A study is reported testing two hypotheses about a close parallel relation between indicative conditionals, if A then B , and conditional bets, I bet you that if A then B . The first is that both the indicative conditional and the conditional bet are related to the conditional probability, P(B|A). The second is that de Finetti's three-valued truth table has psychological reality for both types of conditional— true , false , or void for indicative conditionals and win , lose (...) , or void for conditional bets. The participants were presented with an array of chips in two different colours and two different shapes, and an indicative conditional or a conditional bet about a random chip. They had to make judgements in two conditions: either about the chances of making the indicative conditional true or false or about the chances of winning or losing the conditional bet. The observed distributions of responses in the two conditions were generally related to the conditional probability, supporting the first hypothesis. In addition, a majority of participants in further conditions chose the third option, “void”, when the antecedent of the conditional was false, supporting the second hypothesis. (shrink)

Brouwer's demonstration of his Bar Theorem gives rise to provocative questions regarding the proper explanation of the logical connectives within intuitionistic and constructivist frameworks, respectively, and, more generally, regarding the role of logic within intuitionism. It is the purpose of the present note to discuss a number of these issues, both from an historical, as well as a systematic point of view.

The appropriateness, or acceptability, of a conditional does not just ‘go with’ the corresponding conditional probability. A condition of dependence is required as well. In this paper a particular notion of dependence is proposed. It is shown that under both a forward causal and a backward evidential reading of the conditional, this appropriateness condition reduces to conditional probability under some natural circumstances. Because this is in particular the case for the so-called diagnostic reading of the conditional, this analysis might help (...) to explain some of Douven and Verbrugge’s empirical observations. (shrink)

It has recently been argued that a non-Bayesian probabilistic version of inference to the best explanation (IBE*) has a number of advantages over Bayesian conditionalization (Douven [2013]; Douven and Wenmackers [2017]). We investigate how IBE* could be generalized to uncertain evidential situations and formulate a novel updating rule IBE**. We then inspect how it performs in comparison to its Bayesian counterpart, Jeffrey conditionalization (JC), in a number of simulations where two agents, each updating by IBE** and JC, respectively, try to (...) detect the bias of a coin while they are only partially certain what side the coin landed on. We show that IBE** more often prescribes high probability to the actual bias than JC. We also show that this happens considerably faster, that IBE** passes higher thresholds for high probability, and that it in general leads to more accurate probability distributions than JC. (shrink)

Four experiments investigated uncertainty about a premise in a deductive argument as a function of the expertise of the speaker and of the conversational context. The procedure mimicked everyday reasoning in that participants were not told that the premises were to be treated as certain. The results showed that the perceived likelihood of a conclusion was greater when the major or the minor premise was uttered by an expert rather than a novice (Experiment 1). The results also showed that uncertainty (...) about the conclusion was higher when the major premise was uttered by a novice and an alternative premise by an expert, compared to when the major premise was uttered by an expert and the alternative by a novice (Experiment 2). Similarly, the believability of a conclusion was considerably lower when the minor premise was uttered by a novice and denied by an expert, as opposed to when an expert uttered the minor premise and a novice denied it (Experiment 3). Experiment 4 showed that the nature of the uncertainty induced by a denial of the minor premise depended on whether or not the context was a conversation. These results pose difficult problems for current theories of reasoning, as current theories are based on the results of experiments in which the premises are treated as certain. Our discussion of the results emphasises the importance of pragmatics in reasoning, namely, the role of general knowledge about the world in assessing the probability of a premise uttered by an expert or a novice and the role of interpretations of the premise based on pragmatic inferences in revising these initial probabilities. (shrink)

We investigate how the perceived uncertainty of a conditional affects a person's choice of conclusion. We use a novel procedure to introduce uncertainty by manipulating the conditional probability of the consequent given the antecedent. In Experiment 1, we show first that subjects reduce their choice of valid conclusions when a conditional is followed by an additional premise that makes the major premise uncertain. In this we replicate Byrne. These subjects choose, instead, a qualified conclusion expressing uncertainty. If subjects are given (...) a third statement that qualifies the likelihood of the additional premise, then the uncertainty of the conclusions they choose is systematically related to the suggested uncertainty. Experiment 2 confirms these observations in problems that omit the additional premise and qualify the first premise directly. Experiment 3 shows that the qualifying statement also affects the perceived probability of the consequent given the antecedent of the conditional. Experiment 4 investigates the effect of suggested uncertainty on the fallacies and shows that increases in uncertainty reduce the number of certain conclusions that are chosen while affirming the consequent but have no effect on denying the antecedent. We discuss our results in terms of rule theories and mental models and conclude that the latter give the most natural account of our results. (shrink)

Conditionals somehow express conditional beliefs. However, conditional belief is a bi-propositional attitude that is generally not truth-evaluable, in contrast to unconditional belief. Therefore, this article opts for an expressivistic semantics for conditionals, grounds this semantics in the arguably most adequate account of conditional belief, that is, ranking theory, and dismisses probability theory for that purpose, because probabilities cannot represent belief. Various expressive options are then explained in terms of ranking theory, with the intention to set out a general interpretive scheme (...) that is able to account for the most variegated usage of conditionals. The Ramsey test is only the first option. Relevance is another, familiar, but little understood item, which comes in several versions. This article adds a further family of expressive options, which is able to subsume also counterfactuals and causal conditionals, and indicates at the end how this family allows for partial recovery of truth conditions for conditionals. (shrink)

More than a decade of research has found strong evidence for P(if A, then C) = P(C|A) (“the Equation”). We argue, however, that this hypothesis provides an overly simplified picture due to its inability to account for relevance. We manipulated relevance in the evaluation of the probability and acceptability of indicative conditionals and found that relevance moderates the effect of P(C|A). This corroborates the Default and Penalty Hypothesis put forward in this paper. Finally, the probability and acceptability of concessive conditionals (...) (“Even if A, then still C”) were investigated and it was found that the Equation provides a better account of concessive conditionals than of indicatives across relevance manipulations. (shrink)

The aim is to motivate theoretically a relevance approach to conditionals in a comparative discussion of the main alternatives. In particular, it will be argued that a relevance approach to conditionals is better motivated than the suppositional theory currently enjoying wide endorsement. In the course of this discussion, an argument will be presented for why failures of the epistemic relevance of the antecedent for the consequent should be counted as genuine semantic defects. Furthermore, strategies for dealing with compositionality and the (...) perceived objective purport of indicative conditionals will be put forward. (shrink)

Suppose that two competing norms, N1 and N2, can be identified such that a given person’s response can be interpreted as correct according to N1 but incorrect according to N2. Which of these two norms, if any, should one use to interpret such a response? In this paper we seek to address this fundamental problem by studying individual variation in the interpretation of conditionals by establishing individual profiles of the participants based on their case judgments and reflective attitudes. To investigate (...) the participants’ reflective attitudes we introduce a new experimental paradigm called the Scorekeeping Task. As a case study, we identify the participants who follow the Suppositional Theory of conditionals (N1) versus Inferentialism (N2) and investigate to what extent internally consistent competence models can be reconstructed for the participants on this basis. After extensive empirical investigations, an apparent reasoning error with and-to-if inferences was found in one of these two groups. The implications of this case study for debates on the proper role of normative considerations in psychology are discussed. (shrink)

The two main psychological theories of the ordinary conditional were designed to account for inferences made from assumptions, but few premises in everyday life can be simply assumed true. Useful premises usually have a probability that is less than certainty. But what is the probability of the ordinary conditional and how is it determined? We argue that people use a two stage Ramsey test that we specify to make probability judgements about indicative conditionals in natural language, and we describe experiments (...) that support this conclusion. Our account can explain why most people give the conditional probability as the probability of the conditional, but also why some give the conjunctive probability. We discuss how our psychological work is related to the analysis of ordinary indicative conditionals in philosophical logic. (shrink)

Two experiments were conducted to investigate the roles of covariation and of causality in people's readiness to believe a conditional. The experiments used a probabilistic truth-table task (Oberauer & Wilhelm, 2003) in which people estimated the probability of a conditional given information about the frequency distribution of truth-table cases. For one group of people, belief in the conditional was determined by the conditional probability of the consequent, given the antecedent, whereas for another group it depended on the probability of the (...) conjunction of antecedent and consequent. There was little evidence that covariation, expressed as the probabilistic contrast or as the pCI rule (White, 2003), influences belief in the conditional. The explicit presence of a causal link between antecedent and consequent in a context story had a weak positive effect on belief in a conditional when the frequency distribution of relevant cases was held constant. (shrink)

This paper addresses the apparent mismatch between the normative and descriptive literatures in the cognitive science of conditional reasoning. Descriptive psychological theories still regard material implication as the normative theory of the conditional. However, over the last 20 years in the philosophy of language and logic the idea that material implication can account for everyday indicative conditionals has been subject to severe criticism. The majority view is now apparently in favour of a subjective conditional probability interpretation. A comparative model fitting (...) exercise is presented that shows that a conditional probability model can explain as much of the data on abstract indicative conditional reasoning tasks as psychological theories that supplement material implication with various rationally unjustified processing assumptions. Consequently, when people are asked to solve laboratory reasoning tasks, they can be seen as simply generalising their everyday probabilistic reasoning strategies to this novel context. (shrink)

Mental model theory has been used to explain many differing phenomena in adult reasoning, including the extensively studied case of conditional reasoning. However, the current theory makes predictions about the development of conditional reasoning that are not consistent with data. In this article, young children's performance on conditional reasoning problems and the justifications given are analysed. A mental model account of conditional reasoning is proposed that assumes that (1) young children can reason with two models and (2) the fleshing out (...) of conditionals involves activation of information in semantic memory that uses the minor premise as a retrieval cue. (shrink)

Many conditionals seem to convey the existence of a link between their antecedent and consequent. We draw on a recently proposed typology of conditionals to argue for an old philosophical idea according to which the link is inferential in nature. We show that the proposal has explanatory force by presenting empirical results on the evidential meaning of certain English and Dutch modal expressions.

According to the Principle of Conditional Non-Contradiction (CNC), conditionals of the form “If p, q” and “If p, not q” cannot both be true, unless p is inconsistent. This principle is widely regarded as an adequacy constraint on any semantics that attributes truth conditions to conditionals. Gibbard has presented an example of a pair of conditionals that, in the context he describes, appear to violate CNC. He concluded from this that conditionals lack truth conditions. We argue that this conclusion is (...) rash by proposing a new diagnosis of what is going on in Gibbard’s argument. We also provide empirical evidence in support of our proposal. (shrink)

Reasoning with conditionals is central to everyday life, yet there is long-standing disagreement about the meaning of the conditional. One example is the puzzle of so-called missing-link conditionals such as "if raccoons have no wings, they cannot breathe under water." Their oddity may be taken to show that conditionals require a connection between antecedent ("raccoons have no wings") and consequent ("they cannot breathe under water"), yet most accounts of conditionals attribute the oddity to natural language pragmatics. We present an experimental (...) study disentangling the pragmatic requirement of discourse coherence from a stronger notion of connection: probabilistic relevance. Results indicate that mere discourse coherence is not enough to make conditionals assertable. (shrink)

In this article, 41 models of covariation detection from 2 × 2 contingency tables were evaluated against past data in the literature and against data from new experiments. A new model was also included based on a limiting case of the normative phi‐coefficient under an extreme rarity assumption, which has been shown to be an important factor in covariation detection (McKenzie & Mikkelsen, 2007) and data selection (Hattori, 2002; Oaksford & Chater, 1994, 2003). The results were supportive of the new (...) model. To investigate its explanatory adequacy, a rational analysis using two computer simulations was conducted. These simulations revealed the environmental conditions and the memory restrictions under which the new model best approximates the normative model of covariation detection in these tasks. They thus demonstrated the adaptive rationality of the new model. (shrink)

The notion of “the burden of proof” plays an important role in real-world argumentation contexts, in particular in law. It has also been given a central role in normative accounts of argumentation, and has been used to explain a range of classic argumentation fallacies. We argue that in law the goal is to make practical decisions whereas in critical discussion the goal is frequently simply to increase or decrease degree of belief in a proposition. In the latter case, it is (...) not necessarily important whether that degree of belief exceeds a particular threshold (e.g., ‘reasonable doubt’). We explore the consequences of this distinction for the role that the “burden of proof” has played in argumentation and in theories of fallacy. (shrink)

In the form of inference known as inference to the best explanation there are various ways to characterise what is meant by the best explanation. This paper considers a number of such characterisations including several based on confirmation measures and several based on coherence measures. The goal is to find a measure which adequately captures what is meant by 'best' and which also yields the truth with a high degree of probability. Computer simulations are used to show that the overlap (...) coherence measure achieves this goal, enabling the true explanation to be identified almost as often as an approach which simply selects the most probable explanation. Further advantages to this approach are also considered in the case where there is uncertainty in the prior probability distribution. (shrink)

This paper considers an application of work on probabilistic measures of coherence to inference to the best explanation. Rather than considering information reported from different sources, as is usually the case when discussing coherence measures, the approach adopted here is to use a coherence measure to rank competing explanations in terms of their coherence with a piece of evidence. By adopting such an approach IBE can be made more precise and so a major objection to this mode of reasoning can (...) be addressed. Advantages of the coherence - based approach are pointed out by comparing it with several other ways to characterize ‘ best explanation ’ and showing that it takes into account their insights while overcoming some of their problems. The consequences of adopting this approach for IBE are discussed in the context of recent discussions about the relationship between IBE and Bayesianism. (shrink)

According to the Bayesian paradigm in the psychology of reasoning, the norms by which everyday human cognition is best evaluated are probabilistic rather than logical in character. Recently, the Bayesian paradigm has been applied to the domain of argumentation, where the fundamental norms are traditionally assumed to be logical. Here, we present a major generalisation of extant Bayesian approaches to argumentation that utilizes a new class of Bayesian learning methods that are better suited to modelling dynamic and conditional inferences than (...) standard Bayesian conditionalization, is able to characterise the special value of logically valid argument schemes in uncertain reasoning contexts, greatly extends the range of inferences and argumentative phenomena that can be adequately described in a Bayesian framework, and undermines some influential theoretical motivations for dual function models of human cognition. We conclude that the probabilistic norms given by the Bayesian approach to rationality are not necessarily at odds with the norms given by classical logic. Rather, the Bayesian theory of argumentation can be seen as justifying and enriching the argumentative norms of classical logic. (shrink)

According to what is now commonly referred to as “the Equation” in the literature on indicative conditionals, the probability of any indicative conditional equals the probability of its consequent of the conditional given the antecedent of the conditional. Philosophers widely agree in their assessment that the triviality arguments of Lewis and others have conclusively shown the Equation to be tenable only at the expense of the view that indicative conditionals express propositions. This study challenges the correctness of that assessment by (...) presenting data that cast doubt on an assumption underlying all triviality arguments. (shrink)

According to so-called epistemic theories of conditionals, the assertability/acceptability/acceptance of a conditional requires the existence of an epistemically significant relation between the conditional’s antecedent and its consequent. This paper points to some linguistic data that our current best theories of the foregoing type appear unable to explain. Further, it presents a new theory of the same type that does not have that shortcoming. The theory is then defended against some seemingly obvious objections.

Bayesians have traditionally taken a dim view of the Inference to the Best Explanation, arguing that, if IBE is at variance with Bayes ' rule, then it runs afoul of the dynamic Dutch book argument. More recently, Bayes ' rule has been claimed to be superior on grounds of conduciveness to our epistemic goal. The present paper aims to show that neither of these arguments succeeds in undermining IBE.

This article compares inference to the best explanation with Bayes’s rule in a social setting, specifically, in the context of a variant of the Hegselmann–Krause model in which agents not only update their belief states on the basis of evidence they receive directly from the world, but also take into account the belief states of their fellow agents. So far, the update rules mentioned have been studied only in an individualistic setting, and it is known that in such a setting (...) both have their strengths as well as their weaknesses. It is shown here that in a social setting, inference to the best explanation outperforms Bayes’s rule according to every desirable criterion. 1 What Is Inference to the Best Explanation?2 Judging the Rules—By Which Lights?3 From an Individualistic to a Social Perspective 3.1 The Hegselmann–Krause model 3.2 A probabilistic extension of the Hegselmann–Krause model 3.3 Simulations4 Results and Discussion5 Interpretation6 Conclusion. (shrink)

Conditionals whose antecedent and consequent are not somehow internally connected tend to strike us as odd. The received doctrine is that this felt oddness is to be explained pragmatically. Exactly how the pragmatic explanation is supposed to go has remained elusive, however. This paper discusses recent philosophical and psychological work that attempts to account semantically for the apparent oddness of conditionals lacking an internal connection between their parts.

In previous published research (“Conditionals and Inferential Connections: A Hypothetical Inferential Theory,” Cognitive Psychology, 2018), we investigated experimentally what role the presence and strength of an inferential connection between a conditional’s antecedent and consequent plays in how people process that conditional. Our analysis showed the strength of that connection to be strongly predictive of whether participants evaluated the conditional as true, false, or neither true nor false. In this article, we re-analyse the data from our previous research, now focussing on (...) the semantics of conditionals rather than on how they are processed. Specifically, we use those data to compare the main extant semantics with each other and with inferentialism, a semantics according to which the truth of a conditional requires the presence of an inferential connection between the conditional’s component parts. (shrink)