Ian Rumfitt

Many prominent writers on the philosophy of logic, including Michael Dummett, Dag Prawitz, Neil Tennant, have held that the introduction and elimination rules of a logical connective must be ‘in harmony ’ if the connective is to possess a sense. This Harmony Thesis has been used to justify the choice of logic: in particular, supposed violations of it by the classical rules for negation have been the basis for arguments for switching from classical to intuitionistic logic. The Thesis has also had an influence on the philosophy of language: some prominent writers in that area, notably Dummett and Robert Brandom, have taken it to be a special case of a more general requirement that the grounds for asserting a statement must cohere with its consequences. This essay considers various ways of making the Harmony Thesis precise and scrutinizes the most influential arguments for it. The verdict is negative: all the extant arguments for the Thesis are weak, and no version of it is remotely plausible.

[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