Re: interpolation theorem of propositional logic

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.

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

Relevant Pages