Re: Semantics without variable maps

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.

Francois

Received on Friday, 5 May 2006 11:13:10 UTC