[FOM] Non-distributive predication
Tom McKay
tjmckay at syr.edu
Fri Jul 11 09:44:54 EDT 2003
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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