SUO: Re: Building the Hierarchy
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For the use of this working group, I have placed some
excerpts from Chang and Keisler (1st edition) here:
http://suo.ieee.org/ontology/msg04736.html
I am still saving my pennies and pop bottles
toward the $275 price tag of the 3rd edition.
Pertinent to the immediate discussion, one may wish to consider the
following (meta-)excerpt, where a "set of axioms for a theory T" is
defined as a theory that has the same consequences as T. Here, the
set of axioms need not be finite, though, of course, we are avidly
interested in those that are.
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Note 32
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| 1. Introduction
|
| 1.4. Theories and Examples of Theories
|
| A (first-order) theory T of $L$ is a collection of sentences of $L$.
| T is said to be 'closed' iff it is closed under the |= relation.
| In view of Table 1.3.1, this is the same as requiring that T
| be closed under |- . Since theories are sets of sentences
| of $L$, we may apply the expressions:
|
| a model of a theory,
|
| consistent theory,
|
| satisfiable theory,
|
| as introduced in Section 1.3.
|
| A theory T is called 'complete' (in $L$) if and only if its set of
| consequences is maximal consistent. If T is a theory of $L$, with
| $L$ c $L$’ and $L$ =/= $L$’, then T is not a closed theory of $L$’.
| On the other hand, it is easy to see that if $L$’ c $L$, then the
| 'restriction' of a closed theory T to $L$’, in symbols T | $L$’,
| is always a closed theory of $L$’. T is a 'subtheory' of T’ iff
| T c T’. If T is a subtheory of T’, then T’ is an 'extension' of T.
|
| A 'set of axioms' of a theory T is a set of sentences with the
| same consequences as T. Clearly, T is a set of axioms of T, and
| the empty set is a set of axioms of T if and only if T is a set
| of valid sentences of $L$. Every set of sentences !S! is a set
| of axioms for the closed theory T = {p : !S! |= p}. A theory T
| is 'finitely axiomatizable' iff it has a finite set of axioms.
|
| The most convenient and standard way of giving a theory T is by
| listing a finite or infinite set of axioms for it. Another way
| to give a theory is as follows: Let $A$ be a model for $L$;
| then the 'theory of' $A$ is the set of all sentences which
| hold in $A$. The theory of any model $A$ is obviously
| a complete theory.
| Chang & Keisler, 'Model Theory', pages 36-37.
|
| C.C. Chang and H.J. Keisler, 'Model Theory',
| North-Holland, Amsterdam, Netherlands, 1973.
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Robert E. Kent wrote:
>
> John and others,
>
> ----- Original Message -----
> From: "sowa" <sowa@bestweb.net>
> To: "SUO" <standard-upper-ontology@ieee.org>; <cg@cs.uah.edu>
> Sent: Saturday, May 17, 2003 8:59 AM
> Subject: SUO: Re: Building the hierarchy
>
> [snip]
>
> > 5. Robert Kent pointed out the difference between a theory and an
> > axiomatization of a theory. In logic, the word _theory_ is applied
> > to the "deductive closure" or the complete set of implications of
> > some axioms. Since two or more axiom sets might have the same
>
> The IFF uses the term "theory" for any set of sentences (and actually
> extends this to expressions, i.e., formulas), and uses the term "closed
> theory" for theories that are deductively or semantically closed (every
> consequence or semantically entailed expression of the theory is in the
> theory). This accords with both the book _Model Theory_ by Chang and Keisler
> and the book _Information Flow_ by Barwise and Seligman. We are most
> concerned that we are aligned with Barwise and Keisler, since the first
> order notions (FOL) in the IFF extend the information flow (IF) notions in
> Barwise and Seligman in a very coherent way (a quadruple of coreflections).
> In particular, in the IFF the FOL notion of a closed theory adjointly
> corresponds to the IF notion of a regular IF theory. So where you (John S.)
> use "axiom set" and "theory", the IFF uses "theory and "closed theory".
> Perhaps these two synonym pairs could be booknoted for later reference.
>
> Robert E. Kent
> rekent@ontologos.org
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