On the logic of nonmonotonic conditionals and conditional probabilities

I will describe the logics of a range of conditionals that behave like conditional probabilities at various levels of probabilistic support. Families of these conditionals will be characterized in terms of the rules that their members obey. I will show that for each conditional, , in a given family, there is a probabilistic support level r and a conditional probability function P such that, for all sentences C and B, CB holds just in case P[B|C]r. Thus, each conditional in a given family behaves like conditional probability above some specific support level.Chris Swoyer provided very helpful comments on drafts of this paper.     

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.     

The evidential support theory of conditionals

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.     

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.     

Epistemic comparative conditionals

The interest of epistemic comparative conditionals comes from the fact that they represent genuine ‘comparative epistemic relations’ between propositions, situations, evidences, abilities, interests, etc. This paper argues that various types of epistemic comparative conditionals uniformly represent comparative epistemic relations via the comparison of epistemic positions rather than the comparison of epistemic standards. This consequence is considered as a general constraint on a theory of knowledge attribution, and then further used to argue against the contextualist thesis that, in some cases, considering a new counter- possibility can raise the epistemic standard of knowledge attribution. Instead, the paper shows that considering a new counter-possibility can only lower the epistemic position of a putative knower. Moreover, since the comparison, by the nature of conditionals, is free from any commitment to the truth-values of specific knowledge attributions, my conclusion is free from the debate between contextualism and invariantism on whether the truth-value of a knowledge attribution can actually vary with context.     

Reasoning with conditionals

This paper reviews the psychological investigation of reasoning with conditionals, putting an emphasis on recent work. In the first part, a few methodological remarks are presented. In the second part, the main theories of deductive reasoning (mental rules, mental models, and the probabilistic approach) are considered in turn; their content is summarised and the semantics they assume for if and the way they explain formal conditional reasoning are discussed, in particular in the light of experimental work on the probability of conditionals. The last part presents the recent shift of interest towards the study of conditional reasoning in context, that is, with large knowledge bases and uncertain premises.     

Conversation and conditionals

I outline and motivate a way of implementing a closest world theory of indicatives, appealing to Stalnaker’s framework of open conversational possibilities. Stalnakerian conversational dynamics helps us resolve two outstanding puzzles for a such a theory of indicative conditionals. The first puzzle—concerning so-called ‘reverse Sobel sequences’—can be resolved by conversation dynamics in a theory-neutral way: the explanation works as much for Lewisian counterfactuals as for the account of indicatives developed here. Resolving the second puzzle, by contrast, relies on the interplay between the particular theory of indicative conditionals developed here and Stalnakerian dynamics. The upshot is an attractive resolution of the so-called “Gibbard phenomenon” for indicative conditionals.     

The development of reasoning with causal conditionals.

A total of 512 children in Grades 1 through 6 received a conditional inference task using causal conditionals (If cause P, then effect Q) and a generation of alternatives task. The inference task used premises for which there were few or many possible alternative causes. Results show a steady age-related increase in uncertainty responses to the two uncertain logical forms, affirmation of consequent (AC) and denial of antecedent (DA), and an increase in production of disabling conditions for modus ponens. More uncertainty responses were produced to AC and DA with premises with many possible alternatives. Individual differences in inference production were related to numbers of alternatives produced in the generation task. Results support the idea that both developmental and individual differences in reasoning can be at least partially explained by differential access to knowledge stored in long-term memory.     

The negations of conjunctions,conditionals, and disjunctions

How do reasoners understand and formulate denials of compound assertions, such as conjunctions and disjunctions? A theory based on mental models postulates that individuals enumerate models of the various possibilities consistent with the assertions. It therefore predicts a novel interaction: in affirmations, conjunctions, A and B, which refer to one possibility, should be easier to understand than disjunctions, A or B, which refer to more than one possibility; in denials, conjunctions, not(A and B), which refer to more than one possibility, should be harder to understand than disjunctions, not(A or B), which do not. Conditionals are ambiguous and they should be of intermediate difficulty. Experiment 1 corroborated this trend with a task in which the participants selected which possibilities were consistent with assertions, such as: Bob denied that he wore a yellow shirt and he wore blue pants on Tuesday. Experiment 2 likewise showed that participants' own formulations of verbal denials yielded the same trend in which denials of conjunctions were harder than denials of conditionals, which in turn were harder than denials of disjunctions.     

Bivalence and subjunctive conditionals

Writers such as Stalnaker and Dummett have argued that specific features of subjunctive conditional statements undermine the principle of bivalence. This, paper is concerned with rebutting such claims. 1. It is shown how subjective conditionals pose a prima facie threat to bivalence, and how this threat can be dissolved by a distinction between the results of negating a subjective conditional and of negating its consequent. To make this distinction is to side with Lewis against Stalnaker in a dispute about possible worlds semantics for such conditionals, and reasons are given for doing so. 2. These arguments are extended to answer Dummett's claim that behaviourist and phenomenalist analyses in terms of subjunctive conditions violate bivalence. This answer is shown to be compatible with the principle that hypothetical statements are true only in virtue of categorical facts.     

Causal conditionals and counterfactuals

Causal counterfactuals e.g., 'if the ignition key had been turned then the car would have started' and causal conditionals e.g., 'if the ignition key was turned then the car started' are understood by thinking about multiple possibilities of different sorts, as shown in six experiments using converging evidence from three different types of measures. Experiments 1a and 1b showed that conditionals that comprise enabling causes, e.g., 'if the ignition key was turned then the car started' primed people to read quickly conjunctions referring to the possibility of the enabler occurring without the outcome, e.g., 'the ignition key was turned and the car did not start'. Experiments 2a and 2b showed that people paraphrased causal conditionals by using causal or temporal connectives (because, when), whereas they paraphrased causal counterfactuals by using subjunctive constructions (had…would have). Experiments 3a and 3b showed that people made different inferences from counterfactuals presented with enabling conditions compared to none. The implications of the results for alternative theories of conditionals are discussed.