- 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 Lowenheim-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.
- 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.
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