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wlammen committed Jan 21, 2024
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Expand Up @@ -15574,10 +15574,10 @@ only postulated (that is, axiomatic) rule of inference of predicate
set, ~ ax-pr postulates the existence of at least two sets, and finally
~ ax-inf postulates the existence of an infinite number of sets. It is
nevertheless interesting to shortly develop this case because the results are
exactly meeting our intuition. Every universally quantified wwf is provably
true, and every existentially quantified wwf is provably false. Otherwise,
exactly meeting our intuition. Every universally quantified wff is provably
true, and every existentially quantified wff is provably false. Otherwise,
nothing can be said. Reciprocally, in the case that every universally
quantified wwf is true, then ~ ax-6 is negated, but ~ ax-gen , ~ ax-4 ,
quantified wff is true, then ~ ax-6 is negated, but ~ ax-gen , ~ ax-4 ,
~ ax-5 , ~ ax-10 , ~ ax-11 , ~ ax-12 and ~ ax-13 are automatically true, and
so are also every universal quantification of ~ ax-7 , ~ ax-8 and ~ ax-9 .
$)
Expand All @@ -15587,13 +15587,13 @@ only postulated (that is, axiomatic) rule of inference of predicate
nax6ex $p |- ( -. E. x T. -> -. E. x ph ) $=
( wex wtru trud eximi con3i ) ABCDBCADBAEFG $.

$( In an empty domain the for-all operator always holds, even when applied to
a false expression. This theorem actually shows that ~ ax-5 is provable
there. Also we cannot assume that ~ sp generally holds, except of course
in the form of ~ sptruw . Without the support of an ~ sp like theorem it
seems difficult, if not impossible, to arrive at a theorem allowing to
change the bounded variable in the antecedent. Additional axioms need to
be postulated to further strengthen this result.
$( In an empty logical domain the for-all operator always holds, even when
applied to a false expression. This theorem actually shows that ~ ax-5 is
provable there. Also we cannot assume that ~ sp generally holds, except
of course in the form of ~ sptruw . Without the support of an ~ sp like
theorem it seems difficult, if not impossible, to arrive at a theorem
allowing to change the bounded variable in the antecedent. Additional
axioms need to be postulated to further strengthen this result.

A consequence of this result is that ` E. x ph ` is not true for any
` ph ` . In particular, ` E* x ph ` does not hold either, a somewhat
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