botocudo at gmail.com
Thu Nov 29 13:13:10 EST 2012
> > In Section 15 of his Introduction to Mathematical
> > Logic, Alonzo Church uses the word "contradiction"
> > for a propositional formula that is false under
> > all assignments to its variables. This terminology
> > seems perfectly satisfactory to me.
> In a wider context this terminology is not completely happy,
> unfortunately. By a tautology is usually meant a sentence that is true
> by virtue of its truth-functional structure, a substitution instance
> of a validity in propositional logic. But there are contradictions
> e.g. in first-order logic -- (x)(Ey)P(x,y) & (Ex)(y)~P(x,y) for
> instance -- that are not (substitution instances of) logical
> falsehoods in propositional logic, that are not false by virtue of
> their truth-functional structure.
Another perspicuous example that does not depend on the duality of the
quantifiers, but on equality instead, would be:
(Ex)Px & (Ex)~Px & (x)(y)x=y
No need for a very populated universe, in such a case!
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