Solomon Feferman
Stanford University

"The "Logic" Question"

Abstract:

In Alfred Tarski's posthumously published lecture, "What are logical notions?", he proposed a permutation invariance criterion in answer to that question. The permutation invariant operations on any given domain were later characterized by Vann McGee as exactly those definable in the language L_{\infty,\infty} allowing disjunctions, conjunctions and quantifier strings of arbitrary cardinality; McGee also characterized in terms of the same language the operations that are isomorphism invariant, as proposed by Gila Sher. In my 1998 article "Logic, logics and logicism", I critiqued the Tarski-Sher thesis on several grounds, including that it assimilates logic to mathematics (specifically to set theory) and that the notions involved in the characterization are not robust. Imposing set-theoretical absoluteness as one additional criterion leads me--by contrast--to characterize the logical operations as exactly those definable in the language L_{\omega,\omega} of ordinary classical first-order predicate calculus.

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