ON H O W N O T TO I NV A L I DA TE DI SJ UNCTI VE SYLLOGI SM
J OHN WO O DS
1. Preliminaries . I s hall say of any theory of entailment, T, t hat it
is paradox-free just i n case,
first, none of Lewis ' paradoxes (I) is a t heorem of T.
second, a l l proofs o r arguments sustaining Lewis ' paradoxes mus t
employ o r presuppose a t least one inf erenc e-rule o r statement or princ iple inc ompat ible wi t h at least one t heorem or
rule o r princ iple o f T; a n d
third, ev ery occurrence o f ' - 3 ' i n f ormulae expressing t he paradoxes, and of 'entails' i n T, is int erpret ed t o mean 'entails'
in Moore's sense (
T wi l l be2 said t o exclude the paradoxes i f and only i f it is paradox free.
).
It is evident, I trust, t hat on the assumption that the paradoxes are
false, being paradox-free i s o n l y a necessary a n d n o t a s uf f ic ient
condition of t he t enabilit y o f a t heory of ent ailment .
To c laim f or entailment a paradox-free analysis is to c laim satisfaction (implic it ly ) o f wh a t I c all t he inv alidat ion-c ondit ion. Th e i n validation-condition imposes upon any T, f o r wh i c h t he paradoxes
do not hold, t he necessity of ref ut ing Lewis ' independent proof s o f
them (
3
) • (') Th e paradoxes: a n impos s ible propos it ion ent ails a n y propos it ion; a
necessary
propos it ion i s ent ailed b y a n y propos it ion; a n y t w o necessary
T h
propositions ent ail eac h ot her; a n d a n y t w o impos s ible propos itions e n t a il
u s ot her.
each
, (
if rom
2 p « w h e r e f ollowing f ro m a n d being deduc ible f r o m a re ex emplif ied
f )the relation in whic h the conclusion of an argument in Barbara stands t o
by
theFc onjunc tion of its premisses and by the relat ion in wh ic h «x is coloured»
T
t o «x is red». I t is c lear t hat f o r Lewis t he o n ly differenc e between
istands
o
r implic at ion i n h is sense a n d ent ailment i n Moore's sense is t e rmin o sstrict
logical.
p Ma Indeed Lewis insists t hat his system o f s tric t implic at ion, S2, giv es
theoordinary meaning of proof, of inference, and of deduc ibility . If, then, we
r a
areo t o read S2 as Lewis intended i t t o be read we s hould giv e t o ' _ 3 ' t h e
d
o
Moorean'
interpretation.
r
x (e
fLogic
3,r (Ne w York , Dov er Public ations, Inc ., 1932), pp. 250-2.
e )«e
,312
Tp1
i he
t en
i pt
s ra
o
i i
o
n l
f
c ss
q
o m
»
ma
i
p
one statement or rule of any argument of f ered as prov ing t he paradoxes true; and among such wi l l be at least one propos it ion or rule
invoked by Lewis ' arguments.
To c laim t hat a c ondit ion is met is not t o meet it. The creation of
a paradox-free theory does not, o f itself, constitute f ulf ilment o f t he
invalidation-condition. A t mos t i t reveals a n inconsistency bet ween
theorems o f t he t heory and t he inference-rules of Lewis ' proofs. To
meet the inv alidat ion-c ondit ion one mus t establish that the statements
of the theory, wit h whic h the rules of the proofs are inconsistent, are
true. This is not always easy. The question is: is i t alway s possible ?
As anyone knows, wh o has had a reductio argument go sour on him,
a statement of the theory init ially might be accepted as t rue only t o
be rejected upon t he discovery t hat it is inconsistent wi t h some rule
of t he proofs. Th e s y mmet ry of being inconsistent wi t h allows f or,
and even encourages, the appeal to one and t he same fact as evidence f o r each o f mut ually inc ompat ible views. Thus i t c ould happen
that one philos opher takes t he inconsistency between a paradox -f ree
theory and Lewis' proofs as evidence of the inadequacy of the theory,
and anot her philos opher takes this as s howing t he inv alidit y of t he
proofs. Clearly , bot h would beg the other's question; and this s hould
provoke us to ask whet her and how the ris k of petitio in such matters
can be eliminat ed. More of this shortly. Let us say f or n o w t hat an
unarguably essential adjunc t of any t heory c laiming t o be paradox free, is an accompanying independent defense of t he implied c laim
that t he t heory does i n f ac t satisfy t he inv alidat ion-c ondit ion. A n y
theory, paradox-free i n t his way , and whos e independent defense is
sound, is, as we s hall say, properly paradox-free. Alt ernat iv ely , such
a t heory properly excludes t he paradoxes.
2. The Anders on-Belnap Theory . I t i s t he ma i n purpos e o f t his
paper t o inquire int o a t heory of ent ailment — a j o i n t c ont ribut ion
by Alan Ross Anders on and Nu e l D. Belnap, J r — whic h purport s
to be properly paradox-free. While I concede, at least f or the moment ,
that t heir theory excludes t he paradoxes, I hav e grave doubts about
whet her t heir exclusion is proper. I hope t o be able t o ex plain my
doubts and, perhaps even t o persuade the reader to share them.
Our inquiry wi l l be limit ed t o an ex aminat ion o f Anderson's and
Belnap's putative refutation of Lewis' independent argument in behalf
of the f irs t paradox, t he paradox, namely , whic h asserts t hat an i mpossible proposition entails any proposition whatever. For convenience, t he argument is here reproduced:
313
Assume
P
-— P
( 1 )
(1) e n t a i l s
p
(
2
)
(2) e n t a i l s
p
yq (
(1)
— • P
( 4 )
3 ent ails
(3)
(
) a n d (4) ent ails q
5
)
Anderson
and Belnap are agreed that if, wi t h respect to the relevant
truth-functional connectives, t he princ iples of s implif ic at ion, addit ion
and disjunctive syllogism are v alid modes of inference, t he argument
stands and t he f irs t paradox is proved. Since, howev er, t hey do not
believe the paradox to be true, they are c ommit t ed (and acknowledge
that t hey are c ommit t ed) t o inv alidat ing at least one o f these rules
of inference. Th a t is, Anders on a n d Belnap openly accept t he i n validation-condition; t h e ques t ion f o r us t o settle i s whet her t hey
manage t o satisfy it.
Ominously, t h e i r dec is ion i s t o dis c redit dis junc t iv e-s y llogis m.
Writes Belnap, (, We do hold t hat t he inference f r o m p and p
to q is an error: i t is a s imple inf erent ial mistake. Such a n inference
commits not hing less t han a Fallac y of Relevance» (
1 sense i n whic h t he real f l a w i n Lewis ' argument is not a Fallac y
a
) . Relevance
of
Y e t
but
« rat
t hherea rFallac
e y of Ambiguit
i
s y : t he passage f rom (2)
to (3) i s v alid o n l y i f t he ' v ' i s read t rut h-f unc t ionally , wh i l e t he
passage f r o m (3) a n d (4) t o (5) is v alid o n l y i f t he ' y ' is tak en i n tensionally» (').
Now, a t f irs t blus h c ert ainly , t h e repudiat ion o f dis junc t iv e s y llogism is at least as self-evidently preposterous as Lewis' f irs t paradox
is t hought t o be. Wh a t c ould be logic ally more s olid t han t hat i f at
least one of p and q is t rue and p is not true, then q is t rue ? Is it possible, one wonders , f u l l y t o appreciate h o w Anders on a n d Belnap
could possibly jus t if y so heroic a stand ?
Central t o t heir position is t he c laim t hat t he inference f r om — p
and (p o r q) is v alid only f or some int ens ional int erpret at ion of 'or'
and, not, i n general, f or its extensional o r t rut h-f unc t ional int erpretation. Let us unders t and ' OR' as expressing t hat int ens ional sense
of 'or' f or whic h disjunctive syllogism (hereafter 'IDS') holds, and let
us reserve f or 'y ' its customary t rut h-f unc t ional interpretation. Giv en
(
no.4 7 , Offic e o f Na v a l Research Cont rac t n o . S A R/ Nonr - 609 (1 6 ) ( N e w
Haven:
Interac tion Laboratory , Y ale Univ ers ity , 1960), p. 33.
)
(N) ibid. , p.34.
u
314
e
l
D
.
B
E
L
N
A
P
,
this contrast, we may n o w add that, in t he opinion of Anderson and
Belnap (
princ
6
iple o f addit ion holds f o r extensional o r mat erial dis junc t ion,
) , not
but
p f or its int ens ional analogue. ( ' N' is t o be read as 'ent ails ).
N then, t heir attack upon Lewis ' argument exposes i t t o three posSo,
sible
( p and allegedly fatal objections:
y first, i f the proof asserts o r implies t he t rut h of bot h p N ( p y q)
and (—p • (p v q) ) N q, t hen the f ormer is t rue but the lat t er
1
is not;
7
) second, i f i t asserts o r implies p N (p OR q) a n d ( — p • (p OR .1))
N q, t hen the lat t er is t rue but t he f ormer is not ;
i
and
s
t third, i f i t asserts or implies p N (p v q) a n d (
both are true, but there is a Fallacy of Ambiguit y on step (3)
r ,
of t he argument ; as a v alid consequence o f (2) i t mus t be
u read t rut h-f unc t ionally , b u t t he o n l y legit imat e rôle i t c an
e ,
• play( i np DS requires
O
Ri t t o be read intensionally.
b p
)
)
N
u q
3. I nt ens
y ional
, dis junc t ion. Th e wh o l e f orc e o f t he ref ut at ion o f
t t
Lewis'
argument
turns on a conception of intensional dis junc t ion f or
p
whic
h
DS
is
guaranteed
to hold and f or whic h addit ion is guaranteed
N
to
f
all.
Wh
a
t
is
there,
t
hen,
i n t he meaning of «p O R q» ov er and
(
above i t s t rut h-f unc t ional v i s
q
meaning
of 'OR' t hat resists addit ion and tolerates DS, and about the
O
à
v
i
s
;
meaning o f a' y ' nwh idc h s imply switches t his order o f resistance a n d
R
toleration
w
h
a? t
q
i Notvvithstandig
s
t h e i r relat iv e inat t ent ion t o s uc h questions, A n )
tderson
h andeBelnap
r doe mak e t wo suggestive comments (
i
a
p OR q,
o then uif p were not the case, q would be the case. That
7 I. I f b
s
is,
t) : t he t rut h of any intensional disjunction guarantees t he t rut h of a
n
corresponding
t
h subjunctive
e
c onditional t o the effect that if all but one
o
of the disjuncts were false, the remaining one would be true.
t
IL Someone asserts t hat p O R q when he asserts t hat eit her p is
;
the case or q is the case but that he does not k now whic h. He cannot
t
mean t o assert s imply t hat p y q because t here are cases i n wh i c h
h
haying asserted that p y q solely on the strength of p, the subsequent
a
discovery that p is false would require h i m t o wit hdraw his assertion
t
of p y q; i n whic h case he would not be prepared t o assert t he c ori
s(
, (6
ments»,
Philos ophic al Studies, X I I I (1962), 9-24, P. 22.
t 7)
I
)
h
d
315
eA
l
e
lm
.
a
n
R
o
s
s
A
N
D
responding subjunctive conditional to the effect that had q been false
p wo u l d hav e been t rue, i n v iolat ion o f I . Presumably, t hen, o n e
correctly asserts t hat p OR q only wh e n k nowledge o f t he t rut h of
p y q is c laimed independent ly of t he k nowledge o f t he truth-values
of p and q.
I int erpret I and I I as constituting the heart of the attack upon DS:
p l r q does not generally license t he corresponding subjunctive c ondit ional that if p where false q would be true. I t does not license its
corresponding s ubjunc t iv e c ondit ional because t here a r e s it uat ions
where t he only av ailable grounds upon wh i c h t o assert t hat p v
is t he t rut h of p. I f then, p where t o be false, one wo u l d not any
longer, i n those circumstances, be prepared t o assert the subjunctive
conditional, and this s imply because one would not have the warrant
to assert that p v q.
Just such a situation may be t hought t o obt ain wi t h i n t he context
of Lewis ' argument f or t he f irs t paradox. Giv en t hat p • p , we are
entitled to mov e to p; and f rom p we may mov e to p y q• No w where
q i s a n indet erminat e, a r b i t r a r i l y selected propos it ion, t h e o n l y
grounds we hav e f or mat erially dis joining i t t o p is t he t rut h of p.
But giv en p • —p, we mus t als o mov e t o p , i n wh i c h case w e
deprive ourselves of the only av ailable grounds upon whic h t o assert
the disjunction of p wit h any arbit rary q. This is just t o say t hat we
thereby depriv e ourselves of the only av ailable grounds f or asserting
— disjunctive s y llogis m therefore fails.
What, t hen, is t he dif f erenc e i n meaning bet ween ' OR' a n d ' y ' ?
Simply that p OR d o e s , but p y q does not, license the corresponding
subjunctive conditional.
Why is it that 'OR' does not, but 'v' does, tolerate addit ion ? Simply
because it is not the case t hat the t rut h of the result of dis joining t o
a giv en t rue p any arbit rary q i s alway s k nowable independent ly
of k nowledge o f t he truth-values o f it s disjuncts.
And why is it that 'OR' does, but 'v ' does not, tolerate DS ? Simply
because ' O R' a l o n e generates t h e c orres ponding s ubjunc t iv e c ondit ional (whic h encapsulates, af t er all, the whole logical force of DS);
on t he other hand, cannot generate the corresponding c ondit ional
because i n those cases where p v q is af f irmed s olely on the strength
of p, one c ould not say: «p v q is true, and p is false, so q is t r u e ,
just because one c ould no longer say t hat p y q.
4. Properly paradox -f ree ? I f I a m right i n t hink ing t hat t his i s
what it is f or DS t o f ail, i t is an elementary, but gross, error t o sup316
pose t hat susceptibility t o such f ailure, t o f ailure o f t his k ind, establishes the inv alidit y of DS. Not hing we have interpreted Anderson
and Belnap as hav ing meant i n any wa y defeats t he c l a i m t hat i f
p y q were true and if p were false, then q would be true. At most, i t
has been shown that where p is the only available grounds for asserting
p y q, t hen DS cannot establish t he t rut h o f q s imply because i n t he
very applic at ion o f i t I depriv e my s elf o f t he right t o say t hat my
deduction is sound (i.e. v alid and possessed exclusively of t rue premisses). The c ompound categorical inference (p, so (p y q); and since
—p, therefore q) is self-defeating; i t cannot be ot her t han unsound.
It is unsound because it embodies the categorical assertion of bot h p
and — p ; t he premisses of t he inference are int ernally inconsistent.
But unsoundness, as Anderson and Belnap should we l l k now, is not
invalidity. A v alid argument can liv e wi t h false o r inconsistent premisses, a sound argument cannot. Suppose, then, (per impossible, of
course) t hat p is true and — p is true. The t rut h of p wo u l d prov ide
f ull aut horit y f or asserting p y q, and the t rut h of t o g e t h e r wit h
p y q, wo u l d prov ide f ul l aut horit y f or asserting q. So i f i t were t he
case t hat p p , w e s hould hav e f u l l aut horit y f o r asserting t he
truth o f a n y q we please: Th i s jus t i s Lewis ' f irs t paradox — a
paradox whose t rut h is ent irely c ompat ible wi t h t he unsoundness of
Lewis' argument i n its behalf.
Anderson a n d Belnap hav e, t hen, c onf us ed inv alidit y w i t h u n soundness — a n egregious h o wl e r i f ev er t here was one; a n d t he
very enormit y of t he error, i n c onjunc t ion wi t h t he f ac t t hat those
who mak e it are logician's of high rank , is liable t o produce second
thoughts about the correctness of our interpretation of t heir dismissal
of DS. Indeed one might be tempted to say that, strictly speaking, Anderson and Belnap are not alleging the inv alidit y of DS, but rather, if
you lik e, t he ,
; s y hs the
whic
t e only
m m basis f or asserting p y q is negated. Suc h temptations
aare
t ibetter
c
resisted. I f it is the systemmatie unsoundness of Lewis' argument
Anderson and Belnap, they need not have t roubled
u n that
s obothers
u
themselves
n d n e wi
s t h DS, f o r t he argument i s rendered uns ound a t t he
s » outset, at it s f irs t step, p — p . Wh a t is more, s t ric t ly speaking
very
o
that
argument is not uns ound; i t takes t he f o r m o f a c ondit ional
fproof, i n whic h t he «premise» i s not asserted, i n whic h, t hat is t o
say,
p — p does not occur as a prendss, but only as an assumption.
a
Any
assessment, therefore, i n t erms o f soundness a n d unsoundness
n
yis irrelev ant ; i t misconstrues t he v ery nat ure o f t he argument t hus
aassessed.
p
p
l
i
c
317
a
t
i
o
o
f
i
t
i
n
n
It is arguable, of course, t hat i f it were t rue bot h t hat p and —p,
it wo u l d be t rue t hat p y q. But t he joint t rut h o f p y q a n d — p
would not y ield q unless —p were to negate p, unless, that is, it were
not the case bot h that p and —p. Since b y hypothesis, it is t he case
that p and d o e s not negate p, and q cannot f ollow f rom the
joint t rut h of (p y 47) and —p.
Ho w is this t o be answered ? (
conjunction
8
o f p y q and — p presupposes t he t rut h o f — (p — p ) .
Yet) t he
T hf irs
e t s t ep
v o
e f trhey argument giv es t he c ont radic t ory o f this.
Surely,
a p then,
p l tihec applic
a t ati ion
o ofn DS is inconsistent wi t h t he f irs t step
of ot he prooff. I t hink not ; a n d t his because t he f irs t step o f Lewis '
proof
ev en though the t rut h of t he denial of the f irs (
D is not asserted,
S
step,
t b y t he applic
o
at ion o f DS, i s asserted, b y implic at ion at least.
Sot not hing whic
h
h Lewis'
e proof c ommit s us t o asserting is inc ompat ible wi t h any t hing else t he proof asserts. I n apply ing DS I implic it ly
assert — (p. —p); i n supposing or assuming t he f irs t step, I suppose
or assume, but do not assert, t he contradictory of this. My supposing
what I suppose is c ompat ible wi t h my asserting wh a t I assert, a l though wh a t I suppose is not c ompat ible wi t h wh a t I assert. I d o
not contradict my s elf when I say: «Suppose, although it is false, t hat
p»; neit her do I contradict my s elf when I say «Suppose, per impos -
sible, that p and — p.»
In the absence, then, of a viable defense of it, the most to be said
of the case o f t he inv alidit y of DS is t hat i t merit s a Scotch v erdic t
of «Not proven». Because t hey hav e n o t adequat ely jus t if ied t h e i r
repudiation of DS, Anderson and Belnap cannot maint ain satisfaction
of t he inv alidat ion-c ondit ion. A s i t stands, t heref ore t heir t heory is
not properly paradox-free.
5. Consistency and Petitio. I have, i n effect, argued t hat Anders on
and Belnap have not s hown DS t o be inv alid; but this alone does not
allow me t o say t hat I hav e s hown i t t o be v alid. Surely I c an d o
better. I wonder, f rank ly , whet her I c an. I t c an b e demonstrated,
although I shall not t rouble y ou wit h such matters here, that, on t he
basis of principles eit her explicitly accepted by Anderson and Belnap
or deriv able f rom those they accept or so obviously tenable that there
could be no good reason why they would not be accepted, t he denial
of DS begets an ex plic it self-contradiction. I t is here t hat problems
(
minator»,
8
Analysis, x x v (1965), 49-53.
)
F
31$
o
r
a
f
u
l
l
e
r
d
i
arise. Two of these principles, bot h «so obv ious ly tenable t hat t here
could be t it good reason why they would not be accepted' are these:
1 I f ( p N q) t hen ( p o q ) whatever
.
T h athet correct analysis of consistency might be, p is conof q.
i sistent
s , wi t h tihe denial
f
p I f (p o q) t hen, i f M( p v q) t hen M( p q ) . That is, i t is a neces2.
d sary,
o bute notsperhaps a sufficient c ondit ion of (17 is consistent
n wit hoq», t hat
t i f eit her p o r q is logic ally possible, b o t h a r e
e logically
n
t possible.
a
Not
i ic e t h a t t his c onc ept ion o f consistency
l
departs mark edly f rom Lewis'. For Lewis, t wo propositions are
q mut ually ,consistent i f and only i f t heir c onjunc t ion is logic ally
t possible.
h
O nethis v iew an impossible proposition is inconsistent
n wit h ,every proposition, i n v iolat ion of our strong int uit ion that,
notwithstanding t he impos s ibilit y of «7 bachelors are married»,
it, manifestly, is consistent wi t h «At least 5 bachelors are married». I f only because i t avoids t his consequence, t he int erpretation o f consistency wh i c h I hav e of f ered seems dif f ic ult t o
quarrel wit h.
Here, t hen, i s my problem. I f y ou giv e me these t wo princ iples ,
the rejec t ion o f DS i s open t o a conclusive reductio. A n d i t seems
that y ou s hould giv e me these princ iples f or what , pray , c ould be
more obvious t han they ? But if y ou do allow me t heir use, y ou aid
and abet me in begging a key question against Anderson and Belnap.
For o u r t wo princ iples , s t raight away establish t he t r u t h o f Lewis '
paradox, t he v ery paradox t hat t he denial o f DS was t o r i d us of .
This i s obvious. I f - ( ( p . - p) N (7) t hen, b y o u r f irs t princ iple,
(p - p) o then
q . M( ( p — p ) — q ) , wh i c h , lat t er i s f als e o n ins pec t ion; f r o m
whic
T h he f ollows
n ,
the t rut h of (p . - p) N
b It isy possible, o f course, t hat we might at t empt t o soften o u r c onception
o
u of consistency i n an ef f ort to, as it were, «unbeg» t he question.
Such a sense of consistency would hav e t o capture at least one
r
essential
s
e ingredient : i f p is consistent wi t h q, t hen t here is not hing
in
c t heo meaning of 'p' and of 'q' whic h makes it impossible f or p and q
to hav e t he same truth-values.
n Thus,
d
p 2*.r I f (i p o .7) t hen M( p I t
i s a necessary, b u t n o t sufficient
n
c condition
i
of «p is consistent wit h q» t hat the mat erial equiv ap
l lence
e of p and q be logic ally possible.
, Surely, then, 2* squares very nic ely wi t h our int uit ions about consistency,
and it certainly does not appear t o giv e us t he paradox - i
m((p.
-p )
f
M
(
p
q
)
o
q
)
q )
is obviously true.
319
Could w e not , t hen, substitute i n t o o u r reduc t io defense o f DS
this weak ened c onc ept ion o f consistency a n d t hus v indic at e D S
wit hout begging t he issue against Anders on and Belnap ? We c ould
not. I n t he f irs t place, such a substitution inv alidat es t he reduc t io;
the inferences wh i c h t h e s t ronger sense o f consistency generat ed
are f lat ly inv alid f or its weak er counterpart. A n d secondly, whereas
the weak er sense o f consistency does not immediat ely generate t he
first paradox, it does immediat ely generate the f ourt h f rom whic h t he
first direc t ly f ollows . I t is obv ious ly not possible t hat ( p p ) same
the
h at rutv h-v
e alue as (—q y q), f rom whic h it f ollows that (12 p )(11 q ) , and hence also t hat (p N
p Both
)
N
t he strong
q and
.
weak conceptions of consistency generate t he
paradoxes and render otiose t he need f or an independent defense of
DS. What , t hen, are Anders on and Belnap t o mak e of this? I f t hey
deny t he t rut h of eit her (2) o r (2*) t hen I want t o say t hat t hey are
not t alk ing about consistency save i n some non-standard sense; and
if t hey deny (1), t hen I wa n t t o say t hat t hey hav e ceased t alk ing
about ent ailment i n Moore's sense, and henc e t hat t heir t heory o f
entailment violates condition-three of a paradox-free analysis of entailment. This is what I want to say, but I confess that I do not k now
how t o prove what I wa n t t o say wit hout begging t he question. So
perhaps, i n t he end, w e s hall hav e t o c ont ent ourselves wi t h t h e
earlier verdict of N o t proven».
John WOODS
University of Toront o
320