Re: Question about Frege's Theorem

lugita15@xxxxxxxxx wrote:
Now that we have the whole Henkin Semantics/faithful model thing out of
the way, I have another question. If the set of models under
consideration in Henkin Semantics are restricted to the "faithful
models," or those which have all definable subsets of the domain as
predicates, does there exist a formal deductive system that is both
sound and complete with respect to this semantics.

Yes. Pick any of the standard deductive systems for second order logic, it will be both sound and complete w.r.t. the class of (faithful) Henkin models. (If you include the axiom of choice as a logical principle you have to take some care, and require faithful models to contain also choice functions, but even then the completeness proof is pretty much a straightforward adoption of the ordinary Henkin style completeness proof for first order logic.)

--
Aatu Koskensilta (aatu.koskensilta@xxxxxxxxx)

"Wovon man nicht sprechen kann, daruber muss man schweigen"
- Ludwig Wittgenstein, Tractatus Logico-Philosophicus
.

Relevant Pages

  • Re: since AK was reviewing Endertons article,
    ... HE got HIS take on Henkin semantics from the horse's mouth. ... am opposed to model theory in principle. ... let's just say that that moots a great many higher considerations. ...
    (sci.logic)
  • Re: Question about Freges Theorem
    ... consideration in Henkin Semantics are restricted to the "faithful ... sound and complete with respect to this semantics. ... soundness/completeness theorems are proved in Shapiro (who's book ...
    (sci.logic)
  • Re: Question about Freges Theorem
    ... The unary quantifiers in a Henkin interpretation run over some ... So yes, a particular Henkin ... have to conclude that some sentences that are sound with respect to ... Henkin Semantics do not satisfy all Henkin Models. ...
    (sci.logic)