On Feb 8, 2010, at 10:43 AM, Cory Casanave wrote:
> ...
> Considering the "Pat Axiom": That if 2 theories can be shown to be
>incompatible they must share some concepts - intuitively obvious but I have
>never seen it made explicit, thanks! (01)
Actually, the "Pat axiom" needs a couple small qualifications. First, each of
the two theories in question has to be consistent. An inconsistent theory is
incompatible with every theory, regardless of any concept overlap. Second, the
theories must not put incompatible conditions on the number of things that
exist. In first-order logic (with identity), it is possible to express that
there are only N things, for any natural number N. So if T1 says "There are
exactly three things" and T2 says "There are exactly four things", they will be
incompatible even if they share no concepts (though I suppose one could say in
this case that they share the concept of identity). (02)
Yours in excruciating correctness, (03)
Chris Menzel (04)
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