HO W NO T TO DEFINE TRUTH-FUNCTIONA L ITY
James A . MARTIN
V irt u a lly all modern logic books, and most philosophers, have
as p a rt o f th e ir conceptual apparatus the notion o f t r u t h
-tional compoundedness, i.e., o f a sentence's being truth-funcf u n c - compound. ( M y use o f 'sentence re f le ct s n o philosotionally
phical vie w, o r prejudice; t h e reader is in vit e d t o substitute
'statement', 'proposition', 'judgement', o r whatever, throughout
— w i t h a f e w cle a r exceptions.) L o g ic te xts, f o r example,
typ ica lly ma ke a d istin ctio n between simp le a n d compound
sentences; and among those that are compound a distinction is
made between th o se w h i c h a re tru th -fu n ctio n a l a n d th o se
which a re not. No r a re examples o f philosophical disputes in
which the notion o f truth-functionality is a focal p o in t hard to
find. A n y d isp u te co n ce rn in g t h e va rio u s p u t a t ive co u n te r
examples t o t h e thesis o f exte nsionality (i.e., t h e v i e w t h a t
every meaningful compound either is, or can be "analyzed into",
a truth-functional compound) w i l l do. Such controversies in clude, f o r example, th o se fo cu sin g o n sentences in v o lv in g
modalities, nomological and general sentences, and sentences
about propositional attitudes (e.g., 's believes p', and 's is ce rtain that p l. Unfortunately, discussions o f the notion o f tru th functionality are often marred b y a la ck of cla rity and, indeed,
of accuracy, wh ic h sca rce ly b e fits i t s apparent importance.
What I should like to do in this paper is to indicate where the
lack of cla rity (and of accuracy) lies, and to make some progress
toward getting things right.
I begin wit h some b rie f remarks o n compoundedness. Mo st
logic books te ll us that a sentence is compound if and o n ly if it
contains with in it another, different sentence. 'I t is false that p'
is compound because it contains 'p', wh ich is a different sentence fro m 'I t is false that p'. 'p and cr, 'S believes that p', and 'I f
(I) I hav e been helped a great deal in my t hink ing o n t he topic s o f this
paper b y discussions w i t h Da v id Sanford.
HO W N O T T O DE FI NE TRUTH-FUNCTI O NA L I TY
4
7
7
p had happened, then q wo u ld have happened' are other examples wh ich are compound f o r simila r reasons. So far, so good.
But consider something o f the f o rm 'p o r p', e.g., 'Eith e r Susie
has freckles o r Susie has freckles'. He re is a case that shows
that i t makes a difference whether we couch o u r re ma rks o n
logical th e o ry in terms o f sentences, statements, propositions,
or whatever. Fo r cle a rly 'Susie has freckles' is a different sentence fro m the longer one of wh ich it is a disjunct. But it is b y
no means obvious that it is (o r makes, o r expresses) a different
statement or proposition. On the other hand, the sentence 'Susie
and Anne were both there' makes o r expresses wh a t seems an
obviously co mp o u n d statement o r proposition, b u t sin ce i t
contains n o (g ra mma tica l) sentence d iffe re n t f ro m th e wh o le
sentence it does n o t satisfy the requirement (as stated above)
for being a compound sentence.
Cla rity in the concept of compoundedness is p a rtly a function
of the outcome of disputes over the relative merits of the above
mentioned " lo g ica l units", as we ll as o f the decisions o r discoveries we make concerning such matters as the crite ria o f in dividuation, identity, and so on, t o be employed in connection
with the various competing notions (sentence, proposition, etc.).
We cannot solve the lit t le puzzle I 've raised u n t il we have a
solution to some b ig ones.
There are other problems with compoundedness. Consider the
notion o f " co n ta in me n t" e mp lo ye d i n i t s d e fin itio n . O n a
straightforward understanding of this word, the sentence 'I will
play yo u t o mo rro w afternoon o n the f ie ld behind the lib ra ry
at t wo o 'clo ck' contains (n o t co u n tin g itse lf) a t le a st seven
different sentences: ' I w i l l , ' I w i l l play', ' I w i l l p la y you', and
so on. I t does not, however, seem to be a compound. No r can
we avail ourselves of an easy wa y out by simp ly defining 'compound' in te rms o f the truth-functional connectives, and 'simple' a s it s complement; th is wo u ld ma ke i t too transparently
analytic that a ll compounds are truth-functional, wh e n in fact,
this is a philosophically controversial doctrine.
A p la u sib le suggestion, f o r wh ich I a m indebted t o Da vid
Sanford, is that a sentence is compound if and only if it contains
component sentences different f ro m i t wh ich can be replaced
478
J
A
M
E
S
A. MA RTI N
by a n y in te llig ib le sentence wit h o u t yie ld in g nonsense. Thus,
'I will' is not a component, in the required sense, o f 'I will p la y
you to mo rro w' because replacing it wit h T w o plus t wo equals
four' yields 'Two plus two equals four play you to mo rro w' a b it
of nonsense. The in tu itive p la u sib ility o f this suggestion is enhanced b y an interesting parallel wit h truth-functionality. Just
as i n a tru th -fu n ctio n a l compound, a component ca n b e re placed by any other sentence with the same truth value without
altering the tru th value o f the compound, so in any compound
a component ca n b e re p la ce d b y a n y in t e llig ib le sentence
without decreasing the in t e llig ib ilit y of the compound. A truthfunctional compound is a mo re re stricte d kin d o f compound,
and so here we e mp lo y a test in wh ich not o n ly in t e llig ib ilit y
but t ru t h va lu e mu st be preserved. Some d ifficu ltie s re ma in :
again, h o w sh a ll we handle cases lik e 'Susie and An n e we re
both there' ? But such questions seem to spring from the deeper
problems th e sentence-statement-proposition issue, a n d so d o
not, as far as I can see, constitute a legitimate complaint against
the suggestion mentioned.
Assuming th a t we have an adequate grasp o f the notion o f
compoundedness, wh a t sh a ll w e sa y about b e in g tru th -fu n ctio n a lly compound ? Again, ma n y lo g ic books sa y something
like this: A sentence is truth-functionally compound ju st in case
its t ru t h o r fa lsity depends on (is a function of, is determined
by) th e tru th o r fa lsity o f the component sentences (
definition
is defective, however, i n th a t i t can be taken in a t
2
) . S t wo
least
u cways,
h
onea o f wh ich is quite wrong. First, i t ma y be
taken i n such a wa y as t o re q u ire th a t the t ru t h va lu e o f a
component o f a truth-functional sentence be a determinant o f
the truth value of the compound. This, as I shall show, is a common b u t wro n g interpretation. Second, th e d e fin itio n ma y be
read as asserting that o n ly the tru th value of a component of a
truth-functional compound can be relevant in determining the
truth value o f the compound. Th is wa y is right.
The f irst ve rsio n is exceedingly co mmo n and is o fte n e m(
bolic
2 Logic , 2nd edition, p. 10, o r BARKER'S The Elements o f Logic , p. 102.
)
S
e
e
,
f
o
r
e
x
HO W N O T T O DE FI NE TRUTH-FUNCTI O NA L I TY
4
7
9
ployed in discussions in vo lvin g sentences about propositional
attitudes, e.g., b e lie f sentences. Wh e n we focus o n sentences
like 'S believes p', we may be tempted to suggest that the truth
value of the component must play some role in determining the
truth value of the compound; such sentences seem to be something more or less than truth-functional because the truth value
of p can be altered without altering the truth value of the whole
sentence. Fo r example, Urmson, in h is Philosophical An a lysis
says that 'if p is an element in the complex proposition then the
truth or fa lsity of the complex proposition will be in part determined b y the truth o r fa lsity of p'. The problem wit h sentences
like 'S believes p', he goes on to say in the same place, is that
the tru th o f the b e lie f sentence i n no wa y depends on the
truth of p' (p. 72). The same point is urged in both of the above
cited Copi books (in the vicin it y of the definitions of 'truth-functional') and in numerous other places (
this
3 ve rsio n b y wh ich it s e mp lo yme n t ca n b e recognized, i s
something
). A
slike
t a'beliefs,
n d aafter
r dall, can be true or false'. The point
of
s thlis pronouncement
o g a n seems to be th a t since the tru th va lu e
of
sentence is independent o f the tru th value o f the
o the belief
f
sentence believed, b e lie f sentences a re n o t truth-functional.
This wa y o f construing the definition is wrong. I f belief sentences were truth-functional this wo u ld not entail that the truth
values o f such sentences a re even p a rt ly determined b y th e
truth values o f the believed sentences. Fo r ma n y tru th -fu n ctional sentences a re independent i n t h is w a y o f t h e ir co mponents. I do not mean just that in some circumstances the value
of the component is irre le va n t to the value o f the compound.
It is true that if q is false, the value of p is irre le va n t in determining the value of (p and 11), i.e., one ma y alter the truth value
of p with o u t altering the value o f the compound. But there is a
line in the tru th table for (p and q) where the value of p makes
(3) Some prominent examples: J ohn PASSMORE'S A Hundred Years of Britis h
Philosophy, p. 235; Ma x BLACK'S A Companion t o Wittgens tein's Trac tatus ,
p. 298; A lb e r t E. BLUMBERG'S a rt ic le ent it led «Modern Logic » i n v o l. 5 o f
The Enc y c lopedia o f Philos ophy , P a u l Edwards , (ed.), p . 15; a n d G eorge
PITCHER'S The Philos ophy of Wittgens tein, p. 149.
480
J
A
M
E
S
A . M A R TI N
all the difference. Ob vio u sly the interpretation taken b y Copi,
Pitcher, Urmson, and others, ought to be understood in this way.
Let us suppose that in saying that a sentence is not truth-functional i f the va lu e o f a component is irre le va n t t o th e va lu e
of the compound, wh a t is meant is th a t the compound is n o t
truth-functional if on no assignment of truth values to the components does the altering of the value of some component alter
the value of the compound. But this is st ill wrong.
Consider (p o r not-p). On no assignment of values to the components o f th is sentence does th e a lte rin g o f th e va lu e o f p
alter th e va lu e o f the compound. I f we ma ke o u r crite rio n a
matter of the alteration of the value of the compound b y altering
the value o f the component, the consequences w i l l include the
absurd result that there are no truth-functional tautologies (o r
self-contradictions). There does seem to be a sense in which the
value o f p determines the value o f (p o r not-p), i n wh ich we
want to say that the truth of this compound is somehow p a rtly
constituted by, o r a function of, the value o f p. I wo n 't bother
here t o explicate th a t sense; t h e need can be skirte d wit h a
different example. Consider the compound ((p o r not-p) o r q).
Here, the value of q is completely irre le va n t to the value of the
compound. A lt e rin g it makes no difference; n o r does the tru th
of the wh o le depend, as fa r as I can see, in a n y wa y upon it :
it is not "constructed" fro m it.
It is possible, o f course, th a t one mig h t be able to extricate
the problematic interpretation f ro m th is predicament b y constructing mo re complicated definitions o f 'truth-functional i n
which tautologies and self-contradictions are treated differently
from o th e r so rts o f sentences. Sin ce th e sp o ilin g examples I
have thus far employed are all lo g ica lly necessary propositions,
it might seem that a definition in which such propositions me rit
a special clause could succeed at least in the min imu m task o f
putting th e d istin ctio n in question we re we wa n t it . B u t th is
appearance is misleading; such a n approach wo u ld be ra th e r
more complicated than m y examples thus f a r wo u ld suggest,
for i t is e a sy t o f in d examples o f truth-functional, lo g ic a lly
contingent propositions which contain " irre le va n t" components,
e.g., ((p o r not-p) and q), and ((p and not-p) o r q). Ne ith e r corn-
HO W N O T T O DE FI NE TRUTH-FUNCTI O NA L I TY
4 8 1
pound is lo g ica lly true, o r false; ye t in neither does th e tru th
value o f p p la y a role. Thus, th e relevance o f a component's
truth value to the tru th value o f the compound is n o t a necessary condition of truth-functionality.
It is wo rth noting in vie w of the common definitions of truthfunctionality, that it is n o t a sufficient condition either. To see
this, consider such sentences as
(A) ' S kn o ws th a t p'
(B) 'S is under the mistaken impression that r'.
These d iffe r f ro m the standard b e lie f examples i n p re cise ly
the a lle g e d ly cru cia l respect: ( i f t h e y a re t ru e ) w e ca n n o t
change the tru th values o f p and r with o u t altering the values
of (A) and (B). I f we substitute any false sentence f o r (true) p,
(A) is thereby falsified; the same remarks hold mutatis mutandis
for (B). Nevertheless neither sentence is truth-functional.
Again, i t is possible th a t some v e ry complicated p rin cip le
will wo rk. But a fa r easier wa y o f dealing wit h the problem is
available. W e mu st abandon th e f irst in te rp re ta tio n and re ly
on the second, viz., that if a component does p la y a ro le in determining the tru th value of a compound, it is o n ly in virtu e of
its tru th value, i.e., i t is in virtu e of o n ly its tru th value. Some
standard remarks about truth-functional components reflect just
this p o in t. Some examples: I n such compounds a component
sentence can be replaced b y any other sentence wit h the same
truth value with o u t altering the value of the compound, o r: We
can t h in k o f the components a s t ru t h values. Th e imp o rta n t
point wh ich is captured in such remarks is that nothing other
than the t ru t h va lu e o f a component is re le va n t to th e t ru t h
or f a lsit y o f a truth-functional compound.
Let us p u t it in terms o f sentences like 'S believes p'. I t isn't
the irrelevance of p's truth value to the value of the compound
that makes th e b e lie f sentence non-truth-functional, i t is th e
relevance of something besides truth value. I t is not that beliefs
can be false as we ll as true, but rather that we cannot substitute
for the believed sentence another sentence wit h identical tru th
value, and be certain that the truth value of the compound will
482
J
A
M
E
S
A . MA RTI N
remain as it was. S imila rly, the cru cia l point about (A) and (B),
the fa ct in virt u e o f wh ich neither is truth-functional, is th a t
the tru th value o f neither compound is necessarily preserved
through substitution of like tru th values. I f 'S knows p' is true,
p is true; and although, as noted earlier, we can falsify the compound b y substituting f o r p a n y fa lse sentence, w e ca n n o t
substitute f o r p ju st a n y tru e sentence, n o r f o r r in the (true)
sentence 'S is under the mistaken impression that r' ju st a n y
false one. A person wh o kn o ws one th in g doesn't necessarily
know everything, and the ma n wit h one mistaken belief need
not have them all.
The reason wh y (A) and (B), and more o rd in a ry belief sentences, are non-truth-functional is that something about the components o th e r than t h e ir t ru t h va lu e is re le va n t t o th e t ru t h
value o f the compound; wh a t th e y say is relevant. (Indeed, in
some cases one might wa n t to say that h o w wh a t is said gains
expression ca n be important. I f ' w
but
l S doesn't kn o w this, i t wo u ld be misleading, o r even false,
to
' say
a nthat
d S knows
' w
that x is w
it
is
wo,
and
when
he
i2 w h e n
w h wo
a u ld
t deny, o r refuse to assent to, ' x is
WI
h'
a
e
r
e
w
s To
y osunm uup:
o nIlt isy dcommmo
o n ly
u thought that wh a t I have called
the
ss ,'relevance'
a
yo f a component's tru th value to the value o f a
compound
i
sis both a necessary and a sufficient condition o f the
compound's
being
On this vie w, a component
t
h
a truth-functional.
t
occurs i n a compound tru th -fu n ctio n a lly ju st in case it s tru th
value so me h o w p a rt ly constitutes, o r p a rt ly determines t h e
truth value o f the compound. I have shown that th is provides
neither a necessary n o r a su fficie n t co n d itio n o f tru th -fu n ctionality. Wh a t i s necessary i s t h a t n o th in g o th e r th a n t h e
truth value of a component can be relevant to the tru th o r falsity o f the compound. Perhaps the best wa y to p u t it is this: I f
a compound contains a component tru th -fu n ctio n a lly then, i f
anything about that component is relevant to the truth value of
the compound, it is its tru th value.
Dartmouth College
J
a
m
e
s
A . MARTIN