WH A T IS A TRUTH FUNCTIONA L COMP ONE NT?
David H. SANFORD
Discussions wit h James Ma rtin have convinced me that many
published definitions of truth functionality are defective. Someone mig h t in sist th a t because th e d e fin itio n s a re stip u la tive
they cannot possibly be defective. Since the meaning o f 'tru th
functional component' is given b y the definitions, there cannot
be a counterexample wh ich shows th a t the definitions f a il. I
could re tre a t a b it and sa y I wa n t t o suggest a n a lte rn a tive
stipulation, but this so rt of response wo u ld concede to o much.
A man might be content with an inadequate definition of 'courageous act' even though he can te ll when an act is courageous
and when it is not. No one wh o has read a Socratic dialogue
needs to be told that a person can use a wo rd co rre ctly without
being able t o g ive a co rre ct account o f it s use. Th is ancient
moral applies t o ma n y technical te rms a s w e l l as i t applies
to more important words and phrases used in everyday life. I t
applies i n particular, I th in k, t o 't ru t h functional component'
We are clearer about what is o r is n o t a tru th functional component of a given sentence than we are about how to say what
it is f o r one sentence t o b e a t ru t h fu n ctio n a l component o f
another. I shall t ry to provide a definition wh ich is free o f the
defects that Ma rtin and I have noticed.
Here is a definition wh ich is cle a re r and more precise than
most.
A compound of given components is a truth function of them
if its tru th value remains unchanged under a ll changes o f the
components so lo n g a s t h e t ru t h va lu e s o f t h e components
remain unchanged (
1 Let us suppose that we kn o w what it is for one sentence to be
a component o f another. A n d le t us re st rict o u r attention to
).
declarative sentences. Th e d e fin itio n ca n be re writ t e n as f o llows: P is a truth functional component of Q if and only if
(
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P is a component of Q, and
(S) (if S has the same truth value as P, then the sentence which
results fro m replacing P in Q b y S has the same tru th value as
Q)
This will not do. Consider
(1) He n ry knows that beavers la y eggs.
According to the definition, 'Beavers la y eggs' is a truth functional component of (1). (1) mu st be false because 'Beavers la y
eggs i s false. An d the truth value of (1) remains unchanged no
matter wh a t sentence is substituted f o r 'Beavers la y eggs' so
long as the truth value of the sentence remains unchanged, that
is, so long as the substituted sentence is false.
Although th e sentence that results f ro m replacing 'Beavers
la y eggs' in (1) b y one false sentence has the same tru th value
as the sentence that results from replacing 'Beavers la y eggs' in
(1) b y a n y other false sentence, sentences wh ich re su lt f ro m
replacement by different true sentences need not have the same
truth value. Th is is w h y we wa n t n o t to regard 'Beavers la y
eggs' as a tru th functional component o f (1). Th e fo llo win g is
a necessary condition o f 'P is a tru th functional component o f
Q):
P is a component of Q, and
(S)(T) ( i f S and T have the same truth value, then the sentence
which results fro m replacing P in Q by S has the same
truth va lu e a s t h e sentence wh ic h re su lts f ro m re placing P in Q b y T).
'Beavers la y eggs' is not a tru th functional component of (1)
because it fails to satisfy this condition.
Unfortunately, t h e co n d itio n i s n o t sufficient. Co n sid e r
(2) He n ry believes t h a t beavers b u ild d a ms D H e n r y b e lieves something.
'Beavers b u ild dams' is a component o f (2). Since (2) is tru e
no matter what sentence is substituted for 'Beavers build dams',
when any two sentences with the same tru th value are respective ly substitued f o r it , th e re su ltin g sentences a re both tru e
and thus have the same tru th value.
The f o llo win g p rin cip le seems t o b e imp lic it in o u r understanding o f tru th functionality:
W H A T I S A TR U TH FUNCTI O NA L CO MP O NE NT?
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If P is a component of R wh ich is in tu rn a component of Q,
then P is a tru th functional component of Q o n ly if P is also
a truth functional component of R.
Example (2) shows that satisfaction of our necessary condition
does not insure that this p rin cip le is satisfied. We wa n t not to
count 'Beavers b u ild dams' as a tru th functional component o f
(2) because i t is n o t a tru th functional component o f 'He n ry
believes th a t beavers b u ild dams'. Let us emend o u r necessary
condition accordingly. The task will be somewhat simpler if we
stipulate t h a t e ve ry sentence i s a component o f itse lf. Th is
stipulation w i l l le a d t o t h e conclusion, wh ic h I f in d n e it h e r
welcome n o r unwelcome, t h a t e v e ry (d e cla ra tive ) sentence
is a truth functional component of itself. If the conclusion seems
anomalous, it can easily be avoided by an increase of complexity. P, then, is a tru th functional component of Q o n ly if
P is a component o f Q, and
(R) ( i f P is a component of R and R is a component of Q, then
(S)(T) ( i f S and T have the same truth value, then the sentence wh ich results fro m replacing P in R by S has
the same truth value as the sentence wh ich results
from replacing P in R b y T)).
Since we co u n t sentences a s components o f themselves, t h is
condition i s satisfied o n l y i f th e f irs t co n d itio n i s satisfied.
'Beavers b u ild dams' is not a tru th functional component of (2)
because it fails to satisfy this n e w condition.
Unfortunately, th is also is not sufficient. Consider
(3) He n ry kn o ws that (dogs liv e in trees and beavers ch e w
wood).
'Beavers c h e w wo o d ' i s a component o f (3 ). Sin ce (3 ) i s
false no matter what sentence is substituted fo r 'Beavers ch e w
wood', wh e n a n y two sentences wit h the same tru th value are
respectively substituted fo r it, the resulting sentences are both
false and thus have the same truth value. Furthermore, the sentence 'Dogs l i v e i n trees a n d beavers ch e w wo o d ' satisfies,
as i t should, a ll o u r conditions f o r containing 'Beavers ch e w
wood' as a tru th functional component.
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There seems t o be st ill another p rin cip le imp licit in o u r understanding o f truth functionality:
If P is a component o f R wh ich is in tu rn a component of Q,
then P is a truth functional component of Q o n ly if R is also
a tru th functional component o f Q.
Example (3 ) sh o ws th a t satisfaction o f o u r re vise d necessary
condition does not insure that this second principle is satisfied.
We want not to count 'Beavers chew wood' as a truth functional
component of (3) because it is a component of 'Dogs live in trees
and beavers ch e w wood' wh ich is n o t itse lf a tru th functional
component o f (3). A lso , example (2) shows that satisfaction o f
the second p rin cip le does n o t in su re satisfaction o f the first.
Further re visio n and complication o f o u r condition are called
for.
I believe th a t the f o llo win g is both a necessary and a sufficient condition o f 'P is a tru th functional component of Q':
P is a component of Q, and
(R)(if P is a component of R and R is a component of Q, then
(S)(T)(if S and T have the same tru th value, then
((the sentence wh ich results f ro m replacing P in R
by S has the same truth value as the sentence wh ich
results fro m replacing P in R b y T) and
(the sentence wh ich results f ro m replacin g R in Q
by S has the same truth value as the sentence wh ich
results fro m replacing R in Q b y T)))).
I would be pleased, but not surprised, if someone could provide
a simp le r fo rmu la tio n wh ich does t h e same jo b . I wo u ld b e
rather less pleased, b u t s t ill n o t surprised, i f someone co u ld
think o f an example wh ich shows that the jo b cannot be done
adequately wit h o u t fu rth e r complication.
Duke Un ive rsity
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H. SANFORD
Note added in proof : I neglec t t o mak e ex plic it t hat I wa n t t o s ay when a
partic ular oc c urrenc e o f a s entenc e i s t r u t h f unc t ional. Separat e oc c urrences mus t b e handled s eparately . Th u s t h e s ec ond oc c urrenc e o f 'Dogs
bury bones ' i n ' (He n ry k nows t hat dogs b u ry bones ) D dogs b u r y bones "
is t r u t h f unc t ional, b u t t h e f irs t oc c urrenc e i s not .