[FOM] re Plural Logic/Foundations

[Stewart Shapiro]

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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