Standard accounts of semantics for counterfactuals confront the true–true problem: when the antecedent and consequent of a counterfactual are both actually true, the counterfactual is automatically true. This problem presents a challenge to safety‐based accounts of knowledge. In this paper, drawing on work by Angelika Kratzer, Alan Penczek, and Duncan Pritchard, we propose a revised understanding of semantics for counterfactuals utilizing machinery from generalized quantifier theory which enables safety theorists to meet the challenge of the true–true problem.
That believing truly as a matter of luck does not generally constitute knowing has become epistemic commonplace. Accounts of knowledge incorporating this anti-luck idea frequently rely on one or another of a safety or sensitivity condition. Sensitivity-based accounts of knowledge have a well-known problem with necessary truths, to wit, that any believed necessary truth trivially counts as knowledge on such accounts. In this paper, we argue that safety-based accounts similarly trivialize knowledge of necessary truths and that two ways of responding to this problem for safety, issuing from work by Williamson and Pritchard, are of dubious success.
It is evident that the differences between intuitionistic logic (and mathematics) and its classical counterpart(s) can be traced back at least in part to their differing treatments of negation. The intuitionistic failures of double negation elimination and classical reductio ad absurdum are the most obvious facts supporting this. As a result, the definition and/or rules formulated for negation are crucial in comparing these competing logics. One quite common method of handling intuitionistic negation, however, turns out to be untenable.The method in question is the definition of intuitionistic negation in terms of '0 = 1' and the conditional. In Elements of Intuitionism Michael Dummett writes that 'surprisingly, negation is definable in intuitionistic arithmetic ' (1977: 35), giving essentially 1 the following definition:This definition of negation is common among intuitionists, and its origin can be traced to Heyting's first formulations (1930) of intuitionistic reasoning.Before giving our objections, it should be pointed out that others have argued against this same definition. For example, Neil Tennant objects to defining negation in terms of '0 = 1' because this definition makes negation dependent on arithmetic, raising the question of 'how one would justify inferential passage from patent absurdities in a non-arithmetical discourse to the supposedly primal absurdity 0 = 1' (1999) in accounting for the behaviour of negation. On Tennant's account, if P and Q are two contrary sentences of some non-arithmetic discourse, say 'The ball is red' and 'The ball is green', then we would need to be able to justify (possibly non-logical) inferences from P and Q to '0 = 1', otherwise we cannot conclude ¬Q (i.e. Q → 0 = 1) from P. It is far from clear how the justification of this inference would proceed, and Tennant observes that for a relevance logician (such as himself) no justification seems possible. This is a subtle philosophical argument 2 for rejecting this manner of 1 Dummett actually writes this as a biconditional instead of using = df .2 A related philosophical criticism has to do with the use of '0 = 1' in the definition. One wonders why we should not just define ¬A as (A → 0 = 3) or (A → 0 = 247) or (A → 47 = 396). None of these seem any less arbitrary than using '0 = 1', yet one may worry that they impose different meanings on the negation operator. The objections raised in this paper, however, demonstrate that all definitions of this sort are inadequate, and thus that these quibbles are irrelevant.
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