**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 , it is true for any
specific (that
would typically occur as a free variable in the wff
substituted for ). (A free variable is one that does not occur in
the scope of a quantifier: and are both
free in ,
but only is free
in .) This is
one of the axioms of
what we call "pure" predicate calculus (ax-4 2079
through ax-7 1711 plus rule
ax-gen 1535). 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 1535. Conditional forms of the converse are given
by ax-12 1870,
ax-15 2087, ax-16 2088, and ax-17 1605.
Unlike the more general textbook Axiom of Specialization, we cannot choose
a variable different from 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 1971.
An interesting alternate axiomatization uses ax467 2112 and ax-5o 2080 in place
of ax-4 2079, ax-5 1546, ax-6 1706,
and ax-7 1711.
This axiom is obsolete and should no longer be used. It is proved above
as theorem sp 1719. (Contributed by NM, 5-Aug-1993.)
(New usage is discouraged.) |