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Re: [ontolog-forum] type free logic and higher order quantification

To: Christopher Menzel <cmenzel@xxxxxxxx>
Cc: "[ontolog-forum]" <ontolog-forum@xxxxxxxxxxxxxxxx>
From: Pat Hayes <phayes@xxxxxxx>
Date: Fri, 19 Aug 2011 16:30:50 -0500
Message-id: <3A167C03-87F1-4E39-AF0A-230ACFD9A8E8@xxxxxxx>

On Aug 19, 2011, at 2:56 PM, Christopher Menzel wrote:    (01)

> On Fri, 2011-08-19 at 10:31 -0400, John F. Sowa wrote:
>> Rick,
>> As Pat emphasized, there is a huge semantic difference between
>> quantifying over relations (which can be done in Common Logic with a
>> first-order style of semantics) and the kind of higher-order logic
>> that logicians have been kicking around.
>> In fact, many logicians have now come to believe that the infinite
>> hierarchy of relations, relations of relations, etc., which are
>> assumed for higher-order logic, should *not* be considered part of
>> logic.  They should be considered a version of set theory.
> Well, the issue of whether higher-order logic really deserves the name
> is a difficult one dating back at least to a squabble between Zermelo
> (yea) and Skolem (nay) and the debate is in fact still lively and the
> issue largely unsettled.  (For the Zermelo/Skolem controversy, see the
> two papers by the terrific historian of logic Gregory Moore, "The
> Emergence of First-order Logic" and "Beyond First-order Logic: The
> Historical Interplay between Mathematical Logic and Axiomatic Set
> Theory"; use The Google for citation details; also Matti Eklund, "On How
> Logic Became First-order".  There is Stewart Shapiro's excellent
> "Foundations without Foundationalism," an extended defense of the thesis
> that second-order logic is indeed logic.)    (02)

Ah, but Stu *defines* HO logic as any logic which quantifies over properties, 
so on his criterion CL would be a higher-order logic. In fact, he refers to the 
Henkin model theory "for second-order logic", which would be an oxymoron if one 
takes the view (as you do, and I do on even-numbered days of the month) that 
Henkin logic is essentially first-order.    (03)

All of which goes towards my point, that one has to be very careful, when 
talking about the 'order' of a logic, to know exactly what one means. I have 
been taken to task by some very competent logicians for calling CL first-order, 
as they are using a purely syntactic criterion of order. Nobody is right in 
debates like this, but clearly the terminology lacks precision.    (04)

OK, no more from me on this fascinating, but ultimately purely 
historical/terminological discussion.    (05)

>> PH
>>>> Ask yourself whether [axioms (R a) and (Q b)] entail the following:
>>>> (exists (p)(and (p a)(p b) )
>>>> i.e. that there is a property that applies both to a and to b.
>>>> 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.
>>>> It really does matter which way you choose, as you will be
>>>> interested in very different logics.
>> RM
>>> Understood. I've been heads down on functional programming lately, so
>>> I'm getting more comfy with higher order thinking.
>> But the usual versions of functional programming are not higher-order
>> in the way logicians use the term.
>> For Pat's example, the axioms (R a) and (Q b) can be satisfied in
>> a CL domain that has only four entities:  {R, Q, a, b}.
> Note I had assumed that Pat meant his premises to be (R a) and (R b),
> not (R a) and (Q b).  I therefore thought his "test" for higher-order
> thinking was whether one thought one could existentially generalize on
> predicates, but perhaps I was hasty and misunderstood him.    (06)

Yup, you did :-) The point of the example is to show up what a difference the 
comprehension principles (or axioms) make.     (07)

Pat    (08)

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