[SCL] Quantifying over propositions

[Bill Andersen]

John,

Seems you have a problem, unless I'm missing something.  A sentence  
can be treated as a 0-ary relation if it appears at the top level

     (city nyc)

but if you have instead, say

     (believes john (city nyc))

then it doesn't seem as if you have any way to tell if you intend  
'(city nyc)' to denote a proposition or to denote some non- 
propositional object.  Plus, wasn't this point about the denotation  
of 0-ary relations settled in the model theory - in the way that you  
suggest?  I may be misremembering.

     .bill


On Aug 10, 2005, at 2010, John F. Sowa wrote:

> Pat,
>
> In any earlier note, I asked why there was a need
> for a special syntax for propositions, since they
> could be treated as 0-adic relations.
>
> After pondering the idea, I thought that there
> might be a problem with the model theory:
>
>   1. An expression of the form (R x y) is true
>      iff the denotation of R includes a tuple
>      with the two values <x,y>.
>
>   2. But since a 0-adic relation would not have
>      any tuples in its denotation, there would be
>      no way to evaluate an expression (p) to T or F.
>
> However, it dawned on me that the expression (p)
> would be true iff the denotation of p included a
> tuple of the same length as the list of arguments,
> namely, the tuple < > of length 0.  If (p) were
> false, then the tuple < > would not be in the
> denotation of p.
>
> This seems to be a very natural generalization of
> the current CL model theory, and it also seems to
> be necessary to support sequence variables:
>
>   1. We need to support seqvars of length zero.
>
>   2. There may be some cases where the only argument
>      of a relation is a seqvar.
>
>   3. Therefore, expressions of the form (R ...x)
>      will occur, and the model theory must be able
>      to handle the case where ...x has length 0.
>
> This seems to imply that 0-adic relations must be
> supported by the CL model theory.
>
> A final point to note is that Boolean operators with
> 0 arguments are already supported, namely (and) and (or).
> There doesn't seem to be any reason not to support 0-adic
> relations, which could then be interpreted as propositions.
> (And 0-adic functions would be constants.)
>
> John
>
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