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Re: [ontolog-forum] Inconsistent Theories

To: cmenzel@xxxxxxxx, "[ontolog-forum]" <ontolog-forum@xxxxxxxxxxxxxxxx>
From: "John F. Sowa" <sowa@xxxxxxxxxxx>
Date: Mon, 08 Feb 2010 21:40:54 -0500
Message-id: <4B70CB36.8050408@xxxxxxxxxxx>
Chris M. and Pat C.,    (01)

Here's another way to state the point:    (02)

For two consistent theories T1 and T2 to be inconsistent, there
must be something on which they disagree (e.g., some statement S
is derivable from T1, and ~S is derivable from T2).    (03)

If all the nonlogical symbols (names of predicates, entities, etc.)
are different in T1 and T2, there is very little that they could
disagree about.  But Chris made the following point:    (04)

CM> In first-order logic (with identity), it is possible to express
 > that there are only N things, for any natural number N.    (05)

To illustrate Chris's point, suppose that a theory T1 says that
there exists exactly one x.  It can do so by the following
statement in predicate calculus:    (06)

    (Ex)(Ay)(y=x).    (07)

Literally, this says that there exists an x, and for all y,
y is equal to x.    (08)

The theory T2, however, might say that there exists an x
and a y and they are not equal:    (09)

    (Ex)(Ey)~(x=y).    (010)

These two theories are contradictory.    (011)

One could quibble about whether the symbol "=" is a logical or
a nonlogical symbol.  I would claim that identity is part of
the logical apparatus (partly because I agree with Wittgenstein
that x=y is a statement about the names x and y, rather than a
statement about the subject domain).  In any case, I don't want
to get into metaphysical arguments about the nature of identity.    (012)

John    (013)

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