Tit trni•FUNCTIONS
Peter M. SIMONS
It is sometimes said that truth-functions are functions from propositions to propositions, again that they are functions from trtith-values
Io truth.values, yet again that they are functions forming compound
sA:nteni-vs (oi propositions) from simpler ones. Even the names given
to the classical propositional calculus vary: one hears also of scoter).
liai calculus, while the German itris5agen4aMal means 'statement
cakulus'. and i t has even hten proposed to call i t a truth-value
eakultis. (
' ) The whol e ter mi nol ogy o f t r u t h
T -h i s
propositions
and truth-values as abstract objects, evert i f this is
c f ou nnc tas
i oultimately
ns
regarded
afacirn de parier. The sense of 'proposition
u modern
p p oone
w originating
s
un rs eis lthe
.f ip
qucaion
with Holzano's Salz an Jich
to reinvented
e
aby Frege
t
i as
n hisgGedankel propositions are objects
iand
nr
c
which
aare timelessly true or false. the archetypal truth-bearers, 'this
l Of
use
l 'proposition' differs from that found in medieval and ancient
authors,
as has frequently been pointed oui.() The truth or falsehood
s
f a (modern) proposition is conceived as independent o f which
of
expressions
are used to express it, who uses them. and under what
o
circumstances.
Propositions are thus different flout both sentences
r
Ithe
c labstract obyects investigated by linguists) and from statements
a r
(concrete
speech acts). But a sentence used in an assertion (making a
statement)
very often. perhaps with other factors, determines a
i f i
proposition,
and thereby a truth•value. It is not our intention here to
c a
tsettle
i the question of priority. i.e. whether a sentence or statement
gels
o na owl-value because it expresses a proposition, or whether
propositions
arise by abstraction from sentences or statements which
.
are themselves primarily to be teckoned true or false. As long as the
existence of propositions and truth-values is accepted. that is enough
for my present purposes.
One reason for the contli%ion between propositions. sentences and
statements is that for truth-tunctional logic the difterettee$ arc of little
moment, I f we consider 'eternal' sentences. which are true or false
208
t
SI S I MUNS
whoever utters them. in whatever (rionnal) conditions. etc., then such
sentences determine a proposition alone, and we are entitled to speak
of the ptoposition expressed by the sentence. GiVen such SenICDCCS S,
01special functor "that f o r m s a name of this proposition.(} From
propositions with names like •p• w e may gain a sentence which
expresses a true pmposiiion just in case p is true by predicating a
qxcial predicate o f it. namely * . . . i s the case'. This syntactic
switching from sentences to names o f propositions and back is
governed by two rUICS:
RI f o r all S! S ( t h a t S) is the ease
R2 f o r all p: p t h a t (,p is the case)
Such switching was not unknown in the young Frege. before he
arrived at the idea that sentences were a kind or name : i n his
Begriffstehrift (though not later) he treated the content stroke as we
have treated 'that' and the judgement stroke as we have treated 'is the
case'.(
between
the order of bracketing and the difference between the
4
identity
predicate
and a strong biconditkmal connective
) I
,h i es why
a n the
o t difference
h e r
reason
between sentence and proposition is
overlooked.
and especially so when sentences are taken as names of
s i
truth-values,
and identity replaces equivalence.
m i l
like 'and', 'or', 'if • then a r c r.rpre.isieinx forming
a Corinrctives
r i
compound
sentences. Grammatically they are functors of category
t y
Os.
Such functorial expressions must be distinguished from the
o
sentence-pi:now
in which they typically occur. of the form S and S .
f
the
variables
here
being placeeholder dummies showing where sentenR
ces
are
inserted
to
make the pattern with the aid of the word 'and'.
I
Functor
entiret%iuns arc sometimes called 'incomplete' oi
i
Frege)
*unsaturated'.
but this latter expression is better reserved for
n
the
sentence
pattern
the
connective serves to form, since the pattern,
i
hut
not
the
word,
is
incapable
or isolated or separate existence_ (
J
5
The
expression
of
sentential
negation
in English is the ksgicians"it
R
is
)2 not the case that T h i s has ihc desired category sh, but is itself a
complex
expression involving the name-forming functor 'that', the
,
special
predicate
Is the cam' and the usual English negation particle
m
'not% which modifies verbs and not sentences.
a
r We can use connectives and our special syntactic adaptors to form
k
e
d
o
n
l
y
b
1141:Tti-FUNCTIO4S
2
.
0
9
expressions similar to connectives. but allowing proposition names as
inputs andior outputs. The complex expression 'that t... a n d - 4 has
category nisi. while expressions having gaps for proposition names
ate i s the case and --- is the case, of category .T m i.e . a binary or
relational predicate of propositions, and the expression 'that (... is the
case and - - is the easel' forms a name of a proposition from the WOWS
of two propositions, and has category Out. I t is this last nominal
functor which may be said in express conjunction as a truth-function
of propositions. l r
The
p
4 result
a r of
e the functor is a name of a proposition, the conitmolon of
p
(which, being the propositional equivalent of a
p and
r oq.p Moe sfiincior
i
CtinnettiVe.
t i o n s we
. may call a jrarekor), expresses an operation, a function
from
to propositions. which we may call a jemetitm
w propositions
e
Conjunction
is
i
hus
one particular binary junction. Both junctions and
e
x
p
junctort
are
to
be
contrasted
with connectives.
r
e
s
s For proposltiOnS we may always ask I s truth-value?'
i t t r u Wh-questions
e
o
r are given answers which employ
iis its
concepis
f
a
l
I
call
clamillert
Among
classifiers are the concepts o f
t
height.
s
e
age.
r
direction.
,
number.(
o
r
h
e
o
a
' iculat
W
h of the
a general
t
properly wa nt by a classifier. c_g. 2m
r pail
e ratite
tall,
)b T 32
h years
e
aold.
n W
s w e r
ity
which
ointerests us is the
a cocieept expressed by 'the truth-value of-tw
which
h
.
is
s
abbreviated
i
x
"TV 4-- Cla s s ifie rs may be considered as
h
A
function-like,
in
that
they
e
t
c
.
q
n arguments and yield values. The values
t u e s t i o take
N
are
oi
w v properties.
In the case of the Pbelassificr ihcy are
g
e
p hypostatised
fsthe abstract
o
rtruth-values T and F. When it is said of a particular
junction
p r othatp (inoconstrast to others) it is a tretth-liviulion then what is
meant
s i ist that
i the
o truth-value of its outpui depends only or. better. al
most.
on
the
truth-values
of its inputs, and not on other factors. This
n s
is
t an instance
h of a general kind of relation holding among certain
elassficrs
which I call re gedurity( ) For the special case of conjunce
tion
c we
l have;
a
s p q sr
i
Y
fi ' V i
e
r
Since t r a t h (( p )
func tions
T
r wVas
such
junctions 1 alohology and Ctintrailieflon are also
h the
i monadic
c
(counted
as truth-functions, in the w ide sense familiar from matheniiiu hq
)t
a
r
A
he
P
- n
o
o
v t
ra
d e p
(l r
e n d
)u
e n t
T
e o
V
s n
210
P
.
M
.
SIMONS
tics, we cannot always happily speak o f
extent
that functionality is captured by the notion of regularity, the
terminology
e h - p o w /ofr truth-functions
are.
is justified. I f we represent the comH o wof ec l avs s ie r
pounding
of
t a n d os y m bol i c a l l y by t h e same sign as used to represent
fi
t r s ( hs composition,
e
flanetional
u c eh
then we can say that the regularity represena above
ted
s
is that of T V A (itself binary) on TV . In general, for any
junction
' t ha.. iteis a truth-function ill Tv" re g TV.
t neClitiSe
r u t however
h - we can indicate the effect of truth-functions with
truth-tables
v a l u orematrices, it is tempting to take them as functions from
truth-values
to truth-values. But, whik there ate indeed such simple
o
functions.
they
are not the same as the truth-functions we have
f
described,
which
are junctions. i.e. functions from propositions to
t
h
propositions.
The connection may be made clear again using the
e
example
Let K be the binary function on truth-values
c
o ofnconjunction.
j
to
truth-values
given
by
K(T . T ) = T. KtF. T) = F. W
u
n
c
t
K(F.F)
=
F.
Then
-i
o
n
.1
P4P-0) V ( p T W O /
7
) The=two functions.
F
. K and •s, are hmtiornorphically related under the
fkinction TV : K is the homomorphie transform of A in I he domain of
values o f TV . namely truth-values, This ckme relationship may
explain the tendency to lake truth-functions now as one, now L5 the
other, hot dues not excuse it. For the monadic truth-function o f
negation, with junctor a n d its homomorph N . the inversion
function between the truth-values. we have precisely a commutative
composition diagram
p
TV!
TV(p) •
•
-1.0)
I
Tv
T V ( - - ( p ) )
NI (TV (p)1
TRUTI CRI NCTI O NS
2
1
1
That we are entitled to take both truth-functions and their holmmorphs as functions becomes clear when wc consider which singlevalued relations w e c a n de fine a s equivalent t o the m, I f
- I", then clearly i s single-valued. and the same
goes for a triadic relation defined similarly in terms of K • On the other
hand it is possible also to consider the truth-function* to correspond to
cerlain relations among propositions, or again, among truth-values
.The latter alternative has been suggested by Dr Steen Olaf Welding.(') though provided we keep ckar as to the general priority of
relations over functions there seems to be no reason to accept
Dr Welding's criticisms o f the functional view o f truth-functions.
which GUI, as we have demonstrated
,cleared
b e ofdconfusion.
e f e nIndthee case
d of truth-functions
we
thepfunctional
e p de n rather
than the relational view first.
,p i have
tr soarrived
o v hi atad
s
and
perhaps
t
ti have
h
a
t found it more natural to proceed that way. But the
i
other
way is
s equally acceptable: for instance we can define a relation
bei ween propo%it ions k such that p k q iff both p and q wie trim. It also
has a homomorphic image under TV in truth-values. namely that
relation which holds only from I to 1 and not otherwise. Since the
Fregean concepts corresponding to these relations are just the functions A and K. Dr Welding's suggestion may be seenSS a way of doing
propositional logic without functions at all, reversing the Fn:gean
procedure. As such it is legitimate. but neither the only possihk
viewpoint. nor the most natural or easiest to use. As elsewhere in
mathematics. the use of functions makes things run smoothly. although it is theoretically dispensable.
University of Soichorg
P
e
t
e
r
M. ShioNs
NOTE'S
C
1I
Duckworth
2O
, 4?)
1M c f Am wition. A.R. I* BrodAr. NM. , &Naimoli% Vol. I . Prinizelon Princeton
1C
9
lqr.t.
, Appendix. where i 7 6
C In1 FRLot,
11 1. 1 2 1G.,
' li i e%t e d
Olml.
1 cu. i1964.
c u tCI.l 1124.
e t !
at f at w i t t n f o
q m
u h
a 'n e e t
.m s' o
A
o cO i r n" ,, k z
.r e
N
, A u f f i
.A t z e ,
d
P
. e
aR .
A
n
k. L
e
k
*. g
212
P.M. SIMONS
Cf. ell) . 1 .
1The term •Panstafilairer • comes from Grating, K t)pr e nhe i m , P . 4)er Get.
talthegriff
. n mam
a Lichle
t o t der neuen Logik•, EWA-Flamm I 119380, 211 I t. The espliontion
terms akitt euc
. I eI . . l ni is not thew,. they porter to we classifiers straightforwardly as
d .i. n
functions
t this iWPsWilet COMMitt us straight Away to a h e r ontoksgy of aNst met
s s .Sisz
,
atilects,
I
prefer
to leave open whether we can reduce such apparently ah-stracium de( i r
minding
like • Wing is the height cif this 7
o questmos
z e
1The sne
r of
, the flitaction-lakc notiust sit argument and value enables a unified treatment
(in terms of enualit4, and difference of their values) in be gisen„ Nu this
.et classifiers
/
nothang,
,tells
i ntis
t
h
s athemonteloitioti
p l e tpeionitics
n k te atszott
vistitidlce
H t'i fl
o
wl
I r ci ptiAti r with respect to the memadic classifier jr tilt kw a
i i camonadic
D there implicit)
idomain
p 41(hsvalues
i
1
1
6
7
.
s c h •••• %
Sly)
t
W
e
Ns rite e . p
e
SinceSthis it true w hene ver fi t croissant, or VII heriever g always takes different VAIlliet, it
does mat
r in our
i stew suffice to define the notion orfunctiontal deptokid nc r I n Grains,
K., . A/ k i ta 1 Ihcaiyaf depenikricc•• 1:thiiiiitia I R 3 9 : unpubinheil LvCCAA15( W a l
elDilditdOlin) Witll
we zliti reputiesty is termed •equideperkience•
i
e't Wit
e tni•46_, S.O., • Logic a% l i e d ors truth-value relations-. Rem( internartonafe
de Phiroboldlie
itt I( I 916P, 15I-11156
m
/
4
(
1
9
/
1
1
)
.
7
3
9
6
.