SOME REMARKS CONCE RNING STATEMENTS,
TRUTH-VALUES, A N D CATEGORIES O F PREDICATES
JORGEN JORGENSEN
In many textbooks of logic statements are spoken of i n such ways
that i t seems reasonable t o define: a statement is any t hing t hat c an
significantly be said t o be t rue o r false. I n t his paper I t ak e i t f or
granted that this def init ion (whet her expressly f ormulat ed or not) i mplies the f ollowing presuppositions: a) statements may be linguis t ic ally f ormulat ed sentences, b) statements are s omet hing t hat is eit her
true or false, and c) one must k now the meaning of the words 't rue'
and 'false' in order to k now what a statement is. To these presuppositions I would make a f ew comments whic h are of some importance, I
believe.
ad a ) St at ement s n e e d n o t b e linguis t ic ally f ormulat ed. T h e
linguistic f ormulat ion of a statement is inessential f rom a logical point
of v iew. One and t he same statement may be expressed i n dif f erent
languages and even i n dif f erent sentences wi t h i n the same language.
Besides, one may often be in doubt as t o h o w t o f ormulat e a certain
statement in the most appropriate way, even when k nowing very well
what statement one wishes to formulate. This indicates that the linguistic f ormulat ion is inessential to a statement, and that the essential thing
is t he meaning connected wi t h t he possible f ormulat ions o r expressions, i.e. the apprehension or conception (often called the proposition
or t hought ), wh i c h t he apprehending o r judging pers on has i n his
mind. O n e may , f o r instance, recognize a t hi ng o r a c olour o r a
melody wit hout f ormulat ing any designation of t hem or any sentence
describing them. On the ot her hand, many sentences, f or instance interrogative, imperat iv e and exclamatory sentences, are not expressions
of statements. Thus, t o state s omet hing and t o use a language seem
to be t wo processes t hat are i n princ iple independent of each other,
although t hey are v ery of t en connected i n organis ms t hat hav e a n
extensive t raining i n us ing a language.
But what then is a statement ?
To answer this question seems to be very dif f ic ult , and I s hall t herefore restrict myself to a very short description of what I consider the
most essential characteristics of a statement. First, I t hink a statement
is t he res ult o f a psychological process t ak ing plac e i n t he person
mak ing the statement and nowhere else. Next, this process is directed
towards an object in the wide sense of this word i n whic h it may be
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said that a concept is a concept of something, a fear fear of something,
hate hate of something etc. Furt her, t he process makes one o r more
of the objects features appear, or appear more distinctly t han before.
The object towards whic h the process is directed is called the subject
of the statement, and the features emphasized are called the predicate(s) o f t he statement. The statement does not contain the subject as a
part, but it 'has' t he subject, i.e. i t is directed towards t he subject. I n
so f ar t he process is a process o f analysis: a giv en content o f c onsciousness is articulated i n such a way t hat certain features of it are
emphasized as features or parts of the giv en totality. Whic h features
are emphasized depend part ly on the apprehending person and part ly
on the object apprehended. The judging indiv idual cannot arbit rarily
decide what features are t o appear, i.e. t o be f ound by the analysis.
Alt hough t he indiv idual may wis h t hat t he object s hould posses certain features he w i l l be unable t o create such features b y analysis,
since he can thereby only ascertain the existence of features t hat the
object actually possesses. I t may therefore, perhaps, be best to define:
to fudge (or to mak e a statement) is t o predicate, i.e. by means of a
more or less t horoughgoing analysis t o ascertain t hat a giv en objec t
contains, o r has c ert ain features, e.g. a c ert ain qualit y , a c ert ain
magnitude, a c ert ain pos it ion i n space a n d t ime, c ert ain relat ions
between its parts, etc. etc.
If one wants to communicate one's statement to other persons, then
one has to give expression to it by naming whic h object the statement
concerns and what is asserted about t hat object. Of t en the most convenient way to do this is to use the usual linguis t ic expressions designating the object and its ascertained features. Since these designations
are in princ iple different f rom the object and its features, t hey should
never be c alled 'subject' a n d 'predicate' b u t alway s 'subject-designation' a n d 'predicate-designation'. I n t h e s t at ement ' t h e eart h is
round' t he subject is the eart h it s elf and the predicate a propert y of
the earth, wh i l e t he wo r d 'eart h' is subject-designation and ' round'
predicate-designation. This is, of course, obvious, but as we s hall see
later on, i t is a t hing whic h i t is import ant t o remark and bear i n
mind.
ad b). I t is generally agreed t hat statements ma y be t rue o r false.
Nevertheless it seems incorrect t o def ine statements as any t hing t hat
may be t rue o r false. I ndeed, a general def init ion o f a statement
ought t o state t he characteristics t hat are t o be f ound i n any statement. But t rut h c annot be f ound i n f als e statements, a n d falseness
not i n t rue statements: i t i s me r e l y t h e alt ernat iv e characteristic
'true or false' t hat can be f ound i n any statement. Just because these
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alternatives a r e dif f erent ( a n d moreov er mu t u a l l y exclusive) t h e y
merely giv e a n 'extensional' def init ion o f statements, namely as an
aggregate of t wo dif f erent classes, each of whic h is t o be def ined i n
its special way . Thus t rut h and falseness are in-essential characteristics o f statements (ev en i f t hey p l a y a n import ant part i n logic ).
Therefore I f ind i t better t o define: a statement is an ascertainment
of s omet hing (t he subject) c ont aining s uc h a n d such features (t he
predicates). Suc h statements ma y t hen b e eit her t rue o r false. B u t
what does t hat mean ? Under whic h conditions is a statement true,
and under whic h is i t false ?
ad c). Us ually a statement is said t o be true, i f and only i f it agrees wi t h realit y . Wh a t t hen is realit y ? Th e wo r d 'realit y ' has, as
is well k nown, various senses, t he most import ant of whic h seem t o
be t he f ollowing: I n everyday-life i t signifies t he us ual realit y , and
it is t rue t o state t hat grass is green, t he heaven blue, c hurc h bells
noisy, roses flagrant, sugar sweet, etc. But in science the word 'realit y '
signifies t he objective (physical) realit y , and i t is t rue t o state t hat
things are composed o f colourless atoms, t hat light ray s a r e elec tromagnetic waves, t hat sounds are a i r waves, etc. I n psychology,
however, t h e wo r d ' realit y ' s ignif ies t h e ps y c hologic al o r phenomenological reality (i.e. what is immediat ely given), and it is t rue t o
state that the f ull moon is larger on t he horiz on t han i n zenith, t hat
a thing's qualit y of being wa r m or cold varies wi t h my temperature,
that my images are colourless, etc. Further, i n the wor l d of literature,
art, o r imaginat ion, t he ' imagined' fantasies o r day-dreams are t he
'real' work s of art o r imaginat ion, and it is t rue t o say t hat Hamlet
k illed Polonius, t hat Ophelia we n t mad, t hat Pegasus is a winged
horse, etc. A n d f inally , i n t he s phere o f mat hemat ic al objects t he
word 'realit y ' signifies t he mat hemat ic al concepts, and i t is t rue t o
say t hat i n an Euc lidean t riangle t he s um of t he angles equals t wo
right angles, t hat i n a square the diagonal is inc ommens urable wi t h
the s ide, t h a t c ont inous f unc t ions t h a t c annot b e dif f erent iat ed
exist, etc.
If des iring to giv e the word 't rue' an unambiguous sense one thus
cannot define t he t rut h of a statement as its c onf ormit y or agreement
wit h realit y . But , i f des iring t o f o l l o w c ommon usage, one c annot
deny t hat a l l t he abov e examples o f statements are t rue, ev ery one
of t hem expressing a correct or true apprehension of the object about
whic h i t predicates s omet hing, a n d it s negat ion expressing a n i n correct or false apprehension of the same object. The question t hen
arises whet her it is possible t o f ind a def init ion o f 't rut h' t hat c omprises a l l t he v arious examples o f t rue statements.
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This question can, i n my opinion, be answered i n t he af f irmat iv e.
Common t o all t he statements ment ioned is t he fact, t hat the object
about wh i c h t hey state s omet hing ac t ually c ont ains t h e predic at e
whic h they ascribe t o it , i.e. t hat t hat predicate can be f ound by an
analysis of the object: grass is something that is green, physical things
are composed of atoms, t he phenomenon o f t he f u l l moon is larger
on t he horiz on t han i n z enit h, t he imaginary a n i ma l Pegasus has
wings, a n d t he s um of angles i n an Euc lidean t riangle equals t wo
right angles, etc. I n one word: i n every t rue statement the predicate
is contained i n t he subject o f t he statement, — n o mat t er whet her
the subject is an everyday t hing, a phy s ic al object, an immediat ely
given content of an experience, an imaginary object, o r a conceptual
object. And, conversely, a statement cannot be true, if the object does
not contain the features s ignif ied by the designation of the predicate.
If t he subject of a statement does not contain the features predicated
of it, t hen the statement is false. I f so, the result of a t horoughgoing
analysis wi l l s how t he falseness o f t he statement. Theref ore, t rut h,
respectively falseness, may be def ined t hus :
A statement is t rue, i f and only i f t he predicate of t he statement
is contained in the subject of the statement: and a statement is false,
if and only i f t he predicate of t he statement is not contained i n t he
subject of t he statement.
If c alling statements whos e predicates are contained i n t heir subjects 'analy t ic statements' o n e ma y assert t hat a n y analy t ic statement is a t rue statement, and t hat only analy t ic statements are t rue
statements. I n t his connection i t mus t be emphasized t hat wh a t we
are h e r e c onc erned w i t h i s t h e s ubjec t a n d t h e predic at e o f
the statement, a n d neit her w i t h t h e des ignat ion o f t h e s ubjec t
or t h e predic at e, n o r w i t h t h e j u d g i n g indiv idual' s c onc ept o f
the subject or the predicate. The fact t hat a predicate of a statement
is contained in the subject of the same does not imply that the concept
of t h e predic at e i s als o c ont ained i n t h e c onc ept o f t he subject.
Whet her t his is t he case or not depends o n h o w ample and correct
a concept of the subject the indiv idual has formed. Usually, howev er,
the properties, relat ions o r ot her features o f t he subjects o f statements are not f ound by an analysis of the concepts of these subjects,
but by an analysis of the subjects themselves, i.e. t he objects about
whic h s omet hing is predicated. Where, as, f o r instance, i n mat hematics, s omet hing is stated about objects t hat only 'exist' b y v irt ue
of a definition, the subjects of the statements are concepts, the properties of whic h are t o be f ound by a conceptual (logic al) analysis of
the def init ion ( n o mat t er whet her t his def init ion i s f ormulat ed ex 128
plic it ly in a def init ional equation, implic it ly i n a postulate-system, o r
merely understood).
According t o Albert us Magnus t he abov e-ment ioned def init ion o f
the t rut h o f statements was f irs t proposed b y t he Arabian philos ophers Avicenna, Alf arabi, and Alhaz en ab. 1000 A. D. and was lat er
taken up by Hobbes and Leibniz. As these thinkers did not, however,
distinguish c learly bet ween t he subjects of statements and t he c oncepts o f these subjects, t h e statements we r e conceived as relat ions
between the concept of the subject and t he concept of the predicate,
and t he analysis was conceived as a purely conceptual one. I n c onsequence, it lay near at hand f or Kant lat er to int roduc e the pres umably unt enable dis t inc t ion bet ween wh a t h e c alled analy t ic a n d
synthetic statements t hat play s o import ant part i n his wh o l e p h i losophy. I n my sense of t he wo r d t here are no t rue synthetic statements. A l l t rue statements are analytic, but the analysis f rom wh i c h
they s pring varies i n character as t o t he sphere of objects t o wh i c h
the subjects o f t he statement belongs. I f t he subject belongs t o t he
sphere of everyday objets, t he analysis wi l l be a us ual observation;
if it belongs t o the sphere of physical objects, t he analysis wi l l be a
more o r less c omplic at ed inv es t igat ion b y means o f v arious a p paratuses; if it belongs t o t he phenomenologic al sphere, t he analysis
wi l l consist i n an awareness of what is immediat ely giv en; i f it belongs to t he sphere of imaginat ion, the analysis consists i n observing
the characteristics o f t he imaginary beings o r occurrences ( b y t he
way, one ma y here eas ily f ind ex amples of non-mat hemat ic al, u n decidable statements, as f or instance, ' Ophelia wore a pigt ail'); and
finally, i f i t belongs t o t he conceptual sphere, t he analysis w i l l be
of a logic al character and often consists in a proof or disproof based
on the giv en or presupposed def init ion.
The f iv e spheres of objects here ment ioned overlap, of course, and
it may be v ery dif f ic ult t o f ind precise c rit eria of each of them. But
this is not essential f o r my present argument , n o more t han is t he
question as t o whet her they may be reduced t o a s maller number or
not. Wh a t matters is merely, t hat the decision of t he question as t o
whether a giv en statement is t rue o r not depends on some k ind o f
analysis of the subject of the statement, no mat t er how many spheres
of objects are concerned.
The nex t question that arises is: Ho w many kinds of statements can
be dis tinguished ? This seems a v ery dif f ic ult ques t ion t o ans wer,
and I s hall therefore restrict myself t o ment ion a f ew examples t hat
are o f import anc e i n t his connection. Firs t , i t ma y be convenient,
however, t o dis t inguis h bet ween statements o n t h e one h a n d a n d
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designatory conventions, that are not statements, on the other. Such
conventions are, o f course, necessary presuppositions f o r linguis t ic
formulations of the statements, and, unfortunately, thcse conventions
are themselves f ormulat ed i n sentences t hat are ident ic al wi t h sentences expressing statements, e.g. 'this is a tree', or 'this is Mr. N. N.'.
Being conventions such sentences are, howev er, not expressions o f
ascertainments and can neit her be t rue n o r false. They may alway s
be conceived as ans wers t o t he question: ' Wh a t is t his c alled ?' o r
'What is the name of t his ?', and t hey s hould alway s be f ormulat ed
by means of the words 'called' o r 'named': 'This is c alled a tree', or
'N. N. i s t he name o f t his person' ( i f conceived as statements c oncerning t he usage of the language t hey are statements, but t hey are
not statements c onc erning t h e des ignat ed objects, wh i c h r e ma i n
unchanged independently of any arbit rary change of the designation).
Returning n o w t o the real statements a mos t import ant dis t inc t ion
seems to be the one between primary statements and secondary statements. Statements the subject of which is not statements, but some
other kind of objects, are called primary statements. And statements
the subjects o f which is one o r more primary statements are called
secondary statements. Consequently, t he statement 't his is red' i s a
primary statement, a n d ' t he s t at ement ' t his i s r e d ' i s f als e' i s a
secondary statement. I f necessary, t h i s classification m a y b e c ontinued by t he int roduc t ion af tertiary, quaternary, etc. statements.
Of o t h e r pos s ible f undament al f undament a div is ionis m a y b e
mentioned: a) the object-sphere to whic h the subject of the statement
belongs; b ) t he number of possible predicate categories of t he subjects (about whic h s omet hing wi l l be s aid f urt her on), and c ) t he
extension of the subject.
These dis t inc t ions ma y b e crossed s o t hat a rat her c omplic at ed
system of different kinds of statements results. Here, however, I shall
confine my s elf t o s ome remark s concerning t he f irs t -ment ioned dis tinction, occasionally adding a lit t le about the ot her ones.
The s imples t a n d mos t f undament al o f t h e p r i ma r y statements
are pres umably those, t he subject of whic h is a present object being
analysed i n s uc h a wa y t hat eit her dif f erent features, o r dif f erent
parts o f i t and relat ions bet ween s uc h part s are ascertained. Statements i n wh i c h t he features are predicated o f t he subject ma y be
called at t ribut iv e statements, whi l e statements i n whic h relations between parts of the subject are predicated of the same may be c alled
relational statements. Us ual l y t h e linguis t ic expression o f a n a t tributive statement designates t he s ubjec t b y a s ingle designation,
while the several parts of the subject of a relat ional statement mus t
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have separate designations i n the linguis t ic expression of such a statement. Bot h k inds o f statements have, howev er, o n l y one subject,
i.e. t he predicate(s) i s (are) as c ribed t o one a n d o n l y one s ubjec t
irrespective of this hav ing parts or not.
These t wo k inds o f p r i ma r y statements ma y als o res ult f r o m a
process of recognition, i n whic h a present object is ascertained t o be
the same as an earlier experienced object, o r t o be s imilar t o such
object. Such statements o f recognition are necessary presuppositions
for the introduction of designations of objects at all. Without recognit ion of objects one cannot introduce and apply designations of them.
But lat er on, wh e n t he applic at ion o f t he object-designations has
been learnt , these designations of t en substitute t he subject i n l i n guistic f ormulat ions o f statements, a n d t hes e t h e n appear t o b e
purely linguis t ic af f airs . T h e c onnec t ion w i t h t h e non-linguis t ic
reality t h a t alone giv es t h e sentences a me a n i n g i s int errupt ed,
and t he designation of t he subject or the concept of the subject may
then eas ily b e mis t ak en f o r t h e r e a l subject, wh i c h mis t ak e has
caused serious errors i n logic al theory. Th e int roduc t ion o f subjectdesignations is, o f course, a n enormous practical advantage, because
by means of t hem one c an t alk about objects t hat are not present,
— t heir names represent t hem. But as t he names do not generally
resemble t he designated objects, one is, as a rule, f orc ed t o f ound
one's predications o n one's memories o r o n one's concepts o f t he
objects concerned; a n d as one's memories o r concepts a r e us ually rat her vague o r meagre (and moreov er of t en inc orrec t ) o n e is ,
of course, easily led t o rat her unimport ant o r mis leading statements.
The name or t he concept easily comes t o f unc t ion as t he subject of
the statement ins t ead o f merely repres ent ing it . Un d e r s uc h c onditions t he argument , a t wors t , dev elops i n t o a p u r e l y v erbal af f air or, at best, int o a purely conceptual construction (as is t he case
in a l l f ormal systems). Conversely, t he subject-designation ma y be
omitted f r o m t he linguis t ic f ormulat ion o f t he statement i n cases
where t he subject is present t o t he conversing persons, o r it may be
referred t o b y a pronoun. Suc h statement-expressions wi t h o u t a
subject-designation are i n German c alled 'Subjektlose Urt eile' (s ubject-less statements), b u t t his designation is mis leading, since, as a
matter o f course, ev ery statement has a subject — whet her des ignated o r not. A predic at ion c annot be made wit hout a subject. The
sentence ' i t rains ' is merely a linguis t ic s hort ening of anot her sentence t hat ma y be f ormulat ed s omewhat as f ollows : ' my out -door
env ironment is i n a rainy state'. O n t he ot her hand a pluralit y o f
predicates o f earlier f ormulat ed statements ma y be condensed i n a
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single subject-designation o f a n e w statement, such as f o r instance,
'the o l d ma n t hat liv ed i n t his hous e has died', wh i c h statement
presupposes t he t rut h o f t he statements 't he ma n was old', and 'the
man liv ed i n t his house'. A n d here i t i s immat erial whet her t he
man belonged t o t he sphere of everyday t hings o r t o t he sphere of
imagination.
Returning t o t h e classification o f statements i t i s import ant t o
remark t h a t p r i ma r y statements (as w e l l as statements o f higher
order) ma y be af f irmed o r negated. Af f irmat ion o f a primary state-
ment is called a positive (o r affirmative) statement, and negation
of a primary statement is c alled a negative statement. Positive and
negative statements are o f t he order nex t above t he statements af f irmed o r negated. Linguis t ic ally a negat ing statement may be s hortened b y means o f t he wo r d 'not ' as, f or instance, 't his rose is not
red'. This shortened f orm of t he linguis t ic expression has led t o the
untenable conception t h a t negat iv e statements as c ribe a 'negat iv e'
predicate t o t he subject. But 'negative predicates' seem unt hink able,
and t hus all statements ascribe a positive predicate t o some subject.
If, howev er, t he subject does not hav e t he ascribed predicate, t hen
the statement is false, and t he negat ion o f i t is t hen t rue.
A pos it iv e (secondary) statement is t rue, i f and o n l y i f t he predicate ' t rue' c an b e f o u n d b y analy s is o f it s subject, wh i c h i s a
primary statement. Th e pos it iv e statement ' t he statement 't his ros e
is red' is t rue' is it s elf true, i f and only i f t he statement 't his rose
is red' is true, and this last-mentioned statement is true, i f and only
if t he present ros e ac t ually has t he qualit y red. Th e t rut h o f t he
secondary statement is t hus ascertained b y a n analysis o f t he c orresponding primary statement, a n d t he t rut h o f t his p r i ma r y statement by an analysis of its subject. O n t he ot her hand, t he propert y
'true' can, o f course, nev er be f ound b y analy s ing merely t he l i n guistic expressions o f statements, s inc e these linguis t ic expressions
have b u t t he propert y ' t rue' i n relat ion t o t h e i r subjects, t hat c an
never oc c ur as a part o f t he linguis t ic expressions (unless, exceptionally, t he statement is a statement about a linguis t ic expression,
such as, ' t he wo r d 'statement' contains nine letters').
Analogously, a negative (secondary) statement is true, i f and only
if the predicate 'false' c an be f ound by an analysis of the corresponding primary statement. I f so, the subject of the negative (secondary)
statement (i. e. t h e p r i ma r y statement) h a s t h e propert y falseness,
and i t is consequently t rue t o ascribe t his propert y t o it . Th i s i s
done i n t he negative statement, and therefore t his statement is true.
True positive o r negat iv e statements presuppose t rue, resp. false,
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primary statements. A n d t he f ac t t hat t he t rut h o r falseness of primary statements depends o n t he properties of t heir subjects, wh i c h
properties are independent of the person mak ing the statements, presumably s how that t rut h and falseness cannot be int roduc ed by con•
egously, wh e r e f als e pos it iv e o r negat iv e statements a r e concerned.
n In s o f a r as t he t r u t h o r falseness o f secondary statements detpends o n t he t rut h o r falseness o f t he c orres ponding pr i mar y s t aitements, t he f irs t ment ioned may be said t o be truth-functions of the
olatter. Th u s t rut h-f unc t ions a r e a s pec ial k i n d o f secondary statenments. No t a l l secondary statements are truth-functions. Th e state,ment ' t he statement p is complicated' is , f o r example, a secondary
bstatement, but i t is not a t rut h-f unc t ion. I n passing i t may als o be
uremarked, t hat not ev ery statement c ont aining anot her statement as
tpart of it, is a truth-function. For example, t he statement ' A believes
p'
a is not a truth-function, since it is not even a secondary statement.
rIts subject is not a statement, but the belief of A whic h is a psychoegical f ac t t hat may c ont ain t he statement p, but t hat is not it s elf a
statement.
b
a The relat ion bet ween p r i ma r y statements o n t he one hand, a n d
positive,
resp. negative, statements o n t he ot her ma y be expressed
s
in a so-called t rut h-t able:
e
d If and only if the primary statement p is true, resp. false, t hen the
positive
s t at ement ' p i s t rue' i s t rue, resp. false, a n d t he negat iv e
o
statement
' p is false' is false, resp. t rue.
n
In
t
his
wa
y we get a t rut h-t able f o r af f irmat ion corresponding t o
t
the
us
ual
t
rut
h-t able f o r negation, a n d i t is s hown t hat t he t rut hh
value
o
f
t
he
t
rut h-f unc t ion ev ent ually depends o n t he 'content' ( o r
e
'intension')
o
f
t he pr i mar y statement, v iz , o n t he properties o f t he
a
subject,
whic
h
properties
are designated by the predicate-designations.
s
If one does n o t t ak e t his 'c ont ent ' i n account, t hen o n e does n o t
c
k now t he sense of the words 't rue' and 'false' at all, and t he whole
e
procedure change int o a meaningless play.
r
t Af f irmat ion and denial need not concern merely single statements,
as
a was t he case i n t he examples giv en above. Such statements may
be
i c ombined eit her c onjunc t iv ely o r disjunctively, and by af f irming
or
n deny ing these combinations one gets new kinds of truth-functions,
customarily c alled 'conjunctions', resp. rdisjunctions'.
m
As regards t he ot her usual t rut h-f unc t ions t he mat t er seems t o be
e
more
complicated. Howev er, space do n o t a l l o w me t o g o f urt her
n
into
t
his
question here and I, therefore, wi l l conclude by mak ing a
t
f ew remark s c onc erning what I c all categories of predicates.
o
f
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f
a
c
t
s
.
A
n
d
Consider a s imple everyday thing, e.g. a table, and let the analysis
of same res ult in the f inding of the f ollowing properties: rectangular,
four-legged, b r o wn , glossy, hard, wooden, c ombus t ible, large, et c .
All t he properties f ound by analysis of t he v arious tables may n o w
be classified i n dif f erent groups whic h I s hall c all categories of predicates, because eac h propert y corresponds t o t he predic at e i n one
of t he statements t hat ma y be made c onc erning one o r anot her of
the tables. The princ iple according t o whic h the classification is c arried Out is , t hat predicates t hat are mu t u a l l y exclusive belong t o
one category. A category of predicates thus comprises t he predicates
that ex c lude one anot her i n statements c onc erning t he s ame s ubject. N o subject has mo r e t han one predic at e f r o m a n y category.
Experience s hows wh i c h predicates are inc ompat ible.
In t h e abov e-ment ioned statements c onc erning t he tables, predicates f r om t he f ollowing categories are j. alia contained:
the category o f f orm: rectangular, round, t riangular;
the category of c olour: brown, whit e, grey ;
the category o f leg-number: four-legged, three-legged, one-legged;
the category o f ref lec t ion: glossy, d u l l ;
the category of mat erial: wood, marble, stone.
If we d o not c onf ine ourselves t o t he analysis o f tables, b u t i n clude a l l possible objects, we s hall f ind a lot of qualit ies belonging
to v arious categories o f wh i c h t here i s a n immens e number: t h e
category o f size, o f weight , o f durat ion, o f s t abilit y , o f sound, o f
taste, of various k inds of value, etc. etc.
Every subject (objec t ) ma y hav e predicates belonging t o v arious
categories, but t here hardly exists a n object hav ing predicates f r o m
all categories. Quadratic numbers, f or example, cannot be red or hot,
tables not hard-hearted, at oms not green, etc. Predicates belonging
to a category the members of whic h cannot be ascribed t o a c ert ain
subject ma y be s aid t o b e f oreign t o t his subject. A n d statements
ascribing s uc h f oreign predicates t o a s ubjec t ma y b e s aid t o b e
meaningless, i n contradistinction t o false, false statements as c ribing
a predicate to a subject to whic h it is not foreign, but merely actually
not belonging. O f a quit e concrete subject i t is alway s meaningless
to predicate properties t hat i t ac t ually does not possess (a concrete
swan c annot be bot h wh i t e a n d blac k allov er). B u t o f a mo r e o r
less abstract subject i t is alway s meaningf ul t o predic at e t he v arious properties belonging to a category of predicates of whic h a member c an be as c ribed t o t he c orres ponding concrete s ubjec t (swans,
generally, may be eit her whit e o r black).
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Of statements c onc erning abstract subjects t he f ollowing seem t o
be true: predicates belonging to one and the same category may be
dis juncted, but not con juncted, i.e. of the same subject it wi l l always
be t rue t o assert t hat i t has eit her t he f irs t , o r t he second, o r t he
third, etc. predicate of one o f its possible categories, ev ery category
comprising also t he negation of the ot her predicates wit hin the category, but this negation not being identical wit h any predicate outside
the category. For instance, 'shapeless' is neither identical wit h 'colourless', n o r wi t h ' immat erial' , n o r w i t h 'thoughtless', n o r wi t h a n y
other predicate belonging t o another category t han t he one t o whic h
the predicate 'shape' belongs. Ev ery category f orms so t o say a universe o f predicates, and i t is such a predicate-universe t hat is s plit
up wh e n represented as t he logic al s u m o f a t e r m i n i t a n d t he
negation of t hat t erm (t his negat ion being the dis junc t ion of all t he
other t erms i n t he universe). Thus t he universe o f truth-values, f or
instance, i s exhausted b y t he predicates ' t rue ' a n d 'false', because
'false' i s t he negat ion o f 't rue' and v ic e versa. Ev ery statement t he
predicate of whic h is the disjunc tion of all possible predicates wit hin
a c ertain category is a tautology i n Wittgenstein's sense. Such statement mus t alway s be t rue, because one of t he predicates mus t belong to the subject. I f this was not the case, all the predicates would
be f oreign t o t he subject, and t he statement consequently meaningless. So is t he nat ure o f our conceptual univ ers e and o f our c orresponding significations o f predicates. Th e f ac t t hat t he predications
must b e res t ric t ed t o a def init e c at egory , Wit t gens t ein, howev er,
seems not t o hav e observed. But i t c an hardly be denied t hat i t is
highly art if ic ial t o predicate of a giv en subject a dis junc t ion of predicates f r o m dif f erent categories, as e.g. ' t he t able is eit her rectangular, o r brown, or four-legged, etc.'. Indeed, these predicates do not
exclude one another, and i n c ommon usage t hey are not disjuncted.
Corresponding t o the above-mentioned statement concerning t he disjunc t ion of predicates it may thus be stated: predicates belonging t o
different (b u t possible, i.e. n o t subject-foreign) categories ma y b e
con juncted, but not disitincted. Wh i l e i t is c ont rary t o usage t o say
'the t able is eit her rectangular, o r brown, o r four-legged, etc.', i t is
quite i n order t o say 't he t able is rectangular, and brown, and f ourlegged, etc.'. This last-mentioned statement would, howev er, be c ontradictory, i f t he negat ion o f a predicate was not restricted t o t he
corresponding category of predicates, i.e. i f 'four-legged', f or instance,
belonged t o t h e negat ion o f 'rec t angular'. Wh e t h e r t h e las t -mentioned conjunctive statement is t rue or false mus t be decided by an
analysis o f t he s ubjec t concerned. Statements, howev er, i n wh i c h
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a conjunction o f predicates belonging to the same category is predicated of a subject, are always false and thus have a certain affinity to contradictions in Wittgenstein's sense.
Actually, the characterization here given of tautologies and contradictions seem to show that the present concepts of same are generalizations o f the corresponding Wittgensteinian concepts. According
to Wittgenstein tautologies and contradictions are respectively disjunctions and conjunctions o f statements, and what matters are the
truth-values o f these statements, wh ile here we have t o d o wit h
disjunctions and conjunctions o f predicates generally o f which the
truth-values are special instances. And, o f course, statements ma y
have predicates belonging to other categories than those of truth-values; they may, for instance, be more or less complicated, be primary
or secondary, etc., be attributive o r relational, etc. But in the statement 'any statement must be either true o r false', merely a category
of truth-value is considered. As this category contains merely the two
predicates 'true' and 'false', it may properly be said to be divalent,
and its two members to be contradictory. Other categories of predicates may be trivalent, tetravalent, o r generally polyvalent, and i n
all these cases the members of the categories are said to be contrary.
Contrary predicates exclude one another, but merely taken a ll to gether they exhaust the category: A n inference fro m the falsity o f
a statement to the truth of its negation therefore becomes more and
more indefinite, the more comprehensive the category o f the statement's predicate is. l i t h e category is divalent it is possible to infer
from the absence of one of its members to the presence of the other
member, e.g. from the absence of the predicate 'false' to the presence
of the predicate 'true', and vice versa (tedium non datur). But if the
category is trivalent, then the absence o f one o f its members w i l l
merely allow an inference to the presence o f the one o r the other
of the two remaining members. And in case the number o f predicates in the category increases, the indefiniteness also increases.
As t o polyvalent propositional logics, th e y are, i n m y opinion,
either purely fo rma l (uninterpreted) games, o r dealing wit h predicates not belonging to the category true-false. I f a th ird tru th value, e.g. 'indefinite', is introduced, the two other values should presumably not be named 'true' o r 'false', b u t rather 'definitely true'
and 'definitely false', and the subject o f these predicates w i l l then
be the judging person's knowledge o f the truth-values o f the statements, not these statements themselves. Similarly, in case a propositional logical formalism with infinitely many truth-values is interpreted as a logic o f probability, the extreme members o f the cate136
gory o f probabilit y are not 't rue' and 'false', but 't rue wi t h t he probabilit y l ' , a n d ' t rue w i t h t h e probabilit y 0' , a n d bet ween t hes e
two ex t reme v alues a l l t h e o t h e r probabilit y -v alues a r e situated.
That the statement p has the probabilit y mi n ( m - n ) means t hat the
predicate o f p may be ascribed t o n subjects, but t hat merely m of
these subjects hav e t his predicate, i. e. t hat p is t rue merely i n m
of t he n possible cases. O r more exactly: wh e n I ascribe t he probability-predicate m / n t o a statement p, t hen I a m merely predicating o f p t hat i t belongs t o a group of n statements o f whic h m
and not more t han m are true. I f m = n , t hen p has t he probabilit y
1, a n d i t is possible t o inf er t hat p is t rue, because p mus t t hen
belong t o t he group o f t rue statements. B u t nevertheless t he c oncept 't rue' a n d t he concept ' t rue wi t h t he probabilit y 1' are quit e
different, a n d t he las t -ment ioned o f these concepts c an merely be
defined b y me a n s o f t h e f irs t -ment ioned. A n d analogous ly a s
regards t he concepts 'false' and 'false wi t h t he probabilit y 0'.
Finally, I may add t hat t he v arious categories o f predicates mentioned i n t his paper are but a s mall selection o f a vast number of
such categories wh i c h hav e mu t u a l l y numerous a n d c omplic at ed
relations. A s f ar as I k now, t his vast f ield has u n t i l n o w been but
lit t le investigated. Pres umably t he linguis t ic semanticists hav e done
more i n t his direc t ion t han t he logicians. A n d possibly such inv es tigations c an t h r o w s ome light on t he Hegelian-Marx is t 'dialectics',
whic h are ac t ually a c ont inuat ion o f s ome o f Aristotle's ideas t hat
have been wi d e l y neglected b y lat er logicians. I f my modes t s uggestions i n t his paper c ould s t imulat e t he interes t i n t he problems
here t ouc hed on, t hey wo u l d not hav e been writ t en i n v ain.
University o f Copenhagen (Kobenhav n)
P i r g e n j d E GE N S E N
Professor o f Philos ophy
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