[FOM] Plural Logic/Foundations

[Neil Barton]

Dear Harvey and FOM,

Apologies for the slow reply to this, it's been a busy month.

I don't think there's any unique account of what plural resources brings to
the foundations of mathematics. It's certainly not (necessarily) the case
that plural talk and its logic intends to provide a `foundation' in the
same way that say set theory, category theory, or homotopy type theory does
(though it's unclear to me that those three are aiming at the same
enterprise). I'll say something *very* short about plural resources below
(which will be common knowledge to lots of people), and then I'll paste the
abstract for my talk at that event.

Consider the sentence:

``The rocks rained down.''

Now, such a sentence doesn't make any mention of the set of rocks, nor is
it satisfied by any one rock in isolation (a single rock can't `rain
down'). It rather refers in the *plural* to the rocks, which are raining
down together.

Once we've got this very coarse natural language motivation in place, it's
natural to introduce a logic for it. Plural First-Order Logic (or PFO)
contains singular variables, x, y, z and so on, but also plural variables
xx, yy, zz and so on, that range *plurally* over the domain (so `xx' can be
interpreted as some things xx). We also have plural quantifiers \exists xx
and \forall yy, to be read as ``for some things xx'' and ``for any things
yy'' respectively. There's then several natural looking axioms, including
an extensionality axiom and a plural comprehension scheme.

The key interest for plural logic and foundations is that it opens the
possibility of a nominalism about entities, whilst retaining the expressive
resources that those entities were supposed to underpin. For, when I refer
plurally, it's unclear that I thereby commit myself to the existence of
some reified `collection' (the status of this question is hotly contested
in the philosophical literature; see Linnebo's SEP article below). Further,
it's possible to encode a significant amount of second-order content using
plural resources. Boolos [1] showed how to interpret monadic second-order
logic plurally, Hewitt [2] has shown that with a further expansion of
logical resources this can be extended to full SOL, and Uzquiano [3]
provides an interpretation of Morse-Kelley class theory in plural logic
(over ZFC). For more of the details, including an extensive bibliography,
I'd highly reccommend Øystein Linnebo's excellent SEP article on this:

http://plato.stanford.edu/entries/plural-quant/

As for what I'll be talking about, my interest is largely in the prospects
of using plural logic to maintain a nominalism about proper classes whilst
studying mathemtical languages (such as MK) that appear to presuppose them.
Generally, the focus has been on whether or not such an interpretation is
satisfactory (lots of people think that plural resources just presuppose a
background set theory). I'll be talking about how much class theory we can
get out, *assuming* that the paraphrase is satisfactory. Here's an abstract:

In my talk I'll build on work that tries to motivate class theories through
the use of plural logic. I'll argue that if you accept the use of plural
logic, then you have good reason to accept some use of non-definable
classes in class theory. I'll then trace the different levels of
non-definability one might admit, from admitting individual non-definable
classes, through restricted forms of class comprehension, and finally up
through extensions of Morse-Kelley with class choice principles. I'll
finish by discussing some applications and surprising results, in
particular just how much you can code up plurally within a strong class
theory.

With Best Wishes,

Neil


[1] Boolos, George (1984). “To Be Is To Be a Value of a Variable (or to Be
Some Values of Some Variables),” Journal of Philosophy, 81: 430–50; repr.
in Boolos 1998a.

[2] Hewitt, Simon (2012). ``The Logic of Finite Order.'' _Notre Dame
Journal of Formal Logic_ 53 (3):297-318.

[3] Uzquiano, Gabriel (2003). “Plural Quantification and Classes,”
Philosophia Mathematica, 11(1): 67–81.

On 23 March 2016 at 22:04, Harvey Friedman <hmflogic at gmail.com> wrote:

> Can anybody give a brief clear generally understandable account of
> what plural foundations of mathematics is for the FOM readership?
>
> This meeting was announced on the FOM recently:
>
>
> https://www.eventbrite.co.uk/e/plural-foundations-plural-logic-and-the-foundations-of-mathematics-tickets-22579344427
>
> Harvey Friedman
> _______________________________________________
> FOM mailing list
> FOM at cs.nyu.edu
> http://www.cs.nyu.edu/mailman/listinfo/fom
>
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