Re: Antonelli’s Logicism

[Joe Shipman]

Is even countable choice necessary?

Suppose C is just AUB, and neither “most C-are-A” nor “most-C-are-B” are true, then it must also be the case under the intended interpretation that A-B is not smaller than B and B-A is not smaller than A, so we can still define H(A,B) without any copying.

— JS 

Sent from my iPhone

> On Aug 27, 2020, at 10:50 PM, Richard Kimberly Heck <richard_heck at brown.edu> wrote:
> 
> On 8/27/20 8:01 PM, Joe Shipman wrote:
>> 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?
> 
> If you were sure you could do this 'tagging', then that would work. But
> the possibility of doing such tagging in general requires that the
> universe be embeddable into a proper part of itself. I.e., it requires
> (at least) that the universe be Dedekind infinite. Without (countable)
> choice, we do not know that.
> 
> In "Cardinal Arithmetic in Weak Theories", Visser considers a theory he
> calls COPY, which is a weak second-order theory whose characteristic
> axiom is precisely that there are two functions that map the universe to
> two disjoint copies of itself. Visser shows that this theory interprets
> Q even though (like Q) it does not have a pairing function.
> 
> Riki
> 
> 
> 
>>>> 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
>