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Capturing the relationship between conditionals and conditional probability with a trivalent semantics

2014; Taylor & Francis; Volume: 24; Issue: 1-2 Linguagem: Inglês

10.1080/11663081.2014.911535

1958-5780

Daniel Rothschild,

Bayesian Modeling and Causal Inference

AbstractAssigning truth-conditions to conditional statements leads to problems in assigning probabilities to those statements (Lewis, 1976). This note presents and assesses a trivalent semantics of conditional sentence, arguing that this semantics does well at capturing the probabilities of conditional statements. The major problems and prospects for this view are reviewed.Keywords: conditionalstrivalenceprobabilitytriviality AcknowledgementsI am grateful to Chris Barker, Denis Bonnay, Paul Égré, Danny Fox, Kai von Fintel, Linda Rothschild, James Shaw, Robbie Williams, Seth Yalcin and two anonymous reviewers for many helpful comments on earlier drafts.Notes1 Kratzer et al. (Citation1981); Kratzer (Citation1981, Citation1986, Citation2012) denies this syntactic parsing, claiming that the function of 'if'-clauses is to restrict higher up modal quantifiers in the sentence; in the case of (1-b) the natural choice is the probability operator it is likely that. While there is much to be said for her view, I think that, generally speaking, her strategy will not easily explain all the recalcitrant facts about probability and conditionals that the proposal presented below aims to. For example, many judge (7) to have the same truth conditions as (1-b):However, in this case it seems rather implausible that the antecedent of the conditional embedded under it is true that pops out to restrict it is likely. Other cases can be found of this sort: such as cases where there is no linguistically present modal operator at all and speakers or audiences merely make judgements of the probability of various conditional sentences without expressing them out loud. But I will leave further discussion of these points to another occasion, so that this discussion may be considered as presenting an alternative to Kratzer's account of (1-a) and (1-b). See von Fintel (Citation2007), Égré and Cozic (Citation2010), and Rothschild (Citation2011, Citationin press) for further discussion.2 This is a variation on Lewis's first 'triviality theorem' (Lewis, Citation1976).3 One can construct for instance by making it the result of conditionalising on . It will then follow that , since but .4 Some have argued that the proposition expressed by a conditional sentence varies with the epistemic state of the speaker – this is, for instance, a direct consequence of Kratzer's theory of epistemic modals and indicative conditionals. In this case, the argument I gave in the previous paragraph would have no force. There are, in fact, further problems with maintaining 'the equation' even if one allows conditionals to express different propositions relative to different credal states, but they will not be discussed here. See Edgington (Citation1995) and Bennett (Citation2003) for discussion and citations to the major results.5 See also Huitink (Citation2008); HuitinkCompass and Cantwell (Citation2008) for more in this vein.6 This is the same trivalent truth table for conditionals given by McDermott (Citation1996) and de Finetti (Citation1936). In McDermott's account, however, the probability of a conditional is not its conditional probability (p. 20), rather it is simply the probability it is true.7 As follows: is true in all worlds in which and are true, false in all worlds in which one of or is false, and undefined in all other worlds. is false in all worlds in which and are false, true in all worlds in which one of or is true, and undefined in all other worlds is true in all worlds in which is false, false in all worlds in which is true, and undefined in all other worlds. Since we will mostly talk about conjoining bivalent formulas with and we can usually just think about the connectives as classical.8 Cantwell (Citation2006) proposes this same way of extending probability functions to trivalent formulas, though he does not use it for the same purposes as I have done here.9 These kinds of cases lead McDermott (Citation1996 p. 2) to argue for the trivalent account.10 As noted above, however, these rules do not work for other forms of trivalence.11 Of particular use here is the literature on quantified conditionals (e.g., *FintelQuantifiers, FintelIatridou, LeslieIf, KlinedinstQuantified).