[FOM] Non-distributive predication

[Tom McKay]

I would like to get comments from fom readers about the significance 
for the foundations of mathematics of allowing non-distributive 
predication in logic. What follows is an account of the key ideas and 
some thoughts about what the significance might be.
A draft of chapters of a book on plurals and non-distributive 
predication are on my page in my department's website 
(http://philosophy.syr.edu/ ). (Click on "Faculty & Staff" and on my 
name.) Chapters 1 and 4 contain the material that will be of most 
interest to fom readers.

Non-distributive predication
	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.
	They come from many different countries
	They met together yesterday
	They are surrounding the building
	They outnumber us
	They encircled the soldiers with their trucks
	Reforming first-order logic to allow for non-distributive 
predication undercuts some (though probably not all) of the 
motivation for set theory and mereology, which 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). (This way of describing the problem is due to Oliver 
and Smiley, "Strategies for a logic of plurals," The Philosophical 
Quarterly  51: 289-306.)
	When I say 'They come from many countries,' or 'They are 
surrounding Adams Hall,' or 'They are seven in number (or numerous),' 
I am talking about some individuals who non-distributively have these 
properties, not some individual things (sets or mereological sums) 
that have some corresponding distributive property. Note also that 
the inference from "Some students are surrounding the building" to 
"Some single thing is such that it is surrounding the building" is 
not logically valid.
	If we start with only distributive predication, we cannot get 
non-distributive predication out of it.
	I propose instead that we allow in logic what we have in 
English: some individuals may satisfy a predicate together without 
satisfying it individually. More generally, an argument place of an 
n-place predicate may have several individuals associated with it in 
satisfying the predicate.
	X come from many different countries
	X met together yesterday
	X are surrounding the building
	X outnumber Y
	X encircled Y with Z
Singular predication and distributive predication are then just 
special cases. An argument place can be associated with just one 
individual:
	John encircled his house with a fence.
A predication can be distributive, which means that it has a 
distributing universal associated with it.
	They are doctors. [ALLx: x is among them] x is a doctor
A predication can be ambiguous about whether it is distributive.
	They weigh over 400 pounds.
X (together) weigh over 400 pounds
	[ALLy: y is among X] y weighs more than 400 pounds
The distributive reading is special in that it has an implicit 
distributing universal. It is a lexical feature of each argument 
place of a verb that it allows, does not allow, or requires a 
distributing universal. 'X are human' requires distribution, 'X weigh 
400 pounds' allows it but does not require it, 'X are seven in 
number' does not allow it. The important predicate 'Z are among Y' 
requires distribution in its first argument place but not it in the 
second.
	We can introduce individual names and compound terms.
	individual constants: a, b, c, a1, a2, etc.
	compound terms: \a, b/, \a, b, c/, \X, Y/, \a, X/
to be used in representing sentences like these (where 'MX' is a 
monadic, non-distributive predicate):
	Alice and Betty met together. M\a, b/
	Alice, Betty and Carla met together. M\a, b, c/
	Some students and some teachers met together.
	[SOMEX: [ALLz: z is among X] Sz] [SOMEY: [ALLz: z is among Y] 
Tz] M\X, Y/
	Alice and some students met together.
	[SOMEX: [ALLz: z is among X] Sz] M\a, Y/
A distributive predication of several individuals can be represented 
in the following way, employing a distributing universal:
	Alice, Betty and Carla are teachers
	[ALLz: z is among \a, b, c/] Tz
This will of course have to be provably equivalent to the usual 
conjunction used in representing such a sentence.
	The relation 'X are among Y' has central importance and will 
be written 'XAY'.

Logic with non-distributive predication (with limited quantifiers)
(Byeong-Uk Yi has developed a slightly different but similarly 
motivated logic of plurals in Understanding the Many, Routledge 2002.)

The syntax is like first-order logic except that variables are upper 
case and we have compound terms (as indicated above). ALL is defined 
as ~SOME~.
I sometimes use restricted quantification.
	[SOMEX: F] G can be understood as SOMEX (F & G)
	[ALLX: F] G can be understood as ALLX (F -> G)
(I will not employ intermediate quantifiers in this note, so the 
unrestricted form of quantifiers can here be taken as basic. In my 
book ms., I include intermediate quantifiers and so must there take 
the restricted form as basic.)
We can define singular variables by introducing the predicate X is an 
individual ('IX'), defined in the following way:
	IX  =df  [ALLY: YAX] XAY
Then a clause with a singular (lower case) variable x:
	Š x Š
is defined by a clause that involves only plural variables:
      IX & Š X Š	(where the context Š xŠ does not already have occurrences of X)
Our sole "real" variables then are the plural variables, and 
distribution is explicitly represented by the presence of a 
distributing universal.

Logical axioms
All truth-functional tautologies
A -> SOMEY AT/Y (where T is any term and Y in AT/Y is substituted for 
one or more occurrences of T in A)
Rules: MP and Universal closure of theorems

Some relations are defined in terms of the fundamental relation among (A)
X == Y  =df  XAY & YAX 	(X are the same things as Y)
x = y  =df   IX & IY & X == Y
XOY =df  SOMEZ (ZAX & ZAY) 	(X overlap Y)

Axioms for among
Two choices for the axiomatic base:
(AX 1)	ALLX ALLY (XAY ´ ALLW ((IW & WAX) Æ WAY))
(E)	ALLX SOMEY (IY & YAX)
or
(M1)	ALLX ALLY ALLZ ((XAY & YAZ) Æ XAZ)
(M2)	ALLX ALLY (ALLW (WAX Æ WOY) Æ XAY)
(E) 	ALLX SOMEY (IY & YAX)

I mention both possibilities because the relationship is interesting. 
AX 1 simply formulates the fact that the relation A is distributive 
in its first position: XAY is always equivalent to ALLZ ((IZ & ZAX) Æ 
ZAY). M1, M2 and E are fundamental axioms of a mereology with 
individual atoms. (Given any set of atoms, the intended 
interpretation is isomorphic to the field of subsets of that set, but 
excluding the empty set from the field. An important difference from 
the set-theoretic or mereological interpretation: we are talking 
about relations among the things that are in the subsets or that are 
"atomic parts" of a mereological sum, rather than relations among the 
subsets or the sums. We will not take the subsets or sums as 
additional individuals. Loosely, we just drop all the set brackets.)

We need axioms for terms as well. If c is any individual constant:
AST:	Ic
Thus any sentence Fc with an individual constant c can be written 
equivalently as Ic & Fc. We will also want an axiom for compound terms
ACT:	ALLX ALLY ALLW ((WOX Ž WOY) ´ WO\X, Y/)

These follow (where c is any individual constant and ti is any term):
cA\ŠcŠ/
erasure of inner brackets:  \ Š \t1, Š, tn/, Š/   ==  \ Š t1, Š, tn, Š/

If we define the sum of some things X and some things Y, then we can 
restate ACT.
SXYZ   =df  ALLW ((WOX Ž WOY) ´ WOZ)	(X and Y sum to Z)
ACT*:	ALLX ALLY SXY\X, Y/

We will want to add axiom schemas for identity and comprehension:

AS 1:	ALLX ALLY (X == Y Æ (F ´ FX/Y))
where F does not contain Y and one or more occurrences of X in F is 
replaced by Y in FX/Y

AS 2: SOMEY (IY & FY) Æ SOMEX ALLY (IY Æ (YAX ´ FY))
where FY can be any sentence in which Y occurs
(There is nothing corresponding to the empty set, so the restriction 
in the antecedent of AS 2 is necessary.)

We should note that the following is an immediate theorem:
SOMEX XAX Æ SOMEX ALLY YAX
(If some things exist, then there are some things such that all 
things are among them.)

Relationship to second-order logic
	As Boolos argued, we can use the logic of plurals as a basis 
for understanding monadic second-order logic. Consider a second-order 
quantification:
	SOMEF Fx
In the version of our system with singular variables, this would be:
	SOMEY xAY
Monadic second-order quantification corresponds to quantification 
into the second argument place of the among relation. 
Correspondingly, we can state the significant principles that can be 
formulated in second-order logic. For example, we can state the 
general principle of mathematical induction as an axiom rather than 
as an axiom schema. Assume that we have 0 and the successor relation 
Sxy: x is a successor of y, and we have the other Peano axioms. Then 
the induction schema of first-order Peano arithmetic can be replaced 
by the following plural axiom:
	ALLX ((0AX & ALLy (yAX Æ ALLz (Szy Æ zAX)) Æ ALLy yAX)
(If we introduce a functional notation, say 'sx' for the successor of 
x, then the clause 'ALLz (Szy Æ zAX)' can of course be replaced by 
'sy'.) Thus we have a categorical characterization of arithmetic, as 
in second-order logic. Accordingly, we cannot have a complete 
axiomatization of among theory, and the axioms we have laid out must 
be seen as an attempt to get at the main principles of the among 
relation.
	Similarly, we can state the axiom of completeness of real 
analysis (that whenever there is a an upper bound among some things, 
there is a least upper bound among them):
ALLX (SOMEx ALLy (yAX Æ y ¾ x) Æ SOMEx (ALLy (yAX Æ y ¾ x) & ALLz 
(ALLy (yAX Æ y ¾ z) Æ x ¾ z)))
Since second-order analysis is categorical and has only uncountable 
models, it appears then that the Löwenheim-Skolem theorems also fail 
for among theory.

Relationship to set theory
	The presence of the theorem
SOMEX XAX Æ SOMEX ALLY YAX
is of course a key difference that reflects the fact that among 
theory is not subject to the Russellian paradoxes. Broadly, 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, and the Russellian paradoxes do not come into play. 
Whether we use plural or singular quantifiers, non-distributive or 
distributive predicates, we are still talking about whatever are the 
individuals of our domain.
The set theoretic hierarchy is itself an item of mathematical 
interest, and it is a tool for modeling in mathematics, so we cannot 
expect among theory to have the mathematical reach that set theory 
has. Nevertheless, it is plausible that among theory can serve as the 
basis for modeling much of elementary mathematics  --- probably the 
same as what is modeled in full (impredicative) second order 
arithmetic. Since second-order arithmetic involves only sets that are 
not members of further sets, plural talk can replace the 
set-theoretic in second-order arithmetic. The only difference then is 
in the absence of anything corresponding to an empty set in among 
theory. Is this right?
  	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.


-- 
Tom McKay
Philosophy Department
Syracuse University
Syracuse  NY  13244
315 443 2536
tjmckay at syr.edu
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