Substance Logic

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ERIC WALTHER AND EDDY M ZEMACH

SUBSTANCE LOGIC

I. INTRODUCTION: THE GENESIS OF THINGS

The system of Substance Logic we expound in this article is a logical

system which includes no predicates, using only (a) terms denoting sub

stances and (b) logical connectives. Substance Logic (henceforth, 'SL')

is designed to be a perspicuous nominalistic system, offering the philoso

pher a canonical notation for an ontology (and ideology) in which one

can countenance substances only. Its primitive non-logical expressions

are, therefore, 'this Socrates', that cat', 'this love' and the like. This man

is a substance, a thing. One must resist the temptation to regard a man as

metaphysically decomposable into two components, the particular sub

stratum and the property Manhood. A man' is to be construed as a substantive, not a predicative, expression. Thus, in SL, statements like 'this

is a man' will be understood literally, with the 'is' functioning not as the

'is' of predication but as an identity sign. X is a man' does not say that

X exemplifies Manhood but that he is identical with a certain man; that

man.

However, before we can talk about substances like this man or that

cat, we have to explain how can these substance terms be learned. The

Platonist's usual argument is that what makes it possible for us to identify

this thing and that thing as cats is the presence of the same property,

felinity, in both substances. To offer an alternative explanation, we use the

following two pre-thing notions: the notion of types and the notion of

events. One ofus has argued elsewhere that it is a mistake to regard types

as abstract entities or universals. Rather, types are particulars recurring

both at many different times and in many different places. Mr. Jones is a

material thing, a particular, although he can be, all of him, in two distinct

spatia-temporal locations (e.g., in his officeat9 AM and at home at 8 PM).

Similarly, the type-entity The Cat can be said to be, all of it, in manydistinct spatiotemporal locations. One can, therefore, learn to recognize

the type-entity The Cat whenever and wherever it is to be found. It is the

A. Kasher ed.}, Language in Focus, 55-74. All Rights Reserved.

Copyright© 1976 by D. Reidel Publishing Company, Dordrecht-Holland.



56 ERIC W LTHER ND EDDY M. ZEM CH

same type-entity The Cat which is seen first on the mat, then on the couch,and at the same time climbing a tree in the yard. The type-entity The Cat

is a particular, which one can spot on many distinct occasions, and in

many distinct places.

In The Problems o Philosophy Russell claimed that even if all proper

ties are analyzed away, one cannot be eliminated: the relation, simil r to

How else can we say why we consider and b to fall under the same sortal

if not by saying that they are similar to each other? But then this similari

ty, and the similarity manifested by another coupleof

objects, must beinstances of the same universal, similar to; otherwise we shall have an

infinite regress of similarities. This move, however, is now avoided. The

re-identification of the same entity (i.e., the type-entity The Cat) does not

involve the relation of similarity. If one has learned to identify something,

one can reidentify it. What is reidentified is not similar to The Cat but is

the type-entity The Cat itself. One can properly talk of similarity only

when there are two things which resemble each other in certain ways.

When only one thing is in question, there are neither ontological nor

epistemological reasons to presuppose that it must maintain a similarityrelation to itself.

Next, let us introduce a new ontological category of particulars, that of

cat-events. Whenever and whenever The Cat is present there is a particular

cat-event present. Thus instead of saying that we have seen The Cat on

many occasions, we can say we have seen so many cat-events. Note again

that cat-events are not defined as occurrences of entities of feline nature,

but rather as the various phases, or occurrences, of The Cat. Every

appropriate chunkof

space-time which can be said to host The Cat hosts acat-event.

Finally, we make the move from cat-events to individual cats. This

move is performed when a criterion for cat-identity is applied. Suppose

that, in the present case, the criterion is that if xis a certain cat-event, then

x is the same cat as y if x is cat-continuous withy. That is x and y would

be considered different phases of the same cat if and only if there is a

spatio-temporal route between the location of x and the location ofy such

that each temporal location along this route hosts a cat-event at an

appropriately sized spatial location. Note, incidentally, that this is not tobe understood as if a kitten would be the same cat as its mother.

Although the kitten is bodily continuous with the mother, the two are not



SUBSTANCE LOGIC 57

cat-continuous. The spatia-temporal route which establishes their bodilycontinuity contains a cat-discontinuity at the point where the appro

priately sized spatial location shifts between that of the mother and that

of the (sufficiently cat-like) foetus. (The same result can be achieved by

having an explicit rule to the effect that a kitten-event and a cat-event can

belong to the same cat only if the former precedes the later in time.)

Similar criteria may be used in defining other kinds of individual beings,

and in teaching a student how to use the terms the same F and a differ

rentF

correctly.We

need not turn to universalsat

any stageof

thisprocess. This F is not to be defined as this thing which Fs . Names of

properties are not used in the suggested analysis.

Among the individuals recognized by the present ontology there are

not only cats and dogs, but reds and blues (i.e., red things and blue things)

as well Here we must join hands, at least for a while, with Russell and

other logicians who insist, Aristotle to the contrary, that a man and

pale are to be treated equally. The sky is blue (to use an oft repeated

example) does not do more justice to the fact it purports to describe than

This blue skies . However, Russell, Quine, and their followers used thisequal status in order to reduce names of substances to predicates (e.g., by

saying that man is a predicate true ofmen). We wish, on the other hand,

to exploit this equal status in the opposite direction, and claim that

predicates denote substances (e.g., by saying that pale is a name of each

pale thing). Thus Socrates , man , and white will indeed all be of the

same logical type: Socrates is a substance, a man is a substance, and a

white (e.g., this white thing, that white thing, etc.) is a substance too.

II. IDENTITY

Things such as Socrates, this cat, that white, a certain car and a given

lover are located, as we have just said, in space and time. However, in a

certain man-shaped spatio-temporallocation one may find Socrates, and

also a certain philosopher, and also a certain Greek. Some theorists say

that this location includes more than one thing; according to them, infi

nitely many things can comfortably occupy the same place at the same

time. But things, thus understood, have a spooky ontological status. Theonly substances we wish to recognize are material things in the ordinary

sense of the term, and hence we would say that if two things occupy the



58 ERIC WALTHER AND EDDY M. ZEMACH

same spatio-temporallocation, then, at that location, they are one and the

same thing. A location is not distinct from the entity in it (A,;,L);

hence no two things can occupy the same location. At one location, there

can be only one thing(A,;,L B,;,L-=>A,;,B),

Identity at different locations is non-transitive. For example: suppose

that Plato is sick at one time. At that time, then, Plato is identical with a

certain sick; symbolically, p S. If at a later time Plato got well we would

say that at that location Plato is identical with a certain healthy thing, i.e.,

P;,H, One cannot, however, draw the conclusion that a certain sick

thing is identical with a certain healthy thing (S=H) for the location at

which P is identical with S is not the one at which P is identical with H.

The same conception of identity applies to space as well as to time. A

certain foot, for example, is identical with Socrates at one location (that

is, both Socrates and this foot are to be found in this location and, within

its boundaries, they are one and the same thing: S,;,F), and a certain

head is identical with Socrates at another location (S;,H). But it is

nowhere the case that a certain foot is identical with a certain head. On

the other hand, if Socrates is identical (at a certain location) with a cer

tain tall, and Socrates is identical (at that same location) with a certain

philosopher, we may conclude that, at that location, a certain philosopher

is identical with a certain tall.

But, one may ask, if the basic form of SL sentences is X,; Y , are we

not in fact using predicates - i.e., the predicate 'identical with ... at .. .'?

And if so, are we not committed to the existence of the property, identity?

This we wish vehemently to deny. t is indeed true that in PM, ' ... = .. . 'isconsidered as a predicate, and identity a relation. But even within the

PM framework this strange relation causes a lot oftrouble. We cannot

but agree with Wittgenstein, who says Tractatus Logico Philosophicus,

5.05303 to say of two things that they are identical is nonsense, and to

say of one thing that it is identical with itself is to say nothing at all.

X= Y says nothing about X (i.e., about Y and ascribes no property

to it.

Some may hold (with the early Frege) that X= Y is really about X

and Y , and what it says is that X and Y denote the same object. But

this view cannot be true. In X= Y , X and Y are used, not mentioned.

X= Y does not mention any way of referring to X (i.e., Y . Moreover,



SUBSTANCE LOGIC 59

that X and Y denote the same object is a contingent fact about the(English) language, but (as argued by Kripke) if X= Y is true, it is

necessarily true; everything is necessarily identical with itself.

But if X= Y predicates nothing of X and ' ... = ... ' is no predicate,

neither is ' ... ... . L here attributes no quality to = but rather identi

fies the location in which the statement as a whole is said to be true. L

just indicates the location, or thing, about which this statement, X= Y ,

is made, and is not to be taken as a modifier of the (impossible) relation

of identity''.Finally, a few words ought to be said about an objection to the notion

of localized identity. Richard Sharvy2 argues that the notion is incoherent

since it would make (1x) Fx referentially opaque. For example, if road A

and road B are identical at L the term 'this road' said at L would be

ambiguous; does it refer to A or to B? The problem is genuine, but it is

not serious nor peculiar to localized identity. I t is similar to the famous

puzzle-case of Theseus' Ship, which is ship-continuous with two ships.

As the story goes, the old planks, which had been removed from Theseus'

ship and replaced one by one, were re-assembled to form another ship,

which can also be claimed to be Theseus' ship). However, as Chisholm

points out,3 the problem can be solved by stipulation only. The problem

is pragmatic, not logical; it might, for example, be decided in a court of

law. 'This road' and 'This ship' are open-textured since the entities denot

ed by these terms bifurcate - the first in space, the second in time. If we

wish to close this open-texturedness we can do so only by stipulation.

This concludes the informal part of the paper, showing SL's philo

sophical and didactical underpinning. We proceed now with the exposition of the system itself.

III SIMPLE EQUALITIES

In the ordinary predicate calculus, the elements consist of capital letters

(representing predicates) combined with constants (functioning as pure

names of individuals): Fa, Gb, Hcb, etc. A substance logic which rejects

predicates needs elements of a different type: equalities as we shall call

them). An equality consists of an equals sign with a term on either side.Terms are simple or complex; equalities whose terms are simple will be

discussed first. Every term is singular, particular, or general; it is the



60 ERIC WALTHER AND EDDY M ZEMACH

simple singular terms which constitute the basis relative to which theothers are to be understood.

A simple singular term indicates an object, recognized to be a token of

a specific type: this cat, that white thing, Socrates. An equality in which

both terms are singular asserts that the thing identified by the one term is

the same thing as the thing identified by the other term. Where there is no

explicit indicator of the location within which the equality is to hold, the

location may be assumed to be the universal (i.e., all-inclusive) one; the

formal calculus here introduced is designed for equalities of that sort

exclusively.

(1) C W This cat is this white thing.

(2) C1= C This cat is the same as that cat.

(3) S = M Socrates is that man.

(4) S1 =S2 This Socrates (here now) and that Socrates (there

& then) are the same object.

Where it may be necessary to distinguish individuals of the same type,

they are to be represented by the same capital letter with differing sub

scripts. A capital letter without subscript may denote the type-object, or

it may denote an individual in an understood context where only one

individual of that type is being referred to. If both S1 ' and S2 ' are under

stood to refer to whole objects - objects taken to be the same object as

every object with which they are spatia-temporally and type continuous-

then (4) above must be true: there is only one Socrates. The same would

not hold for (2).

Truth-functional compounds whose elements are equalities areto

begiven their usual truth-functional meaning. Logical axioms (and rules) of

SL will be expressed using the letters X Y Z, and are to be understood as

schemata; the result of substituting terms is an axiom (or a permitted

assumption). The first axioms express transitivity and symmetry for

equality when all terms are singular.

(A-1) X= Y Y=Z) :=l X Z(A-2) X= Y=: Y X

Since our symbols for individuals do not carry a bare referring function(isolated from a characterizing function), as do the traditional constants

and variables, the traditional quantifiers seem out ofplace. (Also, the open



SUBSTANCE LOGIC 61

sentences to which quantifiers are designed to apply would be too much

like predicates for our tastes.) The effects ofquantification are obtained by

the device ofparticular terms and general terms (occurring in the positions

which might be occupied by singular terms). The notation is as in these

examples:

(5)

(6)

m=S

m=S

Some man is Socrates.

All men are Socrates.

The first is to be understood as asserting that k =S for some k in other

words, that some man is the same man as Socrates. The second is to be

understood as asserting that k =S for each and every k in other words,

that any man whatever (however chosen) willbe the same man as Socrates.

It will be observed that the symbols m and m are only place holders apt

for instantiation. They do not denote individuals directly, and the equali

ties in which they occur are not elementary sentences. Their significance

derives from that of the elementary sentences which stand in consequence

relations to them: relations which follow the conventional quantificationalpatterns. Thus we posit the following axioms:

A-3)

(A-4)

X Y=>Xk= y

Xk= Y=>x= Y

(for any k

(for any k

The rules of UI and EG amount to the procedure of introducing such an

axiom and applying Modus Ponens. Instantiation and generalization are

always to be performed upon the first term of an equality (a limitation

which we introduce in order to distinguish (11) and (13) below); but weextend the axiom of symmetry to all cases in which t le st one of the

terms is singular:

(A-2 ) X=y=y=X) (X=y ::y =X

The rules of El and UG specify items which are to be included as

assumptions in proofs which satisfy certain conditions. A proof of a

theorem is a sequence of items ending with the theorem, where each item

is (a) a tautologous truth-functional compound of equalities; (b) anaxiom; (c) a consequence, by Modus Ponens, of two previous items; (d)

an assumption conforming to the stipulations ofEI or UG.




(E.I.) x= Y=>Xk= Y may occur as an assumption in any proof in

which Xk does not occur in the last item (of the proof), in the

prior items (vz. items prior to the assumption), or in the ante

cedent of the assumption.

(U.G.) Xk= Y=>x= Y may occur as an assumption in any proof in

which Xk does not occur in the last item, in the prior items,

or in the consequent of he assumption.

The final condition has been stated here with a generality which becomes

necessary later, when the rule is extended to apply to equalities containingcomplex terms. But in another respect, E.I. and U.G., and axioms three

and four, have been stated in too limited a way. Note that the term which

is not affected by the instantiation or generalization (the Y ) is singular.

This limitation is to be removed; in place of Y , one may also have y

throughout, and y throughout.

f one starts with the equality A =B and considers all equalities which

arise as one substitutes particular and general terms for A and B and

reverses the order of terms in the equality, one arrives at a list of elevenbasic (non-equivalent) equalities.

The list is arranged below in a rough hierarchy (weaker preceding

stronger), so that implication relations (in upward-running sequences)

can be read off easily (following parenthetical indications). Some relevant

theorems follow the list.

(7) a=b Some A is a B. (weakest statement)

(8) a=B This B is an A. (implies 7

(9) A=b This A is a B. (implies 7)(10) a=b EveryA is a B. (implies 9, etc.)

(11) b=a EveryB is an A. (implies 8, etc.)

(12) A=B This A is this B. (implies 8 and 9, etc.)

(13) a=b Some A is (identical with) every B (implies 11, and

that there is only one B)

(14) b=a Some B is (identical with) every A. (implies 10, and

that there is only one A)

(15) A=b Every B is (identical with) this A. (implies 13 and 12,

etc.)(16) a=B Every A is (identical with) this B. (implies 14 and 12,

etc.)



(17)

(T-1)

(T-3)

SUBSTANCE LOGIC 63

a=b Every A and every B are identical. (implies all statements here listed)

a b b a

A1=a=Ak=a

(T-2) a=b=b=a

(T-4) a=a=a=a

Since negation is a truth-functional operator, the meaning to be attri

buted to negations of equalities has already been specified. Still, some

intuitive advantages are gained by introducing non-equalities (using the

s i g n · ~ ) as abbreviations for negations of equalities. Thus, A ~ B , A is

non-equal with B , amounts to - (A= B) , as one would expect. But when

terms are not singular, the intuitive advantage mandates a kind of inver

sion. For example, -(a=b) is the negation of Some A are B , which

amounts to No A are B . When the latter is read as a non-equality, it says

every A is non-equal with every B ; hence, a ~ b . Similarly, -(a=b) ,

the negation of All A are B , amounts to Some A are not B , or some A is

non-equal with every B : a ~ b The formal principle is this: a non-equali

ty between two terms is (by definition) equivalent to the negation of the

equality between the opposite terms (where general and particular termsare opposites, while singular terms are regarded as their own opposites),

(A-5) x ~ y - x=y )

x ~ Y= - x= Y

X ~ y - X=y )

x ~ y - x=y) etc.

What lies behind the intuitive convenience of this convention is that non

equalities may be instantiated and generalized in the same circumstances

as equalities. (This fact can be expressed as a set of provable theorems.)

From a ~ b , for example, one may derive A ~ B ; and a ~ B - : : J a ~ b is a

permitted assumption under the usual conditions.Co-ordinate with the ability to recognize something as an A, would be

the ability to recognize that something is not an A: that it is a non-A.

Accordingly, the term B indicates this nonbeautiful object; B = S

is read this unbeautiful object is Socrates , and Socrates is unbeautiful

is S= - b . It must be remembered that B , as a term, has nothing to do

with B , and that -B 27 , indicates the twenty-seventh non-B, which has

nothing to do with the twenty-seventh B Of course, no object can be both

a Band a non-B, and every object is one or the other:

(A-6) X = y = - X = -y )

T 5 s b =s = -



64

(T-6)(T-7)

(T-8)

ERIC WALTHER AND EDDY M. ZEMACH

A = b v A = b

A=A

a¥= a

The logic here presented is Aristotelian, in the sense that All A are B

implies Some A are B , and that Some A are B or some A are non-B (in

other words: there are A s ) is logically true for any term A which is used.

Nothing much remains to be said about syllogistic logic, since all of the

usual results are derivable using the axioms already introduced.

(T 9) a ¥ b a = - b Some A are notB is equivalent to

Some A are non-B .

(T-10) a¥= b:::: a= b All A are notB is equivalent to

All A are non-B .

T-11) a=b = a=l= b Obversion: All A are B and

(T-12)

(T-13)

(T-14)

(T-15)

No A are non-B .

a = b - b = - a Contraposition: All A are B and

All non-Bare non-A .

(a=b b = c ) = : ~ a = c(a= b b = c - = : ~ a= c (Some typical syllogisms)

a¥= b c a - = : ~ c ¥ b

A further comment onAxiom A-6 is in order. Whereas the other axioms

would hold for equalities in limited locations as well as in the universal

location, A-6 does not. This oddity is easily discovered. Socrates is not a

foot; he is a non-foot. But as we said earlier, in a location which contains

only his foot, he (Socrates; that non-foot) is identical with that foot. Ofcourse (as we said a moment ago), no foot can be identical with a nonfoot:

but in this intuitively obvious assertion (as in most ordinary assertions),

it is identity in the univers l loc tion which is meant. The non-foot (Socra

tes) and the foot are identical in a limited location because, within its

limits, the non-foot and the foot are the same thing. Outside those limits,

the one exists where the other does not; in the all-inclusive location, they

are therefore not identical. I t is exactly this sort of situation - where

things identical in one location are not identical elsewhere - which the

notion oflocalized or partial identity is meant to permit.

The point made here gives us occasion to answer a challenge to the

notion of localized identity, offered by Bernard Williams. 4 Suppose



SUBSTANCE LOGIC 65

that 'A ::B' is false, while A ~ B is true. By "a bil , we will meana thing which is B-identical-at-L: that is, identical with B at L.

By our supposition, A is a non-bil. B, on the other hand, is a bil

(assuming, as we shall to avoid further complexity, that B does exist in

L), since B=B Moreover, when we say that A is a non-bil and B is a bil,

we mean that they are these in the universal location: for the non-bil

which A is, is simply A. What they are in the universal location, they must

be everywhere: specifically, in M. Williams argued that, "by Leibniz's

law , if A and B are identical at M, then whatever is true of A there, must

be true of B there. Now A is a non-bil there; soB must be one too; which

contradicts the fact it is a bil there. We merely deny that there is anything

contradictory in the results. The bil B) and the non-bil A) are indeed

identical in loc tion M, though not elsewhere. Note that we c ll A a non

bil only because of the parts of it lying outside of M. In conclusion, we

maintain that a bil can be a non-bil in a limited location, but not in the

universal location. Indeed, being a non-such in a limited location is entire

ly without significance (trivial). Any object X, existing wholly within the

limited location L is also a non-X there: take, for the non-X, the compound object consisting of X together with some heterogeneous object

which exists wholly outside ofL.

Where F is a limited location, (a) s :. f and (b)-(S :.f) are not equi

valent. In a limited location a thing (e.g., S) may be identical with many

things that are not identical with each other in wider locations. (a) says

that some of those things are non-fs. (b) says that none of those things is an

f. To make (a) true and (b) false we just conceive that one of those things is

a non-f, andone

of them is an f. These two are, of course, not identical inthe univers l location, but they may be identical in the limited location F.

IV. COMPLEX TERMS

Complex terms are devices for achieving multiple reference to one and the

same individual, when that individual is not identified by a singular term.

The expression

(18) a= b a= c

means 'Some A are B, and some A are C', because it is simply a conjunc

tion of two complete and independent components. To express the state-




ment 'Some A are Band C', we need an expression similar to the first inwhich both occurrences of 'a' must be replaced by the same singular

term. Such an expression will not be a conjunction, but rather a single

(particular) term, apt for instantiation. The expression which we will use

for that purpose consists of a term to be instantiated (in particular form),

immediately followed by the expression in which the instantiation is tooccur enclosed in parentheses:

19) a(a=b a=c)

This expression is a complex term It occurs, for example, in the following

equality:

(20) a(a=b a=c)=a Some A (which is a B and a C) is

anA.

The meaning of (20) becomes clear in relation to the following permitted

assumptions (satisfying conditions ofE. I. :

(21) a(a=b a=c) =a=>AtCAt = b At =c) = a

(22) a =At(At = At =c)=> A2 =At CAt= b At= c)

The consequent of the latter assumption is understood to be equivalent

to:

Thus the effect of (20) is that there is an A (denoted both by 'At' and by

A2 in the proof context) which is both a Band a C. The same effect is

achieved more economically by:

24) a= b(b =c) Some A is a B which is a C.

The logical equivalence of (20) and (24) is a theorem: (T 16).

The device illustrated here:

(25) a= a(a = b =>a= c) All A s which are B s, are C s

shows that general complex terms are not needed. We therefore simplify

matters somewhat by permitting only particular or singular complexterms in our calculus. These are defined in such a way that nested com

plexities are permitted, as in the following examples:



SUBSTANCE LOGIC 67

26) a= a(a = b=> a= c(c= d))All A s which are B s, are C s which are D s.

(27) a= a(b = b((a = c =c)=> a= b))

Every A is an A such that every B is a B such that i f he A is a

C and the B is a C then the and the Bare identical.

(28) b =b(b = c=> a= a(a=c=> a= b))

Every B which is a C is identical with every which is a C.

f the expression within parentheses in a complex singular term is re

presented P(Y) (where Y represents the relevant simple term), then

the axiom which was suggested at numbered line 23 above may be

written as follows:

(A-7) X= Y P Y)): :X= Y(lf (Y))

The rest of the derivation system is extended to apply to equalities with

complex terms in the obvious way. Every result obtained by substituting

equalities containing complex terms for the corresponding equalities with

simple terms in an axiom or an assumption, is also to count as an axiom

or an assumption.

Hoping that the given indications are sufficient to enable an interested

reader to work out for himself the remaining details of the formal system,

we close with a few more translation examples and theorems.

29)

(30)

(31)

(T-17)

(T-18)

(T-19)

b = b(b =a=> b =c) All B s which are A s, are C s.

a= a(a = b =>(a= d :::>a= c)) All A s which are B s, are

C s if they are D s.

a= b(b =d => a= a(a= b =>a= c) All A s which are B s,

are C s ifany of them are D s.

a=a a=b)=a=b

(25) is equivalent to 29).

(27) is equivalent to (28).

V. RELATIONS

John loves Mary; Mary loves Joe; John does not love Joe. From facts

like these, PM logicians are wont to abstract the property x loves y . InSubstance Logic, we reject entities of that order. This love, that love, and

that non-love must be substances; so are this taller-than, that swifter-



68 ERIC W LTHER ND EDDY M. ZEM CH

than, and so forth. Just as this man is the same object as this hungry, sothis loving may be the same object as that taller-than. But obviously there

is a more basic type of equality involved, which must be explained first.

This loving is John and Mary; it is a substance consisting of substances.

If John loves Mary, one points at ohn and Mary and says: This is a

love . On our view, that is literally true.

t first blush one might object, saying that a thing is simple while a

relation is a complex state of affairs - a dyadic relation, for example,

essentially includes two components which are so-and-thus-ly related in a

particular way, etc. But is the case really different with things? After all,

for something to be a cat, it must have, say, a head and a tail, and the head

must be so-and-thus-ly related to the tail, etc. A certain love, like a certain

cat, is a quite complex material thing, whose parts are related to each

other in the particular way which makes the whole thing what it is, i.e., a

love (rather than, for example, a cat). Thus, when we write L=JM ,

what we say is that there is a certain thing, i.e. a love, which is the same

thing as the two things, John and Mary, put together.

The complex material body which is John and Mary is, by virtue of

having the parts that it has, a certain love, and a certain taller-than

too, and a certain heavier-than, and a certain co-habiting, and ...

Hence the implication L=JM and T=JM implies L=T holds

if of course, we assume that the location indicator over the equal

sign . s identical throughout this context. Let us repeat that L =T does

not mean that love is nothing but the relation of being taller-than, any

more than S R means that to be square is nothing else than to be red.

What the latter equality means isthat

this square, S is that red; similarly,what the earlier equality means is that a certain love is a certain taller-than,

too.

Other difficulties go deeper, however. John s loving Mary is logically

independent of Mary s loving John; yet the objects JM and MJ are iden

tieal. When we refer to that object as a love (meaning that John loves

Mary), we cannot mean that all of the Iovings, which might occur there, do

occur there, but that one of them does. How is that one discriminated

from the others? The basic factor in such discriminations must be a speci

fication of the components. When we say that John loves Mary, we domore than characterize Min some indistinct way as a loving. Each loving

has a uniquely appropriate division into components (lover and beloved),



SUBSTANCE LOGIC 9

whose positions in the loving are comprehended and distinguished by

anyone who understands what a loving is. Not only, then, do we charac

terize JM as a loving, but we add that in that loving, John is the lover, and

Mary the beloved. Where Lk denotes an individual loving, Ll will de

note the lover-component and will denote the beloved-component. If

that loving is John loving Mary, our complete expression of that fact is:

Lk=JM L l=J L ~ = M . The explanation already given shows why

the first conjunct is insufficient by itself. We regard the first conjunct as

necessary because and are dependent in significance upon Lk ;

though it does appear that an axiom yielding L k = L ~ L ~ would make

sense, intuitively, and would make the first conjunct derivable from the

others. In general, both of the second and third conjuncts are necessary

(not just one), because of the possibility, for some relations, that the ob

jects corresponding to J and M overlap.

The statement 'John loves Mary' does not specify the individual love

(e.g., £ 74) which is John loving Mary; it amounts to 'John with Mary

constitues a (some) love'. t is essential, of course, that the particular

L-terms be linked with respect to instantiation: John must be the lover inthe same love which is JM, etc. The device of the complex term can be

exploited to that end, as follows:

(32) l= l l=JM 11 =J 12 =M John loves Mary

The same insights are to be followed in representing all relational facts.

The following is an example of a triadic relational fact:

(33) g=g(g=JRM g1 =J g2 = R g3 = M)

John gives that ring to Mary

In the formal calculus, we adopt an abbreviated formulation of such

statements. We introduce the ordered symbol X, Y to represent the

organized relational substance with components X and Y Thus 'John

loves Mary' and 'Mary loves John' can be expressed quite simply by

I=J, M and ' l=M, J respectively, and J, M=FM, J. In general, we in

tend the following axiom to hold:

(A-8) X, Y=Z, W=:(X=Z Y= W

We call a term like J, M dyadic, one like M, R, J triadic, and so forth;

in general, we call such terms polyadic, and contrast them with the mona-



70 ERIC WALTHER AND EDDY M ZEMACH

die terms which appear as their components. For convenience, we willalso refer to the substances represented by such terms as dyadic, triadic

and so forth. Axiom number eight, as just stated, concerns equalities

whose terms are dyadic; but we intend no such limitation.

Because there is no similar way of explicating, without ambiguity, an

expression like A, B= C D E in terms of components, such an equality

is to be regarded as necessarily false. Note, however, that polyadic sub

stances may be represented by single letters ( L for Iovings; G for

givings).

Substitution of particular and general components for the singular

components in L=M, W ( This love is this man with that woman )

yields a number of useful statements. But to avoid ambiguities,we must be

careful and explicit about extending the previous axioms to equalities

containing polyadic terms. A polyadic term will be called singular if all

of its components are singular. Axiom one holds for equalities in which

all three terms (be they monadic or polyadic) are singular. Axiom two

holds for equalities in which at least one of the two terms (be they mona

dic or polyadic) is singular. Axioms three and four, and assumptionsE.I. and U.G., are used to achieve the effect of instantiation and

generalization in derivations. As before, they are to occur only with

respect to the left-hand term of an equality. With respect to the right-hand

term, one must first instantiate the left-hand term; conversion by axiom

two is then possible, and makes what was originally the right-hand term

available on the left. If the term on the left is polyadic, instantiation of its

components must proceed from left to right, one at a time; generalization

must proceed in the opposite order. More precisely: a component whichis to the right of a non-singular component m y not be instantiated or

generalized. These conventions are contained in the formal restatements of

the axioms and assumptions given below.

A new abbreviatory convention is helpful at this point. When a capital

letter is bold face, that is to mean that it might be replaced by the partic

ular or general forms of that component (as a plain capital, of course, it

is singular). Thus axioms three and four, when written as below, include

the extension of applicability which was originally explained verbally:

(A-3) x=Y=>Xk=Y (A-4) xk Y > x Y

First, we now understand that a polyadic term may replace the y in these



SUBSTANCE LOGIC

axioms. Next, we also consider the following to be axioms:

(A-3/2) x, Y =z > xk Y = z(A-3/2') X y =Z =>X Yk =Z

(A-3/3) x, Y z w > xk Y z w(A-3/3') X y, Z = W > X Yk Z W

(A-3/3 ) X Y z =W > X Y Zk =W(etc. for A-3/4, A-3/5, etc.)

(A-4/2) Xk Y=Z=>x Y Z(A-4/2') X Yk Z =>X y Z(A-4/3) Xk Y Z W > x, Y Z W

(A-4/3') X ko Z W =>X y, Z W

(A-4/3 ) X Y Zk W =>X Y z W

(etc. for A-4/4, A-4/5, etc.)

The types ofpermitted assumptions are expanded similarly.

(E.I., orig) x Y > xk Y

(E.I./2) x, Y z > xk Y z(E.I./2') X y = Z > X Yk Z

(etc. for E.l./3, E.l./4, etc.)

(U.G., orig) xk = y :::>X= y

(U.G./2) xk Y =z > x, Y = z(U.G./2 ) X Yk Z =>X y Z

(etc. for U.G./3, U.G./4, etc.)

71

With the instantiation principles in mind, unambiguous interpretations

are obtainable for polyadic equalities whose terms contain particular and

general components. Starting from the equality already mentioned,

L=M, W , if one considers all of the equalities which arise when one (1)

replaces singular components by particular or general components, and/or

(2) reverses the order of terms in the equality, provided that the result of

such reversal is not equivalent to the original, one arrives at a list of

thirty-seven basic equalities. The following selection includes most of the

useful ones:

(34) I m, w Every love is that of a man with a woman.

(35) m w I Every man loves some woman or other.(36) 1= m w Some love is that of the man (there is only one)

with a woman.



7

(37)

(38)

(39)


m, w =I There is a man who loves every woman.I = M, w This man loves a woman.

m W = I Every man loves this woman.

Non-equalities may be introduced, as before, as abbreviations for corre

sponding negations of equalities.

(40) I =f m w Some love is not that of any man with any woman.

(Negation of 34)

(41) m w =f.l For any given man and woman, there is some love

which is not he with she.

(42) l =f m w No man loves any woman.

(43) m, w =f I There is a man who fails to love some woman.

(44) m, w =f l Some men don t love any women.

(45) m, w = =I Most comprehensibly paraphrased: If all men are

identical and all loves are identical, then the love is

not the man with a woman.

The reader may have noticed that there is a certain type of statement

which has been omitted from the list. t is often pointed out that the state

ment Every man loves some woman is ambiguous; it may mean (35)

above, or it may mean the stronger statement, There is a woman whom

every man loves . A supplementary device is required in order to express

that stronger statement. A rather perspicuous way of doing it takes its clue

from the grammatical device of passive voice. Where Every man loves

some woman is ambiguous, the smoothest way of deciding the ambiguity

in favor of the stronger alternative is to say, Some woman is loved by

every man . Indeed, every loving is a being-loved-by, with appropriate

re-ordering of the components. If we represent John loves Mary by the

equality J, M =I , it makes sense to represent the equivalent statement,

Mary is loved by John , by an equivalent equality: M, J=Il . The equiva

lence would be expressed by a new axiom:

(A-9) X Y=z= Y X=zz

We may now express the statements which were omitted from the previous

list as follows:

(46) w m II Some woman is loved by every man.

(47) w m =II Every woman is loved by some man or other.



SUBSTANCE LOGIC 73

With the aid of the new axiom, it is easy to prove that (39) implies (46) and

that (46) implies (35), and to observe that (35) does not imply (46). Like

wise, (47) is to (37), as (35) is to (46), so far as implication relations go.

All that is required to extend the device to higher-order polyadic terms

is an established pattern for systematically listing all reorderings of a

given set of components. In the case of triadic terms, the most likely

pattern is: (1) ABC (as given); (2) ACB; (3) BAC; (4) BCA; (5) CAB;

(6) CBA. Then, it being understood that K, B, F=g means K gives B to

F , the following translations are unambiguously determined:

(48) a, b c = g There is an A which gives every B to some

Cor other.

(49) b c, a gggg Every B is given to a C by an A.

(50) b a, c=ggg Every B is given, by some A or other, to a C.

(51) a b, c g Every A has a B which he gives to each and

every C.

(52) c a b = ggggg Each Cis given, by each A, at least one B.

(53) a c b= gg Every A gives, to each C, at least one B.

(54) c b, a = gggggg Every C receives a Bas a gift from all of the

A s.

There are only two equivalent pairs on this list: (49) with (50), and (52)

with (53). Moreover, if the number of g s in any of the given equalities

were changed, without changing anything on the left, the result could not

be equivalent to any given equality (for the effect of doing so would be to

redistribute the roles of gift, giver, and recipient among the A s, B s and

C s). Our final example shows a convenient way of expressing the factthat whenever one thing is bigger than a second, that second is smaller

than the first, and vice versa:

(55) b = ss s = Every big is a backwards-small; every

small is a backwards-big.

(T-20) (55) implies J, M = b =M, J = s .

There seems to be an alternative to the entire device; complex terms

may be exploited to the same end. Compare the following with (46), (47),

( 49), and (52):

(56) w=w(m, w=l)



74

(57)58)

59)


w=w(m, w=l)b = b(c = c(a, b, c =g))

c=c(a, b,c=g)

We conclude with a series of examples in which both dyadic terms and

complex terms are involved. For convenience, a single table of letters has

been used, including 0 for owner-of s, N for married-to s, G for off

spring-of s, C for capitatings (head-of s), and K for neck-supported

body-parts (i.e., heads); the rest are alphabetically obvious.

(60)

(61)

(T-21)

(62)

(63)

(T-22)

64)

m = m(m, S = f=> m = j

k =k(k, h = c > k, a= c)

h =a implies (61)

All of Socrates men-friends

are jealous.

All heads of horses are heads

ofanimals.

p = p(d = d(p, d = o > p, d =b)) Every person who owns a

donkey, beats it.

= d(p = p(o = o(p, d = o > o =b)))

(62) is equivalent to (63)m = m(w = w p =p((m, w = n m, w = b):::> (p, m = g=> p,

m 1 No man who beats his wife is liked by any of his

children.5

C. W Post College, Long Island University

The Hebrew University of Jerusalem

NOTES

E. M. Zemach, 'Four Ontologies , Journal ofPhilosophy 67 (1970), 231-247.2 Things , The Monist 53 (1969), 488-504.3 The Loose and Popular and the Strict and Philosophical Senses of Identity , in:N. S. Care and R. H. Grim (eds.), Perception and Personal Identity, Case WesternReserve University Press, Cleveland, 1969, pp. 82-106.4 In a lecture, Southampton, July 1968.

5 We wish to thank our friend John Bacon for his detailed criticism ofan earlier versionof this paper. We also wish to acknowledge, with gratitude, the assistance of the C. W.Post Research Committee n defraying the costs of typing the manuscript.

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Substance Logic