Most of the reluctance to engage seriously with normative considerations comes from an understanding of norms as "external" to one's reasoning, that is, as set by someone other than the participant herself (often researchers). Objections to normativity disappear as soon as attention shifts to norms that are constitutive of one's own reasoning, meaning that they help define reasoning for what it is4. We do not deny that norms 'set by other people' (social norms) are important. But if it is only such norms that are objectionable the debate has been ill-specified, and the objections to norms should be suitably diluted. A way to trace "internal" norms is to identify the goals that underlie and drive one's reasoning process. Goals are highly complex and not easy to specify as they stem from various sources. They are not observable and they interact with each other in complicated ways. In reasoning experiments, for example, the participant has to decide how to go about solving the task, which depends on the participant's interpretation of what is asked of her, which in turn depends on pragmatic goals influencing natural language processing of instructions, how much is underdetermined by the experimenter's design and so on. But whatever the underlying goals turn out to be, it has to be recognized that they heavily influence the type of reasoning participants engage in. In the next section we discuss concrete examples of how different goals trigger different reasoning processes, and we show this by varying the context in order to generate different types of reasoning (and thereby different reasoning norms) and study the effects of this variation on the experimental data.

With the understanding of normativity that we propose as "internal" and not "external" to reasoning, the discussion of human rationality can be set on new grounds. Consider the following:

What seems to set apart normative rationality from other types of rationality is the "ougthness" involved in normativism. Bounded rationality, for example, is not bounded because it "ought" to be so. Instead, there are just biological limits to how large brains can

2For what Elqayam and Evans (2011) argue against, the term "normativism" seems to us more appropriate than "normativity." This is indeed the term that these authors use, while many of the commentators talk about "normativity." This is not to say that the differences of opinion are merely terminological; it is rather the choice of key terms that is influenced by the theoretical positions adopted.

3The authors seem to take issue with the concept of "error" because it evidences the use of norms: 'While the term "normative" has been dropped, the term "error" has not: A recent book (Stanovich, 2009) presents an extensive discussion of the source of reasoning and decision-making errors, implying norms'. (Elqayam and Evans, 2011), p.242

4We discuss constitutive and regulative norms and their relations also in Achourioti et al. (2011).

grow and how much information and how many computational algorithms they can store and execute.' (Elqayam and Evans, 2011), p. 236

As mentioned above, even this is contentious in the literature: there may be distinct advantages to limited systems, and there is much evidence that human brain-size is under selective pressure from both directions. But we accept that resource bounds are a fact. Resource constraints certainly influence the reasoning that participants engage in; this is one of the reasons that may render classical model theoretic thinking intractable and force naive participants to resort to nonmonotonic example construction through preferred models, that leads to more manageable computational processes. But notice that participants are switching reasoning subgoals, not attempting the same goal with a different tool.  They lie, for example, at the heart of the difference between monotonic and nonmonotonic systems. And justifying one model rather than another is clear evidence of normative status, even if the norms in this case could not be otherwise because of resource bounds. Elqayam and Evans (2011) follow Evans and Over (1996) in setting apart "normative" rationality from "instrumental," "bounded," "ecological" and "evolutionary" rationality. The way we understand normativity, it is integral part of all of these four types of rationality. In fact, most of the present paper discusses norms that are part of so-called "instrumental rationality." Hence, we take issue with remarks as the following:

'Some researchers have proposed that we should adopt alternative normative systems such as those based on information, probability, or decision theory (Oaksford and Chater, 1991, 1998a,b, 2007), while others proposed that at least some forms of rationality need not necessarily require a normative system at all (e.g. Evans, 1993, 2002; Evans and Over, 1996; Gigerenzer and Selten, 2001). By this position, organisms are rational if they act in such a manner as to achieve personal goals, and such rationality need not involve any normative rule following.' (Elqayam and Evans, 2011), p.234

The message here is that achieving personal goals need not involve normative rule following. It must be clear by now that we take reasoning goals to be intrinsically normative in that they play a big role in the choice of one reasoning mode rather than another (without claiming that some conscious decision-making process of selection takes place, or that they are necessarily constituted as such in "rules"). Pragmatic goals of relevance, for example, are essentially normative when in some contexts they exclude the interpretation of a natural language "or" as the classical logic disjunction, ∨. Just as with the selection task, examination has to reveal these hidden normative systems behind, for example, ecological rationality. Martignon and Krauss (2003) argue that Gigerenzer's heuristics require Bayesian methods for their population with content in context. And Martignon et al. (in preparation) give an account of this same process based on nonmonotonic logic. Ecological rationality is up to its ears in normativity.

We have so far proposed an understanding of normativity as applying to the use of formal systems rather than attaching to the systems themselves and as involving questions of correctness that do not have evaluative connotations but refer to norms which are internal to human reasoning and constitutive of it. To clarify these points even further, we now discuss the status of competence theories and the "is-ought" fallacy which normative approaches are said to commit. Here is an interesting quote:

'. . . arbitrating between competing normative systems is both crucial and far from easy. This is where the difference between normative and competence theories becomes critical. Competence theories are descriptive and can hence be supported by descriptive evidence. In contrast, can one support normative theory with descriptive evidence? Can one infer the *ought* from the *is*?' (p. 240)

We do not agree that competence theories can be supported by descriptive evidence without normative considerations. It is especially competence theories that have to see beyond the data in order to account for the discrepancy between theory and observation. And at the same time it is a truism that the further one moves away from observable data the more difficult it becomes to actually test the theory. So how is it possible at once to model competence and stay as close as possible to actual performance? Competence theories have constitutive norms, and these norms generate regulative norms once their reasoning is embedded in action. Our examples in the next sections show how the various constitutive norms participants adopt for syllogistic and probabilistic reasoning (competence theories) generate regulative norms once embedded in actual reasoning. A proper understanding of the data depends on the choice of logical norm.

Elqayam and Evans (2011) argue that much of the experimental cognitive research is liable to the "is-ought" fallacy (or naturalistic fallacy as it is often called by philosophers). However, in order for this transition from "is" to "ought" to make sense, "is" and "ought" must be clearly separated, and we show in this paper that descriptive and normative matters cannot be so neatly set apart. A purely descriptive approach is simply unattainable, since what the participants "do" already depends on the theoretical framework within which one performs the observation and this theoretical framework must take into account the reasoning goals at hand, the latter clearly creating normative demands. The dependence of description on formal theory is clearly seen when incompatible descriptions match the same data; when, as we discuss, for instance, the same answer to a reasoning task could be generated by reasoning processes that are as different as monotonic and nonmonotonic logics.

Interestingly, Elqayam and Evans (2011) take the "is-ought" fallacy to be especially triggered in cases where more than one theory matches the data, which then lends support to descriptive theories in their approach5 . But we believe that it is precisely the need to select among equally matching theories that proves descriptivism to be impossible, on the one hand, and what saves the psychologist from the homogeneity trap, on the other. There we think, is the real danger when studying human reasoning without making explicit the norms and goals involved; namely, the

5It must be clear by now that we do not subscribe to a distinction of formal systems into normative and descriptive; it is rather the use we put these systems to in accounting for human reasoning that can be labeled as such.

idea that a single theory can play the role of setting the basis, descriptive or normative, over which to design and assess all experimental work.

Having to arbitrate between formal models is not in itself a problem we should want to eliminate, but it becomes such a problem if it means having to choose between theories that claim to explain human reasoning as a whole. This is where a multiplelogics approach as advocated here offers an improvement in the way formal models are used: in order to account for differences between participants' reasoning within a particular task, we ask ourselves how we can modify the task so that these differences become apparent. This we find the most interesting experimental challenge, which relies, however, on being open to different formalizations sensitive to participants' underlying norms and goals. Formalizing involves representation of reasoning norms (which are goal-sensitive) as much as empirical engagement. And here is where a single descriptive framework, even if that were possible, is bound to fail: it offers no way to account for pervasive participant differences flowing from different goals, if all one is allowed to do is to "describe" participants' micro-behavior.

## **3. THE SYLLOGISM AS ILLUSTRATION**

#### **3.1. REASONING GOALS AS NORMS EMBODIED IN FORMAL SYSTEMS**

The earliest paper on the psychology of the syllogism by Störring (1908) does not address the relation between logic and psychology at all, but employing great logical and psychological insight gets on with describing a small number of participants' responses to syllogistic problems. It identifies Aristotle's *ekthesis* as a good guide to participants' reasoning processes. This itself is remarkable, coming so soon after the "divorce" of logic and psychology, and the establishment of the latter as experimental science. By mid-century, Wason (1968) argues strongly against the very idea that logic bears any useful relation to human reasoning, claiming to demonstrate this fact experimentally with Piaget's theory as his target.

It was a further half century before Wason's interpretation of his experiment was prominently challenged in psychology (Chater and Oaksford, 1999; Stenning and van Lambalgen, 2001; Evans, 2002; Stenning and van Lambalgen, 2004) (but see also Wetherick, 1970) by showing how it rested on the assumption that classical logic had to be the goal of participants' supposedly failed reasoning in Wason's Task, for any of his arguments for irrationality to succeed. But it behooves someone so vehement that logic contributes nothing to understanding human reasoning to perhaps find out what constitutes a logic. This simultaneous coupling of explicit denial of the relevance of classical logic, with its under-the-counter adoption as *the* criterion of correct reasoning, stems directly from an avoidance of the issue of participants' goals in reasoning, and this in turn is a direct result of the suppression of formal specifications of reasoning goals, in favor of a proposed descriptivism treating "human reasoning" as an activity with a homogeneous goal. Wherever descriptivism is espoused we find tacit appeal to homogenous normativism.

As we shall see in our example of the syllogism, it is a difficult experimental question to even specify what empirical evidence is required to distinguish between monotonic and nonmonotonic reasoning in the syllogistic fragment. It has been assumed that merely instructing different reasoning criteria is sufficient to discriminate. The empirical problems of discriminating these goals has been largely ignored or denied, and their neglect stems directly from conflict of this difficulty of observation with the descriptivism which we lament. Once a formal specification of an alternative interpretation of the task is available, it is possible to launch a genuine empirical exploration of what participants may be trying to do.

It is not difficult to see why a multiple-logics stance defuses accusations of prescriptive normativism. As soon as there is explicitly acknowledged plurality, then the need for specification of appropriateness conditions for the different logics is clear for all to see. Fortunately, multiplicity brings with it the materials for an answer. Why is classical logic a good model for adversarial reasoning such as the settlement of dispute? Well, it is bivalent, admitting no intermediate truth values. It is extensional, which means the relevant questions of meaning are easily identified, if not necessarily decided, in agreeing premises. It is truth functional, with similar consequences—no hidden meanings can obscure the connection intended by an intensional conditional. It reasons from identified premises with fixed interpretations. Wandering premises are not good for dispute resolution. But above all, its concept of validity requires the preservation of truth in conclusions from true premises under *all* assignments of truth values.

Why is Logic Programming a good logic for cooperative reasoning about the effect on our preferred model of knowledge rich interpretation of new information? Well, the knowledge-base of conditionals corresponds to the long term regularities in the environment, along with the numerous exceptions to these regularities. Working memory holds the representation of the current preferred model of the focal situation (the "closed world"). The closure of the world is made possible by the restriction of expression which allows the rapid settlement of whether a particular proposition can be derived from the large knowledge base. And so on. Even these partial descriptions of the differences between the logics are enough to explain for many contexts whether classical or a nonmonotonic logic is appropriate. The norm can be seen to be appropriate to the goal. It is when human reasoning is assumed to be logically homogeneous, lack of adequate justification is inevitable. For example, there is a pervasive though not universal view in the psychology of reasoning that monotonic and nonmonotonic logics are two ways of "doing the same thing," where the nonmonotonic logic is seen as a poor man's approximation to classical logic. For example, Mental Models theory correctly asserts that to achieve classical reasoning, participants should consider all models of the premises in syllogistic reasoning. But when it is clear that they mostly actually only consider one model, this is considered a performance error (forgetfulness): not a symptom of nonmonotonic goals to identify a preferred model. This is accompanied by separate experimental demonstrations that participants *can* successfully search for counterexample models when explicitly instructed to do so, in a quite different task. This is taken as supporting that indeed the failure to look for them in solving syllogisms is a performance error. At no point is it questioned whether the participants' goal is different in these two tasks. Just because people can do counterexample reasoning sometimes, does not mean that this is always their goal.

The LP machinery may often operate below awareness; this does not mean that the participant who adopted the goal that it performs does not "have" the goals under which it operates. And plurality is absolutely required for other reasons. There is no way that any logic can provide a model of both dispute and exposition because the logical properties listed above are incompatible6 .

From these arguments it follows that pure descriptivism is impossible in situations where both CL and LP are live options for participants' interpretation (most laboratory reasoning tasks) because choice of logic, and with it reasoning goals, is required for interpretation of the data. There is no alternative to seeking evidence for which goals the participant has adopted (usually inexplicitly). Merely varying the instructions is not an adequate tool for discovery.

#### **3.2. DESCRIPTIVIST APPROACHES TO THE SYLLOGISM CANNOT DISCRIMINATE THESE GOALS**

There are 64 pairs of syllogistic premises which can be enumerated with their valid conclusions.  The conventional task for studying "syllogistic reasoning" is defined by the goal of "getting these answers" to the question "What follows from these premises?" For example, if the premises are *All A are B. All B are C* then *All A are C* is a valid conclusion. So participants who answer with this conclusion score a point. This is OK as far as it goes as an denationalization, but if it is all we can offer, then it makes the syllogism an uninteresting pursuit for the researcher and participant alike. Who says these ones are valid? So it is generally further assumed by the experimenter that these right answers are given by classical logic—was not Aristotle, the author of the first logical theory of syllogisms, thereby the inventor of classical logic?—but pure descriptivism is already out the window. CL has constitutive norms, and with them its users and uses acquire regulative norms.

Troubles compound. These participants have been selected for not knowing explicitly what the syllogism, or classical logic, are. It is true that they know the natural language of the premises, and it is easy to suppose that this determines the reasoning goal. But it is the *discourse* that they have trouble understanding out of context. And they often complain about the bizarreness of the discourse in ways that make one think they in fact adopt a goal quite different to the one the experimenter stipulates. For example, given *Some A are B. Some C are B* they frequently complain that "it doesn't tell me whether the Bs are the same or different." This complaint makes no sense if the premises are understood "classically." Classically it is absolutely clear that they could be either the same or different unless the quantifiers force them to be related, and in this case they "obviously" do not. Yet about 60% of participants claim that there is a valid conclusion here8 On a "story-understanding" LP interpretation, they are of course right that the discourse is "defective" and there are ways of fixing it so that there are valid conclusions based on preferred models—several ways.

So we do not yet know what the participants' goals are at any level beyond assuming they are to please the experimenter, who has not been good enough to divulge his goals in a way that the participant can interpret them. Just saying "I want what logically follows" or "what must be true" is not helpful, since "logically" has many meanings in the vernacular ("reason carefully" is often a good gloss), and any participants who have taken intro logic have been weeded out. "Logically" also has many technical meanings. In LP, a conclusion *must* be true (in the current context) if it follows in the current context from the preferred model. The psychological effects of this kind of emphatic instruction are congruent with the idea that participants take a little more care with whatever goals they happen to have.

Why should we care? What clarification of the goals of the participants would make the syllogism more interesting? We should care about the syllogism because it is a suitable microcosm for seeking the psychological foundations of classical logical reasoning, if any, and that is interesting because classical logic is a crucial mathematical model of dispute or demonstration. So we should be interested in how we can characterize reasoning in this task in a way that it will bear some useful relation to reasoning outside this tiny domain, in say first-order classical logic, or even the much smaller, monadic first-order logic. This would be interesting. Tasks are not themselves interesting if there is no way of connecting them outside the laboratory or across domains. Small fragments are good for satisfying the exigencies of experiment, but they are of little interest in themselves. A good fragment generalizes—and for that one needs to know the goals (and norms) of the participant. There are also significant practical educational gains in understanding exactly why it is that participants have trouble differentiating the discourses of two logics. These problems are close to well known problems of mathematics education in distinguishing generation of examples from that of proofs (Stenning, 2002, chapter 5).

The real problem in this example is that there is more than one systematic reasoning goal that participants might adopt in doing the task as set—that is, more than one logic that might apply. The complaint quoted above is one clue here, though there are many others. The complaint is consistent with the idea that participants are adopting what might be called a "story understanding" task: roughly "What is the model of these premises which their author (presumably the experimenter) intends me to understand by them?" In nonmonotonic logics that capture this reasoning process, these are usually referred to as the *preferred* model (Shoham, 1987). This is cooperative nonmonotonic reasoning *to* a unique minimal model (i.e., *one* interpretation of the premises), as opposed to the adversarial monotonic reasoning from an interpretation, to conclusions true in *all possible models*, that classical logic specifies.

6Logicians produce "embedding theorems" which prove that one logic can be "embedded" within another, often when the two look rather incompatible. It does not follow that the more encompassing logic is an appropriate cognitive model for the encompassed systems' cognitive applications.

7These "glitches" turn out to be at the heart of some of the psychological issues about CL: more below.

8Percentage responses here and following are taken from the metanalysis by Khemlani and Johnson-Laird (2012).

The proposal that cooperative communication worked through the contruction by speaker and hearer of what is now known as a "preferred model" appeared in Stenning (1975) and was condensed in Stenning (1978). Nonmontonic logic was new (McCarthy, 1980), and preferred models had to wait several more years (Shoham, 1987), but what was proposed informally was a direct route to cooperation for psychological process accounts (rather than an indirect Gricean pragmatics founded on adversarial classical logic). Stenning and Yule (1997) showed how subtle is the empirical discrimination of reasoning in classical logic and reasoning in nonmonotonic logic in the microcosms of the syllogism. The "Source-Founding Model" described there is a "shell" for capturing syllogistic reasoning processes, and it demonstrated that adopting a "guess the intended model" reasoning goal could actually yield all and only valid classical logical conclusions if the right model (roughly the "weakest") was chosen, without any conceptual change to a new logic. The interesting psychological conceptual problems are about bald conceptual differences, but are actually difficult to resolve experimentally because the syllogism is so inexpressive. There is considerable evidence that most of the success participants achieve in syllogistic reasoning is achieved by preferred model construction. This is an example of the central importance of the empirical study of goals to the psychology of reasoning. Evans (2002) picks up the point about monotonic and nonmonotonic goals and about interpretation, but suggests no empirical approach other than variation in narrow instructions (rather than tasks) which Stenning and Yule (1997) showed to be inadequate.

It is an immediate consequence that merely observing scores on the 64 syllogisms under different instructions in the conventional draw-a-conclusion task, will not tell us what logic a participant is reasoning with. We have to address the logical concepts that they have (for example, attitudes to conditionals with empty antecedents—more presently) and with them their processes of reasoning. We beg the reader's patience with some details which are important for understanding the role distinct goals (embodying distinct norms) play. We will use the diagrammatic methods this reference uses, though it also supplies analogous sentential ones. So for example, the syllogism *All A are B. Some C are not B* is represented by **Figure 1**.

In the final diagram, the single cross marks an element which is C but not A or B, which must exist in any model where the premises are true9. The choice of preferred models in the diagrams of each premise, combines with this construction of all consistent sub-regions, and with the rules for retaining or deleting the crosses, to ensure the result that any remaining cross represents an arbitrary individual with the properties defined by its subregion. The surprise is that this individual classically must exist if the premises are true. That is, the rules for choosing the nonmonotonically "preferred" model can conspire, in this tiny fragment of classical logic, to choose a model for the premises

which has to exist in any situation where premises are true i.e., is a classically valid conclusion. This is of course not to say that participants who adopt a generally nonmonotonic goal for the task will automatically adopt the particular procedures required for getting classically valid preferred models: there are many parameterizations of the tweaking of nonmonotonic strategy. Informally, participants have to prefer the "weakest" model.

Stenning and Yule (1997) also provides a sentential algorithm which mirrors this graphical algorithm, as well as a "Source-Founding method" which is an abstract algorithm which captures what is in common between nonmontonic and classical methods. It shows the equivalence of the model manipulations in the diagrams with Aristotle's *ekthesis*. So it will be impossible to empirically distinguish participants' with classical norms from those with these "correctly tweaked" nonmonotonic reasoning norms by merely inspecting input premises and output conclusions. Yet identifying these norms is just what we argued psychology has to do to establish what implicit grasp of classical logic its participants have.

But help lies at hand. What has happened, in our nonmonotonic alternative method, to all those paradoxical properties of classical logic that bother every introductory logic student so much? For example, the paradoxes of material implication, whereby, from ¬*p* it follows that *p* → *q*; and from *q* it also follows that *p* → *q*. Or, for a related example, the conclusion that the King of France has been bald since the Revolution because there has been no King of France?: the problem of existential presuppositions. Besides, if the nonmonotonic tweaks get the classical answers, who needs to put up with these crises of classical logic?

So what is the psychological bottom line? The psychological half-way line, is that who needs classical logic is anyone who wants to go beyond the syllogism into the vastly more expressive first-order logic, and needs this still important model of

9The diagrammatic system is described in more detail in the reference above and also in Stenning and Oberlander (1995), e.g., Figure 2. In the variant used here, existential presuppositions are made for universals, because that assumption is commonplace in the psychology literature. Below we see that it is not clearly the right assumption when the task context changes to dispute.

demonstration and rational dispute (e.g., for mathematics, science, politics or the law). An experimenter might be tempted to the conclusion that this was just a bad fragment to pick, and progress to the psychological study of first-order or at least monadic first-order logic. There are formidable obstacles on that path, and no one has ventured down it far. But there is an alternative strategy within the syllogism. How can we get data richer than input-output pairings of premise-pairs and conclusions? If the conventional psychological task of presenting a pair of premises and asking whether any, and which of, the eight conclusions follows, brings forth nonmonotonic norms (albeit sometimes refined ones) from most participants, then perhaps what is needed is a new task and task context (dispute perhaps)? And what about getting participants to perform not just inferences, but also *demonstrations* of those inferences (by producing counterexamples)? This would provide evidence beyond input-output functions.

What are the quintessential features of classical reasoning that we should focus on in the data? The clues are in the paradoxes, though it requires some digging to unearth them. We are claiming, as is commonplace in traditional logical discussion, that classical logic is a model of dispute. What does this mean? Its concept of validity is that valid conclusions must be true in all models of the premises. What this means is that there must be no counterexamples (or "countermodels"). So classical logical demonstration is a doubly negative affair. One has to search for the *absence* of *counter*examples, and what is more, search exhaustively. A dispute starts from agreed and fixed premises, considers all situations in which these are all true, and wants to be certain that inference introduces no falsehood. The paradoxes of material implication immediately disappear. If *p* is false, then *p* ⊃ *q* cannot be false (its truth-table reveals that it can only be false if both *p* is true and *q* is false. (And truth tables is all there is to truth-functions). And the same if *q* is true. So given that *p* is false or *q* is true, we cannot introduce falsehood to true premises by concluding *q* from *p* ⊃ *q*. Everything follows from the nature of this kind of dispute, in which the premises must be isolated from other knowledge because they must be explicitly agreed, and in which no shifting of interpretation can be hidden in implications, or indeed in predicates. This latter is ensured by extensional and truth-functional interpretation. The "paradoxes" are thus seen as paradoxical only from the vantage point of nonmonotonic reasoning (our usual vantage point), whose norms of informativeness they violate. In dispute, proof and demonstration, the last thing one wants is the informativeness of new information smuggled in. And if you are engaged in telling a story, *failing* to introduce new information in each addition to the story will invoke incomprehension in your audience. Tautologies do little for the plot. This contrast is what we mean by each logic having its own discourse, and these two are incompatible.

Bucciarelli and Johnson-Laird (1999) earlier presented counterexample construction as an explicitly instructed task using syllogisms, though with a different partly graphical presentation of situations. Their purposes were to refute the claims of Polk and Newell (1995) that in the conventional draw-a-conclusion task, participants do not search for counterexamples, as mental models theory claimed that they understood that they *should*: 'If people are unable to refute conclusions in this way, then Polk and Newell (1995) are certainly correct in arguing that refutations play little or no role in syllogistic reasoning' (Bucciarelli and Johnson-Laird, 1999, page 270). Whilst their investigations of explicit countermodeling do, like ours, establish that participants can, when instructed, find countermodels above chance, they certainly do not counter Polk and Newell's claim that participants do not routinely do this in the conventional task on which mental models theory is based. Other evidence for Polk and Newell's skepticism now abounds (e.g., Newstead et al., 1999). But nowhere do any of these authors explicitly consider whether the participants' *goals* of reasoning in countermovement diverge from their *goals* of reasoning in the conventional task, even less whether they exemplify two different logics. At this stage, Mental Models theory was seen by its practitioners as the "fundamental human reasoning mechanism." Another example of our dictum that it is exactly where homogeneity of reasoning is proposed, that normativism goes off the rails.

Searching for an absence of counterexamples then, is the primitive model-theoretic method of proof in the syllogism classically interpreted. The whole notion of a counterexample to be most natural, and best distinguished from an exception, needs a context of dispute. How do we stage one of those in the lab? Well, we tried the following (Achourioti and Stenning, in preparation). A nefarious character called Harry-the-Snake is at the fairground offering bets on syllogistic conclusions. You always have the choice of refusing the bets Harry offers, but if you think the conclusion he proposes does not follow from his premises (i.e., is invalid), then you should choose to bet against him. If you do so choose, then you must also construct a counterexample to his conclusion. Evidently we also have to explain to participants what we mean by a counterexample (a situation which makes both premises true and the conclusion false); what we mean by a situation (some entities specified as with or without each of the three properties A, B and C; and how to construct and record a counterexample. (In fact we use contentful material that does not affect likelihoods of truth of premises). Two features of this situation are that Harry-the-Snake is absolutely not to be trusted, and that it is adversarial—he is trying to empty your wallet. Another is that you, the participant, have chosen to dispute the claim Harry has made. You do not have to ask yourself "What if I thought this did not follow?" It has a vividness and a directness which may be important. Our selection of 32 syllogisms (unlike Bucciarelli and Johnson-Laird's) was designed to concentrate on the "no valid conclusion" problems which are at the core of understanding CL, and to allow analysis of the "mismatching" of positive and negative middle terms.

Our most general prediction was an increased accuracy at detecting non-valid conclusions. In the conventional task this is extremely low (37%): highly significantly worse than chance: in the new task it is 74%, significantly better than chance, and valid problems are 66% correct, which is also above chance. Valid problems are now harder, but the task now focusses the participant on the task intended. We also made some more specific predictions about a particular class of syllogisms which we call "mismatched," in which the B-term is positive in one premise and negative (i.e., predicate-negated) in the other. Mismatching middle-term double-existential problems (e.g., *Some B are A, Some C are not-B*) "obviously" do not have single-element models, and so no valid conclusions. Compare a corresponding matched case *Some B are A, Some C are B* which yields as a unification model the singleelement: (ABC). The most popular conclusion is *Some C are A*, drawn by 39% of participants. Note that this unification model is *not* a countermodel of this conclusion. With the mismatched example above, one cannot get a 1-element model. This difference between matched and mismatched double-existential problems and their most popular conclusions is systematic, as we describe below.

One might suppose that absence of valid conclusions is a general property of mismatching syllogisms because of the unification barrier to 1-element models, until one thinks about what happens if the first premise was instead *All B are A*. This universal premise would be satisfied by a single element model (such as *A not-B C*). But only if the negated B term is accepted as making the universal premise true by making its antecedent empty. That is, by the very same model which countermodels the existential case. Here is one place where the connection between CL's "paradoxes" and matching/mismatching shows up. Participants accepting the empty antecedent conditional as true can produce this one-element model.

So mismatching may serve as a tracer for issues with emptyantecedents. To find 1-element models for these mismatched problems requires accepting empty-antecedent conditionals as true. Now comes the question, do any of these syllogisms have valid conclusions? They can have 1-element models if one accepts empty antecedent conditionals, but are these models ones that establish valid conclusions? This model does not establish a valid conclusion anymore than the model (ABC) establishes a conclusion for *Some A are B. Some B are C*. In fact the problem does have a different valid conclusion *Some A are not C*. In summary, these mismatched problems provide a way to gain information about participants' intuitive grasp of empty-antecedent conditionals. And accepting empty-antecedent conditionals as true is a special case of accepting the paradoxes of material implication—the essential example of CL's "weirdness"—in the context of dispute. This is what we mean by looking for its "weirdnesses" as being the best evidence of implicit grasp of a logic. CL is weird in disputes; only from the non monotonic perspective, even for "logically naive" subjects.

If a participant has some implicit grasp of the one-element model generalization, and is happy with models satisfying conditionals by making their antecedent empty, then mismatched problems could behave differently than matched in this modeltheoretic search-for-counterexample method: the striking logical feature (empty-antecedent conditionals being true) connects directly to an unexplored psychological feature. Mismatched problems, when we do the analysis, are actually observed to be slightly but significantly *harder* than matched ones in the conventional task of constructing a conclusion. To see how they might behave differently in countermodel search, one also needs to consider what the favorite conclusions are in the conventional task. For our example, the favorite response is *No C are A*. Now, we observe, that the model one gets by unifying the premises is (A not-B C) is immediately a *countermodel* of this popular conclusion (ie. some C *are* A in this model). If we take the matched and the mismatched problems in our experimental sample of 32, each paired with its favorite conclusion (from the meta-analysis), we find all the mismatched problems have this property that the unification model countermodels the favorite (and usually invalid) conclusions; whereas with the matched problems, the unification model is, in each case a *model* of the erroneous but favorite conclusion. This is evidently an empirical psychological generalization (favorite conclusions in a particular task have no logical status), though we clearly need the CL modeltheory to even notice this piece of psychology. We predicted that *when looking for countermodels* (ie. doing CL), mismatched problems should be easier than mismatched ones.

What actually happens when Harry shows up? To cut a long story short, participants experience disputing with Harry-the-Snake as a much more arduous task than the conventional drawa-conclusion task. They slow down by a factor of about three, an observation that already casts doubt on claims that this countermodel search takes place in the conventional task. Countermodel reasoning is hard work. Their overall accuracy of judgment of validity is not hugely increased, but it does not suffer from the extreme asymmetry of the conventional task. Both VC and NVC problems are done at a better than chance level. The control group in our conventional task control group are also much better than the literature average (these are highly selected students), but they are still asymmetrical in their success in the same way with VC easier than NVC problems. So we find the predicted improvement in detecting invalid conclusions, and we find that indeed whereas mismatched problems are somewhat harder than matched ones on the conventional task, they are substantially *easier* in countermodel reasoning in dispute with Harry, and that participants show evidence of accepting empty antecedent conditionals as true in the dispute task.

The pattern of errors in countermodel construction is consistent with a process by which participants first try to construct a premise model, then check to see if it is a countermodel, and if it is not, then adjust it to try to achieve a falsification of the conclusion. The problem appears to be that the adjustment often yields a model that falsifies the conclusion but is no longer a model of the premises. Mismatched models are more accurately countermodeled, and this is because the models that result from the unification of their premises are already countermodels of Harry's proposed conclusions, as illustrated above. This pattern that mismatched problems are actually easier for countermodel construction whereas they are harder in the conventional task strongly suggests that the majority of participants in the conventional task are operating proof-theoretically, probably by the nonmonotonic methods discussed above.

The countermodel construction data provides rich evidence that empty antecedent conditionals can be treated as true in this context. If the data is scored requiring existential presuppositions, most of the models produced for problems with one positive and one negative universal (i.e., no explicit existential premise) are not even models of the premises, let alone countermodels of the conclusion. A final observation that supports this general interpretation of a change of process invoked by dispute with Harry is that the orders of difficulty of problems in the conventional and in the Harry tasks are actually uncorrelated—an extremely strong result in support of the claim that here is the first task in the literature that produces substantial classical reasoning conducted on a classical conceptual basis. But even here, there are still many errors in countermodel reasoning. The usual justification of the conventional task is that the order of the difficulty of problems is systematic and always the same. The first time anyone makes a comparison with a context designed to invoke a different logic, one finds this order of difficulty changes radically.

Clarifying the intended goals of reasoning (norms to adopt) for participants is one of the few ways we have of pursuing the question whether there are contexts in which participants intuitively understand the concepts of a logic. One can imagine the objection that we have told them to do countermodel reasoning and so it is not surprising that they appear to reason classically. But this is a psychologically bizarre idea. It's no use telling these participants to reason in classical logic because they do not explicitly know what that means. They do have some grasp of what a dispute is, and the role of counterexamples therein—the discourse of dispute. We are merely negotiating a common reasoning norm with our participants. If they did not understand these things, the negotiation would not succeed. We doubt it succeeds with all our participants. But we certainly do not instruct them about what to do with empty antecedent conditionals. And sure enough, we see the peculiarities of classical logical reasoning in their performance. This is just what the psychological foundations of classical logic are: an inexplicit intuitive grasp of dispute. These empirical conceptual questions such as "What do participants 'know' about classical logic?" have far more psychological reach than questions about how many syllogisms do participants get "right" in any particular contextualized task where the goals are not understood the same way by participant and experimenter, or across participants.

Participants are, unsurprisingly, not tactically expert. But here at least is the beginning of an empirical program to study this kind of reasoning in contradistinction to various kinds of nonmonotonic reasoning. Although the two may overlap within the syllogism, outside the syllogism they diverge. And even within the syllogism, here is evidence that the two very different reasoning goals are operative in different contexts, and lead to radically different mental processes, each incomprehensible without an understanding of the different logical goals, and of the participants' informal contextual understandings of their logical goals.

## **4. REASONERS' GOALS IN THE NEW PROBABILISTIC PARADIGM**

Classical logic has been found wanting as a complete model of human inference for many reasons, some of which we have already covered. The "new paradigm" of subjective probabilities aspires to become its replacement (Over, 2009; Oaksford and Chater, 2013). A central question has been whether people's interpretation of indicative conditionals, 'if *A*, then *B*', is given by the material conditional *A* ⊃ *B* (see **Table 1** for a reminder of its truth values) or the conditional probability *P*(*B*|*A*). There is evidence that in some circumstances participants do indeed reason that the probability of 'if *A*, then *B*' is given by *P*(*B*|*A*), both when dependencies between antecedent and consequent are expressed in the

**Table 1 | Truth values of the classical logic material conditional (***A* **⊃** *B***), conjunction (***A* **∧** *B***), and semantic values of the conditional event (***B***|***A***) and biconditional event (***B***|***A***) ∧ (***A***|***B***), where 1 denotes "true," 0 denotes "false," and** *u* **denotes "undefined."**

task through joint frequencies about patterned cards (Evans et al., 2003; Oberauer and Wilhelm, 2003) and when dependencies are derived from causal beliefs (Over et al., 2007). These interpretations also extend to conditional bets such as "I bet you 1 Euro that if the chip is square then it is black" (Politzer et al., 2010), a result which is predicted by foundational work on subjective probability by Bruno de Finetti (Milne, 1997, gives an overview).

The conditional event, *B*|*A*, is often defined only for conditional probabilities in terms of the ratio formula,

$$P(B|A) = \frac{P(A \land B)}{P(A)}$$

under the condition that *P*(*A*) *>* 0. Coherence-based probability logic (CPL), proposed as a competence model for how people reason (Pfeifer and Kleiter, 2009), makes this a primitive, *B*|*A*, which is "undefined," "void," or "undetermined" when the antecedent is false, matching how participants often interpret the conditional when reasoning under certainty (Johnson-Laird and Tagart, 1969). Although this interpretation is often called the "defective" conditional, there is a long history of justification suggesting that there is nothing defective about it. CPL derives a semantics for conditional probabilities, providing a bridge between certainty and uncertainty. This explains why people who use a "defective" conditional when reasoning about certainty also reason using conditional probabilities for uncertain conditionals (Evans et al., 2007, show this empirical link): it's the same underlying conditional.

Hailperin (1996) provides a further analysis of this conditional event (he calls it the "suppositional") in terms of a more primitive operator in an extension of classical propositional logic, "don't care" logic. We present this in a some detail here as it shows clearly the relationship with classical logic. Let 1 denote "true," 0 denote "false," and *u* denote "undefined." The ordering on these semantic values is 0 ≤ *u* ≤ 1. This leads to natural *min* and *max* functions for deciding the minimum and maximum of two values which are used to define conjunction and disjunction, respectively. Let *min*(*x, y*) = *z*; then *z* is either the *x* or *y* and chosen such that *z* ≤ *x* and *z* ≤ *y*, i.e., the value is less than or equal to both *x* and *y* according to the ordering above. Let *max*(*x, y*) = *z*; then again *z* is one of the *x* or *y* and *z* ≥ *x* and *z* ≥ *y*, i.e., the value is greater than or equal to both *x* and *y*. Some examples to illustrate: *max*(0*,* 1) = 1 and *min*(0*,* 1) = 0. If *x* and *y* are the same value then the answer is that value for both *min* and *max*. When the *u* value is included then *max*(0*, u*) = *u* (since *u* is greater than or equal to both 0 and itself) and *min*(0*, u*) = 0 (since 0 is less than or equal to both *u* and itself). Finally 1 − *u* = *u* (this is used for defining negation). *U* is a semantic function from formulas to semantic values, i.e., one of 0, 1, or *u*, as follows:

$$\begin{aligned} U(\neg A) &= 1 - U(A) \\ U(A \land B) &= \min(U(A), U(B)) \\ U(A \lor B) &= \max(U(A), U(B)) \end{aligned}$$

If we use only 1s and 0s, this is also the semantics of classical propositional logic. When *u* is included, then the semantics is equivalent to Kleene's strong 3-valued logic (Kleene, 1952) which turns out to be useful for the semantics for logic programming (Fitting, 1985). Hailperin (1996) introduces an additional "don't care" unary connective, , with a semantic value defined:

$$U(\triangle A) = \begin{cases} 0, & \text{if } U(A) = 0\\ u, & \text{otherwise} \end{cases}$$

If *A* is true then *A* evaluates to *u*; otherwise it has the same semantic value as *A*. This allows *B*|*A* to be defined ¬*A* ∨ (*A* ∧ *B*), giving the same semantic values as CPL. Note the similarity with the disjunctive expression of the material conditional, ¬*A* ∨ *B*, which is equivalent to ¬*A* ∨ (*A* ∧ *B*). Both have the same semantic value when the antecedent is true, equivalent to a conjunction. The disjunct highlights the difference when the antecedent is false: for the conditional event the conditional is undefined but for the material conditional it is true. (This is one of many non-classical truth semantics; Baratgin et al. (2013) provide other interesting examples of further logical components which are useful for psychological theorizing.) Individuals and quantifiers are missing from this semantics, which limits its ability to model discourse; for instance it is not clear how to model an interpretation of "most logicians who develop a logic love it."

Returning to the psychology, there are interesting twists to the new paradigm story. It turns out that the experimental data also require us to model a defective biconditional, what Fugard et al. (2011b) named the biconditional event. This is expressed as (*B*|*A*) ∧ (*A*|*B*) (see **Table 1** for its semantics values) and is equivalent to *A* ∧ *B*|*A* ∨ *B*. Developmental studies show that 12 year olds respond mostly with conjunctions, then by age 16 biconditional event interpretations appear before disappearing again in adults (Gauffroy and Barrouillet, 2009). In adults, it is well replicated that nearly half of participants interpret the conditional as a conjunction, *A* ∧ *B*. Shifts of interpretation have also been found within adults: many participants who begin with a conjunction interpretation change that interpretation (without feedback) to a conditional probability (Fugard et al., 2011b; Pfeifer, 2013). Participants occasionally are explicit about this, describing their reasoning about what they think they are supposed to do and changing their goals, occasionally swearing as they do so, a sure sign of norms awry.

Gauffroy and Barrouillet (2009) explain the developmental trend in a revision of mental models theory. Essentially the idea is that more slots of memory are required as one moves from conjunction—produced by heuristic processes immune to strong developmental changes' (p. 274)—through biconditional event, to conditional event. All reasoners are assumed to have the same reasoning goals, they just fail if they have insufficient memory. Fugard et al. (2011b)instead argued that there are two main stages to reasoning about these sorts of conditionals when the dependencies are expressed in the stimulus, for instance as colored cards. First one has to visually perceive the dependencies, which requires attending to all cases. If you are reasoning about new evidence then you first have to examine the evidence. All evidence is initially relevant, even those cases where the antecedent is false, as you can only tell it is false once you have seen it. The developmental trend can be seen as strategic ignorance when all the evidence has been examined: first from no narrowing of hypothetical scope for conjunctions (*A* ∧ *B*), to focusing on only those cases where either antecedent or consequent are true (*A* ∧ *B*|*A* ∨ *B*), finally to only those cases where the consequent is true, (*A* ∧ *B*|*A*) which is equivalent to the conditional event *B*|*A*. Further support for this model is that conjunctions seem to disappear in Experiment 1 by Over et al. (2007) where instead of reading dependencies from the stimulus, they were taken from beliefs, e.g., that "If nurses" salaries are improved then the recruitment of nurses will increase. There is no need to consider evidence when you are asked your opinion. This hypothetical narrowing could be for many reasons. Perhaps there are variations in pragmatic language function which affect the interpretation of what the experimenter wants. Another explanation is that working memory and reasoning processes have competing goals: represent everything that one sees versus reason about top-down goals concerning the present task (Gray et al., 2003). The two could well be related and influence reasoning about goals. People can switch goals for resource reasons.

The "new paradigm" is often presented as providing the semantics for the conditional as illustrated by 'the Equation': *P*('if *A*, then *B*') = *P*(*B*|*A*). But interpretation is required for probabilities too. Fugard et al. (2011a) showed that a relevance pragmatic language effect, well replicated for non-probability problems in the classical logic paradigm, also affects probabilistic theories of conditionals. Consider the following sentence about a card.

If the card shows a 2, then the card shows a 2 or a 4.

In the old binary paradigm, people tend to think this sentence is false (though with the usual individual differences) since the possibility that the card could be a 4 seems irrelevant if you know it is a 2. Fugard et al. (2011a) found that when participants were shown four cards, numbered 1 to 4, and told that one has been chosen at random, many thought the probability of this sentence is 0. Probability logic (with the simple substitution interpretation) predicts that they would say the probability is 1. Given the same cards but instead the sentence

If the card shows a 2, then the card shows an even number,

most participants give the probability 1 which is now consistent with the Equation. The new paradigm of transforming 'if's into conditional events does not predict this different in interpretation. Here, as for much of the psychology of reasoning, there are differences between participants in interpretation and not all reasoners have the goal to take relevance into consideration. Fugard et al. (2011a) found no association between irrelevance aversion and tendency to reason to a conjunction probability, suggesting that the two processes are logically and psychologically distinct.

The problem for the probability story, as the semantics above shows, is that the disjunction in probability logic is the same as the disjunction in classical logic, so this provides a clue for a solution. Schurz (1991) provided an extension of classical logic for interpretations like these: sentence *ϕ* is a *relevant* conclusion from premises if (a) it follows according to classical logic, i.e., *ϕ* holds, and (b) it is possible to replace any of the predicates in *ϕ* with another such that *ϕ* no longer follows. Otherwise *ϕ* is an irrelevant conclusion. Take for instance the inference *x* = 2 *x* = 2 ∨ *x* = 4. Since *x* = 4 can be replaced with any other predicate (e.g., for the synesthetes *red*(*x*)) without affecting validity, the conclusion is irrelevant. However for the inference *x* = 2 *even*(*x*), not all replacements preserve validity, for instance *odd*(*x*) would not, so the conclusion is relevant. Fugard et al. (2011a) propose adding this to the probability semantics.

Reasoners still have goals when they are reasoning about uncertain information. There are competing processes related to working memory and planning, which could explain developmental processes and shifts of interpretation within participants. Goals related to pragmatic language, such as relevance, are also involved in uncertain reasoning. The investigations above highlight the importance of a rich lattice of related logical frameworks. The problems of classical logic have not gone away since, as we have shown, much of classical logic remains in the 3-valued semantics. Rather than only examining whether or not support is found for the probability thesis, instead different norms are needed through which to view the data and explain individual differences. These norms need to bridge back to the overarching goals reasoners have.

We finish this section with a comment on the treatment of this same problem by Bayesian modeling. The probability heuristic model (PHM) of Chater and Oaksford (1999) was one of the first to protest against the idea that classical logic provided the only interpretation of syllogistic performance. A protest with which we evidently agree. This Bayesian model certainly changes the measures of participants accuracy in the task. For the present argument, two observations are relevant. Firstly, PHM is probably best interpreted as a probability-based heuristic *theorem prover* for classical logic. The underlying logic is still in classical logic and even includes first-order logic statements. The truth of the propositions is assessed classically. This means that despite the rejection of the formal model of classical logic, it has not departed very far. PHM does not propose an alternative interpretation of the goal of reasoning as we do here. Secondly, once the Bayesian model is in place, the psychology stops. There is no motivation for seeking other models of other qualitatively different kinds of reasoning, because probability based models are supposed to account for all reasoning. This may be a consequence of at least a whiff of poor prescriptivism here, and bears out the claim we made that this problem is found wherever one framework is seen as sufficient. In contrast, in a multiple-logics approach, contrast between logics is a rich source of insight and guidance as to how to find the relevant psychological evidence. It should be evident from this example that logic can make empirical experimental analysis much richer. Instead of hundreds of experiments on essentially the same design, one gets a vista of empirical questions to explore.

## **5. CONCLUSIONS**

A variety of formal systems, with their different constitutive norms, and their different consequences for the regulative norms of their users, will be required for modeling the different goals of human reasoning. The main goal of the experimental program of psychology of reasoning and decision at this point should be to find contexts in which participants will exhibit their maximum grasp of each system. Exploration can then spread out to investigate how the logics work together in more complex tasks; how participants can generalist from these focal points; and how teaching affects what they can do. If we win our bet on Harry as a good teacher of an explicit grasp of the logical differences between disputes and stories, and we can show the rudiments of classical logic in a good proportion of participants' performances, then that does not mean that CL "won" over nonmonotonic logics such as LP, or over probability logics, or whatever other logics can be shown to have their contexts. It means we know a little more about where to look for classical logic's psychological roots. We can ask how do these cognitive foundations develop, and what individual and social experiences affect them. We can ask how people at different stages of development and education experience the phenomenology of their reasoning. We can ask how best to achieve educational goals of making explicit students' knowledge of logics. And so on.

In many cases, the empirical discriminations between logics are surprisingly hard. Natural languages often do not provide adequate (or indeed any) cues to intended reasoning goals. People are good at recognizing the goals in customary rich social contexts (few mistake a dispute for a story), but the lab removes all these cues, as do many real-world professional contexts. Much effort is currently going onto the issue of what probability theory is good for, but little into where nonmonotonic logics are to be preferred. Deep knowledge of the logical and computational properties of these systems is available outside psychology but often shunned. Formal systems such as logics and probability are still conventionally seen as competing with psychology for explanations of reasoning. A recent prominent example of this attitude (here to probability rather than logic) is Jones and Love (2011).

Bayesian modeling of cognition has undergone a recent rise in prominence, due largely to mathematical advances in specifying and deriving predictions from complex probabilistic models. Much of this research aims to demonstrate that cognitive behavior can be explained from rational principles alone, without recourse to psychological or neurological processes and representations.

Bayesians would dispute whether they claim to explain in rational terms *alone*. We would disagree with many of their "rational explanations." One might certainly feel disappointed if rational explanations were all of psychology. One of the reasons for our detailed examples is to show that logical bases for explanations do not mean they cannot reveal psychological processes. A huge amount of research in a descriptivist style has failed to make the most important empirical distinctions about which interpretations of the tasks are adopted. But having said all this, to challenge the idea that rational explanations are part of psychology is truly extraordinary. What is needed is more attention to norms, and to the way the constitutive norms of formal systems give rise to regulative norms for their use, and above all, on participants' access to these norms of both kinds.

There is no alternative to a psychology of reasoning which has a rich theoretical vocabulary of reasoning norms, which constitute different goals, and a fine nose for finding the contexts of reasoning that call for the goals, based on the norms of the logical models. Descriptivism never worked in any science.

#### **ACKNOWLEDGMENTS**

The authors would like to thank the editors, Shira Elqayam and David Over, and Laura Martignon for their insightful comments which we feel have greatly improved the paper.

#### **REFERENCES**

Evans, J., and Over, D. (1996). *Rationality and Reasoning*. Hove: Psychology Press.

Stich, S. (1990). *The Fragmentation of Reason*. MIT Press.

**Conflict of Interest Statement:** The authors declare that the research was conducted in the absence of any commercial or financial relationships that could be construed as a potential conflict of interest.

*Received: 31 January 2014; accepted: 24 September 2014; published online: 20 October 2014.*

*Citation: Achourioti T, Fugard AJB and Stenning K (2014) The empirical study of norms is just what we are missing. Front. Psychol. 5:1159. doi: 10.3389/fpsyg. 2014.01159*

*This article was submitted to Cognitive Science, a section of the journal Frontiers in Psychology.*

*Copyright © 2014 Achourioti, Fugard and Stenning. This is an open-access article distributed under the terms of the Creative Commons Attribution License (CC BY). The use, distribution or reproduction in other forums is permitted, provided the original author(s) or licensor are credited and that the original publication in this journal is cited, in accordance with accepted academic practice. No use, distribution or reproduction is permitted which does not comply with these terms.*

**Exploring normative models**

# Development and necessary norms of reasoning

# *Henry Markovits\**

*Département de Psychologie, Université du Québec à Montréal, Montréal, QC, Canada*

#### *Edited by:*

*Shira Elqayam, De Montfort University, UK*

#### *Reviewed by:*

*Jonathan St. B. T. Evans, University of Plymouth, UK Andy Fugard, University College London, UK*

#### *\*Correspondence:*

*Henry Markovits, Département de Psychologie, Université du Québec à Montréal, C.P. 8888, Succursale "A", Montréal, QC H3C 3P8, Canada e-mail<EMAIL_ADDRESS>
The question of whether reasoning can, or should, be described by a single normative model is an important one. In the following, I combine epistemological considerations taken from Piaget's notion of genetic epistemology, a hypothesis about the role of reasoning in communication and developmental data to argue that some basic logical principles are in fact highly normative. I argue here that explicit, analytic human reasoning, in contrast to intuitive reasoning, uniformly relies on a form of validity that allows distinguishing between valid and invalid arguments based on the existence of counterexamples to conclusions.

**Keywords: development, reasoning, logic, norms, communication, epistemology**

The question of the potential usefulness of normative models in understanding human reasoning is a complex one, something that underlies some of the more important debates in the psychology of reasoning. Some of the earliest debates about the nature of human reasoning were explicitly framed around the question of whether human reasoning is essentially "logical" (e.g., Henle, 1962). In these debates, the logical position essentially claimed that humans possessed an inferential apparatus that would (mostly) invariably lead to inferences that corresponded to those found in elementary logic textbooks, reprising Boole's view that Boolean logic simply described human reasoning. A more nuanced approach to this question was given by Braine's (1978) theory that claimed that humans possessed certain limited syntactic reasoning procedures that invariable led to "logically correct" inferences (see also Rips, 1983). These inference rules were the product of biological evolution. Finally, Piaget's theory (Inhelder and Piaget, 1958) made a different claim, suggesting that while children went through stages where their reasoning was constrained by physical and concrete parameters, their development led more or less invariably to the stage of formal reasoning, where logical reasoning is the norm. In fact, Piaget explicitly proposed propositional logic (albeit a modified version of this) as a competence model for formal thought.

Unfortunately for these approaches, empirical research has clearly shown that human inferential performance is highly variable (Markovits, 1985; Overton et al., 1987; Cummins et al., 1991). Many studies have shown that when even educated adults are given what appear to be formally identical arguments, they give difference conclusions. Judgments of deductive validity differ as a function of premise content (e.g., Markovits and Vachon, 1990; Thompson, 1994; Cummins, 1995), and in response to factors such as conclusion believability (Evans et al., 1983). There is little surface evidence that the use of classical propositional logic as a consistent basis for inferential reasoning is very wide-spread, even among highly educated populations. One reaction to these studies has been an attempt to reject the idea that human reasoning is logical at all, by suggesting that much of the inferential apparatus

is dominated by biologically based forms of inference. For example, the heuristics described by Tversky and Kahneman (2004) and Gigerenzer and Selten (2001), although differing in many respects provide simple, context-specific forms of rapid inferential reasoning. These heuristics are context dependent, and their use can account for at least some of the variability in human reasoning. However, they do not correspond to a clear model of logic of any kind, although one might suppose (as Gigerenzer explicitly argues) that they are biologically efficient. Similarly, the probabilistic model proposed by Oaksford and Chater (2003, 2007) and Evans et al. (2007) suggests that inferential procedures model the (Bayesian) statistical properties of people's knowledge of their environment. Such models propose that people process relations in a way that explicitly reflects their personal beliefs, which in turn is at least partly determined by real-world knowledge stored in long-term memory (Oaksford and Chater, 2012). Inferences are thus basically probabilistic, and essentially variable, and translate the real nature of people's underlying knowledge. The question of whether reasoning of this kind can be cast in terms of a normative model is open, partly because there is not a strong consensus about the way that probabilistic models function (Elqayam and Evans, 2013; Oaksford and Chater, 2013).

Nonetheless, it is worth making one specific point in this context. Probabilistic models propose that people's inferences are determined by their individual estimations of conclusion likelihood. Since there is no mechanism by which such estimations can be judged as being more or less accurate, a normative model that depends on some external criteria might seem to be impossible to verify. It might, however, be possible to model standard deductive inferences within a Bayesian framework. Deductive reasoning can be seen as an attempt to construct a representation of premises for which there is a shared attempt to maintain some consistent level of internal probability, e.g., a shared belief that the probability of *q*|*p* for a given major premise is close to 1 (Oaksford and Chater, 2012). In this case, it might be possible to use a normative model in order to evaluate the way that people reason in this constrained system. However, since such an exercise is clearly artificial, and does not generally reflect the nature of real world information, norms of this kind will be correspondingly artificial.

The key point is that probabilistic models of inference essentially depend on what must be idiosyncratic representations of real world probabilities, since they depend on information stored in long-term memory. This is quite critical, since it makes Bayesian norms almost by definition undetectable. Bayesian models are used to understand how people can detect environmental regularities, something that is clearly biologically useful, since it allows some level of anticipation of the specific properties of a person's immediate environment (Tenenbaum et al., 2006). However, environments can be variable and individual experience will reflect this variability. Thus, probabilistic models produce by their very nature variable outputs that cannot be compared since this variability reflects variability in inputs. Inferential reasoning can be applied across the whole range of experience and probabilistic approaches to inference must then reflect the wide variety of individual experience. Thus, it could be argued that these approaches suggest that human reasoning cannot, even in principle, be described by a normative model (Elqayam and Evans, 2011).

In the following, I will nonetheless attempt to argue that despite variability and the undoubted influence of many forms of heuristics, human reasoning in its conscious component does indeed depend on a simple normative form of basic logic (which does not necessarily correspond to a specific logical model), for both epistemological and developmental reasons.

### **WHAT IS A NORMATIVE MODEL?**

Before attempting a more specific analysis, it is important to make some initial distinctions. Normative models can be considered in very different perspectives (Elqayam and Evans, 2011). In the following, I will consider that a normative model is a prescriptive description of the *optimal* way that a system should function in order to accomplish its basic goals. It is important to distinguish between such models and descriptive models, which are attempts to describe the actual workings of a given system in a real-life situation. Simple variability is not an indication that a normative model is inapplicable to a given system. However, variability requires showing that the system's functioning tends towards a normative model when conditions are optimized.

There is one further critical part of any analysis of normative models. Normative models are mathematical or logical abstractions that aim to capture the essential functioning of what are necessarily messy and complex systems. Such models are, by definition, the product of human reasoning, since they are the result of people trying to understand the basic parameters of a specific system. The role of normative models in the understanding of human reasoning becomes double-edged, since such models must not only describe the way that people can optimally reason, but importantly these models must also be able to account for the ability of people to construct these models in the first place.

With this in mind, it is useful to note that many normative models have epistemological underpinnings that are essentially based on standard bivalent deductive logic. Given the increasing importance of Bayesian models, as discussed previously, it is particularly useful to note that Bayesian statistics are derived using such logic. In fact, careful analysis of the *arguments* for probabilistic models clearly shows that these are based not on Bayesian inferences, but on classical logical arguments. If human inferences were uniformly Bayesian, then one would expect arguments to be phrased specifically in terms of degree of belief. However, conclusions that explicitly leave open the possibility that alternative theories have a clear probability of being correct are rarely encountered. In other words, it is important to distinguish between the characteristics of the output of a given model, and the epistemological underpinnings of these models. In most fields, the second part of this equation is basically irrelevant. When discussing characteristics of normative models of human reasoning, this becomes fundamental. In fact, one key component of the argument that will be presented is that a minimal normative model for human reasoning is necessary in order to account for the ability to produce normative models in the first place.

#### **DUAL-PROCESS THEORIES**

There is one final distinction that is important in this context. One response to the clear evidence that people's reasoning does not consistently conform to logical norms is the idea that there exist two separable inferential systems. One of these is meant to be a major source of variability in reasoning, while the other has at least the potential to reason more logically. Dual process theories (Sloman, 1996; Stanovich and West, 1998; Evans, 2007) postulate that people have two major inferential systems that interact with each other. Such theories have multiple forms and use different criteria to attempt to distinguish between these two systems. There is, unfortunately, no real consensus as to the characteristics or the definition of these two systems (see Evans and Stanovich, 2013, for a recent discussion). However, roughly speaking, these postulate a basically heuristic form of inference, which we is often referred to as System 1, which is presumed to be an evolutionarily primitive system that makes rapid inferences that are automated, contextual, use surface properties of problems, and rely extensively on stored knowledge. Such inferences are low-cost and do not involve working memory capacity. The second system, which postulates a more analytic form of inference, referred to as System 2, by contrast, is conscious, slow, and relatively costly in its use of working memory.

Although there are variable descriptions of the dual process framework, Evans and Stanovich (2013) suggest that one minimalist approach to this distinction is to suggest that heuristic inferences correspond to rapid, autonomous inferential processes, while analytic inferences are characterized by working memorybased processes that support hypothetical thinking. The latter are particularly characterized by cognitive decoupling, allowing inferences that are not necessarily tied to existing knowledge structures (Stanovich and Toplak, 2012). For our purposes, a key characteristic of System 1 inferences is they are intrinsically variable, since they necessarily reflect the idiosyncratic nature of people's internal representations. Given this intrinsic variability, there is no reason to think that a single normative model would ever be able to capture its properties, at least not within a relatively straightforward model. However, System 2 allows at least

the possibility of hypothetical thinking involving some degree of conscious processing of information. If this basic distinction is reasonably accurate, then the ability to even consider the possibility that normative models could exist must be the result of System 2 processing.

## **DEVELOPMENTAL EPISTEMOLOGY**

With these distinctions in mind, there are two forms of argument for a normative model for System 2 reasoning. The first is an epistemological argument that partly owes its form to Piaget's work on cognitive development. Piaget in fact referred to his field of study as genetic epistemology. The basic argument, which was derived from Kant (see Henle, 1962) can be stated as follows. In order to adequately process the infinite variety of information that is potentially accessible, the human mind requires some basic categories. Kant assumed that the basic categories were a priori, that is, that they are a basic component of the human cognitive system and are essentially biological. Such an essentialist view of the basis of human cognition is actually quite common in many developmental theories. For example, studies examining people's understanding of categories and concepts (Gelman and Markman, 1986), and object permanence (Baillargeon et al., 1985) have indeed claimed that people have biological underpinnings that allow them to consistently extract specific categories or object qualities from complex forms of information. These are determined by the biological niche humans have constructed over evolutionary time. For example, understanding that a physical object retains its basic identity even when it changes shape or disappears would be a critical component of a cognitive system evolved to survive in a world in which there is are constantly moving objects. Similarly, and more in line with our current problem, understanding that objects that move by themselves can be considered to correspond to a category (which we refer to as implicitly living). Objects in such living categories are considered to have shared invisible attributes, which is a very useful way of conceptually dealing with the world in which separating out living from nonliving categories is a vital component, and understanding the specific properties of the latter can be particularly useful. Such a basically biologically based approach has in fact been proposed for human reasoning. As stated previously, Braine's natural logic approach (Braine, 1978) takes just such a stance. There is one problem with this, however. The distinction between living and nonliving categories, and the ability to understand primitive transformational consistencies, such as those required to understand object permanence are found in very young children, which at least suggests empirical support for a biological hypothesis. This is simply not the case for inferential reasoning, and the idea that there is an essential biological basis that corresponds to some form of internal rules of inference appears to be empirically untenable.

Piaget's approach to this problem was both biological and developmental (Piaget, 1971). In line with Kant, he assumed that the human mind did indeed require basic categories in order to adequately process information about an inordinately complex world. However, he did not assume that these categories were biological in origin. Instead, he postulated that biology provided the basic processes that allow systematic cognitive change. He also postulated that such changes were essentially systemic. In other words, he clearly made a distinction between the accumulation of knowledge, which could lead to a piecemeal and unconnected body of knowledge, and the development of the basic categories of mind that allowed people to process such knowledge. This latter can be considered the basis of the epistemology of the mind. In this perspective, the idea of a normative model of the mind can be seen as having the same basic function for cognition as the idea of a universal grammar has for language. More specifically, if human minds had essentially different epistemologies; that is, if they used different basic forms of categorization and reasoning, then the problem of just how people could communicate efficiently would arise.

Piaget added one component to this analysis. He started from empirical results that showed that children's understanding of the world appears to progress through different levels or stages. He early on remarked that young children appear to have a more primitive epistemology than adults that is the underlying basis of their thinking relied on basic categories that were less consistent than those that appear to underlie adult reasoning. For example, young children have variable notions of the basic concept of quantity, being unable to consider that quantity is an invariant property of mass (Piaget et al., 1997). The lack of such invariance makes their thinking inherently unstable. A parent who is faced with a child who does not want to eat a meal because there is too much food, and who mushes up the food into a smaller area, is using this instability to successfully manipulate his or her child. In contrast, adults have no problem understanding invariance of quantity, in fact most consider the questions used to examine this notion to be simply stupid, since the answers are self-evident.

It is this form of basic difference in epistemology that Piaget attempted to describe in his research program, something that has often been lost in the debates over details about the age at which specific abilities appear, etc. This approach thus assumed that there was change and development in such a basic epistemology. However, the course of this development is not really variable, but was meant to mirror both the physical and biological properties that are critical components of the basic cognitive system. Thus, Piaget postulated that there was an invariant developmental sequence by which the very primitive cognitive categories present at birth, combined with whatever innate tendencies might drive early information processing, would gradually transform into the adult version. Piaget also supposed that epistemological development would tend towards the same basic normative model. A critical point is understanding just why this would be true. One reason, which underpins Piaget's basic hypothesis is that epistemological structures are generated by interactions with the physical world. These start by interactions based on action and perception. Over time, and repeated interactions, children develop a logic of actions, which follows a coherent and nearly universal sequence (Piaget,1965). In fact, this same sequence has been observed in primate species and some mammals (Scarr-Salapatek, 1976). Thus, it could already be argued that the end-point of sensori-motor development can be described by a normative model, one that reflects the deep structure of the way that humans structure physical action in the real world.

To make this distinction more explicit, when it is claimed that most 2 year-olds have the same basic epistemology, one that corresponds to a clear normative model, this does not mean that they have learned the same things. Specific learning clearly depends on the concrete environment in which children are raised. Thus a child might learn that cookies are good to eat in one context, and another might learn that candies are good to eat, depending on what is available. However, when trying to get cookies or candies or anything else that is desirable, that is hidden by an obstacle, all children will use the same action logic. Once again, the actual actions can vary (for example, one child might bat the obstacle away, while another might reach for it to move it away), but the logic of the action sequence is the same (action 1 is directed towards the obstacle with the aim of displacing it, action 2 is directed towards the goal). In fact, debates about sensori-motor cognition are mostly about whether sensori-motor logic is developed faster or slower, but there is little debate about the form of this logic. There are thus quite solid grounds for suggesting that development at this level can indeed be described by a normative model (Dasen, 1972).

Piaget's explanation of development considers that more abstract forms of conceptually based cognition are derived by processes of representation and symbolic manipulation of the logic of actions. Thus, for example, basic categorization is derived from the process of perceptually based similarity relations, causal categories are derived from direct causality, etc. More specifically, reasoning reflects the basic structure of conditional action schemas. For example, there are many ways to get a biscuit, thus understanding the uncertainty of such actions directly reflects what children have already learned about the physical world (Byrnes and Overton, 1986). Increasingly abstract forms of reasoning require a long and complex process of representation and restructuring of more abstract concepts, but these still reflect the underlying structure of the physical world. This point of view would thus suggest that the end-point of epistemological development would be essentially the same. Once again, if one examines studies looking at the ability of pre-adolescents to make inferences about the concrete world, empirical results strongly suggest that the same level is attained by most children, although there is a great variation in age. Thus, most children can understand that liquid quantities are conserved over transformations, allowing them to consistently infer that a simple change in container will not alter the quantity.

The same is found with transitive inferences, the logic of which most children understand by adolescence when the content is concrete and clear (Markovits et al., 1995). Once again, similarly to studies of sensori-motor development, debate about the form of such concrete logic is not about the form of such logic, but about whether the corresponding abilities are developed more or less rapidly. Thus, there is also very clear evidence that the logic of pre-adolescents, which allows understanding of many forms of conservation, and basic forms of transitivity, causality, categorization, etc. when these are applied to concrete, perceptible problems consistently described the abilities (and performance) of most pre-adolescents (although age of acquisition is highly variable). In other words, a normative model can be claimed to exist to describe the concrete reasoning of most children by adolescence.

Since development on the more abstract, formal level is derived from reasoning structures developed previously, this suggests that formal reasoning should indeed correspond to a single form of normative model. This is indeed the underlying rationale for Piaget's claim that there is a single normative model for formal thinking.

## **COMMUNICATION AND NORMATIVE REASONING**

Having a shared epistemology is certainly useful in order to allow different members of a species to process variable information in ways that are internally consistent. There is one further argument for the existence of a single normative model of human reasoning. If this normative model is a model of the underlying epistemology of the conscious component of the cognitive system, then the ability to communicate must require sharing the same basic epistemology. In fact, it could be argued that normalization of System 2 reasoning is particularly critical in this context. In order to understand this point, it is useful to consider communication with System 1 (intuitive) reasoning. Although much of just what is involved in such reasoning remains mysterious, we can fruitfully speculate about some aspects of this. Intuitive inferences can reflect (at least) two forms of information. The most critical of these from our point of view is the internal structure of experiences stored in memory. Theories such as probabilistic models of inference (Oaksford and Chater, 2003) assume that intuitive inferences reflect stored knowledge about the world. A useful example is the belief-bias effect, which is the tendency of people to accept conclusions that are judged as believable as being logically valid (Evans et al., 1983). Although experiments examining this effect use conclusions that are highly believable for a large number of people, believability is clearly personal. In addition, there is certainly an emotional component to the force with which believability acts on System 2 processing. One excellent example of this is given by a study by (Klaczynski, 2000). He found that interference with System 2 processing was much stronger when the beliefs that were being examined were related to a domain with very high emotional valence (religion) than when these were related to a domain with lower emotional valence (class). There are increasing numbers of studies that link emotional experiences to inference-making (Blanchette and Richards, 2010). Thus, it is reasonable to assume that Intuitive processing tends to be highly personal, in that it reflects idiosyncratic personal experience, which conditions not only understanding of the underlying structure of experience and events, but is complicated by emotional valences that clearly reflect individual experiences. Intuitive processing is by definition unconscious, and certainly experience shows us that idiosyncratic and emotionally driven inferences do not generate much in the way of metacognitive awareness.

In other words, intuitive processing can clearly serve individual purposes by allowing people to make rapid, low-cost inferences that reflect their past experience. Doing so increases the chances that future behavior will mirror past circumstances, which allows individuals to profit from experience in a very immediate way. If that were the end of the story, there would be no need for any other inferential system. However, there is another component to human behavior, and that is the fact that humans live in social groups. This makes intercommunicability a critical component of any form of reasoning, since group behaviors require reconciling many divergent individual agendas in a way that allows group cohesion. Further complicating this dynamic is that fact that group complexity increases exponentially with numbers of group members. This explosion of social information has been hypothesized to be a major evolutionary driver for human cognition (Dunbar, 1993). Recently, Mercier (Mercier, 2011; Mercier and Sperber, 2011) has proposed a general evolutionary theory for the development of System 2 reasoning abilities that suggest that these abilities have evolved in order to regulate communication in complex social groups.

This perspective views reasoning as a form of argumentation which allows people both to present overt reasons for actions in an attempt to convince others and to allow others to evaluate arguments. While reasoning might have other functions, it is a useful hypothesis to see it as a means for exchanging explicit reasons for action, since this would have the potential to allow groups to make decisions that were more efficient than a simple majority rule would provide. Seeing reasoning as a form of communication, or even as a useful underpinning for communication, makes an even stronger case for the existence of a common epistemological core, since the essence of argumentation requires a sufficiently strong common basis.

Now, there are (at least) two ways that effective communication can be insured. The first involves the use of common biologically based intuitive schemas. In the absence of explicit language or symbolic thought, such schemas characterize the social cohesion of many social animals. There is reasonable evidence that humans also have such implicit social schemas. For example, we have recently shown that humans share an intuition about coalition formation as a function of individual power that is similar to what has been found behaviorally in chimpanzees (Benenson et al., 2009). Another form of implicit inference is that underlying the Gricean view of linguistic communication, in which pragmatic interpretations of language acts are underpinned by common assumptions that are derived from shared experience (Grice, 1981). However, such intuitive schemas can only work well when social behavior is relatively constrained. Human social behavior, while certainly sharing many aspects with more biologically constrained social species, is, however, very flexible. Flexibility, while allowing greater ability to adapt to changing circumstances, has a clear effect on the possibility of ensuring effective communication solely on the basis of shared intuitive schemas. This puts a greater functional burden on overt, language based communication to ensure that social interactions do not degenerate into conflicts based on differing individual intuitions and perceptions. But, in order for such communication to serve this function, it is imperative that there exist a shared epistemology, i.e., that the ideas shared overtly are underpinned by some common basic principles of reasoning. This again provides theoretical weight for the idea that explicit human reasoning should have a normative core.

### **WHAT WOULD A NORMATIVE MODEL OF REASONING LOOK LIKE?**

There are thus reasons related to the basic epistemology of human cognition and to the importance of reasoning to communication that suggest the necessity of a single normative model for explicit human reasoning. There are some basic considerations that can give clues to just what this model entails. The first is an argument from conceptual power. The last couple of hundred years have led to a proliferation of models of logic that have radically different underpinnings. Each of these models is a product of the human mind, individually or in concert with others. One important constraint for a normative model that does indeed correspond to the workings of the explicit, analytic mind is that it should allow the construction of multiple forms of models of logic. In addition, such a model should be readily understandable by most people, exactly because of the premise of communicability that we have claimed previously. Both of these constraints, along with historical and developmental considerations suggest that the normative model of the mind should involve some basic principles that underpin the notion of validity.

Secondly, one of the key components of cognitive development is that change goes towards increased complexity. A good example of this is well-documented in language learning, the phenomenon of over-generalization. Young children are faced with a complex variety of linguistic forms, with many idiosyncratic formulations which have historical roots, but often violate what are more frequent forms. Children's strategies for learning language is to identify (by whatever process this is done) the most frequent pattern, and generalize this as a rule that is used in all occurrences of a given class, even when this involves generating words that have never actually been encountered (Onnis et al., 2002). Young children do the same with cognitive categories, picking out simple rules and extending these to concepts that are not actually instances of these categories. In other words, the developing human mind has a clear strategy, which requires generating simple rules and extending these to a wide variety of instances. It is only through continued interactions and reflection that these initial simple rules are extended to more complex concepts. What this in turn suggests is that if basic analytic reasoning relies on core normative principles, it will take a form that reduces the cognitive load required to reason. Thus, while there are a multitude of logical systems that take into account the true complexity of human experience, this argument suggests that normative principles will be less complex than any of these logics.

In this perspective, developmental studies can provide some very useful information. As we have seen, variability is a key characteristic of the reasoning of adults. However, while such variability is suggestive of a system of thought that has no common epistemology, the addition of an intuitive component of the functioning of the human mind makes variability more readily explicable in terms of the combination of a personal form of intuition which reflects individual experience and explicit, analytic thought that despite some variability has similar underpinnings.

The problem here is to determine exactly what are the critical components of normative thinking. The key to this is to distinguish between a normative model that is determined by its outcome, and one that is described by the nature of the underlying processes. Most studies that have looked at whether people reason normatively have examined outcomes, specifically whether the responses given to inferential problems are the same as whatever norm is being compared to. However, unless it is believed that inferential rules are constructed directly in the mind (as some theories do indeed suggest, e.g., Braine, 1978; Rips, 1983), we can rephrase the question of normality by trying to specify what kinds of underlying analyses must be implied by any system of thought that can in principle produce "correct" answers.

#### **CHILDREN'S UNDERSTANDING OF VALIDITY**

In this perspective, we can distinguish at least two major components of basic logic that are necessary to produce a form of reasoning that can in principle become powerful enough to generate complex normative models. One important use of such a logic is the ability to explicitly examine the consequences of intuitive forms of reasoning in a way that allows people who do not share the same intuition to communicate. One key component of this ability is understanding the distinction between belief and some form of validity. In other words, before people can explicitly examine and compare the consequences of divergent personal experience, they must be able to distinguish, at least in principle, inferences that are derived directly from experience and those derived by some process of "logical reasoning".

If this corresponds to a basic component of human reasoning, then it should be evident, in some form, in children. In fact, there is clear evidence that this distinction is fairly primitive. For example, Moshman and Franks (1986)found a clear developmental trend so that by early adolescence, most children can spontaneously understand the distinction between belief and validity, well before the level of schooling in which this distinction is taught. More strikingly, Morris (2000) found that children as young as 5-years of age, when given an appropriate content can generate this distinction. In other words, understanding the distinction between belief and validity is an early developmental acquisition, a critical one if explicit reasoning is indeed a counterpoint to intuitive reasoning. Of course, this is not really news to parents of young children, who despite the real difficulties of doing so, are nonetheless able to "reason"with children in a way that confronts the child's intuitions with some form of logic.

# **POSSIBILITY AND NECESSITY: COUNTEREXAMPLES IN REASONING**

If children can understand the distinction between validity and belief, the next question is just how validity is determined. This of course goes to the question of just what kind of "logic" is available to children, and how this is related to the logic of adults. I have claimed (Markovits, 1993) that the key component of understanding this question derives from one of Piaget's later works, on the relation between possibility and necessity (Piaget, 1987).

Before examining this, it is useful to make an important distinction underlying Piaget's approach. Piaget proposed standard propositional logic as a competence model for advanced adult reasoning. This has often been interpreted as implying a rule-based form of reasoning, which would invariably lead to standard logical responses. However, this is a mischaracterization. What was specifically proposed was that the underlying epistemology that characterized advanced adult reasoning would allow the ability to generate such responses. The work on possibility and necessity was an attempt to specify the nature of this epistemology. The

basic question that was raised concerns the kinds of factors that can explain how children and adults can conclude that a potential conclusion is necessary. The analysis makes no mention of rules, instead it places such logical conclusions within the more general context of the range of information that can be generated by the reasoner. In this, one critical component of reasoning is the range of possibilities that are processed by children in the context of a given problem. A given inference is necessary if it excludes whatever possibilities are generated by the child at the moment of reasoning. This corresponds to what one could refer to as local necessity, since the actual generality of a conclusion depends critically on the range of possibilities that are generated. Critically, if a child or adult is aware of a possibility that is not excluded, they will reject a given conclusion. This interaction between possibilities and necessity is modulated by the degree of abstraction of the processes used to analyze a given problem.

A similar idea underlies the mental model analysis of reasoning (Johnson-Laird, 2001; Johnson-Laird and Byrne, 2002). Mental model theory considers that people construct internal representations of possible states of the world (models) that characterize the major premise of a given inference. Possible states must be generated by a reasoner, based on combinations of semantic and pragmatic factors (Byrne, 2005). Critically, an inference is considered to be valid if there are no explicit counterexamples in the reasoner's representation. The presence of a counterexample is sufficient to render a putative conclusion invalid. An inferential judgment is thus an on-line consequence of a reasoner's ability to (1) generate a more or less full range of possibilities consistent with premises and (2) determine whether these possibilities contain a counterexample.

One important distinction between the Piagetian analysis and mental model theory is that the latter postulates that internal representations are derived from a semantic analysis of logical connectors, albeit one that is modulated by pragmatics. Since the semantics of logical connectors are assumed to be generally invariant, variation in reasoning performance is accounted for by such individual difference factors as memory capacity. Mental model theory does not have a very clear developmental component, although Barrouillet and colleagues (Barrouillet and Lecas, 1999; Barrouillet et al., 2008) have suggested that working memory limitations can affect children's ability to actually represent the full semantics of logical connectors. Development will necessarily tend towards the same forms of logical reasoning that are determined by the shared semantics of logical connectors. Thus, although mental model theory presents an analysis of reasoning that is consistent with the interaction between possibility and necessity, the developmental component has a very different focus (see Markovits and Barrouillet, 2002 for a version of mental model theory that has a developmental focus that is more consistent with the Piagetian model).

Piaget's work indicated that one of the key factors in development is the ability to generate increasingly abstract forms of possibilities (Gauffroy and Barrouillet, 2011). Young children start by considering possibilities that are more concrete and related to the situational factors for a given problem context. With development, these possibilities become more extended and abstract, and less tied to situational constraints. However, by 6- or 7-years

of age (Markovits, 2000), children can understand that a conclusion that eliminates all other possibilities is necessary. Of course, since this judgment of necessity depends on the range of possibilities that are eliminated when this conclusion is made, it is subject to revision even if exactly the same form of reasoning is applied, since the generation of possibilities can vary from one moment to the next. In other words, reasoning at this level is sequentially defeasible (Pollock, 1987), that is the same person can arrive at a different conclusion for the same inference, simply because the domain of possibilities accessed during reasoning might change, seeMarkovits (1985) andByrne (1989)for examples of defeasible reasoning in adults. However, the *processes* by which judgments of validity are made are in principle general. More importantly, it is possible to overtly challenge any such inference by comparing possibilities. This in turn is realistic only because judgments of validity depend, not on an accumulation of data, but on the presence or absence of a counter-example to a given conclusion.

There is one further point that can be made in this context. As noted previously, it has been argued that most relations are, in real life probabilistic (Oaksford and Chater, 2007). If the point of reasoning was to faithfully reflect the real characteristics of the real world, then one would expect reasoning to be essentially probabilistic. However, the analysis of "logical" reasoning that I use here does not generate conclusions such as "it is probably true that...." It simply allows judgments of validity or not, in other words, it allows concluding that something is certain or that it is not. Why should this form of judgment be useful when trying to reasoning about phenomena that are inherently probabilistic? There are some good reasons for this, but the chief one is that it is cognitively much simpler. While a probabilistic judgment or any other kind of intuitive judgment that relies on stored knowledge about the world can conceivably be made very rapidly by associative processes, explicitly communicating the basis for such judgments would in theory require explicitly processing a large quantity of information. In fact, in many cases most people are unable to do such explicit processing. In this, case comparing conclusions would simply result in people fighting over personal beliefs. Even if we assume that a reasoner does (remarkably) have conscious access to the relevant information, and is consciously aware that conclusion X is probable because of data set Y, the problem of how someone else, whose personal data base does not contain the same information, can process this conclusion.

A useful, because somewhat more real example, can be taken from research on aggressive behavior. One of the underlying mechanisms of such behavior concerns the expectation (mental model) that a given social interaction will have an aggressive outcome (Dodge et al., 1990). Children who develop a strong expectation that interactions will be aggressive, tend to infer that most actions will have an aggressive outcome. Now, imagine that two children with differing expectations are interacting, and attempting to determine the outcome of a given course of action. In order to make a reasonable comparison, both children would have to recount the many kinds of interactions that form the basis of their expectations. Then each would have to attempt to integrate the other's information into their own internal data base, a daunting task at best, and one that would require not only a great deal

more cognitive resources that most children possess, but a great deal of time. The fact that such discussions do not really take place might well account for the difficulty in lowering aggressive behavior (Dishion et al., 1999). In contrast, since many arguments consist in deciding on taking a given course of action or not, a simple counterexample strategy would be quick and useful, precisely because of its inherent limitations. If this is indeed characteristic of basic reasoning, then once again, developmental studies should provide evidence.

# **THE DEVELOPMENT OF ELEMENTARY CONDITIONAL REASONING**

We can illustrate this evidence by examining simple conditional (if–then) reasoning. Conditional reasoning is one of the key components of propositional logic, and is certainly one, if not the most, frequently studies forms of reasoning. Piaget considered that conditional reasoning was one of the key competencies of formal operational thinking. The basic approach that we are taking here suggests that understanding the development of conditional reasoning will rest upon the basic understanding of the interplay between possibility and necessity and the way that this can be used increasingly abstract context. The key component of this suggestion is identification of just what the range of possibilities are required in order to make "logical" conditional inferences, and whether children are able to process these in a way that allows them to make simple judgments of validity.

To simplify this discussion, we focus on two conditional inferences. The Modus ponens (MP) inference involves reasoning that "If P then Q, P is true." The affirmation of the consequent (AC) inference is "If P then Q. Q is true." The logical conclusion to a MP inference is that "Q is true". Now studies with adults have shown that the ability to conclude that the MP inference is valid depends on the extent to which disabling conditions (Cummins et al., 1991; Cummins, 1995) are incorporated into people's representations of premises. A disabling condition (disabler) is a condition that is associated with a given P then Q premise, one that could invalid the link between antecedent and consequent terms. The strongest form of disabler is given when reasoning with empirically false premises, for which the true relation is a disabler. Very young children have little problem in considering MP inferences to be valid for a variety of contents. The one major exception is reasoning with empirically false premises, for which young children have great difficulties in accepting the MP inference. However, interventions that allow them to inhibit retrieval of the true disabler, such as embedding premises into a fantasy context (Dias and Harris, 1988, 1990) dramatically improve their ability to accept this inference. With increasing age, children are more able to spontaneously inhibit potential disablers when given standard logical instructions (Markovits and Vachon, 1989).

In contrast, there is no logical conclusion to an AC inference, although typically, many people respond to AC inferences by concluding that "P is true" (Cummins et al., 1991; Thompson, 1994). The reason for the theoretical lack of any logical conclusion is that a true conditional allows for the possibility of what have been referred to as alternative antecedents, that is cases where alternate conditionals "If A then Q" are, or might be true. Considering such alternatives (which comprise the domain of possibilities in

this case), allows producing a counterexample to what is the usual AC conclusion. In fact, there is strong empirical evidence that the availability of such alternative antecedents in memory is a strong determinant of the conclusions given to AC inferences (Markovits and Vachon, 1990; Cummins et al., 1991; Thompson, 1994; Quinn and Markovits, 1998). In other words, when people can access such alternatives when reasoning, they tend to reject the certain conclusion for AC inferences.

Analysis of both the MP and the AC inference shows that, in both cases, children's judgments of validity are determined by the kind of information that is incorporated into the representation of premises, during the online process of reasoning. Although the specific information varies (disablers and/or alternative antecedents), the basic dynamic is the same. Even young children will accept a conclusion as valid if their representation of the premises does not include a potential counterexample, otherwise the conclusion will be considered to be invalid. Finally, although considering these two inferences is particularly useful, it is worth briefly examining the two other inferences that define conditional logic. Studies have generally shown that responses to the denial of the antecedent (DA) inference responds to the presence or the absence of alternative antecedents in the same way as AC inferences. The developmental pattern is also very similar. Similarly, the modus tollens (MT) inference corresponds somewhat similarly to the MP inference to the presence of disablers. However, the use of negation in both these inferences somewhat complicates analysis. Understanding the basic notions underlying the understanding of logical validity is more easily presented by concentrating on the MP and the AC inferences.

I have in fact argued that the best elementary definition of "logical" reasoning is the ability to understand the certainty of the MP inference and to simultaneously understand the uncertainty of the AC inference (Markovits and Lortie Forgues, 2011). As I have stated, the key factor in this form of reasoning is the kind of information that is incorporated into children's representations of premises. Constructing a representation of premises that does not include potential disablers, but that does include potential alternative antecedents requires inhibiting the former while retrieving the latter. The tension between these two contradictory cognitive processes can explain why even educated adults find it difficult to reason logically in the limited sense that we use (see Markovits, 2014). However, there is strong empirical evidence that children as young as 6–7-years of age are indeed able to understand both the certainty of the MP inference and the uncertainty of the AC inference, when the content of conditional premises allow for very ready access to alternatives. For example, when reasoning with propositions such as "If an animal is a dog, then it has four legs. An animal has four legs. Is it a dog?," young children will readily reject this inference (Markovits, 2000). More tellingly, they will do so by explicitly citing the existence of a counterexample to the implied conclusion ("cats have four legs"). When given problems where the alternatives are explicitly presented, even younger children will reject the AC inference while accepting the MP inference (Markovits and Thompson, 2008).

In other words, very young children are able to reason "logically" with simple, direct forms of propositional logic when the content allows appropriate inhibition and retrieval of relevant

information. Given the strong tendency of children to accept inferences, the most striking part of these results is that they are capable of processing the fact that there are alternatives to a putative conclusion, and using this as a basis for rejecting a putative conclusion. I would then argue that this is exactly the nature of the kind of logical reasoning that is the "norm" for most people, which involves constructing a simple representation of an inference and deciding on the validity of a potential conclusion based on the existence of a possible counterexample.

Does this model imply that children or even educated adults will consistently give the standard propositional logic answer to (for example) all AC inferences. Not at all. Since the exact response to any inference depends on the range of counterexamples that are generated while reasoning, which is in turn related to retrieval and inhibitory processes, variability is expected. This is an important point to make. The increasing ability to generate potential counterexamples can in fact produce variable responses. For example, children who are more efficient in simply producing potential causal alternatives tend to reject the MP inference more frequently (Janveau-Brennan and Markovits, 1999). This corresponds to the notion of local necessity, in which validity is the result of application of the basic principle related to counterexample detection to an online process of possibility generation. Thus, this model makes no presuppositions about the nature of the underlying rules or algorithms. Such a basic epistemology is thus potentially consistent with a wide variety of reasoning systems, including for example the incomplete logic proposed by Gauffroy and Barrouillet (2009).

The important part of this conception of reasoning is the idea that people can recognize the presence or the absence of a counterexample and use this to make a judgment of validity that is internally consistent. Critically, especially for its usefulness in communication, people can adjust their inferences based on externally communicated counterexamples. Evidence shows that both children (Rumain et al., 1983) and adults (Markovits, 1985) will revise conclusions to AC inferences when given additional information about potential alternative antecedents. Similarly, explicit information suggesting disablers results in adults rejecting the MP inference (Byrne, 1989). The effect of providing explicit information about the existence of potential counterexamples is to reliably increase rates of rejection of conclusions that were previously accepted. In other words, both children and adults are able to revise conclusions that are generated by their internal inferential processes simply by considering alternatives furnished by other people. In a similar vein, Klaczynski (2001) has shown that children are able to recognize logical arguments that rely on the presence of counterexamples as being superior to arguments that rely on other kinds of processes.

In this context, basic logical reasoning can be seen as the ability to use a counterexample to invalidate an otherwise acceptable conclusion. Empirical results show that this ability is present at a very early age. Critically, there is also evidence that both children and adults can use externally presented counterexamples to modify their own inferences. Thus logical reasoning has both an internal function that allows for judgments of validity and an external function that provides a basis for modifying personal judgments by considering specific arguments that present potential counterexamples.

If such an ability does indeed represent the kernel of logical reasoning, then development should reflect not any change in this basic form of logic, but should be tied to the increasing ability to generate and inhibit potential counterexamples in an increasingly abstract way. I have indeed argued that this is a good description of the development of simple conditional reasoning between early childhood and later adolescence (Markovits, 2013). To summarize this pattern, young children can consistently reject the AC premise and accept the MP premise for premises that use if–then relations that link classes and properties (Markovits, 2000). The ability to do the same with causal conditionals ("If cause P then effect Q") is found only in pre-adolescents (Janveau-Brennan and Markovits, 1999). Reasoning this way with contrary-to-fact premises is a later development (Markovits and Vachon, 1989). Finally, understanding the certainty of the MP inference and the uncertainty of the AC inference with completely abstract premises is a much later development, and is only found with educated adults (Venet and Markovits, 2001; Markovits and Lortie Forgues, 2011). In other words, the developmental pattern is completely consistent with the idea that the basic mechanism underlying simple logical reasoning is available to quite young children, and that subsequent

and abstract forms of content. Importantly, extending basic reasoning to abstract content (Markovits and Lortie Forgues, 2011) gives the ability to reason logically even with content that has no concrete referents or for which the referents are unintuitive. Such a form of reasoning is the historical basis for the various models of logic that have been constructed, and which correspond to a wide variety of different forms of reasoning. In other words, the kind of abstract reasoning that develops from the primitive base found in young children can become powerful enough to allow people to construct normative models of many types.

development extends this same process to increasingly complex

#### **REFERENCES**

**Conflict of Interest Statement:** The author declares that the research was conducted in the absence of any commercial or financial relationships that could be construed as a potential conflict of interest.

*Received: 01 February 2014; accepted: 05 May 2014; published online: 23 May 2014. Citation: Markovits H (2014) Development and necessary norms of reasoning. Front. Psychol. 5:488. doi: 10.3389/fpsyg.2014.00488*

*This article was submitted to Cognitive Science, a section of the journal Frontiers in Psychology.*

*Copyright © 2014 Markovits. This is an open-access article distributed under the terms of the Creative Commons Attribution License (CC BY). The use, distribution or reproduction in other forums is permitted, provided the original author(s) or licensor are credited and that the original publication in this journal is cited, in accordance with accepted academic practice. No use, distribution or reproduction is permitted which does not comply with these terms.*

**OPINION ARTICLE** published: 30 January 2014 doi: 10.3389/fpsyg.2014.00049

# In search for a standard of rationality

#### *Emmanuel M. Pothos 1 \* and Jerome R. Busemeyer 2*

*1 Department of Psychology, City University London, London, UK*

*2 Department of Psychological and Brain Sciences, Indiana University, Bloomington, IN, USA*

*\*Correspondence<EMAIL_ADDRESS>
#### *Edited by:*

*Shira Elqayam, De Montfort University, UK*

**Keywords: decision making, classical probability theory, quantum probability theory, rationality, conjunction fallacy**

The debate on human rationality goes to the heart of fundamental questions about human existence. When can we say that a decision is correct? What is the basis for the achievements of the human intellect? What is the most important cognitive distinction between humans and non-human organisms? Proposals of rationality have an interesting status as psychological theories. They are not quite theories of decision making in practice—such theories are referred to as descriptive theories, to imply that they describe what goes on. Rather, proposals of rationality are normative theories, to imply theories of how people ought to reason, if they seek decision outcomes, which are deemed to be correct, on the basis of some absolute standard (here, we are simplifying a complex debate; arguments have been expressed against a distinction between normative and descriptive rationality as we make above, e.g., Elqayam and Evans, 2011, 2013). Of course, a normative theory must be partly a descriptive theory as well, since it is assumed that humans can, in principle, sometimes, reason on the basis of the normative prescription (they may just not do so, in typical situations, perhaps due to process demands or time or other constraints).

Currently, the dominant approach to normative rationality is based on classical probability (CP) theory. This approach was established after a major shift in conceptual thinking about rationality. Before (effectively since antiquity), it was believed that the standard for correctness in decision making was classical logic. But this position came under intense scrutiny, with experimental results showing that naïve participants, even in simple tasks, would not reason in a way consistent with classical logic (Wason, 1960). One reaction to such results was to develop dual theories of reasoning, which would, broadly speaking, involve a rational component and a heuristics one (cf. Sloman, 1996). However, a priori arguments emerged against any kind of role of logic in human practical decision making (i.e., decision making exempting mathematical/ scientific etc.; Chater and Oaksford, 1993). By contrast, a theory of rationality (and decision making) based on probability theory seemed (and seems) to align itself closely with the intuition we have about what it means for decision making to be successful. Such a theory is about the use of available information from the environment, so as to optimally predict the probability of future events (Anderson, 1991; Oaksford and Chater, 2007; Tenenbaum et al., 2011). The fact that the classical prescription appears to be consistent with human cognition in many cases (e.g., Griffiths et al., 2010, and the above references) also corroborates the psychological relevance of CP theory.