Proofs Without Syntax

Details Proofs Without Syntax Proofs Without Syntax

2004-08-20

arxiv.org/abs/math/0408282v3

Annals of Mathematics, 2006

Mathematics

Logic

Appears in Annals of Mathematics, 2006. 5 pages + references. Version 1 is submitted version; v3 is final published version (i

Scientific paper

"[M]athematicians care no more for logic than logicians for mathematics." Augustus de Morgan, 1868. Proofs are traditionally syntactic, inductively generated objects. This paper presents an abstract mathematical formulation of propositional calculus (propositional logic) in which proofs are combinatorial (graph-theoretic), rather than syntactic. It defines a *combinatorial proof* of a proposition P as a graph homomorphism h : C -> G(P), where G(P) is a graph associated with P and C is a coloured graph. The main theorem is soundness and completeness: P is true iff there exists a combinatorial proof h : C -> G(P).

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