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Exercise 5 . 1 . 3 . Mimic the proof of Generalization Theorem to show the fol - lowing Generalization on Constant Theorem, which will
Exercise Mimic the proof of Generalization Theorem to show the fol
lowing Generalization on Constant Theorem, which will be used in the proof of
Completeness Theorem. Let and be a formula, a constant and a variable
respectively. We abuse : to denote the formula obtained from by
substituting for all occurrences of
a Show that if is an axiom, then : is also an axiom. Hint: Check
that for any term :: is ::: where
: is the term obtained from by substituting for all occurrences
of
b Generalization on Constants Theorem Assume that and that
is a constant symbol which does not occur in Then there is a variable
which does not occur in such that : Furthermore, there
is a deduction of AAy: from in which does not occur.
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