An investigation into the online processing of counterfactual and indicative conditionals

The ability to represent conditional information is central to human cognition. In two self-paced reading experiments we investigated how readers process counterfactual conditionals (e.g., If Darren had been athletic, he could probably have played on the rugby team) and indicative conditionals (e.g., If Darren is athletic, he probably plays on the rugby team). In Experiment 1 we focused on how readers process counterfactual conditional sentences. We found that processing of the antecedent of counterfactual conditionals was rapidly constrained by prior context (i.e., knowing whether Darren was or was not athletic). A reading-time penalty was observed for the critical region of text comprising the last word of the antecedent and the first word of the consequent when the information in the antecedent did not fit with prior context. In Experiment 2 we contrasted counterfactual conditionals with indicative conditionals. For counterfactual conditionals we found the same effect on the critical region as we found in Experiment 1. In contrast, however, we found no evidence that processing of the antecedent of indicative conditionals was constrained by prior context. For indicative conditionals (but not for counterfactual conditionals), the results we report are consistent with the suppositional account of conditionals. We propose that current theories of conditionals need to be able to account for online processing differences between indicative and counterfactual conditionals.     

Indicative conditionals,conditional probabilities,and the “defective truth-table”: A request for more experiments

While there is now considerable experimental evidence that, on the one hand, participants assign to the indicative conditional as probability the conditional probability of consequent given antecedent and, on the other, they assign to the indicative conditional the “defective truth-table” in which a conditional with false antecedent is deemed neither true nor false, these findings do not in themselves establish which multi-premise inferences involving conditionals participants endorse. A natural extension of the truth-table semantics pronounces as valid numerous inference patterns that do seem to be part of ordinary usage. However, coupled with something the probability account gives us—namely that when conditional-free ? entails conditional-free ψ, “if ? then ψ” is a trivial, uninformative truth—we have enough logic to derive the paradoxes of material implication. It thus becomes a matter of some urgency to determine which inference patterns involving indicative conditionals participants do endorse. Only thus will we be able to arrive at a realistic, systematic semantics for the indicative conditional.     

Indicatives,concessives, and evidential support

This paper discusses the issue of categorical acceptability of indicative and concessive conditionals. It presents experimental results in favour of two claims concerning the role of the evidential support relation for acceptability (or otherwise) of conditionals of both types. In particular, the results show that, contrary to fairly standard philosophical theorising, high probability of a conditional's consequent given its antecedent is necessary but not sufficient for the acceptability of that conditional, and that the antecedent being evidence for the consequent is a further acceptability condition. The results further show that the evidential support relation is crucial in differentiating between the acceptability of an indicative conditional and the acceptability of the corresponding concessive conditional: typically, the use of a concessive conditional signals that the corresponding conditional probability is high in spite of the fact that the antecedent is evidence against the consequent, or in any case is not evidence for the consequent.     

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.     

Betting on conditionals

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.     

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.     

Could it be the Case that if I am Right My Opponents will be Pleased? A Rejoinder to Johnson-Laird,Byrne and Girotto

I take up the four issues considered by Johnson-Laird, Byrne and Girotto in their reply to Politzer (2007). Based on the conceptual clarification which they adduce, it seems that the disagreement can be settled about the first one (truth functionality) and can be attenuated about the second one (the paradoxes of material implication). However, I maintain and refine my criticisms on the last two (negation and the probability of conditionals), backed up by considerations borrowed from the perspective of the conditional probability semantics for conditionals.     

Abstract reasoning and the interpretation of basic conditionals

Studies examining the interpretation that is given to if–then statementstypically use what are referred to as basic conditionals, which give contextless relations between two unrelated concrete terms (If the ball is blue, then the shape is square). However, there is some evidence that basic conditionals require a more abstract form of representation. In order to examine this, we presented participants with truth-table tasks involving either basic conditionals or conditionals referring to imaginary categories (If it is a bori, then it has red wings), and standard conditional inference tasks with abstract and familiar premises. As expected, fewer typical defective conditional interpretations were given to basic conditionals. In addition, partial correlations showed a unique relationship between the interpretation of basic conditionals and abstract inferential reasoning. Results suggest that people process basic conditionals as a form of abstract reasoning, and that the interpretation of conditionals must consider the semantic context.     

NTV观点引发的逻辑哲学问题——直陈条件句能嵌套吗

NTV观点的核心思想是直陈条件句不是命题,它没有真值条件并且无真值,其既不为真也不为假,而只表现为一个相应的概率值。这就引发了一个有争论的逻辑哲学问题——直陈条件句能否嵌套。如果把直陈条件句视为无真值的,那么嵌套后的复合条件句就不能判断其真假（真值表不能使用）。对于直陈条件句能否嵌套的问题,学界有不同的观点,我们认为要完全解决真之条件主义与条件概率主义两者之间的矛盾是非常困难的,然而,完全接受条件概率论题的哲学后果是非常危险的,同样,暗示放弃真之语义观也是非常危险的。     

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.     

Indicatives,Subjunctives, and the Falsity of the Antecedent

It is widely held that there are important differences between indicative conditionals (e.g., “If the authors are linguists, they have written a linguistics paper”) and subjunctive conditionals (e.g., “If the authors had been linguists, they would have written a linguistics paper”). A central difference is that indicatives and subjunctives convey different stances toward the truth of their antecedents. Indicatives (often) convey neutrality: for example, about whether the authors in question are linguists. Subjunctives (often) convey the falsity of the antecedent: for example, that the authors in question are not linguists. This paper tests prominent accounts of how these different stances are conveyed: whether by presupposition or conversational implicature. Experiment 1 tests the presupposition account by investigating whether the stances project—remain constant—when embedded under operators like negations, possibility modals, and interrogatives, a key characteristic of presuppositions. Experiment 2 tests the conversational-implicature account by investigating whether the stances can be cancelled without producing a contradiction, a key characteristic of implicatures. The results provide evidence that both stances—neutrality about the antecedent in indicatives and the falsity of the antecedent in subjunctives—are conveyed by conversational implicatures.     

Relevance differently affects the truth,acceptability, and probability evaluations of “and”, “but”, “therefore”, and “if–then”

In this study, we investigate the influence of reason-relation readings of indicative conditionals and “and”/“but”/“therefore” sentences on various cognitive assessments. According to the Frege–Grice tradition, a dissociation is expected. Specifically, differences in the reason-relation reading of these sentences should affect participants’ evaluations of their acceptability but not of their truth value. In two experiments we tested this assumption by introducing a relevance manipulation into the truth-table task as well as in other tasks assessing the participants’ acceptability and probability evaluations. Across the two experiments, a strong dissociation was found. The reason-relation reading of all four sentences strongly affected their probability and acceptability evaluations, but hardly affected their respective truth evaluations. Implications of this result for recent work on indicative conditionals are discussed.     

Embedding If and Only If

Some left-nested indicative conditionals are hard to interpret while others seem fine. Some proponents of the view that indicative conditionals have No Truth Values (NTV) use their view to explain why some left-nestings are hard to interpret: the embedded conditional does not express the truth conditions needed by the embedding conditional. Left-nestings that seem fine are then explained away as cases of ad hoc, pragmatic interpretation. We challenge this explanation. The standard reasons for NTV about indicative conditionals (triviality results, Gibbardian standoffs, etc.) extend naturally to NTV about biconditionals. So NTVers about conditionals should also be NTVers about biconditionals. But biconditionals embed much more freely than conditionals. If NTV explains why some left-nested conditionals are hard to interpret, why do biconditionals embed successfully in the very contexts where conditionals do not embed?     

Something amazing about the Peripatetic of Pallet: Abaelard's development of Boethius' account of conditional propositions

Mediaeval logicians inherited from Boethius an account of conditional propositions and the syllogisms which may be constructed using them. In the following paper it is shown that there are considerable difficulties with Boethius' account which arise from his failure to understand the nature of compound propositions and in particular to provide for their negation. Boethius suggests that there are two different conditions which may be imposed for the truth of a conditional proposition but he really gives no adequate account of how such propositions may be obtained. The true greatness of Peter Abaelard as a philosophical logician is revealed in what he is able to do with the material which he found in Boethius. It is shown that he developed a precise theory of conditionals giving an account of how true conditionals may be obtained and principles which may be used to reject others as false. Unlike Boethius Abaelard properly appreciates that conjunctions must be treated as logical units. Even he, however, falls victim to difficulties which arise when this connective is brought into contact with negation and the conditions which he lays down for the truth of a conditional.     

A process model of the understanding of uncertain conditionals

To build a process model of the understanding of conditionals we extract a common core of three semantics of if-then sentences: (a) the conditional event interpretation in the coherencebased probability logic, (b) the discourse processingtheory of Hans Kamp, and (c) the game-theoretical approach of Jaakko Hintikka. The empirical part reports three experiments in which each participant assessed the probability of 52 if-then sentencesin a truth table task. Each experiment included a second task: An n-back task relating the interpretation of conditionals to working memory, a Bayesian bookbag and poker chip task relating the interpretation of conditionals to probability updating, and a probabilistic modus ponens task relating the interpretation of conditionals to a classical inference task. Data analysis shows that the way in which the conditionals are interpreted correlates with each of the supplementary tasks. The results are discussed within the process model proposed in the introduction.     

Conditional reasoning in context: A developmental dual processes account

The way individuals interpret “if p then q” conditionals varies with content and context, often resulting in a biconditional reading. Surprisingly, truth table tasks reveal the existence of two different types of biconditional interpretations: equivalence, as for promises and threats, and defective biconditional, as for causal conditionals or indicative conditionals involving binary terms. The aim of this study was to determine how the interpretation of indicative conditionals is affected in children, adolescents, and adults, by restricting their context of enunciation to only one possible alternative for both the antecedent and the consequent. Moreover, we wanted to determine what is the exact nature of the biconditional interpretation induced by these restricted contexts. For this purpose, third, sixth, and ninth graders and adults performed a truth-value task on indicative conditionals presented either in restricted on non-restricted contexts. Restricted contexts had no effect on children who have a conjunctive interpretation of the conditional, but elicited a predominant defective biconditional reading in adolescents and adults. These results corroborate the developmental dual process account of conditional reasoning proposed by Gauffroy and Barrouillet (2009).     

Conditioning against the Grain

This paper discusses counterexamples to the thesis that the probabilities of conditionals are conditional probabilities. It is argued that the discrepancy is systematic and predictable, and that conditional probabilities are crucially involved in the apparently deviant interpretations. Furthermore, the examples suggest that such conditionals have a less prominent reading on which their probability is in fact the conditional probability, and that the two readings are related by a simple step of abductive inference. Central to the proposal is a distinction between causal and purely stochastic dependence between variables.     

Partitions and Conditionals

The literature on conditionals is rife with alternate formulations of the abstract semantics of conditional logic. Each formulation has its own advantages in terms of applications and generalizations; nevertheless, they are for the most part equivalent, in the sense that they underwrite the same range of logical systems. The purpose of the present note is to bring under this umbrella the partition semantics introduced by Brian Skyrms in (Skyrms, 1984).     

Conditional Excluded Middle in Systems of Consequential Implication

It is natural to ask under what conditions negating a conditional is equivalent to negating its consequent. Given a bivalent background logic, this is equivalent to asking about the conjunction of Conditional Excluded Middle (CEM, opposite conditionals are not both false) and Weak Boethius' Thesis (WBT, opposite conditionals are not both true). In the system CI.0 of consequential implication, which is intertranslatable with the modal logic KT, WBT is a theorem, so it is natural to ask which instances of CEM are derivable. We also investigate the systems CIw and CI of consequential implication, corresponding to the modal logics K and KD respectively, with occasional remarks about stronger systems. While unrestricted CEM produces modal collapse in all these systems, CEM restricted to contingent formulas yields the Alt2 axiom (semantically, each world can see at most two worlds), which corresponds to the symmetry of consequential implication. It is proved that in all the main systems considered, a given instance of CEM is derivable if and only if the result of replacing consequential implication by the material biconditional in one or other of its disjuncts is provable. Several related results are also proved. The methods of the paper are those of propositional modal logic as applied to a special sort of conditional.