**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
∀*y**x* = *y*.) This is one of the axioms of
what we call "pure" predicate calculus (ax-4 2135
through ax-7 1734 plus rule
ax-gen 1546). 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 1546. Conditional forms of the converse are given
by ax-12 1925,
ax-15 2143, ax-16 2144, and ax-17 1616.
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 2024.
An interesting alternate axiomatization uses ax467 2169 and ax-5o 2136 in place
of ax-4 2135, ax-5 1557, ax-6 1729,
and ax-7 1734.
This axiom is obsolete and should no longer be used. It is proved above
as theorem sp 1747. (Contributed by NM, 5-Aug-1993.)
(New usage is discouraged.) |