The probability of conditionals: A review.

A major hypothesis about conditionals is the Equation in which the probability of a conditional equals the corresponding conditional probability: p(if A then C) = p(C|A). Probabilistic theories often treat it as axiomatic, whereas it follows from the meanings of conditionals in the theory of mental models. In this theory, intuitive models (system 1) do not represent what is false, and so produce errors in estimates of p(if A then C), yielding instead p(A & C). Deliberative models (system 2) are normative, and yield the proportion of cases of A in which C holds, i.e., the Equation. Intuitive estimates of the probability of a conditional about unique events: If covid-19 disappears in the USA, then Biden will run for a second term, together with those of each of its clauses, are liable to yield joint probability distributions that sum to over 100%. The error, which is inconsistent with the probability calculus, is massive when participants estimate the joint probabilities of conditionals with each of the different possibilities to which they refer. This result and others under review corroborate the model theory.

Reasoning about properties: A computational theory.

We present a theory of how people reason about properties. Such inferences have been studied since Aristotle's invention of Western logic. But, no previous psychological theory gives an adequate account of them, and most theories do not go beyond syllogistic inferences, such as: All the bankers are architects; Some of the chefs are bankers; What follows? The present theory postulates that such assertions establish relations between properties, which mental models represent in corresponding relations between sets of entities. The theory combines the construction of models with innovative heuristics that scan them to draw conclusions. It explains the processes that can generate a conclusion from premises, decide if a given conclusion is necessary or possible, assess its probability, and evaluate the consistency of a set of assertions. A computer program implementing the theory embodies an intuitive system 1 and a deliberative system 2, and it copes with quantifiers such as more than half the architects. It fit data from over 200 different sorts of inference, including those about the properties of individuals, the properties of a set of individuals, and the properties of several such sets in syllogisms. Another innovation is that the program accounts for differences in reasoning from one individual to another, and from one group of individuals to another: Some tend to reason intuitively but some go beyond intuitions to search for alternative models. The theory extends to inferences about disjunctions of properties, about relations rather than properties, and about the properties of properties. (PsycInfo Database Record (c) 2022 APA, all rights reserved).

Recursion in programs, thought, and language.

This article presents a theory of recursion in thinking and language. In the logic of computability, a function maps one or more sets to another, and it can have a recursive definition that is semi-circular, i.e., referring in part to the function itself. Any function that is computable - and many are not - can be computed in an infinite number of distinct programs. Some of these programs are semi-circular too, but they needn't be, because repeated loops of instructions can compute any recursive function. Our theory aims to explain how naive individuals devise informal programs in natural language, and is itself implemented in a computer program that creates programs. Participants in our experiments spontaneously simulate loops of instructions in kinematic mental models. They rely on such loops to compute recursive functions for rearranging the order of cars in trains on a track with a siding. Kolmogorov complexity predicts the relative difficulty of abducing such programs - for easy rearrangements, such as reversing the order of the cars, to difficult ones, such as splitting a train in two and interleaving the two resulting halves (equivalent to a faro shuffle). This rearrangement uses both the siding and part of the track as working memories, shuffling cars between them, and so it relies on the power of a linear-bounded computer. Linguistic evidence implies that this power is more than necessary to compose the meanings of sentences in natural language from those of their grammatical constituents.

Beliefs and emotions about social conventions.

Deontic assertions concern what people should and shouldn't do. One sort concern moral principles, such as: People should care for the environment; and another sort concern social conventions, such as: People should knock before entering an office. The present research examined such deontic assertions and their corresponding factual assertions, such as: People care for the environment and People knock before entering an office. Experiment 1 showed a correlation between emotions and beliefs for both sorts of deontic assertion, but not for their factual counterparts in which the word "should" had been deleted (as in the preceding examples). Experiment 2 showed that changing the pleasantness of participants' emotions about social conventions changed their strength of belief in them. Experiment 3 showed conversely that changing the participants' strength of belief in social conventions changed the pleasantness of their emotions about them. These results corroborate the mental model theory of deontic assertions, which postulates that emotions and beliefs about deontics depend on parallel systems that interact with one another.

Causal conflicts produce domino effects.

Inconsistent beliefs call for revision-but which of them should individuals revise? A long-standing view is that they should make minimal changes that restore consistency. An alternative view is that their primary task is to explain how the inconsistency arose. Hence, they are likely to violate minimalism in two ways: they should infer more information than is strictly necessary to establish consistency and they should reject more information than is strictly necessary to establish consistency. Previous studies corroborated the first effect: reasoners use causal simulations to build explanations that resolve inconsistencies. Here, we show that the second effect is true too: they use causal simulations to reject more information than is strictly necessary to establish consistency. When they abandon a cause, the effects of the cause topple like dominos: Reasoners tend to deny the occurrence of each subsequent event in the chain. Four studies corroborated this prediction.

Reasoning about Durations.

A set of assertions is consistent provided they can all be true at the same time. Naive individuals could prove consistency using the formal rules of a logical calculus, but it calls for them to fail to prove the negation of one assertion from the remainder in the set. An alternative procedure is for them to use an intuitive system (System 1) to construct a mental model of all the assertions. The task should be easy in this case. However, some sets of consistent assertions have no intuitive models and call for a deliberative system (System 2) to construct an alternative model. Formal rules and mental models therefore make different predictions. We report three experiments that tested their respective merits. The participants assessed the consistency of temporal descriptions based on statements using "during" and "before." They were more accurate for consistent problems with intuitive models than for those that called for deliberative models. There was no robust difference in accuracy between consistent and inconsistent problems. The results therefore corroborated the model theory.

Possibilities and the parallel meanings of factual and counterfactual conditionals.

The mental model theory postulates that the meanings of conditionals are based on possibilities. Indicative conditionals-such as "If he is injured tomorrow, then he will take some leave"-have a factual interpretation that can be paraphrased as It is possible, and remains so, that he is injured tomorrow, and in that case certain that he takes some leave. Subjunctive conditionals, such as, "If he were injured tomorrow, then he would take some leave," have a prefactual interpretation that has the same paraphrase. But when context makes clear that his injury will not occur, the subjunctive has a counterfactual paraphrase, with the first clause: It was once possible, but does not remain so, that he will be injured tomorrow. Three experiments corroborated these predictions for participants' selections of paraphrases in their native Spanish, for epistemic and deontic conditionals, for those referring to past and to future events, and for those with then clauses referring to what may or must happen. These results are contrary to normal modal logics. They are also contrary to theories based on probabilities, which are inapplicable to deontic conditionals, such as, "If you have a ticket, then you must enter the show."

Reasoning about epistemic possibilities.

Reasoning about epistemic possibilities - those based on knowledge - is fundamental in daily life. It is formalized in modal logics, of which there are infinitely many, based on the semantics of 'possible worlds'. An alternative psychological theory postulates that possibilities (and probabilities) in daily life are based on the human ability to construct mental models of finite alternatives, which can each be realized in an indefinite number of different ways. This account leads to three main predictions that diverge from normal modal logics. First, the assertion of an epistemic possibility, A, presupposes the possibility of not-A, in default of knowledge to the contrary. Second, reasoners condense multiple possibilities into one, contravening modal logics, but reducing the load on working memory, e.g.: When knowledge shows that this condensation would be inconsistent, reasoners resist it. Epistemic possibilities are akin to non-numerical probabilities, forming a scale that runs from impossible to certain. In contrast, epistemic necessities state a necessary condition for some other proposition, e.g.: "It is necessary that it rains tomorrow for the plants to survive." The article reports five experiments corroborating these predictions. Their results challenge current conceptions of human reasoning.

If and or: Real and counterfactual possibilities in their truth and probability.

The theory of mental models postulates that conditionals and disjunctions refer to possibilities, real or counterfactual. Factual conditionals, for example, "If there's an apple, there's a pear," parallel counterfactual ones, for example, "If there had been an apple, there would have been a pear." A similar parallel underlies disjunctions. Individuals estimate the probabilities of conditionals by adjusting the probability of their then-clauses according to the effects of their if-clauses, and the probabilities of disjunctions by a rough average of the probabilities of their disjuncts. Hence, the theory predicts that estimates of the joint probabilities of these assertions with each of the four cases in their partitions will be grossly subadditive, summing to over 100%. Five experiments corroborated these predictions. Factual conditionals and disjunctions were judged true in the same cases as their counterfactual equivalents, and the sum of their joint probabilities with cases in the partition ranged from 240% to 270% (Experiments 1a, 1b). When participants were told these probabilities should not sum to more than 100%, estimates of the probability of A and C, as the model theory predicts, were higher for factual than counterfactual conditionals, whereas estimates of the probability of not-A and not-C had the opposite difference (Experiment 1c). Judgments of truth or falsity distinguished between conditionals that were certain and those that might have counterexamples (Experiment 2a), whereas judgments of the likelihood of truth reflected the probabilities of counterexamples (Experiment 2b). We discuss implications for alternative theories based on standard logic, suppositions, probabilistic logic, and causal Bayes networks. (PsycInfo Database Record (c) 2020 APA, all rights reserved).

The Analytic Truth and Falsity of Disjunctions.

Disjunctive inferences are difficult. According to the theory of mental models, it is because of the alternative possibilities to which disjunctions refer. Three experiments corroborated further predictions of the mental model theory. Participants judged that disjunctions, such as Either this year is a leap year or it is a common year are true. Given a disjunction such as Either A or B, they tended to evaluate the four cases in its 'partition': A and B, A and not-B, not-A and B, not-A and not-B, as 'possible' or 'impossible' in ways that bore out the difference between inclusive disjunctions ('or both') and exclusive disjunctions ('but not both'). Knowledge usually concerns what is true, and so when participants judge that a disjunction is false, or contingent, and evaluate the cases in its partition, they depend on inferences that yield predictable errors. They tended to judge that disjunctions, such as follows: Either the food is cold or else it is tepid, but not both, are true, though in fact they could be false. They tended to infer 'mirror-image' evaluations that yield the same possibilities for false disjunctions as those for true disjunctions. The article considers the implications of these results for alternative theories based on classical logic or on the probability calculus.

Possibilities as the foundation of reasoning.

Reasoning about possibilities is fundamental in daily life. Yet, it has been little studied in psychology. We present a psychological theory in which it is the foundation of human reasoning. The theory explains how possibilities have distinct interpretations (deontic, epistemic, and alethic), how people represent them in models, and how these models yield inferences. Key principles are that the semantics of possibilities are the same finitary alternatives underlying probabilities, that speech acts can create obligations inexpressible as probabilities, that compound assertions - conditionals and disjunctions - refer to conjunctions of possibilities holding in default of knowledge to the contrary, and that mental models condense multiple consistent possibilities into one. The theory is incompatible with all normal modal logics and with probabilistic logic. Yet, experiments have corroborated its predictions. The article discusses its precursors, rivals, and potentials.

Emotions and beliefs about morality can change one another.

A dual-process theory postulates that belief and emotions about moral assertions can affect one another. The present study corroborated this prediction. Experiments 1, 2 and 3 showed that the pleasantness of a moral assertion - from loathing it to loving it - correlated with how strongly individuals believed it, i.e., its subjective probability. But, despite repeated testing, this relation did not occur for factual assertions. To create the correlation, it sufficed to change factual assertions, such as, "Advanced countries are democracies," into moral assertions, "Advanced countries should be democracies". Two further experiments corroborated the two-way causal relations for moral assertions. Experiment 4 showed that recall of pleasant memories about moral assertions increased their believability, and that the recall of unpleasant memories had the opposite effect. Experiment 5 showed that the creation of reasons to believe moral assertions increased the pleasantness of the emotions they evoked, and that the creation of reasons to disbelieve moral assertions had the opposite effect. Hence, emotions can change beliefs about moral assertions; and reasons can change emotions about moral assertions. We discuss the implications of these results for alternative theories of morality.

How to infer possibilities: A reply to Oaksford et al. (2018).

Individuals draw conclusions about possibilities from assertions that make no explicit reference to them. The model theory postulates that assertions such as disjunctions refer to possibilities. Hence, a disjunction of the sort, A or B or both, where A and B are sensible clauses, yields mental models of an exhaustive conjunction of possibly A, possibly B, and possibly both A and B, which each hold in default of information to the contrary. Oaksford, Over, and Cruz (this issue) are critical of the model theory and defend a probabilistic approach to reasoning. In this reply, we deal with their three main claims: (a) Our results concern only the periphery of their probabilistic theory. We show that they refute their theory insofar as it applies to possibilities. (b) The model theory leads to logical absurdities. We rebut this criticism as it applies to the model theory in Hinterecker, Knauff, and Johnson-Laird (2016), and explain why standard modal logics, which concern possibilities, do not set appropriate norms for inferences about them. (c) The algorithm for reasoning based on models needs a normative theory. In fact, it has such a theory, but the demand for "a specification of a sound, complete, and decidable normative system" is chimerical for everyday reasoning. (PsycINFO Database Record (c) 2019 APA, all rights reserved).

The Truth of Conditional Assertions.

Given a basic conditional of the form, If A then C, individuals usually list three cases as possible: A and C, not-A and not-C, not-A and C. This result corroborates the theory of mental models. By contrast, individuals often judge that the conditional is true only in the case of A and C, and that cases of not-A are irrelevant to its truth or falsity. This result corroborates other theories of conditionals. To resolve the discrepancy, we devised two new tasks: the "collective" truth task, in which participants judged whether sets of assertions about a specific individual, such as: If A then C, not-A, C, could all be true at the same time; and one in which participants judged the truth of conditional predictions about specific future events. The results consistently matched the three possibilities, thereby corroborating the model theory. They also showed a massive violation of the probability calculus in estimates of the probabilities of the four cases in the partition of conditionals (A and C, A and not-C, not-A and C, and not-A and not-C), which summed to over 200%.

The Relation Between Factual and Counterfactual Conditionals.

What is the relation between factual conditionals: If A happened then B happened, and counterfactual conditionals: If A had happened then B would have happened? Some theorists propose quite different semantics for the two. In contrast, the theory of mental models and its computer implementation interrelates them. It postulates that both can have a priori truth values, and that the semantic bases of both are possibilities: states that are possible for factual conditionals, and that were once possible but that did not happen for counterfactual conditionals. Two experiments supported these relations. Experiment 1 showed that, like factual conditionals, certain counterfactuals are true a priori, and others are false a priori. Experiment 2 replicated this result and showed that participants selected appropriate paraphrases, referring, respectively, to real and to counterfactual possibilities, for the two sorts of conditional. These results are contrary to alternative accounts of conditionals.

Simulation in children's conscious recursive reasoning.

When do children acquire the ability to understand recursion-that is, repeated loops of actions, as in cookery recipes or computer programs? Hitherto, studies have focused either on unconscious recursions in language and vision or on the difficulty of conscious recursions-even for adults-when learning to program. In contrast, we examined 10- to 11-year-old fifth-graders' ability to deduce the consequences of loops of actions in informal algorithms and to create such algorithms for themselves. In our experiments, the children tackled problems requiring the rearrangement of cars on a toy railway with a single track and a siding-an environment that in principle allows for the execution of any algorithm-that is, it has the power of a universal Turing machine. The children were not allowed to move the cars, so each problem's solution called for them to envision the movements of cars on the track. We describe a theory of recursive thinking, which is based on kinematic simulations and which we have implemented in a computer program embodying mental models of the cars and track. Experiment 1 tested children's ability to deduce rearrangements of the cars in a train from descriptions of algorithms containing a single loop of actions. Experiment 2 assessed children's spontaneous creation of similar sorts of algorithms. The results showed that fifth-grade children with no training in computer programming have systematic abilities to deduce from and to create informal recursive algorithms.

The modulation of disjunctive assertions.

The theory of mental models postulates that disjunctions of the sort, A or B, where A and B are sensible everyday clauses, have a core meaning that allows an inclusive interpretation, referring to three possibilities: A and not-B, not-A and B, and A and B. The meaning of the clauses and knowledge can modulate this meaning by blocking the construction of at least one model of a possibility-for example, "Rui is playing tennis or he is surfing" blocks the model of Rui doing both activities. This theory is implemented in a computer program. Three experiments investigated the core interpretation and interpretations in which the contents of the clauses should block the model of A and B (as in the preceding example), the model of A and not-B, or the model of not-A and B. In Experiment 1, the participants listed the possibilities for each of the four sorts of disjunction. The results corroborated the predicted modulations. In Experiment 2, these predicted interpretations governed the conclusions that participants accepted from disjunctions and categorical premises. In Experiment 3, the predicted interpretations yielded reliable effects on the conclusions that the participants drew for themselves. We relate these results to theories of reasoning.

Modality, probability, and mental models.

We report 3 experiments investigating novel sorts of inference, such as: A or B or both. Therefore, possibly (A and B). Where the contents were sensible assertions, for example, Space tourism will achieve widespread popularity in the next 50 years or advances in material science will lead to the development of antigravity materials in the next 50 years, or both. Most participants accepted the inferences as valid, though they are invalid in modal logic and in probabilistic logic too. But, the theory of mental models predicts that individuals should accept them. In contrast, inferences of this sort­A or B but not both. Therefore, A or B or both­are both logically valid and probabilistically valid. Yet, as the model theory also predicts, most reasoners rejected them. The participants' estimates of probabilities showed that their inferences tended not to be based on probabilistic validity, but that they did rate acceptable conclusions as more probable than unacceptable conclusions. We discuss the implications of the results for current theories of reasoning.

Logic, probability, and human reasoning.

This review addresses the long-standing puzzle of how logic and probability fit together in human reasoning. Many cognitive scientists argue that conventional logic cannot underlie deductions, because it never requires valid conclusions to be withdrawn - not even if they are false; it treats conditional assertions implausibly; and it yields many vapid, although valid, conclusions. A new paradigm of probability logic allows conclusions to be withdrawn and treats conditionals more plausibly, although it does not address the problem of vapidity. The theory of mental models solves all of these problems. It explains how people reason about probabilities and postulates that the machinery for reasoning is itself probabilistic. Recent investigations accordingly suggest a way to integrate probability and deduction.