[FOM] re Plural Logic/Foundations

[Richard Heck]

On 04/20/2016 06:33 PM, Harvey Friedman 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?

The systems of "plural logic" known to me are usually analogous, in
fairly straightforward ways, to the systems of second-order logic.
Indeed, from a syntactic point of view, they are often
indistinguishable, modulo a bit of "syntactic sugar" here and there.

This was largely the point of the papers by Boolos that launched the
study of such logics in the first place: "To Be Is To Be the Value of a
Variable (or Some Values of Some Variables)" and "Nominalist Platonism".
The original idea was to use plural language as a sort of informal
interpretation of the formalism of second-order logic. Boolos's specific
intent---this is perhaps most visible in the earlier paper "On
Second-order Logic"---was to provide some way of making sense of
second-order ZFC *without* having to assume either that the second-order
variables range over sets or even classes. Boolos thinks the former is
bad because then the first-order quantifers can't range over all sets,
which seems to be what we intend them to do. And he thinks the latter is
bad because (i) set theory aims to be a theory of *all* "collections"
and (ii) classes are a sort of "collection". I.e., classes are really
sets, but we're afraid to call them such, on pain of inconsistency.
Plurals are supposed to be the way out of this bind.

Since then, things have evolved, and some people have suggested an
alternative view according to which second-order logic and plural logic
might live side by side. Many of the issues here are more conceptual, or
even metaphysical, than formal, and have to do with how different sorts
of comprehension axioms, etc, can be motivated depending upon how the
formalism is being interpreted. E.g., some people have thought that the
usual sorts of motivation for predicativity have little (or less) bite
in the plural case. Francesca Boccuni, Oystein Linnebo, and Fernando
Ferreria have all explored ways in which the marriage of these two sorts
of systems can be used to great effect.

Boccuni has shown, e.g., that you if you (carefully!) combine a
predicative version of Frege's infamously inconsistent system (the
predicative version is consistent) with an impredicative version of
plural logic, you get a system that is as strong as second-order PA.

Linnebo's SEP article on plural quantification
    http://plato.stanford.edu/entries/plural-quant/
is a good guide to the basics, and to the literature.

Richard Heck

-- 
-----------------------
Richard G Heck Jr
Professor of Philosophy
Brown University

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