Near-equational and equational systems of logic for partial functions. I

Abstract

Equational logic for total functions is a remarkable fragment of first-order logic. Rich enough to lend itself to many uses, it is also quite austere. The only predicate symbol is one for a notion of equality, and there are no logical connectives. Proof theory for equational logic therefore is different from proof theory for other logics and, in some respects, more transparent. The question therefore arises to what extent a logic with a similar proof theory can be constructed when expressive power is increased.The increase mainly studied here allows one both to consider arbitrary partial functions and to express the condition that a function be total. A further increase taken into account is equivalent to a change to universal Horn sentences for partial and for total functions.Two ways of increasing expressive power will be considered. In both cases, the notion of equality is modified and nonlogical function symbols are interpreted as ranging over partial functions, instead of ranging only over total functions. In one case, the only further change is the addition of symbols that denote logical functions, such as the binary projection function Ae that maps each pair ‹a0, a1› of elements of a set A into the element a0. An addition of this kind results in a language, and also in a system of logic based on this language, which we call equational In the other case, instead of adding a symbol for Ae, one admits those special universal Horn sentences in which the conditions expressed by the antecedent are, in a sense, pure conditions of existence. Languages and systems of logic that result from a change of this kind will be called near-equational. According to whether the number of existence conditions that one may express in antecedents is finite or arbitrary, the resulting language and logic shall be finite or infinitary, respectively. Each of our finite near-equational languages turns out to be equivalent to one of our equational languages, and vice versa.