MA NY -V A L UE D L OGIC A N D FUTURE CONTINGENCIES
Kenneth R. SEESKIN
Those wh o t h in k that the existence o f future contingencies
entails a denial o f the p rin cip le o f bivalence mu st g ive some
account o f h o w a proposition wh ic h a t a p a rt icu la r t ime i s
neither tru e n o r false can, a t that time, be meaningful and be
combined w i t h o th e r propositions i n a syste m o f deductive
inference. This paper attempts to examine some of the alternatives t o o rd in a ry, two -va lu e d lo g ic th a t a rise i n th is connection, and to explore the logical and metaphysical consequences
that result from them.
The f i rs t s u c h a lte rn a tive i s t h e th re e -va lu e d syst e m o f
Lukasiewicz (hereafter L
I
value
a s w e l l a s t h e reasons f o r accepting i t a re g ive n b y
) . T h e h imse lf i n t h e f o llo win g passage: (l)
Lukasiewicz
d e f i n i t i o n
o
I can
f assume with o u t contradiction th a t m y presence in
t
Wa
h rsa w at
e a certain moment of next year, e.g. at noon on
21
t
h De cei mb e r,
r i s a t p re se n t t i m e d e te rmin e d n e it h e r
positively n o r negatively. He n ce i t is possible, b u t n o t
d
necessary,
t
r
u that I shall be present in Wa rsa w at the given
time.
O
n
th is assumption th e proposition ' I sh a ll b e i n
t
h
Wa rsa w at noon on 21 December of next year', can at the
present time be neither true n o r false. Fo r if it we re tru e
now, m y fu tu re presence in Wa rsa w wo u ld h a ve t o b e
necessary, wh ich is contradictory to the assumption. I f it
were false now, on the other hand, my future presence in
Warsaw wo u ld h a ve t o b e impossible, w h i c h i s a lso
contradictory to the assumption. Therefore the proposition
considered i s a t th e mo me n t n e ith e r t ru e n o r false and
must possess a th ird value, different fro m '0' o r fa lsity and
(') J a n LUICASIENVICZ, "P hilos ophic al Remark s o n Ma n y -V a lu e d Sy s tems
of Propos itional Logic ," in Polis h Logic 1920-1939, ed. Storrs Mc Call (Ox ford:
Clarendon Press, 1967), p. 53.
760
K
E
N
N
E
T
H
R. SEESKIN
'1' o r truth. This value we can designate b y '
sents
'the possible', and jo in s 'the tru e a n d 'the false' as
1
a
/ 2third
' . Ivalue.
t
r e p r e This passage has t wo hidden b u t cru cia l assumptions. First, i t
assumes t h a t " I sh a ll b e present i n Wa rsa w a t n o o n o n 21
December o f next year" is a meaningful proposition now. Fe w
would question this assumption, but it is wo rth noting because
it is d ire ct ly related to a second and mo re questionable one:
a proposition mu st have a tru th value to be meaningful. Th is
assumption w i l l be considered la te r; f o r n o w, i t is su fficie n t
to re a lize th a t i t is cru cia l t o L u ka sie wicz' argument. H e i s
claiming t h a t because f u t u re co n t in g e n t p ro p o sitio n s a r e
meaningful b u t n e ith e r t ru e n o r false, t h e y " mu st possess a
third value."
L
usei Lukasiewicz' own notation here.)
i
s
1 /2
0
d
e
1
1
1
0
0
f
/
2
i
1/
1
1
1/
1/
2
2
n
2
e
1
1
1
1
d
a
(Column
o n th e e xtre me le f t g ive s antecedent, to p mo st r o w
c
gives
consequent.)
c
The
o cases deserving special attention are those in wh ich one
of rthe terms takes the t h ird value. I f the antecedent takes 1
/ 2 d the consequent 1 o r the antecedent 0 and the consequent
and
1 i
indeterminate
co n stitu e n t ca n n o t a ffe ct t h e t ru t h v a lu e o f
/ n
the
wh
o
le
implication.
Yet, i f the antecedent takes 1 a n d the
2 g
consequent
0, th e re su ltin g imp lica tio n takes 1
, t
/t 2o s i n
become
e ith
c eer tru ei o r tfalse. Soc f a raso good.
n
Bu t consider th e
case
h t wh e re b o t h antecedent a n d consequent t a k e 1
/e2h. T h i s
r e
e f
s o
u l
l l
t o
i w
n i
MA NY -V A L UE D LO G I C A N D FUTURE CO NTI NG ENCI ES
7 6 1
proposition must take 1 f o r the simp le reason that if it did not,
L
no
I fo rmu la (n o t even P P ) co u ld e ve r ta ke 1 i n a ll cases.
Unfortunately,
assigning su ch propositions 1 i n e v e ry ca se
w
avoids
one
se
t
o f d ifficu ltie s o n ly t o create another. W e a re
o
left
u wit h a situation in wh ich any two contingent propositions
imp
ly each other. A t th is point, i t is imp o rta n t to understand
l
the
d deeper problem of wh ich this odd result is symptomatic.
h The problem arises fro m the apparently harmless fact that if
it
a is not ye t determined whether an event w i l l occur, it is n o t
yet
v determined wh e th e r i t w i l l n o t o ccu r either. Th is fa ct is
reflected
i n L u ka sie wicz m a t r i x f o r negation: t h e negation
e
of
a
n
indeterminate
proposition is a lso indeterminate. B u t i f
n
an
indeterminate
p
ro
p o sitio n a n d i t s n e g a tio n a lwa ys t a k e
o
the
same
value,
t
h
e
y
w i l l b e lo g ica lly interchangeable. L e t
t
us
th
e
n
reconsider
t
h
e
case o f a n imp lica t io n wh o se t e rms
a
both
take
1
u
/t 2 .
would
say that such imp lica tio n s are true. We wo u ld not, f o r
C
example,
l
e
a
rexpect
l
anyone to wa it to see whether
o
yl ,
to (1)
h I f ethere w i l l be a sea-fight to mo rro w, then there w i l l be
rg e a sea-fight tomorrow
a
i
returns o u t to be true. Bu t neither wo u ld we expect anyone t o
e
say
s that
s.
o
I (2) I f there w i l l be a sea-fight to mo rro w, th e n i t is n o t the
m
case that there will be a sea-fight to mo rro w
f
e
s
iuis tonbe counted as true me re ly because the occurrence o f the
scsea-fight
t
is n o w contingent. Th e p ro b le m is th a t Lukasiewicz
a
had
to
n
decide
wh e th e r to ma ke a ll such imp lica tio n s t ru e o r
h
to ma ke none o f them true, wh ile o u r in tu itio n s t e ll
cpwhether
e
srus that some ought to be true and some not.
io Similar p ro b le ms a rise i n re g a rd t o d isju n ctio n a n d co n junction. Th e tables f o r both operations a re g ive n b e lo w:
n
p
w
o
h
s
ii
ct
h
i
w
o
e
n
s
t
o
762
KENNETH R. SEESKIN
V
I
1
1
1 /2
0
1
1
1
/
21
1
/
2
1
/
2
0
A
1
1
1
1
1
/
2
0
1
/
2
0
1
/
2
0
1
/
1
2
/
2
1
0
0
0
/
2
0
0
In keeping with the belief that the value of a disjunction should
be th e ma ximu m va lu e o f it s d isju n cts wh ile th a t o f a co n junction should b e t h e min imu m va lu e o f it s conjuncts, L
Iassigns 1
/take
2 tth
o i s va lu e . T h e d if f ic u lt y i s t h a t b o t h P V —P a n d
d (PA
i s —
j P) f a il t o become theses o f th e system. Th is is mo st
unfortunate
sin ce A risto tle , w h o m L u ka sie wicz cit e s a s h is
u n c t
i o n as u t h o rit y i n re je ct in g bivalence, mo s t ce rt a in ly b e major
lieved
in excluded middle. (
a
2
temptation
to sa y that the va lu e o f a d isju n ctio n made u p o f
n
)d F indeterminate
two
o l l o w i n disjuncts
g
is also indeterminate except when
A
one
s denial
t o to f lthe
e other
,
— — — its va lu e then being 1.
c rois ithe
Similarly,
wee co urld sa y th a t wh e n b o th conjuncts a re in d e tn j h
terminate,
the resulting conjunction takes 1
e
u n
is
denial
/ic2 the
e
x
c
es po f t the other,
w h i n ewh ich
n case i t takes O. Yet, t h is
t
o
avenue
n
of
escape
e
is
closed
so
long
as we keep to the p rin cip le
a
i o
of
truth
functionality.
n s
w Looking a t L
tautologies
a re tautologies o f cla ssica l lo g ic. O n t h e o t h e r
h I a s
hand,
th
e
converse
is h a rd ly th e case, and a f a ir n u mb e r o f
a
e
w
h
o
l
standard
argument
patterns
(in clu d in g re d u ctio ad absurdum)
n
fail
to
hold
in
L
e
,
b
Io i
t
.t (3)
t hh PeDP
r )DP
sO (—
eh (4)
au DmPl ) D —P
ox (P
pt (5)
e Ds (QA — C))) D —P
dl (P
ae (6)
r A (P DQ))DQ.
b (P
er e :
l
m (c
e
s 2
)a
r
At
R
h
I
Sa
Tt
O
a
Tl
Ll
E
MA NY -V A LUE D LO G I C A N D FUTURE CO NTI NG ENCI ES
7 6 3
C.I. L e wis points o u t that classical tautologies n o t va lid in L
Ican often b e made th e consequents o f imp lica tio n s t h a t a re
valid in L
ieasily
b y seen since i f a classical ta u to lo g y takes 1
P
p/ 2does,
r w
e hfthen
e nan, imp lica
s tio
a n ythat, has P or —P as the antecedent
and
t
h
a
t
f
o
rmu
la
a
s
t
h
e consequent w i l l b e a n imp lica t io n
a c i n
whose
terms both take 1
g
/t 2 . hH
posit
a lin
e kr between
e
L
formula
te a
h ne dva
r lide in classical lo g ic but n o t in L
Icim
consequent
m lica
a t iodn bey ma kin g P o r — P t h e
lc aas sn s iobfc aaev la lid imp
antecedent
i
f
the
fo
rmu
la
is
n o t satisfied i n L
tla
h
e
g
i
c
w o
Ib
1
w
h
e
n
P
t
a k e s
si
t
ry o
n
/sg
a
y
i
t
2
th ( eg n
n
d
e
,tn
7
c h y
e
B
There
is t also a strong tendency t o l i n k th e t wo systems b y
ti )
a
u
saying
o
a
t — that nthose formulas that are va lid in classical lo g ic but
tyh ( in L
not
IJe (
L
validate
Ic.r (a n it o r assign it 1
/ship
. e between
I f
t hthei tswo systems wo u ld be greatly simplified; and
C
cn
P2 —
ca
it
m that both P rio r (
v.o must
Pel a bei noted
w
5
maintained
e
r
t
h aet i t i s t ru e , a lth o u g h re ce n t ly Re sch e r h a s
rC
n
P
r
acknowledged
)tn
s c h e( r
.— )a rn d u R heeis mistake.
1
see
is
th
a
t
th
e
ir
cla
im is preceisely wh a t wa s a t issue in the
,P
C
I
MD
a
o
)thesis
W
h
a
t
tkc. P Mc
a
th K in s e y re fu te d . I f P rio r' s a n d Re sch e r's o rig in a l
b
claim
o
we
t re htrue — — — i.e. i f classical tautologies n o t va lid
e
o
n
K
a
)
(
m
in
L e ei l nma
trc0
t e
i) D
P
o
r
P
when
P
takes
1
ih
a
d
o
n
tin —
T
/cfu
2
w
o
u
l
d
a
i
l
e
sh (
n
( a
h
v e
d
o
e
L
a
i 3( York : Dov er Publications, 1959), pp. 220-222.
New
tu
o
l
o
lIys ()—
yd
i
e
l
dLogic, V (1940), p. 149.
Logic
,"
Jour.
Sy
mbolic
4
f CP
ilb
(
a
)
n
tyu
.
o
s 5JD ed. P. EDWARDS ( Ne w Y o rk : Mac millan, 1967), V , 3.
sophy,
ia
l
i
c
ftu I()Pm p
.
a
i
o
n
ka
n
o
) t
Publis
5
A
C
L hing Co., 1968), pp. 82-83.
e
d ().ED
ln
p.
o
N
7
sla 27.
C
P
W
n
iyn )..I)
R
N
M
S)
lf1
eP
.E
a
yy/x iC
S
R
tK•
n
1
c E
2
a
C
iId(
/c.e S
H
o
N
C4
E
2
n
S
lIp C
.)
R
,E
n
H
t U
n
a
,".
E
o
i Y
so
T
R
L
tsto ,L
764
K
E
N
N
E
T
H
R. SEESKIN
that i s v a lid i n L
Itrue
s iwh
n ecne P, a n d Q b o t h t a ke 1
McKinsey
t
s an
i nexception
c e
t o th is ru le , there mu st also
a/ 2 . s B u showed,
be
t ahn exception
r
et o th e thesis o f P rio r and Rescher. O r, p u t
w
e e
another
hi
as way,
v ,the formula:
s
ea
s (8) Pe D
e
n ,- ,
is
P a, classical tautology wh ich takes 0 when P is V2. Two other
exceptions
are:
D(((
,
Qi ,(9) — (— (PA —P)D(PA —P))
P D P () ( P V —P)D — (PV—P)).
s (10)
D P )
a D
l
The
I have belabored th is p o in t is th a t i f Prio r's a n d
w ,- reason
a
Rescher's
y ( ( s o rig in a l c la im we re tru e , i t wo u ld b e possible t o
P
argue
th a t L
I D
classical
lo g ic simp ly b y treating both I a n d 1
P
)
values.
/c2 o
D au sl d d e s i g n a t e d
P
b Another
e)
point wo rt h y of attention is that L
)
sible
mI m
ai n ageneral
k e s t o ini f e rt th a t II— A DB i f All— B; t h is appears
implausible
d i em p since
o s i n- cla ssica l lo g ic w e ca n in f e r e it h e r one
from
t h e o th e r. B u t co n sid e r t h e f o rmu la P A (PDQ). F r o m
t
this
o f o rmu la w e c a n d e d u ce Q b y u s in g in fe re n ce s t h a t
Lukasiewicz
wo u ld accept. Bu t as we have seen, (P A (PDQ))
i
n
cDC) lis n o t va lid in L
O.
s ndi ce
n cmoere , w e h a ve co me across a n unfortunate re su lt
ui O
since
ei t i t forces Lukasiewicz e ith e r t o abandon o r to se rio u sly
modify
at a kt h ee sesma n tic a n a lo g u e o f t h e cla ssica l d e d u ctio n
theorem.
O n th e o th e r hand, L u ka sie wicz ca n s t ill ma in ta in
l1
that
for
all
A and B, if If—A DB, then A 11—B.
2
l/
h
e
tw L
possible
that
P" and taking M as p rimitive , we get the f o llo wn
I
h
ing
table,
where
P
a
e
ti l a IP —s —MP; SP = — M —1
u1 s t 3
Q P
o/ o l ,2 a n d
M
P
A
a
c
n
o
g
id o e
sQ n
oi t
fs a
i
n
s
m
o
MA NY -V A L UE D LO G I C A N D FUTURE CO NTI NG ENCI ES
P
MP
IP
SP
QP
1
1
0
1
0
1
0
0
1
1
/
2
7 6 5
1
From t h is t a b le i t ca n b e p ro ve d t h a t S W D Q)D (SP D SQ),
SP DP, S P D SSP, a n d P DS MP a re a l l theses o f t h e syste m.
As P rio r notes, th e fa ct th a t the mo d a l functions n e ve r ta ke
1
/hold MP V —MP as a thesis. (
2
8
that
p o ssib ilit y ca n b e defined i n t e rms o f imp lica t io n a n d
)i F u r t ahs ef orllo
negation,
mws
o :r M
e P, = — P P . T h i s a p p e a rs co u n t e rs tu itive
in
lo g ic P is ma t e ria lly equivalent t o
T
a
rsince
s ink classical
i
ch—PDP, ab u t in L
s
the
as MP.
sot same
h trutho value w
— P
theDstandpoint o f the problem o f future contingencies,
n, From
sitP i s sig n ifica n t t h a t i n L
from
iIi wsthe
e tru th
c oaf Pnand th e imp o ssib ility o f P f ro m it s falsefsi na Nf ol west h is
hood.
r does n o t mean th a t i f P is tru e , i t mu st b e
lo
g
ica
lly
true.
tte
h
e Th e necessity that Lukasiewicz has i n min d is
to
f c e
eni bee understood
salong
s the lin e s o f the definiteness we assonciate
if t w yi t h p a st a n d p re se n t events. W h i l e i t i s n o w n e in the
tcessary
oP
f sense o f being unalterable that Caesar crossed
the
Rubicon,
t h a t h e d i d i s h a rd ly a lo g ica l t ru t h n o r, a s
i
w
P
far
a
s
w
e
kn
o w, wa s h e co mp e lle d t o d o so . He n ce i t i s
is
today whether there w i l l be a sea-fight to mo rro w;
f
a
tcontingent
but
lh th e
s d a y a fte r t o mo rro w i t w i l l b e necessary e ith e r th a t
wa s a sea-fight th e d a y before o r necessary th a t there
e
othere
was
not.
Prior, however, wa rn s that the fact that we can in fe r
ua
the
necessity
o f P f ro m th e t ru t h o f P cannot b e fo rma lize d
rn
as
P
D
SR
(9)
Wh
e n P ta ke s 1
id
w i l l
n/t 2 , S P
b
e
th
(
l
s
e
u 8 ap. 249.
uf1962),
a
n
d
is )
h
u
s
tth A
ia .
os N
.
n P
s R
a I
n O
d R
766
K
E
N
N
E
T
H
R. SEESKIN
P D SP w i l l also take 1
/ 2 a Pn DdI P w i l l ta ke 1
wise,
correct
/n2 ow thfo rma
e nliza tio n—s a re P D (PDSP) a n d —P D(—P DI P ).
b Weemu st n o w a sk wh e th e r a n yth in g can b e done t o L
P
eliminate
a I ot o e its smo re
d
. objectionable features. The f irst suggestion
that
comes
t
o
min
l
C
oa n w
s e dq is the introduction o f a fo u rth tru th value.
By
h
a
vin
g
f
o n e co u ld h o ld th a t a proposition is
o e nf t o ulr values,
u
y
contingent
t
h h b u t eth a t i t and it s denial ta ke d iffe re n t values. I t
should
s
e
y be noted
s
t that Lukasiewicz h imse lf rejected L
of
wh ich validates a ll the tautologies o f
Ie a
i nfour-valued
f
a
v
osystem
r
m
.
classical
lo
g
ic.
(
L
i
I
L e
kafter
but
' ) is,
T hI think,
e
a reasonably close approximation to it. (")
-2
)f First,
o u fr o -r m o rd e re d p a irs o f t h e t ra d it io n a l va lu e s t h u s:
iv(1,sa1),l (1,
u 0e), d(0, 1 ), a n d (0, 0 ). Ne x t define t ru t h values f o r
implication
s
n
y
s and negation in the fo llo win g manner:
t te
o
m
t (11) (P,
h Q)D(R, S) ( P D Q , RDS).
a (12) —t (P, Q) = (—P, — Q).
h
f
e
o
l
o
l accept the fo llo win g abbreviations: (1, 1) = 1, (1, 0)
Finally,
o (0, 1)w = 3, and (0, 0) — O. This gives us the fo llo win g ma trix,
n
2,
which
s
e
i s t h e d ire ct p o we r M
2t( trix
ma
o ffor
h implication
t h e
and negation:
cHere
e l a" 1 s"r asn di "c0 " aa gla in
h
, stand f o r t ru t h a n d fa lsity, a n d " 2 "
tand
e w" 3 " o-stand
a
- f o r alternative species o f indeterminacy. Fro m
vthe
t
ma
a trix
l itu is e
cleard that the negation of one kind of indeterminacy
L
is not that same kind but the other; hence one "objectionu
able"
aspect of L
when
Ik h athe
s antecedent and the consequent o f an imp lica tio n both
a e e n
b
s l i m i n
e
i (t e d .
a
e 9(") J an uLUKASIEWICZ, A ris t ot le's Sy llogis t ic , 2 n d ed. (1951; rp t . O x f o rd :
B
Clarendon
)
Press, 1957), pp. 154-180.
tw I(") Fo r t he s ak e o f c larit y and s implic it y , I hav e modif ied Luk as iewic z •
iown
n
o
b system
b y n o t int roduc ing modal func tions c apable o f t a k in g v alues
tc i than
other
i
1 co r 0 and b y a llo win g " 2 " and " 3 " t o des ignate indet erminat e
values.
zd
e
a.
w
,
cp
h
t p
a
tu .
a2
h
a
l 4
p
p
8
l
e
n
sy 2
p
MA NY -V A L UE D LO G I C A N D FUTURE CO NTI NG ENCI ES
D
1
2
3
0
--,
1
1
2
3
0
0
2
I
1
3
3
3
3
1
2
I
2
2
0
1
1
1
1
1
7 6 7
take the same indeterminate value. The resulting imp lica tio n is
true — — — precisely that aspect of L
I defense
In
w e
ho f a
the dfo u rth value, though, one co u ld argue th a t
h ore suplt ise not
this
d quite as distressing in L
t least
2
at
a so cases
i t lik ew(2) aabove
s have been eliminated. A co n tin gent
ia
n
vproposition
o
i L and
d its
. denial n o w mu st have different indeterminate
ma kin
lik e (2) e ith e r 2-indeterminate
I
s values
i
n
c g cases
e
or 3-indeterminate b u t n e ve r true. On ce more, h o we ve r, o u r
intuitions t e ll u s th a t i f both th e antecedent and consequent
are indeterminate in the same wa y, th e re su ltin g imp lica tio n
ought t o b e t ru e i f i t is a classical ta u to lo g y lik e P D P a n d
indeterminate otherwise. I n general, g ive n a syste m o f lo g ic
with n truth values where n is finite, a n y assignment of values
to n + 1 contingent propositions w i l l h a ve t o g iv e a t le a st
two the same value. In other words, the question of whether to
count imp lica tio n s wh o se t e rms b o t h t a k e t h e sa me in d e terminate value as true o r as indeterminate cannot be avoided
merely b y introducing a fin ite number o f additional values.
Turning to disjunction, we get the fo llo win g ma trix, keeping
in mind that P V Q ( P D Q) D Q:
The most startling aspect of this ma trix is that excluded middle
is a thesis o f th e system. Un fo rtu n a te ly, t h is " lu x u ry " wa s
purchased a t a v e ry high price. Consider the case wh e re th e
disjuncts t a k e d iffe re n t in d e te rmin a te va lu e s; t h e re su lt in g
disjunction i s a lwa ys tru e . Indeed, t h e f a ct th a t su ch cases
are counted as tru e is w h y excluded mid d le is n o w a thesis.
But it is st ill hard to accept the fact that in a ll such cases the
disjunction is true. It will be remembered that in L
l L u k a s i e w i c z
768
K
E
N
N
E
T
H
R. SEESKIN
-P
P
Q
0
1
1
1
1
0
1
2
1
1
J
1
3
1
1
0
1
0
1
1
:I
2
1
1
1
3
2
2
2
1
3
2
3
1
1
3
2
0
2
1
2
3
1
1
1
2
3
2
1
1
9
3
3
3
1
2
3
0
3
1
0
1
1
1
1
0
2
2
1
1
0
3
3
1
1
0
0
0
1
P V Q PV - P
MA NY -V A L UE D LO G I C A N D FUTURE CO NTI NG ENCI ES
7 6 9
had t o decide between ma kin g excluded mid d le a thesis o r
making the value of a disjunction the ma ximu m value of either
of its disjuncts. He chose the la tte r and assigned disjunctions
composed o f t wo indeterminate disjuncts 1
/2. N
value
o f oa disjunction
w ,
s can
i n bec greater
e
than the value o f either
t its hdisjuncts,
of
e
the fo rme r has been chosen. B y introducing a
fourth value, we h a ve n o t resolved th e d ile mma b u t me re ly
opted fo r a different side o f it. The fourth value has gotten us
nowhere, a n d i t i s c le a r t h a t a f i f t h v a l u e in d e te rmin a te
between 2 and 3 wo u ld fare no better. I n addition, i t is easy
to see that a quandary exactly parallel to this one wo u ld arise
with conjunction and noncontradiction.
It is true, o f course, th a t a ll classical tautologies a re tautologies o f L2 and vice versa. Yet, th e superficial characteristics
of t h is syste m a re a g a in deceptive. T h e wh o le re a so n f o r
introducing indeterminate values wa s t o a vo id fa ta lism. B u t
just a s cla ssica l lo g ic ca n n o t assign a p ro p o sitio n a va lu e
indeterminate between tru th and fa lsity, L2 cannot assign one
a value indeterminate between its t wo types o f indeterminacy.
Consider th e f o llo win g f o u r arguments wh e re P p re d icts a n
event coming before the one predicted by Q:
(13) A ssu me t h a t P is 3 a n d Q i s 3 . Th e n P D Q i s tru e .
Then if P turns out true, Q must tu rn out tru e also.
(14) A ssu me th a t P is 3 and Q is 2. Th e n P D —C) is tru e .
Then i f P tu rn s o u t true, Q mu st t u rn o u t false.
(15) A ssu me th a t P is 2 and Q is 3. Th e n —P DQ is tru e .
Then if P turns o u t false, Q must tu rn out true.
(16) A ssu me that P is 2 and Q is 2. Then — P — Q is true.
Then if P turns out false, Q must turn out false as we ll.
What these arguments sh o w is t h a t we ca n b e g in wit h a n y
contingent proposition and f o rm a n imp lica tio n connecting i t
with a n y subsequent, contingent proposition, a n d regardless
how th e f irst one turns out, deduce h o w the second one mu st
turn out simp ly b y kn o win g what indeterminate value the second one takes. No r is this d ifficu lty re a lly avoided b y pointing
out th a t a ll these arguments sh o w is th a t we ma y n o t k n o w
770
K
E
N
N
E
T
H
R. SEESK1N
what value the second proposition takes. I f we cannot kn o w
what indeterminate va lu e a proposition had u n t il i t becomes
either t ru e o r false, th e n we cannot reason w i t h contingent
propositions when they are contingent and one o f the p rima ry
advantages o f t h is syste m wo u ld b e lo st. I n addition, e ve n
though w e ma y n o t k n o w wh a t indeterminate va lu e a co n tingent p ro p o sitio n has, unless w e in tro d u ce a ca te g o ry o f
having n o t ru t h va lu e a t a ll, i t w i l l h a ve t o h a ve one in d e terminate va lu e o r t h e o th e r, ma k in g t h e e ve n t i t p re d icts
fated e ve n th o u g h w e a re n o t a wa re o f h o w. I n sh o rt, L ,
commits us t o much the same typ e o f universe th a t classical
logic does.
Let us n o w consider a system that (a) does n o t re st on the
tacit assumption that meaningful propositions mu st have tru th
values and (b) is o n ly p a rtia lly tru th functional. No doubt, an
adequate treatment o f the b e lie f that meaningful propositions
must have tru th values is f a r beyond the scope o f this paper.
On the other hand, it is safe to say that although this assumption is b y n o means obvious, L u ka sie wicz a n d ma n y o f h is
followers se e m t o re g a rd i t a s self-evident. F o r instance,
Steven Cahn, in Fate, Logic, and Time sounds v e ry much like
Lukasiewicz in saying: (
12
)
But if it is not tru e that a sea-fight w i l l o ccu r tomorrow,
and i f it is also n o t false th a t a sea-fight w i l l o ccu r t o morrow, w h a t tru th -va lu e d o e s t h e p ro p o sitio n " t h e re
will be a sea-fight to mo rro w" have ? I t must have a tru th value to be meaningful, and i t ce rta in ly is a meaningful
proposition I f it is not tru e and it is n o t false, i t must
have a t h ird tru th -va lu e , a n d i t i s su ch a tru th -va lu e
which three-valued logic provides.
Since truth and fa lsity are the values we assign to propositions
about the past and present, i t is misleading to g ive contingent
propositions a value that is somehow intermediate between the
(
Press,
1 1967), pp. 125-126.
9
S
t
e
v
e
n
M
.
C
MA NY -V A L UE D LO G I C A N D FUTURE CO NTI NG ENCI ES
7 7 1
two. Th a t is, t o sa y that 1
/ 2 i sseem t o re q u ire u s t o a lt e r o u r tra d itio n a l notions o f
would
i n t and
truth
e r fm
a lsit
e ydsince
i a tp aert o f wh a t we mean b y " t ru t h " a n d
"bfalsity"
e
is
t thwa t there
e
eis n o intermediate status. I f a proposition
has a t ru t h va lu e , t h e tra d itio n a l v i e w t e lls u s t h a t i t
n
must
be e ith e r tru e o r false. I t is, then, less co n tro ve rsia l t o
I
argue
th a tn future contingent propositions ta ke no t ru t h va lu e
a
than
t o p o sit a t h ird value o n a f u ll semantic p a r wit h t ru t h
d
and
f a ls it y y e t mi d w a y b e twe e n t h e m. A cco rd in g t o t h i s
0
approach, f u t u re co n tin g e n t p ro p o sitio n s d o n o t o ccu p y a
mysterious h a lf wa y house b u t, ra th e r, a re n o t y e t t ru e o r
false. (
and
4 th is fact, I wo u ld argue, i n n o w a y prevents th e m f ro m
being
meaningful.
')
W As
h a re su lt o f confusing n o t y e t b e in g t ru e o r fa lse w i t h
having
an intermediate value, Lukasiewicz was le d to believe
e n
that
L
t
Ihother t ru t h va lu e ra t h e r t h a n a t e mp o ra ry d e n ia l o f t ru t h
value
ce o a lto g e th e r, L u ka sie wicz, i t seems, t h o u g h t t h a t i t s
introduction
in t o t h e o rd in a ry, two -va lu e d ca lcu lu s d i d n o t
u
y l
have
t
o
a
f
f
e
ct t h e w a y t h a t simp le f o rmu la s co mb in e t o
d
a
determine
the
values o f complex ones. Acco rd in g ly, th e tru th
b
r
values
o
f
complex
formulas i n L
e
0,
and
1
Itc a r e
a
/sro2 i Lm p l e
both
Irfu
ness
c dilemmas,
t i o as
n we ll as o u r in a b ility to solve them,
n a ut ohf nthese
suggests
th
a
t
the
admission
o f a t h ird alternative does affect
e
a
r
n
o
f
t i
the
wa
y
that
values
are
assigned
to complex formulas —
t1
d
h
,
h
n
that
1
a
L
fg n
1 a n d 0 b u t a determination o f a d iffe re n t kin d . Mo re o ve r,
/2
o
u
e2
Aristotle
h imse lf appears t o have a rrive d a t a simila r concluia
fn s
n
rjctsion, i n Ch a p te r 9 o f De Interpretatione. A lt h o u g h th e re i s
o
u
e
troom
it f o r doubt, A rist o t le appears t o b e d e n yin g b iva le n ce
while
a ffirmin g excluded middle. (
jp
so
h
3
4
u
ltn
e (
)sa
T h u s
P
1
a
yThree-Valued
Logic ," Phil. Rev iew, L X I V (1955), 264-274.
1
V
—
tlg
a
h 3. Logic ," Jour. of Phil., LX I I I (1966), 481-495.
Free
P
i
s
a
n
u
S
a ()
te
u
e
d
iv 1F r
e
4
v
e
n
h
o
O
d
n
e )
r
icb
.n
A
a
ryY
e
o sR
d
ce
"t Ii
ve
t1
m
r S
a
/rw
u Ti
O
l
lte
2
t Ta
u
h
a
"h
772
K
E
N
N
E
T
H
R . SEESKI N
if n e ith e r o f it s d isju n cts is t ru e ye t. But, a s P rio r remarks,
Aristotle's reason f o r a ffirmin g excluded mid d le co u ld h a rd ly
have been th a t a ll disjunctions whose disjuncts a re " n e u te r"
are to be counted as tru e b y fiat. (" ) I t is tru e not because its
truth va lu e i s a fu n ctio n o f it s d isju n cts, b u t because o n e
disjunct is th e denial o f the other. Th a t is, i t is tru e because
it has t h e f o rm o f a ta u to lo g y a n d t h is consideration ta ke s
precedence o ve r th e fa ct th a t n e ith e r o f it s d isju n cts i s y e t
true.
A llo win g " A " a n d " B " t o ra n g e o ve r fo rmu la s a n d u sin g
" V (A )" t o designate th e va lu e o f A , I propose a syste m i n
which a va lu a tio n V g ive s so me sentence parameters t ru t h
values, and leaves others undefined. (" )
(17) V ( — A) = T if f V (A ) = F o r A is a classical contradiction.
F if f V (A ) = T o r A is a classical tautology.
Undefined otherwise.
(18) V ( A B ) = T i f f V (A ) F o r V (B ) T o r A D B i s a
classical tautology.
— F if f V (A ) = T and V(B) = F, o r A DB is a
classical contradiction.
Undefined otherwise.
It should be cle a r fro m these definitions th a t th is system w i l l
validate a ll those and o n ly those tautologies o f classical logic.
Following Aristotle, P — P is tru e although both its disjuncts
may be undefined. In addition, the system allows us to employ
our normal argument schemes to contingent propositions since
even though — P is undefined if P is, i t is not true, as it was in
L
I If m y a rg u me n t i s co rre ct , t h e lo g ic a l ch a ra cte r o f t h e
,
future
is u n like th a t o f the past o r present n o t o n ly because
some
propositions predicting future events are neither true nor
t
hfalse, b u t because t h e lo g ic o f th e contingent fu tu re i s n o t
truth
functional wh ile (excluding cases like category mistakes
a
t (
e 1(
Free
v 5)1 Logic ," J our, of Phil., LX I I I (1966), 481-495.
e A6
r .)
N
C
y .f
c P.
R
o IB
n O.
C
t R
,
i F.
V
no
MA NY -V A L UE D LO G I C A N D FUTURE CO NTI NG ENCI ES
7 7 3
and presupposition failures) t h a t o f th e past a n d present is.
In other words, a t e ve ry instant in time , a t o t a lly n e w set o f
propositions acquires truth values and thus can be incorporated
into a truth functional logic. It follows that temporal passage is
a genuine feature o f re a lit y a n d cannot b e analyzed a s a n
illusion o r an appearance. It also fo llo ws that propositions wit h
explicit te mp o ra l references ca n n o t b e co n ve rte d in t o e q u ivalent "tenseless" propositions i n t h e ma n n e r suggested b y
Quine (" ) and Willia ms. (")
Yale Un ive rsity.
(") W . V. O. QUINE, E lement ary Logic , re v . e d . (1941; r p t . N e w Y o r k :
Harper and Row, 1965), p. 6.
(
Meaning,
1
eds. Paul HENLE, Horac e M. KA,LLEN, and Susanne K. LANGER (Ne w
York
8 : The Liberal Arts Press, 1951), pp. 282-306.
)
D
.
C
.
W
I
L
L
I
A
M
S
,
"
T
h
e
S
e
a
F
i
g
h
t
T
o
m
o
r
r
o
w
,
"