FOM: The logical, the set-theoretical, and the mathematical: Reply to Ketland

Joe Shipman shipman at
Tue Sep 12 09:28:54 EDT 2000

Jeffrey Ketland wrote:

> Joe said:
> >My position is as follows.  Comprehension axioms are "logical".
> I think that can't be right. Suppose we are interested in whether an
> interpreted sentence S is logically true. Then I would propose these
> necessary conditions:
>     (i) S should be recognizably true (in a finite time) by any reasonable
> person, perhaps after careful deliberation, and possibly re-checking.

This seems to make "logically true" dependent on the power of our brains.  I
think "logically true" is closely related to "necessarily true", or "true in all
possible worlds", or "true in all possible interpretations", none of which
notions contain any such dependency.  In the particular case of FOL, the set of
validities is r.e., so you can identify "logically true" with "recognizably true
in a finite time", but before Godel discovered this in 1929 I don't think it
would have been regarded as a necessary property.

>     (ii) S should be devoid of ontic commitments

This is reasonable; see the discussion below on Infinity.

>     (iii) determining the truth value of S should (somehow) be independent
> of contingent knowledge or empirical information.

No, the truth value of S should be non-contingent, but that is different from
"determining the truth value of S" which is a humanistic process.

> These conditions are met for simple FO logical truths like,
>     (1) All wise Greeks are Greeks
>     (2) For any object x, if x is a wise Greek then there is a y such that y
> is a Greek
>     etc.
> (more precisely, these English sentences have a FOL logical form, which is a
> theorem of any standard system).
> It seems that our brains are built to see the truth of sentences like (1),
> (2), etc. In some sense, our human brains contain FOL machines (and if
> they're not damaged or intoxicated, then they work properly). FOL exhibits
> the basic structure of human mental reasoning.
> Perhaps our mind/brains also contain the rules for modal logic, tense logic
> and provability/epistemic logic and the (lowest level) truth predicate, as
> well. It would be interesting to see empirical evidence concerning how
> logical rules like these are psychologically implemented in the human
> mind/brain.

So far this seems correct

> But the conditions I stated above don't seem to hold for the comprehension
> axiom:
>     (3) There is a set X such that for any y, y is in X iff y is an integer
> There are philosophers who, after many years of deliberation, think that the
> sentence (3) is *false* (because sets don't exist). E.g., contemporary
> nominalists, like Hartry Field and Charles Chihara. For that reason---the
> existence of human minds who do not find (3) to be self-evidently logically
> true---I cannot count (3) as a truth of logic. A truth of mathematics maybe,
> but not a truth of logic.

Again, this depends on your premise that it matters what human minds can agree
on.  Anyway, in the context of second-order-logic rather than set theory,
comprehension is not really problematic -- (3) becomes "there is a property P
such that for any y, y has property P iff y is an integer", which is practically
tautological.  And I was referring to comprehension schemes within SOL or HOL,
not set-theoretic comprehension.

> 2. Logical validity and consequence
> If V is a set of *logical* validities, then I do not see how V can be non
> r.e. And if R is a *logical* consequence relation, then I do not see how it
> cannot be matched up with a deductive system which verifies all and only the
> valid consequences.
> (Sometime in the 1960s, Hilary Putnam suggested that it is probably
> impossible for the human mind to come to know, a priori, a set of sentences
> which is not r.e.. How could the set of self-evident logical truths not be
> r.e.?)

Careful --  you have moved from "logical truths" to "self-evident logical

> I would argue (like Quine - see his Philosophy of Logic 1970) that the very
> notions of "logic" and "complete proof procedure" are intimately tied
> together. The reason is that "logos" means "reason": and thus the very
> notion of logic is inseparable from what human minds can (in principle)
> recognize to be the case by actual reasoning. In particular, human reason
> seems to be bounded by certain finiteness constraints. Perhaps there are
> other creatures in the universe whose physical structure means that they can
> perform mental supertasks (e.g., determining the truth values of any Pi^0_1
> sentence by running through all the positive integers). But we're not them.

OK, this makes some sense, but not only is it possible to alternatively identify
"logic" with "necessity", it is also not true that "logos" means "HUMAN
reason".  I can think of some ancient Greek texts where the word is used with
absolutely no connotation of finiteness....

> The set of (full) second-order validities is highly non-computable (as we
> have seen, with the messages from Solovay) and, by Godel's 1st
> Incompleteness Theorem, we all know that there is no complete proof
> procedure (deductive system) for the full second-order consequence relation.
> I can only conclude that the full second-order consequence relation is not a
> *logical* consequence relation, but is an intrinsically mathematical notion.
> It makes perfect mathematical sense to talk about this relation (as it does
> to talk about many other computationally intractable relations).

If you're going to insist that "logic" entails computability, then obviously
there must be second-order validities which are not "logical", but again this is
just a dispute about the definition of "logical".

> 3. Axiom of Infinity
> Joe:
> >(Mossakowski insists on an axiom of Infinity as well, but Jones argues
> >that this axiom is purely ontological and is only necessary in a
> >metaphysical sense.   However, I am willing to accept a modified
> >logicist thesis that (ordinary) mathematics = logic plus Infinity.)
> I agree. I cannot see how anything like an axiom of infinity could count as
> logically true. Plenty of people don't believe that there are infinitely
> many objects, but they understand logical reasoning. One cannot develop
> (much) mathematics without an axiom of infinity.

This is certainly wrong.  See work by Simpson and Friedman -- a huge amount of
mathematics can be developed in a system which is conservative over Peano
Arithmetic and requires no ontological commitment to infinite objects.

> So, the thesis that maths = logic + infinity cannot count as a logicist
> thesis.

Let T be an arithmetical theorem which requires an axiom of Infinity to prove
(for definiteness, Con(PA)).  T can be stated without any ontic commitments
because it is arithmetical, but (unlike, say, the Prime Number Theorem) it
cannot be regarded as a logical truth unless you accept some axiom of infinity
as "logical".  On the the other hand, S-->T where S is an axiom stating that the
set of integers exists is indeed a logical truth, and the ontic commitment
accepting the set of integers is sufficiently fundamental that I think it is
reasonable to call "maths=logic+infinity" a form of logicism.

> Better to say,
>     maths = logic + set-theoretic comprehension + infinity
> (where the latter 2 are extra-logical).
> But this brings us back to set-theoretic foundations - i.e., anti-logicism!

But if you use SOL or HOL you don't need set-theoretic comprehension, because
you are dealing with properties not sets.

> (I seem to recall that Hintikka has recently argued that AxC is logically
> true. Is it "Principles of Mathematics Revisited" (1997, I think)?).

I'd like to learn more about this.  My favorite form of AC is Russell's
"Multiplicative Axiom" but there may be others which appear even more "logical".

> 4. "Logical" revisited
> My proposal is vague. In general, if a sentence S is considered *false* by
> well-informed rational people, who have thought about the question for many
> years, then I cannot count S as logically true. And the comprehension axioms
> fit into this category. The comprehension axioms say that sets exist. The
> comprehension axioms are false if there are no sets (or properties, etc.),
> and this is what nominalists say. (I think they're wrong: but the mere
> empirical fact that they can argue this way strongly suggests that
> comprehension axioms are not *logically* true).

It seems strange to argue in favor of a set-theoretical foundation and against a
logical foundation on the grounds that we have to account for people who don't
believe in sets!

An important distinction here -- the set-theoretic Separation axiom seems
significantly less "mathematical" and more "logical" then the Replacement
axiom.  So as far as Zermelo set theory is concerned, there may not be much
difference between the set-theoretical and the logical.  (The Powerset axiom
seems less "logical", but without replacement you can only apply it finitely
many times anyway and can't get "above" type theory, which is fine for
logicists--the validities of SOL are recursively equivalent to the validities of
type theory if I understand recent postings correctly.)

-- Joe Shipman

More information about the FOM mailing list