Re: interpolation theorem of propositional logic



David C. Ullrich wrote:
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.

If you restrict FOL to 0-ary predicate symbols,
even not allowing equality, you arrive a 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.

A propositional variable is a sentence symbol.
Because a propositional variable in essence can
represent a full propositional formula. This
can be done either by using biimplication, i.e.
for example:

p <-> q & ~r.

Now p stands for q & ~r. Or by explicit substitutional
rules and/or lemmas.

For example many natural deduction systems
come with the rule, that if A is an axiom,
the one can use A[S] where S is a substition
from propositional variables to propositional
formulas.

Also there are lifting lemmas, that say for
example if A is a tautology then A[S] is
also a tautology.

Etc..
.

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