The Psychology of Uncertainty and Three-Valued Truth Tables.

Psychological research on people's understanding of natural language connectives has traditionally used truth table tasks, in which participants evaluate the truth or falsity of a compound sentence given the truth or falsity of its components in the framework of propositional logic. One perplexing result concerned the indicative conditional if A then C which was often evaluated as true when A and C are true, false when A is true and C is false but irrelevant" (devoid of value) when A is false (whatever the value of C). This was called the "psychological defective table of the conditional." Here we show that far from being anomalous the "defective" table pattern reveals a coherent semantics for the basic connectives of natural language in a trivalent framework. This was done by establishing participants' truth tables for negation, conjunction, disjunction, conditional, and biconditional, when they were presented with statements that could be certainly true, certainly false, or neither. We review systems of three-valued tables from logic, linguistics, foundations of quantum mechanics, philosophical logic, and artificial intelligence, to see whether one of these systems adequately describes people's interpretations of natural language connectives. We find that de Finetti's (1936/1995) three-valued system is the best approximation to participants' truth tables.

Understanding Conditionals in the East: A Replication Study of With Easterners.

The new probabilistic approaches to the natural language conditional imply that there is a parallel relation between indicative conditionals (ICs) "if s then b" and conditional bets (CBs) "I bet $1 that if s then b" in two aspects. First, the probability of an IC and the probability of winning a CB are both the conditional probability, P(s|b). Second, both an IC and a CB have a third value "void" (neither true nor false, neither wins nor loses) when the antecedent is false (¬s). These aspects of the parallel relation have been found in Western participants. In the present study, we investigated whether this parallel is also present in Eastern participants. We replicated the study of Politzer et al. (2010) with Chinese and Japanese participants and made two predictions. First, Eastern participants will tend to engage in more holistic cognition and take all possible cases, including ¬s, into account when they judge the probability of conditional: Easterners may assess the probability of antecedent s out of all possible cases, P(s), and then may focus on consequent b out of s, P(b|s). Consequently, Easterners may judge the probability of the conditional, and of winning the bet, to be P(s) ∗ P(b|s) = P(s & b), and false/losing the bet as P(s) ∗ P(¬b|s) = P(s & ¬b). Second, Eastern participants will tend to be strongly affected by context, and they may not show parallel relationships between ICs and CBs. The results indicate no cultural differences in judging the false antecedent cases: Eastern participants judged false antecedent cases as not making the IC true nor false and as not being winning or losing outcomes. However, there were cultural differences when asked about the probability of a conditional. Consistent with our hypothesis, Eastern participants had a greater tendency to take all possible cases into account, especially in CBs. We discuss whether these results can be explained by a hypothesized tendency for Eastern people to think in more holistic and context-dependent terms than Western people.

Utilitarian Moral Judgment Exclusively Coheres with Inference from Is to Ought.

Faced with moral choice, people either judge according to pre-existing obligations (deontological judgment), or by taking into account the consequences of their actions (utilitarian judgment). We propose that the latter coheres with a more general cognitive mechanism - deontic introduction, the tendency to infer normative ('deontic') conclusions from descriptive premises (is-ought inference). Participants were presented with vignettes that allowed either deontological or utilitarian choice, and asked to draw a range of deontic conclusions, as well as judge the overall moral rightness of each choice separately. We predicted and found a selective defeasibility pattern, in which manipulations that suppressed deontic introduction also suppressed utilitarian moral judgment, but had little effect on deontological moral judgment. Thus, deontic introduction coheres with utilitarian moral judgment almost exclusively. We suggest a family of norm-generating informal inferences, in which normative conclusions are drawn from descriptive (although value-laden) premises. This family includes deontic introduction and utilitarian moral judgment as well as other informal inferences. We conclude with a call for greater integration of research in moral judgment and research into deontic reasoning and informal inference.

Corrigendum: Bayesian reasoning with ifs and ands and ors.

[This corrects the article on p. 192 in vol. 6, PMID: 25762965.].

Uncertain deduction and conditional reasoning.

There has been a paradigm shift in the psychology of deductive reasoning. Many researchers no longer think it is appropriate to ask people to assume premises and decide what necessarily follows, with the results evaluated by binary extensional logic. Most every day and scientific inference is made from more or less confidently held beliefs and not assumptions, and the relevant normative standard is Bayesian probability theory. We argue that the study of "uncertain deduction" should directly ask people to assign probabilities to both premises and conclusions, and report an experiment using this method. We assess this reasoning by two Bayesian metrics: probabilistic validity and coherence according to probability theory. On both measures, participants perform above chance in conditional reasoning, but they do much better when statements are grouped as inferences, rather than evaluated in separate tasks.

Bayesian reasoning with ifs and ands and ors.

The Bayesian approach to the psychology of reasoning generalizes binary logic, extending the binary concept of consistency to that of coherence, and allowing the study of deductive reasoning from uncertain premises. Studies in judgment and decision making have found that people's probability judgments can fail to be coherent. We investigated people's coherence further for judgments about conjunctions, disjunctions and conditionals, and asked whether their coherence would increase when they were given the explicit task of drawing inferences. Participants gave confidence judgments about a list of separate statements (the statements group) or the statements grouped as explicit inferences (the inferences group). Their responses were generally coherent at above chance levels for all the inferences investigated, regardless of the presence of an explicit inference task. An exception was that they were incoherent in the context known to cause the conjunction fallacy, and remained so even when they were given an explicit inference. The participants were coherent under the assumption that they interpreted the natural language conditional as it is represented in Bayesian accounts of conditional reasoning, but they were incoherent under the assumption that they interpreted the natural language conditional as the material conditional of elementary binary logic. Our results provide further support for the descriptive adequacy of Bayesian reasoning principles in the study of deduction under uncertainty.

Deontic introduction: A theory of inference from is to ought.

Humans have a unique ability to generate novel norms. Faced with the knowledge that there are hungry children in Somalia, we easily and naturally infer that we ought to donate to famine relief charities. Although a contentious and lively issue in metaethics, such inference from "is" to "ought" has not been systematically studied in the psychology of reasoning. We propose that deontic introduction is the result of a rich chain of pragmatic inference, most of it implicit; specifically, when an action is causally linked to a valenced goal, valence transfers to the action and bridges into a deontic conclusion. Participants in 5 experiments were presented with utility conditionals in which an action results in a benefit, a cost, or neutral outcome (e.g., "If Lisa buys the booklet, she will pass the exam") and asked to evaluate how strongly deontic conclusions (e.g., "Lisa should buy the booklet") follow from the premises. Findings show that the direction of the conclusions was determined by outcome valence (Experiments 1a and 1b), whereas their strength was determined by the strength of the causal link between action and outcome (Experiments 1, 2a, and 2b). We also found that deontic introduction is defeasible and can be suppressed by additional premises that interfere with any of the links in the implicit chain of inference (Experiments 2a, 2b, and 3). We propose that deontic introduction is a species-specific generative capacity whose function is to regulate future behavior.

The probability of iterated conditionals.

Iterated conditionals of the form If p, then if q, r are an important topic in philosophical logic. In recent years, psychologists have gained much knowledge about how people understand simple conditionals, but there are virtually no published psychological studies of iterated conditionals. This paper presents experimental evidence from a study comparing the iterated form, If p, then if q, r with the "imported," noniterated form, If p and q, then r, using a probability evaluation task and a truth-table task, and taking into account qualitative individual differences. This allows us to critically contrast philosophical and psychological approaches that make diverging predictions regarding the interpretation of these forms. The results strongly support the probabilistic Adams conditional and the "new paradigm" that takes this conditional as a starting point.

Heuristic thinking and human intelligence: a commentary on Marewski, Gaissmaier and Gigerenzer.

Marewski, Gaissmaier and Gigerenzer (2009) present a review of research on fast and frugal heuristics, arguing that complex problems are best solved by simple heuristics, rather than the application of knowledge and logical reasoning. We argue that the case for such heuristics is overrated. First, we point out that heuristics can often lead to biases as well as effective responding. Second, we show that the application of logical reasoning can be both necessary and relatively simple. Finally, we argue that the evidence for a logical reasoning system that co-exists with simpler heuristic forms of thinking is overwhelming. Not only is it implausible a priori that we would have evolved such a system that is of no use to us, but extensive evidence from the literature on dual processing in reasoning and judgement shows that many problems can only be solved when this form of reasoning is used to inhibit and override heuristic thinking.

Valuing intervention and observation.

Understanding causal relations is fundamental to effective action but causal data can be confounded. We examined the value that participants placed on data derived from a hypothetical intervention or observation. Our materials involved a possible cause ("bottled water"), a possible confound ("food"), and a context ("a restaurant"). We supposed that participants seek to draw as specific a causal inference as possible from presented data and value information sources more highly that allow them to do so. On this basis, we predicted that in circumstances where an intervention removed the confounding causal factor but observation did not, participants would prefer data derived from an intervention when the possible cause was present (the bottled water was drunk) but show the reverse preference when the possible cause was absent (the bottled water was not drunk). Experiment 1 confirmed this prediction. Using a between-subjects design, Experiment 2 tested for a difference in confidence in causal judgements given identical data, including data on the confound, as a function of method of data collection (intervention or observation). There was no significant difference in confidence ratings between the two methods but confidence ratings were sensitive to the probability of an effect (illness) given the cause. Using a within-subjects design, Experiment 3 revealed systematic individual differences in preference for the two methods. Participants were divided between those who considered intervention more confounded and those who considered observation more confounded. Our experiments point to the subtleties of participants' evaluation of data from studies of human beings.

When can we say 'if'?

In this study, we focus on the conditions which permit people to assert a conditional statement of the form 'if p then q' with conversational relevance. In a broadly decision-theoretic approach, also drawing on hypothetical thinking theory [Evans, J. St. B. T. (2007). Hypothetical thinking: Dual processes in reasoning and judgement. Hove, UK: Psychology Press.], we predicted that conditional tips and promises would appear more useful and persuasive and be more likely to encourage an action p when (a) the conditional link from p to q was stronger, (b) the cost of the action p was lower and (c) the benefit of the consequence q was higher. Similarly, we predicted that conditional warnings and threats would be seen as more useful and persuasive and more likely to discourage an action p when (a) the conditional link from p to q was stronger, (b) the benefit of the action p was lower and (c) the cost of the consequence q was higher. All predictions were strongly confirmed, suggesting that such conditionals may best be asserted when they are of high relevance to the goals of the listener.

Thinking about conditionals: a study of individual differences.

Recent studies have shown the existence of two qualitatively distinct groups of people based on how they judge the probability of a conditional statement. The present study was designed to test whether these differences are rooted in distinctive means of processing conditional statements and whether they are linked to differences in general intelligence. In the study, each of 120 participants completed three separate cognitive tasks involving the processing of abstract conditional statements--the probability-of-conditionals task, the conditional truth table task, and the conditional inference task--in addition to completing a test of general intelligence (AH4). The results showed a number of predicted effects: People responding with conditional (rather than conjunctive) probabilities on the first task were higher in cognitive ability, showed reasoning patterns more consistent with a suppositional treatment of the conditional, and showed a strongly "defective" truth table pattern. The results include several novel findings and post challenges to contemporary psychological theories of conditionals.

On the basis of belief in causal and diagnostic conditionals.

According to the suppositional theory of conditionals, people assess their belief in a conditional statement of the form "if p then q" by conducting a mental simulation on the supposition of p in which they assess their degree of belief in q. This leads to them to the judge the probability of a conditional statement to be equal to the conditional probability, P(q|p). Evidence for this conditional probability hypothesis has been adduced in earlier studies for abstract, causal, and counterfactual conditionals. For the realistic conditionals, it is natural to assume that people perform such mental simulations by building causal mental models from prior causes to later effects. However, in the present study we show that the conditional probability hypothesis extends to diagnostic conditionals, which relate effects to causes. This new finding presents a major challenge for theoretical accounts of the mental processing of conditional statements.

Iffy beliefs: conditional thinking and belief change.

The ability to entertain possibilities and draw inferences about them is essential to human intelligence. We examine the hypothesis that conditional if-then statements trigger a mental simulation process in which people suppose the antecedent (if statement) to be true and evaluate the consequent (then statement) in that context. On the assumption that supposing an event to be true increases belief that the event has occurred or will occur, this hypothesis is consistent with the claim that evaluating a conditional will heighten belief in its antecedent more than in its consequent. Two experiments, employing conditionals of the form If animal A has property X, then animal B will have property X, in which X was a property that people could not readily relate to the animals, supported this claim. The effect was stronger following the evaluation of conditionals with dissimilar animal categories.

The probability of causal conditionals.

Conditionals in natural language are central to reasoning and decision making. A theoretical proposal called the Ramsey test implies the conditional probability hypothesis: that the subjective probability of a natural language conditional, P(if p then q), is the conditional subjective probability, P(q/p). We report three experiments on causal indicative conditionals and related counterfactuals that support this hypothesis. We measured the probabilities people assigned to truth table cases, P(pq), P(p notq), P( notpq) and P( notp notq). From these ratings, we computed three independent predictors, P(p), P(q/p) and P(q/ notp), that we then entered into a regression equation with judged P(if p then q) as the dependent variable. In line with the conditional probability hypothesis, P(q/p) was by far the strongest predictor in our experiments. This result is inconsistent with the claim that causal conditionals are the material conditionals of elementary logic. Instead, it supports the Ramsey test hypothesis, implying that common processes underlie the use of conditionals in reasoning and judgments of conditional probability in decision making.

Suppositions, extensionality, and conditionals: a critique of the mental model theory of Johnson-Laird And Byrne (2002).

P. N. Johnson-Laird and R. M. J. Byrne proposed an influential theory of conditionals in which mental models represent logical possibilities and inferences are drawn from the extensions of possibilities that are used to represent conditionals. In this article, the authors argue that the extensional semantics underlying this theory is equivalent to that of the material, truth-functional conditional, at least for what they term "basic" conditionals, concerning arbitrary problem content. On the basis of both logical argument and psychological evidence, the authors propose that this approach is fundamentally mistaken and that conditionals must be viewed within a suppositional theory based on what philosophical logicians call the Ramsey test. The Johnson-Laird and Byrne theory is critically examined with respect to its account of basic conditionals, nonbasic conditionals, and counterfactuals.

Conditionals and conditional probability.

The authors report 3 experiments in which participants were invited to judge the probability of statements of the form if p then q given frequency information about the cases pq, p not q, not pq, and not p not q (where not = not). Three hypotheses were compared: (a) that people equate the probability with that of the material conditional, 1 - P(p not q); (b) that people assign the conditional probability, P(q/p); and (c) that people assign the conjunctive probability P(pq). The experimental evidence allowed rejection of the 1st hypothesis but provided some support for the 2nd and 3rd hypotheses. Individual difference analyses showed that half of the participants used conditional probability and that most of the remaining participants used conjunctive probability as the basis of their judgments.

Background beliefs in Bayesian inference.

We report five experiments in which the role of background beliefs in social judgments of posterior probability was investigated. From a Bayesian perspective, people should combine prior probabilities (or base rates) and diagnostic evidence with equal weighting, although previous research shows that base rates are often underweighted. These experiments were designed so that either piece of information was supplied either by personal beliefs or by presented statistics, and regression analyses were performed on individual participants to assess the relative influence of information. We found that both prior probabilities and diagnostic information significantly influenced judgments, whether supplied by beliefs or by statistical information, but that belief-based information tended to dominate the judgments made.