[SCL] proposition variables in SCL??

[Pat Hayes]

>Hi, Pat
>
>See below.
>
>>Here's how it would go: a name P can now have *three* kinds of 
>>extension: a truth-value, a relation(al extension) and a function; 
>>and they all can be 'folded' onto the denotation of the name. You 
>>CAN write things like
>>(iff P (P))
>>but you aren't obliged to specify any connection. And you can (that 
>>is, it makes semantic sense to) write things like
>>(exists (P)  P)
>>which could even be false in some interpretations. As far as I can 
>>see, this introduces no new tautologies.
>
>I could certainly use something like this.  If one wants to do 
>things like a 'holds in' kind of temporal
>model.  For example...
>
>      (=> (and (holdsIn T Phi) (subInterval T T'))
>            (holdsIn T' Phi))
>
>where Phi is some proposition, this could be useful.  We'd need 
>propositions to provide signatures
>for predicates like holdsIn.  A general capability like this would 
>have to extend to propositions that
>can't be represented by a simple constant.


Right, and thats where it seems to get messy, since a term denoting a 
truthvalue seems to be indistinguishable from an atomic sentence, but 
they have different semantic conditions.  We would need to either 
impose some rather tight internal-coherence distinctions on the model 
theory, or else somehow distinguish a term denoting a proposition 
from an atomic sentence in the syntax.  To see why, try doing a bit 
of FO reasoning on this combination:
(not p)
(= p (f a))
(f a)

This would be satisfiable in the straightforward extension to the SCL 
MT, because f might have a relational extension with <I[a]> in it but 
a functional extension and folded propositional assignment which 
takes I[(f a)] = I[p]  to false, since these two semantic constructs 
have nothing to do with one another. Equality reasoning would enable 
you to derive

(not (f a) )

from the first two, presumably.  And still it is consistent.  The two 
occurrences of 'f' in this syntax are a function/relation pun, but 
there's no way to tell this from looking at the sentence/term syntax.

Pat


>
>    .bill


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