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Re: [ontolog-forum] Inconsistent Theories

To: "[ontolog-forum]" <ontolog-forum@xxxxxxxxxxxxxxxx>
From: "John F. Sowa" <sowa@xxxxxxxxxxx>
Date: Mon, 08 Feb 2010 23:56:06 -0500
Message-id: <4B70EAE6.7030705@xxxxxxxxxxx>
Chris    (01)

JFS>> 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.    (02)

CM> Good thing, John, since you're wrong.    (03)

Please note that I was making a comment about the expression "x=y",
not about the following two statements, which are definitely
about the the subject domain:    (04)

    Theory T1:  (Ex)(Ay)(y=x).    (05)

    Theory T2:  (Ex)(Ey)~(x=y).    (06)

Each of these theories makes a claim about some domain.  For each
of them, it is possible to find a model that satisfies that theory.
But it is not possible to find a model that satisfies both.    (07)

My reference to Wittgenstein was to the passages in the _Tractatus_
quoted below.  As he notes, "a=b" is a statement about the symbols.
But such statements can be used in formulas such as the above to
make assertions about the domain.    (08)

The point is even clearer in Peirce's existential graphs (EGs),
which express full FOL with identity without using named variables.    (09)

Instead, Peirce uses a line (which he calls a line of identity)
to represent each individual (thing or entity or whatever you
want to call it) that is asserted to exist.  For an intro to EGs,
see his tutorial:    (010)

    http://www.jfsowa.com/peirce/ms514.htm    (011)

The theory T1 would be represented by an EG of the following form:    (012)

    One line represents some x [but there is no label "x" on the line].
    A second line inside a conditional says that if there is a y
    [but there is no label "y" on this line] then it is connected
    to the previous line.    (013)

The theory T2 would be represented by an EG with lines for each of
the two individuals and a negation of the connection between them.    (014)

____________________________________________________________________    (015)

4.241 If I use two signs with one and the same meaning, I express this
by putting between them the sign "=".    (016)

"a=b" means then, that the sign "a" is replaceable by the sign "b".    (017)

(If I introduce by an equation a new sign "b", by determining that it
shall replace a previously known sign "a", I write the equation --
definition -- (like Russell) in the form "a=b Def.".  A definition is
a symbolic rule.)    (018)

4.242 Expressions of the form "a=b" are therefore only expedients in
presentation:  They assert nothing about the meaning of the signs "a"
and "b".    (019)

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