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Semantics without variable maps

From: Chris Menzel <chris.menzel@gmail.com>
Date: Thu, 4 May 2006 14:24:58 -0500
Message-ID: <3a0f63b30605041224gddab82aje371052ddbc27fe5@mail.gmail.com>
To: public-rif-wg@w3.org

On 5/4/06, Francois Bry <bry@ifi.lmu.de> wrote:
> ...[PFPS:]
> > Variable maps are needed to assist in giving meaning to free variables.
> >
> Not only to free variables, also to quantified variables. In fact, the
> very idea of Tarskian models does not need formulas free variables. Such
> formulas are considered only so as to define the interpretation function
> (assigning a truth value to a formulain an interpretation) recursively
> on the formulas structure.
> As a consequence, one finds in the logic literature both
> "interpretation" offormulas  with free variables:
> - their free variables are considered existentially quantified (this is
> usual in computer sceince)
> - their free variables are considered universally quantified (this used
> to be usual in German logic of the 19th and beginning of the 20th century).

There's a third option, and that is not to distinguish free variables
semantically from constants.  There is then no need for variable maps
at all.  Modifying Peter's nice exposition, the semantics on this
approach looks like this:

Given a vocabulary, an interpretation is a pair I = < D, II > where D, the
domain of the interpretation, is a non-empty set and II, the interpretation
function of the interpretation, is a function from the n-ary predicates of
the vocabulary to sets of n-ary tuples over D, from constants AND VARIABLES
of the vocabulary to elements of D, and from n-ary functions of the
vocabulary to
mappings from n-ary tuples over D to elements of D.  If the equality symbol
is in the vocabulary, II maps it into the binary identity relation over D.

Given an interpretation, I=<D,II>, let II* be the (unique) extension
of II such that, for complex terms f(t1,...,tn),

  II*(f(t1,...,tn) = II(f)(II*(t1),...,II*(tn))

Given an interpretation, I=<D,II>, a variable x, and an element d of
D, let Ix/d be the interpretation <D,IIx/d>, where IIx/d(k) = II(k)
for constants and variables k other than x, and IIx/d(x) = d.

An interpretation I=<D,II> supports a formula as follows:
1. I supports P(t1,...,tn) precisely when II(P) contains the tuple
2. I supports p precisely when I does not support P.
3. I supports (p ^ q) precisely when I supports both p and q.
4. I supports (p v q) precisely when I supports either p or q.
5. I supports Ex p precisely when there is some domain element d such that
  Ix/d supports p.
6. I supports Ax p precisely when Ix/d supports p for all domain elements

This is more or less the Common Logic semantics for traditional
first-order languages ("segregated dialects" in CL-speak:

Variable maps appear to be a formal reflection of the idea that free
variables correspond more or less to pronouns and that pronouns do not
have the sort of fixed meanings that names have.  But (a) names
themselves vary in meaning from context to context and (b) for small
enough contexts, anyway, the meaning of a pronoun is typically no less
fixed than that of a name.  Granted, since Tarski, variable maps are
more or less how it's been done, but the idea made somewhat more sense
in Tarski's own (semi-formal) account, as, by an interpretation, he
was really thinking of (some slice or other of) the real world as
providing a sort of distinguished semantic backdrop.  But since
Tarski's original account was turned into the model theory we know and
love today in the 40s and 50s by, notably, Carnap and, later, Kemeny
(J of Symbolic Logic 21(1) 1956), to my eyes it really seems to exist
as an unnecessary artifact.

Chris Menzel
Received on Thursday, 4 May 2006 19:25:31 UTC

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