The dynamics of belief: Contractions and revisions of probability functions

Conclusion Using probability functions defined over a simple language as models of states of belief, my goal in this article has been to analyse contractions and revisions of beliefs. My first strategy was to formulate postulates for these processes. Close parallels between the postulates for contractions and the postulates for revisions have been established - the results in Section 5 show that contractions and revisions are interchangeable.As a second strategy, some suggestions for more or less explicit constructive definitions of the revision process (and indirectly also of the contraction process) were then presented. However, the results in Section 6 are less conclusive than in the earlier ones. This problem area still awaits further development.     

Conditionals and Curry

Curry’s paradox for “if.. then..” concerns the paradoxical features of sentences of the form “If this very sentence is true, then 2 + 2 = 5”. Standard inference principles lead us to the conclusion that such conditionals have true consequents: so, for example, 2 + 2 = 5 after all. There has been a lot of technical work done on formal options for blocking Curry paradoxes while only compromising a little on the various central principles of logic and meaning that are under threat. Once we have a sense of the technical options, though, a philosophical choice remains. When dealing with puzzles in the logic of conditionals, a natural place to turn is independently motivated semantic theories of the behaviour of “if... then...”. This paper argues that a closest-worlds approach outlined in previous work offers a philosophically satisfying reason to deny conditional proof and so block the paradoxical Curry reasoning, and can give the verdict that standard Curry conditionals are false, along with related “contraction conditionals”.     

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).     

Conditionals and consequences

We examine the notion of conditionals and the role of conditionals in inductive logics and arguments. We identify three mistakes commonly made in the study of, or motivation for, non-classical logics. A nonmonotonic consequence relation based on evidential probability is formulated. With respect to this acceptance relation some rules of inference of System P are unsound, and we propose refinements that hold in our framework.     

Conditionals and Actuality

It is known that indicative and subjunctive conditionals interact differently with a rigidifying “actually” operator. The paper studies this difference in an abstract setting. It does not assume the framework of possible world semantics, characterizing “actually” instead by the type of logically valid formulas to which it gives rise. It is proved that in a language with such features all sentential contexts that are congruential (in the sense that they preserve logical equivalence) are extensional (in the sense that they preserve material equivalence). For a subjunctive conditional, the natural conclusion to draw is that it is non-congruential. It is much harder to defend the claim that an indicative conditional is non-congruential. The pressure to treat the indicative conditional as truth-functional is correspondingly greater. The implications of these results for attempts to interpret the indicative conditional as an epistemic or doxastic operator are assessed.

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Conditionals are material: the positive arguments

A number of papers have argued in favour of the material account of indicative conditionals, but typically they either concentrate on defending the account from the charge that it has counterintuitive consequences, or else focus on some particular positive argument in favour of the theory. In this paper, I survey the various positive arguments that can be given, presenting simple versions where possible and showing the connections between them. I conclude with some methodological considerations.     

Conditionals and indexical relativism

I set out and defend a view on indicative conditionals that I call “indexical relativism”. The core of the view is that which proposition is (semantically) expressed by an utterance of a conditional is a function of (among other things) the speaker’s context and the assessor’s context. This implies a kind of relativism, namely that a single utterance may be correctly assessed as true by one assessor and false by another.     

Indicative and Subjunctive Conditionals

In any plausible semantics for conditionals, the semantics for indicatives and subjunctives will resemble each other closely. This means that if we are to keep the possible-worlds semantics for subjunctives suggested by Lewis, we need to find a possible-worlds semantics for indicatives. One reason for thinking that this will be impossible is the behaviour of rigid designators in indicatives. An indicative like 'If the stuff in the rivers, lakes and oceans really is H₃O, then water is H₃O' is non-vacuously true, even though its consequent is true in no possible worlds, and hence not in the nearest possible world where the antecedent is true. I solve this difficulty by providing a semantics for conditionals within the framework of two-dimensional modal logic. In doing so, I show that we can have a reasonably unified semantics for indicative and subjunctive conditionals.     

Conditionals and Supposition-Based Reasoning

Case-based reasoning is a familiar method of evaluating sentences. But when applied to conditionals, it seems to lead to implausible conclusions. In this paper I argue that the problem arises from equating the probability of a conditional sentence on the evidential supposition of some condition with the conditional probability of the former, given the latter.     

Conditionals Right and Left: Probabilities for the Whole Family

The fact that the standard probabilistic calculus does not define probabilities for sentences with embedded conditionals is a fundamental problem for the probabilistic theory of conditionals. Several authors have explored ways to assign probabilities to such sentences, but those proposals have come under criticism for making counterintuitive predictions. This paper examines the source of the problematic predictions and proposes an amendment which corrects them in a principled way. The account brings intuitions about counterfactual conditionals to bear on the interpretation of indicatives and relies on the notion of causal (in)dependence.     

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.     

Vagueness, Conditionals and Probability

This paper explores the interaction of well-motivated (if controversial) principles governing the probability conditionals, with accounts of what it is for a sentence to be indefinite. The conclusion can be played in a variety of ways. It could be regarded as a new reason to be suspicious of the intuitive data about the probability of conditionals; or, holding fixed the data, it could be used to give traction on the philosophical analysis of a contentious notion—indefiniteness. The paper outlines the various options, and shows that ‘rejectionist’ theories of indefiniteness are incompatible with the results. Rejectionist theories include popular accounts such as supervaluationism, non-classical truth-value gap theories, and accounts of indeterminacy that centre on rejecting the law of excluded middle. An appendix compares the results obtained here with the ‘impossibility’ results descending from Lewis (1976).

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Indicative Conditionals and Graded Information

Journal of Philosophical Logic - I propose an account of indicative conditionals that combines features of minimal change semantics and information semantics. As in information semantics,...     

Kaufmann on the Probabilities of Conditionals

Kaufmann has recently argued that the thesis according to which the probability of an indicative conditional equals the conditional probability of the consequent given the antecedent under certain specifiable circumstances deviates from intuition. He presents a method for calculating the probability of a conditional that does seem to give the intuitively correct result under those circumstances. However, the present paper shows that Kaufmann’s method is inconsistent in that it may lead one to assign different probabilities to a single conditional at the same time.