 = (V (C1 ), τ C1 , V C1 ).
Note that V (A1 ) 6∈ τ C1 (w). Similarly, the model MC1 will be updated by the
succeeding surprising facts C2 and C3 . Then we get the updated model MC1 ,...,C3
in which V (A2 ) and V (A3 ) are eliminated from τ C1 ,...,C3 (w). At the final stage,
A4 supports C4 . Hence in the model MC1 ,...,C3 ,C4 , V (A4 ) is the only set in ! theneighborhood of w, and the conjecture A4 gains plausibility as to the functioning of
the pole.
This argument thus demonstrates that M, w |= hC1 ihC2 ihC3 ihC4 i2A4 , that is,
A4 is still conjectured after all the relevant abductive steps have been followed
through.
As to the second example, we note that earlier expositions of the Kepler case (notably, by [11]) have indeed argued that Kepler’s reasoning is an example of Peirce’s
1903 schema of abduction. However, all previous expositions have used Peirce’s 1903
schema as if that would have been the definite formulation and the one that could
carry out its duties in full. In reality, that schema has nothing to suggest how totake into account the actual dynamics of the process of revision of the model that
Kepler was making along the course of his investigation.
Here is how Kepler’s reasoning looks like in our dynamic logic of abduction DC.
Example 6* . Let M = (W, τ, V ) be a neighborhood model and w ∈ W satisfying
the following conditions:• τ (w) = {V (Ai ) | V (Ai ) 6= ∅ & 1 ≤ i ≤ 4};• V (F ) 6= ∅, V (Ci ) 6= ∅ for 1 ≤ i ≤ 4.
At the beginning, we have M, w |= Ai ≺ F for 1 ≤ i ≤ 4. Hence in the updated
model MF , we have τ F (w) = τ (w) ∩ V (F ). Later, more surprising facts C1 , . . . C4
occurred in the process of discovery. We have M, w 6|= Ai ≺ Ci for 1 ≤ i ≤ 3, but
M, w |= A4 ≺ C4 . In the process of updating, V (A1 ) is eliminated from τ F (w), andsimilarly, V (A2 ) and V (A3 ) are eliminated from τ F,C1 (w) and τ F,C1 ,C3 (w). In the
final stage, τ F,C1 ,...,C4 (w) = {V (A4 )}. The argument therefore demonstrates thatM, w |= hF ihC1 ihC2 ihC3 ihC4 i2A4 , that is, A4 is the most plausible conjecture after
the abductive process has been carried out.
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A Dynamic Approach to Peirce’s Abductive Logic
The treatment of the examples is a semantic one. Interestingly, the two examples come out as quite similar. Far from making them redundant, the similarity inquestion points at an intrinsic connection between the two, which is interesting precisely because they come from two different contexts: the first from a commonsense,
everyday observation and the second from a lifelong scientific inquiry of a famousscientist.
Our logic of abduction thus captures those aspects of the interrogative-style
abductive reasoning that the 1903 schema masks behinds its broad strokes andterse formulation: the dynamics of the investigand that evolves from a tentative,
probationary idea or hypothesis towards a plausible conjecture. Such dynamics arepart and parcel of actual inquiry, including the Kepler case which for Peirce wasan instructive case highlighting the workings of the real discovery in science. But
we also find the phenomenon in the Deweyan experimental logic of an everyday
reasoner.

8

Discussion

That the logical form of abductive reasoning has in Peirce’s work as its conclusion an
investigand mood can be likened to that of requests for further information. If this isso, then our findings vindicate Hintikka [14], who proposed that abductive inferences
aim at answering the inquirer’s questions put to some definite source of information.
It remains to be investigated to what extent the abductive logic of support agrees
with the interrogative model of inquiry, and how to modify and extend them inorder to investigate the suggested allegiance.13 We expect the interrogative modelof inquiry not to fully rationalize all the critical steps of inquiry, however, among
them the guessing element at those stages in which uncertainty is in its highest.
Our novel construal of the logic of abduction also vindicates the Gabbay–Woods
approach to abduction [9, 36]. Their schema takes abduction to be an ignorancepreserving process that does not advance the epistemic state of the inquirer. Abduction starts from the facts and aims at a plausible conjecture, even when thefacts have run out. In our model for abductive logic, a plausible conjecture is justa proposition in the the mind of an agent, or on the draft version of a conference
paper by a research team, that can serve as a possible reason for the occurrence ofa surprising fact. It need not explain it, although it can well turn out to do so later13[12] proposes a certain complementarity between dynamic epistemic logic and the interrogativemodel of inquiry. As often is the case, the mere formal connections, or precisifications, are insufficientto establish the requisite deeper conceptual connections. What is also needed is the reality check thattakes those connections through what is done in actual scientific practice and what the experimental
condu! ct of the scientists is, including the all-important controlled experiments.

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Ma and Pietarinen
on. Abduction is a process from a surprising fact to the invitation to investigate.
New facts come to pass as the investigators look for new sources of information, andold conjectures are deleted from the minds, clouds and preliminary drafts.
Many important issues are left for the future research. For one thing, how does
an investigator or a group of investigators gravitate to a limited set of most plausible
conjectures? This difficult question cannot be answered as yet. We can can only
point out a couple of preliminary suggestions. First, we should avoid the temptation to define the meaning of ‘the most plausible conjectures’ too simplemindedly
as those that are more ‘truth-like’ or ‘increasingly better approximations’ to truth.
For such approximations are normally measured by prior likelihoods that nevertheless fail to capture essential aspects of scientific practice in discovery. For it is in
the nature of scientific inquiry, we believe, that even those hypotheses, models andconjectures that score high on plausibility, are raw, that is, they may have to be
dumped overnight, if new and surprising evidence, facts, observations and experiments so happen to mandate.
Plausibility orderings are used in [1] to define the concept of belief in the dynamic
approach to belief revision: an agent believes that a proposition P is true at thecurrent state, iff P is true at the most plausible state with respect to the plausibility
ordering for the agent at the current state. Hierarchies of belief are interpretedby hierarchies of plausibility order in the model. This approach is a qualitative
approach to belief. Degrees of beliefs are studied by a quantitative approach in [33],for instance. In game theory, hierarchies of beliefs are introduced for an epistemic
analysis of solution concepts in dynamic games (e.g., [2]).
Our approach to the abductive search for plausible conjectures suggests thatsome hierarchies or orderings of plausible conjectures can indeed be introduced between conjectures. But the creation of those orderings is not to be taken in any
belief-revisionary fashion due to their deflationary, Bayesian, or even the IBE-related
flavours. If one were to ground such ranks on probabilistic methods, many novel
guesses (for which the misnomer of ‘high-risk-high-gain’ is in these days often used)that fall outside of standard deviations would be weeded out. This would merely
serve to block the growth of inquiry. To aid narrowing down very large spaces ofplausible conjectures and to so eliminate uncertainty concerning the myriad of outcomes that abductive reasoning is prone to produce, one could instead conceive of
various plausibility measures and orderings between conjectures with such weighings. In those cases the more plausible hypotheses could be taken to be those thatsurvive ! longer, or more precisely speaking are the ones that survive even when
faced with a larger number of or more weighty surprising facts than many other hypotheses do. Such plausible hypotheses are those that scientists continue to upholdby putting their hopes on ones that are anticipated to turn out to be the case in100

A Dynamic Approach to Peirce’s Abductive Logic
the future, albeit perhaps only in the distant posteriority. But the crucial point isthat the increase in plausibility need not imply any increase in the inquirer’s belief
in them.14 The ranking between plausible conjectures would follow from the proportion of hypotheses kept on board to those that are dropped, given the amountof different but connected surprising facts encountered during the related abductive
reasoning processes, plus the all-important economic and strategic factors relevantto the proportional selection to test those hypotheses.15 No strong epistemic element
is involved here: we do not work with knowledge or belief concerning hypotheses
and conjectures. What is crucial is how plausibility relates not to any single guessbut to the network of related, connected guesses and their anticipated outcomes.
To increase plausibility o! f one’s guesses one could for instance look for possible
connections between the outcomes of a number of other, related guesses in a numberof other, related abductive tasks. We leave deeper analyses of these issues for further
work.

9

Conclusion

In his late theory, Peirce called abduction “reasoning from surprise to inquiry”. In
his unpublished 1905 letter he attributes a novel logical form to it which is that of
the Modus Tollens and in which the conclusion is put in the interrogative (“investigand”) mood. The conclusion does not present a given hypothesis for contemplation;
it merely suggests that it would be reasonable and prudent, given the overall goals,
background theories and assumptions, common experience and the context of investigation, to inquire whether a suggested hypothesis could turn out to be the case ornot. Peirce’s novel schema proposes adoption of a hypothesis as a reasonable thing
to do in the sense of promising specific strategic advantages to those who follow
their intellectual courage to guess at that hypothesis.
In this paper, we provided a new interpretation of Peirce’s insights concerning
the logical form of abduction, and developed those insights further as detailed in
the dynamic logic of support. That logic incorporates the indispensable dynamic
14

Poignant remarks from Peirce attest this: “Belief has no place in science at all . . . Nothing isvital for science; nothing can be. Its accepted propositions, therefore, are but opinions at most;and the whole list is provisional. The scientific man is not in the least wedded to his conclusions.
He risks nothing upon them. He stands ready to abandon one or all as soon as experience opposes
them” (CP 1.635).
15
Fear no failures, since the cost of questions is small: “In pure abduction, it can never bejustifiable to accept the hypothesis otherwise than as an interrogation. But as long as that condition
is observed, no positive falsity is to be feared; and therefore the whole question of what one out ofa number of possible hypotheses ought to be entertained becomes purely a question of economy”(CP 6.528, c.1901; cf. [19]).

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Ma and Pietarinen
aspects of abductive reasoning and analyzes what the conjecture-making in Peirce’s
sense could logically speaking mean. After all, the goal of Peirce’s 1905 abductive
schema is not to abduce any singular proposition to serve as an explanation of facts.
Its aim is to conclude which overall and general lines of investigation and research
agendas would be the most reasonable ones for scientists to adopt. In that very
concrete sense, abduction has a much wider reach than what has ordinarily beenrecognized. It suggests some preliminary pointers and directions towards what could
evolve into a complicated process of interlocking procedures for theory formation.
It does not give closure on the sets of propositions that one is to believe in at theexpense of some others.

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Received January 2015

Abduction and Beyond
Methodological and Computational
Aspects of Creativity in Natural Sciences
Andrés Rivadulla∗
Complutense University Madrid, Spainarivadulla@filos.ucm.es

Abstract
My first aim in this paper is to complement abduction as a standard discovery
practice in the methodology of science. I do this in two ways: firstly by meansof the introduction of preduction as a deductive discovery strategy. I term
preduction the method of reasoning, used predominantly in theoretical physics,
which starting from the available theoretical background as a whole, allows forthe deductive anticipation of previously unknown results, provided that themathematical combination of already accepted results of different theories ordisciplines is compatible with dimensional analysis. This is the method by which
many hypotheses, laws, and theoretical models are introduced into physics.
Preduction complements Charles Peirce’s view that it is only via abductionthat new ideas come to science. But, secondly, since standard abduction is
not always sufficient to provide best explanations, the methodology of sciencesometimes has to proceed more preductivo: abduction resorts to preduction. I
call this form of abductive reasoning sophisticated abduction.
Moreover, the automation of ampliative inferences such as induction andabduction in the field of computational discovery of scientific knowledge encourages the hypothesis of computational preduction. Computational preduction
extends the possibilities of machine learning, which in the past forty years has
facilitated the possibilities of computational systems both to rediscover empirical laws and to automatically discover equations in data bases. Automatedpreduction should also facilitate scientific creativity in theoretical physics.
Keywords: Scientific Discovery, Standard Abduction, Theoretical Preduction, Sophisticated Abduction, Automated Discovery, Computational Preduction
I very much thank the four anonymous referees for their interesting comments on an earlier versionof this paper.
∗
Complutense research Group 930174 and research project FFI2014-52224-P financed by the
Ministry of Economy and Competitiveness of the Kingdom of Spain.
Vol. 3 No. 3 2016
IFCoLog Journal of Logic and its Applications

Rivadulla

1

Introduction

Throughout the twentieth century, philosophers of science have to a great extent
abandoned the issues relating to the context of scientific discovery. Popper’s negationof the methodological relevance of the ways by which new ideas enter into science has
contributed to ignorance of the context of discovery in the methodology of science.
Most philosophers have consequently disregarded the creative role played in naturalscience by ampliative inferences such as induction and abduction. Moreover, theconsolidation of the hypothetic-deductive method in the methodology of science inthe twentieth century has ironically contributed to the neglect of the significance ofdeductive reasoning in the context of discovery. As Peter Medawar acknowledgedin [14], his contribution to Schilpp’s volume dedicated to Karl Popper, “disclaiming
any power to explain how hypotheses come into being” was a sign of the weaknessof hypothetic-deductivism.
In spite of this negative attitude, abduction has played a fundamental role in thehistory of Western science for the introduction of highly significant hypotheses, bothin observational and in theoretical natural sciences, from the postulation of modelsof the planetary movements in ancient astronomy to the most recent hypotheses ondark matter and dark energy in theoretical astrophysics. Section 2 is intended toshow the importance of standard abduction in the methodology of natural sciences.
In Section 3 I argue that deductive reasoning can be extended to the context
of discovery of theoretical natural sciences such as mathematical physics. I use theterm theoretical preduction to describe the method of reasoning which consists in
the implementation of deductive reasoning in the context of scientific discovery. My
aim will be to show the role that preduction plays in the methodology of physicalsciences. I claim that preduction is a method of reasoning which, starting from theavailable theoretical background as a whole, allows for the deductive anticipationof new results, provided that the combination and mathematical manipulation of
the previously accepted results of different disciplines of theoretical physics - thosewhich have been taken as premises of preductive reasoning - are compatible withdimensional analysis. I affirm that this is the way in which many factual hypotheses,
laws, and theoretical models are postulated in physics. Since the results, which
are the premises of preductive reasoning, derive from different theories, preduction
represents transverse or inter-theoretical deduction. This is what makes it possible
to anticipate new ideas in physics.
In Section 4 I tackle the role of abduction in the context of explanation. It is
widely known that during the 1960s, abduction was identified as a form of inference
to the best explanation. Nonetheless, the explanation of many natural phenomenarequires more than spontaneous acts of creativity such as those that one might106

Abduction and Beyond
imagine in the cases of Kepler, Darwin or Wegener. Much mathematical work often
has to be done in order to advance justified theoretical explanations. In thosecases, the explanation takes place more preductivo. I term sophisticated abduction
the corresponding inference to the best explanation, and I illustrate this fact ofthe methodology of science by resorting to an interesting case study of theoreticalastrophysics: stellar pulsation.
Finally in Section 5 I address the question of discovery from the point of view ofmachine discovery theory. Since the 1970s, despite the neglect of scientific discovery
by most official philosophers of science, the field of computer-supported scientificdiscovery has attracted the attention of scientists working in the domain of artificial intelligence and knowledge acquisition, meaning that the field of computational
discovery of scientific knowledge has become increasingly important and fruitful.
Indeed, according to [30, 1]: “Whereas early work in the philosophical traditionemphasized the evaluation of laws and theories (e.g., Popper 1965 [Conjectures andRefutations]), recent research in the paradigm of cognitive science has emphasized
scientific discovery, including the activities of theory formation, law induction, andexperimentation. Moreover, the early philosophical approaches focused on the structure of scientific knowledge, whereas recent work has focused on the process of scientific thought and on describing these activities in computational terms”.
Once ampliative inferences such as induction and abduction have been implemented in the field of machine learning, the next step will encourage the computational hypothesis of theoretical preduction. Computational preduction equates
to automating preduction mechanisms. This possibility complements any alreadyexisting computational methods for the automation of scientific discoveries. Automated preduction would facilitate the anticipation of new results in theoreticalphysics. The philosophy of science invites discovery science to start with the possibility of automated preductive discovery in theoretical sciences such as mathematicalphysics.
A complete analysis of abduction – abductive reasoning has many faces, and isa very polyhedric instrument – should make reference to the underlying logic of abduction – an enterprise which is being tackled by many contemporary logicians in avery competent way– as well as to the use of abduction at the meta-methodologicallevel. Indeed scientific realists argue nowadays that realism is grounded in a second order or grand abduction. But since both these questions constitute separate
domains of contemporary philosophy, they fall outside the scope of this article.
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2Standard abduction. The methodology of discovery
in natural sciences

Is the question of how it happens that a new idea is introduced into science relevantto the methodology of science? There are two different answers to this question:
1. The negative one, the Popper-Reichenbach approach, according to which thephilosophy of science is concerned only with questions of justification or validity, i.e. only how scientists test their theories, hypotheses or conjectures ismethodologically relevant. How they find them is irrelevant.
2. The positive one, the Peircean approach
Following Popper-Reichenbach’s view, the contemporary official methodology ofscience has focused exclusively on synchronic or systematic aspects of science (thejustification context), and has disregarded for decades the issues relating to themethods of conceiving new ideas (the discovery context). According to the official
view, science applies exclusively the deductive testing of hypotheses (Popperiantests).
Nonetheless, in some natural sciences, for instance in geology, we find the following viewpoints. Dan McKenzie, one of the creators of the plate tectonics model,
confesses in [13, p.185] that: “hypothesis testing in its strict form is not an activity familiar to most earth scientists”, and John Sclater [28, p.138] affirms: “Earthscientists, in most cases, observe and describe phenomena rather than conducting
experiments to test hypotheses”. And in palaeoanthropology, James Noonan, Pääbo
et al. [16] in a paper on the comparison of the genomes of Neanderthals and modern
humans, claim that “Our knowledge of Neanderthals is based on a limited numberof remains and artefacts from which we must make inferences about their biology,
behavior, and relationship to ourselves”. For his part, Carlos Lorenzo [10, p.103]
affirms that “Philogenetic trees are only evolutionary hypotheses built upon a continuously changing empirical basis. It is usual that these hypotheses are tested, andmodified, if necessary, on the ground of new data”. Thus deductive testing is not
the exclusive methodology in natural sciences. It may not even be the predominantone.
Thirty years before Popper and Reichenbach’s neglect of the philosophical relevance of the processes of conceiving or inventing scientific hypotheses, Charles Peirce
took up a position on the processes of forming or devising explanatory scientific hypotheses. Peirce’s basic idea was very simple: more often than not, scientists stumble over unexpected, surprising, striking facts or phenomena. Instead of passively
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wondering about them, it seemed reasonable to Peirce to adopt an approach both
logical and philosophical about how to propose or to introduce new hypotheses: anapproach which, if true, would account for the facts. Peirce gave the name abduction to the logical operation which introduced new ideas into science. For instance:“All the ideas of science come to it by the way of abduction. Abduction consists instudying facts and devising a theory to explain them” [18, 5.170]
Popper-Reichenbach’s negation of the importance of the context of discovery was
one of the major errors of contemporary philosophy of science. Indeed, abduction has
been widely applied in the natural sciences, from ancient astronomy to contemporary
theoretical astrophysics, evolutionary biology, the earth sciences and palaeontology,
among others. Let me illustrate some of them in more detail.
From a methodological viewpoint, palaeoanthropology behaves like a typical empirical science: recognition of surprising facts, abduction by elimination of mutually
exclusive hypotheses, hypothesis revision in the light of novel data, and the beginning of a new cycle. For instance, the postulation of a new hominin species, Homoantecessor:
1. The surprising fact was the discovery on 1995 at Sierra de Atapuerca, Spain,of a part of the facial skeleton of a young man – the Gran Dolina Boy – withan antiquity of nearly 800 kiloyears.
2. This fossil did not show either the primitive features of Homo ergaster or thederived characters of Homo heidelbergensis – 500 kiloyears old – which were
inherited by Homo neanderthalensis.
3. Moreover, the skull capacity of the Gran Dolina Boy – about 1000 cc, considerably bigger than that of the best preserved skulls of Homo ergaster – provided
an excellent reason for not considering it to be a member of Homo ergaster.
4. Furthermore, the Gran Dolina Boy could not be a representative of Homoerectus, since this is mainly distributed throughout Asia (also in Israel and inGeorgia), and these fossils are considerably older.
5. Conclusion: The Gran Dolina Boy could only belong to a new species, Homoantecessor: “a common predecessor of both the evolutionary line that in Europe led to Homo neanderthalensis and of the evolutionary line that in Africa
led to the modern populations of Homo sapiens”. [2, p.35]
As for the case of the continental drift hypothesis in the earth sciences, the postulation of this hypothesis has proceeded – admits Alfred Lothar Wegener [34, p.167]– purely empirically: “by means of the totality of geodetic, geophysical, geological,
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biological and palaeoclimatic data. . . . This is the inductive method, one which
the natural sciences are forced to employ in the vast majority of cases”. (Both thename and the logic and philosophy of abduction were completely unknown in 1915,
the year of the first edition of Die Entstehung der Kontinente und Ozeane). A huge
number of facts which suggested this hypothesis to Wegener is summarily presented
here (Cf. also [25]):
1. Geodetic data: Observation, on the basis of astronomical, radiotelegraphic andradio-emission measures, of the continuous separation of Europe and America.
2. Geophysical data: Compatibility of the Fennoscandian rebound and theisostasy hypothesis with lateral continental displacements.
3. Geological data: Affinities between the plateaus of Brazil and Africa, andbetween the mountains of Buenos Aires and the Cape region, etc.
4. Palaeontological data: The distribution of the Glossopteris flora – a fern fossil
register – in Australia, South India, Central Africa and Patagonia, and of
Mesosaurus in Africa and South America
5. Palaeoclimatical data: The Spitzberg(en) Islands, nowadays affected by a polar
climate, must have enjoyed a much warmer climate in the Mesozoic and in the
Palaeozoic.
As these examples show, observational natural sciences like palaeoanthropology andthe Earth sciences are predominantly empirical. They rely on experience, on observation, they apply abduction as a discovery practice. They do not implement
any Popperian deductive testing of hypotheses but only, when required, additional
evidence tests.
Nonetheless, as the following example shows, abduction is also applied in theoretical sciences as the only form of introducing new ideas into science. In astrophysics,
the Newtonian version of Kepler’s Third Law is a very useful instrument. It permits,
for instance, the calculation of the Sun’s mass and the masses of the planets of oursolar system; it allows the calculation of the masses of the stars that constitute abinary system; and last but not least, it allows the determination of the number ofstars of our Galaxy. It is a very successful law indeed. Moreover, Kepler’s Third Law
demands that any heavenly body situated far away from its rotation centre movevery slowly. Thus, according to the law, stars located far away from their galaxy
centres must have very small orbital velocities.
In the 1930s, Jan Hendrik Oort and Fritz Zwicky discovered to their surprisethat this was not the case. Instead of declaring Kepler’s Third Law to be wrong
110Abduction and Beyond– and therefore deducing that Newtonian mechanics is mistaken – Zwicky, in an
undoubtedly abductive manner, proposed that there must be inside galaxies much
more matter than can be observed. Since he had no idea how this hidden matter
was constituted, he coined for it the now-famous term dark matter. As we will see
according to the nomenclature to be introduced in Section 4, the hypothesis of theexistence of dark matter was the best explanation of the (surprising) fact that stars
located on the boundaries of their galaxies have an orbital velocity that is higher
than expected. I illustrate further abductive inferences in theoretical cosmology in
[26].

3

The boundaries of abduction. Preductive practices ofcreativity in theoretical natural sciences

For the sake of argument I return to Charles Peirce [18, 5.145]: “Induction cannever originate any idea whatever. No more can deduction”. Nonetheless I do notcompletely agree with him that “All the ideas of science come to it by the way of
abduction” [18, 5.145]. Thus my main question is: can deductive reasoning be usedin the context of scientific discovery? My answer will be: Yes, it can.
I maintain that in the methodology of theoretical physics, we can implement
deductive reasoning in the context of discovery, beyond its ordinary uses in thecontext of justification. This is possible because theoretical physics uses mathematics
as an indispensable tool. Theoretical physicists apply at will Leibniz’s principle ofsubstitutio salva veritate: dimensional analysis guarantees substitutio salva legalitate,i.e. the legitimacy of the undertaken substitutions. Thus a new form of reasoning inscientific methodology, which I call theoretical preduction or simply preduction, canbe identified.
Peter Medawar [14, p.289] pointed out the weakness of deductive reasoning inscientific methodology: “The weakness of the hypothetico-deductive system, insofar
as it might profess to offer a complete account of the scientific process, lies in its
disclaiming any power to explain how hypotheses come into being”. Later, Thomas
Nickles [15, p.446] conjectured about the applicability of deductive reasoning in scientific discovery: “it is worth noting that even an ordinary deductive argument neednot be sterile: it may be epistemologically ampliative even though it is not logically ampliative, for we are not logically omniscient beings who see all the logicalconsequences of a set of propositions”. But it was Elie Zahar [35, 244-245] who unambiguously advanced the thesis of the deductive character of scientific discovery:“the process of discovery . . . rests largely on deductive arguments from principles
which underlie not only science and deductive metaphysics but also everyday de111

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cisions” (my italics). On pages 249-250 Zahar further argues that “the logic ofscientific discovery is not inductive; neither does it resemble artistic creation, but
is, instead, largely deductive. . . . Unlike Popper, I think that deduction constitutes
the most important element in the process of invention”. But since Zahar’s thesisof the deductive character of scientific discovery is based on the metaphysical assumption of several philosophical principles and meta-principles, which, as a kind ofpremise for deductive reasoning, allow for the creation of theories – the Principle ofCorrespondence, the Principle of Identity, the Principle of Sufficient Reason, and
the Principle of Proportionality of Cause and Effect – his proposal becomes a metaphysical theory about the deductive nature of scientific discovery which cannot bemade congruent with both ampliative and anticipative strategies in the methodology
of natural sciences.
1. Preduction, is a method of reasoning that begins with previously accepted
results of the available theoretical background. It consists in resorting to
the available results of theoretical physics as a whole, in order to anticipate
new ideas by mathematical combination and manipulation, compatible withdimensional analysis, of the available results, although not every combinationneed be heuristically fruitful.
2. The results postulated methodologically as premises of an inferential procedure
proceed from differing theories, and any accepted result can serve as a premise– on the understanding that accepted does not imply accepted as true.
3. This suggests the notion of a hypothetic-deductive method. Indeed, preduction
is an implementation of the deductive way of reasoning. The specificity of
preduction is that it is an extension of deductive reasoning into the contexts
of scientific discovery (and explanation).
4. Since the results, which are the premises of the preductive way of reasoning,proceed from different theories, preduction is transversal or inter-theoreticaldeduction. This is what makes it possible to anticipate new ideas in physics,
i.e. deductive creativity in science.
5. Preductive reasoning differs from standard abduction, and from any ampliative
inference, in that the preduced results do not proceed from empirical data, butrather follow from deductive reasoning starting from the available accepted
theoretical background taken as a whole.
Abduction is certainly a very important form of reasoning to the methodology ofscience. But it is not enough, for new ideas: hypotheses, laws, theoretical models,112

Abduction and Beyond
etc. may also be advanced deductively. So abduction must be complemented by
preduction as forms of scientific discovery. Abduction and preducción are complementary strategies of scientific creativity. Other characterizations of the conceptof theoretical preduction, together with illustrative examples from physics, I havepresented in Rivadulla [22, 24, 23, 25].
Physicists apply preductive reasoning in a natural way in order to anticipate as
yet unavailable ideas. Stellar astrophysics can be considered paradigmatic for theapplication of preductive reasoning. The British physicist Arthur S. Eddington [3,1] affirms, for instance, that although “At first sight it would seem that the deep
interior of the sun and stars is less accessible to scientific investigation than any
other region of the universe. . . , the interior of a star is not wholly off from such
communication. A gravitational field emanates from it . . . ; further, radiant energy
from the hot interior after many deflections and transformations manages to struggle
to the surface and begin its journey across space. From these two clues alone achain of deduction can start, which is perhaps the more trustworthy because it is only
possible to employ in it the most universal rules of nature –the conservation of energy
and momentum, the laws of chance and averages, the second law of thermodynamics,
the fundamental properties of the atom, and so on. There is no more essential
uncertainty in the knowledge so reached than there is in most scientific inferences”(my italics). And Dale A. Ostlie & Bradley W. Carroll [17, p.590] claim, withregard to the Chandrasekhar limit formula for the mass of white dwarfs: “Thisformula is truly remarkable. It contains three fundamental constants –~, c, and G –representing the combined effects of quantum mechanics, relativity, and Newtonian
gravitation on the structure of a white dwarf” (my italics). These quotations confirm
that, if we wish to postulate theoretical models in stellar astrophysics, we have to
combine available relevant results from different disciplines of theoretical physics, so
as to derive an equation, or a set of coupled equations, relating to the phenomenon
under investigation.
As a matter of fact the discovery of the interior structure of main sequencestars amounts to preducing a theoretical model of the stellar interiors. This model
consists of five basic differential equations: pressure, mass, luminosity and temperature (twofold) gradients (Cfr. [17, ch.10]). From a methodological viewpoint themost interesting equations of this theoretical model are the equations of hydrostatic
equilibrium and of temperature gradient.
The idealisations needed for hydrostatic equilibrium are that of a sphericallysymmetric and static star. The corresponding preductive procedure consists in the
combination of three theories: Newtonian mechanics (second and third laws, and theuniversal gravitation law), classical statistics mechanics (the Maxwell-Boltzmann
distribution of the ideal gas pressure), and quantum physics (Planck’s radiation law
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Rivadullaof the radiative pressure of a black body).
The idealisations, assumed for the obtaining of the temperature gradient, are alsothat of a static sphere with black body conditions plus the conditions of adiabatic
expansion. The corresponding preductive procedure consists in the combination of
three disciplines: classical physics for the temperature gradient of radiative transfer(combination of the equation of radiative transfer with the equation of the black
body radiation pressure); classical statistical mechanics; and thermodynamics ofadiabatic processes (for the obtaining of the temperature gradient of a gas expanding
adiabatically).
A full account of the theoretical model of stellar interiors should be completedwith the mass conservation equation and the equation of the luminosity gradient,with the latter depending on the energy generated both by nuclear and gravitational
processes.

4

Getting across the frontiers. Sophisticated abduction

Abduction has to two sides. I completely agree with Lorenzo Magnani [12, p.294]
that abduction both generates plausible hypotheses and that successful abduction
provides best explanations of facts. Magnani [11, p.17-18] anticipates this idea as
he recognizes that “Theoretical abduction is the process of inferring certain facts
and/or laws and hypotheses that render some sentences plausible, that explain or
discover some (eventually new) phenomenon or observation; it is the process ofreasoning in which explanatory hypotheses are formed and evaluated”.
Indeed according to Peirce [18, 7.202] abduction, as “the step of adopting ahypothesis as being suggested by the facts”, also supplies an explanation for novelor surprising phenomena: “The explanation must be such a proposition as would
lead to the prediction of the observed facts, either as necessary consequences orat least as very probable under the circumstances. A hypothesis then, has to be
adopted which is likely in itself and renders the facts likely”. Gilbert Harman [7,p.88-89] and [8, p.165] prefers to name abduction inference to the best explanation
to point to the fact that abduced hypotheses provide the best explanation for theevidence than would any else.
Since the late 1970s, abduction has been, for the methodology of science, inference to the best explanation. Paul Thagard [33, p.77] claims that “Inference
to scientific hypotheses on the basis of what they explain was discussed by such
nineteenth-century thinkers as William Whewell and C. S. Peirce . . . To put itbriefly, inference to the best explanation consists in accepting a hypothesis on the
grounds that it provides a better explanation of the evidence than is provided by114

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alternative hypotheses”. Thagard [33, p.77-78] illustrates this viewpoint resorting to
Fresnel’s wave theory of light, which “explained the facts of reflection and refraction
at least as well as did the particle theory, and . . . there were other facts,. . . , which
only the wave theory could simply explain. (. . . ) Hence the wave theory should
be inferred as the best explanation”. And Hilary Putnam [19, p.198] confesses that
“we accept the Darwinian theory of evolution by natural selection as what Peirce
called an ‘abduction’ or what has recently been called an ‘inference to the best
explanation”’. In the contemporary theory of abduction John Josephson [9, 5] for
instance claims that “Abduction, or inference to the best explanation, is a form of inference that goes from data describing something to a hypothesis that best explainsor accounts for the data”.
In Section 2 I have focused mainly on the creative aspect of abduction. As a formof scientific discovery, abduction has made it possible to introduce very significant
hypotheses in science during the last two and a half millennia, hypotheses that also fit
best to – i.e. best explain – the facts. Nonetheless sometimes creative/explanatory
acts are not as simple as it might seem.
Creativity in science is not always spontaneous. If for the sake of illustration
we restrict ourselves to theoretical physics, then very frequently hard physicalmathematical work is needed when the available empirical data do not directlysuggest an attractive explanation. Since moreover theoretical physicists nowadays
generally combine theoretical and empirical results accepted in differing branchesof physics, preductive reasoning unavoidably becomes subsidiary to abduction whenproviding satisfactory explanations for as yet unexplained constructs. This is why I
say (cf. [24, p.361]) that in such cases the theoretical explanation takes place morepreductivo. Since in such cases the inference to the best explanation depends on theimplementation of preductive reasoning on the context of theoretical explanation, I
name this procedure sophisticated abduction.
In order to understand this proposal as unambiguously as possible, it is important
to note that explanation in theoretical physics encompasses more cases than are supposed to be subsumed under Popper-Hempel’s law-covering nomological-deductivemodel of scientific explanation. As I claim in [21, p.166-167] “any physical construct:facts, laws, hypotheses, etc., can only receive a theoretical explanation, when it canbe deduced mathematically in the framework of another physical construct of higher
level. Thus not only facts, but also empirical generalizations, abstract laws and eventheories themselves admit of explanations in this sense”. For instance, Kepler’s empirical laws of planetary movements was theoretically explained only after it wasincorporated into Newtonian celestial mechanics; Stefan’s Law of black body radiation and Wilhelm Wien’s displacement law received a theoretical explanation when
they could be deduced mathematically from Max Planck’s radiation law; and Bohr’s
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atomic model for the first time provided a theoretical explanation of the spectra ofelements etc. But even the principles of phenomenological thermodynamics could
be satisfactorily explained, as they were justified within the framework of statistical
thermodynamics. This concept of theoretical explanation realizes Einstein’s idealof hypothetic-deductive physics as he develops it in different articles reprinted in
Einstein’s [4].
Since abduction belongs not only to the context of discovery but also to thecontext of scientific explanation – they are the twin faces of abduction – explanatory
abduction is committed to the norms of theoretical explanations. In [24, §4] I havegiven two examples of sophisticated abduction in the realm of quantum mechanics.
In the following I am going to present, very summarily, a different example takenfrom the recent history of theoretical astronomy.
At the end of the sixteenth century, between 1595 and 1596, some fifteen years
before Galileo’s astronomical observations, the German astronomer David Fabricius(1564-1617) observed a variation in the brightness of star o Ceti in the constellation
Cetus. Had he been during his lifetime a better known astronomer, this discovery
would have contributed to the refutation of the Aristotelian theory of the immutability of heavens. The star o Ceti was named Mira, admirable in Latin, by the Polishastronomer Johannes Hevelius (1611-1687). Finally, in 1667 the French astronomer
Ismael Boulliau (1605-1694) determined the period of eleven months of Mira Ceti.
Later, in 1783, John Goodricke (1764-1786) discovered the variability of δ Cephei,
and Edward Pigott (1753-1825) in 1784 the variability of ηAquilae. These variable
stars came to be known as Cepheids. At the end of the nineteenth century, Henrietta
Swan Levitt (1868-1921) discovered more than two thousand variable stars, most ofthem in the Magellanic Cloud. But the phenomenon of the variability of Cepheids
demanded an explanation.
Several hypothetical explanations were suggested. The first ones made use of ananalogy argument: the variability could be due to the existence of spots like sunspots.
Later, Goodricke, Belopolsky and even Schwarzschild proposed that the variability
was due to the binary character of the stars. The tidal effects in the atmospheresof the stars might be responsible for the variable brightness observations. But in
his The Internal Constitution of the Stars, 1926, Arthur Eddington stated that theAmerican astronomer Harlow Shapley (1885-1972) had put forward, in [29], “thechief arguments against the binary theory and in favour of the pulsation theory”.
According to [3, 184], “the binary model is a geometrical impossibility”. The reason
was that, according to the already accepted astrophysical background, the radii ofthe binary stars should be larger than the radii of their own orbits. Three years after
Shapley’s proposal Eddington himself – in two articles published in Observatory 40,
1917 and in Monthly Notices of the Royal Astronomical Society 79, 1918 – provided a116Abduction and Beyond
solid theoretical argument on behalf of the pulsation hypothesis. Henceforth variable
stars became pulsating stars. The issue is that, from a methodological viewpoint, theexplanation of stellar pulsation cannot be conceived in terms of standard abduction.
Much more than a creative jump is involved here. On my view the explanationof stellar pulsation is a typical process of sophisticated abduction, since it requires
the preductive construction of a theoretical model for a pulsating variable star,and this is only possible when suitable results from different branches of theoreticalphysics are taken into account and mathematically combined with one another. This
procedure is by no means simple.
The first stimuli came from a paper, already old at the time, by the German
astrophysicist August Ritter, “Investigations on the height of the atmosphere andon the constitution of gaseous heavenly bodies”[20]. In this article, equation (245),
Ritter claims that the pulsation period of a supposed uniformly dense gaseous sphereis inversely related to the square root of its density. Modern theoretical models ofstellar pulsation start with Eddington’s suggestion in [3, p.186] that, for reasons ofsimplicity, “the oscillations through the greater part of the interior are approximately
adiabatic. We therefore start by considering adiabatic oscillations of a sphere of gas”.
The resulting model is, like in Ritter’s case, an equation expressing that the pulsation
period of a variable star is inversely related to the square root of its average density.
In order to achieve this result, it is necessary to combine (Cf. [17, §14.2] or [1,§16.4] classic thermodynamics (first law) with acoustics and wave theory (adiabatic
velocity of sound) and classical mechanics (hydrostatic equilibrium). According to
Ostlie & Carroll [17], the resulting equation provides a quantitative agreement withthe observed periods of Cepheids that is “not too bad”, and for Erika Böhm-Vitense[1], this equation “explains the observed period-luminosity relation”. The bigger thestar, the lower its density, and since the period is inversely related to the density,
the star’s oscillation will be quicker. This is a typical phenomenon with big stars.
This explanation is the best explanation astrophysicists can admit for stellar
pulsation. It is an abductive one. But since it proceeds in a preductive way, it is notstandard abduction, it is sophisticated abduction. Preduction lies at the other sideof the border line from abduction. Sophisticated abduction, which uses preduction,is an excellent example of the collaboration of both kinds of reasoning, abduction
and preduction, in scientific creativity.

5

Computational models of scientific discovery

The existence of computational studies of science makes evident the lack of imagination of many academic philosophers of science, including both the pro-positivist
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Hans Reichenbach and the anti-positivist Karl Popper, who neglected the importance of the context of scientific discovery in favour of the context of justification.
The ignoring of scientific discovery left the field open for researchers in cognitive science and artificial intelligence to develop the computational study of science. Indeed,
Shrager and Langley [30, 20] attempt “to define a new field of study – the computational modeling of scientific behavior. Despite its relatively recent development,
this research area has already made significant progress on issues that philosophersof science have traditionally ignored. In particular, the field has emphasized thenature of discovery rather than evaluation, and it has dealt with the processes that
underlie science as well as the representation of knowledge. The result has been
a rapidly growing set of computational models that deal with many facets of thescientific enterprise”.
Computational science is a branch of cognitive science and of artificial intelligence that aims at constructing science by computational means. This defines
the field of computational research on scientific discovery, which means “researchon computational models of scientific discovery”, i.e. the construction of “detailed
computational models of the acquisition of knowledge in scientific domains” ([30,p.1-2]).
The implementation of AI methods for the discovery of scientific knowledge permits the automation of both inductive and abductive inferences. Over the past thirty
years, machine learning scientists have provided us with computational systems that
implement the rediscovery of many empirical laws of chemistry and physics, the automated discovery of equations in data bases, etc. Systems such as Bacon, PI,
Dendral, Echo, IDS, Lagramge, Fahrenheit, Dalton etc. have been designed andapplied for empirical discoveries. The cognitive scientists Herbert Simon and Paul
Thagard and the computational scientists Pat Langley, Jeff Shrager, Bernd Hordhausen, Brian Falkenhainer, Jan Zytkow, D. Poole, A. C. Kakas, among others, havecontributed to a general optimism about the possibility of automated inductive andabductive scientific discovery. For instance, under the theoretical assumption thatchemical substances are made up of molecules, and molecules of atoms of the elements, and that atoms are conserved in reactions and that volumes of gasses, underconstant temperature, are proportional to the numbers of their molecules, etc., dalton “deduces the chemical formulas of the molecules involved. For example, on being
told that three volumes of hydrogen and one of nitrogen produce two of ammonia,
it concludes, correctly, that hydrogen and nitrogen are H2 and N2 , respectively, andthat ammonia has the formula NH3 ” [?] 10-11[]Sim1992.
Computational scientists repeatedly assert that they have filled the gap left behind by official philosophers of science, and they reproach them for abandoning thecontext of discovery ([31, p.23] and [32, p.3], [5, p.3, p.23, p.37-38], [30, p.20], [27,118Abduction and Beyondp.3]) Is the moment now ripe for the philosophers of science to regain the lost groundand to advance new possibilities in computational science? In order to answer thisquestion, it is necessary to extend the spectrum of possibilities of scientific discovery.
In this respect, both a wider concept of ‘scientific discovery’ and a closer attention
to real scientific methods are unavoidable. In particular, we must investigate thepossibility of new strategies for the introduction of novel ideas into science, beyond
standard ampliative inferences such as induction and abduction.
As to the first issue, Dzeroski [27, p.3-4] define scientific discovery – briefly,
and without further reference to exclusive methods of research – as “the process
by which a scientist creates or finds some hitherto unknown knowledge, such as aclass of objects, an empirical law, or an explanatory theory. . . . A defining aspect of
discovery is that the knowledge should be new and previously unknown” (my italics).
Let me announce the computational hypothesis of theoretical preduction. Computational preduction is tantamount to assuming the possibility of automating preduction mechanisms. The achievement of computational preduction is a task thatmust be carried out by computational scientists. The role of the philosopher ofscience is to signal an as yet unexplored possibility, namely that the research on
computational models or systems of scientific discovery must be extended beyond
ampliative inferences such as induction and abduction, and into the realm of anticipative inferences like theoretical preduction. This is the content of the computational
hypothesis of theoretical preduction. If this hypothesis is worthy of consideration,then it is down to computational scientists whether or not they develop powerful
deductive reasoners capable of putting it into practice.

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and data. Cambridge: University Press, 1989.
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[3] A. Eddington. The Internal Constitution of the Stars. Cambridge: University Press,
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[4] A. Einstein. Ideas and Opinions. London: Souvenir Press, 1973.
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[6] N. R. Hanson. Patterns of Discovery. An inquiry into the conceptual foundations ofscience. Cambridge: University Press, 1958.
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Rivadulla[8] G. Harman. Knowledge, inference and explanation. American Philosophical Quarterly,5(3):164–173, 1968.
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[26] A. Rivadulla. Abduction in observational and in theoretical sciences. Some examplesof ibe in palaeontology and in cosmology. Revista de Filosofía, 4(2):143–152, 2015.
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Scientific Knowledge, number 4660 in LNAI. Berlin-Heidelberg: Springer, 2007.
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1983.

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122

Course of Action Recommendations andTheir Place in Developmental Abduction
Donna E. West
State University of New York at Cortland
westsimon@twcny.rr.com

Abstract
This inquiry identifies affective, cognitive, and linguistic evidence that, inpoint of fact, imperatives are responsible for the truly originary hypotheses
clearly driven by instinctual abductions, given that the source for these instincts is often an external one. This account demonstrates how such imperatives beckon generation of proposition-making – recommending a course ofaction for self or other. This proposal likewise uncovers how children’s event
judgments acquire an action-based individual and social force – recommending
a course of action for another. This is illustrated by children’s need to recognize
social and conversational roles prior to suggesting that others take those roles inparticular event profiles (cf. [46] for further elaboration). It is obvious that children’s imperative-making demonstrates the power to frame recommendations
for future modes of conduct. Such includes: internalizing “pure” Secondness
and arriving at percepts, translating them to perceptual judgments, and finallysuggesting an entirely novel remediating path.

1

Introduction

This account demonstrates how recommendations for new courses of action constitute quintessential abductions, and how such recommendations are grounded (early
on in ontogeny) in practical habits to feel, think, and act in particular ways. It
asserts that in ontogeny a two-tiered process characterizes abductive rationality:
receipt of unbidden guesses, and a stage in which their viability is determined illustrated by recommendations for would-be action habits. Accordingly, what drives
inferential reasoning at the outset is an internal call to change an action habit (often
The author wishes to express gratitude to the anonymous referees for their energy and insightfulcomments.
Vol. 3 No. 3 2016
IFCoLog Journal of Logic and its Applications

West
motivated by instinct from external sources) rather than internally driven interrogative strategies constructed on the fly. The present model proffers that abductive
reasoning emerges consequent to an impulse to resolve social and scientific problems
via mental habits which explain inter- and intra- event relations. In turn, mental
habits (regularities and their changes) inform abducers whether premises qualify as
candidates to explain puzzling consequences. It is critical to note that the sign most
instrumental in ascribing mental habits to relevant action correlates are indexes,
since they naturally draw attentional paths between participant roles and the event
structure in which they are likely to be imbedded. These attentional paths highlight
regularities of states and action; and inferences are born with implicit suggestions ofnovel relations. Eventually, children’s event repackaging constitutes implicit abductions, since new modes of logical syntax are applied to action performance. Children
can implicitly draw inferences once iconic overlays are applied to indexes, bringing
about the “habitation of events.” At this stage in hypothesis-making, children candetermine the suitability of particular inferential relations – whether to approve/dismiss the premise or whether to revise it. Linguistic directives, (in line with Peirce’s
and Vygotsky’s models) propose explicit course of action recommendations, codifying and validating extended applications of courses of action in diverse context fordifferent participants filling the same function. Tracing inferential reasoning from its
inception as an instinct, through to its status as an organizing device is critical, if weare to recognize its significance in ontological pursuits. In short, primary to proposing novel courses of action is the impulse to express integral knowledge of “event
morphology”—instinctual guessing regarding the participant-holder’s orientations(epistemic, deontic).
Abductive endeavors require proposing successful solutions for potential participants in light of their past habits (preferences/conduct).1 These factors determine
the recommendation or suggested strategy for action in the face of unanticipatedhappenings; and their issuance via language imperatives serves to facilitate reaching
the goal.
This account offers an alternative approach to best explanation rationality – itposits that anticipatory social logic and an imperative to act (how discrete persons
should behave within event profiles) has its source in implicitly learned competencies(not in-born capacities) to guess correctly [42], and is inscribed upon ontological,
more objective eco-based knowledge (in line with Magnani’s [13, 15] and in press,
and Gibson’s 1979 models [6]), for novel plausible proposals to be formulated and1

While both Abduction and Retroduction appear for the first time in “Lessons of the Historyof Science” (c. 1896), and seem to be used interchangeably, Peirce deliberately abandons abduction
as a term in 1907, preferring retroduction in a footnote in MS 318 “Pragmatism,” cf. Bergman andPaavola’s [2, p.1907] textual gloss in the Commens Dictionary of Peirce.

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articulated at any point in the inquiry process. At certain junctures in the inquiry– impelled by “forced choice” [3, p.154], the function of abductions as imperatives
surfaces – suggestions are proffered to improve another’s approach to a problem. As
Woods [51, p.367] intimates, the suggestions are subjunctively informed, in that they
are constructed upon an appreciation for the legitimacy of individual perspectives,
despite their uniqueness or lack of similarity to those of the abducer. Ontologically, forced choice relies upon a natural unfolding of potential causes and effects
associated with events, particularly the unexpected consequence, in light of theimposition of distinctive physical and psychological phenomena for different event
participants/observers.
These emerging, encapsulated recommendations are imperatives, materializing
in a flash, resulting in retroductions – consolidations of past event memories andanticipatory propensities of participants and their roles. Retroductions which integrate how event morphology and social factors inform each other illustrate morethan rudimentary, originary abductions; rather, they offer a novel avenue to inform
others how to legitimately solve problems. Recommendations constitute an all-atonce suggestion (imperative) for future conduct for another, to be entertained andperhaps discarded in favor of a more fitting paradigm – ordinarily bearing some ultimate kernel of truth. Consonant with Magnani’s [15, p.353-357] revisionary modelof abduction, modifications to proffered hypotheses are effectuated consequent tosalient but often tacit factors particular to each eco-cognitive framework, and may
not represent the best strategy or explanation. In fact, “a best explanation” may
never exist. The very existence of best explanations appears to violate Peirce’s notion of the Final Interpretant, because best explanations already offer the ultimate
operation, vitiating any need for a Final Interpretant. Under the best explanation
paradigm, each abduction already represents the Final Interpretant – any further
inquiry is tossed to the wind. The best explanation approach to abduction actually blocks the way of inquiry and the way to pragmatic relevance, which Peirce
adamantly cautions against.
This account illustrates how abductions (spontaneous, plausible projections –revisionary in nature) are not best explanations, but the creative kind to which
Magnani [16, p.267-268] and [14, p.28] refer. Findings which demonstrate the implicit nature of children’s constructions of event profiles and their management ofsystems of imperative and subjunctive rationality will be examined to trace theontogeny of abductive reasoning.
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2

Originary Hypotheses- Their Application to Peirce’s
Categories

Especially with respect to originary hypotheses, Peirce introduces the issue of insight— sudden propositions developed in a flash2 . This sudden aha-based instinct appears to be more in line with idiosyncratic compulsivity than are instincts groundedin physiological capacities (which are universal in nature). The former are Originary
hypotheses rooted in the convergence of Firstness and Secondness, the intersectionof possibility and brute force. Since, for Peirce, abduction is “the spontaneous
conjecture of instinctive reason” [25, V.6 S.475], it necessarily goes beyond a physiological impetus—incorporating idiosyncratic percepts and discoveries derived fromunique events in Secondness. The guessing instinct, which might better be named
the guessing imperative (by which abductions are created), has its root in affectiveFirstness and Secondness; any selective or dimissory imperatives are rooted in logic.
[] Tschaepe seems to conflate deliberate with conscious, as an abduction can beboth unconscious and deliberate. More conscious abductions appear not to be as
originary, but “revisionary,” in Magnani’s sense of the term. Peirce’s later model
further supports this distinction: between originary hypotheses (those generated
out of sudden, unconscious insight); and those emergent from more regenerative,
constructive and conscious effort. “Abduction moves from uncontrolled [automaticand originary] to controlled [conscious] [42, p.122]. The former kind of guessing(unconscious, instinct) translates into a habit of such proportion that it is realizedin conduct which is less able to be regulated – propositions derived from these abductions are so automatic that they are tantamount to instincts. The more dynamic
kind of abduction emanates from dynamic, deliberate and revisionary propositions,
often conscious and less constrained by automatic processes.
Originary guessing illustrates Peirce’s assertion that we have a propensity toguess correctly: it is tantamount to Firstness-based proposals intrinsic to perceptual judgments, and is formulated in the course of knee-jerk imperative responses;
the remaining kinds of perceptual judgments are conscious, and require greater workmanship on the part of the inventor, which may likewise be guided by the imperative
to guess right. According to Peirce, this “ability of guessing right is neither blind
nor infallible, but is an instinctive ability, similar to the animal instinct of flying or
nest-building of ordinary birds” ([25, V.6 S.476]).
Originary hypotheses can likewise be prompted by Thirdness-based imperatives.
The impulse of doubt is just one of these, though without the ordinary face ofThirdness. Doubt can often dispel conventional propositions (elements defining
2“The abductive suggestion comes to us like a flash. It is an act of insight. . . ” [25, V.5 S.181]

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Course of Action Recommendations
Thirdness), as in suspicions that a particular theory/algorithm no longer leads tothe presumed end nor captures elements which later have augmented the original
concept. Even young children display a propensity for this Thirdness-based imperative to take hold, e.g., in joint attentional linguistic exchanges – a child of 1;3 uses
a banana as a telephone [21] and a child of 2;2 refers to a pencil as a hammer [50,p.319]. In fact, in the naming process, children regularly overextend the use of bothnouns [18, 20, 19, 36, 37] and verbs [41] to incorporate unconventional members of
the event type; and still later, they increase the number of slots within events tocreate event types [40, p.151-152], e.g., from “go,” to “put” to “give.”3 “Go” encodes
two slots (one for an animate self-start, migratory participant, the other for a spatial
destination point); “put” accounts for three slots: an animate manually dexterous
participant capable of intentionality, an inhabitable location with a surface, and apatient (ordinarily an object) typically incapable of self-start skills. “Give” invites
still additional slots: ordinarily two manually dexterous participants with intentionality, a destination point (most often the location of one of the participants, and atransferable (tangible/intangible) commodity. The latter verb type profiles the dynamic, transcendent and reciprocal process of imposing goods or an attribute uponanother – consonant with Gibson’s [4, p.31] and [5, p.164-165] process of reafferent
flow: “The living animal is stimulated not only from sources in the environment
but also by itself. . . .Action-produced stimulation is obtained, not imposed—that is,
obtained by the individual not imposed on him” [5, p.31]. Both afferent and efferent
processes represent the character of the reafferent and on-line (instantaneously updated) perceptual account of a potential abducer: “Instead of entering the nervous
system through receptors it re-enters. The input is not merely afferent, in the terminology of the neurologist, but re-afferent—that is, contingent on efferent output”
Gibson [5, p.31]. Tracing the perceptual account can uncover the beginnings anddevelopment of perceptual judgments, in that it monitors dynamic changes in fields
of output and reception within a system: “It is intrinsic to the flow of activity, notextrinsic to it; dependent on it, not independent of it” Gibson [5, p.31]. See [46] [47]for further discussion of the semiosis of abductive reasoning and event profiles.
These extensions and overextensions constitute abductions impelled by Thirdness, except that the imperative in Thirdness (to expand the original, conventional
category) is not, in fact, conventional as Thirdness ordinarily is. Rather, it extends
and alters conventional means/uses – reaching abductive status by virtue of newofferings: application to different movement valencies, participant attribute modifications (adding/deleting characteristics such as animacy/self-start), or simply by
diversifying slot possibilities. Similarly, Thirdness is obviated in change of belief
3

For further discussion, cf. [48].

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West
processes, which (although latent) constitute subjunctive advances, integral to thesemiosis of generating novel event profiles. Augmenting subjunctivity in event profiles motivates increased comprehension (albeit tacit) of degrees of certainty as tothe likelihood of events to transpire, the likelihood of certain someones to take anactive/passive slot in events, etc. To illustrate the issue of event certainty, doubt
constitutes whether the other contributory events have materialized, whether theflow of contemporaneous events is sufficient to prompt other contingent events, or
simply whether outlying participants/objects can fit into the original event profile– in view of the number and kinds of slots attributed to it. Hence, doubt can beassuaged (within the mind of the abducer or via the abduction itself) by inclusionof what may appear, at first glance, to be anomalous with respect to previously
established propositional assumptions. As such, these extensions act in compliancewith the imperative – to amplify original profiles. This form of revision represents a
departure from Woods’ treatment of “ignorance preservation,” since the emergenceof doubt and the introduction of new experience matrices, together with instantaneous reflection on them initiated by recommendation-based imperatives, result insignificant affirmative changes to propositional logic. Such affirmative changes entail
reorganization of mental schemes to accommodate novel, but not hasty assertions.
The influence of imperatives unquestionably energizes the semiosis of abductive
reasoning – from their beginning in Firstness as prescinded elements of Secondness,to propositional reorganization in Thirdness. Accordingly, Peirce makes plane that
abductions are frequently instigated by factors beyond Firstness and Secondness,despite his early misuse of “induction” (testing hypotheses) to define the process. In
1898, Peirce represents the operation of abduction as follows: “. . . [Induction] to bevalid must be prompted by a definite doubt or at least an interrogation; and what
is such an interrogation but first, a sense that we do not know something; second
a desire to know it; and third an effort – implying a willingness to labor for thesake of seeing how the truth may really be” ([29, p.171]. Peirce’s characterization of
abduction here certainly reaches beyond posing questions and ignorance preservation to directives to produce novel templates. Since production of novel templates
results (according to Peirce) in course of action recommendations, imperative based
inducements are far from inconsequential, because their results are often tangible
and can illustrate underlying inferential strategies. In fact, enacting a course of action recommendation can supply the impetus to organize behavioral steps within aframework to validate the strength of the recommendation in actual event scenarios.
As evidenced above, knowledge seeking is intrinsic to imperative directives whichrefine event structure. This is especially credible in view of Peirce’s claim ([28,p.447]) that the propensity for “guessing right” (triggered by the imperative toinvestigate and codify kinds of events) is a universal capacity, even in some non128

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primate species. Nonetheless, what is particularly noteworthy in the latter passage
is his accentuation of the effect of Secondness upon the operation of abductions in
Thirdness. His use of “interrogative” and “effort and . . . willingness to labor for thetruth,” unequivocally demonstrate the relevance of Secondness to abductive processes in Thirdness; and the effort expended, coupled with a “willingness” supports
Peirce’s constructivist account. It is through the expenditure of energy via effort
in Secondness (imposed by the imperative to notice particular components of experience in Secondness), together with the affect (in Firstness) which triggers thewillingness to create and recreate novel hypotheses.
Peirce makes emphatic the originary nature of abductive reasoning – its idiosyncratic beginnings, out of the substance of assertions absent planning and deliberation. Despite the revisionary character of abductions (modified or dismissed) consequent to additional knowledge from anomalous C events, the instantaneous emergence of originary hypotheses should not be underestimated. In line with Tschaepe’s[42, p.129] position, abductions would never materialize were potential abducers toengage in inquiry (interrogative processes) alone, without exacting some impulsive,
truly insightful hunch flowing from the inner core of the individual abducer. Ina word, the originary nature of abductions distinguishes them from interrogative
phases; and the fact that Peirce brands them as originary unquestionably demonstrates both the influence of Firstness in the bargain, as well as the element ofSecondness in constructing/fashioning explanatorily adequate guesses – the persistent labor which may be required to guard against hasty assertions. In fact, it is theimperative phase of abductions which precludes abducers from making hasty assertions when they are permitted to intervene to determine that the set of premises
offered as a potential abduction has sufficient validity – that they are not hasty.
The imperative serves as an index of the degree of accuracy present in the originary
strategy. It communicates when the hypothesis is sufficiently well-formed to offer
a viable course of action for the involved players. The imperative then has a regulatory function – indicating to the abducer whether the hypothesis is ready to bearticulated. This regulative device limits defective judgments, whose premises were
issued outside of our good guessing instinct or absent abeyance to the natural acumen of the guessing instinct (the imperative module). In short, with this imperative
device (hastening guessing by determining when best to guess), fewer revisions arenecessary, which lends economy, systematicity, and reliability to originary hunches
in Thirdness.
It is as part of the system of originary guessing that Firstness “bridges the logical with the psychological,” as Tschaepe [42, p.129] intimates. Firstness likewise
encroaches upon Thirdness, bridging the phenomenological with the ontological, andthe affective with the cognitive; new renditions are primarily fueled by Firstness, re129

Westjuvenating propositions and retroductions in Thirdness. Additionally, to reiterate:
the influence of Secondness upon the semiosis of abduction is unparalleled – theimpulsivity of imperatives, together with the expenditure of labor in repackaging
retrospections into retroductions represents a quintessential illustration of its relevance. Peirce’s concept of guessing (as originary hypothesis-making) is not founded
upon flimsy assertions, independent of a search for the Final Interpretant; rather
concerted effort to offer a viable, innovative protocol elevates productivity and work
product. The purpose which drives the effort consists in elements of Firstness andSecondness – such that percussiveness gives rise to personalized output overlaying
components of the stream of ever-flowing context. Here the confluence of Firstness and Secondness generates and regenerates “works of art” which often servenon-subjective ends. As such, guessing entails a deliberate process of choice andchance, typically without lengthy planning. This supports Tschaepe’s [42] claim:“guessing is the initial deliberate originary activity of creating, selecting, and dismissing potential solutions. . . ” [42, p.117]. Paavola [23, p.152] aptly describes theguessing instinct (development of originary hypotheses) as unconscious instances ofproblem solving. It appears obvious that although some intent to pursue the truth
via our own competencies is operational and some awareness of how retrospections
become retroductions, consciousness is ordinarily not inherent to this initial process. Accordingly, because of the lack of conscious intervention, abducers are often
unable to reproduce/rearticulate the action-based recommendation; and even theireffects (Interpretants) may evade notice. This evidences the fact that knowledge
upon which abductions rest is often tacit in nature as Magnani [13, p.8] suggests.
As such, originary abductions are frequently deliberate but unconscious primarily
because they represent abductions in their initial stages. What Peirce explicitly attributes to an originary abduction is a primary or elementary novel set of premises
holding together logically. It constitutes: “the simpler Hypothesis in the sense of
the more facile and natural, the one that instinct suggests, that must be preferred”[25, V.6 S.477].

3

Further Application to Peirce’s categories

Peirce characterizes reasoning from a pure Secondness perspective as “compulsive”([25, V.2 S.268]). In fact, Peirce explicitly makes compulsivity a necessary component of action based and scientific discovery: “But how is it that all this truth
has ever been lit up by a process in which there is no compulsiveness nor tendency
toward compulsiveness” ([25, V.5 S.172])?
He articulates this same element, compulsivity to be a primary characteristic of130

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all kinds of abduction, those effecting individual and collective ends. This compulsivity takes flight from the appearance on the scene of unexpected consequences. Partof the unexpected consequence is likewise spontaneous reactions which unforeseen
circumstances impose. For Nubiola [22], abducers must resolve the doubt intrinsic tosurprising events, such that they “regularize a surprising phenomenon to make thesurprise disappear through the creation of a new habit” ([22, p.124]. Hence, Peirce’s
component of surprise, i.e., the surprising event, becomes a foundational imperative
factor through Peirce’s habit in the abductive turn. Orienting action within eventsfor different players (inherent in recommending a course of action) demonstrates
an obvious and sustained influence of Secondness upon hypothesis formation ([25,V.8 S.41]. “. . . [volition] does involve the sense of action and reaction, resistance,
externality, otherness, pair-edness.” Pure Secondness does not merely result in reaction to stimuli, but surfaces as “volition”, having their foundation in Firstness– self initiated action. “Volition” here illustrates the import of imperative operations in abductive reasoning, in that volition provides the impetus to determine
which responses from interrogative endeavors will be adopted and incorporated into
retroductions and course of action recommendations.

To formulate a recommendation suitable to the particular complexion of theother in the specific context, children must ultimately supersede Secondness, andcreate an internal, virtual reality. They must assume event roles other than thosewhich they have experienced (acting out the recommendation) and must apprehend
the diverse nature of each event profile, e.g., agent, instrument, receiver, and thelike by determining how others in distinct roles act out the recommendation (cf. [46,p.127], [45, p.164-165]). Nonetheless, Secondness is accorded a pivotal role in theoperation of abductive reasoning, particularly when it is infused with an imperative– the volitional force driven by Firstness. Peirce accords direct experience a pivotal
role in the emergence of higher reasoning skills ([25, V.8 S.266]: “The practicalexigencies of life render Secondness the most prominent of the three. This is nota conception, nor is it a peculiar quality. It is an experience.” The exigencies
pregnant in experiences (both internal volitional and external ones intrinsic to theexperience itself) provide the raw material upon which percepts and perceptual
judgments are grounded. Hence, Secondness constitutes the rudimentary foundation
for later decisions of what to attend to and how to direct others’ attention, pivotal
to recommending courses of action.
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4

The Percept and the Perceptual Judgement

According to Peirce, “percepts come with beliefs, preconceptions, and prejudicesleading to perceptual judgments; thus there is no hard and fast line of demarcation between perception, conception, interpretation and knowledge” ([25, V.5 S.184].
Likewise, while perceptual judgments are subject to criticism, percepts are not Short[35, p.512]. This is so since the latter is beyond an agent’s control, agents of abductions can control the former. Although percepts may be far less subject to humancontrol/intervention, they are not impervious to such agency. Because percepts are
grounded in first impressions of sense [35, p.511] and are grounded in Secondness,they cannot qualify as abductions. Even when percepts rise to the level of interpretations – noticing similarities/differences with already experienced events, they fallshort of abductive status, since although they may qualify as propositions, they lackany explanatory adequacy inherent in abductions ([35, p.517]. Elements of Secondness actively impinge upon perceptual judgements, suggesting the effort imposed by
the agent in the abduction. Percepts can emerge consequent to deliberate attentional effort, contrary to Short’s [35, p.518] claim, that perceptual judgments are
made involuntarily, not requiring justification, insofar as they are predicated on a“look” ([25, V.7 S.627]). Since, according to Peirce, judgments are instinctual, they
are not subject to justification since they are based on a priori knowledge ([25, V.1
S.118]). In fact, Short [35, p.520] capitalizes on two kinds of propensities which particularly represent instinctual judgments: “original knowledge in two instincts – theinstinct of feeding, which brought with it elementary knowledge of mechanical forces,
space, etc., and the instinct of breeding, which brought with it elementary knowledgeof psychical motives, of time, etc.” ([25, V.1 S.118]). This appears to dovetail with
Hintikka’s claim that abductions do not provide probabilistic support. In fact, anobject/feature can be noticed as a result of particular preferences/salience for individual agents, or percepts may be couched in unconscious/automatic surveillance of
certain arrays in particular ways. The fact that percepts/perceptual judgements areoften unconscious does not relegate them to involuntary status, so as to disqualify
them from consideration as potential abductions. In fact, the simple attentional actof looking toward a particular stimulus need not be involuntary as Short asserts.

5

The Emergence of Course of Action Rationality

According to Peirce: “All mental operations are of an inferential character” [?, 1:37].
This includes percepts, as well as perceptual judgements. In 1877 ([30, V.1 p.112]),Peirce takes this a step further by identifying the process which converts percepts andperceptual judgements to hypothetical status – accounting for how one inference (set132

Course of Action Recommendationsof logically connected propositions) is ratified over others, namely, the governanceof habit over thought and conduct. Kilpinen (forthcoming) identifies habit to be
the “guiding principle” of abduction. In “How to Make Our Ideas Clear” ([30, V.1
p.131], Peirce illustrates the further application of habit to action and articulates
that the “whole function of thought is to produce habits of action.” Peirce’s objective
was not to refer to thoughts and actions which have already materialized, but which
are about to materialize, those for which spontaneous conjecture is incipient. In thesame passage, Peirce traces the process of hypothesis-making – proposing a set ofaffairs likely to be effectual in a context which has not yet materialized, but for whichsome potentiality exists: “[T]he ideal of the habit depends on how it might lead us
to act, not merely under such circumstances as are likely to arise, but under suchas might possibly occur, no matter how improbable they may be.” The inception of
abduction as course of action recommendations can be traced to Peirce’s insistencethat retroductions are not complete unless they “recommend a course of action,”([24, 1909 637:12]). To reach status as retroductions then, thought habits must betransferred into action habits.
The merit of this approach lies in its establishment of on-line proposals for discrete strategies within causative events and their sequence – to advance to the unanticipated consequence. In fact, without enactment, anticipation of unexpected consequences may well be stymied. The approach satisfies Peirce’s Pragmatic Maxims,
so as to make necessary action-base sequences to recreate the C event. It capitalizes
on: the purpose for the recommended course of action, the means (via sequenced
steps) to invoke the original surprising consequence. At the same time, recommending a course of action clarifies the problem-solving direction for the abducer,
while providing increased resolve for those implementing the strategy, to follow thesuggested course.
To successfully propose courses of action which lead diverse others to reproduce
the unexpected consequence, children must initially rely upon subjunctive skills,
such that reactions of distinct participants are anticipated to certain event types.
Creative inferencing depends supremely upon two realizations: that events havedifferent profiles/slots for participants which define social roles, and that a consequence of the same event affects others in different ways. In human ontogeny,
recommendations for courses of action are prompted by subjective considerations;afterward they rely upon more objective perceptual judgments – how a reasonable
person in the respective culture would react, or how a known unreasonable person
might respond. Here Peirce informs us as to the ultimate purpose for propositions–the explanation for their existence – to “recommend a course of action” ([24, 1909
637:12]) functional to set the stage and trigger the C event. The recommendation
is just that – a plausible recommendation, not an ultimate suggestion for another’s
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conduct. In the same year [25, V.6 S.324], Peirce clarifies that the action which flows
from recommendations is of the existential kind, which includes states of being andtransitively based actions (acting upon something in a practical sense).
Peirce’s characterization of abductions as “recommending a course of action”
makes plain their dependence on perceptual judgments. Perceptual judgments entail
inferences which qualify as abductions, in view of their propositional and classificatory nature. As such, to generate well-conceived hunches, children must rely upon
consolidations of embodied experiences because they unite single actions, such that alaw or habit is formed and propositions experience logical connections. These habits
are aggregates of habit or as Stjernfelt [38, p.118] terms them, “action habits,” which
rest upon perceptual judgments, not merely single experiences guided by a percept
or a percipuum. These “action habits” consist of implicit abductions (either of theinstinctual or inferential kind); they constitute perceptual judgments, since their regularized schemes and embodiment of different persons in distinctive roles illustratecommonalities across similar event types. The influence of each contributing factor
to the end-state is weighed; and a novel, adequate strategy is proposed, excluding
immaterial factors and embracing factors according to their degree of influence on
the event.
Moreover, the relevance of spontaneity and insight in forming new habits is obviated in recommending a course of action. Here Peirce illustrates how thought habit
translates into action habit, completing retroductions by recommending a course ofaction. Peirce recounts a particularly poignant family incident in which his brother,
Herbert, in a “flash” of “insight” made a determination to instantaneously cover a
woman’s burning dress with a rug to smother the flames, saving her from peril ([24,c.1902 5.538; c.1906 5.487 n1]. Herbert needed a course of action (a habit), to settle
upon the best means of salvaging the person and dress after little opportunity toconsider the effect of each factor in isolation. Possible competing but less effective
remedies include: pushing the enflamed person onto the carpet, dousing her withwater, removing her dress, etc. It is obvious that some form of instinct/insight initially served as the catalyst for the action recommendation, and that the insightwas driven by a thought habit and a suggestion for action change. Although theinference need not amount to the best explanation or the best course of action, it
must represent a plausible remedy. The explanation needs to convincingly evince
a successful result given the surprising event (the igniting of the dress). The inference (to quickly cover the ignited item with a rug) qualifies as a viable abduction,
likewise because it is not an outgrowth of empirical support/ deliberation (albeit
fallible). Despite its fallibility, the premise qualifies as an abduction, given the likely
success of its intervention for virtually any agent: “It is an act of insight, althoughof extremely fallible insight” [24, 1903 5.181]. The upshot is that abductions are134

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subject to alteration/reformulation, given the potential of fallibility – they constitute abductions despite their incompleteness/overextension provided that they rest
upon sound/plausible logic.
Without superseding elements of haecceity, inexperienced abducers would be
hard pressed to generate abductions – to transcend self-participatory memories, andto (by virtue of their own impulses) go beyond and perhaps ignore irrelevant factors
materializing in the context. Many irrelevant factors (which may not appear to
be inconsequential from the outset) distract and mislead children; they constituteevents which simply happen to co-occur in the context with the unanticipated event(which strikes children’s notice). These event distractors are but one illustrationof how salient contextual factors can compromise inference-building prior to 7;0(Piaget and Inhelder [32, p.364]. It is this habit of attention to and reliance upon
contextual exigencies that interferes with the semiosis of abductive reasoning. In
short, children’s recommendations regarding courses of action (however implicit andunfounded) would provide clear evidence of hasty assumptions, and would illustrate
when abduction revisions are underway at early stages in ontogeny.
Peirce’s explicit directive that abductions must “recommend a course of action”[24, 1909 637:12], although late in his attempts to communicate his concept of abduction, nonetheless is in line with his increasing pragmatic emphasis. “It will be
remarked that the result of both Practical and Scientific Retroduction is to recommend a course of action,” [24, 1909 637:12]. This recommendation is not generated
consequent to extensive deliberation; rather it arises spontaneously [25, V.5 S.181].
This spontaneous reasoning emerges in a “flash” to preclude any contrivance from
infecting the recommendation: “The abductive suggestion comes to us like a flash.
It is an act of insight. . . ”
There exists a double-tiered propositional scheme determining the recommendation for another based upon the eco-cognitive contexts which propel certain responses. This is consonant with Magnani’s [15] approach, and provides an ignorance preserving character for the originator of the abduction – each reactor andeach context actuates distinctive modes of action which include integration of affective, social, and logical factors relevant to an innovative action-centered proposal(cf. Magnani, Arfini, and Bertolotti forthcoming). As Magnani, et al. claim aboutaffordance theory, “[it is] a conceptual tool, not only for investigating . . . humanperception, but also. . . human manipulation and distribution of cognitive meanings
in suitable environmental supports.” These proposals are quintessential illustrationsof how ignorance is preserved because how another will respond in the same context
as another and how the same individual will react in a different context remains undetermined. In these cases, abductions are integrative hypotheses to an ignorance
problem—how a given unresolved issue is to be remedied.
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Likewise paramount in recommending a course of action is subjunctive appreciation – a convincing suggestion which is functional for a known other or for an
unknown, objective other (a reasonable person standard). Lakoff and Johnson’s [12,p.34-35] stage theory, especially their characterization that “bodily projections” arenecessary to more advanced applications of lived experience becomes particularly
relevant to proposing plausible courses of action. This is so given that to propose
behavior sequences capable of being successful, the abducer initially needs to insert
self into the place of the other – measuring the combinatorial effect of all of theother’s attributes. Without projecting one’s self into the place of another, an outgrowth of lived experience, courses of action for potential experiencers are unlikely
to be effectual. Later in development, once perspective-taking skills are substantially advanced and the reciprocal event roles are in place (West, 2013, 2014), theself need not be substituted for another; simply projecting the other into the newsituation via imaginative skills suffices to predict that individual’s response and to
propose a recommendation accordingly. In short, projecting one’s self into the situation links individual epistemological and deontic issues to social ones, perpetuatinginterpsychological and intrapsychological advances. Recommending a useful courseof action particular to another (a remedy which is likely to function for another),requires metacognitive skills– to presuppose what others know, viz. the idiosyncraticemotional and informational-base of another.
Nevertheless, such expectations of another’s affective and/or cognitive reactions
give rise to creative, affirmative forces – capable of keeping others from harm. Essentially, well-formed expectations, predicated upon believable and well-founded retroductions, stimulate individual cognitive/epistemic growth (EP 2:192), and perhaps
scientific advances, as well. This growth demonstrates a primary advance in intrapsychological reasoning, in that it materializes in the sudden synthesis of heretofore unforeseen connections/relations, to offer a novel rendition of or a projective
account of events driven by instantiations of Firstness and unconventional Thirdness, without being subject to taboo or cultural sanctions. In fact, affect (which isan artifact of experience) is so crucial to epistemic development that, absent its influence, the inception of novel propositions which inhabit Peirce’s sense of abduction
are unlikely to come to fruition.

6

Peirce’s Constructivist Approach

For Peirce, the abductive suggestion “is the idea of putting together what we had
never before dreamed of putting together” [25, V.5 S.181]. While Peirce is silent as
to the procedure employed to “put together” what we “never dreamed of putting
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together, he implies that the process is constructive, while at the same time, instinctual – generated in a flash.” It is constructive, in that it is not an invention out of
nothing, but is predicated upon forced selection of discrete elements never previously
related logically; yet, connections are conceived of utilizing in-born capacities (naturally predisposed to individual discovery techniques) which develop into creativecompetencies.
Rather than advancing a purely innatist model, Peirce posits a more constructivist account of the ontogeny of logical rationality – contending that while in-born
capacities are propensities common to the species, competencies are the skills developed consequent to the degree of capacity innately given. In other words, hunches
which are grounded in inherent capacities, although universal, nevertheless are refined consequent to constructive endeavors/competencies—they (capacities) are provided in different degrees and kinds to each individual; and absent the operation of“putting together” pieces of implicit knowledge, via effort (competencies) in Secondness, propositions could never acquire sufficient novelty to qualify as abductions. If
all abductions were required to be sequences of moves, already fabricated prior to
engaging in the conduct which leads to the C event (as Hintikka indicates), many
abductive moments would be overlooked, since the possibility of accounting for theimpulses/capacities (internal, external) which initiate them would be excluded. Instead, Peirce’s model of abduction incorporates opportunities to modify the sequenceof behavioral advances, as well as to reject the initial form of the hunch in favor ofa more workable hypothesis. In this context, the imperative is at work – taking
advantage of the feel of enactments to charge the abducer with additional “whatifs.” In Peirce’s constructivist account, abductive reasoning is, unquestionably, an
active process of stops and starts toward eventually hitting the target – the C event.
This process toward hitting the target is demonstrated by Labra-Sproehnle [11, p.10]
active, on-line reformation and dismissal of viable guesses is the substance of generating abductions. He describes an experiment in which subjects (ages beyond 7;0)
play the game “Battleship,” where the objective is to be the first to sink any of fourships given a set number of tries. Gleaning location information after each attempt
is crucial for success; in fact, generating a strategy of moves prior to play enactment(according to Hintikka’s approach) might well contribute to inferior performance.
Of supreme import is the order in which each move is orchestrated with respect tothe results of previous attempts to strike the battleship: “. . . it is assumed that thenature of the problem-solving behavior is connected intimately with the dynamic
configuration of the continuum of results of the active thinking processes performed
to solve the game.”
Actively putting together fibers to solve previously baffling consequences (asLabra-Sproehnle illustrates) is consonant with Peirce’s later approach. Peirce gave
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significant place to interrogative pursuits around 1900; but, thereafter he emphasizes active guessing –imperative paradigms directing agents toward the eventualityof viable hypotheses. In 1903, Peirce comments that abduction “merely suggests
something that may be” [25, V.5 S.171-172]; indicating that the suggestion is an impulse, not a floundering sequence of questions. Playing the battle ship game requires
a decision to venture forth with a proposition – that a portion of the opponent’s
craft is, in fact, at the point on the board where the missile strikes. This decision
to strike at particular locations represents not a binary or open-ended inquiry, but
an impulse that “it must be there.” Peirce’s characterization of a pheme4 as “having a compulsive effect on the interpreter. . . ” ([25, V.4 S.538], illustrates Peirce’s
commitment to imperatives as a type of command over the interpreter and perhaps
over the consequences of the event as well: “Such a sign [pheme] intends or has the
air of intending to force some idea either an interrogation or some action [in a command, or some belief in an assertion], upon the interpreter of it, just as if it were thedirect and unmodified effect of that which it represents. . . ” (c[24, 1906 295:6]. It isevident that a hunch determining the location of the opponent’s craft in the battle
ship game can easily surface as a “command,” especially obvious given its spatial
nature. Determining the location of an object (as in the case of the battle ship),relies upon substantive inferences regarding defining attributes, since knowledge ofinherent iconic characteristics, e.g., shape, is necessary to strike the ship. Hence, in
scenarios in which perspectives and placement are targeted, indexical and iconic inferences are the basis upon which resultant courses of action are founded. Symbolic
representations of battle-shipping are far less material, such as knowledge of classificatory features; rather, indexical functions are primary, especially those driven
by iconic representations. In short, the imperative function becomes heightened in
hypothesis-making when the role of index in the showing enterprise supersedes thatof other sign types. In fact, Peirce’s increased recognition of the force of index inproposition-making and in illustrating the Dynamical Interpretant appear to coincide with his characterization of imperatives as signs advancing novel propositions.
Accordingly, the imperative function becomes increasingly critical when attention
to new assertions needs to be highlighted, as when targeting particular locations.
It is obvious that retroductions (gathering and consolidating previously unrelated
4

Peirce initially employs pheme nearly interchangeably with Dicisign in MS [24, c.1906 295:6].
The substitution was intended to emphasize index’s role as proposition tracer in the operation ofinferential reasoning. It is that this time that Peirce launches index into its ultimate role as coordinator of perspectives, events, and episodes. As such, index traces the trajectory of modal propositions– indicative, to subjective imperative, to subjunctive (advocating diverse points of view), and finallyto an objective imperative (advocating viable courses of action for the scientific/social community
at large).

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memories of past events) supersede the operation of simple inquiry – they convert
single pieces of implicit knowledge (individual memories with their semanticallyladen associations) into a never-before-created fabric – an impulsively assembled
mosaic, never before dreamed of according to Peirce. The imperative again rears
its head by exacting delivery of individual retrospections from extinction. Their
contribution to abduction constructs novel schemes informed by binding past event
memories into a purposive consolidation (cf. [1] for a more complete account ofbinding). This account of retroduction illustrates that Peirce’s teleological approach(every process has a purpose) is pervasive, even into the fabric of abductive logic;
it entails combining individual retrospections into retroductions – creating a mosaic
of increasing relevance of each retrospection to the others. In short, the process ofretroduction (abduction) creates a kind of natural affinity between binding event
memories and the means to guess correctly (the guessing instinct, as Peirce puts it).
Some additional clarity regarding Peirce’s concept of instinct is in order, since hedirectly associates it with abduction, and since he uses the term to qualify guesses(abductions) as compulsive potentialities. Peirce is explicit that abductive reasoning
is predicated upon triggering the flow of plausible hunches; and it is guessing which
serves as the active manifestation of an imperative – that a proposition is about
to surface. Essentially, guessing serves as the linguistic prompt from Vygotsky’s
perspective to comply with the command that retrospections are ready to intelligibly
be bound with other retrospections into a retroduction (an abduction). In short, theguessing instinct represents a behavioral index that the imperative indeed has madeits mark, indicating the point when the process of inquiry has reached momentary
sufficiency.
Like the present approach, Paavola [23, p.152] distinguishes between abductive
instinct and abductive inference – the former a rapid decision emanating from direct experience and the latter an organized search for hypotheses – but he appears
to ascribe less credibility to instinct, which he assumes to be unconscious in all of
its manifestations. Additionally, Paavola contends that there is “no sharp line” between them. Although Paavola [23, p.150-152] recognizes that instinct plays a rolein the abductive process, he indicates that not every abduction relies upon instinct;
abductions may, instead, derive wholly from a more organized strategy of analytic
abductive inference in which agents explicitly search out premises. It is unclear
what Paavola intends by “explicitly,” whether he refers to a planned search toward
identifying relevant antecedents, or whether he means that the knowledge base uponwhich abductive inferences are formulated is explicit. Moreover, Paavola does notprovide additional hints as to its application – he is silent about any contrast between explicit and implicit knowledge/learning – the latter characterizing instinctual
abductions. He does posit that despite their independent application, both types
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of abduction can be employed together in problem-solving pursuits [23, p.135]; but,he fails to demonstrate just how the two function in tandem – integrating instincts
which may consist of flashes of insight from sources outside the self, in which personal experience is virtually irrelevant. Accordingly, Paavola’s working definition of
abductive instinct appears to frustrate a primary tenet of Peirce’s intent, his agapistic principle5 which is alive and well in abductive genres and especially potent inearly childhood when experience is meager. If, in fact, abductive instinct were todepend wholly upon previous experience, as Paavola contends, the insight which,(according to Peirce, NEM III: 206) flows from “our dear and adorable creator” mayin fact be thwarted.
The upshot is that Paavola fails to recognize that instinctive abductions cannot
merely be unconscious, but likewise may begin from a seed of originary insightplanted by a source outside the self. This regenerative impact of instinctual guessing
upon abductive reasoning has the benefit of furthering Peirce’s agapistic principle
and his concept of the continuum—to imprint shared features across members of thecontinuum. This transfer of image function of instinctual abduction provides the key
to how children transition from mere inquiry to developing plausible inferences. The
process is not purely internal, nor does it always involve any conscious knowledge,
however abbreviated. Rather, in this Peircean agapistic sense abductive instinct canemanate from something other than self-generated hunches; it can operate afterward
to guide the abducer to select inferences which are not hasty, but which further theprocess of guessing right.
This interpretation supports Peirce’s derivative claim that foundational to abductive reasoning is affective and subjunctive knowledge, materializing in one ofthree constructions: revised retroductions (consonant with Piaget’s concept of accommodation), establishment of new propositions through the process of uniting
unassociated retrospections, or simply a consequence of Firstness-based inventions,
dreams/creative hallucinations/imaginings (cf. West in press for a further analysis
of dreams and hallucinations as instinctual abductions). In short, it appears obviousthat Peirce did not confound competency with capacity; without question, his use of
instinct refers to competencies – supremely developmental and constructivist in nature, that admit unconventional but not counterintuitive nor internally inconsistentvisional realizations. Nevertheless, Peirce’s intent was not to disregard the necessity
for universal capacities, while emphasizing the role of the constructivistic process
for abduction. But, the latter process is more characteristic of novel logical assem5

Peirce’s Agapistic Principle is derived from his Kantian based assumption that man is madein the image of God (cf. [34, p.464]. This agapistic perspective naturally applies to abductive
instinct and the propensity to guess right, in that not merely the conception of causal relations,but originary, novel insights spring unbidden.

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blages. It entails gathering retrospections and reforming them into a significant, newpathway/habit via the process of retroduction. Essentially, this novel inferencing isa manifestation of what Peirce refers to as “habituescence” or “the consciousness oftaking a habit,” [24, 1913 930]. In short, Peirce’s concept of instinct nicely incorporates the immediacy (response to an imperative) of the conjecture with the creative –capitalizing on regenerativity, while recognizing that instinctual guesses are likewise
grounded in natural, perhaps unconscious propensities. In short, as a competency,
original abductions rise to immediate productions of retroductions, while the lessoriginary brand of abductions warrant a more deliberative process—application ofa more dynamic, sequential system of constructive strategy-making.

7

Abduction as Imperative

In keeping with Peirce’s repeated claim that abductions derive from natural instinctual hunches, this imperative-centered approach characterizes their initial phase as
having a distinctly instinctual function, while in their latter phase, they require moredeliberate (and perhaps conscious) consideration. In the first phase, the imperative
functions to start the process of inquiry consequent to an insight which derives from asurprising or unexpected consequence; whereas, the imperative function in the latter
phase involves exercising the “tendency to guess right” through self-talk. The latteris akin to Vygotsky’s concept of inner speech, such that by intersubjective thought
articulations (questions/possibilities) abducers formulate and direct hypothesis consideration by “guessing the right answer” Hintikka [7, p.527]. Hintikka [7, p.530]
explains that abducers need to open up the probability of a question whose answer
is anticipated.6 Like the present approach, Hintikka recognizes two phases in abductive reasoning: guessing, which is “the only source for new information in anargument” [7, p.522], and abductive inference; but he gives special status to thelatter, since it is in the latter process only, that the answer to interrogatives guides
the hypothesis in a strategic manner. Nonetheless, Hintikka cautions that “such a
strategic justification does not provide a warrant for any one particular step in theprocess,” because every step may not “aid the overall aim of the inquiry” [7, p.530].
Hintikka’s distinction between guessing (or as Peirce calls it, the ”guessing instinct”)
and abductive inference is quite insightful, especially in recognizing that each stepof the inquiry may not discernibly contribute explanatorily to the substance of theinquiry; but procedurally, each step is necessary insofar as it may “[open] up the
6

As Hintikka further illustrates, “From the strategic vantage point we can say that thus any
seriously asked question involves a tacit conjecture or guess” (1998:527). This abduction, founded
upon less educated guesses, characterizes Peirce’s “guessing instinct.”

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possibility of a question whose answer does so” Hintikka [7, p.530]. The present
approach perceives the mechanics of opening up new possibilities both in its instinctual and inferential phases as an imperative, rather than an interrogative function,
since it is structuring inner dialogue on the part of the abducer toward the perceived answer which organizes the steps toward making plausible hypotheses, notthe questions themselves:
The act of observation is the deliberate yielding of ourselves to that force
majeure, - an early surrender at discretion, due to our foreseeing that
we must, what we do be borne down by that power, at last. Now thesurrender which we make in Retroduction, is a surrender to the Insistenceof an Idea. The hypothesis, as the Frenchman says, c’est plus fort que
moi. It is irresistible; it is imperative. We must throw open our gates
and admit it at any rate for the time being [25, V.5 S.581] [29, p.170].
In fact, the ideas which impose themselves as imperatives upon our consciousness
contract the questions which are eventually formulated. Both of these operations
Peirce refers to as experiments, but experimentation of a particular kind—internallysurrendering to ideas which compel themselves. Some of these ideas may be basedon actual experience, while others may not. As Peirce clearly indicates, knowing
the likely answer through intrasubjective (as opposed to intersubjective) dialogue
constitutes abductions compelled by an imperative.
Unlike the present model, abductions are characterized by Hintikka [7, p.523]
and Hookway [9, p.110], as question-answer steps, ignoring how interrogatives canbe driven, in their originary state, by an imperative function, especially within an
inferential process. Hintikka ultimately characterizes abductions as a series of perpetual inquiries (questions), and indicates that the sequence of interrogative steps
is critical to reach success and to arrive at conclusions from surprising events. Although the order of questions within the inquiry is critical to assert a plausible
explanation for the unexpected event as Hintikka contends, what impels the natureof the questions and their sequence in the first place, appears to be under-recognized– as a “goal” [7, p.527]. Despite the fact that Hintikka rightly acknowledges the needfor a goal to direct the inquiry toward abductive closure, he fails to set forth theprocedural paradigm for how questioning results in tenable hypotheses. The interrogative process itself is but a part of the over-all tendency to seek knowledge, animperative. Peirce’s 1908 letter to Lady Welby outlines in the trichotomies which attend to interpretants that “imperative” represents the second member within which“interrogative” is subsumed ([30, 1908 2:481]). In the draft of the 1908 letter, he
specifically ascribes the function of the imperative to the Dynamic Interpretant ([30,1908 2:483]). Peirce’s stance regarding the ultimate plan for opening up new ground142

Course of Action Recommendations
via the semiosis of interpretants measures imperative functions as higher in the hierarchy than are interrogative functions. As such, imperative functions have thepower to impel the abducer by providing attentional and logical direction (like anindex) – to envision the goal, together with some tacit means to ascertain such goal.
In short, imperatives are ultimately responsible for the form and effectiveness ofinterrogative proposals, and hence for the plausibility of abductions. These imperatives are likewise responsible for determining whether hunches based upon instinctsor retroductions qualify as abductions – if their results recommend a practical orscientific course of action. In fact, Peirce suggests that supplying a viable course ofaction recommendation is vital to success for abductive reasoning [24, 12:637]. In
short, it is not inquiry or its sequence that determines the adequacy of explanationsin the abductive turn, but determining when question-asking has momentarily satisfied/sated the new line of thought – when a workable hypothesis has been evoked.
The abductive process proffered herein departs from Woods’ [51, p.365] ignorance preservation paradigm, in that it features imperatives to serve as gate-keepersof plausible hunches. In this role, imperatives preclude unfinished/incomplete hypotheses, and hasty assumptions from materializing as serious antecedents. Although Woods [51, p.370] defines abductions as suggestions for new actions, his
claim that abduction is, unquestionably, an ignorance preserving proposition, actually subverts emergence of course of action recommendations. This is so sincelittle impetus to proffer courses of action which could result in successful reproduction of the puzzling consequence are hardly likely to surface in the face of hasty
and truncated hypotheses. Accordingly, Woods departs from Peirce’s concept of
abduction by failing to recognize the import of Peirce’s Dynamical Interpretant tocompel agents toward plausible hunches/good guesses. Woods’ characterization of
abduction as necessarily an ignorance preserving operation is a consequence of two
misclassifications: underdetermining its ampliative effect, and implicitly assumingthat all abductions are retroductions. He asserts that abductions must entail reasoning backwards from an unexperienced and surprising consequence to antecedent
events which have necessarily been experienced by the abductor [51, p.370]. Woods[51, p.368] further manifests his mischaracterization of abductions as retroductions
when he subsumes them along with two other obvious ignorance preserving operations: Subduence (accessing answers from sources other than internally driven
hypotheses), and surrender (conscious substitution of inferior answers for less exacting ones). Gabbay and Woods’ explicit assumption that abductions can only
partially advance toward determining the answer [51, p.370], short-circuits any realeffectiveness; abductions can never uncover the whole key to the inquiry – nor canthey preserve truth.
In recognizing two distinctive but integratable abductive processes (instinct and
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inferential abductions), the present imperative-based model admits to the affirmative promise intrinsic to abductive reasoning – its unique capacity to “open up new
ground” [27, V.III p.206 1907]. In fact, it is the imperative function which affords
the abductive reasoning agent directional guidelines to select from among the poolof antecedents and to associate such with the natural process/flow toward the unexpected consequence. This process is guided by directionalized propositions. It
assumes that hunches do not reach truly abductive status until the second phase,when retrospections are adequately integrated with one another to assert retroductions, the most uncomplicated, but weakest and least reliable form of reasoning:” I
consider retroduction. . . to be the most important kind of reasoning, notwithstanding its very unreliable nature, because it is the only kind of reasoning that opens
up new ground.” [27, V.III: 206 1907]. Absent the indexical affordance provided by
the imperative function (within both phases), questioning ad infinitum often end inlogical and emotive quagmires which otherwise never become resolved. Without theimpulse supplied by imperative operations, to “guess right,” which as Peirce deems,
leads us to the truth [30, 1902 2:250],7 revising prior hypotheses to arrive at a more
viable form of abduction is likely to be thwarted, since any new hunch consequent
to revisionary hypotheses serves merely to diversify the state of ignorance. Moreover, it is the imperative function which triggers the revision of abductions, not,as Woods [51, p.367], and Thagard [39, p.244].8 indicate, the subjunctive function,
because while subjunctive functions can amplify hunches by providing diverse logical perspectives, they cannot supply the necessary orientation to determine whichapproaches need to be discarded in favor of more promising ones.
Furthermore, Hintikka’s characterization of what constitutes abductions (unlike
the imperative model proposed herein) overlooks another primary component ofPeirce’s paradigm – recommending a course of action. The force within the abducer
to inform others how their habits of conduct are to increase the likelihood of the
C event, is short circuited if the effect of proposing adequate strategies/approaches
is unrecognized. Although Hintikka’s contention has merit – that abductions are
not truth-preserving in that they cannot provide even probabilistic support for their
output [7, p.505] – his approach lacks adequacy to account for semiosis in the abductive process. His model fails to address dynamic change (advances) within the7“. . . unless a man had a tendency to guess right, unless his guesses are better than tossing up
a copper, no truth that he does not already virtually possess could ever be disclosed to him, while
if he has any decided tendency to guess right, as he may have, then no matter how often he guesses
wrong he will get at the truth at last” [30, 1902 2:250]
8“In sum, abduction is multimodal in that [it] can operate on a full range of perceptual as wellas verbal representations. It also involves emotional reactions, both as input to mark a target as
worthy of explanation and as output to signal satisfaction with an inferred hypothesis” Thagard[39, p.244].

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question-asking process (which the imperative function supplies)—especially prominent in ontogeny, critical to determine Peirce’s Final Interpretant. Inquiry Approachneglects the necessary eventuality of working toward refinement of abductions, tocraft an hypothesis which can reliably (after intervals of inquiry and periodic invitations to express novel propositions) elicit the C event. Developing a series ofrevisionary hypotheses would be truncated, so too would be the semiosis of hunches,
were abducers not induced to frame the proposal at each stage in the inquiry.
In fact, Vygotsky [43] is adamant that articulating proposed methods to ascertain
an outcome is a major determinant in problem-solving success. Hintikka’s model
falls short of articulating the method; it merely validates judgments via particular
sets of strategic principles. Hintikka’s point is legitimate – that piecemeal “move by
move rules” are not sufficiently systematized to lead to viable proposals [7, p.513];
but it ignores the impetus for continued inquiry. Despite the adequacy of Hintikka’s
claim, it fails to recognize abductions which glean insight from their implementation– the process of revising hypotheses after framing versions of the original proposal.
As such, even the most revisionary abductions are governed by manipulative schemas(Magnani [15, p.374] which entail “thinking through doing and not only in the pragmatic sense about doing.” Physical manipulation of action schemes (although they
continue to influence abductive reasoning at later developmental stages) graduate tomore mental manipulative schemes in which linguistic directives command coursesof action. Magnani supports this premise: “The various procedures for manipulating
objects, instruments, and experiences will be in their turn reinterpreted in terms ofprocedures for manipulating concepts, models, propositions, and formalisms” [13,p.55].
This process materializes via expressive language or simply by means of what
Vygotsky [43, p.16-17] refers to as “egocentric speech” and “inner speech.” The latter is a more advanced form of articulated speech; it constitutes an internal formof proposition development which transcends the need to self-regulate via external,
audible directives from others (cf. [44, p.4-6] for an extended discussion). As applied
to abduction (whether implicit or explicit), self-regulation and that from others’ isessential, in that recommending courses of action are predicated upon articulated
strategies impelling the enactment of interventions in problem-solving scenarios.
When abductions emerge, they are often implicit, e.g., when gaze trajectories unite
antecedents and their consequences; as such, they represent unarticulated propositions. Soon thereafter, inferences become explicit with the onset of language; andstill later in development (when conditional terms are productive), inferences take ona more assertion-like character. Children’s inferences are facilitated by articulated
affirmations, because syntactic and semantic choices require them to package assumptions into some argument structure. As such, children are compelled to choose
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a particular antecedent or combination of antecedents to ensure production of the
puzzling consequence. By mentally articulating a course of action, children selectthe more plausible premises and are set on a path to formulate a connection with therelevant consequence. This process does not enter the realm of induction, because
children are not experimenting, merely formulating hypotheses via hunches which(like indexes) compel their attention. In short, Vygotsky’s approach is consonantwith that of Peirce, in that enactment, together with facilitating linguistic modesof directing remedial action represent the proposed courses of action, the essence ofPeirce’s abductive rationality [30, 1909 637: 12]. Additionally, when Peirce determines what children need to reach the “Supreme Art,” commanding the self is theultimate skill:
7th , how the facilitation asserted in the 4th point, where it is caused byattention to feelings, where the attention is of the nature of an inward
exertion of power, is perhaps even greater when a different kind of exertion is substituted for the attention to feeling, this is different kindof exertion being describable to a person who has experienced it as anact of giving a compulsive command to oneself. Some books call it ‘selfhypnotization,’ whatever that may signify. This [is] effective whether
there be any ‘disposition,’ i.e., any imperfectly developed or otherwise
imperfect habit, or not” ([30, 1911 674:14].
This “inward exertion of power” to give “a compulsive command to oneself” is askill without which the result of practical and scientific retroductions could not cometo fruition, because without self-control or self- direction, embodiments for coursesof action would be short-circuited. Consequently, the primary effect of retroduction
would be unrealized without this imperative skill. The power of spontaneous propositional pointers in the form of these attentional commands to consolidate propertiesof events into propositions mapped onto the behaviors which embody them constitutes a powerful tool to generate viable assertions which propose an effective courseof action, while revising strategies along the way. In this way, abductions are notpiece-meal or fabricated sets of strategies, but hypothesis up-dates to be orchestrated
on-line, in the stream of commerce. Such action streams may consist of consciousor unconscious organized bundles of goal-driven behaviors which coalesce to bringabout an unexpected event. Although both conscious and unconscious strategymaking are deliberate, only the former is accounted for in Hintikka’s model. Thisis so since to qualify as pre-formulated strategies (steps developed prior to exercising imperative enactments and language directives), generation of the set of steps
before proceeding to reproduce the C event is required. Conversely, abductions
derived from unconscious strategy-making often begin implementation before their
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combinatorial effect materializes. This unconscious kind typically consists of Burton’s [3, p.153] notion of “forced choice" – guessing independent of experience, andis more effective than is guessing in the face of pure chance [3, p.153]. This typeof operation is characteristic of children’s early problem-solving endeavors, in thatinferential reasoning materializes even in the absence of anticipated consequences,and often even without a relevant antecedent, despite their dependence upon thepresence of tangibles.

8

Conclusion

Were these interrogative imperatives to stop there, the way to inquiry would beblocked, because conclusions and revised explanations would be thwarted. As such,inquiry might continue ad infinitum – without any impulse/compulsion to encapsulate new/surprising events into perceptual judgments, and to fold them into recommendations for future conduct. Despite the need for question-posing (perpetual knowledge-seeking) in the operation of abductive reasoning, it is insufficientto qualify as abductions, because the questioner must be impelled to feel that an
information threshold has been reached upon which plausible hypotheses and suggestions for behavioral interventions can be drawn. Hence, imperatives surface tofacilitate transcendence from interrogative processes to the formulation and expression of novel states of affairs from which recommendations for courses of action canissue. It is obvious then that abductive reasoning requires both interrogative andimperative operations. The latter is obviated by the fact that establishing foundations for courses of action for self or others entails a purposive boost – a reason
for providing explanatory rationale and for designing action schemes tto implement
pragmatic outcomes.
Nubiola’s [22, p.125] position implicitly supports the need for an imperative
module to restore well-formed guessing – to provide the push (affective and cognitive)
necessary to have abducers readily share creative propositions with potential event
participants. According to Nubiola, surprise within a surprising event is just thatelement; it “forces us to seek an abduction which converts the surprising phenomenon
into a reasonable one” [22, p.125]. This surprise, coupled with the feeling of readinessto repackage events into behavioral strategies, together comprise the imperative
module.
An imperative, rather than an interrogative triggers initial abductions because
questions can be posed ad infinitum – perhaps without generating, selecting or
settling upon a suitable premise. This imperative basis for abductions is constructedupon Peirce’s requirement that abductions be spontaneous, such that they qualify
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as “forced guesses/decisions” [30, c. 1907 p.687]. The element of “force” upon which
Burton [3] relies serves a managing function – that of expressing propositions which
otherwise would remain hidden, never expressed, and ultimately lost.
Accordingly, this inquiry has identified affective, cognitive, and linguistic evidence that, in point of fact, imperatives are responsible for the truly originary
hypotheses clearly driven by instinctual abductions, given that the source for these
instincts is often an external, divine, one. This account demonstrates how such imperatives beckon generation of novel propositions – recommending a course of action
for self or other. This proposal has uncovered how children’s event judgments acquirean action-based individual and social force – recommending a course of action foranother. This is illustrated by children’s need to recognize social and conversationalroles prior to suggesting that others take those roles in particular event profiles (cf.
West 2014 for further elaboration). It is obvious that children’s imperative-making
demonstrates the power to frame recommendations for future modes of conduct [46,p.165-172]. Such includes: internalizing “pure” secondness and arriving at percepts,
translating them to perceptual judgments, and finally suggesting an entirely novel
remediating path [45, 46].

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152

Abduction: from the Ignorance Problem tothe Ignorance Virtue
Tommaso Bertolotti
University of Pavia, Italy
bertolotti@unipv.it
Selene Arfini
University of Chieti-Pescara, Italy
selene.arfini@unich.it
Lorenzo Magnani
University of Pavia, Italylmagnani@unipv.it

Abstract
The ignorance issue afflicting abduction is usually intended as to what extent it can be seen as an “ignorance-preserving” (or “ignorance-mitigating”)
inference, and what this exactly means. In this paper, we will approach theissue of abduction and ignorance by leveraging the interplay of logical, epistemological and cognitive views on the matter. In detail, after introducing theambiguous relationship between ignorance and achievement, we will deal withthe presence of the ignorance element in the traditional logical formalizationsof abduction, which configures the ignorance problem. Then, we will reframe
the situation in an epistemological and cognitive perspective, making room forthe view of ignorance as a virtue enabling the emergence of successful creative
abductions.
Keywords: Abduction, Ignorance, Epistemology, Agent-Based Logics

1

Introduction

Scientia Potestas Est: Francis Bacon’s famous quote, dating back to 1597, assessed afundamental belief concerning the relationship between knowledge and power (firstly
understood as a kind of enablement). Since the dawn of modern minds, humanVol. 3 No. 3 2016
IFCoLog Journal of Logic and its Applications

Bertolotti, Arfini and Magnani
beings have been gathering and sharing knowledge because of its empowering role.
Better and more extended knowledge concerning everything from the migrationsof herds to the components of a smartphone have afforded better predictions andsubsequently a better performance over the world.
Not only knowledge about objective external realities, such as scientific ones,
plays an empowering role, but also knowledge about human beings gathered throughexperience or gossip and, for instance, through the fictional examples provided byliterature helps making our social world more predictable – thus favoring those who
actually posses this knowledge [16]. In brief, the claim that knowledge empowers
is quite straightforward and there are not many occasions in which a human agent
would claim that it is better to know less than to know more – obviously leaving
aside all of those occasions in which, for emotional reasons, “we wish we hadn’t
known.”
Yet, even before Gigerenzer’s cognitive exploration of heuristics [7] and the emphasis on the catchy “less is more” effect (describing situations in which less knowledge actually affords a better performance than more knowledge), literature displayed a surprising number of instances seemingly arguing against the fact that
Scientia Potestas Est. Such sometimes ambiguous relationship between knowledge,
ignorance and achievement makes a quick appearance in Plato’s Theaetetus, when
the greek philosopher describes Thales falling in a well as he would stargaze andspeculate while walking, so that a servant (the epitome of ignorance) makes fun ofhim.
More pregnantly, the Biblical book of Ecclesiastes reads as follows: “And I set
my heart to know wisdom and to know madness and folly. I perceived that this also
is grasping for the wind. For in much wisdom is much grief, And he who increases
knowledge increases sorrow” (1:17-18). Sure the Bible refers to an existential kindof sorrow, but is it really that hard to interpret, à la Gigerenzer, “sorrow” in thesense of a failed achievement?
Another example: in a way that goes much beyond the epistemological virtue of
simplicity (most appreciated by mathematicians and physicists), poet John Keatsat the end of his Ode on a Grecian Urn stresses that “Beauty is truth, truth beauty,–that is all / Ye know on earth, and all ye need to know.” To recommend the identity
between beauty and truth as the only thing one has to know demarcates a very small
subset of the equation between knowledge and power spelled out by Bacon more than
two centuries earlier. To mention a more recent example, George Orwell’s Nineteen
Eighty-Four managed to depict a fictional, dystopian and yet disturbingly coherent
society in which one of the governing maxims claims that Ignorance is strenght.
The fact that knowledge, ignorance and the achievement of a target do not necessarily come in constant proportions has recently acquired a new interest among
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Abduction: from the Ignorance Problem to the Ignorance Virtue
logicians and epistemologists as it connects with what Hintikka [10] classified as the“fundamental problem of contemporary epistemology,” that is abduction. Abductive reasoning has a crucial importance inasmuch as it is one of the few ampliative
reasonings, as it allows the inferential expansion of the agent’s knowledge beyond
what she already knows.1
To make a straightforward example, if I know that ‘all Romans are mortal’and that ‘Caesar is mortal’, I could try to expand my knowledge by abducing that
‘Caesar is Roman’: this is not necessarily true, because Caesar might very well be apampered Chihuahua dog in Beverly Hills. Since the conclusion is not included in
the premises (indeed it expanded my knowledge), it is not warranted by them either:to obtain it, I produced some knowledge out of something that was not knowledge
yet, and that could be defined as ignorance. What about the relationship between
the previous, underlying ignorance and the newly produced knowledge?
The ignorance issue afflicting abduction, as framed by [25] and [6], is about towhat extent abduction can be seen as an “ignorance-preserving” – or “ignorancemitigating” – inference, and what this means exactly.2 In this paper, we will approach the issue of abduction and ignorance by leveraging the interplay of logical,
epistemological and cognitive views on the matter. In detail, the next section willdeal with the presence of the ignorance element in the traditional logical formalizations of abduction, which configures the ignorance problem. The third section will
reframe the situation in an epistemological and cognitive perspective, making room
for the view of ignorance as a virtue enabling the emergence of successful creative
abductions.

1

Abduction can be said to expand the agent’s knowledge when 1) the knowledge-enhancing
effect is at play and so the fruit of abduction is not potential knowledge but just knowledge (thinkof the Galilean thought experiment concerning falling bodies), and 2) when the guessed hypothesis(so potentially endowed with knowledge content) is accepted because it is evaluated (for example
empirically).
2[18] explains that abduction represents a kind of reasoning that is constitutively provisional,
and it is possible to withdraw previous abductive results (even if empirically confirmed, that is
appropriately considered “best explanations”) in presence of new information. From the logicalpoint of view this means that abduction represents a kind of nonmonotonic reasoning, and in thisperspective we can even say that abduction interprets the “spirit” of modern science, where truths
are never stable and absolute. Peirce also emphasized the “marvelous self-correcting property ofreason” in general [21, 5.579]. So to say, abduction incarnates the human perennial search of new
truths and the human Socratic awareness of a basic ignorance which can only be attenuated/mitigated. In sum, in this perspective abduction always preserves ignorance because it reminds us we
can reach truths that can always be withdrawn; ignorance removal is at the same time constitutively
related to ignorance regaining.

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Bertolotti, Arfini and Magnani

2

Abduction formalized and the emergence of the ignorance problem

The following schema is the basic structure of abduction provided by C. S. Peirce[21, 5.189]:
1. The surprising fact C is observed.
2. But if A were true, C would be a matter of course.
3. Hence there is reason to suspect that A is true.
Needless to say, the problem we are exposing in this paper is nested in the second
step: if A were true. How to produce A if we assume that it was not already within
our knowledge? With the characteristic intellectual élan, Peirce would also contend
that human beings’ capacity to make plausible abductive hypotheses is ultimately
based on instinct. Albeit convincing from a number of perspectives, this description
is question begging if one is trying to formalize the process at stake. How can we
account for this? Instinct seems too non-explanatory, and all reasoning must happen
according to some (leading) principles, but instinctive reasoning does not work thatway, otherwise it would not be instinctive.
Such a conundrum about this seemingly strange reference, proposed by Peirce,
to instinct as the ultimate basis of abduction can be easily clarified considering thefollowing passage, where Peirce philosophically speculates taking advantage of anearly “evolutionary” perspective:
How was it that man was ever led to entertain that true theory? You
cannot say that it happened by chance, because the possible theories, if
not strictly innumerable, at any rate exceed a trillion? – or the third
power of a million; and therefore the chances are too overwhelmingly
against the single true theory in the twenty or thirty thousand years
during which man has been a thinking animal, ever having come into any
man’s head. Besides, you cannot seriously think that every little chicken,that is hatched, has to rummage through all possible theories until itlights upon the good idea of picking up something and eating it. On thecontrary, you think the chicken has an innate idea of doing this; that is
to say, that it can think of this, but has no faculty of thinking anythingelse. The chicken you say pecks by instinct. But if you are going to think
every poor chicken endowed with an innate tendency toward a positive
truth, why should you think that to man alone this gift is denied?” [21,5.591, added emphasis].
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Abduction: from the Ignorance Problem to the Ignorance Virtue
Furthermore, as shown by [17, chapter 5], the Peircean reduction of abduction
to instinct is connected to his reduction of perception to a form of abduction. Peirce
explains to us that perceptions are abductions, and thus that they are hypotheticaland withdrawable. Moreover, given the fact that judgments in perception are fallible
but indubitable abductions, we are not in any psychological condition to conceivethat they are false, as they are unconscious habits of inference. Unconscious cognition legitimately enters the abductive processes (and not only in the case of some
aspects of perception, as we will see). The same happens in the case of emotions,which provide a quick – even if often highly unreliable – abductive appraisal/explanation of given data, which is usually anomalous or inconsistent. Peirce also
contends that perception is the fruit of an abductive “semiotic” activity that is inferential in itself. The philosophical reason is simple: Peirce stated that all thinking
is in signs, and signs can be icons, indices, or symbols. The concept of sign includes
feeling, image, conception, and other representation: inference is in turn a form ofsign activity, that is, the word inference is not exhausted by its logical aspects andrefers to the effect of various sensorial activities. We objected above that instinct
seems too non-explanatory and surely all reasoning happens according to some (leading) principles, but instinctive reasoning does not take place according to some suchreason: the problem is that instinct (and perception) do not perform abductions inthe same way reasoning does, because Peircean theory of cognition is not restricted
to inferential activities but refers to the effect of various other sensorial/semiotic
activities.3
In summary, Peirce’s idea that human beings’ capacity to make plausible abductive hypotheses is ultimately based on instinct, which is in itself abductive, isextremely remarkable and tell us a lot about his modern philosophical perspective
on human (and animal) cognition, informed by a kind of wide “cognitivism” avant
la lettre: he says “It is a primary hypothesis underlying all abduction that the human mind is akin to the truth in the sense that in a finite number of guesses it willlight upon the correct hypothesis” [7.220, added italics]. Albeit convincing from anumber of perspectives, this description is question begging if one tries to formalize
the process at stake.
Consequently, the best way to understand, and appropriately frame, the problem3

The multifarious character of cognition is also testified by Peirce’s conviction that iconicity
hybridates logicality: the sentential aspects of symbolic disciplines like logic or algebra coexist
with model-based features – iconic. Sentential features like symbols and conventional rules areintertwined with the spatial configuration, like in the case of “compound conventional signs” (written
natural languages are concerned by iconic aspects too). What is called sentential abduction is inreality far from being strongly separated from model-based aspects: iconicity is always present in
human reasoning, even if often hidden and implicit.

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Bertolotti, Arfini and Magnani
atic relationship between abduction and ignorance is to briefly review how abduction
can be formalized. Whereas abduction can be of many kinds – visual, multi-modal,
model-based and so on [17] – it is easier to set off from the kind of abduction defined
as sentential, that is the one dealing with meaning expressed by a symbolic language
to which propositions are associated, and then move to other kinds of abduction andsee how ignorance affects them.

2.1

Framing the ignorance problem

The simplest way to understand abduction is the so-called “syllogistic” model, whichsees abduction as the fallacy known as “fallacy of affirming the consequent.”
P1: If A then B
P2: B
C: Then A
This model, albeit extremely straightforward, is relatively meagre in giving off
details about how abduction actually works. The GW-model, proposed by Gabbay
and Woods, is far more eloquent.
From this perspective the general form of an abductive inference can be formallyrendered as follows. Let α be a proposition with respect to which you have an
ignorance problem. Putting T for the agent’s epistemic target with respect to the
proposition α at any given time, K for his knowledge-base at that time, K ∗ for
an immediate accessible successor-base of K that lies within the agent’s means to
produce in a timely way,4 R as the attainment relation for T ,as the subjunctive
conditional relation, H as the agent’s hypothesis, K(H) as the revision of K upon
the addition of H, C(H) denotes the conjecture of H and H c its activation. The
general structure of abduction can be illustrated as follows, by recurring to the
GW-schema (Gabbay & Woods):

4 ∗K is an accessible successor of K to the degree that an agent has the know-how to construct
it in a timely way; i.e., in ways that are of service in the attainment of targets linked to K. For
example if I want to know how to spell ‘accommodate’, and have forgotten, then my target can’t
be hit on the basis of K, what I now know. But I might go to my study and consult the dictionary.
This is K ∗ . It solves a problem originally linked to K.

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1. T !α2. ¬(R(K, T )3. ¬(R(K ∗ , T )4. H 6∈ K
5. H 6∈ K ∗6. ¬R(H, T )7. ¬R(K(H), T )8. If H
R(K(H), T )9. H meets further conditions S1 , ....Sn
10. Therefore, C(H)11. Therefore, H c[setting of T as an
epistemic
targetwith respect to a
proposition α][fact][fact][fact][fact][fact][fact][fact][fact][sub-conclusion, 1-9]
[conclusion, 1-10]

A few notes about the G-W schema: basically, line 9. indicates that H has
no more plausible or relevant rival constituting a greater degree of subjunctive attainment. Characterizing the Si is the most difficult problem for abductive cognition, given the fact that in general there are many possible candidate hypotheses.
It involves for instance the consistency and minimality constraints, corresponding to lines 4 and 5 of the standard AKM schema of abduction. The classicalschematic representation of abduction is expressed by what [6] call AKM-schema,
which is contrasted to their own (GW-schema). For A they refer to Aliseda [1;2], for K to Kowalski [12], Kuipers [13], and Kakas et al. [11], for M to Magnani [14] and Meheus [19].5 Finally, C(H) is read “It is justified (or reasonable) toconjecture that H” and H c is its activation, as the basis for planned “actions”.
It is easy to see that the distinctive epistemic feature of abduction is capturedby the schema. It is a given that H is not in the agent’s knowledge-set. Nor is it
in its immediate successor. Since H is not in K, then the revision of K by H is
not a knowledge-successor set to K. Even so, H(K(H), T ) . So we have an
ignorance-preservation, as maintained since the beginning of our discussion (cf. [25,chapter ten]).
Indeed, what matters for the preservation of ignorance, in the GW-schema, isthat T cannot be attained on the basis of K. Neither can it be attained on the basisof any successor K ∗ of K that the agent knows then and there how to construct.
H is not in K: H is a hypothesis that when reconciled to K produces an updated
K(H). H is such that if it were true, then K(H) would attain T . The problem is
5

A detailed illustration of the AKM schema is given in [17, chapter two, subsection 2.1.3].

159

Bertolotti, Arfini and Magnanithat H is only hypothesized, so that the truth is not assured. Accordingly Gabbay
and Woods contend that K(H) presumptively attains T . That is, having hypothesized that H, the agent just “presumes” that his target is now attained. Given thefact that presumptive attainment is not attainment, the agent’s abduction must be
considered as preserving the ignorance that already gave rise to her (or its, in thecase for example of a machine) initial ignorance-problem. Accordingly, abduction
does not have to be considered the “solution” of an ignorance problem, but rather
a response to it, in which the agent reaches presumptive attainment rather thanactual attainment. C(H) expresses the conclusion that it follows from the facts ofthe schema that H is a worthy object of conjecture. It is important to note thatin order to solve a problem it is not necessary that an agent actually conjecturesa hypothesis, but it is necessary that she states that the hypothesis is worthy of
conjecture.
It is remarkable that in the above schema[. . . ] R(K(H), T ) is false and yet that H(K(H), T ) is true. Let us
examine a case. Suppose that your target T is to know whether α is true.
Suppose that, given your present resources, you are unable to attain thattarget. In other words, neither your K nor your K ∗ enables you to meetyour target. Let H be another proposition that you don’t know. So
K(H) is not a knowledge-set for you. On the principle that you can’t
get to know whether α on the basis of what you don’t know, K(H) won’t
enable you to attain T either. This is a point of some subtlety. Pages
ago, weren’t we insisting that there are contexts – autoepistemic contexts– in which not knowing something is a way of getting to know somethingelse? No, we said that not knowing something was a way of getting topresume something else. But just to be clear, let us point out that in the
GW-schema α and H are not candidates for the autoepistemic inferenceof α from H or K(H). So R(K(H), T ) is false. H(K(H), T ) isdifferent. It says, subjunctively, that if H were true, then the result of
adding H to K would attain T . Clearly this can be true while, for thesame H, K and T , R(K(H), T ) is false [25, chapter eight].
It should be observed that the GW schema aims at illustrating the inferential
structure of abduction, considering the agent’s attempt to attain an epistemic target,
her background knowledge and its possible (timely accessible) updated versions.
Albeit the schema seems to suggest a fully explication of any abductive reasoning,
it remains an explication of theoretical abduction at the sentential level; other typesof abductive reasoning hardly fit this plain description (as manipulative abduction,
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model-based one, etc). Indeed there are traits of abduction, even at sentential level,that fall outside of a logical level of investigation and that need an eco-cognitive
investigation to be analyzed. In this sense, the ignorance preservation remains within
the background knowledge of the agent, because the activation of the hypothesis
H does not modify K or K∗, but simply helps to find an answer to one specificignorance problem.
Finally, considering H justified to conjecture is not equivalent to considering it
justified to accept/activate it and eventually to send H to experimental trial. H cdenotes the decision to release H for further premissory work in the domain ofenquiry in which the original ignorance-problem arose, that is the activation of H
as a positive cognitive basis for action. Woods observes:
There are lots of cases in which abduction stops at line 10, that is,with the conjecture of the hypothesis in question but not its activation.
When this happens, the reasoning that generates the conjecture does
not constitute a positive basis for new action, that is, for acting on that
hypothesis. Call these abductions partial as opposed to full. Peirce
has drawn our attention to an important subclass of partial abductions.
These are cases in which the conjecture of H is followed by a decision tosubmit it to experimental test. Now, to be sure, doing this is an action.
It is an action involving H but it is not a case of acting on it. In a full
abduction, H is activated by being released for inferential work in thedomain of enquiry within which the ignorance-problem arose in the firstplace. In the Peircean cases, what counts is that H is withheld fromsuch work. Of course, if H goes on to test favourably, it may then be
released for subsequent inferential engagement [24].
We have to remember that this process of hypothesis evaluation (and so of hypothesis activation) is not abductive, but inductive, as Peirce contended. Woods
adds: “Now it is quite true that epistemologists of a certain risk-averse bent might
be drawn to the admonition that partial abduction is as good as abduction ever getsand that complete abduction, inference-activation and all, is a mistake that leaves
any action prompted by it without an adequate rational grounding. This is not an
unserious objection, but I have no time to give it its due here. Suffice it to say that
there are real-life contexts of reasoning in which such conservatism is given shortshrift, in fact is ignored altogether. One of these contexts is the criminal trial atcommon law” [24].
To summarize, we can say that the core of the ignorance-preservation problem
of abduction is that the abducted hypothesis H is at best minimal and consistent161

Bertolotti, Arfini and Magnaniwith the rest of the agent’s knowledge base K (in the rest of the article we will show
how even this requirement is often rather wishful). H, though, did not belong to K
nor to K∗ in the beginning, and at the end even a satisfactory solution of the initialproblem does not say why H should be derived from K or K∗. Perhaps, as shownby [18], the initial ignorance is somehow mitigated but not totally removed.
Whereas the GW-model nicely captures “sentential” abduction, it is relatively
obscure as to how H is generated (since it does not derive from K, it is hard toformulate a general mechanism for its production) and so it partially delineates anignorance problem that is common to abduction in general. The issue regarding thegeneration of H is strictly related, as we will better see in the next section, to thedifference between selective and creative abductive inferences: in the first case, theagent simply picks a hypothesis among a range of pre-formed or pre-selected ones,
while in the second the abductive process accounts for the whole production of the
hypothesis, as there are no previous ones – or they are unsatisfactory. This clearly
opens up a necessary distinction in the kind of ignorance that is preserved, or mitigated: it can be either the ignorance affecting the selection of the correct hypothesis(e.g. “I don’t know what train he caught” or “The doctors don’t know if he’s got asevere bronchitis or a mild case of pneumonia”), or an ignorance of a superior epistemological level, which cannot be reduced to an ignorance-that, concerning wholly
uncharted epistemic domains.6 To obtain a better view of this issue, and what is
the role assigned to ignorance, we can turn to examine it from an epistemologicaland cognitive perspective.

3

The ignorance virtue in the cognitive epistemology of
abduction

In order to get there, we must understand how a pivotal distinction between theoretical and manipulative abduction extends the application of the concept beyond
a sentential dimension, but still referring to the agent’s knowledge-content, or lackthereof. According to the point of view of C.S. Peirce, all thinking is in signs,
and signs can be icons, indices or symbols. Moreover, all inferences are a form ofsign activity, where the word sign includes “feeling, image, conception, and otherrepresentation” ([21, 5.283]) and, in Kantian words, all synthetic forms of cognition.
In order to clarify this concept, we should mention that the epistemological dis6

As we will delineate better in the following section, what we called “ignorance-that” refers
to specific ignorances the agent is aware of, against a broader kind of ignorance the agent is notaware of. In spite of the similarities, it does not relate to the difference between knowing-that andknowing-how.

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tinction between theoretical and manipulative abduction is based on the possibilityof (ideally) separating two aspects in real cognitive processes, resorting to the differentiation between theoretical/cognitive ones, where only “inner-neural” aspects areat stake, and manipulative ones, in which the interplay between internal and external
aspects is fundamental. Theoretical abduction illustrates much of what is importantin creative abductive reasoning, in humans and in computational programs: the objective of selecting and creating a set of hypotheses (diagnoses, causes, hypotheses)that are able to provide good (preferred) explanations of data (observations).
A more specified distinction divides this category in the aforementioned “sentential” abduction – which is related to logic and to verbal/symbolic inferences – and“model-based abduction” – which refers to the exploitation of internalized models
of diagrams, pictures, etc.7 Manipulative abduction, instead, accounts for manycases of explanations, occurring in science and in everyday reasoning, displaying akind of “discovering through doing” in which the exploitation of the environmentis crucial. Through manipulative abduction, new and yet unexpressed informationis codified by means of manipulations of some external objects (epistemic and, ingeneral, cognitive mediators). Manipulative abduction captures a large part of scientific thinking where the role of “acting” is central, and where the features of this“scientific acting” are implicit and hard to isolate: actions can provide otherwise
unavailable information that enables the agent to solve problems by starting andperforming a suitable abductive process of generation or selection of hypotheses([17]).

The distinction between off-line and on-line thinking is analyzed in detail in [17, subsection
3.6.5, pp. 189–193]. Some authors have raised doubts about the on-line/off-line distinction on the
grounds that no thinking agent is ever wholly on-line or wholly off-line. We think this distinction
is at least useful from an epistemological perspective as a way of theoretically illustrating different
cognitive levels in human and animal cognition. It must be kept in mind that the theoreticaldistinctions between types of abduction are meant to frame the main traits of each type, butthey are not necessarily mutually exclusive and different analyses may highlight different kinds of
abduction at play in a same process. Sentential abduction refers to the possibility of working on
sentences, be them expressed in logical or verbal language, and it is hence more closely connected
to traditional logical studies. Nevertheless, the iconic dimension is never that far out: whereas
the semantic understanding and appraisal of a sentence concern sentential abduction, the visualor auditive recognition of the signs expressing it rather involves a model-based approach to theprocess. Not to mention the language-forming capabilities afforded by diagrams, dicisigns, andicons. Peirce himself robustly exploited diagrammatic aspects of reasoning in his own research on
logic: his invention of existential graphs is very well-known.
7163

Bertolotti, Arfini and Magnani

3.1

Abduction and lower cognitive processes

Our claim is that within the vast topic of abductive inference it is possible to exploit
connections between high-level and low-level inferential patterns to obtain a better
understanding of the different kinds of abductive cognition and thus to gain a bettergrasp on whether role of ignorance can be acknowledge as more than something thatis necessarily “preserved” or at best “mitigated”.
It is very hard for us, as human beings, to exit our language-based, propositionalized framework. Furthermore, as we engage in the attempt to convey some meaning
to each other (as we are doing right now, writing a paper), we cannot abstain fromrelying on a symbolic language endowed with propositional meaning. The “problem” is that we must recur to this kind of language also to describe events that
occur in non-propositional terms (for instance at physiological, or neuro-chemical,
or perceptive level). Such awareness, albeit raised to a slightly different scope, wasalready clear to Peirce himself who, speaking about perception, stressed the fact thatwhen we think about perception we immediately turn our perceptual judgments into
propositions, but this way we are not reflecting on raw perception anymore:
Looking out of my window this lovely spring morning I see an azalea infull bloom. No no! I do not see that; though that is the only way I can
describe what I see. That is a proposition, a sentence, a fact; but what
I perceive is not proposition, sentence, fact, but only an image, which I
make intelligible in part by means of a statement of fact. This statement
is abstract; but what I see is concrete.8
It is clearly not possible to transcend the propositional level in theoretical communication, it would be very hard to communicate this argument making use ofhormones and unmediated electric impulses, nevertheless we must be very careful
not to let our “perceptual view” taint every conception of abductive inference.
Many kinds of abduction, in fact, do happen below the sentential-propositional
threshold, which could be considered as a relatively new acquisition. Animal abduction [15], to make a clear example, involves forms of abductive inference thatare clearly pre-sentential (as pre-linguistic) and can be individuated even in the“cognitive” faculties exhibited by bacteria reacting to their environments [3], notto mention the toolmaking ability displayed by crows which could be identified as
manipulative abduction [23].
8

Cf. the article “The proper treatment of hypotheses: a preliminary chapter, toward an examination of Hume’s argument against miracles, in its logic and in its history” [1901] (in [22, p.
692]).

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In these cases, following [8], we can say that cognition “shades off” into otherkinds of biological processes. It is not usually considered as genuine cognition, forexample, when some bacteria adjust themselves to changing circumstances aroundthem by using little internal magnets to distinguish north and south and thus movetowards water or when they use external clues, through tactile exploration, to adjust
their metabolic processes: in this sense, it can be suggested that basilar forms of
abductive inference indeed preceded the development of proper forms of cognition.
The reason for the warning we were calling for a few paragraphs above, against
the overeagerness to turn everything into propositional language, is clear in thiscase. Even if we state that non-human animals and other organism are able to makemore or less complex forms of abductions, we must be careful about the meaning
we give to words such as explanation, knowledge, and subsequently ignorance.
We can nevertheless suggest that such inferential processes operate on kinds ofrepresentation that are produced abductively, and are most probably unapparent to
the organisms making use of them.9
Of course, this whole argument could be affected by the cumbersome issue of theexistence and nature of representations in the animal mind: we believe that fromour purpose a deflationary approach can be fruitfully assumed. We could consider
as a representation of the outside world any modification in the inner system that
more or less corresponds to a modification of conditions in the outer world, andthat can serve as a base for future behavior. Since cognitive agents are endowedwith some kind of communication system (be it nervous and/or chemical), we candefine mental representations as “patterns of neural [or chemical, or even genetic]
activation,” coherently with Clark’s connectionist view [17, p. 160]. As for the(pragmatic) content of such representations, we believe that Millikan’s insight is themost useful.
Millikan suggests that internal representations of animals might mostly consistsof PPR (“Push-me pull-you” representations), meaning they are both aimed at representing a state of affairs and at producing another, often suggesting a chance forbehavior as received by the Gibsonian/affordance tradition [20]. The indicative content of a PPR mental representation about external agent will therefore never beof the kind Oh, look at that organism PERIOD but rather Look at that organism:
9

Although the notion of representation can be philosophically considered as “emptied” to acertain extent, we decided to maintain it for two main reasons: the contingent one, is to adhere toMillikan’s authoritative lexicon as far as animal cognition is concerned; the more essential one isthat this reflection belongs to the model-based reasoning framework, by which a model is used inorder to achieve a goal. Representations, as models, do represent a target, and even if the sameprocess can be conveyed by concepts such as “structural coupling of inner and outer systems,” wefeel that the notion of representation better depicts its instrumental role.

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Bertolotti, Arfini and Magnani
should I attack/avoid/hurt/kill/eat it/mate with it?:
An animal’s action has to be initiated from the animal’s own location.
So in order to act, the animal has to take account of how the things tobe acted on are related to itself, not just how they are related to one
another. In the simplest cases, the relevant relation may consist merelyin the affording situation’s occurring in roughly the same location and at
the same time as the animal’s perception and consequent action. Moretypically, it will include a more specific relation to an affording object,such as a spatial relation, or a size relative to the animal’s size, or aweight relative to the animal’s weight or strength, and so forth [20, p.
19].
Millkan’s contention is that animal representations are bases for action. This
comes as no striking news, because any cognizant is wired so to proceed from representation to action in order to survive. The presence of a central controller is notneeded to explain why some abductive representations are followed by actions andsome not: we can hypothesize that while perception stimulates the activation of aneural network, only if the electrochemical signal reaches a certain threshold it can“fire” the activation of a distinct motor-related neural network, triggering aggression
or escape. Such process can easily happen without the presence of a central intelligence that assesses the representation and decides when action should be enacted.
In this sense, many kinds of low-level abductions could be seen as self-performing
abductions, handling ignorance to their agents’ unawareness.
This is not true of animals alone, but also many of the abductive inferences
human beings operate are of this kind, from perception (of inner and outer statesof things) upwards. Conscious, sentential abductive inferences are just the highest
steps of the pyramid. Indeed, to say that an organism can detect the presence ofanother organism does not compel us into affirming that it has consciousness of theother organism’s presence: it suffices to imagine that the states of neural activation
originate a mental representation fit to guide its behavior.
With the same reasoning, we cannot always speak about “ignorance” as a matterof unattained sentential propositions about the world: this is all the more true incase of the ignorance preserved in abductive inferences performed by agents withor without the capacity to explain their behavior at a sentential level. It will be
sufficient to sat that they lack the comprehension of how some facts affect theiraction-reaction behavior. But it is worth mentioning that the ignorance-preserving
trait remains unaltered by the different formal expressions with which we can express
the inferences. The relationship between abduction and ignorance is “locked” in166

Abduction: from the Ignorance Problem to the Ignorance Virtue
both its formal expression and in its cognitively relevant occurrences; the difference
between abductive processes (and their conditions of effectiveness) are connectedto the different interpretations of ignorance the agent is preserving, and exploiting,during the abductive process.
In order to investigate the specific forms of ignorance mitigated (and preserved)
while performing an abductive inference, we will follow the division between selective and creative abduction we mentioned before, inasmuch as it concerns the
epistemological dynamics and the generation of hypotheses within or outside theknowledge of the agent. In the next section, we will provide an eco-cognitive exposition of hypothesis-generation presented through the two kinds of abduction and theforms of ignorance preserved and/or mitigated during the processes. This scheme
will show the specific difference between the simple preservation of ignorance theselective abduction implies (which is the cause of its broad usefulness but also of its
unquestionable handicap in the scientific practice), and the enhancement of knowledge through ignorance brought about by the generation of a new hypothesis (daring
but functional) in a creative abduction.

3.2

Creative abduction and the virtuous prerogative of ignorance

Abduction, as an inferential activity, is obviously performed when the agent is embedded in a constant dynamic of action-reaction with her surroundings. This canalso be seen as a negotiation of signs and data that she is catching and diffusing
throughout the epistemic process. In the cognitive economy of the agent10 abduction does not only concern a certain amount of known information, but also theendowment of some signs with a practical activation (as the PPR representations
and affordable values). Thus, adbuctive processes causes the development of theagent’s cognitive environment through the transformation of some unexploited data
into something unexpectedly useful. These performances are strictly related to what
the agent knows better about, and what is far from her natural, usual or establishedfield of expertise.
Simplifying, we can see the complex of data the agent possesses and manages,
together with those that are within her reach in her cognitive environment, as anagent-centered system. Her topics of expertise, the knowledge she usually employs,
correspond to the central information: she can easily attain them, and she has aminimal ignorance about them. Instead, the information that is still within theagent’s cognitive system but that is not in her dominion of expertise, or that she
is vaguely ignorant about, correspond to the peripheral information: she knows
10“An economy is an ecology for the generation and distribution of wealth. A cognitive economy
is an ecology for the generation and distribution of knowledge” [25, p. 85].

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Bertolotti, Arfini and Magnani
something about it but it is not part of her practical knowledge field. Abductive
inferences are performed upon and within the cognitive economy of the agent, acting
on these two parts of her system, albeit in two very different manners and making
use of two very different kinds of ignorance.
The first type of ignorance is set within the limits of the agent’s cognitive environment and it is rooted in her own central information. It involves a part of delusion
about the actual knowledge the agent has on her field of expertise (which is kind ofnatural to expect). In order to overcome it, all it takes is a specific question, but also
the agent’s awareness about the information she lacks. When the agent knows what
is missing, she can obtain the answer through some targeted questions; when her
knowledge does not cover the amount of data she thinks it does, something is missing in her central information. Performed in this context, abduction placatesthe ignorance that the agent recognizes: this prompts the inference inthe first place, and maintains what she cannot expect there. The richnessof this kind of inferential activity depends on the agent’s interest and competence.
By doing this, the agent enacts a selective abductive inference. It gives the agent
the possibility to inquire into her specific ignorance and find the best explanation[17], selecting it from a counted number of choices (or still within a type of possible
choices): the hypothesis still preserves the ignorance about the unforeseen possibility
that it could be less than the best possible explanation. Think of a doctor struggling
with her ignorance on whether the patient is affected by a severe bronchitis or a mild
pneumonia: a new symptom can solve this ignorance, help the doctor formulate her
diagnosis and start the treatment, but of course the diagnosis could still be wrong,
and the doctor could shift from a small ignorance between two possibilities to awrong course of action.11
As contended by [18], if we say that truth can be reached through a “simple” abduction (bothselective or creative) where simple means that it does not involving an evaluation phase, whichcoincides with the whole inference to the best explanation, fortified by an empirical evaluation,then it seems we confront a manifest incoherence. In this perspective it is contended that even a
simple abduction can provide truth, even if it is epistemically “inert” from the empirical perspective.
Why? We can solve the incoherence by observing that we should be compelled to consider abduction
as ignorance-preserving only if we consider the empirical test the only way of conferring truth to ahypothetical knowledge content. This clause being accepted, in the framework of the formal modelof abduction introduced above the ignorance preservation appears natural and unquestionable.
However, if we admit that there are ways to accept a hypothetical knowledge content different
from the empirical test, simple abduction is not necessarily constitutively ignorance-preserving:in the end we are dealing with a disagreement about the nature of knowledge, as Woods himself
contends. Those who consider abduction as an inference to the best explanation – that is as atruth conferring achievement involving empirical evaluation – obviously cannot consider abductive
inference as ignorance-preserving. Those who consider abduction as a mere activity of guessing are
more inclined to accept its ignorance-preserving character. On this matter, also refer to footnote 2.
11

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It is obviously possible to go beyond one’s ignorance within the central information, but in order to do that the agent has to face what is missing and perform an
abduction like the one just described. Obviously it is an enhancement in the management of her eco-cognitive structure, but it is again a more or less ignorance-preserving
inference: as we saw, the agent “only” becomes aware of what she recognizes to be
missing.
Conversely, the second type of ignorance is harder to manage than the first. It
does not require just a specific question to be inquired, and so discovered. Indeed,
it does necessitate more than the agent’s ordinary expertise in order to become
apparent: rather, it requires more patience and resources to be integrated within
the core of her knowledge base. In order to discover a way to attain one’s target
inside this kind of ignorance, it becomes necessary to change the agent’s eco-cognitive
system and enhance it with the perspective that even in the most peripheral partof her knowledge base there are still plenty of useful answers to her questions. It
also involves a change in the direction of the interest that it supposed to guide theabductive process.
Here it is not possible to use a selective abduction to direct the inquiry withinsuch a vast and problematic ignorance. The method that can shed some light is
Magnani’s aforementioned creative abduction ([17]), and in particular the transparadigmatic abduction ([9]).
In these cases the hypotheses “transcend” the vocabulary of the evidence language, as opposed to the cases of simple inductive generalizations: the most interesting case of creative abduction is called by [9] trans-paradigmatic abduction. Thisis the case where the fundamental ontological principles provided by the backgroundknowledge are violated, and the new hypothesis transcends the immediate empirical agreement between the two paradigms, for example in the well-known case of
the abductive discovery of totally new physical concepts during the transition fromclassical mechanics to quantum mechanics [17].
Creative abduction – and its trans-paradigmatic form above all [9] – does notprovide a simple selection of hypotheses but, through the change of the eco-cognitive
paradigm, it provides a brand new field to investigate. When the agent cannot afforda specific question, or method of enquiry, because she cannot describe what she does
not know – which is indeed unaffordable for her – it becomes necessary to perform
a creative context-shift for example through an almost serendipitous creation of analternative pattern.12
12

By advocating serendipity we do not refer to a total randomness in the process of context
shifting: conversely, we refer to its partially unpredictable and emergent nature. Even serendipity
itself refers to the background, skills (and peculiar ignorance) of an agent – consider Louis Pasteur’s
famous saying that “In the fields of observation luck favors only the prepared mind.”169

Bertolotti, Arfini and Magnani
Thus, enquiring within the second type of ignorance opens the possibility to
discover a whole hypothesis-cluster, leading to the possibility of attaining the target
in one or more new ways. Creative abduction, in this sense, exploits the ignoranceof the agent rather than eliminating it. It is a way to claim knowledge and data byexerting a powerful epistemic drive through the ignorance of the agent.
The development on a new hypothesis-cluster signifies that, out of vast ignorance,
one will not just draw a particular abductive answer (for instance whether San
Diego is larger than San Antonio), but develop a new vocabulary, a new syntax
and a new grammar. That was the case in the big example, already mentioned, ofEinstein developing the Relativity Theory, but one can think of different examples
too. Consider a case in history of technology: many readers will be acquainted
with the (somewhat frustrating) history of handheld and PDA keyboards. Until thefirst first half for the years 2000s, the arm (or finger!) race was about developing
the best typing accuracy, with on-screen keyboards with or without stylus, physicalkeyboards (folding or sliding) with hard or soft keys, optional keyboards in different
sizes and so on. Yet, the typing accuracy deriving from fitting twenty-something keys
in an area slightly bigger than a tiny notepad could not be overly improved. The newcluster-hypothesis, so far nested in the developers’ ignorance, was to concentrate noton the typing but on the processing of that typing, developing predictive softwarethat could overcome the mistakes caused by physical constraints and learn from thehabits of the users. That meant moving from a typewriter mindset to an artificialintelligence one. It was not one single hypothesis (for instance involving the best
shape of a thumb-keyboard), but an hypothesis-cluster opening up a new field of
research and development, unforeseen a few years earlier but from which there wasclearly no going back.
This kind of solution to an ignorance issue provides an enhancement of theagent’s perspective and knowledge but its consequences are wider than in the cases
previously described, that is when one’s precise ignorance is concerned (overcoming
my ignorance about the capital of California, and learning that it is Sacramento,
might improve my local performances but it is unlikely to constitute a systemshattering hypothesis-cluster). Obviously, the epistemic risk and opportunity atstake involved by a daring hypothesis, in the Popperian sense, are high. But the
opening of a new field of research, of development or simply of reasoning is alreadya possibility toward compensating errors, misevaluations, or to further improve themost promising components.13
13

Even if the issue is clearly related to the topic at stake, we are not especially dealing with thepragmatic relationship between ignorance and courses of action where ignorance can be a synonymof uncertainty, as in the case of economics. A specific investigation about that is due soon, setting
off from some reflections introduced in this paper. In the meanwhile, some other exploratory ideas

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4

Conclusion

Abduction, as the ignorance-preserving inference, has a complicated connection withknowledge and ignorance. The history of the concept, as well as its employment asa coherent tool for explaining lower and higher level of cognition, testifies to therichness of this entanglement and surely advocates for the benefits it produces. As
we repeatedly said, and coherently with the Peircean heritage, abduction feeds uponsigns but also upon the unexpected possibilities that those signs can provide to the
cognitive prepared agent. Highlighting the epistemic value of ignorance not just as
something that is preserved or mitigated is a crucial step before one can achieve,
and then implement, the computational modeling of creative inferences based onclusters of hypotheses produced from within an agent’s ignorance – not considering
the latter as a simple lack of knowledge anymore.

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173

Received April 2015