[FOM] Second-order logic and neo-logicism

Richard Heck richard_heck at brown.edu
Wed Mar 25 12:44:15 EDT 2015

On 03/24/2015 12:54 PM, Joe Shipman wrote:
> That is one of the two points I was going to make. My other point is that there is a more cogent argument against "second-order logicism" than that SOL entails strong mathematical theorems, namely:
> "Accepting your full semantics for SOL doesn't get us new mathematics without a stronger deductive calculus. The most you can do in general is to say that a for a strong mathematical statement X, either X or ~X is a logical truth, without specifying which is the case. Please give me some axioms and inference rules to go along with your SOL semantics, which are themselves justifiable as logical principles rather than mathematical ones."
> Personally, I think this is an objection which can be well met up to a point--probably a system as strong as Maclane Set Theory can be justified as purely "logical".

In fact, people interested in the so-called neo-Fregean program almost 
always think about the logic axiomatically, not semantically, for much 
this sort of reason. The interest is in certain axiomatizable fragments 
of SOL, and then of course in what justifies thinking of those fragments 
as logical (which is itself an epistemological notion in this context). 
It's dialectically important that that *not* depend upon assumption of 
the standard semantics but have some other basis.

As it happens, there has been a great deal of discussion of this sort of 
issue, as well as of technical issues connected to the question just how 
much deductive power one actually needs here, and then of further 
philosophical questions that emerge once that question has been 
answered. A great deal is also known about exactly what role HP plays in 
the proof of Frege's Theorem and what role it does not play. To point 
out that, if you start with Q, HP doesn't do any work in getting you the 
rest of PA is to point out something that has been known for *a very 
long time*. In particular, it has been known since the beginning (and is 
heavily emphasized in work by Boolos) that HP plays *no role whatsoever* 
in the proof of induction, and the exact relation between how much 
induction you get and how much comprehension you assume has been known 
since work by Linnebo published in 2004.

In fact, something even stronger---also pointed out by Boolos---is true. 
There is a sense in which HP plays no role even in the proof of the 
"crown jewel" of neo-Fregean logicism: the proof of the existence of 
successors. Given Frege's definitions of 0, predecession, and natural 
number, the existence of successors follows from the fact that 
succession is a function, i.e., from:

(*)    Pxy & Pxz --> y = z

where "Pxy" means: x immediately precedes y in the number-series. HP 
does play a role in the proof of (*), for which only predicative 
comprehension is needed. But the standard proof of the existence of 
successors from (*) requires impredicative comprehension, in particular, 
\Pi_1^1 comprehension, though a different (if similar) proof can be 
given in ramified predicative SOL. Thus, the real work HP does is in 
establishing the basic facts about predecession. And, as I said, this 
has been known for 20 years or so.

It's only more recently (starting with the same paper by Linnebo in 
2004) that attention has been paid to the role HP plays in 
characterizing addition and multiplication, but much the same turns out 
to be true in those cases. As emphasized by Burgess, it's important here 
that we think in terms of the cardinal definitions of these operations, 
not the ordinal ones.

I don't know whether, as you say, "a system as strong as Maclane Set 
Theory can be justified as purely 'logical'". What I do know is that 
such a claim is *extremely controversial* and would be *philosophically 
significant* if it could be established. I think the same is true of the 
claim that Q is "logical", and even of the claim that R is "logical".

So I disagree rather strongly with the claim to the contrary that is 
made in the paper. It's a common mistake to think logicism must be the 
view that *all* of (ordinary?) mathematics is "logical". But even Frege 
did not hold that view.

> I agree with Ran that the objections as previously stated seemed to count mathematical power as a strike against a logicist development in an unfairly question-begging way, but perhaps my way of framing this will lead to more fruitful and technically interesting discussion.

I for one find it hard to see what is new in this objection, as compared 
to a similar objection due to Boolos (yet again) that is 25 years old or 
so now.

Richard Heck

Richard G Heck Jr
Professor of Philosophy
Brown University

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