Forty years of ``unnatural'' natural deduction and quantification: a history of first-order systems of natural deduction from Gentzen to Copi

Abstract

Systems of natural deduction as a method of theorem proving were first developed in detail by Gentzen and Jáskowski. The complex nature of the rules for quantification in these early systems of natural deduction, and in particular of the parallel rules of Universal Generalization (UG) and Existential Instantiation (El) led to an active research program in the period from the late 1950's to the late 1960's in an effort by Quine, Suppes, Leblanc, and Copi to develop simplified and correct rules of inferences for the quantified formulae of first-order functional logic. We explore the history of these efforts and examine the difficulties found in the "unnatural" natural deductive systems that were developed for first-order logic. Our survey covers roughly forty years, from 1929 to 1971, which includes the early work of Hertz on which Gentzen's work was based through the development of the quantifation rules in Copi's system of natural deduction as presented in the third edition of his textbook Symbolic logic. Also considered is the extent to which O. H. Mitchell made a start at developing a system of natural deduction in his 1883 paper On a new algebra