[FOM] re Plural Logic/Foundations

[Richard Heck]

On 04/22/2016 09:45 AM, Harvey Friedman wrote:
> Here is the kind of thing I have min mind.
>
> Clearly the uses of plural quantification in natural language are
> rather special.

I'm not sure why one would think that, or even what it might mean. Plural
quantification is very common. Consider e.g.:

(1) Some critics hate Shakespeare.

It's true of course that we can formalize this using first-order logic,
i.e.,
without doing anything special about the plural "Some critics". But from
the
point of view of linguistic theory, this isn't terribly relevant.
Moreover, note
that (1) can reasonably be followed by:

(2) Yeah, and they only listen to one another.

Clearly, (2) has the force of a Geach-Kaplan sentence, which means that the
plural pronoun "they" must really be understood as plural. Moreover, it
needs
an antecedent in (1). I.e., there needs to be somewhere in the "logical
form" (or
other relevant level of structure) in (1) where we find an expression
that has the
same reference as the pronoun "they" in (2). This suggests (but by
itself surely
does not prove) that (1) really does involve plural quantification every
bit as
much as Geach-Kaplan sentences do.

> 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.

This is no doubt possible, but I don't know of any such considerations.

Richard Heck