On Jul 7, 2012, at 9:39 AM, Michael Brunnbauer wrote:  (01)
> 
> 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.
>  (02) 
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)
So here is a case which distinguishes CL from Henkin. Do  (04)
(P a)
(Q b)  (05)
together entail  (06)
(exists x)(and (x a)(x b) ))  (07)
?  (08)
CL says no. Henkin says yes: the relevant function is (lambda (y) (or (P y)(Q 
y))).  (09)
Pat  (010)
> Regards,
>
> Michael Brunnbauer
>
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