SUO: Global Time & Local Context
John F. Sowa wrote:
>
> pat hayes wrote:
>
> > Ah, that is what is usually called a (logical) theory.
> > I think it is useful to have a term for a set of sentences
> > accepted as true which is NOT closed under logical inference.
> > Theories tend to be infinite, and therefore tricky to put on
> > computer disc files.
>
> There is a commonly accepted term: "set of axioms",
> or if you prefer a single word, "axiomatization".
>
> John Sowa
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Folks,
| We say that [a sentence] p is a 'consequence'
| of [a set of sentences] S, in symbols S |= p,
| iff every model of S is a model of p.
|
| ...
|
| A set G of sentences is called a 'theory'.
|
| A theory is said to be 'closed'
| iff every consequence of G belongs to G.
|
| A set D of sentences is said to
| be a 'set of axioms for' a theory G
| iff G and D have the same consequences.
|
| A theory is called 'finitely axiomatizable'
| iff it has a finite set of axioms.
|
| Since we may form the conjunction of a finite
| set of axioms, a finitely axiomatizable theory
| actually always has a single axiom.
|
| C.C. Chang & H.J. Keisler, 'Model Theory',
| North-Holland, Amsterdam, Netherlands, 1973,
| pages 11-12.
All in all the relation of axiomset to theory is
analogous to the relation of grammar to language,
formally speaking, id est, many ones to one many.
Jon Awbrey
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