How mathematics isn't logic
Ratio 12 (3):279–295 (1999)
| Abstract | If logical truth is necessitated by sheer syntax, mathematics is categorially unlike logic even if all mathematics derives from definitions and logical principles. This contrast gets obscured by the plausibility of the Synonym Substitution Principle implicit in conceptions of analyticity: synonym substitution cannot alter sentence sense. The Principle obviously fails with intercepting: nonuniform term substitution in logical sentences. 'Televisions are televisions' and 'TVs are televisions' neither sound alike nor are used interchangeably. Interception synonymy gets assumed because logical sentences and their synomic interceptions have identical factual content, which seems to exhaust semantic content. However, intercepting alters syntax by eliminating term recurrence, the sole strictly syntactic means of ensuring necessary term coextension, and thereby syntactically securing necessary truth. Interceptional necessity is lexical, a notational artifact. The denial of interception nonsynonymy and the disregard of term recurrence in logic link with many misconceptions about propositions, logical form, conventions, and metalanguages. Mathematics is distinct from logic: its truth is not syntactic; it is transmitted by synonym substitution; term recurrence has no essential role. The '=' of mathematics is an objectual relation between numbers; the '=' of logic marks a syntactic relation of coreferring terms. | |||||||||
| Keywords | mathematical truth logical truth logical form synonymy identity equality | |||||||||
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Similar books and articlesJohn MacFarlane, Logical Constants. Stanford Encyclopedia of Philosophy.
Kevin C. Klement, Propositional Logic. Internet Encyclopedia of Philosophy.
Christopher Menzel (1998). Logical Form. In Edward Craig (ed.), Routledge Encyclopedia of Philosophy. Routledge.
Shannon Dea (2006). "Merely a Veil Over the Living Thought": Mathematics and Logic in Peirce's Forgotten Spinoza Review. Transactions of the Charles S. Peirce Society 42 (4):501-517.
Roger Wertheimer, Synonymy Without Analyticity. International Philosophical Preprint Exchange.
Roger Wertheimer (2000). The Synonymy Antinomy. In A. Kanamori (ed.), Proceedings of the 20th World Conress of Philosophy, Vol Vi , Analytic Philosophy and Logic. Philosophy Document Center.
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