Re: [RIF] Agenda 2 May RIF Telecon

From: Francois Bry <bry@ifi.lmu.de>
Subject: Re: [RIF] Agenda 2 May RIF Telecon
Date: Thu, 04 May 2006 09:07:22 +0200

> 
> 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"

Yes, in addition they do do that.  And, yes, that is their purpose.

> > 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.

True.

> As a consequence, one finds in the logic literature both
> "interpretation" of formulas  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).

Yes, it is possible to given interpretation-only meaning to formulae with
free variable.  This is indeed often done.

> > 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).

Yes, true.

> > ------------------------------------------------------------------------
> > 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

peter

Received on Thursday, 4 May 2006 10:50:14 UTC