Logicism

[Joe Shipman]

This has been a helpful discussion. Tennant’s work also confirms for me that it is a defensible position that theorems of Peano Arithmetic are analytic propositions that have as strong an epistemic justification as one could hope for.

Next questions:
1) does “logic alone“ give good grounds to believe that arithmetical propositions are meaningful with definite truth values even if Peano Arithmetic does not decide them?
2) for what kinds of propositions Y and recognizable axioms of infinity X can “X->Y” be put on as solid an epistemic base as the theorems of PA?

— JS

Sent from my iPhone

> On Aug 12, 2020, at 7:38 PM, Richard Kimberly Heck <richard_heck at brown.edu> wrote:
> 
> 
>> 
>> 
>>> On Aug 12, 2020, at 9:54 AM, Richard Kimberly Heck
>> <richard_heck at brown.edu> wrote:
>> 
>> And Boolos denies, surely rightly, that this is a system of logic.
>> On 8/12/20 3:28 PM, Deutsch, Harry wrote:
>> Yes, Boolos denies that FA is a system of logic, but others such as
>> Crispin Wright (last I looked) maintain that it is. The issue turns on
>> whether “Hume’s principle” is a principle of logic—and yes, it’s a
>> kind of axiom of infinity. 
> 
> Crispin has never claimed, even in FCNO, that HP is a principle of
> *logic*. What he's claimed is that it has similar epistemological
> virtues. The original view, roughly speaking, was that it is
> 'analytic'). His later view has taken a variety of forms, but the
> consistent idea is that we have some kind of free-standing epistemic
> right to HP. This is maybe clearest in the 'foundations for free' paper.
> 
> 
>> In any case, the Fregean derivation of the Peano axioms in FA is very
>> much in the logicist vein. 
> 
> Certainly! Ultimately, my own view (which was also George's, I think) is
> that the question what's 'logic' and what's not isn't sufficiently clear
> to be useful here. What is worth discussion, or so I have argued (and
> Crispin seems to agree), is what kinds of epistemological
> presuppositions HP has, and the same question applies to the 'logic'
> needed for the derivation of the Peano axioms from HP. In any event,
> those two questions have seen quite a bit of discussion over (holy crap)
> the 37 years since FCNO was published.
> 
> 
>> But how about Church’s simple theory of types? He thought of this as a
>> system of “logistic”.
> 
> Of course the difficulty remains the axiom of infinity, which you still
> need here.
> 
> Riki
> 
> 
> -- 
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