What is required for the truth of a general conditional?

General conditionals, if p then q, can be used to make assertions about sets of objects. Previous studies have generally found that people judge the probability of one these conditionals to be the conditional probability of q given p, P(q|p). Two experiments investigated the qualitative relation between the exhaustive possibilities, p & q, p & ¬q, ¬p & q, and ¬p & ¬q, and truth and possibility judgements about general conditionals. In Experiment 1, for truth judgements, people evaluated a general conditional as "true" in sets containing p & q cases but no p & ¬q, and "true" judgements depended only on P(q|p). In Experiment 2, for possibility judgements, people's responses implied that only p & q cases have to be possible in a set for a general conditional to be true of the set. Our results add to earlier findings against representing a general conditional as the material conditional of extensional logic, and they provide novel disconfirmation of two recent proposals: the modal semantics of revised mental model theory and certain inferentialist accounts of conditionals. They supply new support for suppositional theories of conditionals.

Paradigms, possibilities, and probabilities: Comment on Hinterecker, Knauff, and Johnson-Laird (2016).

Hinterecker, Knauff, and Johnson-Laird (2016) compared the adequacy of the probabilistic new paradigm in reasoning with the recent revision of mental models theory (MMT) for explaining a novel class of inferences containing the modal term "possibly." For example, the door is closed or the window is open or both, therefore, possibly the door is closed and the window is open (A or B or both, therefore, possibly(A & B)). They concluded that their results support MMT. In this comment, it is argued that Hinterecker et al. (2016) have not adequately characterized the theory of probabilistic validity (p-validity) on which the new paradigm depends. It is unclear how p-validity can be applied to these inferences, which are anyway peripheral to the theory. It is also argued that the revision of MMT is not well motivated and its adoption leads to many logical absurdities. Moreover, the comparison is not appropriate because these theories are defined at different levels of computational explanation. In particular, revised MMT lacks a provably consistent computational level theory that could justify treating these inferences as valid. It is further argued that the data could result from the noncolloquial locutions used to express the premises. Finally, an alternative pragmatic account is proposed based on the idea that a conclusion is possible if what someone knows cannot rule it out. This account could be applied to the unrevised mental model theory rendering the revision redundant. (PsycINFO Database Record (c) 2019 APA, all rights reserved).

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.

New normative standards of conditional reasoning and the dual-source model.

There has been a major shift in research on human reasoning toward Bayesian and probabilistic approaches, which has been called a new paradigm. The new paradigm sees most everyday and scientific reasoning as taking place in a context of uncertainty, and inference is from uncertain beliefs and not from arbitrary assumptions. In this manuscript we present an empirical test of normative standards in the new paradigm using a novel probabilized conditional reasoning task. Our results indicated that for everyday conditional with at least a weak causal connection between antecedent and consequent only the conditional probability of the consequent given antecedent contributes unique variance to predicting the probability of conditional, but not the probability of the conjunction, nor the probability of the material conditional. Regarding normative accounts of reasoning, we found significant evidence that participants' responses were confidence preserving (i.e., p-valid in the sense of Adams, 1998) for MP inferences, but not for MT inferences. Additionally, only for MP inferences and to a lesser degree for DA inferences did the rate of responses inside the coherence intervals defined by mental probability logic (Pfeifer and Kleiter, 2005, 2010) exceed chance levels. In contrast to the normative accounts, the dual-source model (Klauer et al., 2010) is a descriptive model. It posits that participants integrate their background knowledge (i.e., the type of information primary to the normative approaches) and their subjective probability that a conclusion is seen as warranted based on its logical form. Model fits showed that the dual-source model, which employed participants' responses to a deductive task with abstract contents to estimate the form-based component, provided as good an account of the data as a model that solely used data from the probabilized conditional reasoning task.

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.

The influence of cognitive ability and instructional set on causal conditional inference.

We report a large study in which participants are invited to draw inferences from causal conditional sentences with varying degrees of believability. General intelligence was measured, and participants were split into groups of high and low ability. Under strict deductive-reasoning instructions, it was observed that higher ability participants were significantly less influenced by prior belief than were those of lower ability. This effect disappeared, however, when pragmatic reasoning instructions were employed in a separate group. These findings are in accord with dual-process theories of reasoning. We also took detailed measures of beliefs in the conditional sentences used for the reasoning tasks. Statistical modelling showed that it is not belief in the conditional statement per se that is the causal factor, but rather correlates of it. Two different models of belief-based reasoning were found to fit the data according to the kind of instructions and the type of inference under consideration.

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.

Understanding causal conditionals: a study of individual differences.

Previous studies have suggested that a minority of university students, of lower cognitive ability, are inclined to interpret abstract conditional statements, if p then q, as if they were conjunctions: p and q. In the present study we administered the conditional truth table task to a large sample of students (n = 160), but using realistic, everyday causal conditionals. We also measured their general intelligence. While individual differences were found, these were not consistent with some participants adopting a conjunctive interpretation of such statements. Rather, it appears that students of lower cognitive ability are rather likely to assume that a conditional implies its converse, so that it means also if q then p. The results are discussed with reference to the suppositional theory of conditionals and our more general account of hypothetical thinking.

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.