Re: SUO: Theory Query
On Tue, Apr 16, 2002 at 12:29:17PM -0700, John Sowa wrote:
> What I had in mind is the Lindenbaum lattice...
>
> JA> In this context, we have the notion of a first order predicate
> > language $L$, plus the set !L! of its 'sentences' (i.e., formulas
> > with no free variables).
> >
> > According to one standard usage, a 'theory' is any set of sentences,
> > in which case we already have a natural lattice of theories, namely,
> > the power set !P!(!L!) of !L!.
> >
> > Is that what you have in mind?
>
> No, the set of all subsets of FOL sentences is a much bigger lattice.
> What you need to do is to take the quotient space generated by the
> provability relation |- of FOL (or the semantic entailment operator |=
> which is equivalent to provability for FOL).
Actually, these lattices are the same size, so long as you've got at
least denumerably many predicates or constants in your language. Proof
left to reader. :-)
> In effect, the Lindenbaum lattice is the set of equivalence classes
> of axiomatizations....
>
> The top of the lattice is the set of all tautologies ...
<quibble>
You mean "logical truths". Tautologies are the logical truths of
propositional logic.
</quibble>
-chris