[FOM] "Hidden" contradictions

Joao Marcos botocudo at gmail.com
Sat Aug 24 12:23:15 EDT 2013

Hello, Sam:

Sam Sanders wrote:
> Perhaps what Tarski meant was that we would still regard a
> provable inconsistency (in a non-trivial inconsistent system)
> as false, because most of us are so used to thinking classically.

That was precisely my point, and no more.  If your *nontrivial inconsistent
system* is to have anything like a *sound semantics*, the terminology
"false theorem" would seem misleading, to say the very least.

Carl mentioned Wittgenstein in a previous message.  Among other things,
Wittgenstein wrote that "it is one thing to use a mathematical technique
consisting in the avoidance of contradiction, and another thing to
philosophize against contradiction in mathematics", and insisted that it
was important to remove the "metaphysical thorn" represented by that
*attitude* of mathematicians concerning contradictions.  The quotation by
Tarski seems to illustrate this perfectly well.

That said, as a disclaimer I should add that I did not write to advertise
_paraconsistent mathematics_, to say that we should _embrace
contradictions_, or nothing even remotely like that, and I certainly have
no trouble recognizing the _supremacy_ of classical reasoning on our daily
activities as logicians and meta-logicians.

Best, Joao Marcos
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