Re: interpolation theorem of propositional logic

On Wed, 12 Apr 2006 13:29:30 +0200, Jan Burse <janburse@xxxxxxxxxxx>
wrote:

Hi

David C. Ullrich wrote:
On 11 Apr 2006 03:36:16 -0700, "Li Yi" <liyi.cn@xxxxxxxxx> wrote:
If alpha |= beta, then there is some gamma all of whose sentence
symbols occur in both alpha and beta and such that alpha |= gamma |=
beta.
This is obviously false.
Hint: The weaker statement "If alpha |= beta, then there is some
gamma all of whose sentence symbols occur in both alpha and beta"
is obviously false.

Depends on what one understands by sentence symbols.

The subject line specifies _propositional_ logic.
There's a perfectly standard notion of "sentence
symbol" in propositional logic

If for example sentence symbols means variables, function
symbols and predicate symbols,

and none of these exist in propositional logic.

These things do of course exist in predicate logic.
Calling them "sentence symbols" seems like maximally
strange terminology; the things that they "represent"
are not sentences.

then both of them are true.

Let S(.) denote these symbols from ..

The if alfa |= beta, then there should be a gamma with
S(gamma) subset S(alfa) intersect S(beta). Namely take
the gamma=false for example. Here S(gamma)={}.

If additonnaly it should hold alfa |= gamma and gamma |= beta,
you end up with craigs interpolation theorem.

http://www.cl.cam.ac.uk/~tjr22/doc/argTalk20051109.pdf


************************

David C. Ullrich
.

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