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Full Description
Alonzo Church was undeniably one ofthe intellectual giants of theTwenti- eth Century . These articles are dedicated to his memory and illustrate the tremendous importance his ideas have had in logic , mathematics, comput er science and philosophy . Discussions of some of thesevarious contributions have appeared in The Bulletin of Symbolic Logic, and th e interested reader is invited to seek details there . Here we justtry to give somegener al sense of the scope, depth,and value of his work. Church is perhaps best known for the theorem , appropriately called " C h u r c h ' s Theorem ", that there is no decision procedure forthelogical valid- ity of formulas first-order of logic . A d ecision proce dure forthat part of logic would have come near to fulfilling Leibniz's dream of a calculus that could be mechanically used tosettle logical disputes . It was not to . be It could not be . What Church proved precisely is that there is no lambda-definable function that can i n every case providethe right answer , ' y e s ' or ' n o', tothe question of whether or not any arbitrarily given formula is valid .
Contents
Logic, truth and number: The elementary genesis of arithmetic.- Second-order logic.- A representation of relation algebras using Routley-Meyer frames.- Church's set theory with a universal set.- Axioms of infinity in Church's type theory.- Logical objects.- The lambda calculus and adjoint functors.- Atomic Boolean algebras and classical propositional logic.- Improved decision procedures for pure relevant logic.- The "triumph" of first-order languages.- Equivalence relations and groups.- Discriminating coded lambda terms.- ?-calculus as a foundation for mathematics.- Peano's lambda calculus: The functional abstraction implicit in arithmetic.- The undecidability of ?-definability.- A construction of the provable wellorderings of the theory of species.- Semantics for first and higher order realizability.- Language and equality theory in logic programming.- Alternative (1*): A criterion of identity for intensional entities.- Nominalist paraphrase and ontological commitment.- Peace, justice and computation: Leibniz' program and the moral and political significance of Church's theorem.- Tarski's theorem and NFU.- Church's theorem and randomness.- Russellian type theory and semantical paradoxes.- The logic of sense and denotation: Extensions and applications.- Analysis, synonymy and sense.- The very possibility of language.
