Small correction below:  (01)
On Jul 7, 2012, at 9:51 AM, Pat Hayes wrote:  (02)
> 
> On Jul 7, 2012, at 9:39 AM, Michael Brunnbauer wrote:
>
>> 
>> Hello Chris,
>> 
>> On Fri, Jul 06, 2012 at 01:08:57PM -0500, Chris Menzel wrote:
>>>> So someone trying to define OWL FOL would have to be careful to stay in
>>>> first
>>>> order logic because Properties are first class entities ? Would that be a
>>>> difficult problem ?
>>> As long as one adds no special semantic requirement that there must be as
>>> many properties as there are sets of individuals (which, by Cantor's
>>> Theorem, is simply impossible to require if properties are "first-class
>>> entities", i.e., a species of individual), there is no risk of moving
>>> beyond first-order logic.
>> 
>> After some reading, I think I begin to understand this. As long as the 
>> properties I can quantify over are first class entities, I have Henkin 
>> semantics. Only if I can quantify over all possible properties, I get true 
>> second order logic.
>> 
>
> Exactly. It all turns on what assumptions the semantic makes about what 
>higher-order entities (functions, relations) must exist. Classical second 
>order logic assumes that all mathematically possible functions and relations 
>exist. Henkin semantics assumes that all lambda-definable entities exist. 
>Common Logic makes no existence assumptions at all other than that names must 
>denote things with relational extensions.  (03) 
Actually, that all *terms* must denote things with extensions. So if you use 
functions, that use does carry some consequences for the number of entities in 
the universe.  (04)
Pat  (05)
> 
> So here is a case which distinguishes CL from Henkin. Do
>
> (P a)
> (Q b)
>
> together entail
>
> (exists x)(and (x a)(x b) ))
>
> ?
>
> CL says no. Henkin says yes: the relevant function is (lambda (y) (or (P y)(Q 
>y))).
>
> Pat
>
>> Regards,
>> 
>> Michael Brunnbauer
>> 
>> -- 
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>> ++
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>> 
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