How many kinds of reasoning?
Inference, probability, and natural language semantics
Daniel Lassiter, Noah D. Goodman
Stanford University
October 28, 2014
To appear in Cognition: please consult final version at
http://www.journals.elsevier.com/cognition/.
Abstract
The “new paradigm” unifying deductive and inductive reasoning in a Bayesian framework(Oaksford & Chater, 2007; Over, 2009) has been claimed to be falsified by results which show
sharp differences between reasoning about necessity vs. plausibility (Heit & Rotello, 2010;
Rips, 2001; Rotello & Heit, 2009). We provide a probabilistic model of reasoning with modal
expressions such as “necessary” and “plausible” informed by recent work in formal semantics
of natural language, and show that it predicts the possibility of non-linear response patterns
which have been claimed to be problematic. Our model also makes a strong monotonicity
prediction, while two-dimensional theories predict the possibility of reversals in argument
strength depending on the modal word chosen. Predictions were tested using a novel experimental paradigm that replicates the previously-reported response patterns with a minimal
manipulation, changing only one word of the stimulus between conditions. We found a spectrum of reasoning “modes” corresponding to different modal words, and strong support for ourmodel’s monotonicity prediction. This indicates that probabilistic approaches to reasoning canaccount in a clear and parsimonious way for data previously argued to falsify them, as well
as new, more fine-grained, data. It also illustrates the importance of careful attention to thesemantics of language employed in reasoning experiments.
Keywords: Reasoning, induction, deduction, probabilistic model, natural language semantics.

Suppose that you have learned a new biological fact about mammals: whales and dogs bothuse enzyme B-32 to digest their food. Is it now necessary that horses do the same? Is it plausible, possible, or more likely than not? Expressions of this type—known as epistemic modals in
linguistics—have played a crucial, though neglected, role in recent work that argues for a sharp
qualitative distinction between inductive and deductive modes of reasoning. In the paradigm introduced by Rips (2001) and extended by Heit and Rotello (2010); Rotello and Heit (2009), participants are divided into two conditions and are either asked to judge whether a conclusion is1

“necessary” assuming that some premises are true, or whether it is “plausible”. The former was
identified with the deductive mode of reasoning, and the latter with the inductive mode.
These authors asked participants in both conditions to evaluate a variety of arguments, some ofwhich were valid under classical logic and some invalid under classical logic, or contingent. An
example contingent argument might be “Cows have sesamoid bones; Mice have sesamoid bones;
therefore, Horses have sesamoid bones”. An example valid argument might be “Mammals havesesamoid bones; therefore, horses have sesamoid bones.” They found that there was a non-linear
relationship between the endorsement rates of arguments depending on condition: participantsin both conditions generally endorsed logically valid arguments, but participants in the deductive
condition were much less likely to endorse contingent arguments than those in the inductive condition. These results were interpreted as a challenge to theories of reasoning which rely on a single
dimension of argument strength and interpret deductive validity as simply the upper extreme ofthis dimension, guaranteeing maximal strength (Johnson-Laird, 1994; Osherson, Smith, Wilkie,
Lopez, & Shafir, 1990). In particular, Rips and Heit & Rotello argued that non-linearities cannot
be accounted for by probabilistic theories of reasoning, which identify the strength of an argumentwith the conditional probability of the conclusion given the premises (Heit, 1998; Kemp & Tenenbaum, 2009; Oaksford & Chater, 2007; Tenenbaum, Griffiths, & Kemp, 2006). On the other hand,
they claimed that the results are consistent with two-process theories of reasoning (Evans & Over,
1996).
We argue that the key manipulation of previous experiments hinges not on a psychologicaldistinction between two reasoning processes, but rather on facts about the semantics of modal
words such as “necessary” and “plausible”, which can be modeled using a single underlying scaleof argument strength—conditional probability. We propose an account of reasoning with epistemic
concepts, motivated by recent work in linguistic semantics, which predicts non-linear response
patterns while retaining a single dimension of argument strength. The model predicts a spectrumof response patterns depending on the modal language used, not only the two corresponding to“necessary” and “plausible”. It also makes strong monotonicity predictions that would be very
surprising under two-dimensional theories of reasoning.
We tested the claim that the modal word is the crucial factor using a new paradigm that isolates
its effects. Our arguments had the same form as those used in previous research, except that weplaced a modal word of interest in the conclusion:
Premise 1: Cows have sesamoid bones.
Premise 2: Mice have sesamoid bones.
Conclusion: It is {plausible/necessary/possible/likely/probable/certain} that horses
have sesamoid bones.
We will refer to configurations such as “It is plausible/possible/etc. that C” as a modal frame. Ifvarying the modal frame gives rise to the non-linear pattern of responses found in previous work,
this would indicate that an explanation of these results could be derived from the meaning of thesemodal words.
Together, the model and experimental evidence indicate that the negative conclusions of previous work regarding one-dimensional theories of argument strength are not warranted: it is possible
to explain non-linear response patterns with a probabilistic account of argument strength. Indeed,2

non-linearities are to be expected given independently motivated facts about the semantics of epistemic modals. In addition, we confirm the monotonicity prediction, lending direct support to a
one-dimensional, probabilistic theory.

1

Deduction and Induction

There is a long history of reflection on the relationship between inductive and deductive reasoning (Gabbay & Woods, 2009). In philosophy, it has long been assumed that the two are distinct.
Deductive reasoning has generally been thought of as involving mathematically well-defined procedures for drawing out consequences which follow with certainty or of necessity from some bodyof evidence. Induction, on the other hand, involves going beyond the available evidence, drawing
conclusions that are likely or plausible but (often) not certain to be true. The traditional characterization has thus generally taken for granted that induction and deduction are different, with deduction occupying a position of pride in mathematical and logical reasoning. Since David Hume’s
seminal work on the topic, induction has generally been viewed very differently, as a psychologicalprocess which is practically necessary for humans but lacking in formal justification (see Vickers,2012).
Deductive reasoning is frequently assumed to be relevant to psychologically interesting phenomena such as conditional and syllogistic reasoning. For example, many psychological models
of syllogistic reasoning have explicitly invoked some standard mathematical formulation of deductive inference, either syntactic/proof-theoretic (Rips, 1994) or semantic/model-theoretic (JohnsonLaird, 1986) in character.
With the advent of Bayesian approaches to inductive inference in philosophy (Ramsey, 1931;
Earman, 1992) and psychology (Osherson et al., 1990; Oaksford & Chater, 1998; Tenenbaum,
Kemp, Griffiths, & Goodman, 2011), the theoretical gap between these two types of reasoning has
begun to narrow. It has often been observed that the validities of classical logic can be recovered
as a limiting case of standard probabilistic inference, where attention is restricted to arguments inwhich the conclusion has conditional probability 1, given the premises, no matter what backgroundbeliefs a particular individual may have (e.g., Jaynes, 2003).
Oaksford and Chater (2001, 2007) have argued that there is no psychologically relevant distinction between induction and deduction: the reasoning strategies that are employed by participantsin experiments designed to test their deductive reasoning skills are the same that are employed
for everyday inductive reasoning tasks. Oaksford & Chater argue that both kinds of reasoning are
well-modeled if we assume that reasoners maintain probabilistic models of the world around themand reasoning proceeds by conditioning on the data received—i.e., by Bayesian inference. In thesimplest case on which we will focus here (where the premises do not contain conditionals, andare taken as known facts) this account treats deductive reasoning involving a set of premises P in
the following way: participants evaluate potential conclusions C by finding the conditional probability of C in their (subjective) model, on the assumption that the premises are true. Given a choice
between “valid’ and “invalid” participants should respond “valid” if this conditional probability isgreater than the relevant response threshold, in the absence of intervening factors. If the response
threshold is 1, then the choice will mirror the judgments predicted by a theory based on explicit
3

deductive reasoning.1 On this account, then, the distinction between inductive and deductive reasoning is not a qualitative distinction of different modes or processes, but merely a quantitative
difference in response threshold.
Note that the above characterization of probabilistic approaches to reasoning is simplified in
two important ways. First, everyday reasoning critically handles situations in which the inputs
to reasoning, the premises, are themselves uncertain. There are various ways to generalize basic
probabilistic reasoning to situations involving uncertain information, notably Jeffrey conditionalization (Jeffrey, 1965, §11) and hierarchical modeling (Pearl, 1988, §2.2.1). It remains to be seen
which of these alternatives, if any, is empirically adequate: useful perspectives on this issue areprovided by Hadjichristidis, Sloman, and Over (2014); Pfeifer (2013); Stevenson and Over (2001);
Zhao and Osherson (2010). Second, reasoning from conditional premises (Oaksford & Chater,
2007; Pfeifer & Kleiter, 2010) is an important task at the intersection of semantics and psychologyof reasoning, which may require some amendment to the simple conditional probability approach.
It is controversial what semantics is appropriate for conditional sentences, and even whether they
can be assigned truth-values and conditioned on. We focus in this paper on the case of simple
non-conditional sentences presented as facts, with the expectation that the contributions of thispaper will be synergistic with extensions to handle uncertain and conditional premises. (Indeed,
conditional reasoning is one of the core cases discussed by Oaksford and Chater (2001, 2007) inthe context of presenting a model along these lines.)2

Difficulties for a Unified Model

Rips (2001) conducted an experiment designed to investigate whether there is a qualitative distinction between deductive and inductive reasoning. Participants in two groups were asked to judge
arguments either according to whether the conclusion was necessary (assuming that the premises
were true) or whether it was plausible. The mode of judgement was explained in each condition
by extensive task instructions. Most participants in both conditions accepted valid arguments andrejected contingent arguments whose conclusion was not causally consistent with the premises,
such as “Car X strikes a wall, so Car X speeds up”. However, participants differed by conditionin whether they rejected non-valid arguments which were causally consistent with the premises:
those in the inductive condition generally accepted arguments such as “Car X strikes a wall, so Car
X slows down”, while those in the deductive condition did not.
Rips argued that this result falsifies theories of reasoning in which argument strength is a onedimensional quantity such as conditional probability: “[i]f participants base all forms of argument
evaluation on the position of the argument on a single psychological dimension, then induction anddeduction judgments should increase or decrease together” (p.133). However, Rips’ discussion
1

Note however that not all arguments that have conditional probability 1 are classically valid; e.g., the probability
that a real number selected uniformly at random from the interval [2,3] is not π is 1, but this is obviously not a
deductively valid conclusion (Rips, 2001). It is not clear whether this mathematical fact is relevant to everyday
reasoning problems, though. If it is, then the empirical predictions of the two accounts would be slightly different
when the response threshold is 1; however, the model and experiments we present below suggest that enforcing amaximal response threshold in this way may be practically impossible.

4made clear that the argument against a unified model goes through only on the assumption that
subjects are making judgments by comparing the strength of each argument to response thresholds
which are either deterministic or stochastic with equal variance. (This assumption is importantsince the model we sketch below does not have this feature: see the next section.)
Heit and Rotello (2010); Rotello and Heit (2009) extended Rips’ paradigm in a number ofways, providing a new method of quantitative analysis for these effects. Their core finding wasthat d ′ , a standard measure of sensitivity in Signal Detection Theory (SDT), was significantly
higher in the deductive condition across a variety of arguments types and manipulations. d ′ isdefined as z(H) − z(F), the difference between the z-scored hit rate H and false alarm rate F(Macmillan & Creelman, 2005). This difference means that participants in the inductive condition
were less sensitive to argument validity than participants in the deductive condition (see Table
1). Differential sensitivity indicates that the difference between conditions is not simply a shift in
Deduction
.94
.06
3.31

Acceptance, valid
Acceptance, contingent
Sensitivity (d ′ )Induction
.95
.17
2.56

Table 1: Acceptance rates and d ′ in Experiment (1a) of Rotello & Heit 2009 (three-premise arguments only).
response criterion, assuming equal-variance noise for different argument types. Thus we cannot
fit a one-dimensional SDT model to such results. In accord with Rips, these authors claimed thatthe non-linear relationship between validity and condition is a challenge to probabilistic theories
of reasoning. They argued that the results were better captured by a two-dimensional SDT modelwith possibly orthogonal dimensions of inductive strength and deductive validity, in which the
response criterion can vary in both dimensions.

3

The Probability Threshold Model

In a brief comment on the line of work just discussed, Oaksford and Hahn (2007, p.276) suggested
that the results “tell us more about the semantics/pragmatics of the words ‘necessary’ and ‘plausible’, where it has been known for some time that they don’t neatly carve up some underlying
probability scale”. In this section we propose a model of the relationship between probability andepistemic concepts such as certainty, necessity, and plausibility which makes good on this suggestion. The model is a development of recent work in the formal semantics of natural language. It
predicts the possibility of non-linear response patterns and variation in d ′ depending on the modal
language used, and makes a novel monotonicity prediction which we will test in upcoming sections.

5

3.1

Model description

Our probabilistic model of reasoning with epistemic modals is intended to capture the following
intuitions. A maximally strong conclusion C remains maximally strong whether you ask if it ispossible, plausible, likely or necessary; a maximally weak conclusion remains maximally weakunder the same conditions; but there is much more flexibility for uncertain conclusions. If C has
a probability of .4, it presumably will count as possible and perhaps as plausible, but it would notseem to be likely and surely not necessary or certain. Thus the effect of an epistemic modal ona conditional probability should be a transformation that preserves the minimum value 0 and the
maximum value 1.
As it happens, recent work on the semantics of epistemic modals contains all the components ofa model which has these features: putting these components together, we call this the Probability
Threshold Model (PTM). The crucial modals necessary and plausible—among others such as
certain, likely, probable, and possible—fall into the grammatical class of gradable adjectives, as
discussed by Lassiter (2010, to appear); Portner (2009); Yalcin (2007, 2010) among others. Likeother gradable adjectives, they can be used in comparative and degree modification structures: forexample, conclusion C might be more plausible than C′ given some evidence, or C might be very
likely or almost necessary. This corresponds, for instance, to one person being taller than another,and a glass of water being very large or almost full. Gradable expressions are generally treated informal semantics as functions which map objects (individuals, verb phrase meanings, propositions,etc.) to points on a scale and compare them to threshold values (Kennedy, 2007). For example, themeaning of tall is a function which compares an individual to a height threshold, returning True
if the individual’s height exceeds the threshold: Al is tall is true if and only if Al’s height exceeds
threshold θtall . (For linguistically-oriented surveys of adjective semantics see Lassiter, in press-a;
Morzycki, to appear.)
The location of the threshold is often uncertain, a fact which is closely related to the problem ofvagueness (Williamson, 1994). We follow a number of recent proposals (Edgington, 1997; Frazee& Beaver, 2010; Lassiter, 2011; Lassiter & Goodman, 2013; Lawry, 2008; Schmidt, Goodman,
Barner, & Tenenbaum, 2009; Verheyen, Hampton, & Storms, 2010) by assuming a distribution on
thresholds P(θ). Thus, for instance, the probability that Al is tall is equal to the probability thatthe tall-threshold value falls below Al’s height.
height(Al)P(Al is tall) = P(height(Al) > θtall ) = ∫0

P(θtall ) dθtall(1)

We do not make any assumptions here about how threshold distributions are derived, but see
Lassiter & Goodman, 2013 for a proposal to derive them from pragmatic reasoning, which makes
crucial reference to prior knowledge and utterance context.
Recent work suggests that a threshold semantics is appropriate for gradable epistemic modals
as well, and that probability is a good choice for the underlying scale (Lassiter, 2010, in press-b,
to appear; Yalcin, 2010). We can summarize this model by treating epistemic modals such asplausible and necessary as placing conditions on the probability of their sentential arguments, forinstance:
It is plausible that q ⇔ P(q) > θplausible(2)
6

We will assume that the modals employed in our experiment below — plausible, necessary, possible, probably, likely, and certain — all have lexical entries with this schematic form.
(3)

It is M that q ⇔ P(q) > θM

Since these thresholds are uncertain, we assume a distribution over thresholds for each modal.
Thus, for any such modal θM ,P(q)P(It is M that q) = ∫0

P(θM ) dθM .

(4)We extend this analysis from modalized sentences to arguments with modalized conclusions byassuming that the modal takes scope over the underlying argument. That is, if argument a is of
the form “P1 ; P2 ; so, it is M that C”, we interpret it as “It is M that q” where q is “P1 ; P2 ; so, C”.
As described above, we take the underlying probability for such an argument to be the conditional
probability of the conclusion given the premises, P(q) = P(C∣P1 ,P2 ). From Equation 4, we canthen define:P(C∣P1 ,P2 )strengthM (a) = ∫0
P(θM ) dθM .
(5)One elaboration is needed: a speaker whose subjective probability of rain is .5 may not choose
to assert “It’s raining” or “It’s not raining” with equal probability. Similarly, participants in areasoning experiment may reject arguments of the form “P1 ; P2 ; so, C” if P(C∣P1 ,P2 ) is only
.5, rather than accepting or rejecting at chance. That is, the strength of an unqualified argument“P1 ; P2 ; so, C” itself should not be treated as the conditional probability of the conclusion given
the premises. Instead, we assume that there is a silent modal and a corresponding assertibility
threshold θassert (Lewis, 1979; Davis, Potts, & Speas, 2007) and distribution P(θassert ), which
determine strength∅ (a) as above.2

3.2

Predictions

We assume that participants’ choices in a reasoning experiment will reflect the argument strengthwith some decision noise: Participants’ inclination to agree with argument ai in a forced-choicesetting is governed by equation 5 together with a noise parameter ε, interpreted as the proportionof trials in which participants choose a response at random.
P(“agree”∣ai ) = strengthM (ai ) × (1 − ε) +

ε
2

(6)

So interpreted, the PTM makes two predictions that will be important in discussion below.
First, it predicts a monotonicity property that we will call No Reversals: for any two modals M1
and M2 which fit the lexical schema of equation (3), and for any two arguments a1 and a2 , ifstrengthM1 (a1 ) > strengthM1 (a2 ) then strengthM2 (a1 ) ≥ strengthM2 (a2 ). (The Appendix gives
2

Interestingly, in the experiment reported below the pattern of responses to unmodalized arguments is most similar
to the pattern in the necessary and certain conditions, indicating that P(θassert ) is similar in shape to P(θcertain ) andP(θnecessary ): see Figures 3 and 4 below.

7

a proof of the No Reversals property.) For instance, it should not be possible for a1 to be more
likely than a2 while a2 is more plausible than a1 . Notably, a two-dimensional model of argument
strength like that of Rotello and Heit (2009) does not make this prediction: with two orthogonal
dimensions of strength and a response criterion which can vary in both dimensions, there can easily
be arguments a1 ,a2 such that a1 is stronger by criterion c1 but a2 is stronger by criterion c2 . For
two-dimensional theories, it would therefore be a very surprising coincidence if we find monotonic
results under different “kinds of reasoning”: non-monotonicity would be the usual situation.
The second important property of this model is that it predicts the possibility of non-linearitiesof the type described by Heit and Rotello (2010); Rips (2001); Rotello and Heit (2009) and discussed above. To illustrate, we focus on a particularly simple instantiation of the PTM in which
the threshold distribution is a power law:
P(θM ) = αM θαM −1 , αM ∈ R+ .

(7)

It is easy to show that for this power law distribution,
strengthM (ai ) = P(Conclusion(ai )∣Premises(ai ))αM .

(8)

Depending on αM we get a variety of possible curves, a few of which are sketched in figure 1a(setting ε = .1 for illustrative purposes). The power-law model illustrates how the PTM can account for non-linearities: αM does not greatly influence the response probability for very high or
low confidence, but there is significant variation in predicted response probabilities in the middle
range depending on αM . This feature leads to a prediction that the choice of modal will have lessinfluence on response rates for arguments with very high strength (e.g., logically valid arguments)
than on those with intermediate strength.
In particular, d ′ will vary depending on αM . Suppose for illustration that the mean strengthof logically contingent arguments in some sample is .5, and that the strength of logically valid(inclusion or identity) arguments is 1. The d ′ statistic estimated from this data should then be (onaverage)d ′ = z(1αM (1 − ε) + ε/2) − z(.5αM (1 − ε) + ε/2)(9)
= z(1 − ε/2) − z(.5αM (1 − ε) + ε/2)
If ε = .1, we expect the observed d ′ to be related to αM as in figure 1b. The value of the d ′ statistic
is not predicted to be constant in our probabilistic model, but should depend on the choice of M.
Thus, a model with one dimension of argument strength (conditional probability) is able to predict
non-linearities of the type previously claimed to be problematic for probabilistic accounts.

4

Experiment

Our experiment modified the design of Rotello and Heit (2009) in order to test directly the predictions of the Probability Threshold Model: the specific non-linear response pattern observed
should be controlled by the choice of modal frame, and argument strength ordering should bepreserved between modal frames (No Reversals). Previous experiments in this vein attempted
to induce participants to employ different modes of reasoning using extensive task instructions8αM=3/2
αM=1
αM=2/30.2

αM=1/2αM=1/5αM=1/70.0

αM=1/3

0.0

0.2

0.4

0.6

0.8

1.0

3.5
3.0
2.5
2.0
1.5

0.4

αM=2

1.0

αM=3

0.5

predicted d-prime, mean contingent strength=.5

0.6

αM=5

0.0

1.0
0.8

Predicted response probability

αM=7

0

Conditional probability

1

2

3

4

5

6

7

αM

Figure 1: (a): Predicted response probability for various settings of αM under the power-law version of the PTM. (b): Example of relation between αM and d ′ .
that were manipulated between participants. This set-up makes it difficult to isolate the source ofthe observed non-linearities. Because we hypothesize that these effects are due to the semanticsof the modal expressions employed, we also predict that it should be possible to replicate nonlinearities within participants, using a very minimal manipulation within the stimuli and keeping
the instructions constant: we varied only the choice of modal M within the stimulus sentence. This
minimal manipulation allowed us to isolate the effect of the modal frame by examining acceptance
rates across conditions and for individual arguments of varying strengths, regardless of condition.
Because the task instructions were not manipulated it is possible to use a (more powerful) withinparticipants design. We also used more than two epistemic modal expressions, including possible,
likely, probable, and certain in addition to plausible and necessary. The inclusion of additional
modal vocabulary could allow us to observe more than the two “modes” of reasoning studied inprevious work. We maintained other relevant features of the design of Rotello and Heit (2009),
including the division of arguments into a “Contingent” condition and two “Valid” conditions,
namely “Identity” and “Inclusion”.3
3

In Rotello & Heit’s terminology, our Contingent arguments were “Invalid”, and our Identity and Inclusion arguments were sometimes aggregated as “Valid”. We avoid the term “Invalid” here because it sometimes connotes anormative judgment to the effect that such arguments ought to be rejected. Note also that we do not intend to makeany claims about the ecological validity of the somewhat odd Identity arguments or theoretical claims about whether“reasoning” per se is involved in their evaluation: the goal is rather to replicate previous results using a modified
paradigm which clarifies the source of the patterns observed.

9

Contingent:
Cows have enzyme X.
Seals have enzyme X.
So, it is M that horses
have enzyme X.

Identity:
Horses have enzyme X.
Cows have enzyme X.
So, it is M that horses
have enzyme X.

Inclusion:
Mammals have enzyme X.
Cows have enzyme X.
So, it is M that horses
have enzyme X.

Figure 2: Structure of sample arguments in our three conditions.

4.1

Methods

4.1.1

Participants

507 participants were recruited using Amazon’s Mechanical Turk platform, with a restriction toparticipants located in the United States. They were compensated for their participation.
4.1.2

Materials

Overall there were three argument types—contingent, identity, and inclusion—and seven modal
frames—no modal, “possible”, “plausible”, “likely”, “probable”, “necessary”, “certain”. See Figure 2 for an overview of the argument structures.
The properties used in the arguments were chosen from a list of 21 unfamiliar biological andpseudo-biological properties adapted from Osherson et al. (1990), such as “use enzyme B-32 todigest food”, “have sesamoid bones”, “have a blood PH of 7.9”, and “use acetylcholine as a neurotransmitter”. No property was seen twice by any participant.
As in Osherson et al., 1990, the animal in the conclusion was always “horses”. There were nineother animals that could appear in the premises: “cows”, “chimps”, “gorillas”, “mice”, “squirrels”,“elephants”, “seals”, “rhinos”, “dolphins”. Premises in the contingent condition used distinct pairsof these nine animals. Premises in the inclusion condition used one of these animals and either “animals” or “mammals”. Premises in the identity condition used one of these animals and “horses”.
The order of premises was randomized; neglecting order there were thus 63 specific premise patterns.
In each argument, the conclusion was either in the “no modal” frame—“So, horses have X”—
or in the modal frame “So, it is M that horses have X”, where M was one of: “possible”, “plausible”,“likely”, “probable”, “necessary”, “certain”.
4.1.3

Procedure

Participants saw 21 arguments, one with each target property in randomized order. Each participant
saw each modal frame three times: three blocks were generated for each participant, with one
modal frame occurring in each block in random order. Argument types were assigned to trials suchthat each participant saw 16 contingent, 2 identity, and 2 inclusion arguments, and one additional
argument (to fill out the three blocks of modal frames) which was contingent 80% of the time,
identity 10%, and contingent 10%. These argument types were randomly assigned to trials and10were fleshed out by choosing specific premises in a pseudo random fashion. (Specifically, the nine
animals were randomly drawn as needed until exhausted, at which point they were reset to thecomplete 9; this assured that animals would tend to be distributed though out the trials.4 )Participants were instructed to answer each question according to whether they agreed with theconclusion, assuming that the premises were true. For each question participants were asked to
select “agree” or “disagree” and to give a confidence rating on a five-point scale. At the end of theexperiment, participants were asked to state their native language, with the clarification that this is“the language that was spoken at home when you were a child”.
The full experiment can be found at http://www.stanford.edu/˜danlass/experiment/
animals.html.

4.2

Results

We considered data only from the 484 participants who reported that their native language wasEnglish. We removed data from 44 participants who gave the same response to all questions, aswell as 196 individual trials in which subjects responded very quickly (less than 3.5 seconds). The
results reported are from 8710 remaining trials.
4.2.1

Non-linearities

Recall the crucial finding of Heit and Rotello (2010); Rotello and Heit (2009) showing that sensitivity to argument validity is greater when participants are asked to judge whether a conclusion is“necessary” than when they are asked whether it is “plausible” (Table 1). In more neutral terms,
Heit & Rotello found that asking about necessity versus plausibility affects response probabilities
more for Identity and Inclusion arguments than it does for Contingent arguments. We replicatedthis effect with our within-participants manipulation of modal frame (Table 2). If we follow Heit
and Rotello (2010); Rotello and Heit (2009) in grouping together the Identity and Inclusion arguments as “‘Valid”, the difference in d ′ values is significant (p < .05) based on a 10,000-sample
bootstrap test.
Necessary Plausible
Acceptance, Valid.82.94
Acceptance, Contingent .41.82
Sensitivity (d ′ )1.26.67
Sensitivity, 95% CI[1.06,1.47] [.43,.97]
Table 2: Comparing ‘Valid’ (Identity and Inclusion) and Contingent arguments: acceptance rates
and d ′ in our experiment (plausible and necessary only), with 95% confidence intervals.
4

Because of the random selection of premises participants occasionally saw the same premise categories twice,
though with different target properties. The median number of distinct premise patterns was 19 (of 21 trials).

11Disaggregating these conditions, the pattern of results is similar, as in Tables 3 and 4. In the
Inclusion condition there is a significant difference in d ′ values between “necessary” and “plausible” arguments, p < .05. In the Identity condition the difference is not significant, but the failure ofsignificance in this condition can be attributed to reduced statistical power in the disaggregation:
random sampling of conditions led to fewer data points in the Identity & “Necessary” condition
than in the Inclusion & “Necessary” condition. Indeed, it is clear from Tables 3 and 4 that our participants accepted Identity and Inclusion arguments at almost identical rates. Because we found
Necessary Plausible
Acceptance, Inclusion.84.94
Acceptance, Contingent .41.82
Sensitivity (d ′ )1.33.69
Sensitivity, 95% CI[1.08,1.62] [.36,1.18]
Table 3: Comparing Inclusion and Contingent arguments: acceptance rates and d ′ in our experiment (plausible and necessary only), with 95% confidence intervals.

Acceptance, Identity
Acceptance, Contingent
Sensitivity (d ′ )Sensitivity, 95% CI

Necessary
.80
.41
1.19
[.94,1.48]

Plausible
.94
.82
.65
[.33,1.09]

Table 4: Comparing Identity and Contingent arguments: acceptance rates and d ′ in our experiment(plausible and necessary only), with 95% confidence intervals.
no systematic differences between the Identity and Inclusion conditions, we continue to aggregate
the Identity and Inclusion conditions as “Valid” for increased statistical power in the remainder ofthis section.
Comparing Table 1 with Table 2, there are two clear differences between our results and thoseof Rotello and Heit (2009): our participants rejected Identity and Inclusion arguments with necessary more often than with plausible, and they accepted contingent arguments at much higher
rates. These factors contributed to lower d ′ in both conditions. Both differences can be glossed
as less strong responses in our experiment than previous experiments. The previous lack of difference between modal frames for Identity and Inclusion arguments is plausibly a ceiling effect that
does not occur in our paradigm. The higher acceptance rate for Contingent arguments signals lesscertainty that they should be rejected. Two plausible sources for these differences are the minimal, within-participants manipulation in our paradigm, and the differences in the concreteness ofmaterials. Rotello & Heit used the same predicate “have property X” in all arguments, using thevariable X in the stimuli and instructing participants to treat property X as a novel biological property; following Osherson et al. (1990) we used concrete novel properties. While these differences
12

1.0

may be interesting in themselves, the fact that d ′ differed significantly for “necessary” and “plausible” suggests that our within-participants manipulation successfully captured the core features ofbetween-participants manipulations in previous work. That is, the difference previously reported
can be elicited by a difference of a single modal word, and so appears to be triggered by semantic
properties of the modal words.
The PTM predicts two general patterns directly related to the issue of non-linear response patterns. First, the choice of modal should have less influence on acceptance rates with valid (Identity& Inclusion) arguments than with contingent arguments, since the former have conditional probability close to 1 (Figure 1a). This is indeed what we find: see Figure 3.

valid

valid

valid

contingent

contingent

0.6

contingent

valid

contingent
0.4

Proportion acceptance

0.8

contingent

valid

valid

valid

contingent

possibleplausible

probable

likelynone

necessary

contingent

certain

Modal frame

Figure 3: Valid (Identity & Inclusion) and Contingent endorsement rates by modal frame with 95%
CIs.
Second, our model predicts the possibility of a continuous gradient in d ′ values (Figure 1b).
The experimental results confirm this prediction as well: see Figure 4. We conclude that there are
not simply two “modes” of human reasoning—deductive/“necessary” and inductive/“plausible”—
but a spectrum that depends on the modal frame used.
4.2.2

Quantitative fit of power-law model

For a model which makes specific quantitative predictions, such as the power-law model describedabove, it is possible to test quantitative predictions against the acceptance rate across modal frames
13

1.4
1.2
1.0
0.4

0.6

0.8

d'

possibleplausible

probable

likelynecessarycertainnone

Modal frame

Figure 4: d ′ by modal frame with 95% bootstrap confidence intervals.
for each argument type. Although we have no independent estimate of the conditional probabilityof each argument, this model allows us to choose a baseline condition and use it to make predictions
about acceptance rates in the other conditions. The form of the power-law model entails that the
choice of baseline condition should not affect predictions, because of the following relationship:pr(Is it M1 that C∣P) = pr(C∣P)αM1= pr(C∣P)(r×αM2 )= pr(Is it M2 that C∣P)r(10)

In accord with this prediction we found no systematic effect on model fit by varying the choice ofbaseline. The primary effect of this choice, then, is that estimates of αM for the other conditions
are up to multiplication by α0 , the parameter which determines the shape of P(θM ) for the baseline
condition in this version of the model.
Here we use the no-modal condition as a baseline. To be clear, we do not believe that acceptance rates in this condition are an estimate of the true conditional probability of the unmodalized
conclusion given the premises: rather, we suggested above that acceptance rates in the no-modal
condition should be much more conservative than we would obtain by sampling from the conditional probability. This expectation was already confirmed by the results in figures 3 and 4, where
the no-modal condition patterned most closely with the “necessary” and “certain” conditions.
Figure 5 plots the endorsement rate of each argument type in the baseline unmodalized con14

dition against the endorsement rate for the same argument type in various modal frames. We
calculated the best-fit αM for each condition. For each graph in Figure 5 the curve determined bythis αM and equation 6 superimposed, with the overall best-fit noise parameter ε = .1. As the R2
values in Figure 5 and Table 2 show, this model captures much of the variance in the data, but not
all of it.

0.5

0.6

0.7

0.8

0.9

1.0

0.5

0.6

0.7

0.8

0.9

0.8
0.6

1.0

0.4

0.5

0.6

0.7

0.8

plausible

possible

probable

0.6

0.7

0.8

acceptance rate: no modal

0.9

1.0

1.0

0.4

0.5

0.6

0.7

0.8

acceptance rate: no modal

0.9

1.0

0.8
0.6
0.4

acceptance rate: probable

α = 0.43α0
R2 = 0.44

0.2

0.8
0.6
0.4

acceptance rate: possible

α = 0.18α0
R2 = 0.46

0.2

0.8
0.6
0.4
0.2

0.5

0.9

1.0

acceptance rate: no modal

1.0

acceptance rate: no modal

1.0

acceptance rate: no modal

α = 0.23α0
R2 = 0.40
0.4

0.4

acceptance rate: necessary

1.0
0.4

α = 1.47α0
R2 = 0.64

0.2

0.8
0.6
0.4

acceptance rate: likely

α = 0.47α0
R2 = 0.62

0.2

0.8
0.6
0.4

acceptance rate: certain

0.2

α = 1.43α0
R2 = 0.62
0.4

acceptance rate: plausible

necessary

1.0

likely1.0

certain

0.4

0.5

0.6

0.7

0.8

0.9

1.0

acceptance rate: no modal

Figure 5: Endorsement rates by argument and modal frame. Curves represent model predictions
using (6) and the best-fit αM for each modal M.
In order to discern whether the remaining variance was due to systematic factors that the powerlaw model does not capture, we performed a split-half reliability test, randomly dividing the data
into equal-sized halves 10,000 times and testing the correlation between the two halves on thesame measure that was used to fit our model: acceptance rate for each argument and modal. The
model correlations were overall very close to the split-half correlations (Table 5).
The power-law model does seem to capture much of the true structure in the data, though a gooddeal of intrinsic noise remains. These results do not support the power-law model unreservedly,
though: it would be easy to devise other instantiations of the PTM with equal or perhaps slightly
better quantitative fit than the power-law model. It is not obvious how to specify competing models
that rival the power-law model in simplicity, however. With this caveat, we believe that the powerlaw model is at least a strong competitor in this realm, and that the remaining noise is plausibly
captured by diversity between participants in (for instance) intuitions about the novel biologicalproperties used in our stimuli and the causal relationships that are relevant in this domain (Kemp& Tenenbaum, 2009).
15

Table 5: Model correlations vs. split-half reliability results.
Modalcertainlikelynecessaryplausible
possible
probable

4.2.3

model R2
.62
.62
.64
.40
.46
.44

mean split-half R2
.70
.58
.71
.34
.34
.53

Monotonicity

Recall that, unlike two-dimensional models, the PTM predicts a consistent order in the endorsement rates of arguments across modal frames (No Reversals). In order to test this prediction we
considered two tests, a permutation test and a model-based simulation. The average value of Spearman’s rank-order correlation for all pairwise comparisons between modal frames in our experimentwas .64. A 10,000-sample permutation test revealed that this correlation was highly significant,p < .0001.
We then tested the monotonicity predictions directly, using the power law model as the representative of the PTM. As described in section 4.2.2, we used equation 6 with ε fitted globallyand αM fitted by condition against the no-modal baseline. We simulated 50,000 data sets with thesame number of observations per argument/modal frame combination that obtained in our experiment. These simulations represent the distribution of results that we would expect to encounterin an experiment of this size if the power-law model is correct. The PTM predicts that, with an
infinite number of observations, any pair of conditions would have a rank-order correlation of
1; but sampling error introduces the likelihood of less-than-perfect rank-order correlations with afinite number of observations. This method of simulation thus allows us to inspect the model’s
predictions about rank order for any specific, finite N.
Figure 6 plots the model’s predictions for all 21 pairwise rank-order correlations by condition
against the empirical correlations. The model’s predictions and the experimental results are highly
correlated (R = .86, p < .0001), and the observed correlations fall within the simulated 95% confidence interval in all 21 pairwise comparisons between modal frames. These results suggest thatthe model’s monotonicity predictions fit the experimental results well, providing further support
for our claim that argument strength is based on a single scale which is manipulated by modal
expressions.
In contrast, if the relative strength of arguments were not constant across modal frames wewould expect to see lower rank-order correlations in the experimental data: the points in Figure 6
would tend to be in the lower right-hand quadrant, instead of falling along the x = y line. This result
is not directly incompatible with a two-dimensional model, since such a model could replicate ourmodel’s monotonicity predictions by placing no or very little weight on one of the dimensions.
However, it would be a surprising coincidence if the seven conditions tested here simply happened
to agree on the relative strength of arguments, if the correct model of argument strength is two16

1.0
0.8
0.6
0.4
0.0

0.2

Observed rank-order correlation

0.0

0.2

0.4

0.6

0.8

1.0

Mean simulated rank-order correlation

Figure 6: Model-generated vs. empirical rank-order correlations between argument acceptance
rates in all 21 pairwise comparisons among modal frames.
dimensional and does not enforce this prediction.

5

Discussion and Conclusion

We have shown that non-linearities in responses to Valid and Contingent arguments can be explained by a simple, one-dimensional probabilistic model, the Probability Threshold Model. Nonlinearities are thus are not evidence against a probabilistic account of reasoning. In addition, we
uncovered evidence which supports the PTM over a two-dimensional theory: a spectrum of d ′values depending on modal frame and monotonicity of argument strength across modal frames.
The modal language used in the experiments was the crucial factor in creating different response
patterns, and attention to the formal semantics of epistemic modals led directly to the PTM model.
This conclusion does not rule out the possibility that there is a qualitative distinction between
17inductive and deductive reasoning among those with logical or mathematical training. Neither doesit answer the question of whether deductive and inductive reasoning have different teleologicalor normative profiles (see for example Evans & Over, 2013). However, our results do call into
question previous efforts to show that such a distinction is necessary to account for everyday humanreasoning, suggesting instead that a single process may underlie both.
Several additional phenomena have been used to support a two-criterion account of reasoning; we briefly consider the most important of these, leaving deeper exploration to future work.
Argument length tends to affect plausibility judgments more than necessity judgments (Rotello& Heit, 2009). This is not unexpected in a probabilistic theory, because we expect that adding
premises tends to increase the probability of the conclusion given the premises (Heit, 1998). The
non-linearities predicted by equation 6 lead us to expect that the same change in probability will
have different effects depending on the modal used. However, Rotello and Heit (2009) show that
including additional premises can sometimes decrease the endorsement rate of necessity conclusions. This is unexpected under the simple Bayesian model of reasoning that we have employed inthis paper. We speculate that this decrease in endorsement is a pragmatic effect—perhaps reflecting the perceived confidence or reliability of the speaker—that could be incorporated into a moresophisticated probabilistic model.
Contrary to our monotonicity results, Rips (2001) found a crossover effect in which participantsin the “necessary” condition were slightly more likely to endorse logically valid arguments thatwere inconsistent with causal knowledge than participants in the “plausible” condition, but thereverse was true for arguments consistent with causal knowledge. Heit & Rotello’s work did not
find any analogous effect, nor did we in the experiment described here. However, it is possiblethat the effect is real and attributable to the many differences in materials and methods between
these experiments. In particular, Rips used content knowledge in a central way in his experiments,
manipulating the causal-consistency of conclusions as well as the mode of reasoning. Here too webelieve the most plausible explanation of this effect will be found in the pragmatics of causalityviolating premises, but we must leave this issue to future work.
Our probabilistic theory is a computational-level description (Marr, 1982), and hence leaves
open the psychological process by which people evaluate arguments. One possibility is that people
are only able to sample possible worlds in accord with the distribution implied by the premises(Vul, Goodman, Griffiths, & Tenenbaum, 2014), and evaluate the truth of the conclusion in thesesampled worlds. If people take several samples and respond “yes” when the conclusion is true ineach sampled world, we recover a power law. If the average number of samples depends on themodal, we recover the specific case of the PTM model described above. For instance, we would
posit that people tend to take more samples to evaluate “necessary” conclusions than “plausible”
conclusions. This process-level implementation predicts that under time pressure, when people cantake fewer samples, “necessary” would begin to look more like “plausible”. Indeed, this is exactly
the finding of Heit and Rotello (2010). This interpretation also suggests points of connection witha probabilistic theory based on mental models (Johnson-Laird, 1994).
In order to focus on the effect of epistemic modal conclusions, and their implication for modesof reasoning, we have adopted a simple approach to the underlying evaluation of arguments: conditional probability given simple, certain premises. This must eventually be elaborated to account for18

more complex forms of reasoning, for instance those involving uncertain and conditional premises.
Some of these extensions may follow from careful treatment of the semantics of words used to form
premises and conclusions, as we have argued for here, while others may require revision to the underlying model of argument evaluation. As an example of the former, our linguistically-oriented
perspective suggests addressing conditional premises by drawing from the large literature on the semantics and pragmatics of indicative, subjunctive, and other types of conditionals. Unfortunately,
there is little consensus, even on such basic questions as whether conditionals have truth-conditionsand (relatedly) whether the Equation between probabilities of conditionals and conditional probabilities is satisfied (Edgington, 1995; Égré & Cozic, to appear; Bennett, 2003). The psychologicalliterature on conditionals is equally rich (Douven & Verbrugge, 2010; Evans, Handley, & Over,
2003; Oaksford & Chater, 2007, among many others). We believe there is synergy between the
linguistic and psychological progress on conditionals, which could be incorporated into the current
approach, but this complex topic will require further research.
Premises that convey, or are interpreted as, uncertain evidence provide an additional challenge.
One approach would be to generalize probabilistic conditioning to directly addresses uncertain evidence, for instance via Jeffrey conditionalization (Jeffrey, 1965); this proposal has been subject toconceptual objections (Pearl, 1988, §2.3.3) and its empirical status remains unclear (Hadjichristidis
et al., 2014; Pfeifer, 2013; Stevenson & Over, 2001; Zhao & Osherson, 2010). A different approach is to condition not on the facts in the premise but on the fact of receiving the premises,
using a hierarchical Bayesian (Pearl, 1988; Tenenbaum et al., 2011) or probabilistic pragmatics(Frank & Goodman, 2012; Goodman & Stuhlmüller, 2013) approach. Crucially, the core of ourresults (such as monotonicity) would apply to any theory that derives a single dimension of strength
for the underlying argument that is then operated on by the epistemic modal frame.
We have illustrated an approach to reasoning based on an overall probabilistic view of inference, together with careful attention to natural language semantics. The immediate result of thisapproach is a model that explains modes of reasoning in a new way, and a novel experimental
methodology designed to test this model. We have provided evidence that there are many gradations between “necessary” and “plausible” when judging conclusions, and that a monotonicity
pattern emerges which supports the Probability Threshold Model over two-dimensional models. In
addition, we found a reasonably good quantitative fit to a simple power-law variant of the PTM. We
believe that the approach of this research—careful attention to language for development of boththeory and experimental methodology—will prove fruitful in clarifying a variety of phenomena
related to human reasoning.

6

Acknowledgements

We thank three anonymous Cognition reviewers, Evan Heit, and four CogSci 2012 reviewers for helpful
discussion of earlier versions. This work was supported by a John S. McDonnell Foundation Scholar Award
and ONR grant N00014-09-1-0124 to Noah D. Goodman.

19

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7

Appendix: Proof of monotonicity

Theorem 1 (No reversals) For any epistemic modals M1 and M2 with meanings as in equation 3 and anyarguments a1 and a2 , if strengthM1 (a1 ) > strengthM1 (a2 ) then strengthM2 (a1 ) ≥ strengthM2 (a2 ).
Proof: Let pi denote P(Conclusion(ai )∣Premises(ai )). By assumption and the definition in equation 5:
strengthM1 (a1 ) − strengthM1 (a2 ) = ∫p1
p2

P(θM1 ) dθM1 > 0.

(11)

y

For a probability density P(t) over [0,1], ∫x P(t)dt ≥ 0 if and only if y ≥ x. Thus p1 ≥ p2 . From this we canconclude thatp1
(12)strengthM2 (a1 ) − strengthM2 (a2 ) = ∫ P(θM2 ) dθM2 ≥ 0.
p2

This theorem applies to pairs of ‘upward’ epistemic modals fitting the schema in equation 3, with a >sign in their lexical entries. The proof extends in an obvious way to pairs of ‘downward’ modals such as
unlikely, implausible, impossible, and doubtful. Pairs of one upward and one downward modal will precisely

22

reverse the monotonicity pattern: if a1 is more certain than a2 then a2 is more improbable than a1 . Finally,
there are modals such as It is between 40% and 60% likely that which are strictly non-monotonic, for clear
semantic reasons.

23