**Description: **Axiom of Specialization.
A universally quantified wff implies the wff
without the universal quantifier (i.e., an instance, or special case, of
the generalized wff). In other words, if something is true for all
𝑥, then 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 1797. Conditional forms of the converse are given
by ax-13 2379,
ax-c14 36206, ax-c16 36207, and ax-5 1911.
Unlike the more general textbook Axiom of Specialization, we cannot choose
a variable different from 𝑥 for the special case. In our
axiomatization, that requires the assistance of equality axioms, and we
deal with it later after we introduce the definition of proper
substitution (see stdpc4 2073).
An interesting alternate axiomatization uses axc5c711 36233 and ax-c4 36199 in
place of ax-c5 36198, ax-4 1811, ax-10 2142, and ax-11 2158.
This axiom is obsolete and should no longer be used. It is proved above
as theorem sp 2180. (Contributed by NM, 3-Jan-1993.) Use sp 2180
instead.
(New usage is discouraged.) |