[FOM] Second-order logic and neo-logicism

[Panu Raatikainen]

Dear Ran,

What you're missing is that impredicative SOL does not only have some  
mathematical content, but in particular substantial *set-theoretical  
power* - which both Wright and Rossberg, for example, grant is  
problematic in this context.  (see the quotes in the paper)

Best, Panu




Lainaus Ran Lanzet <lanzetr at gmail.com>:

> 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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-- 
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/