[FOM] Second-order logic and neo-logicism

[Joe Shipman]

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". 

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.

-- JS

Sent from my iPhone

> On Mar 23, 2015, at 12:25 PM, "Ran Lanzet" <lanzetr at gmail.com> wrote:
> 
> I am probably missing something here, and will be glad if you could clarify.
> 
> As far as I understand, your main argument against neo-logicism is roughly this:
> 1. The rules of 2nd-order logic (SOL) employed by the neo-logicist are very strong, in the sense of entailing some serious mathematical content. In particular:
>    a. They are provably equivalent to the "basic rules" of SOL plus the unrestricted impredicative comprehension scheme.
>    b. Once we accept those rules as the background logic, we get immediately from the very weak Q+ to the very strong PA2.
> 2. Hence, it does not seem reasonable to accept the neo-logicist's version of SOL as logic.
> 
> Now I believe the neo-logicist would happily accept (1): after all, her basic claim is that, essentially, all of ordinary mathematics is derivable from logic (more precisely: from her favorite version of SOL plus Hume's principle (HP); and I'm sure she will happily accept that SOL and not HP does the majority of work here). She will, though, undoubtedly object to your step from (1) to (2). She might argue as follows: the move from (1) to (2) is unwarranted, unless we accept the following principle: 
> (*)    if a set of rules entails substantial mathematical theorems, then it is unreasonable to regard that set of rules as part of logic. 
> But accepting this principle -- so she might argue -- is to beg the question against logicism.
> 
> Question: what did I miss here? Or, more specifically: why is the suggested reply ineffective against your argument?
> 
> Best,
> Ran        
> 
> -----Original Message-----
> From: fom-bounces at cs.nyu.edu [mailto:fom-bounces at cs.nyu.edu] On Behalf Of Panu Raatikainen
> Sent: Sunday, March 22, 2015 09:16
> To: Foundations of Mathematics
> Subject: [FOM] Second-order logic and neo-logicism
> 
> 
> The following new paper might interest some here:
> 
> Panu Raatikainen: "Neo-logicism and its logic", History and Philosophy of Logic (forthcoming)
> 
> http://philpapers.org/rec/RAANAI
> 
> 
> It has greatly benefited from certain old discussions here in FOM on the second-order logic; special thanks to Martin Davis!
> 
> 
> All the Best
> 
> Panu
> 
> 
> 
> Abstract:
> The rather unrestrained use of second-order logic in the neo-logicist program is critically examined. It is argued in some detail that it brings with it genuine set-theoretical existence assumptions, and that the mathematical power that Hume’s Principle seems to provide, in the derivation of Frege’s Theorem, comes largely from the “logic” assumed rather than from Hume’s principle. It is shown that Hume’s principle is in reality not stronger than the very weak Robinson Arithmetic Q.  
> Consequently, only few rudimentary facts of arithmetic are logically derivable from Hume’s principle. And that hardly counts as a vindication of logicism.
> --
> Panu Raatikainen
> 
> Ph.D., Adjunct Professor in Theoretical Philosophy
> 
> Theoretical Philosophy
> Department of Philosophy, History, Culture and Art Studies P.O. Box 24  (Unioninkatu 38 A)
> FIN-00014 University of Helsinki
> Finland
> 
> E-mail: panu.raatikainen at helsinki.fi
> 
> http://www.mv.helsinki.fi/home/praatika/
> 
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