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.
Solomon Feferman is Professor of Mathematics and Philosophy, Emeritus and the Patrick Suppes Family Professor of Humanities and Sciences, Emeritus at Stanford University. In 2003, Professor Feferman was the recipient of the Rolf Schock Prize in Logic and Philosophy awarded by the Royal Swedish Academy of Sciences. Professor Feferman is editor-in-chief of the five-volume edition of Kurt Gödel: Collected Works. In 2004 he and his wife, Anita Burdman Feferman, published Alfred Tarski: Life and Logic, a biography of the famed logician, Alfred Tarski, with whom he completed his doctoral studies at the University of California, Berkeley in 1957.