[FOM] re Plural Logic/Foundations

[Harvey Friedman]

Here is the kind of thing I have min mind.

Clearly the uses of plural quantification in natural language are
rather special.

Can one find a restriction on plural quantification, motivated by
natural language considerations, for which you get an interesting
associated fragment of second order logic which you can use to define
new and interesting fragments of the usual systems of f.o.m.?

Natural language considerations may point to significant fragments
that are not apparent from the strictly mathematical point of view.

Harvey Friedman

On Thu, Apr 21, 2016 at 1:44 PM, Stewart Shapiro <shapiro.4 at osu.edu> wrote:
> As I understand it, the move to plurals is (or was) meant to get the
> benefits of full second-order languages, but without a commitment to the
> existence of proper classes, properties, or whatever.  The claim is, or
> seems to be, that the natural language constructions using plurals do not
> carry this commitment.  I don't think it makes much difference to f.o.m., as
> understood in the list, for example.
>
> Consider an instance of comprehension, formulated with the plural idiom.
> Let A(x) be a predicate with x free.  Then the instance of comprehension is
> something like this:  "there are some things such that a given thing x is
> one of them just in case A(x)".
>
> What you end up with, supposedly, is full monadic second-order logic.  And
> given that in most of the structures of interest, one can define pairing,
> then one ends up with full second-order logic.  So for arithmetic, the
> plural formulation would be full second-order PA.
>
>
> On Wed, Apr 20, 2016 at 6:33 PM, Harvey Friedman <hmflogic at gmail.com> wrote:
>>
>> Neil Barton wrote
>> http://www.cs.nyu.edu/pipermail/fom/2016-April/019669.html in reply to
>> my inquiry.
>>
>> One role pluralism *could* have for f.o.m. is this.
>>
>> There might be an interesting way of reformulating traditional f.o.m.
>> formalisms such as (fragments of) PA, Z_2, Z_n, RTT, Z(C), ZF(C),
>> etcetera, without resorting to arbitrary quantified formulas as in
>> induction, comprehension, separation, replacement, etcetera, in favor
>> of formulas based on pluralism. And then show that you get systems
>> just as strong interpretation-wise or other-wise, or maybe weaker in
>> various senses, but still of significant strength. E.g., claim that
>> there are legitimately interesting alternative f.o.m. systems based on
>> a more subtle kind of underlying language/logic.
>>
>> A superficial glance at this literature does not yield a trustworthy
>> answer to what I am asking.
>>
>> A start would be any of these:
>>
>> 1. Plural logic, syntax and semantics. What fragment of FOL= does it
>> correspond to, and what does "correspond" mean?
>> 2. Plural arithmetic. What fragment of PA does this correspond to, and
>> what does "correspond" mean?
>> 3. My 1,2 assumes that there are readily appropriate formulations as
>> axioms/rules like FOL= and PA. Is that assumption appropriate?
>>
>> Harvey Friedan
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>
>
>
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