Ontology in the Tractatus

7/21/2019 Ontology in the Tractatus

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7

Notre Dame Journal of Formal Logic

Volume IX, Num ber 1, Jan uary 1968

ONTOLOGY IN THE TRACTATUSOF L. WITTGENSTEIN

ROMAN SUSZKO

Abstract mathematics

may be a genuine philosophy.

The Tractatus Logico-Philosophicus of Ludwig Wittgenstein is a very

unc lear and ambiguous metaphysical work. Previou sly, like many formal

log icians, I was not int ere sted in the metap hysics of the Tractatus. How-

ev er , I read in 1966 the text of a monograph by Dr. B. Wolniewicz of the

University of Warsaw

2

and I changed my mind. I see now that the con-

ceptual scheme of

Tractatus

and the metaphysical theory contained in it

may be reconstructed by formal means. The aim of this paper* is to sketch

a formal system or formalized theory which may be considered as a cle ar,

although not complete, reconstruction of the ontology contained in Wittgen-

stein's

Tractatus.

It is not easy to say how much I am indebted to Dr. Wolniewicz. I do

not know whether he will ag ree with all the orem s and definitions of the

formal system pre sented he re . Ne verth eless, I must declare that I could

not write the present paper without being acquainted with the work of Dr.

Wolniewicz. I learn ed very much from h is monograph and from conversa -

tions with him . However, when pre sen ting in this paper the formal system

of W ittgenstein's ontology I will not re fer mostly eithe r to the monograph

of Dr. Wolniewicz or to the Tractatus. Also , I will not discuss here the

problem of adequacy between my formal construction and Tractatus. I think

that the W ittgenstein w as somewhat confused and wrong in certa in poin ts.

For example, he did not see the clear-cut distinction between language

(theory) and metalanguage (metatheory): a confusion between use and men-

tion of expressions.

Presented in Polish at the Conference on History of Logic, April 28-29, 1967,

Cracow, Poland, cf. footnote 1.

Received Novemb er 16, 1967


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8 ROMAN SUSZKO

Ludwig Wittgenstein attempted in the Tractatus to build a theory of the

epistemological opposition:

Mind (language) - Reality (being)

One may distinguish in the Tractatus the three following components:

1.

Ontology, i.e. , a theory of being,

2.

Syntax, i.e ., a theory of the stru ctu re of language (mind),

3 . Sem antics, i.e ., a theory of the epistemolog ical rela tion s between

linguistic expressions and reality.

I pre sen t below the formalized versio n of W ittgenstein's ontology. The

syntax and semantics contained in Tractatus will be not considered here.

W ittgenstein 's ontology is gen eral and a formal theory of being. It may be

called here sho rtly: ontology. It concerns (independently of time and

space)

3

,

situations (facts, negative facts, atomic and compound situations)

and obje cts. Thus, the ontology is composed of two pa rts :

1.

s-ontology, i .e ., the ontology of situations(Sachlagen),

2.


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ONTOLOGY IN THE TRACTATUS 9

which converts any sentence into its negation: notc/?. To see this, let us

make thefirst step in formalizing (1) and (2). We write:

(3) for somep, Spandnot Fp

(4) for some x

9

Pxand not Lx.

The letter p is a senten tial variable and the let te r x is a nominal variable.

They ar e bound above in (3) and (4), resp ect ively, by the existentia l quant i

fier:

"for so m e". The symbols P and L ar e unary predic ate s and the

let te rs S and F are unary sentential connectives like the connective "n o t ".

The difference between sen ten ces (senten tial varia bles, senten tial for

mulae) and na me s (nominal varia bles, nom inal form ulae)

4

is very deep and

fundamental in

every

language. It mu st be observed in any rigorous think

ing. H owever, nat ur al language lead s somet imes to confusion on th is poin t.

Having in mind the cat egor ial difference mentioned above we consider the

sentence (1) quite as legima te and meaningful as the sent ence (2). M ore

over, both sen ten ces (1) and (2) ar e true beca use:

1. It is a situation that London is a small city but it is not a fact.

2. D r. B. W. is a good ph ilosoph er but he is not a logician .

1.

The language of ontology.

The language

0

of ontology contains vari

ables of two kinds:

1. sent ent ial varia bles: p, q, r, s

9

t, u

9

w, p

ly

p

2

, . . . ,Pk, . .

2. nom inal variables: x, y

9

z, A, B

9

C,R

9

S

9

x

l9

x

2

, . . . , * & , . . .

There

are in

0

some sentential con stant s, i.e., simple sent en ces. Th ere

may be in

0

some nomina l con stan ts, i.e ., simple nam es. The meaningful

expressions of Q

9

or

well

formed formulae of

0

> ar e divided into sen

ten tial formulae and nominal formulae. The variab les may occur in for

mulae as free or bound (by suitable op er at or s) . Any sent en tial or nominal

formula which cont ains no free variable is called a senten ce or nam e, as

t h e case may be. The form ulae of

0

ar e built of variab les and con stant s

by m eans of ce rt ain sync ate gorema tic expre ssions (and pa ren th eses)

like

predicates, functors

5

, sentential connectives, quantifiers, etc.

At

first

we mention the pr edica te of logical ident ity and the connective

of logical identit y. Both ar e denoted by the sam e symbol: = . Thus, if 77,

ar e nominal formulae and

9

ar e senten tial formulae then the expr es

sions:

(1.1) 77 = =

ar e sentent ial formu lae. The language

0

contains the customary compound

sentential formulae: A

9

v

9

9

=

9

N built by mean s of

classical sentential connectives (conjunction, altern ation , ma ter ial im plica

t i o n ,

mat er ial equivalence, negation, respectively). The quan tifiers: v,3

(general and existen tial) ar e allowed to bind sent en tial varia bles and

nominal

variab les as

well.

Thus we have in

0

senten tial form ulae of the

following forms:


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10 ROMAN SUSZKO

(1.2) V*

(x) Ix {x)

(1.3) Vp (p)

1P (P)

Strictly speaking, we have in

0

two kinds of quantifiers: (1.2) and (1.3).

But we do not need to use different symbols for them. The language Q

o

contains also the description operator (unifier in the sense of P. Bernays,

1958) which is allowed to bind nominal and sentential variables. Itwillbe

not used explicitly here however.

All the symbols mentioned above (=, , V, %

=

9

N, V,3 and the un ifier)

are used in

0

with the ir customary meaning. This means that the relation

of logical derivability in

0

is determined by the classical ru les of in te r

ference.

We assum e that a senten tial formula

follows logically (is

logically

derivable) from sentential formulae

9

. . . ,

k

if and only if

may be obtained in finitely many steps from

lf

. . . ,

k

according to the

ru les of classical logical calcu lus. This calculus includes: two valued

sentential tautologies, modus ponens (for materia l implication) , the rules

for introducing and omitting quan tifiers, two rules of substitution for free

variables, the axioms for logical identity and the axioms for the un ifier.

A formula

is called a logical theorem if and only if

follows logically

from the empty set of formulae. We decide to

allow

in the system of

ontology the definitions of equational type only, i.e. , identit ies (1.1) of some

special form.

Our

task consists in determin ing two sets of senten tial formulae: the

set of ontological axioms and the set of ontological definitions. Con

sequen tly, the ontology or the set of ontological th eorems may be identified

with the*set of all sentential formulae which

follow

logically from ontolog

ical axioms and definitions. Clearly, every logical theorem is an ontolog

ical theorem but not conversely. Many theoremswill be formulated below

with free variables. The reader may convert them into sentences by p re

ceding them by general quantifiers. The ontological axioms and definitions

proposed below cannot be considered as the ultimate solution of our prob

lem. To avoid unnecessary derivations, many ontological theorem s are

included here in the set of axioms. Therefore, our axiom system is not

independent.

It may be replaced by another equivalent one which is sh or ter

and more elegant . On the other hand, perhaps there may be some reason s

(other than

simplification) eith er to strengthen the proposed axiom system

(e.g., to rep lace the equivalencies (4.7), (4.8) and (5.6) by correspon ding

equalities) or to change in some way the adopted definitions. I am not here

to system atically study the logical propert ies (independent axiom system ,

the derivations of th eo rems, the proof of consistency) of the formal system

of ontology. I intend only to p resen t a formal theory which is ident ical in

content

with Wittgenstein 's ontology.

6

2. L ogical identity For logical iden tity we have the twofollowing axioms:

(2.1)

Vs{s s)

(2.2)

Vz(z

z)


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ONTOLOGY

IN THE

TRACTATUS

11

an d the twofollowing schemes of axioms:

(2.3)

x

=

y * ( (x) = (y))

(2.4)

P=q~*

( (P)

(q))

T he

axioms for identity belong to the logical ca lculus. They ar e logical

theorems.

It is easy to see that the

following

schemes of formulae:

(2.5)

x = y * ( (x) = (y))

(2.6)

p= q ( (p) = (q))

follow

logically from the axioms for ident ity. Clear ly, we have:

(2.7) P=q~>P=q

Observe that the form ula:

(*) P=q *P=q

is equivalent to the

following

scheme of formulae:

(**) p

q ( (P)

(q))

In

the sequel wewilluse the auxiliary definitions:

(2.8) (P q)= N( />=

q)

(2.9)

(x y) =

N(* =

y)

(2.10)

p q =

N(/> =

q) (p

or

q,

but not both)

(2.11)

Sp= (p = p)

(it is a situat ion that

p)

(2.12)

Ox

=

(x

= #)

(X

is an object)

T he

term "situation" is a universal sentential connective and the term

"object " is a un iversal predica te. We have the th eo rem s:

(2.13)

Vp Sp Vx Ox

T he

logical axioms of identity have in te rest ing consequences in

o

ontology. We use here the connective

L

of necessity which

will

be intro

duced la ter . We obtain imm ediately from (2.5) th at :

x = y *

(L (x)

=

L (y)) . It is a logical theorem. By (4.9) we have:

L(x

=

x). It

follows that:

x

=y > L(x=y)

On

the other hand, let us suppose that our language contains an opera

t o r B

binding no variable and such that the expression

JB

is a sentential

formula for any nominal formula and any senten tial formula

.

Thus, the

symbol

B

is a mixed operato r (predicate connect ive) studied first by J. Los

in

1948. Itfollows from (2.5) and (2.6) that :

x

= y * {zB (x) = zB (y))

In view of th is logical th eorem one may say that the operat or

B

above can

n ot be used to formalize the notion of believingwhich occurs, e.g., in the

true

sentence:

Ibelieve that Wittgenstein was a great philosopher.


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12 ROMAN SUSZKO

3. S ontology

Weassum e the axiom:

(3.1) Np =Nq~>p=q

The

connective "fact" or"positive fact " isintroducedbythe definition:

(3.2)

Fp=p

The

formula Fpm eans: it is a(positive) fact thatp. The formula Npmay

be read:

it is a

negative fact thatp

. It

follows from (3.2) that: Fp =

FFp

=

, . . . , = FF. . .Fpand:

(3.3) FNp =NFp =Np (double equality)

(3.4) F(p q)=FpAFq

(3.5)

F(pv q)

=

FpvFq

(3.6) Vr(Sr > FrDNr)

Let

be an

arbitra ry sentence.

We

assum e the definitions:

(3.7)

l =F vN 0= FGANG

Clearly, OD1. Itfollows from (2.7) that:

(3.8) 10

Of co urse,wehave:

(3.9) Fland, consequently, lp{Sp

A Fp)

(3.10) NOand, consequently,

lp{Sp

Np)

Thus,

there

exist

(positive) facts and negative facts.

The formula *), i.e.,

Vp\fq(p

=q*P q)may becalled the condition

of ontological two valuedness. Toexplain this terminology let usremark

that

theorems (3.9)and(3.10) state theexistence of (positive) factsand

negative facts. On theother hand, the formula *)above is equivalentto

each

ofthe following formulae:

(1) N(lPiq(P q Fp Fq))

(2)

N(iPiq(P q Np Nq))

(3) S p >p =

lvp=0

( 4 )

Fp=(p= 1)

(5) Np=(q=0)

(6) Vr


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ONTOLOGY

IN THE

TRACTATUS

13

4. Modality The s ontology contains the sentent ial connectives of neces

sity Land of possibility M. We assume thefollowing equalities as ontolog

ical axioms for modal connectives:

(4.1)

NNp = p

(4.2)

NL r = MNr NMr=LNr

(4.3.1) LLr=Lr MMr=Mr

(4.3.2) LMLMr=I M r MLMLr

=

MLr

(4.4) LMZr

=MLr MLMr=LMr

(4.5) MLr =Lr LMr=Mr

The

equalities above reduce so called modalities.

8

We have here 6 mod

alit ies as in Lewis' system S5: Fp, Np, Lp, Mp, NLp, NMp. The

follow

ing

laws

of distributivity:

(4.6) L(pAq) = Lp Lq M(pvq) = MpvMq

also serve to simplify the formulae containing modal op erators. The

formulae (4.6) are assumed here as axioms. We assume as ontological

axioms the formulae:

(4.7) Lr={r

=

1) Mr = (r 0)

(4.8) (p = q) = Lip= q)

and

all formulae of the form:

(4.9) L

where the formula is an arb itrary logical theorem. The comprehensive

axiom scheme (4.9) may be considerably simplified if we take into account

that the logical theorems are suitable "axiomat ized " in a logical calculus.

Clearly, we have the theorems:

(4.10)

L p >Fp

Fp Mp

(4.11)

N(Fp Np)

=

(Np

v

Fp)

=

(p

q)

(double equality)

(4.12) L(p^q) >(LP + Lq)

We assume as axioms the

following

important axiom schemes (Barcan

formulae)

9

:

(4.13)

LVp (p) = VpL (p)

(4.14) Mlp (p) = lpM (p)

5. Boolean

algebra

of

situations

A situation p may include another situa

tion q. Then we say also that the situationpentails the situation qor that

the situation q occurs in the situation p and we write simply: pEq.

Note

that:

pEq =

FpEFq. The connective

E

corresponds to Lewis' str ic t impli

cation.

However, we do not consider it as an int erpretation of the condi

tional sentences in natura l language. This would be a confusion. Inclusion

or

entailment is a special relation between situation s in the same sense as

necessity and possibility are p rop ert ies of situation s. They are connected

together as stated in the

following

axioms:


8/27

14 ROMAN SUSZKO

(5.1) L(p q)=pEq=NM(Fp Nq) (double equality)

(5.2)

Lp=lEp NMp=pE0

It is clear that only one of the connectives L, M, E need be taken as a

primitive undefined symbol. We assume as ontological axioms all for

mulaeof theform :

(5.3) E

where is a sentence and the senten tial formu la follows logica lly from

. It may be shown that theaxiom s 4.9) and 5.3) areequivalent. C learly,

we havethetheorem:

(5.4)

pEq (p +q)

The following

axiom s sta te tha t theu niverse of all situations is a

Boolean algebra withtherelationof inclusion E. Thesituation s p v q

9

p q

andNp

may becalled: the sum andproductof situations

p, q

and thecomp

lementof thesituat ionp.

(5.5) pEp

(5.6) pEqAqEp

=

(p

=

q)

(5.7) pEq qEr *pEr

(5.8) (p q

=

p)=pEq = (pv q

=

q) (double equivalence)

(5.9) pEq

= NqENp

pEq={p Nq

=

0)

(5.10)

pEl l=N0 QEp

(5.11) p

A

q

=

q

A

p pvq

=

qvp

(5.12)

p (q A r)=(p

^)

r p v (qvr)=(pvq)v r

(5.13) p* (qvr) =(p* q) V( />Ar) pv(q

A

r) = (p v q) A (p

v

r)

(5.14) p * 1 = p pv0=p

(5.15)

p Np=0

pvNp=l

Remark.

According to the term ino logical convention assume d above

the

formula pEq is to be read :

t

p includes q . In cu stom ary Boolean

terminology it is read : p is includedinq. If we hadreplacedthe equival

encies (4.7), 4.8) and 5.6) by correspon ding equalities then theconnec

tives

, M,E would be definable as

follows:

Lp = (p=1),Mp

=

(p 0),

pEq =((p>q) =1). Let us introducenow thedefinitionof theindepend ence

of situations. Theformula plqis to beread : thesitua tions />,qareinde

pendent.

(5.16)

plq= M(Fp

A

Fp)

A

M(Fp ANq) A M(Np

A

Fq)

A

M{Np

A

Nq)

From thisweinfer t ha t:

(5.17)

Niplq)

=

(pENq)

v(


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ONTOLOGY

IN THE TRACTATUS 15

second one asse rt s the existence of at least two independent situa tion s. It

is formulated as thefollowing axiom:

(5.19)

lqiq(Sp.

Sq

plq)

It

follows from th is that 3#(1 p 0). Consequently,

3 t f

i q ( P

= q ^ P q )

Thus, we have the negation of the formula (*) discussed in section 3.

We assum e that the Boolean algebra of all situat ions is, in the usual

term inology, atom ic (5.24) and complete (6.1), (6.2). Let us introduce the

following

definition:

(5.20) Pw =

M W

Vr(Sr

wEFr

vwENr)

The formula Pw may be read: "th e situationw is a point".

10

Following E.

Stenius (1960) the points may be called the possible worlds. They con

stitute, in the terminology of Tractatus, logical space (logischer Raum). If

Pw then the formulae

wEFr

9

wENr may be read as

follows:

"th e situation

that r occu rs (does not occur) in the possible world w . It is easy to prove

the followingtheorem s:

(5.21) Pw >{wENq=N{wEq))

(5.22)

Pw >(wE(p

q)=wEp

wEq)

(5.23)

pEq = Vw(Pw (wEp

>

^ # ) )

Thefirst axiom announced above (Boolean atom icity) is represen ted by

the formula:

(5.24) Mp >lt(Pt* tEp)

It may be read:

"every

possible situat ion occu rs in some possible world.

If plq then the four situat ions p, q, Np

9

Nqare different and they differ

from 1 and 0. Th erefore, it follows from (5.19) that th ere exist at least

four situations r such that Mr and MNr. There

exist

possible situations.

Therefore, the possible worlds

exist,

too. Clearly, if 0 ^ r ^ 1 then

0 /

NT]

1. Consider a situat ion r such tha t Mr and MNr. The situations

r, Nr cannot occur in the same possible world. Consequently, th ere exist

at

least two possible worlds:

(5.25)

lw u(Pw

Pu

UJ

w)

There

is , of cou rse, the question how many possible worlds

exist.

The

number of them depends on the number of states of affairs and determ ines

the number of situations. This

will

discussed later.

6. Booleancompleteness Let us introduce the sum operator V binding

sentential var iables like the quantifiers. Using it we build formulae of the

form Vr (r). The situation that Vr (r) is the sum (ontological alt erna

tion)

of all situations r such that (v). This is expressed in the

following

scheme of axioms (Boolean completeness):


10/27

16 ROMAN SUSZKO

(6.1) Vp( (p)

~ *pEVr (r))

(6.2) Vp( (p) pEq) * Vr (r)Eq

Th e sum operator V may be defined by means of the unifier (description

operator) if we assume instead of (6.1) and (6.2) the corresponding existen

tial axioms. It is

easy

to prove that:

(6.3) V s( ( s) = s = p

x

vs =p

2

) >Vr ( r) =p

lV

p

2

One may define the product operator binding sentential variables

which co rrespon ds by duality to the operato r V. We assum e the definition:

(6.4)

Ar (r)

=NVr (Nr)

Th e

situation that r (r) is the product (ontological conjunction) of all

situations r such that ( ). This is illustrat ed by the theorem:

(6.5) Vs( {s) = s = p

x

v s =p

2

) r (r) =p

p

2

Of course, two schemes of axioms, being dual to (6.1) and (6.2), hold for the

operator . It is easy to prove thefollowing theorems:

(6.6) tEAr (r)

=Vr( (r) >tEr)

( 6 . 7 ) Vr (r)Et =Vr{ {r) >rEt)

I t follows

from these theorems that if Vr{ (r) = (r))then:

Ar (r)

=Ar (r) and

Vr (r)

=Vr (r) .

It

may be interesting to compare the operato rs ,V with the quan

tifiers V 3. Let us consider only V and 3. It is c lear that : (q)E31 (t).

Consequently, we have the implication:

(6.8) lq(p = (q))>pElt (t)

Suppose now that:

Vp(lfi = (t))

pEq).

It

follows

that:

Vt( (t)Eq),

i.e.

LVt( (t) * q). The formula: Vt( (t) > q) (lt (t) * q)is alogicalthe

orem.

Therefore, we have: (3f ()

q),

i.e.,

t (t)Eq

. Thus, we have

proved the formula:

(6.9) V>(3*(/> =

(t))

PEq)

3t( (t)Eq)

.

If we now compare (6.8) and (6.9) with (6.1) and (6.2) then we see that:

(6.10) lt (t) =

Vr(3( r

=

(t)))

.

Both

opera tors V and 3 are generalizations in a sense of the connective

of altern ation v. However, th ere is a sharp difference between them. We

explain it in the following intuitive but not quite exact way. The formula

3t (t) is the alternation of all sentences of the form ( )where is any

sentence.

The formula Vr (r) is the alternation of all sentences such

that

the sentence ^(

) is t ru e. The reader may compare the operators

and V and rec on struc t a theorem dual to (6.10). It

follows

directly from

(6.1) that:


11/27


(6.11) pi ip) wEp) *wEvr (r)

We prove that the implication (6.11) may be reversed if the situation

thatw is a possible world. Suppose th at :

1. Pw

2. wEVr (r)

3.

Vp( (p) *N(wEp))

According to (5.20) we infer that:

Vp{ (P)

wENp).

Consequently,

Vp( (p) * pENw). Using (6.2) we obtain: Vr (r)ENw. Therefore, by (3):

wENw, i.e. w =0, cont rary to (1). Thus, we have proved the theorem:

(6.12) Pw (wEVr (r) = lp( (p) AwEp)

One

may read this theorem as

follows:

if w is any possible world then the

situation that Vr (r) occurs in w if and only if some situationpsuch that

(p) occurs in w. Using (6.12), (5.23) and (5.6) one obtains

easily

the the

orem:

(6.13)

Vr (r)

v

Vr (r)

=

\r( (r)

v (r))

The reader may formulate three theorems which are dual to (6.11),

(6.12) and (6.13). Take no tice of the last case

We conclude this section with the theorem:

(6.14) p =Vw(Pw AwEp)

which may be easily obtained from (5.23), (6.1) and (6.2). It stat es that

every

situation is the sum (ontological alternation) of all possible worlds in

which it oc curs. The situation

Vw(Pw

wEp)

is composed of points in the

logical space. One may say that every situation is a place (Ort) in the

logical space.

7. Thereal

world

Let us consider now th ree situat ions s

0

, s

l9

andwwhich

are defined as follows:

(7.1) s

o

=

VrFr

s

x

=

irFr

(7.2) w =ArFr

Clearly,

Fs

AJVs

0

.

This is a logical theorem. It

follows

that:

(7.3) s

x

= 1As

0

=0

Using the theorems dual to (6.1) and (6.2) we obtain the equality:

(7.4) ArFr =

r(Fr A

r / 1)

Starting now with the theorem dual to (6.1) we obtain two formulae:

(7.5) Fp^wEFp Np ^wENp

It

follows

from th is that : V>(S/>

vEFp

v wENp). Thus, we have proved

the

remarkable theorem:

(7.6) w =0 vPw


12/27

18 ROMAN SUSZKO

It means that either the situationw is impossible or it is a point (possible

world). The formula:

(7.7) Mw

is assum ed as an axiom. It states that the product (ontological conjunction)

of all facts is a possible situation. Consequently, the situationw is a point

in

the logical space and

will

be called the real world.

Now, using (5.21), we infer from (7.5) the

following

theorem:

(7.8) Fp

=

wEp

Any situation is a (positive) fact if and only if it occurs in the rea l world.

In another words, the real world includes a situation if and only if it is a

(positive) fact.

From(7.8), (5.21), (5.22) and (6.12) we obtain the theorems:

(7.9) Fw

(7.10)

Np=wENp

(7.11) F(p q) = wEFp wEFq

(7.12) F(Vr (r)) ^ lp( (p) Fp)

(Remark, added November

1967.)

As observed by B. Wolniewicz and P. T.

Geach, Wittgenstein would not say that F where is a logical theorem.

This mean s, in my opinion, that Wittgenstein 's notion of a fact may be ex

pressed more adequately by the connective F defined as

follows:

Fp

Fp p 1. However, observe that the rea l world may be defined using F

and

F a s well. See (7.4). In many theorem s we may replace Fby F.

Also,

we have: FFp =Fp. The difference between F and F is not very deep.

Figuratively speaking, it is like that between nonnegative integers and posi

tive in teger s. The theorem (7.4) correspon ds to the equality: 0 + 1 + 2 =

1+ 2 .

8. States of

affairs

The notion of state of affairs or atomic situation is

rep resented in the system of ontology by the senten tial connective SA. The

formula SAp is to be read: the situation th at is a state of affairs. Wit

tgenstein uses the notion of (positive) atom ic fact (the connectiveFA) but he

does not use the notion of a negative atomic fact (the connectiveNA). We

introduce

both in the

following

definitions:

(8.1) FAp =

SAp*Fp

(8.2) NAp

=

SAp

Np

States of

affairs

ar e neither necessary nor impossible. We formulate

this as an axiom:

(8.3)

SAp*Mp

MNp

It

follows from th is (a formula of Wolniewicz):

(8.4) SAp {Fp=(p MNp))


13/27

ONTOLOGY IN THE

TRACTATUS

19

There

exists only one necessary fact

(LF1)

and only one impossible

fact (NMO). The atom ic facts (positive and negative) are contingent situa

tions.

We assume now the axiom:

(8.5)

yrSAr =

1

This axiom is equivalent to each of two following formulae:

(8.6)

rSA(Nr) =

0

L(Vr SAr)

It follows from (6.12) and (8.5) that in every possible world (including

the

real world) there occur at least one state of affairs, i.e .:

(8.7)

Pw * lp(SAp AtvEp)

Points exist (5.25). Th erefore, states of affairs exist too:

(8.8)

3p SAp

The existence of situations is a logical theorem:

ipSp.

The

difference between different possible worlds con sists in the occur

rence of different sta tes of affairs in them . We see from (8.6) that the

product

of all complements of sta tes of affairs equals the impossible situa

tion

0. One may consider also an arbit rary product of certain states of

affairs and of complements of all other ones. Every such product

will

be

called a combination of stat es of affairs. In pa rt icu lar , the product of all

sta te s of affairs is a combination of them. We assume that every positive

combination of stat es of affairs, i.e. every combination except that men

tioned in (8.6), is not equal zero . Moreover, we assum e that every positive

combination is a possible world (8.10). To make it precise , let us suppose

thatwe have a sentential formula

(r)

such that

lr(SAr

A

(r)).

The situa

tion

that

r(SAr

(r))

is the product of all states of affairs

r

such that

(r). On the other hand, the situat ion that

Ar(SA(Nr)

A

N (Nr)) is the

product of all com plements of stat es of affairs # such that

N (q).

To see

this it is enough to check the equality: Ar(SA(jSfr)

A

N (Nr)) = r( 3# (r =

Nq

SAq

N (q))).

Onemay easily

verify

the following equality (compare the dual equality

to

(6.13)):

Ar(SAr

A

(r))

A

r(SA(Nr)

A

N (Nr))

= Ar((SAr

A

(r))

v

(SA(Nr) N (Nr))). Thus, the product

r((SAr

A

(r))

v

(SA(Nr)

AN (Nr)))

is a combination of sta te s of affairs. We assum e the definition:

(8.9)

Dr( (r), (r)) = r(( (r)

A

(

r

))

v

( (Nr)

A

N (Nr)))

Here, a new opera tor 3 binding senten tial variables is introduced. It

follows from above that for any formula

(r)

the situation that

Dr(SAr

9

(r))

is a combination of stat es of affairs. It is a positive combination if

3r(Ar A

(r)).

As announced above, we assume the following scheme of

axioms (for any arbitrary formula

(r)):

(8.10)

MSAr

A

(r)) P(Dr(SAr,

(r)))


14/27

20 ROMAN SUSZKO

This scheme of formulae may be read: toevery positive combinationof

states of affairs corresponds in thesense of identity)a uniquely deter

mined

possible world.

We

assume that th is correspondence

is

biunique.

In

other words, twocombinationsof states ofaffairs which differ insomeof

their factorsaredifferent. It is expressedby theschemeof axioms:

11

(8.11) Dr(SAr,

(r))

=Dr(SAr

9

(r))

>

Vr(SAr

( (r) = (r)))

Suppose now that th ereexistsasituationp such that

SAp

and

SA(Np).

Then,

it

follows

from thetheorem dualto 6.1)thattheproductof allstatesof

affairs,

i.e.,Dr(SAr,Sr)is

equal

to the

impossible situat ion

0,

contrary

to

(8.10). Therefore:

(8.12)

SAp

N ( S A

(Np ) )

SA

( N p )

NSAp

Itfollows

from (6.6) that

(8.13)

tEDr(SAr, (r))

=

Vp(SAp

A

(p)*tEp)

A

vq(SA(Nq) N Qfq)

~>tEq)

Letusintroducenow thedefinition:

(8.14) sa(t)=Dr(SAr,tEr)

Clearly,thecombinationsa(t)is theproductof allstatesof

affairs

occur

ring intand of all complementsof those states of affairs whichdo not

occur

int. Iftis apossible world thenthesituationsa(t)may becalledthe

combinationofstatesofaffairs inthe possible worldt. Inviewof(8.7) this

combination

is apositive one. Therefore,ift is apoint thensa(t)isalsoa

point. It may be inferred from (8.13) that they areidentical. Thus,we

havethe theorem:

(8.15) Pt>sa(t)=t

Every possible world is a positive combinationof states of affairs.

Therefore, thebiunique correspondence in thesenseof identity) between

positive combinations

of

states

of

affairs

and possible worlds

is

concerned

not

only with all positive combinationsbutalso withallpossible worlds.

12

We have seen (6.14) that every situat ion

is the sum

(ontological alternation)

of cer tain possible worlds. This includes also theimpossible situation0

although

it

occurs

in no

possible world. On

the

other hand, every possible

world is a productofcertain statesof affairs and ofcomplementsofother

states

of

affairs.

Inviewof

this,

we say

that

the

states

of affairs are

generators of the Boolean algebra of all situations, i.e., the Boolean

algebra

of all

situations

is

generated

by the

states

of

affairs. Every situa

tionmay beobtained from thestatesofaffairs bymeansof thegeneralized

sum operation

V

generalized product operation

and

complement opera

tionN.

Letus nowdiscuss theproblemof thenumberofstatesofaffairs,of

possible worldsand ofsituations. It is clear that


15/27

ONTOLOGY

IN THE

TRACTATUS

21

(1) if there exist exactly n states of affairs then there exist exactly

2

n

1 positive combinations of them, i.e., there exist exactly 2

n

1 pos

sible worlds. On the other hand,

(2) if the (atomic and complete) Boolean algebra of all situations con

tains exactly m points (possible worlds) then there exist exactly 2

m

situa

t ions .

In the infinite case,nandmare infinite cardinal numbers and we have

only to replace

2

n

1 by

2

n

in (1) above. The number of states of affairs is

determined in some degree by the postulate of their independence. This is

th e

second component of the principle of logical atomism in

Tractatus;

compare (5.19). Notice that the formula (5.19) follows from our assumption

(8.16) below. It doesn't suffice to assume that any two different states of

affairs are independent: vpvq(SAp ASAqAp q>plq). To formulate the

independence

of states of affairs we must introduce the notions of indepen

denceof three, four, . . . , states of affairs. Let us write I

2

(p,q)instead of

plq.

Subsequently, in analogy to (5.16), we may define for each

k

= 3,

4, . . . , the &ary connective 4 such that the formula h(Pi,Pz,. . . ,

Pk )

means: the situations p

9

p

2

, . . . ,Pkare independent. For example, we

define the formula h(pi

9

th

9

Ps) using a formula which is the product of 8

factors:

M(Fp

1

Fp

Fp

3

)

M(Fp

A

Fp

2

A

Np

3

)

, . . . ,

M{Np

1

Np

2

Fp

3

) A

M (N p

A

Np

2

Np

3

)

Now,

we assume as axioms all formulae of the following kind:

(8.16)

(SAfo ,

. . . , A

SAp

k

Ap^p

2

A, . . . ,

Ap

{

p

]

A, . . . ,A

Pk iPk) > hipu . . .,Pk)

They mean that each& different states of affairs are independent;i


16/27

22 ROMAN SUSZKO

(1) Thestates of affairs are configurations of

objects.

It

is an important but

very

difficult point in the philosophy of Wittgenstein.

I cannot give he re an ultimate and complete explication of th esis (1). The

considerat ions in th is section m ust be seen as pre liminary rem arks only.

Firstly, the main difficulty con sists in the prec ise presen tat ion of the

prin ciples of the o ontology. The o ontology has its own serious p roblem s

and

will

be not presen ted here in a definite form . On the other hand, the

notionof configuration of objects and th esis (1)

give

rise to certain special

problems which cannot be solved here. I

will

propose a definition of the

notion of configuration of objects (9.0). The main idea of th is definition i s

quite right, in my opinion. It

will

be applied to form ulate th esis (1). How

ever,

it

will

be shown th at the form ula (9.1) which corresponds to (1) is not

adequate.

Let us reflect for a moment on the notion of configuration. Here, I

consider the con figurations of finite number of objects. When speaking

about configurations of cert ain objects

x

l9

. . . ,

x

n

we take usually into

account

many possible configurations of these objects. In other words,

there

may be many configurations of the

given

objects x

9

. . . ,x

n

. One

may infer from th is tha t the notion of configuration must contain a hidden

parameter

which changes from one configuration of

1?

. . . ,x

n

to another

one. To reveal th is pa ram ete r we may use the ph rase:

(2) the / configuration of objects x

l9

. . . , x

n

where the letter R is a nominal variable . To explain the ro le of the pa

rameter

R we suppose thatevery configuration of objects x

l9

. . . ,Xn con

sists in a cert ain n ary relation holding between the objects x

l9

. . . ,x

n

. In

part icu lar , it is quite nat ura l to use phrase (2) for the situation th at the

n ary relation R holds between x

l9

. . . ,x

n

. Therefore, we assume that the

ph rase (2) means:

(3) the situation tha t

w ary

relation Rholds for x

l9

. . . ,x

n

Instead of (3) one may use the senten tial expression :

(4) the ra ary relation Rholds for x

f

. . . , x

n

or the symbolic sentential formula:

(5) R*x

l9

. . . , x

n

Thus

we have the definition:

(9.0) the Rconfiguration of x

lf

. . . ,x

n

.=R*x

u

. . . , x

n

It allows us to dispense with phra ses of the form (2) and to form ulate

th esis (1) as follows:

(9.1) SAp s 3 3 ^ . . . lx

n

(P= R*x

l9

. . . , Xn)

The

form ula (9.1) should be assumed a s an ontological axiom. It might

read : a situationp is a state of

affairs

if and only if p consists in a cert ain


17/27

ONTOLOGY

IN THE

TRACTATUS

23

(given) relat ion holding between som e (given) objects. Note th at formula

(9.1) is not quite pr ec ise because first of all the number of existent ial

quan tifiers occurr ing in it is indefinite. Th ere ar e

several

subtle points

connected with the formula (9.1). Some of them

will

be mentioned below.

It follows

im mediately from (9.1) th at :

(9.2)

SA(R

* # , . . . ,

x

n

)

for

every

w ary relat ion

R

and objects

x

l9

. . . ,

x

n

.

(A) The ident ity on the right hand side of (9.1) must not be rep laced by

equivalence. Fo r, suppose th at:

(6) (p = R*x

l9

. . . ,x

n

) SAp

If

F(R * x

l9

. . . , x

n

)

then

R

*

x

l9

. . . ,

x

n

=

1. Consequently,

SA1

9

contrary

to

(8.3). On the other hand, one may see that the formula:

(7)

p = q

>

SAp = SAq

does not hold; compare the scheme (**) in the section 2.

(B) Let

R

be an w ary rela tion and

S

be an ra ary relat ion . Consider

arbitrary objects

x

l9

. . . ,

x

n

,

x

n

+

u

. . . ,x

n

+m.

i n+ m

then the states of

affairs:

(8)

R

*

x

l9

. . . ,

x

n

S

*

x

n+

,

. . . ,

x

n+m

ar e not only different, th eir st ru ct ur es are also different. Suppose now that

n = m.

One may say that the stat es of affairs:

(9)

R

*

x

lf

. . . ,

x

n

S

* x

n+l9

. . . , x

n+m

are of the same structure in a

weak

sen se. A stronger version of th is

notion may be introduced in the case of isom orph ic relat ions. Let

C

be a

one to one correspondence between the fields of the relations

R

and S. It

assigns

to the elements

x, y

of the

field

of

R

certain objects

C

*

x

and

C *y

in thefield of S.

14

If we also have:

(10)

x y > C

*

x C*y

(11)

R

* *i, . . . ,

x

n

5 * (C *

xj,

. . . , (C *

n

)

then

the correspondence C is called an isomorphism between

R

and S. We

say that the sta te s of

affairs

(9) are of the same stru cture in the strong

sense and with respect to the correspondence C if and only if C is an iso

morphism

between

R

and S and

x

n

+k = C

*

X&

for

k =

1, . . . ,

n.

(C )

It is clear t hat the state s of

affairs

(9) ar e ident ical if

R = S

and

x

k

=

x

n+ k

for

k =

1, . . .

,n.

Does the inverse implication hold? In other

words, is it true that:

(12) R*x

l9

. . . , x

n

= S * x

nVu

. . . ,

2n

R= S

Xl

=

n+1

, . . . ,

Xn = X2n


18/27

24 ROMAN SUSZKO

(D)Let Idbe the identity relat ion . This means that for all elemen ts x,

y

of some

class

of objects (which does not need to be specified):

(fd

*

x,y)

=

( # =

y). In

view

of (2.2), (4.8), (4.9) and (4.12) we infer that L(ld* x,x).

Consequently, according to (8.3) the situation that Id *x,x is not a sta te of

affairs which contradicts (9.2). On the other hand, suppose tha t the relation

S is the complement of the relation Rin the sense for all elements x, y of

some

class

of objects (which does not need to be specified): (S* x,y)=

N(R *x,y). We infer from (9.2) that SA(S*x

9

y), i.e., SA{N{R*

x,y)).

It

follows from (8.12) thatN(SA(R* x,y))

9

con trary to (9.2). Thus we see that

the formula (9.2) cannot hold in

full

generality. Therefore, the formula

(9.1) must also be suitably modified. The problem is : how to rest r ic t the

range of nominal variables (R

9

#i, . . . ,Xn)bound by the existen tial quan

tifier s in (9.1).

One

may con jecture that the formula (9.2) holds if the objects x

u

. . . ,

x

n

are individuals, i.e., non abstract objects

(Urelemente)

, and the relation

R

is a Wittgensteinean one. I think that Wittgensten believed in the exist

ence

of a certain absolute class of re lat ions between individuals which

generates the states of affairs accord ing to (9.2). Of co urse, onlyDer

Hebe

Gott

could define the class of Wittgensteinean re lat ion s. However, I think,

it is possible to formulate cer tain necessary conditions which must be

satisfied by Wittgensteinean re lat ions. Our considerat ions above (point D)

suggest

two

following

conditions:

(1) No Wittgensteinean relat ion is an invariant under all perm uta tions

of ind ividuals. In par t icular, the relat ion of identity is not a Wittgen

steinean one.

(2) The Wittgensteinean relat ion s ar e mutually independent. Thus, for

example, the complement of a Wittgensteinean relation is not a Wittgen

steinean relation.

Both conditions call for further explanation . Especially, the condition

of independence is somewhat complicated because the Wittgensteinean re la

tions

may be unary, binary, . . . , n ary, . . . . However, I do not enter into

these details here.

Remark,

suggested by D r. Wolniewicz, D ecember 1967.

Anyway

we

define states of

affairs

there exist infinitely many of objects. If ther e were

finitely many objects then there would

exist

only finitely many configura

tions of them, i.e ., finitely many states of affairs, contra ry to (8.17). Note,

however, that the existence of infinitely many objects is not equivalent to

the existence of an infinite

class

of objects.

10. Odontology

.

T radit ional opposition between realism and nominalism

belongs to the

field

of ontology of objects and is concern ed with the problem

of existence of abstrac t objects. We assume here ( ) the realist ic point of

view. N ominalism is followed by poor and mostly negative conclusions. In

par ticu lar, nom inalism (e.g., reism of T. Kotarbi ski) seem s to be unable


19/27

ONTOLOGY

IN THE TRACTATUS 25

to

reconstruct classical mathematics. On the other hand, it is known that

classical mathematics may be reconstructed within realistic ontology (e.g.,

set theory). Starting from the realist ic point of

view

we introduce the

notion

of an abstract object as a primitive one. We assume also certain

specific axioms concerning the existence of abstract objects. Realistic

formal ontology does not formulate any specific assumptions concerning

individuals.

There

are many contemporary versions of realist ic ontology (e.g., the

theory of types, the set theory of Zermelo Fraenkel or of von Neumann

Bernays G del, the systems NF and ML of Quine). Moreover, most of them

offer certain difficult special problems which seem to be solvable by

decision only. Under such circumstances I do not feel inclined to present

here any elaborated system of o ontology. When speaking about realistic

o ontology I do not intend either to choose a particular system known in

mathematical

logic or to construct a new one. Iwill restrict my considera

tions to the so called principle of abstraction (Cantor, Frege) which is the

kernel of every realistic ontology. The principle of abstraction formulated

below is concerned with the abstract objects derived from objects and

situations as

well.

Iwill discuss also a classification of abstract objects

an d

some problems connected with the principle of extensionality.

There

exists, of course, the question as to whether the ontology of

objects contained in the Tractatus is realist ic or nominalistic. Wittgenstein

gives

no direct answer to this question. However, I think that the Tractatus

must be interpreted in a realistic way. This

follows

from the interpretation

of configurations of objects

given

in the section 9. The revealed parameter

R

represen ts relations which, certainly, are abstract objects. There is

also an indirect h istorical argument. Clearly, Wittgenstein was influenced

by Russell's theory of types. Every strict formulation of the theory of

types (Church's simple theory of types, the so called ontology of St. Les*

niewski, Tar

ski's

general theory of

classes)

proves to be a restricted ver

sion of set theory and, thereby, a version of realist ic ontology of objects. In

former times the realism of the theory of types was not obvious because,

mostly, of certain shortcomings in PrncipaMathematica. The theory of

types is a many sorted theory. It uses essentially many sorts of nominal,

i.e., nonsentential variables. This peculiar syntatic feature of the theory of

types (formed sometimes in thestyle of the so called higher order system)

gave rise to certain misinterpretations of this theory in the past. Now,

only simpleminded people believe in the nominalistic character of the

theory of types.

Certainly, the realistic ontology we are speaking about is not the theory

of types. Our formalized language L

o

contains just one kind of nominal,

i.e. , non sentential variable. Therefore, the principle of abstraction for

mulated below states explicitly the existence of abstract objects.

11.

Principle ofabstraction

The formulae which are neither sentences nor

names contain some free variables. For simplicity, we consider only

formulae which contain as free exactly the nominal variables x

9

. . . ,x

m


20/27

26 ROMAN SUSZKO

and sentential variables p

lf

. . . ,p

n

We assum e that m

9

n = 0

9

1,2, . . .

with the exception of the case when m +n = 0. There formulae

will

be

called of rank (m

9

n). Senten tial or nom inal formulae of rank {m)

will

be

called formulae of rank (0;m,n)or (l;m,n), respectively.

One

might use the

followingsuggestive

notation:

X(*l, . . . ,

Xm, px,

. . . ,

pn)

for arbitrary formulae of rank (m,n). But I

will

not use th is notation.

When we use ( ) cer ta in formulae and of rank (0;m,n)and (l;m,n),

respect ively, then we think about indeterm inate object and situation

which depend on indeterminate objects x

u

. . . , x

m

and situationsp

h

. . . ,

p

n

. In other words, to any objects x

l9

. . . ,x

m

and situation sPi, . . . ,p

n

there

correspond an object and situation . This correspondence is

un ivocal. Thus, one may say that every meaningful form ula of rank(rn,n)

determines an univocal correspondence, i.e., assignm ent. Indeed, the for

mula

of rank (l;m,n)

assigns

to arbitrary objects x

l9

. . . ,x

m

and situa

tions piy . . . , p

n

the corresponding object . On the other hand, the

formula of rank (0;m,n)

assigns

to arbitrary objects x

x

, . . . ,x

m

and

situations pi, . . . , p

n

the corresponding situation .

Certainly,

the assignments determin ed by the formulae of some rank

are abst ract objects. One may assum e that every abstra ct object is an

assignment. This does not mean that every abstrac t object is determined

by some formula. The form ulae only refer in an indir ect way to some

abst ract objects. The existence of abst rac t objects determined by the

formulae of any rank is stated in the principle of abstraction.

(Remark,

added November 1967.) The existence of abstrac t objects i s

meant here in the sense of the existential quan tifier binding nominal var i

ables. This applies to the formulation of PA

0

and

?A

l9

below.

When speaking about assignments it may be useful to int roduce a suit

able notation . Let C be an assignment . The symbolic formula:

(*)

C

*

X

U

. . . ,

X

m

,

/> , . . .

,p

n

will

be used for the object or situat ion which is assigned by C to the objects

Xi, . . . ,x

m

and situation s p

ly

. . . ,p

n

. It may be rea d:

the

value of C at

x

l9

. . . ,

x

m

, p

ly

. . . ,

p

n

Thevalue of an assignment Cis an object or situat ion . Strict ly speak

ing there are exactly two cases:

Case 0; for every x

l9

. . . ,x

my

pi, . . . , p

n

the value (*) of C is a situa

tion.

Case 1; for every x

u

. . . , x

m

p

h

. . . , p

n

the value (*) of C is an

object.

The

assignment C is called of kind (0) or (1) correspondingly to the

case 0 or 1, above. Consequently, form ula (*) is either a sentential form ula

(case 0) or nominal formula (case 1). Th erefore, the aster isk * is a syn

tac tically ambiguous symbol. It is, in genera l, an operator (not binding any

variable) which forms a formula (sentential or nominal)


21/27

O N T O L O G Y IN THE

TRACTATUS

27

o * l, , m,


22/27

28 ROMAN SUSZKO

impossible. This reasoning

shows

that the principle PA

X

is incompatible

with the existence of the nominal formula metioned above.

I n

the case of sentential diagonal formula the reason ing is quite ana

logous. Observe, however, th at for every situation p:

Np P

I t

follows

according to the principle of abstraction PA

0

that there exists an

assignment A of kind (0) such that for all objects x: (A*x) = N(x* x).

Consequently, A * A =N(A * A) which is a contradiction (Russel s an

tinomy).

12. Classification of

abstract

objects We have divided abst ract objects

into

two kinds. We may introduce a more precise division into the realm of

abstrac t objects. It correspon ds exactly to the classification of formulae

according to th eir ranks. We say that the assignm ent determined by some

formula of rank (i;m,n) according to the prin ciple of abstract ion is of kind

(i;m,n) where i = 0, 1 and m, n =0, 1, 2 . . . (the case m = n =0 being ex

cluded). Consequently, the formulation of the principle of abstract ion

(PA

0

, PAJ given in the preceding section must be suitable pr ec ise. We

have to put the expression "kind (i;m,n) instead of the expression "kind

( i )

where i = 0,1. Strictly speaking, we have infinitely many notions of

abstract objects of some kind. They ar e unary predicates

Abs7'

w

.

The

notation

Absf

'

n

(x) is to be read:

x is an abstract object of kind (i;m,n)

O n

the other hand, we have the general notion of abst ract object givenby

t h e unary predica te Abs. If we

wish

to state that our classification of

abstract objects is exhaustive

then

we meet a serious difficulty. Indeed, we

have to assert the general statement of the form:

(1) V# (AbsU ) . . . )

where the th ree point blank represen ts an infinite alternation of all for

mulae of the form:

Absf

n

(x). Certainly, th is alternation is somewhat too

long for our purpose. This is the point where the problem of reduction of

abstrac t objects ar ises. It

will

be discussed lat er . The abstract objects

may be grouped into th ree broad kinds. A terminology stemming from

arithmetic may be applied to them. An abstrac t object is called (a)real,

(b)

imaginary

and (c)

complex

if and only if it is of the

following

kind,

respectively:

( a )

(e;ra,0) where m 0

(b) (i;0,n) where n 0

( c )

( i; m , n )

w h e r e

m^O^n

T h e real abstract objects are intensively studied in set theory. The

mathematical

set theory may be considered a s a realistic ontology of

objects but limited to real abstr act objects. Indeed, the abstr act objects of

kind (0;ra,0) a re usually called m ary rela tions (between objects); compare


23/27

ONTOLOGY IN THE

TRACTATUS

29

(4) and (5) in section 9. Especially, the abstract objects of kind (0;l,0) are

identical with

classes

or sets (of objects). In th is case the formula

A

*

x

may be read: object

x

is an element of the

class

A.

On the other hand,

abstract objects of kind (l;m,0) a re usually called functions or operation s

of

m

arguments (on objects); for the case

m =

1 compare (10) and (11) in

section 9.

The

imaginary abst ract objects of kind (0;0,n) may be called rc ary

relations between situations or

classes

of situations, in the special case

when

n

1. These abst rac t objects are taken into account by Lesniewski

inhis protothetics. However, th is theory is an oversimplification because it

states that there exist exactly two situations; compare section 3. Con

sequently, protothetics distinguishes exactly

2

2n

abstract objects of the kind

(0;0,n).

There are also complex abstract objects. For example, formulae

like

the one

(xBp)

mentioned at the end of section 2, determine abstract

objects of kind (0;l,l).

Now, I intend to sketch how to solve the problem of reduction of

abstract objects. Fi rst , let us consider rea l abstract objects. Thus, we

are within set theory. It is known that von

Neumann

(1927) has shown that

all real abst rac t objects may be reduced to those of kind (l; l,0) , i.e. ,

functions of one argument. In other words, von

Neumann

introduced the

predicate AbsJ'

0

as a primitive notion and

then,

defined all remaining

predicates Abs7'. The von

Neumann

system is not popular with mathema

ticians and philosophers, however. The customary system s of set theory

reduce real abstract objects to those of kind (0;l,0), i.e .

classes

or sets.

This reduction is carried over as

follows.

According to Peano's definition

(1911) functions of

m

arguments may be identified with

(m

+ l) ary rela

tions whichsatisfy the condition of univocality:

(2)

R

*

x

u

. . . ,

x

m

y R

*

lf

. . . ,

x

m

,z *y = z

Subsequently, a device of Wiener Kuratowski is to be used. We define

ordered pa irs and, in general, ordered & tuples

(k

^2) of objects. They are

defined as certain

classes

of objects, i.e ., objects of kind (0;l,0). Finally,

the

&ary

relations are identified with

classes

of ordered & tuples. Thus,

we may introduce the predicate AbsJ'

0

as a prim itive notion and,

then,

define all remaining predicates Absf

It

is easy to realize that the reduction procedure sketched above may

be applied to imaginary abstract objects, i.e., abstract objects of kind

(i;0

9

n)

where

n ^

0. The only difference consists in the definition of ordered

pairs and, in general, & tuples

(k

^ 2) of situation s. There are, of cou rse,

certain

abstract objects. It result s that the abstrac t objects of kind (0;0,n)

where

n

^ 2, i.e., relations between situations reduce to certain classes of

objects, i.e. , abst ract objects of kind (0;l,0). On the other hand, abstract

objects of kind (l;0,n ) where

n

^ 2 reduce to those of kind (l; l,0) and,

finally, to classes of abstract objects, i.e ., abstract objects of kind (0;l,0).

Our

reduction procedure does not affect abstrac t objects of kind (0;0,l) and

(l;0, l). All remaining imaginary abstrac t objects reduce to certain real

abstract objects.


24/27

30 ROMAN SUSZKO

The

reduction procedure which uses the notion of ordered tuples of

objects or of situations may be applied also to complex abstract objects.

Abstract objects of kind (0,m

t

n) where m ^ 1 and n ^ 2 reduce to binary

relations between objects and, finally, to

classes

of objects. On the other

hand,

abstract objects of kind (l;m,w) where

m

^ 1 and

n

^ 2 reduce to

functions of two arguments (binary operations on objects) and, finally, to

classes

of objects. The complex abstract objects of kind (0;l, l) or (1;1,1)

remain

unreduced. All other complex abstract objects reduce to them or to

certa in

real abstract objects.

The

reduction procedure sketched above

allows

one to draw the

follow

ing conclusion. It is possible to introduce the predicates:

AbsJ'

0

, Abs?\ Abs?'

1

, AbsJ'

1

,

Abs'

1

as primitive notions and, subsequently, to define all other predicates

Absf

>w

.

It

follows

that we can now formulate and, also, assume as an

axiom, the statement that our classification of abstract objects is exhaus

tive. It is enough to put the alternation:

(3) Abs'U) v Abs M v Abs M v

AbsJ'V)

v

Ab ^W

in

the three point blank in (1).

13

The

principle

ofextensonalty

The reduction of abstract objects may

seem to be guided by the philosophical idea known as Occam's razor. This

idea seems also to be the source of the principle of extensionality.

The

general problem of extensionality arises already in connection

with the principle of abstraction. The last one states, roughly speaking, the

existence of abstract objects determined by the formulae. It is not excluded

thereby that there may

exist

a formula which determines two different

assignments. Occam's razor

suggests

excluding this possibility and to

formulate a stronger version of the principle of abstraction according to

which every formula determines exactly one abstract object.

It

is not necessary to formulate here the stronger version of the

principle of abstraction. The reader

will

easily

verify

that it

follows

from

th e

usual principle (PA

0

, PAj and the principle of extensionaltiy PE for

mulated below. Itwillbe enough to mention here that the stronger version

of the principle of abstraction

serves

as a principle of defining abstract

objects. It is used in the reduction procedure to define at least the tuples

of objects or situations. In general, the following symbol:

{Xl

9

. . . , X

m

, pi, . . , Pn\x]

may be used to denote the only assignment determined in the stronger ver

sion of the principle of abstraction by the formula X of rank {m,n).

(Remark, added November 1967.) Given a situation

q,

we define:

{q}

=

{pi\Pi =#}. Clearly, {q} is the unit class such that the situation qis the

only element of it. Thus, to every situation

q

there corresponds an abstract

object {q}. This correspondence is biunique (one to one). Therefore,

instead of studying situations and their propert ies we may investigate the


25/27


corresponding abstract objects.

Also,

note that the states of

affairs

(com

pare (5) and (9.2) in section 9) may be put into one to one correspondence

with ^ tuples of objects, k

=

n + 1.

The principle of extens tonality PE. Let A and B be any abstract

objects of the same kind, say (i;m,n). If the equality:

(1) A * X

l9

. . . X

m

, / > i , . . . p

n

= B *X

l9

. . . ,X

m9

p

l

. . . , p

n

holds for all , . . . ,

x

m

, pi, . . . ,Pn

then

A =B.

The principle PE

allows

one to strengthen the principle of abstraction.

Th erefore, one may say that the principle PE follows in a sense from

Occam 's principle. On the other hand, PEserves as a necessary and suf

ficient condition (criterion) for identifying and distinguishing abst ract

objects of the same kind. To see th is it suffices to observe that the impli

cation contained in PE may be reversed. It is c lear that the principle of

extensionality PE may be decomposed into two principles PE

0

and PE

X

corresponding to two cases: i 0 or i = 1. The reader may formulate

partial principles PE

0

and PE i, and observe that the symbol of identity in

(1) is either a connective (PE

0

) or a predicate (PEi).

The case when i =0 gives r ise to certain problems concerning ex

tensionality. The symbol of identity serves to identify either objects or

situa tions. However, besides the connective of logical identity of situations

there

is in our language

o

th e connective of mate rial equivalence. It cor

respon ds to the fundamental dyadic division of all situations into positive

and

negative facts (ultrafilter and dual ult raid eal in the Boolean algebra of

situa tions). This correspondence is expressed in the

following

theorem

(compare definition (2.10)):

(13.1) (p^q) = (Fp Fq)D(Np Nq)

Let us replace in PA

0

and PE

0

the connective of logical identity of situ

ations by the symbol of material equivalence. The principles PA

0

and PE

0

are transform ed in th is way into the

weak

principle of abstraction

WPA

0

and strong principle of extensionality SPE

0

. Principles of abstraction and

of extensionality occur in the usual set theory in the form of WPA

0

and

SPE

0

, respect ively. Note that th ere is in the domain of objects nothing

which would correspond to the distinction between positive and negative

facts and to mater ial equivalence. Therefore, there is no analogous pos

sibility to strengthen PE i and weaken PAx.

We may sta te, because of (2.7) th at :

(1) W P A Qfollows from PA

0

and

(2) PEofollows from SPE

0

.

Moreover, PA

0

and SPE

0

reduce toWPA

0

and PE

0

, respectively, in the case

of ontological two valuedness (section 2 and 3).

In

my opinion, the principle PA

0

does not offer any ser ious difficulty

besides the antinomies. Therefore, the prin ciples PA

0

and PA

X

as well,


26/27

32 ROMAN SUSZKO

limited in certain ways, may be assum ed as axiom sch em es. As to the

pr in cip les of extension ality I would

like

to assume the principles SPE

0

and

P E i

as axioms of ontology. If one assum es PE

0

instead of SPE

o

then one

c a n n o t

exclude the possibility that th ere

exist

two different coextensional

objects A, B of some kind (0;m,n), i.e., A / B and for all x

l9

. . . ,x

m

,

Pl

9

9 Pn A * X

l9

. . . , Xm, pi, . . , p

n

= B * X, . . . , X

m

,p, . . . , p

n

.

The abstract objects A, B would be intensional entities.

Our

pre feren ce for SPE

0

against PE

0

rest s in Occa m's principle and

acco rds with usual math ematica l thinking. The prin ciple SPE

0

excludes the

existen ce of int ensional abst ract objects. Th ere may be only one argum ent

against SPE

0

. We will be hindered in assuming SPE

0

only if somebody will

prove tha t it cont rad ict s the postulate formulated at the end of section 3.

I n d e e d ,

th is postu late is the hear t of Wittgenstein 's ontology. The form ula

(*) is som et imes called t he P rin cip le of Extensionality of Situat ions, PES.

Clearly, SPE

0

follows

from PES. I hope, PES does not

follow

from SPE

0

.

N O T E S

1. The original version in Polish "Ontologia w Traktacie L. Wittgensteina," has

been published inStadiaFilozoflczne 1(52), 1968, 97 120, Warsaw.

2. Rzeczy i F kty (Things and Facts). Pa stwowe Wydawnictwo Naukowe (Polish

State Publishers in Science), 1968,

Warsaw,

Poland.

3. There is an opinion that mereology, a formal theory built by St. Lesniewski, is a

suitable basis for the theory of spatiotemporal relations.

4. The predicates 'ph ilosopher", 'logician " are called sometimes nominal predi

cates or, simply names. However, they are not names in our sense.

5. If Q is a fe ary predicate, / is a fe ary functor and

1 }

. . . , ^ are nominal

formulae then the expression:

"(

l f

. . . , e

k

)

is sentential formula and the expression:

/ ( i , . ,

k

)

is nominal formula.

6. For a concise axiom system see my "Non Fregeanlogic and th eories," sub

mittedt TheJournalofSymbolic

Logic.

7. It is clear that the quantifiers play an essent ial role only when there exist

infinitely many situations.

8. If we assume (4.1), (4.2), (4.3.1), (4.3.2) then we have 14 modalities like Lewis'

system S4. The additional assumption of (4.4) reduces the number of modalities

to 10; compare the system S4.2 of M. A. E. Dummett and E. J. Lemmon (and of

P .

T. Geach).

9. We assume also (in o ontology) Barcan formulae with nominal variables:

LVx {x) = VxL {x), M3x (x) = xM (x).

10. Usually, in the theory of Boolean algebras the points ar e called atoms.


27/27

O N T O L O G Y IN THETRACTATUS 33

11 . The

converse

of

(8.11)

is a

logical theorem formulated with defined terms.

It

follows

from definitions (8.9) and(6.4).

12. Onemight considerthe

reflexive,

antisymmetric andtransitive relationof "en

tailment" between points representedby theconnective Wand defined as follows:

pWq means thatp, qarepointsandevery stateofaffairs occurringinpoccurs

also inq. Every point entails inthis sense thepoint:

Or(SAr,Sr)

Remember, however, that every twodifferent possible worlds

t

q

exclude them

selves, i.e. theproductof themis impossible

(p

q

=0). It mayalsobe shown

t hat there

exists

a situation (state ofaffairs) which occurs inoneofthe points

p, q and doesnotoccurinthe other.

13. R.Wojcicki (University of Wroclaw) observed in a letter to thewriter thatthe

axiom system for thestates ofaffairs reduces to theformulas (8.5), (8.10), (8.16).

All th e remaining theorems (8.3, 8.4, 8.6, 8.7, 8.8, 8.11, 8.12, 8.13, 8.14, 8.15,

8.17)

are

provable.

14. Thereis a systematic ambiguity in our use ofthe asterisk *. This pointwillbe

explained later.

The Polish Academy of Sciences

Warszawa, Poland

and

Stevens

Institute of T echnology

Hob ok en,

New Jersey

Documents

Ontology in the Tractatus