Lejewski - Ontology, What Next?

8/19/2019 Lejewski - Ontology, What Next?

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antinomy in the ambiguity of the notion of class as used in ordinary discourse. On distinguish-

ing between the notion of distributive class and that of collective class, he argued that some

of the presuppositions from which the antinomy was derived, were not true of collective

classes although they did not appear to be obviously false. This explained to him why the

argument making up the antinomy carried with it an element of cogency although, in the

end, it led to a contradiction.3 In order to give his concept of collective class a sound found-ation he related it by means of definitions to the notion of proper part, which to his mind

was less controversial than the notion of class. It is this notion that Lesniewski used as a

primitive notion to construct an axiomatized system of a theory which, subsequently, he

called Mereology.4 It was a very general theory of partwhole relations, and it provided aframework within which a number of statements about collective classes could be established.

Unlike arithmetic, Mereology was not derivable from what Lesniewski regarded as logic,

but it presupposed no other theory besides logic. It is significant that when Lesniewski

decided to give the form of an axiomatized system to the logic presupposed by his Mereology,

he called that logic Ontology.5 And ontology it was in the sense of being a most generaltheory of what there is.

It was constructed by Lesniewki in 1920, ten years after the publication of the first volume

of Whitehead and Russell’s Principia. It is not surprising that Lesniewki’s Ontology displays

certain refinements not present in Principia. The most notable concern the grammar of

its language, definitions, and, generally, the directives, that is to say, the rules specifying

the conditions under which a new thesis is allowed to be added to the system. However, all

the improvements Lesniewski incorporated into Ontology can be adapted to Principia,

and the difference between the two systems of logic can be reduced to the difference betweenthe types of logical language used, respectively, by the authors of Principia and by Lesniewski.

The meaningful expressions of the language of Principia can be made to form a hierarchy

founded on propositions and singular referential names, whereas the hierarchy of meaning-

ful expressions of Lesniewski’s Ontology is based on propositions and general names, thatis to say, common nouns. Some of them designate one object each, some designate more

objects than one, while some others designate no objects at all. As a result of this sort of pre-

supposition the quantifiers binding name or noun variables in the language of Lesniewski’s

Ontology have no existential import, and, in this respect, differ from the quantifiers which

bind name variables in the language of Principia.6 However, the differences between these two

types of logical language are irrelevant as regards their power of expression. By and large, what-

ever can be said in the language of Principia can be equally well expressed in the language

of Ontology and vice versa. The translations from the one language into the other will be

truth preserving, but they may not necessarily preserve the identity of grammar.

As theories of what there is, neither the logic of Principia nor Lesniewski’s Ontology are complete. The general description of reality they offer is open to expansion. They offer

us, as it were, only an unfinished first chapter of such a description with further chapters

to follow.

Now, an axiomatized system can be expanded in various ways. We can expand its vocabulary

by means of definitions, which may give rise to new and important theses. Alternatively, we

can expand it by strengthening its axiomatic foundations in terms of the available vocabulary.

Finally, we can expand a system by introducing new vocabulary that cannot be introduced

by means of definitions but involves subjoining to the system new axioms. It is this kind of

expansion that I have in mind when I ask in the title of my paper what comes next after

ontology, I mean, after Lesniewski’s Ontology. To put it in metaphorical terms, the question

is this: what is the second chapter in the description of reality? And the answer is already athand. It is Mereology. In fact, in the case of Lesniewski, the second chapter in the description

of reality had been written before the first. It cannot be denied that the theses of Mereology

describe, in very general terms, everything that there is. And the only theory that is pre-

supposed by Mereology, is logic, in Lesniewski’s sense, that is to say, his Ontology. It is

to be noted that the expansion of Ontology achieved by subjoining to it mereological axioms

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ices enrich the original vocabulary of Ontology but it does not affect the original grammar

t c the ontological language.

The logic of Principia can also be expanded into a system of mereology. This has

niready been done by Tarski7 and by Goodman in collaboration with Leonard.8

Like the first chapter the second chapter in the description of reality is not finished.

Tirski’s system of mereology has been expanded, by Tarski himself, into a system of atomistic

rereology and, more recently, a similar expansion has been discussed in the literature withreference to Lesniewski’s Mereology as well as an expansion that turns Lesniewski’s original

system into a system of atomless mereology.9 In accordance with atomistic mereology every

reject has an atom as its proper or improper part, an atom being an object that has no

proper parts. The characteristic thesis of atomless mereology asserts that every object has

i proper part.

For an object to have proper parts it is sufficient and necessary that it should be extended

n time or in space. If this is the case then we have a clue as to what the third chapter in

me description of reality should be. It should be, so it seems, a general theory of time or

i general theory of space. General chronology,10 which appears to be a suitable name for the

former, is likely to be simpler, from the formal point of view, than general stereology, by

*hich I understand the latter, notwithstanding the fact that timeproperties characterizingrejects and timerelations holding between objects are perhaps less intuitive than the corres-

ponding spaceproperties and spacerelations. It is for this reason that , in my view, general

±ronology deserves to be given priority.

The preliminary, philosophical enquiry into the nature o f time has been going* on for

omturies, and will no doubt continue, but it is also true to say that the stage has been reached

for some researchers to pursue the study of time in either of the two directions analogous

id those distinguished by Russell in the case of mathematics. Indeed the progress in the

direction towards precision in measuring the temporal extent of objects of minute duration,

ind in the direction of greater accuracy in placing various cosmic events on a time scale that

mends for thousands of light years into the past and for a shorter, perhaps, but not lessrepressive distance into the future has been amazing, and the credit for it must go to

physicists and to astronomers.

The study of time in the direction towards greater abstractness and logical simplicity seems

jo have been pursued along two different routes. One of them has been very busy in recent

jears. It has become a favourite choice of many talented researchers inspired by the ideas

ind enthusiasm of Arthur Prior. The results of his and his followers’ researches have reached

me form of axiomatized systems of what has become known as tense logic.11 The other route

las been explored by relatively few, and, at present, appears to be deserted. More about

i in a moment.

The characteristic feature of tense logic is that in its framework certain notions relating

zo time in ordinary discourse are taken to be embedded in propositions forming functors

for propositional arguments. For instance, in the expressions of the form ‘it has been the

case that p ’ or ‘it will be the case that p \ where ‘p ’ stands for a proposition, the expressions

it has been the case that’ and ‘it will be the case th at’ are taken to be functors belonging

jo the same semantic category (z.e., the same part of speech) to which the functor of negation,

it is not the case that’, belongs. I entirely agree with Prior that systems of tense logic developed

on these lines have much in common with modal logic, and like the latter can be regarded

is manyvalued systems of propositional logic,12 but I find it difficult to see with what sort

of contribution manyvalued systems of propositional logic can be credited as regards

ontology. Faced with the choice between twovalued logic and a manyvalued one, I opt

for the former because compared with the world in which I live, systems of the latter strikeme as very interesting and exciting pieces of deductive fiction. This does not amount to the

claim that in ordinary discourse propositions with tensed expressions as their constituent

parts have no role to play in describing reality. They describe things just as tenseless pro-

positions do, but, in addition, they relate those things to the utterer. Russell saw that when

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he included all forms of verbs involving tense among what he called ‘egocentric particulars’.13

Roughly speaking the additional information contained in statements involving tense is this:

a statement with a verb in the past tense tells us that the object (objects) it is about, is (are)

earlier than the utterer, a statement in the present tense tells us that the object (objects) it

is about, is (are) contemporary with the utterer, whereas a statement with a verb in the future

tense tells us that the object (objects) it is about, is (are) later than the utterer of the state-

ment.The notion of the temporal order of objects, the notion which is embedded in expressions

such as ‘is earlier than’, ‘is wholly earlier than’, ‘is later than’, ‘is entirely earlier than’ had

attracted the attention of those scholars who preferred to explore the alternative, less busy,

route in the direction towards greater abstractness and logical simplicity in the study of

time. And in exploring this route Woodger was one of the leading pioneers.14 However, the

study of the temporal order of things was not for him the main objective. He had set out

to establish an axiomatized system of a substantial portion of biology, and he stumbled, as

it were, on the problem of the temporal order of objects when he had turned to examining

the presuppositions of his biological system. For in order to articulate these presuppositions

he found it desirable to set up an axiomatized system of the theory of the temporal orderof things,15 a theory which he called theory T. Now, theory T relied on the theory of part

whole relations, theory P in Woodger’s terminology. The articulation of it took the form

of an axiomatized system, the presuppositions of which were finally accommodated within

the framework of Principia. In choosing axioms for his systems of the theories P and T

^ Woodger had the benefit of Tarski’s advice. Thus the axioms for the system of P .are an

adaptation, to the language of Principia, of Lesniewski’s axioms for a system of Mereology

as simplified by Tarski.16In terms of the language of Principia the axiomatic foundations of Woodger’s systems

of P and T can now be stated explicitly.

The notion of being a proper or improper part is the only primitive, i.e., undefined

notion of the system of P. It is embedded in expressions of the form ‘a is a part of b \ ‘a P b' in symbolism.

It is assumed, axiomatically, that

WPAL for every a, b , and c, if a isapartof b and b isapartof c then a isapartof c

In symbolism:

(a b c): aP b . b P c . D . aP c

It is further assumed, by way of definition, that

WPD1. for every a and cp, a isthesumofallobjectseachofwhich cps if and only if ((for

every b, if b cps then b isapartof a) and (for every b , if b isapartof a then there exist a c

and a d such that c cps, d isapartof b> and d isapartof c))17

(a cp ):: a S


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The primitive notion of Woodger’s system of T is embedded in expressions of the form

‘0 is wholly earlier than b* or ‘0 is entirely earlier than by or ‘0 is before b in time*. The

symbolic translations of such expressions are of the form ‘a T b In terms of the primitive

notion Woodger expands his system of P into a system of T by subjoining to the former

ihe following axioms and definitions:

WTA1. for every 0 and b , if 0 is-wholly-earlier-than b then it is not the case that b is-wholly-

earlier-than 0

(a b): a T b . D . ^ ( b T a )

WTA2. for every cp and ip, (there exist an 0 and a b such that (0 is-the-sum-of-all-objects-each-

of-which cps, b is-the-sum-of-all-the-objects-each-of-which ips, and 0 is-wholly-earlier-than b))

if and only if ((there exist a c and a d such that (c cps and d tps)) and for every c and d , if

c cps and dips then c is-wholly-earlier-than d))

((pip) •• (3 0 b) . 0 S cp . b S ip . a T b . = (3 c d) . and c, if 0 is wholly earlier than b and no part of c is wholly earlier than

b then 0 is wholly earlier than c

(a b e) :: a l b : - ( d ) : d P c . D ̂ ( d T b) D . 0T c

WTD1. for every0

,0 is-momentary if and only if for every b and c, if b is-a-part-of

0 andc is-a-part-of 0 then it is not the case that b is-wholly-earlier-than c

In other words:

for every 0, 0 is momentary if and only if no part of 0 is wholly earlier than any other

part of 0

(a) mom(0) . = : (b c): b P 0 . c P 0 . D . ~(Z> T c)

WTA5. for every 0, there exists a such that (b is-a-part-of 0 and &is-momentary)

In other words:

everything has a momentary part

(0): (3 b) . b P 0 . mom(6)

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Woodger goes on to expand his system of T into an axiomatized system of biology, but

this is outside my present concern.

An alternative and deductively stronger system of the theory of the temporal order of objects

has been proposed by Tarski.18 Tarski’s system of T, like that of Woodger’s, presupposes

the theory of part-whole relations, but again the presupposed system of P is stronger than

the one presupposed by Woodger. In fact, it is an atomistic system of P . The axiomatic

foundations of the system include an additional definition and an additional axiom, which

state, respectively, that

TPD2. for every a, a is-a-momentary-point if and only if for every b, if b is-a-part-of a then

b is-the-same-object-as a

In other words:

for every a, a is a momentary point (or a space-time point or an atom) if and only if

a has no proper parts

(a )m om p( a) . = : (b): b P a . D . b = a

TP A3, for every a, there exists a b such that (b is-a^part-of a and b is-a-momentary-point)

(a): (3 b ) . b P a . momp {b)

For the purpose of symbolizing the primitive notion of his axiomatized system of T Tarski

uses the same letter as Woodger, but he attaches to it a different meaning. For him an

expression of the form 'a T b ’ means that either the whole thing a precedes the whole thing

b in time, or that the last slice of a coincides in time with the first slice of b. Thus ‘a T b’

in Tarski’s sense can perhaps be read as ‘a is wholly or almost wholly earlier than b \ Inorder to avoid possible misunderstanding Tarski’s original ‘a T b ’ will, in what follows, be

replaced by 'a T ' b \ The axiomatic foundations of Tarski’s system of T consist of the following axioms and

definitions.

TTD1. for every a, a is-momentary if and only if a is-wholly-or-almost-wholly-earlier-than a

(a): mom (a) . = . a T ' a

TTA1, for every a, b , and c, if a is-wholly-or-almost-wholly-earlier-than b and b is-wholly-or-

almost-wholly-earlier-than c then a is-wholly-or-almost-wholly-earlier-than c

(a b c): a T ' b . b T ' c . D . o T ' c

TTA2. for every a and b , if a is-wholly-or-almost-wholly-earlier-than b then there exists a

c such that (


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TTA4. for every a, there exists a b such that it is not the case that a is-wholly-or-almost-wholly-earlier-than b

In other words:

there is no such thing as the first thing in time(a): (3 b) . ^ ( a T ' b)

TTA5. for every a and Z?, if a is-momentary and b is-momentary then (a is-wholly-or-almost-

wholly-earlier-than b or b is-wholly-or-almost-wholly-earlier-than a)

(a b) mom(t f) . mom(Z>) . D : a T ' b . V . b T' a

TTA6. for every a and b , a is-wholly-or-almost-wholly-earlier-than b if and only if for every

c and d, if c is-momentary, c is-a-part-of a, d is-momentary, and d is-a-part-of b then c is-

wholly-or-almost-wholly-earlier-than d

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Now, if every objects has a momentary part then every object is identical with the complete

collection of its momentary parts. This conclusion is by no means obvious. Moreover, there

does not seem to be any evidence that there are such things as momentary things and, by

the same token, the hypothesis which suggests that every object is extended in time, is not

likely to be falsified. If I were pressed, I would support the hypothesis, but no harm will

be done to my purpose if on the subject of the existence or non-existence of momentary

things I remain uncommitted.Another important feature of the two systems is that they do not take cognizance of the

temporal extent of things. Some things are wholly earlier than other things but it is also

a fact that some things do not last as long as other things. The temporal extent, or the

duration, or the ‘life’ of some objects is shorter than the temporal extent of other objects,

and a theory of time relations that aspires to comprehensiveness, should take this into

account. A theory that does take this into account, will be called, in what follows, General

Chronology or, simply, Chronology, for short.

Notions which relate the duration of one object to the duration of another are in ordinary

discourse embedded in expressions of the form ‘a does not last as long as b ’ or ‘a lasts

longer than b \ or ‘the duration of a is shorter than the duration of b \ or ‘the temporalextent of a is shorter than the temporal extent of b \ They do not seem to be definable in

terms of the notions available in the theory of the temporal order of things. Thus, the required

expansion of the vocabulary of the latter will have to be achieved not by means of definitions

but by means of additional axioms.

The presuppositions of the system of Chronology to be outlined below consist of Lesniewski’s

Mereology, which, in its turn, presupposes his Ontology. It is in the language of Lesniewski’s

system that the axiomatic foundations of the proposed system of Chronology can now be

set out, in stages, as follows:

I. Ontology (Lesniewski’s general theory of objects)

The primitive notion of the original system of Ontology established by Lesniewski is

embedded in expressions of the form ‘a is a b’ or ‘a is b \ In symbolism: ‘a e b \

A single and only axiom of the system states that

OA1. for all a and b , a is-a b if and only if (for some c, c is-an a, (for all c and d , if c is-an

a and d is-an a then c is-a d)> and for all c, if c is-an a then c is-a b)

[a b] • • a t b . = [3 c] . ce a [c d]\ ce a . d e a . D . c e d •• [c]: c e a . D . c e b 20

II. Mereology (Lesniewski’s general theory of part-whole relations)

The primitive notion of the original system of Mereology established by Lesniewski is

embedded in expressions of the form ‘a is a proper part of b \ In symbolism: ‘a e ppt(&)’.

The axioms and definitions, which are subjoined to OA1, are as follows:

M AI. for all a and b , if a is-a proper-part-of b then (b is-a b and it is not the case that

b is-a proper-part-of a)

[a b]: a e ppt(6) . D . b e b . ~(& e ppt(a))

MA2. for all a, b , and c, if a is-a proper-part-of b and b is-a proper-part-of c then a is-a

proper-part-of c

[a b c\: ae ppt(&) . b e ppt(c) . D . a e ppt(c)

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MD1. for all a and b , a isa partof b if and only if (a isan a and (a isa properpartof b or(a isa b and b isan a)))

[a b] a e pt(b) . = : a e a : a e ppt(&) . V . a e b . b e a

MD2. for all a and b , a isa completecollectionof bs if and only if (a isan a, for all c,

if c isa b then c isa partof a), and for all c, if c isa partof a then for some d and e,

(d isa b , e isa partof c, and e isa partof d))

[a b]'-- a e ccl(b). = a e a [c]: c e b . D . ce pt(a) [c]: c e pt(cr). D . [3 d e] .

d e b , ee pt(c) . e e pt(d)

MA3. for all a, b and c, if a isa completecollectionof cs and b isa completecollectionof cs

then (a isa b and b isan a)

[a b c\: a e ccl(c). b e ccl(c) . D . a e b . b e a

MA4. for all a and b , if a isa b then for some c, c isa completecollectionof bs

[ a b ] : a e b . D . [3 c] .c£ccl(b)2i

III. Chronology (a general theory of the temporal order and the temporal extent of things)

The proposed system of Chronology is constructed with the aid of two primitive notions.

The one is embedded in expressions of the form ‘a is an object wholly earlier than b \ In

symbolism: ‘a e (b ) \ The constant term —1—’ is a noun forming functor for one nominal

argument and, in this respect, it differs from Woodger’s ‘T ’, which is a proposition forming

functor for two nominal arguments. Within the framework of Chronology iLJ—’ and*T’ are inter definable, a proposition of the form ‘a e >—!— (by being equivalent to the corres-

ponding proposition of the form ‘a T ft’. Thus, the two propositions can be described as

syntactical variants of each other.

The other primitive notion is embedded in expressions of the form ‘a is an object whose

duration is shorter than that of b \ In symbolism: ‘a e < X (b)\

The axioms, and the definitions which precede some of the axioms, are subjoined to OA1,

\[A1, MA2, MDly MD2, M A3 , and M A4 , and state, respectively, that

CAL for all a and b , if a isan objectwhollyearlierthan b then (b isa b and it is not the

case that b isan objectwhollyearlierthan a)

[a b]: a e >—1— (b) . D . b e b . ~ (6 e >—*— (ar))

CA2. for all a, b , c, and d> if a isan objectwhollyearlierthan b , c isa partof a, and d

sa partof b then c isan objectwhollyearlierthan d

[a b c d\: ae l -1— (b) . c e pt(a) . d e pt (6 ) . D . ce *—*— (d)

CA3. for all a, b , and c, if a isan objectwhollyearlierthan b and c then (a isanobject

whollyearlierthan c or for some d , (d isa partof c and d isan objectwhollyearlierthan b))

[a b c] a e (b) . c e c . D : a e L1— (c) . V . [3 d] . d e pt(c) . d e L-L— (b)

CA4, for alla and b , if (a isan a, b isa b , and for all c and d , (if c isa partof a and d

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is-a part-of b then for some e and /, (e is-a part-of c, / is-a part-of d, and e is-an object-wholly-earlier-than /))) then a is-an object-wholly-earlier-than b

[a b ]y . a e a . b t b [ c d]: ct p t ( a ) . d t pt (b) . D . [3 e f ] . e t pt(c) . f t p t(c/) .e £ UL_ (/ ); . D . a £ l j — (£)

C45. for all a and b , if a is-an object-whose-duration-is-shorter-than-that-of b then b is-a b

[ab]: a c < X (b) . D.Z>£&

C46. for all a and if a is-an object-whose-duration-is-shorter-than-that-of b then it isnot the case that b is-a part-of a

[a b]: a t < X ( b ) . D . * ( b t pt (a))

C A 7 for all a and b , if j is-an object-whose-duration-is-shorter-than-that of b then it is not

the case that b is-an object-whose-duration-is-shorter-than-that-of a

[ a b ] : a t < X (b) . D . ~(Z?e < X (a))

CAS. for all a, b, and c, if a is-an object-whose-duration-is-shorter-than-that-of b and c is-a c then (a is-an object-whose-duration-is-shorter-than-that-of c or c is-an object-whose-duratioa-is-shorter-than-that-of b)

[a b c\ a £ < X (b ) . e t c . D : a t < X (c) . V . c e < X (b)

CA9. for all a and b , if a is-an object-whose-duration-is-shorter-than-that-of b then for somec, (c is-a part-of b and it is not the case that a is-an object-whose-duration-is-shorter-than-that-of c)

[a b\\ a t < X (b) . D . [3 c] . c t pt(Z?). ~(o t < X (


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CA12. for all a, b , c, d , e, and / , if (a is-an object-whose-duration-is-shorter-than-that-of &,

c is-a part-of d , e is-a part-of d> d is-a complete-collection-of objects-each-of-which-is-an a

or /, and ((c is-an object-wholly-earlier-than e and a is-an object-wholly-earlier-than f ) or

(e is-an object-wholly-earlier-than c and / is-an object-wholly-earlier-than a))) then c is-an

obj ect-whose-duration-is-shorter-than-that-of d

[a be d e f ] a e


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Suppose that we have three objects a, b and c, of which this holds:

(1) a consists of b and c, and (2) b is wholly earlier than c

On these conditions it will be true to say that

(3) b is an object whose duration is shorter than that of a or it is not the case that b is

an object whose duration is shorter than that o f a, and (4) c is an object whose duration is shorter than that of a or it is not the case that c is

an object whose duration is shorter than that of a

The conjunction of the two alternations (3) and (4) yields an alternation of the followingfour cases:

(5) b is an object whose duration is shorter than that of a and c is an object whose durationis shorter than that of a, or

(6) it is not the case that b is an object whose duration is shorter than that of a and c is an object whose duration is shorter than that of a, or

(7) b is an object whose duration is shorter than that of a and it is not the case that c is an object whose duration is shorter than that of a, or

(8) it is not the case that b is an object whose duration is shorter than that of a and itis not the case that c is an object whose duration is shorter than that of a.

Conditions (1), (2), and (6) seem to be sufficient for a to be an object that has no beginning but has an end, in other words, that lasts from eternity but does not last for ever.

• Conditions (1), (2), and (7) seem to be sufficient for a to be an object that has a beginning but lasts for ever.

Conditions (1), (2), and (8) seem to be sufficient for a to be eternal, that is to say, tohave no beginning and no end.

Finally, conditions (1), (2), and (5) are sufficient for a to be an object that lasts but isfinite in time.

The chronological notions just mentioned are all definable within the framework of the proposed system. On subjoining appropriate definitions one can go on to prove, for instance,that nothing is earlier or later than an eternal object, and that nothing lasts longer than aneternal object. One can also prove that objects which have no beginning are all of the sameduration irrespective of when they end, and that objects which last for ever are all of thesame duration irrespective of when they begin. In fact it turns out that objects with no beginning or no end last as long as eternal objects.

But are there eternal objects? No answer to this question is to be found in our system ofChronology. However, when the existence or non-existence of momentary objects wasmentioned earlier in the essay, I did not conceal my sympathy with the view that no suchobjects existed. And if I had to commit myself as regards the existence or non-existence ofeternal objects, I would risk making an affirmative conjecture. Now, if there are any eternalobjects at all then, in accordance with the proposed system of General Chronology, the world,which is the complete collection of objects, that is to say, the totality of what there is, isone of them.

♦ University of ManchesterManchesterEngland

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NOTES

* This essay is dedicated, with respect and affection, to the memory of J. H. Woodger.

1 See B. Russell, Introduction to Mathematical Philosophy (London 1919), p. 1f.2 See H. Scholz, Mathesis Universalis (Basel/Stuttgart 1969), pp. 399—436.3 See S. Lesniewski, “ O podstawach matematyki” , Przeglad Filozoficzny, Vol. 30 (1927), p. 182ff., and

B. Sobocinski, “ L’analyse d el’antinomie Russellienne par Lesniewski” , Methodos,Vo\.2 (1950), p. 239ff.

« See S. Lesniewski, “ O podstawach matematyki” , Przeglad Filozoficzny, Vol. 31 (1928), p. 261 ff ., andC. Lejewski, “ A Single Axiom for the Mereological Notion of Proper Part” , Notre Dame Journal

o f Formal Logic, Vol. 8 (1967), pp. 279—285.3 See S. Lesniewski, ‘‘liber die Grundlagen der Ontolog ie” , Comptes rendus des stances de la Society des

Sciences et des Lettres de Varsovie, Class III, XXII Annee, 1930, pp. Ill—132; and also C. Lejewski,

“On LeSniewski’s Ontology” , Ratio, Vol. 1 (1957— 1958), pp. 150—176.6 In connection with this topic see C. Lejewski, “ Logic and Existence” , The British Journal for the Philosophy o f Science, Vol. 5 (1954), pp. 104—119; and C. Lejewski, “ Logic and Onto logy” , in:

E. Agazzi (ed.), Modern Logic: A Survey (Dordrecht-Boston 1981), pp. 379—398.7 See A. Tarski, “ Appendix E ” , in: J. H. Woodger, The Axiomatic Method in Biology (Cambridge

1937), pp. 161— 172.* See N. Goodman and H. S. Leonard, “ The Calculus o f Individuals and Its Uses” , Journal o f Symbolic

Logic, Vol. 5 (1940), pp. 45—55.9 See B. Sobocinski , “ Atomistic Mereology I” , Notre Dame Journal o f Formal Logic, Vol. 12 (1971),

pp. 89— 103; B. Sobocinski, “ Atomistic Mereology II” , Notre Dame Journal o f Formal Logic, Vol. 12

(1971), pp. 203—213; B. Sobocinski, “ A Note on an Axiom System of Atomistic Mereology” , Notre

Dame Journal o f Formal Logic, Vol. 12 (1971), pp. 249—251; and C. Lejewski, “A Contribution to

the Study of Extended Mereologies” , Notre Dame Journal o f Formal Logic, Vol. 14 (1973), pp. 55—67.w See A. N. Prior, Papers on Time and Tense (Oxford 1968), p. 85 f.» See A. N. Prior, Time and Modality (Oxford 1957); A.N . Prior, Past, Present and Future (Oxford 1967);

Prior (1968); and the literature referred to therein.

a See Prior (1957), p. 8.3 See B. Russell, An Inquiry into Meaning and Truth (London 1940), p. 108ff.; and B. Russell, Human

Knowledge (London 1948), pp. 100 ff.14 Woodger, J. H ., The Axiomatic Method in Biology (Cambridge 1937), p. 55ff.; and J. H. Woodger,

The Technique o f Theory Construction (Chicago and London 1939).

3 The words ‘thing’ and ‘object’ are used in this essay as synonyms.* See A. Tarski, Logic, Semantics, Metamathematics (Oxford 1956), pp. 24—29; and Tarski (1937),

pp. 161— 172.17 ‘cp’ is a variable that takes one place predicates as its substituends. In other words, it is an intransitive

verb variable. ‘

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