Re: [RIF] Agenda 2 May RIF Telecon

Peter F. Patel-Schneider wrote:
> Interpretations are is in essence a way of giving meaning to a vocabulary.
>   
I think the following would be more accurate:

"Interpretations are a way of giving meaning to well-formed formulas"

> 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).

> It turns out that formulas with no free variables (often called sentences)
> are supported independent of the variable map, so it is customary to say
> that an interpretation by itself supports a sentence, 
"satisfies a sentence" is more common than support and relate to the
widewspread notion of "satisfiability" ("Erfüllbarkeit" as it was called
at the origin).
> ------------------------------------------------------------------------
> First-order Logic (with Equality)
>   
I think, the formal definition might no convey the following essential
remark:

Equality cannot be expressed in classical logic only by formulas because
"equality" and "equivalence" cannot be distinguised from each other.
Therefore, the model theory of equality has to explicitely say, "the
equality relation of the logical language", or "vocabulary" is
interpreted by the equality relation on the domain (or universe) of the
interpretation.

That is the key point. The rest is just a mathematical "implementation"
of  this key point.

I the WG goes to inculding such defs in documents, I would striongly
recommend to explicitel;y refer to well established logic text books
instead of citing Wikipeadia, how good it might be, or proposing yet
another  text similar to those found in good logic text books, eg
Kees Doets. From Logic to Logic Programming, The MIT Press, 1994)
http://www.amazon.com/gp/product/0262041421/sr=8-1/qid=1146724851/ref=pd_bbs_1/104-1241285-2467909?%5Fencoding=UTF8**

Francois

Received on Thursday, 4 May 2006 07:07:33 UTC