[FOM] Second-order logic and neo-logicism
Joseph Shipman
joeshipman at aol.com
Thu Mar 26 06:30:45 EDT 2015
I'm not claiming my objection is new. I am merely asking the neo-logicists to specify here, in reply to my post, the technical results that I assume they already know about:
1) what is the strongest deductive calculus for SOL that you accept as "logical" and therefore justified as part of the logicist project?
2) how much set theory do you get from this calculus?
A very strong deductive calculus for SOL is "any statement that ZFC proves is a second-order validity with standard semantics is valid". I assume they have something less strong in mind.
As for my statement about Maclane Set Theory, isn't this the same strength as a simple theory of types? Type theory can be developed logically in a less cumbersome way than Russell did.
-- JS
Sent from my iPhone
> On Mar 25, 2015, at 12:44 PM, Richard Heck <richard_heck at brown.edu> wrote:
>
>> 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
>
>
> -----------------------
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