[FOM] Non-distributive predication

William Tait wwtx at earthlink.net
Sun Jul 13 23:40:03 EDT 2003

On Friday, July 11, 2003, at 08:44  AM, Tom McKay wrote:

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

Let me respond to this. So that others will better understand what we 
are talking about, let me summarize your description of your logic with 
non-distributive predication.

You are considering predicate logic with no function symbols,  a 
special binary relation symbol A and a binary operation symbol

  \ , /

for constructing terms. You use upper case letters X, Y, etc. for 
variables. On the intended interpretation, the variables  range over 
`pluralities'---one may suppose made up of objects from some domain D, 
and the intended meaning of XAY is that the X's are among the  Y's. The 
axioms and rules of inference are those of standard first-order 
predicate logic, except that you add certain axioms for the relation A. 
  These are the universal closures of

  (1)   XAY iff for all unit pluralities Z [ZAX implies ZAY]
(2)    there is a unit plurality Y such that YAX.

X is a unit plurality , IX, is defined by

    IX iff for all Y [ YAX implies XAY].

X=Y is defined by XAY & YAX. It follows from all of this that either 
all pluralities are empty or else they are all non-empty.

Judging by what you say about these axioms, I think you want also an 

exists X exists Y not-XAY

to ensure that all pluralities are non-empty. The remaining axioms are

(3) Ic

where c is any individual constant and the universal closure of

(4)  [W overlaps X or W overlaps Y] iff W overlaps \X,Y/

where W overlaps X iff  there is a Y such that YAX and YAW.

(5) X = Y implies [F(X) iff F(Y)]

(6) exists Y [IY & F(Y) implies exists X for all units Y [YAX iff F(Y)]

End of Summary.

As you note, there is a more or less  obvious translation of 
second-order predicate logic (where the second order variables range 
over subsets of the domain D of individuals) into your system, and 
conversely---the pluralities corresponding to non-empty sets.

But for this reason, how can your theory have any foundational 
significance? One understands your notion of plurality over a domain D 
exactly to the extent that we understand the notion of a non-empty 
subset of  D. And any contradiction arising for the one notion 
immediately translates into a contradiction for the other.

So why not, when we are just dealing with a domain and the 
subsets/pluralities over it, simply opt for the former, since it is 
most familiar and would involve no revision of our customary 
mathematical teminology?

Of course, in set theory, we are often interested in going from D, not 
only to the domain D* of subsets of D, but to the domain D** of subsets 
of D*, to D***, etc. (possibly transfinite iterations even). This 
iteration of the operation * of taking subsets of is unavailable in the 
ideology of pluralities, since the analogues of D**,  D***, etc., just 
collapse to the analogue of D*. But why shouldn't we then just say: so 
much for pluralities?

I should also like to comment on Dean Buckner's remark ( Sat Jul 12, 
2003  9:13:26  AM US/Central) in reply to Tom McKay's posting

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

There has been a lot of talk on the list about ordinary or natural 
language which I think is misleading. Speaking about sets _is_ ordinary 
language in mathematics, and indeed was so in the 1930's, when 
Wittgenstein was treating it as `language on holiday' . Whatever the 
issues are concerning the cumulative hierarchy of sets, obtained as we 
described above by iterating the operation of `set of', this latter 
notion, of passing from D to D*,  itself was well established in many 
areas of mathematics by then.

Often to make sense of what I have read on the list about `ordinary' or 
  `natural language', I have had to understand by it the restricted 
language of those who have not learned the language of mathematics. 
This is a `natural' language in its own right, which (as in the case of 
any language) one has to learn by learning how its terms are used: it 
does not translate via dictionary definitions into non-mathematical 
language. This is ultimately the reason that the demand for formulating 
mathematics in ordinary or natural language (in the narrow sense) 
cannot be met.

So writing about and in ordinary language on the list is just fine: but 
first learn the relevant (ordinary) language.

Best regards,

Bill Tait

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