Re: SUO: Theory Query
Jon,
What I had in mind is the Lindenbaum lattice, as I presented it
in Chapter 2 of my KR book. A summary of that presentation is also
contained in Section 6 of my paper on causality:
http://www.jfsowa.com/ontology/causal.htm#s6
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).
In effect, the Lindenbaum lattice is the set of equivalence classes
of axiomatizations. Any set of FOL sentences may be adopted as a
set of axioms (or axiomatizations). Any two axiomatizations are
equivalent (i.e., in the same equivalence class) iff they have the
same deductive closures (i.e., exactly the same implications).
The top of the lattice is the set of all tautologies (i.e., everything
provable from or entailed by the empty set).
The bottom of the lattice is the collection of all FOL sentences
(i.e., everything provable from any contraction of the form p & not-p).
The partial ordering of the lattice is determined by implication (i.e.,
if p implies q, then q is more general and p is more specialized).
John Sowa