Supervaluationism and Paraconsistency
Achille C. Varzi
Department of Philosophy, Columbia University(Published in Diderik Batens, Chris Mortensen, Graham Priest, and Jean-Paul Van Bendegem (eds.),
Frontiers in Paraconsistent Logic (Studies in Logic and Computation Series, 8), Baldock: ResearchStudies Press, 2000, pp. 279–297.)

Abstract. Supervaluational semantics have been applied rather successfully to a variety of phenomena
involving truth-value gaps, such as vagueness, lack of reference, sortal incorrectedness. On the other
hand, they have not registered a comparable fortune (if any) in connection with truth-value gluts, i.e.,
more generally, with semantic phenomena involving overdeterminacy or inconsistency as opposed toindeterminacy and incompleteness. In this paper I review some basic routes that are available for thispurpose. The outcome is a family of semantic systems in which (i) logical truths and falsehoods retain
their classical status even in the presence gaps and gluts, although (ii) the general notions of satifiability and refutability are radically non-classical.

1.

Introduction

Since its first appearance in van Fraassen’s semantics for free logic [1966a, 1966b], thenotion of a supervaluation has been regarded by many as a powerful tool for dealing
with truth-value gaps and, more generally, with phenomena involving semantic indeterminacy, partiality, deficiency of meaning. Unlike three-valued semantics, whose truthfunctional character induced a proliferation of competing variants, supervaluational semantics appeared to offer a uniform way of keeping these phenomena under control;
and work in the following decades has supported this expectation with a fair deal ofsignificant developments. These include, among others, applications to such diverse
domains as quantum logic, temporal logic, vagueness, presuppositions, sortal incorrectness, or the semantic paradoxes.1
In light of this prosperity, it is remarkable that supervaluation-like methods have
not registered a comparable fortune (if any) in connection with truth-value gluts, i.e.,
more generally, with phenomena involving semantic overdeterminacy or inconsistency
as opposed to indeterminacy and incompleteness. Part of the explanation is, of course,

1

See Van Fraassen [1968, 1969], Skyrms [1968], Lambert [1969], Thomason [1970, 1972],
Fine [1975], Kamp [1975, 1981], Bencivenga [1978, 1980b, 1981], Klein [1980], Pinkal [1983], vanBendegem [1993], McGee and McLaughlin [1994] among others. The discussion of vagueness in Lewis[1970], Grant [1974], Dummett [1975], and Przel- eçki [1976] also contain supervaluational ideas, although the connection is not made explicit. In this sense, the approach seems to trace back to Mehlberg [1958], §29.

1

to be found in the greater hostility that such forms of semantic anomaly have registeredon the whole. However, truth-functional methods have been applied to semantics withgluts, or with gaps and gluts alike. For instance, four-valued generalizations of Kleene’s[1938] three-valued matrices have become rather popular among those who view gaps
and gluts as two sides of the same coin, two complementary ways in which a semantics
may fail to be uniquely determined. 2 One wonders, then, whether and to what extent
similar generalizations are available—at least in principle—to the friends of supervaluationism.3
The purpose of this paper is to show that one can actually go quite far in that direction. There is no intrinsic difficulty in applying the supervaluational insight to interpret a language in the presence of semantic inconsistency. More generally, there is no
intrinsic difficulty in generalizing the concept of a supervaluation so as to deal with bothkinds of semantic anomaly—gaps and gluts. In fact there are many options available;
and in all cases the distinguishing features of van Fraassen’s original technique can bepreserved: (i) the general notions of satifiability and refutability are non-classical, but(ii) logical truths and logical falsehoods (and more generally entailment relations between sentences) retain their classical status.
2.

Supervaluations, Subvaluations, Ultravaluations

There is no question that the notion of a supervaluation as such is unsuitable for dealing with semantic inconsistency. Typically, the supervaluation induced by an incomplete
interpretation, or model, M is construed as a function of the valuations induced by acertain class of complete extensions of M.4 If M is at worst incomplete, i.e., incomplete
but consistent, such extensions are bound to be classical models, since they result from“filling in” some gaps in an otherwise classical model. (This is not trivially true andmay have to be qualified depending on the syntax of the language, but the basic idea isunproblematic.) Accordingly, if a sentence A involves expressions with respect to which
M is undefined, one can just look at the values A takes on such complete extensions(values that can be determined by a classical evaluation procedure) and then register

2 See e.g. Dunn [1976], Belnap [1977], Woodruff [1984], Muskens [1989], Fitting [1992], andGupta and Belnap [1993]. Most semantics for paraconsistent logics can also be viewed in this light(see Priest & Routley [1989] for a survey).
3 Some hints can be found in Grant [1975], Belnap [1977], Rescher & Brandom [1980], Lewis[1982], and Visser [1984]. My own preliminary forays are in Varzi [1991, 1994, 1997]. See also Hyde[1997] for an application to vagueness.
4 In van Fraassen’s original formulation, this construction is actually left on the background, as
the basic valuations are introduced directly by assigning arbitrary truth-values to those atomic formulasthat are left indetermined by the given interpretation. With few exceptions, however, that account has
been abandoned in favor of the construction outlined here. See Skyrms [1968] and Bencivenga [1980a]
for explicit motivations, and Herzberger [1982] for further discussion and comparisons.

2their pattern of agreement: (i) if A is true in every extension, then certify it true in M; (ii)
if A is false in every extension, then certify it false in M; (iii) otherwise leave the value of
A in M undefined, for there appears to be no definite way of making up for the indeterminacy of M. This is roughly how a supervaluation works: it reduces the problem of
evaluating a sentence on a non-classical model to that of evaluating it on a family ofclassical models. However, it is crucial for this method that we start with a model that is“only” incomplete. If we begin instead with a model that is inconsistent, then its extensions will be of no help: extending an inconsistent structure can never yield classicalvaluations, as inconsistency is preserved under extension. And without classical valuations, the main rationale for construing a supervaluation breaks down 5.
Note that the argument does not depend on the specific notion of a model at issue(hence, ultimately, on the relevant notion of a language). Simply, it emphasizes what
Brian Skyrms [1968] once called the “Aristotelian notion of Redemption” offered bysupervaluations in response to the “Fregean notion of Sin” represented by a gapfulmodeling: if we get the same outcome no matter how we fill in the gaps, then the gaps
don’t matter. But, of course, if there is no gap to begin with there is no redemption either. If we therefore want to restore some balance between gaps and gluts we cannot
just look for a straightforward generalization of supervaluations, at least not in the sensein which, say, a four-valued Kleene semantics is a straightforward generalization of thethree-valued version.6 Rather, we must consider extending the initial notion (e.g., byanalogy) in such a way as to preserve the “spirit” if not the “letter” of supervaluational semantics as usually understood.
Now, if we keep at this intuitive level, one obvious solution suggests itself. If an incomplete model M induces a supervaluation defined on the basis of M’s extensions(specifically its complete extensions, each of which corresponds to some way of fillingin the gaps in M), likewise an inconsistent model M may be taken to induce a sort ofsubvaluation defined on the basis of M's restrictions (specifically its consistent restrictions, each of which reflects some way of “weeding out” M’s semantic gluts). This is
intuitive, especially if we emphasize the duality between incompleteness and inconsistency. In particular, if the guiding idea behind the notion of a supervaluation is that anincomplete model is essentially the meet of a series of complete extensions thereof, sothat the supervaluation is construed essentially as the meet of the classical valuations induced by those extensions, likewise an inconsistent model may be viewed as the join of

5 I take ‘classical’ valuations to be gapless and glutless assignments of truth values, leavingroom for the possibility that these deviate locally from classical logic (as for instance in Bencivenga’s
semantics for free logic [1980a, 1981]).
6 A four-valued Kleene semantics can be obtained from the three-valued semantics simply by
dropping the assumption that truth and falsehood are mutually exclusive, just as the three-valued semantics is itself obtained from classical semantics by dropping the assumption that truth and falsehoodare exhaustive. (The usual clauses work as long as we state those for falsity along with those fortruth.) This was the gist of Dunn’s remarks in [1976].

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a series of consistent restrictions thereof, and the subvaluation can be construed as thejoin of the classical valuations induced by those restrictions. In the general case, where amodel may be both incomplete and inconsistent (i.e., underdetermined with respect tosome basic expressions and overdetermined with respect to others), one can then rely
on a suitable combination of these two complementary ideas. If M is such a model,
neither its complete extensions nor its consistent restrictions are classical. However, thecomplete extensions of its consistent restrictions are complete and consistent, henceclassical, and one can use this fact to obtain the desired generalization (the “ultravaluation”, to give it a name): just take the second-order subvaluation based on the admissible supervaluations; or, alternatively, take the second-order supervaluation based on theadmissible subvaluations.7
To illustrate, suppose we follow Belnap [1977] and think of a model as a data base,or as an epistemic state of a computer. (An epistemic conception of gaps and gluts is ofcourse best suited to supervaluationism.) Sam and Elisabeth enter the relevant information. If neither says anything about P, then there is a gap. If the two of them enter contradictory information about P, then there is a glut—not a desirable situation, but certainly a conceivable one. Now consider the following three cases:
A = true
B = false
C = gap.
If you ask the computer about the disjunction ‘A or C’, it should treat it as true in spiteof the gap, for the gap is ultimately irrelevant: no matter how the computer fills in its
lack of information, the disjunction comes out true by virtue of the first disjunct’s beingtrue. However, if you ask the computer about the disjunction ‘B or C’, then the computer would treat it as neither true nor false: opposite ways of filling in the gap yield
opposite outcomes, and there is no way for the computer to choose one outcome over
the other. This is how supervaluations work. On the other hand, consider the following
set up:A = true
B = false
C = glut.

7 This is the suggestion I examine in Varzi [1991, 1994]. The idea of a subvaluation can befound in various formats in the works cited in note 3 and bears some obvious connections with the
non-adjunctive logics pioneered by Jaśkowski [1948] (compare e.g. Da Costa & Dubikajtis [1977],
Kotas & Da Costa [1979], Jennings & Schotch [1984], Schotch & Jennings [1989].) Epistemic semantics exploiting similar intuitions may be found in Levesque [1984, 1990], Konolige [1985] andFagin & Halpern [1987] (see Vardi [1986] for connections). By contrast, the alternative approaches
considered in the sequel have not to my knowledge been considered in the literature, with the only exception mentioned in note 8 below.

4In this case the disjunction ‘A or C’ will still be true, but ‘B or C’ will be treated asboth true and false. For the computer has been told that B and C are both false, so the
disjunction must be false; and it has been told also that C is true, so the disjunction
must be true. There is no way for the computer to disregard one value or the other: both
Sam and Elisabeth are equally reliable. Thus both values must be accepted. And this is
the subvaluation.
Note that other strategies may be considered. For example, one could program thecomputer so as to suspend judgment on ‘B or C’ also in the second case. The computer
cannot choose between Sam and Elisabeth—one could argue—hence it cannot answer
your query.8 Formally, this alternative way of dealing with gluts amounts to treating allcomplete and consistent structures into which a given model M can be sharpened as
being on a par with one another. If there is no precise information, and if this lack ofprecise information turns out to be relevant, then the sentence cannot be evaluated, regardless of whether the imprecision is due to a gap or to a glut. (Think of a certain
customary way of analysing failure of existence and failure of uniqueness in the case ofdefinite descriptions: surely these are opposite semantic accidents, but they both lead
eventually to the same situation of indeterminacy). Accordingly, one could construe thegeneral valuation induced by a model M as the supervaluation registering the pattern ofagreement among all admissible sharpenings of M, be they complete extensions or
consistent restrictions (or complete extensions of consistent restrictions).
Now both strategies—I submit—correspond to some way of implementing a basic
idea that is intrinsically supervaluational, viz. the idea of construing the valuation on amodel as a function of the valuations induced by a family of models strictly related to
the former in terms of semantic content. The two strategies differ as to the specific format of the function (meet versus join), and one can imagine other formats as well. But
precisely this seems to be the core of the supervaluationary approach: from an abstract
perspective, its distinguishing feature is the purely functional characterization of theprocess whereby a language is evaluated—not a function of the compositional structureof the language’s syntax (as in truth-functional semantics), but a function of the relativecomplementarity of the language’s models. Depending on the variety of functions one
considers, a corresponding variety of semantics ensues.
3.

Example: Propositional Logic

Let us illustrate the above in connection with a concrete example. To avoid lengthy definitions, it will be convenient to focus on a simple case study—say a propositional language L built up as usual from a set A of atomic sentences by means of finitary connectives such as negation (~), conjunction (∧), disjunction (∨), etc. It would be instruc-8 See Lewis [1978: 46] for a suggestion in this vein, though Lewis [1983] favors instead a
subvaluational account of the sort discussed above.

5

tive to work with arbitrary or more complex languages, but spelling out the details at theproper abstract level would lead us too far afield. I shall confine myself to some occasional remarks where this choice of syntax may have the effect of hiding something important.9
With respect to such a language, a model as classically understood is of course
nothing but a bivalent map associating each atomic sentence with a definite truthvalue, i.e., a total function from A to, say, 2={T, F}. By contrast, here we are interested
in the possibility of relaxing this notion by allowing a model to be undefined, or even
overdefined, with respect to some atomic sentences. This means that, in general, amodel need not be a total function on the domain A, and it need not be a total function
either:(1)

A model for L is any relation M ⊆ A × 2.

(Alternatively, we could also define a model to be a set-valued function, e.g., a mappingfrom A to the power set of 2: the two policies are isomorphic.)
Clearly (1) subsumes the classical notion of a model as a limit case. More generally, it is apparent that the class of all models, Mod(L), is partially ordered in terms of“definiteness” by the inclusion relation ⊆, and we may single out complete, consistent,
and sharp (i.e., classical) models as distinguished special cases:(2a) M is complete [M ∈Comp(L)] iff M[p] ≠ ∅ for all p∈A;(2b) M is consistent [M ∈Cons(L)] iff M[p] ≠ 2 for all p∈A;(2c) M is sharp [M ∈Shrp(L)] iff it is both complete and consistent(where M[p] is the image of p under M). Even more generally, one may define various
related notions such as, for instance, completeness and consistency relative to single
sentences, or sets of sentences. These generalizations are obvious and would be particularly useful in the context of languages with greater semantic complexity, where
there may be no guarantee that all gaps in an incomplete model can be filled in—or all
gluts in an inconsistent model weeded out—simultaneously. This would be the case, forinstance, if L were a first-order language with definite descriptions treated as bona fidesingular terms, or a language with an abstraction operator. For L as assumed, however,
we may ignore these complications.
Relative to this simple setting, the intuitive ideas outlined above can be made precise as follows. (I shall begin with the first strategy, and then move on to consider thesecond and other strategies in the next sections.) First of all, every unsharp model canbe sharpened in various ways: if it is incomplete (but consistent), its sharpenings are its
complete extensions (completions, for short); if it is inconsistent (but complete), its
sharpenings are its consistent restrictions (or constrictions); and if it is both incomplete

9

I deal with these matters at some length in my [1994].

6

and inconsistent, one may take as sharpenings the completions of its constrictions (orthe constrictions of its completions). More precisely:(3a) M' is a completion of M [M'∈Comp(M)] iff M' is ⊆-minimal in the class {Mi∈Comp(L): M ⊆ Mi };(3b) M' is a constriction of M [M'∈Cons(M)] iff M' is ⊆-maximal in the class{Mi ∈Cons(L): Mi ⊆ M};(3c) M' is a sharpening of M [M'∈Shrp(M)] iff M' is a completion of some constriction of M.
Note that every complete model qualifies as a completion of itself, and every consistent model as a constriction of itself. In fact, the cardinality of Comp(M), Cons(M),and Shrp(M) will always be a power of 2, as an undefined or overdefined atomic sentence can only be sharpened in one of two ways. Note also that ⊆-minimality and ⊆maximality are required for the simple purpose of avoiding undue loss of informationin the process of sharpening a given model. In the present case this only becomes relevant when the model is both inconsistent and incomplete, but the issue could be more
substantial in the context of different model classes. Finally, and more importantly, notethat the sharpenings of any model are always sharp.
Now, every sharp model M': A → 2 extends to a straightforward classical valuation, i.e., a Boolean map V': L → 2. Using the above notions, we then extend every
model M ⊆ A × 2 to a corresponding valuation V ⊆ L × 2 as follows:(4a) if M ∈Cons(L), set V =I{V': M'∈Comp(M)} (supervaluation);(4b) if M ∈Comp(L), set V =U{V': M'∈Cons(M)} (subvaluation);(4c) in general, set V= U{I{V': M'∈Comp(M")}: M"∈Cons(M)} (ultravaluation).
Thus, in general a sentence is certified true (at least) on a model M iff it comes out classically true in every completion of some constriction of M, and it is certified false (atleast) on M iff it comes out classically false in every completion of some constriction ofM. (For simplicity, I am following the radical course of treating all completions andconstrictions as admissible; a more general account would require a relation of admissibility to be defined among models, so as to constrain valuations accordingly. I shallcome back to such generalizations below.)
It is immediate that the notion defined in (4c) subsumes the ones in (4a)–(4b) as
limit cases. That is, relative to models that are consistent or complete, the ultravaluation
reduces to the supervaluation and to the subvaluation, respectively, each of which reduces to the classical valuation when the model is both complete and consistent, i.e.,
sharp. This justifies treating ultravaluations as embodying and generalizing the spirit ofsupervaluationism, and guarantees that the resulting semantics is normal, i.e., reduces toclassical semantics when all the classical requirements are jointly satisfied.
To get a flavor of the sort of generalization afforded by (4a)–(4c), it is now sufficient to look at some instances of the dual behavior of super- and subvaluations. For
7example, it is clear that truth and falsity are neither exhaustive nor exclusive: just as asupervaluation on an incomplete model will leave certain sentences undefined as a resultof conflicting outputs in the model’s completions, likewise a subvaluation on an inconsistent model will leave certain sentences overdefined as a result of conflicting outputs
in the relevant constrictions. On the other hand, just as a supervaluation may have theeffect of dismissing (or “redeeming”) a gap in case it turns out to be irrelevant, likewise a subvaluation may have the effect of dismissing an irrelevant glut (think again of
the computer’s behavior in response to Sam’s and Elisabeth’s inputs). Here are someillustrative examples:(5a) If M[p]={T} and M[q]=∅:(5b)

If M[p]={F} and M[q]=2:
V [p]={F}V [q]=2
V [~p]={T}V [~q]=2
V [p ∨q]=2
V [p ∧q]={F}V [p ∨~p]=V [q ∨~q]={T}V [p ∧~p]=V [q ∧~q]={F}V [p]={T}V [q]=∅
V [~p]={F}V [~q]=∅
V [p ∨q]={T}V [p ∧q]=∅
V [p ∨~p]=V [q ∨~q]={T}V [p ∧~p]=V [q ∧~q]={F}In particular, note that just as a supervaluation on an incomplete model has the effect of making a tautology such as ‘p ∨ ~p’ true even if ‘p’ is neither true nor false (i.e.,even if neither disjunct is true), a subvaluation on an inconsistent model has the effect ofmaking a contradiction such as ‘p ∧ ~p’ false even if ‘p’ is both true and false (i.e., evenif both conjuncts are true). For just as in the first case the process of filling in the gap
guarantees that either ‘p’ or ‘~p’ is true in every possible completion of the model, in
the latter case the process of weeding out the glut guarantees that either ‘p’ or ‘~p’ isfalse in every possible constriction. This restores a perfect symmetry between one of
the most characteristic features of the supervaluationist semantics for incomplete models—the distinction between the logical law of excluded middle and the semantic principle of bivalence—and what seems to be its natural counterpart for inconsistent models.10
More generally, here is how the classical truth conditions for compound sentences
are weakened on the present account:

10 The distinction was first emphasized by Mehlberg [1956], §29, and (independently) by
McCall [1966] and especially van Fraassen [1966a], §8. The case for its counterpart can be traced backto Ja śkowski [1948] and Przel- eçki [1964], but see also Rozeboom [1962]; more recent sources are Belnap [1977], Rescher and Brandom [1980], Lewis [1982], and Urchs [1994]. I discuss it at some lengthin Varzi [1997].

8

(6a) T∈V [~A] iff F∈V [A]F∈V [~A] iff T∈V [A](6b)T∈V [A ∧ B] only if T∈V [A] ∩ V [B]F∈V [A ∧ B] if F∈V [A] ∪ V [B](6c)T∈V [A ∨B] if T∈V [A] ∪ V [B]F∈V [A ∨B] only if F∈V [A] ∩ V [B].

The supervaluational treatment of gaps is responsible for the weakening in the second
half of (6b) and the first half of (6c); the subvaluational treatment of gluts is responsible
for the weakenings in the other two halves.11
It is then easy to see that these symmetries extend throughout the framework. In
particular, a simple dualization of the usual argument establishing the adequacy of supervaluational semantics to classical propositional logic will show a corresponding adequacy of subvaluational and, more generally, ultravaluational semantics. To be more
precise, where K is any set of models and A any sentence, define:(7a) TK(A) = {M∈K: T∈V [A]} (the K-truth set of A)(7b) FK(A) = {M∈K: F∈V [A]} (the K-falsity set of A).
Thus, the K-truth set of a sentence A is the set of all K-models in which A is at least true,and similarly for its K-falsity set. This allows us to generalize the ordinary notions oflogical truth, falsity, etc. in the obvious way:
(8a)
(8b)
(8c)
(8d)

A is K-true iff TK(A) = K
A is K-false iff FK(A) = K
A is K-satisfiable iff TK(A) ≠ ∅A is K-refutable iff FK(A) ≠ ∅.

Then the following is easily seen to hold whenever Shrp(L) ⊆ K ⊆ Mod(L):(9)

A sentence A is K-true/false/satisfiable/refutable iff A is Shrp(L)-true/false/
satisfiable/refutable, respectively.

For if Shrp(L) ⊆ K and A is K-true, then obviously A is also Shrp(L)-true. On the other
hand, if Α is not K-true, then there must exist some model M∈K with the property thatevery constriction M"∈Cons(M) has some completion M'∈Comp(M") in which A is
not true, i.e., such that T∉V' [A]. But completions of constrictions are complete andconsistent. Hence the above amounts to saying that there must exist a model

11 The strengthening obtained by restoring biconditionals throughout (6b) and (6c) would correspond to the above-mentioned four-valued generalization of Kleene’s three-valued scheme.

9

M'∈Shrp(L) in which A is not true, which means that A is not Shrp(L)-true. The proofs
for the other equivalences is similar.
Thus, the account preserves and generalizes another important feature of supervaluational semantics. It retains holus bolus the classical notions of logical truth andfalsity as well as the classical notions of satisfiability and refutability—not only in thepresence of gaps, but also in the presence of gluts, or both.
Of course, these results do not necessarily carry over to languages of greater semantic complexity. More importantly, they cannot be extended immediately to arbitrary
sets of L-sentences so as to cover other important logical or model-theoretic notions. It
is here that the non-standardness of the semantics shows up. For instance, it follows
from (9) that the classical notion of logical consequence continues to hold if we restrict
it to pairs of sentences; however, this notion becomes rather wild as soon as we go beyond this simplest case. To be more precise, consider the obvious candidates for defining entailment between sets of sentences in the presence of gaps and gluts:(10a) Σ K-entails Γ positively [Σ K/T Γ] iff I{TK(A): A∈Σ} ⊆ U{TK(A): A∈Γ}(10b) Σ K-entails Γ negatively [Σ K/ F Γ] iff I{F K(A): A∈Γ} ⊆ U{F K(A): A∈Σ};(10c) Σ K-entails Γ [Σ K Γ] iff Σ K-entails Γ both positively and negatively.
(In classical logic these notions of entailment coincide, as one verifies by taking
K = Shrp(L); but in the general case they need not agree. This assimilates our semantics
to most general semantics with gaps and/or gluts.12) Then the following clearly holds
whenever Shrp(L) ⊆ K ⊆ Mod(L):(11a) A K B iff A Shrp(L) B.
Moreover, the following hold for arbitrary sets Σ and Γ:(11b) Σ Cons(L) B iff Σ Shrp(L) B(11c) A Comp(L) Γ iff A Shrp(L) Γ.
However, these equivalences may not hold when the sentences A and B are replaced by
two or more sentences. For instance, the entailments corresponding to the rules of adjunction and disjunction may fail:(12a) A, ~A /K/T A ∧ ~A(12b) A, ~A /K/ F A ∧ ~A(12c) A ∨ ~A /K/T A, ~A(12d) A ∨ ~A /K/ F A, ~A

whenever TK(A) ∩ FK(A) ≠ ∅whenever TK(A) ∪ FK(A) ≠ K
whenever TK(A) ∪ FK(A) ≠ K
whenever TK(A) ∩ FK(A) ≠ ∅.

12 See e.g. Blamey [1986] and Muskens [1995]. The significance of this situation has been emphasized in Dunn [1997].

10

It is precisely this that makes the overall framework paraconsistent (and paracomplete). It follows from (11a) that contradictions “explode” consequentially, just like inclassical logic, and tautologies implode. However, (12a)–(12d) show that, in general,inconsistencies and incompletenesses do not metastasize throughout the entire language. For accepting unsatisfiable premiss sets or irrefutable conclusion sets is not tantamount to endorsing unsatisfiable (contradictory) premisses or irrefutable (tautologous) conclusions; and this protects the relations of entailment defined in (10a)–(10c)from logical trivialization. (At present, the exact characterization of the patterns that are
valid in the class of all models is still an open problem.)4.

Up or Down?

Before considering our second strategy for generalizing supervaluations, it is worth
pointing out an important connection between the above account (and the intuitive rationale behind it) and certain structural properties of the notion of a model.
Mod(L) is partially ordered in terms of “definiteness” by the inclusion relation ⊆.
In fact, ⊆ is a complete lattice ordering. This means that if we take any models for L
and put them together, either by union or by intersection, we still get a model for L.
More specifically, the whole class of models can be regarded as being “generated”
from the class of sharp models by closing it under these two operations:(13a) Cons(L) is the closure of Shrp(L) under I ;(13b) Comp(L) is the closure of Shrp(L) under U ;(13c) Mod(L) is the closure of Shrp(L) under both I and U.
Each single model, in turn, can be regarded as being generated from a certain class of
sharp models, viz. the class of its sharpenings:(14a) if M ∈Cons(L), then M =I{M': M'∈Comp(M)};(14b) if M ∈Comp(L), then M =U{M': M'∈Cons(M)};(14c) in general, M =U{I{M': M'∈Comp(M")}: M"∈Cons(M)}.
These facts are illustrated in Figure 1.
Now, (14a)–(14c) correspond very closely to the definitions in (4a)–(4c). And
since a model is essentially an evaluation of atomic sentences, one could take this correspondence very seriously: those definitions may be viewed as a natural extension ofthese very facts to the task of evaluating compound sentences. That is to say, (4a)–(4c)may be viewed as being patterned after (14a)–(14c). Just as every model is themeet /join of a certain class of sharp models, likewise the valuation on any model is themeet /join of the valuations induced by those sharp models.

11

p,~p,q,~qp,~p,~q~p,q,~q

CO

E

ET

~p,~q

M

PL

ET

p,~p,q

E

p,~p

PL

p,q,~q

p,~q

T

~p,q

~p

EN

T
IS

S
ON

M
CO

q,~q

CO

~q

NS

IS

TE

p,q

NT

C

p

q

Figure 1. Lattice of models of a propositional language with only two atomic sentences, p
and q. Each node corresponds to a possible model, with ‘p’ and ‘~p’ indicating the presenceof T and F respectively in the values for ‘p’ (likewise for ‘q’ and ‘~q’). The ordering ⊆ goes
uphill along the dotted lines: models below the curved line in the bottom half (grey region)are consistent; models above the other curved line, in the top half (light grey region), arecomplete. Classical (sharp) models lie in the intersection of these two regions (dark grey region).

I am not sure exactly how much weight to put on this line of argument, though it
does appear to bring out an interesting way of explaining the intuitive rationale behind(4a)–(4c). At the same time, it falls short of providing an exclusive account. For as I
mentioned earlier, there is an obvious variant of (4c). It is obtained by redefining theultravaluation induced by a model M to be the relation(4c') V =I{U{V': M'∈Cons(M")}: M"∈Comp(M)}.
Intuitively, this corresponds to a valuational policy where the things to be done are reversed: here one must first fill in the gaps (i.e., look at the completions of the given
model) and then see whether there is any way of weeding out the gluts (looking at its
12

constrictions) which agree on some common value. In short: a sentence is true (false)on an arbitrary model M iff it is true (false) on some constriction of every completionof M. This policy seems just as natural as the one defined in (4c). And indeed, alongwith (14c) we can also prove the following general equality, matching (4c'):(14c') if M ∈Mod(L), then M =I{U{M': M'∈Cons(M")}: M"∈Comp(M)}.
Thus, if (14c) provides grounds for (4c), (14c') should provide grounds for (4c'); and it
would be natural to expect that the two definitions yield the same semantic relation (thesame ultravaluation).
Rather surprisingly, however, this is not the case. To see this, let VUI be the ultravaluation defined in (4c) (for arbitrary M), and let VIU be its dual defined in (4c') Bothvaluations agree on the partial truth conditions for compound sentences, (6a)–(6c). And
it is easy to verify that both valuations coincide as far as the main logical properties areconcerned—for instance, (9) to (12d) would continue to hold even if we redefined(8a)–(8d) in terms of VIU. However, with respect to the truth values that a sentence may
take on a model that is both incomplete and inconsistent, the two valuations may disagree. For example, if M is overdefined with respect to an atom, p, and undefined withrespect to another atom, q (as in the leftmost node of the lattice in Figure 1), then thebiconditional ‘p ↔ q’ is evaluated in two different ways.13 It is evaluated as neither true
nor false if we use the first schema, VUI. For there are only two constrictions of M, one
where ‘p’ is true and one where ‘p’ is false; and each constriction has two completions,
one where ‘q’ is true and one where ‘q’ is false. Thus, for each constriction there is acompletion where ‘p ↔ q’ is true and a completion where ‘p ↔ q’ is false, leaving thebiconditional truth-valueless. On the one hand, if we use the alternative schema, VIU,
then ‘p ↔ q’ is evaluated as both true and false. For there are two completions of M, one
where ‘q’ is true and one where ‘q’ is false; and in each completion ‘p ↔ q’ is both trueand false, since each completion has a constriction where ‘p’ is true and one where ‘p’
is false. Thus VUI[p ↔ q]=∅ but VIU[p ↔ q]=2. More generally, it turns out that VUI is
always included in VIU (since every completion of any constriction of a model M is extended by some constriction of some completion of M), whereas the converse only
holds if M is either consistent or complete (for in such cases both VUI and VIU reduce tothe same supervaluation and the same subvaluation, respectively).
This result of course weakens the above argument. For if the point of the argument
is that the way a valuation is construed from sharp valuations may be grounded directly
upon the way an arbitrary model can be represented in terms of sharp models, then thequestion arises of what criteria could eventually justify a choice between two equally
grounded valuational policies. And that seems to have no principled answer. Nonetheless, there is a good explanation for this impasse. For in the final analysis this situation

13

The example is based on a remark from Visser [1984: 194].

13

ties in directly with the assumption that inconsistency and incompleteness are two sidesof the same coin. If we take that standpoint, it should come as no surprise that whengluts and gaps occur simultaneously, the strategy we are examining forces us to choose
among two possible policies: we can either weed out the gluts first, or we can start byfilling in the gaps. And although in general these two policies yield equal results, inthose cases when the very expression to be evaluated amounts to a direct comparison
between gluts and gaps (as with ‘p ↔ q’ in the indicated model) the difference must become apparent: on one policy gluts go first, hence gaps prevail; on the other gaps go
first and gluts prevail. (This point becomes clearer if one considers that, in a standard
propositional language like L, a biconditional ‘p ↔ q’ is provably the only type of sentence in two variables—up to logical equivalence—on which the two policies may disagree.)
To be sure, one can now see that for any incomplete and inconsistent model M
there is a whole family of equally legitimate (but increasingly stronger) potential
valuations: a family of ultravaluations forming a complete lattice under the inclusion
relation ⊆, with VUI at the bottom and VIU at the top. The intermediate elements of thislattice need not be very “natural”, as they would somewhat arbitrarily shift between VUIand VIU for different (compound) sentences. Nevertheless, the semantic complexity ofsuch a proliferation has some intrinsic interest and could be a rewarding subject for
further exploration.
5.

Gluts as Gaps

Both variants of the strategy described above exploit the intrinsic duality between gaps
and gluts. Let us now consider the second strategy for generalizing supervaluations
mentioned in Section 2. This is based on an idea that is somewhat opposite to duality,
viz. that semantic gaps and gluts are phenomena of the same sort. As I already mentioned, the intuition operating here is reminiscent of a certain way of dealing with these
phenomena in connection with the semantics of singular terms such as definite descriptions: if there are no objects or more than one object satisfying the identifying propertyof the description, then the description is left without a denotation, i.e., its interpretationis left undefined. Likewise, one could advocate a radical policy with respect to the interpretation of other expressions, including valuation of sentences: if there is either a gap
or a glut in a sentence’s pattern of reference then—one could argue—the preconditions
for a truth-value assignment are violated.
Formally, the shift of attitude corresponding to this view is only a matter ofchanging the final definitions: subvaluations on inconsistent models become standard
supervaluations, i.e., meets of classical valuations, albeit induced by constrictions rather
than completions. More generally, one can obtain a semantics in which gluts are treatedas gaps of a kind by redefining the ultravaluation induced by an arbitrary model M to be
the relation

14

(4c") V =I{I{V': M'∈Comp(M")}: M"∈Cons(M)}.
Thus, on this view a sentence is true (false) on a model M iff it is true (false) on every
completion of every constriction of M; if it is true on every completion of some constriction, and false on every completion of some other constriction, then the sentence isleft truth-valueless.
Let us use VII to indicate the ultravaluation defined by (4c"). The main effects ofsuch an amendment are not difficult to imagine. Gluts are cured with the same medicine
that cures gaps. And since the classical valuations from which VII is construed arefunctions, in the end VII will simply be a (possibly partial) function, i.e., it will
involve no gluts. In other words, unlike VUI and VIU, VII-valuations on inconsistentmodels will never treat any sentence as both true and false, as the presence of conflicting outputs in the relevant constrictions will be treated as a sign of indefiniteness. In fact, it is easy to prove that VII is always bound to be included in the
restriction of VUI to the set of sentences {A: VUI[A] ≠ 2}. (This set is included in {A:
VIU[A] ≠ 2}, so the same could be said of VIU.) As an example, here is a comparison
between the two strategies in the most critical case, where M is both inconsistent andincomplete:(15a) If M[p]=∅ and M[q]=2:(15b) If M[p]=∅ and M[q]=2:
VII [p]=∅VII [q]=∅VII [~p]=∅VII [~q]=∅
VII [p ∨q]=∅
VII [p ∧q]=∅
VII [p ∨~p]=VII [q ∨~q]={T}VII [p ∧~p]=VII [q ∧~q]={F}

VUI [p]=∅
VUI [q]=2
VUI [~p]=∅
VUI [~q]=2
VUI [p ∨q]={T}
VUI [p ∧q]={F}
VUI [p ∨~p]=VUI [q ∨~q]={T}
VUI [p ∧~p]=VUI [q ∧~q]={F}

Another important consequence is that in this case the variant definition corresponding to the reversion of the valuational process does not yield anything new, i.e.,
the following is perfectly equivalent to (4c"):(4c') V =I{I{V': M'∈Cons(M")}: M"∈Comp(M)}.
In fact, both definitions reduce to(4c)''

V =I{V': M'∈Shrp(M)},where the notion of a model’s sharpening may be interpreted equivalently as in (3c) oras its obvious dual, meaning completion of a constriction rather than constriction of acompletion. In other words, if we treat all sharpenings in the same way, it does not mat15

ter whether we get to them via completions or via constrictions: the final class of models is always the same.
This equivalence among possible equally legitimate variants may be regarded as anadvantage of the present strategy over the strategy examined in Sections 3 and 4. On theother hand, the connection with the lattice-theoretic properties of Mod(L) is now much
weaker—in fact, inexistent. For not only are (4c ' )–(4c '') plainly in contrast with(14c)–(14c'). On this account, every model inconsistency results in valuational incompleteness, i.e., M[p] = 2 always implies VII [p] = ∅, which means that valuations on inconsistent models are not direct extensions of the models themselves. More generally,on this account valuations are not monotonic: if M ⊆ M', then VII ⊆ V'II if M' is consistent, while V'II ⊆ VII if M is complete. By contrast, both variants of the first strategy
above are perfectly monotonic, i.e., M ⊆ M' always implies VUI ⊆ V'UI and VIU ⊆ V'IU. If
we stick to the suggested reading of ⊆ as an ordering going up hill in terms of degrees
of definiteness, this is of course an important advantage of that strategy over the one
presently under consideration.
Be it as it may, it will be clear that these differences do not drastically effect thelogical flavor of the semantics. In particular, the general facts expressed by (9) will continue to hold under the present account as well: the basic notions of logical truth andfalsehood, we may say, as well as the notions of sentence satisfiability and refutability,
do not depend on our specific attitude towards gaps and gluts. Still, concerning entailment the situation is slightly different, as the following is now provable (for Shrp(L)⊆ K ⊆ Mod(L)):(16) A set of sentences Σ is K-satisfiable/refutable iff Σ is Shrp(L)-satisfiable/ refutable, respectively
where a set is said to be K-satisfiable/refutable if its elements are all simultaneously Ksatisfiable/refutable. This means that the rule of adjunction is now valid. More precisely,
one can prove that an entailment is (positively or negatively) K-valid in the semantics
based on VII iff it is (positively or negatively) K ∩ Cons(L)-valid in the semantics based
on VUI or VIU. Thus, unlike the previous stategy, the strategy under examination is fully
equivalent—as far as logic goes—to ordinary supervaluational semantics.
6.

Completing the Map

Nothing at this point prevents us from considering further possible ways of generalizing the basic notion of a supervaluation. For instance, generally speaking there is no
definite reason to dismiss the (pairwise equivalent) variants obtained by replacing I
with U in (4c ')–(4c ')' above. For if one can provide arguments to the effect that gluts
should be treated as gaps, then one may be willing to consider the dual attitude and regard gaps as a type of gluts.

16

There are other possible variants. The overall picture is given by the family of ultravaluations generated by the following scheme:(17) Vƒ 1 C1 ƒ 2 C2 = ƒ1 {ƒ2 {V': M'∈C2 (M")}: M"∈C1 (M)},where each ƒi is either U or I, and each Ci is either Comp or Cons (C1 ≠ C2 ). The diagram in (18) shows the ordering of the resulting valuations in terms of relative strength.
(Here I write ‘↑ ’ and ‘↓ ’ to abbreviate ‘Comp’ and ‘Cons’, respectively. Thus, VU↓I↑ isjust the old VUI, and so on.)⊆(18) VI↑I↓ = VI↓I↑⊆

VU↑I↓ ⊆ VI↓U↑
VU↓I↑ ⊆ VI↑U↓⊆⊆

VU↓U↑ = VU↑U↓

As before, each relation of strict inclusion leaves room for an entire lattice of intermediate (but somewhat arbitrary) valuations, which we need not examine here. Andagain, all options turn out to share the same basic properties expressed by (9). Thus, theabove remark to the effect that such properties as K-truth, K-falsity, etc. do not seem todepend on one’s specific attitude towards gaps and gluts (i.e., on the specific cure that
gaps and gluts require) extends to the full spectrum of strategies compatible with thesupervaluational “spirit”. On the other hand, as regards logical entailment, all schemes
induce a departure from classical logic, but in different degrees. The adjunctive pattern
A,~A A ∧~A holds only for the two leftmost (purely supervaluational) schemes, and its
disjunctive dual A∨~A A,~A holds only for the rightmost (purely subvaluational) pair.
There is more room for generalization. For instance, all the strategies discussed so
far and captured by (17) require that in order to evaluate an L-sentence A one considers
the value assigned to A in every sharpening of the given model. However, this is a simplification that one may want to relax. In general, one might want to assume a suitable
relation R to be defined on Mod(L) so that only models in the image R[M] would qualify as admissible refinements of a given model M. For instance, we may suppose thatcertain ways of filling in certain gaps in a model are incompatible with certain ways offilling in other gaps, or with certain ways of weeding out gluts, on account of certain“penumbral connections” (in the terminology of Fine [1975]).14 We may in thesecases speak of R-completions and R-constrictions, respectively. From this point of view,
the policy followed above corresponds to the limit case R = Mod(L) × Mod(L), but it may
be interesting to see what can be gained by considering other cases as well. Moreover,other means for construing valuations may be considered beside meet (I) and join (U).
In principle one may take any pair of functions ƒ1 and ƒ2 , or even any ordered sequence
F = 〈ƒ0, …, ƒn 〉, where each ƒi may be thought of as having a relevant admissibility rela-

14

This point is detailed in Varzi [1997], §3.3.

17

tion Ri associated with it. Thus, taking all this into account, (17) can be further generalized by considering the family of ultravaluations generated by the following scheme:(19) VF R = ƒ0{ƒ1{…{ƒn{Vn: Mn∈Rn[Mn–1]}: …}: M1∈R1[M0]}: M0∈R0[M]}(where Vn is again the relevant classical Boolean assignment).
This yields a sufficiently abstract setting for further investigations. In fact, it emphasizes what is arguably a distinguishing feature of the general philosophy behind thesupervaluational method, viz. the purely functional characterization of the process
whereby a language is evaluated. All of these valuations are not truth-functional, i.e.,they do not assign values as a function of the compositional structure of the language’s
syntax. They do, rather, establish the value of any given sentence as a function of therelative completeness/consistency of the language’s models. Depending on the variety
of functions one considers, a corresponding variety of semantics ensues. And as (19)suggests, the variety may be very wide indeed.15
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