Antonelli’s Logicism

[Joe Shipman]

I strongly recommend this paper by the late Professor Aldo Antonelli:

https://projecteuclid.org/download/pdf_1/euclid.ndjfl/1276284780

He provides the most satisfactory version of logicism that I know of, by using a “Frege quantifier” F that provides a logical representation of a notion that could also be described as “There are at least as many B as A” or “There is an injection of A into B” or “|A|<=|B|”.

My main objection is that in English there is not a nicely simple way to SAY this. He also describes in this paper the “Rescher quantifier”

 R(A, B) iff |A| > |B|

which can be rendered in English as “there are more A than B”, and the related “Most” quantifier

“Most A are B” : Most={(A,B):|A∩B|>|A−B|}

but he doesn’t state whether the Frege quantifier can be replaced by or defined in terms of one of these others to develop arithmetic in a way that would satisfy a logicist (in particular, it would be best to avoid needing some form of AC).

Did Antonelli or anyone else discuss this anywhere?

— JS





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