[FOM] Second-order logic and neo-logicism

Richard Heck richard_heck at brown.edu
Thu Mar 26 21:43:17 EDT 2015

On 03/25/2015 06:58 AM, Alan Weir wrote:
> I don’t think Panu Raatikainen’s objections to neo-logicism (FOM Vol 
> 147 Issue 22) beg the question again the neo-logicist as suggested by 
> Ran Lanzet (Issue 23) by simply assuming that the strong second order 
> logic required to get Frege’s theorem is part of mathematics. The key 
> issue, it seems to me, is that the logic which the neo-logicist 
> assumes entails strong existence claims (the instances of the 
> comprehension axioms for example). Neo-logicists often agree that in 
> order that they themselves not beg the question, they should work with 
> a logic whose first-order component is free. But when you make the 
> same adjustment at second order level then (at least depending on how 
> you finesse the ‘freeing up’ of second order logic) the  logic becomes 
> too weak to generate infinity from Hume’s Principle. See Stewart 
> Shapiro and Alan Weir ‘ “Neo-logicist” Logic is not Epistemically 
> Innocent’, Philosophia Mathematica, (2000), pp. 160-189.

Hi, Alan,

Most of what I'm about to say is stuff you know, but I'll speak as if to 
you since I'm replying to you....

For what it's worth, I think that the objection you and Stewart 
raise---whether it ultimately can be resisted or not---is importantly 
different from "second-order logic has significant mathematical power" 
(or "set-theoretic strength", if there's meant to be a difference 
between those). It's not just the complaint that the logic has 
(second-order) existential implications (Quine et alia) nor that the 
logic is strong (already in Boolos, with maybe the most sophisticated 
discussion in Koellner [1]). The way you present it here, the objection 
turns in part on the idea that there are dialectical reasons that the 
neo-Fregean needs the *first*-order logic to be free, and that then has 
implications for what the *second*-order logic ought to be like. That is 
a *much* more subtle complaint. And, as the title of your paper makes 
clear, it's ultimately an *epistemological* complaint, which is 
essential, since the issue in which neo-Fregean logicists are interested 
is primarily epistemological (though of course it has ontological aspects).

One way to put my point is to say that your complaint actually has 
nothing to do with the question whether 'neo-logicist logic' is really 
*logic* rather than mathematics. Which is all to the good, since the 
question what "logic" really is isn't exactly uncontroversial enough 
that we can just work with the intuitive notion. Most discussions framed 
in those terms therefore end up leaving me feeling like I've got no idea 
what we're supposed to be talking about.

The excessive focus on whether second-order logic is really "logic" 
simply obscures what *I* take the key issue here to be, namely: Does the 
sort of reasoning required for the proof of Frege's Theorem (whatever 
sort that might be) preserve whatever interesting epistemic property one 
might think HP itself has (whatever that might be)? Of course, if one 
doesn't think HP *has* any interesting epistemic property, then one 
might not find this issue terribly gripping. But I am convinced that it 
is the right way to frame this *kind* of issue, anyway, which arises in 
plenty of other contexts.

And one can frame the issue in a more neutral way: Given some 
potentially interesting epistemic property, what sorts of reasoning 
preserve that property? Boolos's complaints about Frege's use of 
second-order logic in "Reading the Begriffsschrift" (as opposed to those 
in some other places) are largely driven by this kind of question, 
specialized to Frege's notion of analyticity---even though he was deeply 
skeptical about that notion (which is half of the point of his discussion).

 From a FOM point of view, the place where there's space for technical 
exploration is around the question: What sort of reasoning *actually is* 
required for the proof of Frege's Theorem? Yes, you can use second-order 
logic, but there are other sorts of systems one could use instead, such 
as the one explored in my paper "A Logic for Frege's Theorem" [2]. One 
might also lower one's ambitions from PA to weaker systems, and then 
various predicative logics will do, as discussed in my paper 
"Predicative Frege Arithmetic and 'Everyday Mathematics'" [3]. Other 
sorts of systems have been explored by Aldo Antonelli [4], Francesca 
Boccuni [5], and Fernando Ferreira [6,7] (and of course there are 
defenders of non-Fregean forms of logicism, too, e.g., Tennant and 
Zalta). So there's lot of room for technical gymnastics here.

Richard Heck

[1] http://philpapers.org/rec/KOESLO
[2] http://philpapers.org/rec/HECTLO
[3] http://philpapers.org/rec/HECPFA
[4] http://philpapers.org/rec/ANTNAV
[5] https://francescaboccuni.wordpress.com/publications/
[6] http://philpapers.org/rec/FERAFG
[7] "Impredicativity and Fregean arithmetic", UConn Abstractionism 
Workshop, 2014

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