**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 1713. Conditional forms of the converse are given
by ax-13 2234,
ax-c14 33194, ax-c16 33195, and ax-5 1827.
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 2341.
An interesting alternate axiomatization uses axc5c711 33221 and ax-c4 33187 in
place of ax-c5 33186, ax-4 1728, ax-10 2006, and ax-11 2021.
This axiom is obsolete and should no longer be used. It is proved above
as theorem sp 2041. (Contributed by NM, 3-Jan-1993.)
(New usage is discouraged.) |