Edited by: David E. Over, Durham University, UK
Reviewed by: Shira Elqayam, De Montfort University, UK; Catarina Dutilh Novaes, University of Groningen, Netherlands
*Correspondence: Andrew J. B. Fugard, Research Department of Clinical, Educational and Health Psychology, University College London, 26 Bedford Way, London WC1H 0AP, UK e-mail:
This article was submitted to Cognitive Science, a section of the journal Frontiers in Psychology.
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This paper argues that the goals people have when reasoning determine their own norms of reasoning. A radical descriptivism which avoids norms never worked for any science; nor can it work for the psychology of reasoning. Norms as we understand them are illustrated with examples from categorical syllogistic reasoning and the “new paradigm” of subjective probabilities. We argue that many formal systems are required for psychology: classical logic, non-monotonic logics, probability logics, relevance logic, and others. One of the hardest challenges is working out what goals reasoners have and choosing and tailoring the appropriate logics to model the norms those goals imply.
Formal systems offer a precise way to characterize people's various reasoning goals. There are many logics for different situations. Some allow reasoners to withdraw conclusions as more information is learned. Others describe the logic of deontic rules about “ought” and “must.” There are logics for relevance and for probabilities. Each logic provides different norms, e.g., for what constitutes a valid logical argument or whether a sentence is true. Elqayam and Evans (
We first remind the reader of the distinction between
Formal systems are instrumental in specifying constitutive and regulative norms, which is in turn necessary in order to understand what participants do in a particular reasoning task. Formal systems are characterized by constitutive norms: doing arithmetic is constituted by complying with the well known constitutive norms of arithmetic. And constitutive norms give rise to regulative norms (Achourioti et al.,
Norms and values are, in the crucial cases for the psychology of reasoning, the least observable features of thinking—the farthest from being fixed by data without system or theory. Participants generally cannot describe their goals in the terms of appropriate systems or theory. Their performances nevertheless can provide evidence for theory-relative normative specification of goals, once a formal analysis is available. In this paper we illustrate these points with experimental examples.
There certainly are abuses of norms to be observed. We propose that these are most evident when any single homogeneous system account of human reasoning is proposed, whether it be classical logic (CL), probability theory, or indeed radical descriptivism with a single description language. As soon as hegemony is proposed, it becomes impossible to study the basis for selection from among multiple systems of reasoning, and it is this requirement to select from multiple possible systems that most clearly dissolves perceived problems of normativity, and connects reasoning goals to instrumental goals. Selecting from multiple possible reasoning goals can be done on instrumental grounds suiting the goals to the problem at hand. We do not believe there is any such thing as “human reasoning” construed as a homogenous system for the simple reason that the demands of different reasoning problems are incompatible, as we illustrate below. The main reasoning goal of this paper it to illustrate this point with examples from past and current practice.
The backdrop to our approach to norms and normativity is the multiple-logics approach to human reasoning outlined in Stenning and van Lambalgen (
Todd et al. (
Stich (
Clearly many authors have proposed many heterogeneities in reasoning, such as what is conventionally meant by the phrase “individual differences” in psychology, individual variation in how “good” some performance is. We are here concerned with a specific type of (in)homogeneity of formal system (e.g., classical logic, probability, nonmonotonic logic, …). Elqayam (
Bounded rationality is a proposal (which we applaud) that rational action has to be understood as governed by the intersection of many systematic constraints. To take one of Simon's examples (Simon,
As yet another orientation point, we recall that more than one logic may operate within an activity. Elsewhere we have proposed that an account of how at least some kinds of argument work, requires an account of how adversarial classical and cooperative nonmonotonic logics have to work together (Stenning,
The plan of this paper is that the first section discusses norms as we understand them, and how they are incompatible with any pure descriptivism. We will concentrate on how participants' very own reasoning goals create variety in
The second section takes the psychological study of categorical syllogistic reasoning as an example to illustrate these points. It argues that the descriptivism prevailing for the last half of the 20th century was exactly what led to a catastrophic inattention to the participants' reasoning goals. It describes the pervasive ambiguity of reasoning experiments for participants, most of whom adopt nonmonotonic reasoning goals where experimenters assumed classical logical ones. It spells out how the contrasting reasoning goals are constituted in the properties of these two logics.
The distinctive properties of classical logic give guidance for design of a context which should improve the chances that we see classical reasoning—in this case a context of dispute. Some results from an ongoing experimental program show how the properties of classical logic which make it suitable for a model of a certain kind of dispute or demonstration are presented as a first indication of the rewards of this kind of empirical program. It provides clear evidence that this context produces more classical reasoning than the conventional draw-a-conclusion task. And perhaps more importantly, it shows how participants have surprising implicit knowledge of some of the peculiarities of classical logic. Psychologically, our goal should be assessing peoples' implicit knowledge and its contextual expression i.e., their implicit logical concepts, rather than their scores on some fixed-context arbitrary task which engenders variable and unspecified goals.
The third section pursues similar themes in the example of probabilistic reasoning. The idea that Bayesianism, or even probability, provides a new homogeneous norm for human reasoning, and for rational action in general, has supplanted the same role that was previously assigned to classical logic in theories of rationality. But probability theory fails to provide reasoning goals at levels comparable to the examples of the previous section. What is argued for is an analogous differentiation of “probability logics” to apply to different reasoning goals, bridging to neighboring logics in a friendly welcoming manner.
Finally we end with some conclusions about the empirical programs that should follow from our arguments for a multiple-logics view of human reasoning, based on the differentiated reasoning goals that this multiplicity affords, together with some comments about the very different view of the relation between logic and psychology which emerges.
The experimental work discussed in the next two sections is intended to emphasis the role of normativity in the psychology of reasoning and should be read as such. It becomes for this reason important that we clarify what we mean by “normativity” and we will do this by reference to Elqayam and Evans (
Logic is often said to be a normative system contrasted with descriptive frameworks that psychologists use. But a logical framework in itself is not descriptive or normative; it is the
The role of normativity in questions such as the one just stated is clearly not of the evaluative kind. Contrast this with the following:
“A normative theory asks evaluative ‘ought’ questions: ‘What
Here the term “normative” takes on almost ethical connotations. To be sure, such questions of prescriptive “goodness” and “badness” are at best outdated and in any case certainly irrelevant to the study of human reasoning. Not so, however, for “right” and “wrong” questions, as witnessed, for example, when participants report “errors” in their own reasoning and correct themselves in the process (we see an example later in how people reason about uncertain conditionals). There is nothing ethically objectionable or evaluative to supposing that humans are not perfect thinking machines and sometimes commit errors or refrain from driving their reasoning all the way to its utmost consequences
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 is
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 grow and how much information and how many computational algorithms they can store and execute.' (Elqayam and Evans,
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. Such limitations are part of what a formal model helps represent. 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 (
‘Some researchers have proposed that we should adopt alternative normative systems such as those based on information, probability, or decision theory (Oaksford and Chater,
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 (
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
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 (
Interestingly, Elqayam and Evans (
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 multiple-logics 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.
The earliest paper on the psychology of the syllogism by Störring (
It was a further half century before Wason's interpretation of his experiment was prominently challenged in psychology (Chater and Oaksford,
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
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
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 incompatible
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.
There are 64 pairs of syllogistic premises which can be enumerated with their valid conclusions. There are a some logical glitches about exactly what ought to be listed as valid
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
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
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,
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
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 (
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
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 true
Stenning and Yule (
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 ¬
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 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
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
Bucciarelli and Johnson-Laird (
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.,
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
So mismatching may serve as a tracer for issues with empty-antecedents. 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
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 model-theoretic 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
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 draw-a-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
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.
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,
1 | 1 | 1 | 1 | 1 | 1 |
1 | 0 | 0 | 0 | 0 | 0 |
0 | 1 | 1 | 0 | 0 | |
0 | 0 | 1 | 0 |
The conditional event,
under the condition that
Hailperin (
If we use only 1s and 0s, this is also the semantics of classical propositional logic. When
If
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. (
Gauffroy and Barrouillet (
The “new paradigm” is often presented as providing the semantics for the conditional as illustrated by ‘the Equation’:
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. (
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. (
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 (
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 (
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 (
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
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.
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.
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.
1For example, one of the prominent accounts of long-term/working-memory interactions (Anderson,
2For what Elqayam and Evans (
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,
4We discuss constitutive and regulative norms and their relations also in Achourioti et al. (
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.
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 (
9The diagrammatic system is described in more detail in the reference above and also in Stenning and Oberlander (