I present some counterexamples to Adams's Thesis and explain how they undermine arguments that indicative conditionals cannot be truth-evaluable propositions.
According to Quine, in any disagreement over basic logical laws the contesting parties must mean different things by the connectives or quantifiers implicated in those laws; when a deviant logician ‘tries to deny the doctrine he only changes the subject’. The standard (Heyting) semantics for intuitionism offers some confirmation for this thesis, for it represents an intuitionist as attaching quite different senses to the connectives than does a classical logician. All the same, I think Quine was wrong, even about the (...) dispute between classicists and intuitionists. I argue for this by presenting an account of consequence, and a cognate semantic theory for the language of the propositional calculus, which (a) respects the meanings of the connectives as embodied in the familiar classical truth-tables, (b) does not presuppose Bivalence, and with respect to which (c) the rules of the intuitionist propositional calculus are sound and complete. Thus the disagreement between classicists and intuitionists, at least, need not stem from their attaching different senses to the connectives; one may deny the doctrine without changing the subject. The basic notion of my semantic theory is truth at a possibility , where a possibility is a way that (some) things might be, but which differs from a possible world in that the way in question need not be fully specific or determinate. I compare my approach with a previous theory of truth at a possibility due to Lloyd Humberstone, and with a previous attempt to refute Quine’s thesis due to John McDowell. (shrink)
It seems beyond doubt that a thinker can come to know a conclusion by deducing it from premisses that he knows already, but philosophers have found it puzzling how a thinker could acquire knowledge in this way. Assuming a broadly externalist conception of knowledge, I explain why judgements competently deduced from known premisses are themselves knowledgeable. Assuming an exclusionary conception of judgeable content, I further explain how such judgements can be informative. (According to the exclusionary conception, which I develop from (...) some remarks in Ramsey, a judgement's content is given by the hitherto live possibilities that it excludes or rules out.) I propose that the value of logic lies in its allowing us to combine different sources of knowledge, so that we can learn things that we could not learn from those sources individually. I conclude by arguing that while single-conclusion logics possess that value, multiple-conclusion logics do not. (shrink)
Timothy Williamson has recently put forward a proof that every object exists necessarily. I show where the proof fails. My diagnosis also exposes the fallacy in A. N. Prior's argument in favour of his modal logic, Q.
In what does the sense of a sentential connective consist? Like many others, I hold that its sense lies in rules that govern deductions. In the present paper, however, I argue that a classical logician should take the relevant deductions to be arguments involving affirmative or negative answers to yes-or-no questions that contain the connective. An intuitionistic logician will differ in concentrating exclusively upon affirmative answers. I conclude by arguing that a well known intuitionistic criticism of classical logic fails if (...) the answer 'No' is accorded parity with the answer 'Yes'. (shrink)
The paper defends the intelligibility of unrestricted quantification. For any natural number n, 'There are at least n individuals' is logically true, when the quantifier is unrestricted. In response to the objection that such sentences should not count as logically true because existence is contingent, it is argued by consideration of cross-world counting principles that in the relevant sense of 'exist' existence is not contingent. A tentative extension of the upward L?wenheim-Skolem theorem to proper classes is used to argue that (...) a sound and complete axiomatization of the logic of unrestricted universal quantification results from adding all sentences of the form 'There are at least n individuals' as axioms to a standard axiomatization of the first-order predicate calculus. (shrink)
[Ian Rumfitt] Frege's logicism in the philosophy of arithmetic consisted, au fond, in the claim that in justifying basic arithmetical axioms a thinker need appeal only to methods and principles which he already needs to appeal in order to justify paradigmatically logical truths and paradigmatically logical forms of inference. Using ideas of Gentzen to spell out what these methods and principles might include, I sketch a strategy for vindicating this logicist claim for the special case of the arithmetic of the (...) finite cardinals. /// [Timothy Williamson]The paper defends the intelligibility of unrestricted quantification. For any natural number n, 'There are at least n individuals' is logically true, when the quantifier is unrestricted. In response to the objection that such sentences should not count as logically true because existence is contingent, it is argued by consideration of cross-world counting principles that in the relevant sense of 'exist' existence is not contingent. A tentative extension of the upward Löwenheim-Skolem theorem to proper classes is used to argue that a sound and complete axiomatization of the logic of unrestricted universal quantification results from adding all sentences of the form 'There are at least n individuals' as axioms to a standard axiomatization of the first-order predicate calculus. (shrink)
