Logicism

[Kapantais Doukas]

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? See, for example, systems PM and P in Godel's incompleteness proofs. System P is in fact a simplification. We just add the Peano Axioms to the logic of the Principia (system PM). But as Godel notes (n. 16): "The addition of the Peano Axioms as well as all other modifications introduced in the system PM, merely serves to simplify the proof and is dispensable in principle".
If PM is a logical system, then some interpretation of some of its "logical" truths are theorems of Peano Arithmetic. At least, this is what note 16 above suggests.

Doukas Kapantais
RCGP
Academy of Athens


-----Original Message-----
From: FOM <fom-bounces at cs.nyu.edu> On Behalf Of Joe Shipman
Sent: Monday, August 10, 2020 7:19 AM
To: Foundations of Mathematics <fom at cs.nyu.edu>
Subject: Logicism

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

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