Counterconventional Conditionals

Some philosophical positions maintain that some aspect of reality depends on human practices, cognitive attitudes or sentiments. This paper presents a framework for understanding such positions in a way that renders them immune to a number of natural but allegedly devastating objections.     

Antiskeptical Conditionals

Empirical knowledge exists in the form of antiskeptical conditionals , which are propositions like [if I am not undetectably deceived, then I am holding a pen]. Such conditionals, despite their trivial appearance, have the same essential content as the categorical propositions that we usually discuss, and can serve the same functions in science and practical reasoning. This paper sketches out two versions of a general response to skepticism that employs these conditionals. The first says that our ordinary knowledge attributions can safely be replaced by statements using antiskeptical conditionals, which provides a way around the standard sort of skeptical argument while accepting its soundness with respect to the usual targets. The second analyzes the objects of our ordinary knowledge attributions as antiskeptical conditionals, which allows us to refute, not just evade, the skeptic's argument. Both versions compare favorably to the best-known current approaches to skepticism, including semantic contextualism.     

Indicative Conditionals

Adams Thesis has much evidence in its favour, but DavidLewis famously showed that it cannot be true, in all but the most trivial of cases, if conditionals are proprositions and their probabilities are classical probabilities of truth. In this paper I show thatsimilar results can be constructed for a much wider class of conditionals. The fact that these results presupposethat the logic of conditionals is Boolean motivates a search for a non-Boolean alternative. It is arguedthat the exact proposition expressed by a conditional depends on the context in which it is uttered. Consequentlyits probability of truth will depend not only on the probabilities of the various propositions it mightexpress, but also on the probabilities of the contexts determining which proposition it does in fact express.The semantic theory developed from this is then shown to explain why agents degrees of belief satisfyAdams Thesis. Finally the theory is compared with proposals for a three-valued logic of conditionals.     

Conditionals in reasoning

The paper presents a non-monotonic inference relation on a language containing a conditional that satisfies the Ramsey Test. The logic is a weakening of classical logic and preserves many of the ‘paradoxes of implication’ associated with the material implication. It is argued, however, that once one makes the proper distinction between supposing that something is the case and accepting that it is the case, these ‘paradoxes’ cease to be counterintuitive. A representation theorem is provided where conditionals are given a non-bivalent semantics and epistemic states are represented via preferential models.     

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 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.     

Conditionals As Representative Inferences

According to Adams (Inquiry 8:166–197, 1965), the acceptability of an indicative conditional goes with the conditional probability of the consequent given the antecedent. However, some conditionals seem to be inappropriate, although their corresponding conditional probability is high. These are cases with a missing link between antecedent and consequent. Other conditionals are appropriate even though the conditional probability is low. Finally, we have the so-called biscuit conditionals. In this paper we will generalize analyses of Douven (Synthese 164:19–44, 2008) and others to account for the appropriateness of conditionals in terms of evidential support. Our generalization involves making use of Value, or intensity. We will show how this generalization helps to account for biscuit conditionals and conditional threats and promises. Finally, a link is established between this analysis of conditionals and an analysis of generic sentences.

     

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.     

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.     

Triviality For Restrictor Conditionals

I present two Triviality results for Kratzer's standard “restrictor” analysis of indicative conditionals (while also clarifying the sense in which Kratzer's semantics might avoid such results). I both refine and undermine the common claim that problems of Triviality do not arise for Kratzer conditionals since they are not strictly conditionals at all.     

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.     

Reasoning About Uncertain Conditionals

There is a long tradition in formal epistemology and in the psychology of reasoning to investigate indicative conditionals. In psychology, the propositional calculus was taken for granted to be the normative standard of reference. Experimental tasks, evaluation of the participants’ responses and psychological model building, were inspired by the semantics of the material conditional. Recent empirical work on indicative conditionals focuses on uncertainty. Consequently, the normative standard of reference has changed. I argue why neither logic nor standard probability theory provide appropriate rationality norms for uncertain conditionals. I advocate coherence based probability logic as an appropriate framework for investigating uncertain conditionals. Detailed proofs of the probabilistic non-informativeness of a paradox of the material conditional illustrate the approach from a formal point of view. I survey selected data on human reasoning about uncertain conditionals which additionally support the plausibility of the approach from an empirical point of view.     

The Probabilities of Conditionals Revisited

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.     

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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