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

To: ontolog-forum@xxxxxxxxxxxxxxxx
From: Christopher Menzel <cmenzel@xxxxxxxx>
Date: Fri, 19 Aug 2011 14:56:14 -0500
Message-id: <1313783774.5073.7.camel@xxxxxxxxxxxxxxxxx>
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.    (01)

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)

> 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}.    (03)

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. (In fact, I
think I must have, as he knows full well that the ability to
existentially generalize on predicates doesn't require full second-order
logic.)    (04)

> In that model, there is no p such that (p a) and (p b) are both true.
> Therefore the statement (exists (p) (and (p a)(p b)) is false.
> But in the logicians' version of higher order logic, for any domain
> of individuals such as {a,b}, there is an infinite hierarchy of
> domains of all possible relations over the domain {a,b}, relations
> of relations over that domain, ...
> Since all possible relations are included, there would have to be
> some relation p such that (p a) and (p b).    (05)

Yes, but it also follows in "second-order" logic that has been given a
Henkin-style "general" model theory and hence is semantically
first-order (hence the scare quotes).  Given (R a) and (Q b), (exists
(P) (and (P a) (P b))) follows from the second-order comprehension
principle    (06)

  (exists (P) (forall (x) (iff (P x) (or (R x) (Q x))))).    (07)

It does *not* follow in CL because CL does not include any comprehension
principles.    (08)

> RM
> > to establish a hierarchy of types to avoid paradox.
> You don't need a hierarchy of types to avoid paradox.    (09)

Very true.    (010)

> In CL, the statement equivalent to Russell's paradox     (011)

I'd say: The sentence that *generates* Russell's paradox...    (012)

> has a stable truth value: false.  No paradox.    (013)

Right. In more detail: the chief culprit in Russell's paradox is the
so-called "naive" comprehension principle that, for any formula φ there
is a set containing (or a property true of) all the things that are φ.
In the CL dialect CLIF, this principle is expressed schematically as:    (014)

  NC  (exists (p) (forall (q) (iff (p x) φ))).    (015)

In CL, where self-predication is permitted, an instance of NC is:    (016)

  R   (exists (p) (forall (q) (iff (p x) (not q q)))).    (017)

A contradiction follows from R immediately. For let r be one of the
things p said to exist by R:    (018)

  (1)  (forall (q) (iff (r x) (not q q)))).    (019)

But CL is type-free, so r itself is among the things being quantified
over; so by universal instantiation:    (020)

  (2)  (iff (r r) (not r r)),    (021)

contradiction.  So, as John notes, in CL, sentence R is false; more
exactly, it is *logically* false; its negation is a theorem of CL.    (022)

-chris    (023)

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