Logicism

[Kapantais Doukas]

-----Original Message-----
From: Richard Kimberly Heck <richard_heck at brown.edu> 
Sent: Wednesday, August 12, 2020 5:54 PM
To: Deutsch, Harry <hdeutsch at ilstu.edu>; Joe Shipman <joeshipman at aol.com>; Kapantais Doukas <dkapa at Academyofathens.gr>
Cc: Foundations of Mathematics <fom at cs.nyu.edu>
Subject: Re: Logicism

On 8/11/20 7:54 AM, Kapantais Doukas wrote:
> Concerning Joe Shipman's post on "logical"/"mathematical".
>
> I might be misinterpreting something in the question, and if so I apologize, but would the system of Principia Mathematica not be a good candidate? 
Even Russell seems to have his doubts whether the axiom of infinity is a truth of logic, which is part of what Joe specified.

DK. I agree. This is why I have not referred straightforwardly to the Principia Mathematica in my post. I have found interesting, though, that Goedel feels the need to specify in footnote 16 that the formal niche in which his incompleteness proofs take place can be either (i) a system that contains Peano Arithmetic as primitive (system P) or (ii) a "logical", according to logicism, system which yields Peano Arithmetic. Now, the "according to..." parameter is, of course, essential. "Logical" according to whom? Probably, the answer to Joe Shipman's question cannot bypass this question.  


On 8/10/20 1:49 PM, Deutsch, Harry wrote:
> How about the system Boolos calls “Frege Arithmetic.” See Logic, Logic, and Logic.

And Boolos denies, surely rightly, that this is a system of logic.

That said, I know of few if any philosophers nowadays who would accept logicism in the form that Joe describes. (Neil Tennant might be an exception; perhaps he'll reply as well.)

At one time, Boolos did defend an answer to Joe's question. The system of logic was axiomatic second-order logic, and the translation from arithmetic to logical truths was: A maps to "If PA, then A", where PA is the conjunction of the axioms of second-order PA. So this was a kind of if-thenism. See the first footnote to "On Second-order Logic" and a paper to be published posthumously, soon, in a volume for Crispin Wright. Boolos had abandoned that view by the late 1970s, however.

The more popular view nowadays is something we might call 'conceptualism' (but is usually called, somewhat confusingly,
'neo-logicism'): that the arithmetical truths (or some core subset
thereof) follow logically from principles that themselves might reasonably be regarded as *conceptual* (roughly, analytic) truths. This is the sort of view Crispin Wright defends in *Frege's Conception of Numbers as Objects* and elsewhere. Here the key result is the one to which Harry alludes: that the axioms of PA can be proven, in FA (axiomatic second-order logic plus 'Hume's Principle'), given 'natural'
definitions of the arithmetical notions.

We can do a bit better: The axioms of Pi-1-n-CA can be proven in second-order logic with Pi-1-n comprehension. Below Pi-1-1, though, one runs into problems, since the ancestral is defined by a Pi-1-1 formula, and one needs, for a large number of reasons, to run inductions in which it is contained. Crucially, this is needed for the proof of the existence of successors. (See the joint paper that Boolos and I wrote for the details.)

But there is a more limited version of the view that I've sometimes found attractive. It can be shown that the axioms of Q, formulated relationally (i.e., Pxy instead of Sy; Axyz instead of x+y...), and without any commitment to the universal existence of successors, sums, and products (but with a commitment to the existence of these in concrete cases), can be proven in predicative FA: predicative second-order logic plus HP. (In fact, you even get commutativity, associativity, etc.) This is a non-trivial theory, in that it interprets the usual version of Q (a result due to, in essentials, to Hájek and
Švejdar) and so is essentially undecidable. Non-trivial arithmetical theories can be proven in even weaker systems still. That leaves a lot unproven, but it's a significant amount of mathematics.

Riki


>> On Aug 10, 2020, at 11:20 AM, Joe Shipman <joeshipman at aol.com> wrote:
>>
>> [This message came from an external source. If suspicious, report to 
>> abuse at ilstu.edu<mailto:abuse at ilstu.edu>]
>>
>> There seems to be agreement that some mathematical propositions can be shown to be equivalent to truths of logic, and others can’t, but drawing a line between what is “logical” and what is “mathematical” is hard.
>>
>> I’ve had trouble finding a simple, streamlined development of BASIC logicism. Can anyone provide a source for a presentation of a logical system (by which I mean, at the very least, a computable deductive calculus that generates truths of logic) , and an interpretation of theorems of Peano Arithmetic as logical truths in this system?
>>
>> — JS
>>
>> Sent from my iPhone


--
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Richard Kimberly (Riki) Heck
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