ontology in the tractatus
TRANSCRIPT
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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
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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
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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:
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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):
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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:
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ONTOLOGY IN THE TRACTATUS 17
(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
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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))
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ONTOLOGY IN THE
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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)))
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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
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ONTOLOGY
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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
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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
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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
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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
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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
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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)
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O N T O L O G Y IN THE
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27
o * l, , m,
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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
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ONTOLOGY IN THE
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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.
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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
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ONTOLOGY IN THE TRACTATUS 31
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,
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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.
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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