[FOM] re Plural Logic/Foundations

[Harvey Friedman]

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