On Fri, Dec 16, 2005 at 10:43:15AM -0500, Lichtblau, Dale wrote:
> Do you have anything that explicates your distinction between a
> "model" and a "theory"? I've always thought of them as pretty much the
> same thing. If they do denote two distinct concepts, what, if any, is
> the nature of the relationship between them? (01)
I've been waiting for John to reply, but since he hasn't, let me point
you to his nice little page on the mathematical background to KR:
http://www.jfsowa.com/logic/math.htm . Note in particular section 13 on
Model Theory. (02)
In a nutshell, in mathematical logic at any rate, a theory is a set of
of sentences in a formal language. (Some definitions require theories
to be deductively closed as well.) The axioms of Peano Arithmetic (PA)
constitute a well known theory. The primitives of the theory are the
numeral "0", a unary function symbol "s", and the binary function symbol
"+" and "*", an the axioms themselves are: (03)
1. (x)~(sx = 0) -- "0 is not the successor of any number."
2. (x)(y)(sx = sy -> x = y) -- "The successor function is one-to-one"
3. (x)(x+0 = x)
4. (x)(y)(x+sy = s(x+y))
5. (x)(x*0 = 0)
6. (x)(y)(x*sy = (x*y)+x) (04)
Induction Schema:
[F(0) & (x)(F(x) -> F(sx))] -> (x)F(x), for any predicate F -- "If 0
has a property F and the successor of x has F if x does, for any x, then
every number has F" (05)
Theories are thus just syntax, sentences in a language. Models, by
contrast, are *semantic* entities. To define the notion properly we
first need that of an *interpretation*. An interpretation of a language
L assigns *meanings* to the basic vocabulary of L (typically,
denotations to names, functions to function symbols, properties and
relations to predicate symbols), and specifies how the meanings of
complex expressions are determined by the meanings of their simpler
parts (a semantic property usually referred to as "compositionality").
In particular, an interpretation specifies how the *truth value* of a
sentence is determined by the meanings of its simpler parts. (For the
standard "Tarskian" definition of truth, see John's page.) A *model* of
a theory, then, is an interpretation of the language of the theory that
makes all the sentences of the theory true. Thus, in particular,
consider the interpretation of the language of PA in which "0" is
interpreted to mean the number 0, "s" the successor function on the
natural numbers, "+" and "*" to mean addition and multiplication on the
natural numbers, and the quantifiers (x) in quantified sentences are
taken to range over the set of natural numbers {0, 1, 2, ...}. Under
that intepretation, all the sentences of PA are true, so it is a model
of PA. (Note that John uses "model" on his page to mean
"interpretation", which is not uncommon.) (06)
There is of course a common and mostly unrelated notion of 'model' that
is typically used to indicate a mathematical or graphical representation
of one thing or another, e.g., a model of a database schema, an
airplane, a business process, etc. (07)
Chris Menzel (08)
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