Point-free Substitution

0 Introduction In modern treatments of predicate calculus [2], no mention is made of states or variables up until the point where substitution is introduced. There, suddenly, the abstraction that reigned before is cast to the winds, and a substitution is defined as the result of a textual replacement of variable names by expressions. It is the purpose of this note to remedy this breach of style by proposing a new characterization of substitution, one that is meaningful also in point-free models and that is equivalent to the classical definition [3] wherever the latter is applicable. More precisely, we prove that a predicate transformer is a substitution according to the classical definition iff it is both universally conjunctive and universally disjunctive. To this purpose, we introduce a number of postulates limiting the set of valid models for the predicate calculus until we finally arrive at a context where variables are available. Our first postulate concerns the existence of a covering set of point predicates. In Section 1 we deduce from this that universal junctivity of f is equivalent to the existence of a point predicate transformer h such that, for every point predicate p and every predicate x , [p ⇒ f.x] ≡ [h.p ⇒ x]. In Section 2 we postulate the existence of a state space and prove that universal junctivity of f is equivalent to the existence of a state transformer g such that, for every predicate x , [f.x ≡ x • g]. Our third and final postulate introduces variables and enables us to prove, in Section 3, equivalence to a classical definition of substitution. Throughout, we assume familiarity with predicate calculus as developed in [2]. In particular , bracketing denotes a special predicate transformer called 'everywhere' that maps every predicate to a scalar. It is postulated to be universally conjunctive and to satisfy Leibniz's rule [x ≡ y] ⇒ [f.x ≡ f.y] for every predicate transformer f .