[FOM] Second-order logic and neo-logicism

Wed Mar 25 02:40:27 EDT 2015

Thank you, Panu!
Rossberg and Wright's point is (if I'm looking at the right quotes, and reading them correctly) that set theory, being the powerful mathematical theory that it is, should not be assumed in the neo-logicist project, and that consequently (since SOL is assumed by neo-logicism) if 
	1. 	SOL is in fact set theory, 
then this would be a serious problem for neo-logicism.
You explain that 
	2. 	substantial set theoretical principles are derivable from SOL. 
But does (2) entail (1)? Does the neo-logicist have to consider (2) problematic for her project if she considers (1) problematic?
The neo-logicist would surely say that PA2 should not be assumed in her project (as this would be question begging). Yet she does not consider the fact that 
	3.	PA2 is "derivable from" (can be interpreted in) SOL+HP
problematic. In particular, she does not consider (3) to entail that SOL+HP is a mathematical (rather than logical) theory. So I ask:
(i) Why should the neo-logicist be bothered by (2) anymore than she is bothered by (3)?
(ii) If (3) is a problem for the neo-logicist to the same extent that (2) is, then can't you simplify your argument by relying on (3) instead of on (2)? If not, why?
Best,
Ran   

-----Original Message-----
From: fom-bounces at cs.nyu.edu [mailto:fom-bounces at cs.nyu.edu] On Behalf Of Panu Raatikainen
Sent: Tuesday, March 24, 2015 23:17
To: Foundations of Mathematics
Subject: Re: [FOM] Second-order logic and neo-logicism

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/

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