Natural deduction rules for English

Abstract

A system of natural deduction rules is proposed for an idealized form of English. The rules presuppose a sharp distinction between proper names and such expressions as ‘the c’, ‘a (an) c’, ‘some c’, ‘any c’, and ‘every c’, where ‘c’ represents a common noun. These latter expressions are called quantifiers, and other expressions of the form ‘that c’ or ‘that c itself’, are called quantified terms. Introduction and elimination rules are presented for any, every, some, a (an), and the, and also for any which, every which, and so on, as well as rules for some other concepts. One outcome of these rules is that ‘Every man loves some woman’ is implied by, but does not imply, ‘Some woman is loved by every man’, since the latter is taken to mean the same as ‘Some woman is loved by all men’. Also, ‘Jack knows which woman came’ is implied by ‘Some woman is known by Jack to have come’, but not by ‘Jack knows that some woman came’.