Re: Semantics without variable maps

Francois Bry wrote:
> Chris Menzel wrote:
>   
>> 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.  
>>     
>
> Chris is right. This 3rd approach is standard in logic and used in some
> logic textbooks. It has the advantage of keeping things simple. It has
> two drawbacks:
>
> 1. A didactic drawback (not too dramatic because the appaoch cvan be
> explained).
>
> 2. It does not corresponds to a widespread tratment of free variuables
> in Computer Science in general, in the theory of relational databases
> and in logic programming in particular.
>   

Also not too dramatic since the approach is more general and the 
"widespread treatment of free variables in CS" is easy enough to deal with.

-Chris

-- 
Dr. Christopher A. Welty                    IBM Watson Research Center
+1.914.784.7055                             19 Skyline Dr.
cawelty@frontiernet.net                     Hawthorne, NY 10532
http://www.research.ibm.com/people/w/welty

Received on Friday, 5 May 2006 19:05:00 UTC