number-theory

6A Intuitionist implicational logic 75As mentioned earlier, the formulae provable in the logic defined below will turnout to be exactly the types of closed A-terms. However, we can also go one stepdeeper than this and get a correspondence between the proofs of these formulae andthe A-terms themselves. But to make this deeper correspondence work we must bevery careful about the definition of "proof" and "deduction" in a Natural Deductionsystem, and we must clarify some features of such deductions that are not oftenemphasised in the standard literature on the topic.Therefore some of the definitions below will involve position-labels and othertedious details that have so far been confined to Chapter 9. These will be neededwhen determining exactly which deductions correspond to which A-terms.6A1 Definition Implicational formulae (denoted by lower-case Greek letters otherthan "A") are built from propositional variables (denoted by "a", "b", "c", ...) usingthe implication connective "->" thus: if a and r are formulae, then so is (a--*r). (Aninfinite sequence of propositional variables is assumed to be given.)6A1.1 Note (i) Of course implicational formulae are exactly the same as types.Parentheses will be omitted using the same association-to-the-right convention asfor types, see 2A1.1.(ii) There are no propositional constants and no other connectives than(Because this book is only concerned with the Curry-Howard correspondence in itssimplest and smoothest form.)6A1.2 Background The intuitionist philosophy of mathematics was developed in theearly 1900's by L. E. J. Brouwer and his group in Holland, and a very readableintroduction is in Heyting 1955. One feature of this philosophy is that some propositionalformulae that are usually accepted as universally valid, and are tautologiesin the usual truth-table sense, are regarded by the intuitionists as having only limitedvalidity. One such formula is(PL)((a-+b)-->a)--*a,which is known as Peirce's law. The logic developed by the intuitionists has attractedcontinuing interest over the years from many logicians and computer scientists quiteindependently of their philosophical views. For example many of the polymorphictype-theories in the current literature have intuitionist logic as their basis, notclassical logic.The formulae provable by the rules in the next definition can be shown tocoincide exactly with the implicational formulae that the intuitionists accept asuniversally valid. (See Troelstra and van Dalen 1988, Ch.2 (System IPC) and Ch.10§5 (separation theorem).)