Possibilities as the foundation of reasoning

Illusory inferences in conditional expressions

A robber points a gun at a cashier and says: "Only one of these two options is true: If you conceal the combination to the safe, then I kill you; otherwise, if you don´t conceal the combination to the safe, then I kill you." Hearing this statement, most people conclude that, in either case, "I kill you." This is an illusory response, in fact; the valid conclusion states "I don´t kill you." The research reported here studied the roles that different expressions of conditionals ("if-then," "only if," and "if and only if") play in the illusory response. Three experiments show that participants inferred the conclusion "I kill you" from the conditional "if-then" and "I may or may not kill you" from the conditional "only if," while selecting both options with similar frequency for the biconditional "if and only if." These results shed light on the main theories of deductive reasoning.

Human verifications: Computable with truth values outside logic

Cognitive scientists treat verification as a computation in which descriptions that match the relevant situation are true, but otherwise false. The claim is controversial: The logician Gödel and the physicist Penrose have argued that human verifications are not computable. In contrast, the theory of mental models treats verification as computable, but the two truth values of standard logics, true and false, as insufficient. Three online experiments (n = 208) examined participants' verifications of disjunctive assertions about a location of an individual or a journey, such as: 'You arrived at Exeter or Perth'. The results showed that their verifications depended on observation of a match with one of the locations but also on the status of other locations (Experiment 1). Likewise, when they reached one destination and the alternative one was impossible, their use of the truth value: could be true and could be false increased (Experiment 2). And, when they reached one destination and the only alternative one was possible, they used the truth value, true and it couldn't have been false, and when the alternative one was impossible, they used the truth value: true but it could have been false (Experiment 3). These truth values and those for falsity embody counterfactuals. We implemented a computer program that constructs models of disjunctions, represents possible destinations, and verifies the disjunctions using the truth values in our experiments. Whether an awareness of a verification's outcome is computable remains an open question.

Norm conflicts and epistemic modals

Statements containing epistemic modals (e.g., "by spring 2023 most European countries may have the Covid-19 pandemic under control") are common expressions of epistemic uncertainty. In this paper, previous published findings (Knobe & Yalcin, 2014; Khoo & Phillips, 2018) on the opposition between Contextualism and Relativism for epistemic modals are re-examined. It is found that these findings contain a substantial degree of individual variation. To investigate whether participants differ in their interpretations of epistemic modals, an experiment with multiple phases and sessions is conducted to classify participants according to the three semantic theories of Relativism, Contextualism, and Objectivism. Through this study, some of the first empirical evidence for the kind of truth-value shifts postulated by semantic Relativism is presented. It is furthermore found that participants' disagreement judgments match their truth evaluations and that participants are capable of distinguishing between truth and justification. In a second experimental session, it is investigated whether participants thus classified follow the norm of retraction which Relativism uses to account for argumentation with epistemic modals. Here the results are less favorable for Relativism. In a second experiment, these results are replicated and the normative beliefs of participants concerning the norm of retraction are investigated following work on measuring norms by Bicchieri (2017). Again, it is found that on average participants show no strong preferences concerning the norm of retraction for epistemic modals. Yet, it was found that participants who had committed to Objectivism and had training in logic applied the norm of retraction to might-statements. These results present a substantial challenge to the account of argumentation with epistemic modals presented in MacFarlane (2014), as discussed.

Possible worlds truth table task

In this paper, a novel experimental task is developed for testing the highly influential, but experimentally underexplored, possible worlds account of conditionals (Lewis, 1973; Stalnaker, 1968). In Experiment 1, this new task is used to test both indicative and subjunctive conditionals. For indicative conditionals, five competing truth tables are compared, including the previously untested, multi-dimensional possible worlds semantics of Bradley (2012). In Experiment 2, these results are replicated and it is shown that they cannot be accounted for by an alternative hypothesis proposed by our reviewers. In Experiment 3, individual variation in truth assignments of indicative conditionals is investigated via Bayesian mixture models that classify participants as following one of several competing truth tables. As a novelty of this study, it is found that a possible worlds semantics of Lewis and Stalnaker is capable of accounting for participants' aggregate truth value assignments in this task. Applied to indicative conditionals, we show across three experiments, that the theory both captures participants' truth values at the aggregate level (Experiments 1 and 2) and that it makes up the largest subgroup in the analysis of individual variation in our experimental paradigm (Experiment 3).

The pupillometry of the possible: an investigation of infants' representation of alternative possibilities

Contrasting possibilities has a fundamental adaptive value for prediction and learning. Developmental research, however, has yielded controversial findings. Some data suggest that preschoolers might have trouble in planning actions that take into account mutually exclusive possibilities, while other studies revealed an early understanding of alternative future outcomes based on infants' looking behaviour. To better understand the origin of such abilities, here we use pupil dilation as a potential indicator of infants' representation of possibilities. Ten- and 14-month-olds were engaged in an object-identification task by watching video animations where three different objects with identical top parts moved behind two screens. Importantly, a target object emerged from one of the screens but remained in partial occlusion, revealing only its top part, which was compatible with a varying number of possible identities. Just as adults' pupil diameter grows monotonically with the amount of information held in memory, we expected that infants' pupil size would increase with the number of alternatives sustained in memory as candidate identities for the partially occluded object. We found that pupil diameter increased with the object's potential identities in 14- but not in 10-month-olds. We discuss the implications of these results for the foundation of humans' capacities to represent alternatives. This article is part of the theme issue 'Thinking about possibilities: mechanisms, ontogeny, functions and phylogeny'.

Reasoning About Want

No present theory explains the inferences people draw about the real world when reasoning about "bouletic" relations, that is, predicates that express desires, such as want in "Lee wants to be in love". Linguistic accounts of want define it in terms of a relation to a desirer's beliefs, and how its complement is deemed desirable. In contrast, we describe a new model-based theory that posits that by default, desire predicates such as want contrast desires against facts. In particular, A wants P implies by default that P is not the case, because you cannot want what is already true. On further deliberation, reasoners may infer that A believes, but does not know for certain, that P is not the case. The theory makes several empirical predictions about how people interpret, assess the consistency of, and draw conclusions from desire predicates like want. Seven experiments tested and validated the theory's central predictions. We assess the theory in light of recent proposals of desire predicates.

Signalling Three-Way Intersections: Is Redundancy Better Than Only Mandatory or Prohibitory Signs?

At intersections, drivers need to infer which ways are allowed by interpreting mandatory and/or prohibitory traffic signs. Time and accuracy in this decision-making process are crucial factors to avoid accidents. Previous studies show that integrating information from prohibitory signs is generally more difficult than from mandatory signs. In Study 1, we compare combined redundant signalling conditions to simple sign conditions at three-way intersections. In Study 2, we carried out a survey among professionals responsible for signposting to test whether common practices are consistent with experimental research. In Study 1, an experimental task was applied (n=24), and in Study 2, the survey response rate was 17%. These included the main cities in Spain such as Madrid and Barcelona. Study 1 showed that inferences with mandatory signs are faster than those with prohibitory signs, and redundant information is an improvement only on prohibitory signs. In Study 2, prohibitory signs were those most frequently chosen by professionals responsible for signposting. In conclusion, the most used signs, according to the laboratory study, were not the best ones for signposting because the faster responses were obtained for mandatory signs, and in second place for redundant signs.

Specificity effects in reasoning with counterintuitive and arbitrary conditionals

When people have prior knowledge about an inference, they accept conclusions from specific conditionals (e.g., “If Jack does sports, then Jack loses weight”) more strongly than for unspecific conditionals (e.g., “If a person does sports, then the person loses weight”). But can specific phrasings also elevate the acceptance of conclusions from unbelievable conditionals? In Experiment 1, we varied the specificity of counterintuitive conditionals, which described the opposite of what is expected according to everyday experiences (“If Lena/a person studies hard, then Lena/the person will not do well on the test”). In Experiment 2, we varied the specificity of arbitrary conditionals, which had no obvious link between antecedent and consequent (“If Mary/a person goes shopping, then Mary/ the person gets pimples”). All conditionals were embedded in MP and AC inferences. Participants were instructed to reason as in daily life and to evaluate the conclusions on a 7-point Likert scale. Our results showed a specificity effect in both experiments: participants gave higher acceptance ratings for specific than for unspecific conditionals.

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.

Does the inclusive disjunction really mean the conjunction of possibilities?

There is an ongoing dispute in the psychology of reasoning about how people interpret disjunctions, p or q. In the original mental models theory (MMT1) people interpret p or q as the disjunction of three possibilities (possibly p¬q, or possibly ¬pq, or possibly pq, where "¬" = not). p or q is true if one disjunct is actually true. In a recent revision of mental models theory (MMT2), people interpret p or q as a conjunction of the three possibilities, and they treat it as true only if each is possible and ¬p¬q is impossible. Two experiments investigated possibility and truth judgments about disjunctions given sets consisting of one or more of the four cases (p¬q, ¬pq, pq, and ¬p¬q). The results showed that in both possibility and truth judgments, participants' interpretations of disjunctions were only consistent with MMT1. Inclusive disjunctions imply the disjunction of the three possibilities, and they are true when one of the three cases (p¬q, ¬pq, and pq) is actual. These findings support MMT1, but not MMT2. In conclusion, the revised mental models theory may be unnecessary for disjunctions.

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.

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."