Remarks on identity and description in first-order axiom systems

Abstract

Hilbert and Ackermann ([1], p. 107) define a first order axiom system as one in which the axioms contain one or more predicate constants, but no predicate variables. Here “axiom” refers to the specific subject-matter axioms and not to the rules of the restricted predicate calculus (quantification theory), which rules are presupposed for each first-order system. It is pointed out by them that an exception could be made for the predicate of identity; for the axiom scheme for this predicate, namelywhich has in (b) the variable predicate F, could nevertheless be replaced, in any given first-order system, by a finite set of axioms without predicate variables. Thus, for example, if Φ[x, y) is the one constant predicate of such a system then PId(b) could be replaced byThus one postulates, in addition to the reflexivity, symmetry, and transitivity of identity, the substitutivity of identical entities in each of the possible “atomic” contexts of a variable (occurrences in the primitive predicates). In this method of introducing identity it has to be taken as an additional primitive predicate and further axioms are consequently needed. In such a system having PId(a), (b1)−(b3) as axioms, the scheme PId(b) can be derived as a meta-theorem of the system, F(x) then being any formula of the system.