Response to Dummett
|i.||that the vocabulary of higher-order logic plus the cardinality operator, 'Nx: . . . x . . .', provides a sufficient definitional basis for a statement of the basic laws of arithmetic;|
|ii.||that when they are so stated, N= provides for a derivation of those laws within higher-order logic;|
|iii.||that someone who understood a higher-order language to which the cardinality operator was to be added would learn, on being told that N= is analytic of that operator, all that it is necessary to know in order to construe any of the new statements that would then be formulable.1|
|iv.||Finally and crucially, that N= may be laid down without significant epistemological obligation: that it may simply be stipulated as an explanation of the meaning of statements of numerical identity, and that-- beyond the issue of the satisfaction of the truth-conditions it thereby lays down for such statements--no competent demand arises for an independent assurance that there are objects whose conditions of identity are as it stipulates.|
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Publication information: Book title: The Philosophy of Mathematics Today. Contributors: Matthias Schirn - Editor. Publisher: Clarendon Press. Place of publication: Oxford. Publication year: 1998. Page number: 389.