[FOM] Non-distributive predication

[Dean Buckner]

Tom:
>I would like to get comments from fom readers about the significance
> for the foundations of mathematics of allowing non-distributive
> predication in logic.

A lot of significance in my view, see any number of my postings, but
particularly my discussion/argument with Harvey Friedman over May 2003
(ACA0, PA, PA' & passim).  To be frank, these views have not enjoyed
universal popularity within FOM.

>Standard systems of first-order logic allow only distributive predication.
> That is, whenever a predicate applies to some things, it applies to
> each of them. In this respect they differ from ordinary language,
> which allows non->distributive predication.

Agree, though be warned it's best to avoid words like "ordinary language" in
FOM.

> ... theory and mereology ...provide a single surrogate for some
> individuals (the set they are members of or the mereological sum
> they are part of).   Standard first-order logic needs such
> surrogates, since each argument position in a verb must be
> associated with a single individual. Non-distributive predications
> are not representable in standard logic. We get substitutes
> for them only by introducing such surrogates, but such substitutes
>"change the subject" (to the surrogate) and change the predicate
> (to something distributive).

Absolutely so.  See my posting of 17 May    "First and a half order logic".

> I propose instead that we allow in logic what we have in English: some
>individuals may satisfy a predicate together without satisfying it
> individually.

>We can introduce individual names and compound terms

Your expression for a compound term, e.g.

    met_together \Alice, Bob/

introduces the comma as a primitive term, doesn't it?  My version of "among"
theory takes reference and the comma as primitive.  We treat "those people
are Alice and Bob" as asserting that "those people" refer to whatever "Alice
and Bob".  I.e. "those people" is an uncompounded name, which refers to the
same thing as the compounded name.  I.e. "and" used in this context is a
device for constructing proper names out of things which are themselves
proper names.  We then define "among" in terms of these primitives

     x is (or are) among S  iff  (E y) S = x & y

Informally, x is one of a set of things S, iff there is one other object S',
or there are other objects S' such that S are the same things as x and S'.

I don't think this matters, except in my version, X cannot be among X.  But
your "among" and mine are inter-definable, I think.

We can explain inferences in ordinary language such as

 Alice is among the people at number 4
 the people at number 4 are among the people in the street
 :. Alice is among the people in the street

>From the definition of "among" we get

 those at number 4 = Alice & the others at number 4
 the people in the street =  those at number 4 & the others in the street

By simple substitution we infer that Alice is one of the people in the
street.

> Loosely, we just drop all the set brackets.

Exactly.  No empty set, and no singleton set.  The question which has
intrigued me every since I read Frege's critique of this sort of theory, is
whether you can get any mathematics out of it.

"[The idea of a class as a collection] is barren, and it is not logic.  Only
because classes are determined by the properties that individuals in them
are to have, and because we use phrases like this: "the class of objects
that are b": only so does it become possible to express thoughts in general
by stating relation between classes; only so do we get a logic."

Frege, G.  "A critical elucidation of some points in E. Schroeder"s
Vorlesungen Ueber Die Algebra der Logik", Archiv fur systematische
Philosophie 1895, pp 433-456, transl. Geach, in Geach & Black 86-106, p.104

I disagree with Frege.  He was unable to see that we can quantify over
plurals.

> The question of how much of mathematics can be grounded in among
> theory, or, more generally, whether and where among theory can
> be a useful tool in place of set theory, needs further exploration.

Entirely agree.  And I also agree that

> among theory differs from set theory in that there is no analogue of the
> empty set, and there is just one relation, among, that is an analogue
> of both the membership and the subset relation. Because among
> theory is not singularist, no hierarchy builds in the way that it
> does in set theory,

Where I may disagree with you is in the possibility of a "set of everything"

    (E x) (y) [ y among x ]

Only true, I think, when there are finitely many things.  The idea of an
infinite set depends on that distinction between "member of" and "subset of"
that among theory obliterates.  Turning to your plural induction axiom, you
need to state somewhere that the successor of any x is not within the set of
predecessors.

    (X among P) (E y among P) [ y succeeds X & ~ y among X ]

But is P among P or not?  If it is, there is an object among P that is not
among P.  There may be a way around this, but I haven't see one.

As I said, I had a long interchange with Harvey about this, his view was
that the onus is on "among" theory to show it can handle "ordinary
mathematics".  I don't know enough ordinary mathematics to do this.



Dean