Re: Antonelli’s Logicism

[Joe Shipman]

The way I explained Frege quantifier isn’t nearly as grammatically smooth and convenient as using “more” or “most”; it’s nice to be able to map the formal language to English straightforwardly.

Is choice really necessary? Suppose I make A and B disjoint by tagging their elements to get A’ and B’, and define C = anything that is A’ or B’, and say H(A,B) iff neither “most C are A’ “ nor “most C are B’ “ ?

Why doesn’t that get the Hartig quantifier?

The Rescher quantifier R(A,B) or |A|>|B| is similarly definable as “most C are A“ after defining C as “A’ or B‘ “ using the “disjointed“ primed versions of A and B.

— JS

Sent from my iPhone

> On Aug 27, 2020, at 7:03 PM, Richard Kimberly Heck <richard_heck at brown.edu> wrote:
> 
> On 8/27/20 11:55 AM, Joe Shipman wrote:
>>  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|”.
> 
> Yes, Aldo did a lot of terrific work, and this is a splendid paper.
> Here's one way to think about what it shows. Most work on neo-logicist
> approaches to arithmetic have worked with 'Hume's Principle':
> 
> HP:    #x:A(x) = #x:B(x) iff A ~ B
> 
> where "A ~ B" means that A and B have the same cardinality. Typically,
> that is defined using second-order logic, which is then also used to
> define the ancestral, the notion of a natural number, and so forth. What
> Aldo shows is that all of that can be done just using the Frege
> quantifier. You do not need any other non-first-order (in the usual
> sense) resources.
> 
> But Aldo does 'posit' HP, just as ordinary neo-Fregeans do, and that's
> where most of the controversy usually is.
> 
> One cool point that Aldo makes along the way, in effect, is that the
> ancestral (transitive closure of a relation) can be defined in terms of
> *Dedekind* finitude (which can be defined in terms of the Frege
> quantifier). This may seem surprising, since the ancestral itself is
> characterized in terms of the other notion of finitude. This fact (which
> I discovered independently, and which Albert Visser also discovered
> around the same time---guess it was in the air) is explored in some
> detail in my paper "Is Frege's Definition of the Ancestral Adequate?" here:
> 
> http://rkheck.frege.org/pdf/published/Originals/FregesDefintionOfAncestral.pdf
> 
> 
>> My main objection is that in English there is not a nicely simple way
>> to SAY this.
> 
> I'm not sure why that is an objection, and didn't you just say it pretty
> simply?
> 
> 
>> 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).
> 
> As Aldo notes, you can define the Härtig quantifier H(A,B) iff |A| = |B|
> from the Rescher quantifier if but only if you have choice. Since it is
> obvious how to define Härtig in terms of the Frege quantifier, without
> choice (as he also notes), it follows that we can't define the Frege
> quantifier from the Rescher quantifier without choice.
> 
> "Most" can clearly be (and even is!) defined in terms of the Rescher
> quantifier, so we can't define the Härtig quantifier in terms of "Most"
> without choice. It follows that we cannot define the Frege quantifier in
> terms of "Most" without choice. This implies that (without choice) we
> cannot use either of these to do 'Frege arithmetic', since we need the
> notion of equinumerosity to formulate HP.
> 
> I suspect that Aldo knew all of that, though it is not fully explicit in
> the paper, I don't think.
> 
> Riki
> 
> PS This paper of mine
> 
> http://rkheck.frege.org/pdf/published/LogicOfFregesTheorem.pdf
> 
> is in much the same spirit as Aldo's. It shows that there is a way to
> get arithmetic without using anything that even looks like a
> second-order quantifier (though it does use free second-order variables).
> 
> 
> -- 
> ----------------------------
> Richard Kimberly (Riki) Heck
> Professor of Philosophy
> Brown University
> Pronouns: they/them/their
>