FOM: second-order logic is a myth
holmes at catseye.idbsu.edu
Wed Mar 17 13:39:02 EST 1999
I repeat the definition I used:
The first definition given is "a science that deals with the canons
and criteria of validity in thought and demonstration and that
traditionally comprises the principles of definition and
classification and correct use of terms and the principles of
correct predication and the principles of reasoning
[Webster's Third New International Dictionary, Unabridged]
I omit the italics I inserted earlier.
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.
I can't see how the admissability of quantifiers over
predicates isn't a question about the principles of correct
predication; it may be one that didn't occur to Aristotle (whose
treatment of quantification was limited), but it is still a question
in the right area. "Can I use a bound variable as a predicate" is a
question about predication. It also can be said to fall under
"correct use of terms".
Your comments that Aristotelean logic and the predicate calculus don't
do this are beside the point; I don't grant that these are the only
"logics", and, moreover, you know that, because that is precisely the
point being debated! Frege's logic (of which predicate calculus is a
descendant) did include quantification over predicates.
Moreover, this allows me to reiterate my point that second-order logic
need not be understood as being about sets: it is about predicates or
properties, and these are topic neutral notions. My opinion is
stronger than this: second order logic _qua logic_ should not be
understood as being about sets (or any specifically mathematical
notions) unless "sets" are understood to be identified with properties
or predicates (which is a viewpoint some have held).
You really should have read the definition I was quoting again
before saying this:
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.
Notice that the definition I quote includes the principles of
classification as part of logic -- not the specific classification
information, which is part of biology, but the topic-neutral methods
of classification which can be applied anywhere. Independently of
this point, my answer is that methods of classification, whether they
are considered logic or not, are topic neutral. The specific
distinguishing characteristics used, of course, are not. I fail to
see how this example scores any points against my position.
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.
I don't dispute that, but that makes no points against my position.
At issue is exactly which pieces of information are content specific.
I can say the following: The structure of the real numbers is
determined completely by a list of properties of which the last is
"for any property P of real numbers for which there is a real number b
greater than or equal to all real numbers with property P, there is a
smallest real number greater than or equal to all real numbers with
property P". In this last property, the notion "greater than or equal
to" is mathematical; but that makes no points against me. The notion
of "property" is not mathematical; it is certainly topic neutral, and
I argue that it is in fact logical. I could say "set" instead, but I
don't have to.
I will remark further that the structure of the real number system (or
any mathematical structure) actually can be described in completely
(second-order) logical terms by binding specific mathematical notions
with existential second order quantifiers. To assert the existence of
a model of the second order theory of the natural numbers is to say
that there is an object 0, a property "is the successor of", and a
property "is a natural number" which satisfy certain well-known
second-order conditions. I'm a logicist, so this doesn't surprise me...
This is in accord with the sophisticated understanding of the
continuum presented by modern f.o.m. research, wherein many questions
about the real number system can be answered only by looking at the
enveloping model of set theory.
I don't see how this makes any points against me, either. The fact
that questions about the real numbers as implemented in a model of set
theory depend on what is true about sets in the model of set theory is
hardly surprising. The real numbers in a particular model of set
theory may or may not be isomorphic to the actual real numbers; what
is true about sets of real numbers in the model of set theory may or
may not match what is actually true about properties of real numbers
in the real world.
I maintain that if there are any models of the second-order theory of
the continuum (I think that there are) that their structure is
uniquely determined by the second-order axioms. I do not claim that
we do or even can know what the answers to some of these questions
are. The situations which obtain in different models of set theory
are interesting as different possible answers to the question of what
is actually true about the standard models. (They are also
interesting in themselves, of course).
I should note that I only consider the real numbers as defined up to
isomorphism; I'm not committed to any specific structure as being THE
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.
I'm not looking for certainty. I am acknowledging (as I must) that
there is a truth of the matter, even when I cannot determine what it
And God posted an angel with a flaming sword at | Sincerely, M. Randall Holmes
the gates of Cantor's paradise, that the | Boise State U. (disavows all)
slow-witted and the deliberately obtuse might | holmes at math.idbsu.edu
not glimpse the wonders therein. | http://math.idbsu.edu/~holmes
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