Re: Antonelli’s Logicism

[Richard Kimberly Heck]

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