# FOM: second-order logic is a myth

Stephen G Simpson simpson at math.psu.edu
Tue Mar 16 19:56:17 EST 1999

```Robert Black 2 Mar 1999 18:33:15 writes:

> Could I recommend that Steve look at chaps 3-5 of Boolos's _Logic,
> Logic and Logic_, particularly the discussion of the topic
> neutrality of logic on pp.44-45?

Thanks for that suggestion.  I have now gotten the book out of the
library.

Boolos's essay `On second-order logic' is indeed relevant to what we
have been discussing, but unfortunately it seems that Boolos mainly
wants to poke holes in other people's arguments and is not really
interested in arriving at conclusions of his own.  Boolos: `It seems
to be commonly supposed that the arguments of Quine and others for not
regarding second- (and higher-) order logic as logic are decisive, and
it is against this view that I want to argue here.'  This seems to be
typical of so many philosophy papers -- always arguing *against*
something, never *for* anything.

I don't think that Boolos deals effectively with the objections to
second-order logic that I have raised.  In my opinion, our discussion
of second-order logic here on FOM is more fruitful and interesting
than Boolos's, because of the give-and-take format.

Pat Hayes 16 Mar 1999 10:51:03 writes:

> [Simpson] emphasises repeatedly that when he says 'second-order
> logic' he means to refer not to Henkin's logic but to full
> classical second-order logic. The trouble is, that this distinction
> makes sense only if we understand 'logic' to be defined
> semantically.

I don't understand this remark.  Maybe you had better explain.

I define logic in the accepted traditional way, as the science of
correct inference.  I don't think logic should be `defined
semantically', although obviously considerations of meaning are
relevant to the question of what is a correct inference.  On the other
hand, I do follow Shapiro in distinguishing `second-order logic with
standard semantics' (in my view this is not really logic, because it
offers no rules,of inference) from `second-order logic with Henkin
semantics' (in my view this is really a species of first-order logic,
with essentially the usual first-order rules of inference).

> It's the same *logic*, though, in Simpson's sense.

Now I'm really confused.  Do you accept Shapiro's distinction between
standard semantics and Henkin semantics, or don't you?

> then 1PC is really a logic

What is 1PC?

> I don't see how one can re-do Tarski without mentioning sets.

I'm not sure what you mean by `re-doing Tarski'.  If you are referring
to the Tarski semantics for predicate calculus, then obviously there
is no way to do it without sets, because the Tarski models are given
by structures consisting of a set D and one or more relations on D.

However, this does not mean that the predicate calculus depends on set
theory.  (Similarly, the Kripke semantics for modal logic is given by
partial orderings, but this doesn't imply that modal logic depends on
the theory of partial orderings.)  On the contrary, the predicate
calculus has a life of its own, which is both historically and
scientifically prior to Tarskian semantics.  The predicate calculus
can and should be understood and motivated as a general method of
set-theoretic structures.

Randall Holmes 16 Mar 1999 14:16:14 writes:

> I agree that second-order logic is not a formal system of
> deduction.

OK, good, I agree, provided we are talking about `second-order logic
with the standard semantics', as Shapiro calls it.

> If one's "principles of correct predication" allow one to quantify
> over predicates (part of one's logic(1)) then a logic(2) (formal

I dispute the idea that `principles of correct predication' (in
definition 1 of `logic' in your dictionary) includes quantification
over predicates.  Aristotelean logic and the predicate calculus deal
with predicates and with quantification over individuals, but not with
quantification over predicates.

By the way, what dictionary are you using?  We really ought to look
this up in the OED!

> The natural numbers and the reals are definable up to isomorphism
> in second-order logic; this is not the case in any first-order
> theory.  Techniques which allow effective definitions are of
> logical interest!

Maybe, but that doesn't imply that such techniques are topic-neutral.
It also doesn't imply that such techniques are properly regarded as
part of logic.

Let me try to make this clear by an example from another field.  In
order to classify butterflies, biologists may need to accumulate a lot
of information about distinguishing characteristics of butterflies,
different types of antennae, parts of wings, etc.  Such information
allows effective definition of classes of butterflies.  Does that mean
that the information is topic-neutral?  No.  Is the information
properly viewed as part of the underlying logic?  No.

My view is that, in a similar way, the categorical definition of the
real number system uses topic-specific information about sets of
reals, and such information is not properly viewed as being part of
the underlying logic.  This is in accord with the sophisticated
understanding of the continuum presented by modern f.o.m. research,
only by looking at the enveloping model of set theory.  Among these
questions are: the continuum hypothesis, Souslin's hypothesis, various
questions in descriptive set theory, etc.  This sophisticated modern
understanding can be summarized (somewhat over-dramatically) by saying
that absolute categoricity of the real number system is an illusion;
the real number system is categorical only within or relative to a
particular model of set theory.  Some people (not me) might call this
a `nihilist' or `postmodern' theory of the continuum.

I sometimes think the advocates of second-order logic are wistfully
longing for a kind of old-fashioned absolute certainty and this leads
them to reject the `nihilist' theory of the continuum.  If this view
of them is correct, then it seems to me they are looking for certainty
in the wrong places.

> Simpson, at least, (qua Aristotelean) ought to take arguments from
> the proper definitions of terms seriously...

Yes, I take such arguments seriously.

-- Steve

```