**Description: **Axiom of Specialization.
A quantified wff implies the wff without a
quantifier (i.e. an instance, or special case, of the generalized wff).
In other words if something is true for all 𝑥, it is true for any
specific 𝑥 (that would typically occur as a free
variable in the wff
substituted for 𝜑). (A free variable is one that does
not occur in
the scope of a quantifier: 𝑥 and 𝑦 are both free in 𝑥 = 𝑦,
but only 𝑥 is free in ∀𝑦𝑥 = 𝑦.) Axiom scheme C5' in [Megill]
p. 448 (p. 16 of the preprint). Also appears as Axiom B5 of [Tarski]
p. 67 (under his system S2, defined in the last paragraph on p. 77).
Note that the converse of this axiom does not hold in general, but a
weaker inference form of the converse holds and is expressed as rule
ax-gen 1425. Conditional forms of the converse are given
by ax-12 1489,
ax-16 1786, and ax-17 1506.
Unlike the more general textbook Axiom of Specialization, we cannot choose
a variable different from 𝑥 for the special case. For use, that
requires the assistance of equality axioms, and we deal with it later
after we introduce the definition of proper substitution - see stdpc4 1748.
(Contributed by NM, 5-Aug-1993.) |