On Aug 18, 2011, at 4:58 PM, Rick Murphy wrote: (01)
> Hey All:
>
> Long time no ont.
>
> I am researching type free logics with higher order quantification. (02)
You need to be achingly precise about what you count as 'higher order'. Common
Logic is type-free and (some would say) uses (what certainly looks very like)
higher-order quantification, in that it allows one to quantify over names in
predicate and function positions, like this: (03)
(forall (f)(iff (transitive f)(forall x y z)(if (and (f x y)(f y z)) (f x z) ))
)) (04)
However, CL has a first-order semantics and its metatheory is much closer to FO
than HO, so many logicians would *not* count it as a higher-order logic. It has
been called FO logic with a HO syntax, or HO logic with a FO semantics. (05)
Here is a test case to hone your intuitions. Consider these axioms: (06)
(R a)
(Q b) (07)
And ask yourself whether these entail the following: (08)
(exists (p)(and (p a)(p b) ) (09)
i.e. that there is a property that applies both to a and to b. (010)
If you intuitively answer "no", then you are thinking first-order, and would
likely find CL congenial. If it seems obviously "yes", then you are thinking in
a genuinely higher-order way. (011)
It really does matter which way you choose, as you will be interested in very
different logics. (012)
Pat (013)
> I
> would like to read some (free) academic research on this topic and
> wondered whether anyone could point me at published papers.
>
> Google only matches six documents on the terms "type free logic" and
> "higher order quantification".
>
> Research here seems thin, probably my ignorance.
>
> Chris I have read your paper. The references don't seem to point at
> direct prior research on these combined subjects.
>
> --
> Rick
>
>
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> (014)
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