- From: Peter F. Patel-Schneider <pfps@inf.unibz.it>
- Date: Thu, 04 May 2006 06:49:57 -0400 (EDT)
- To: bry@ifi.lmu.de
- Cc: public-rif-wg@w3.org
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