Proof-theoretic validity

Abstract

The idea of proof-theoretic validity originated in the work of Gerhard Gentzen, when he suggested that the meaning of each logical expression was encapsulated in its introduction-rules, and that the elimination-rules were justified by the meaning so given. It was developed by Dag Prawitz in a series of articles in the early 1970s, and by Michael Dummett in his William James lectures of 1976, later published as The Logical Basis of Metaphysics. The idea had been attacked in 1960 by Arthur Prior under the soubriquet 'analytic validity'. Logical truths and logical consequences are deemed analytically valid by virtue of following, in a way which the present paper clarifies, from the meaning of the logical constants. But different logics are based on different rules, confer different meanings and so validate different theorems and consequences, some of which are arguably not true or valid at all. It seems to follow that some analytic statements are in fact false. The moral is that we must be careful what rules we adopt and what meanings we use our rules to determine.