| Abstract: | 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.
Of the many questions on which logic is neutral, one is usually supposed to be this: 'How many individuals are there?' On the alternative view defended below, truths about the number of individuals are logically true. They are not contingent logical truths, for it is not contingent what individuals there are. |
