Editor’s note: The letter is in the Prior archive box 3 at the Bodleian Library in Oxford and has been transcribed and commented by David Jakobsen and Simon Graf.
Letter fro
m J.J.C. Smart to A.N. Prior, November 15, 1953
1
15.11.53
Dear Arthur
Before getting down to
brass tacks
one
quibble
!
2
(1) Surely Churchâs Î» conversion isnât the same as
combinatory logic. I know what the former is (though I donât understand it) because we have Churchâs
monograph in the library. Iâm still waiting for you to tell me what combinatory logic is! (2) When I ta
lked
about there being only two values for a truth operator of one argument that was a slip of the pen. What I
mean
t was that Sp had only 2
possible
values. (O
f course S(p,q,r) has only two
possible
values too, so I must
have been
muddled
.) Compare the dou
ble
use of âfunctionâ in maths
. Sometimes by âs
in
xâ we mean
âÎ»x.
sin
x
â but sometimes the y
such
that y=sin x. But I withdra
w quibble no2 as I certainly missed V and F,
and only mentioned S and N.
Half the trouble
is
is
Iâve forgotten what
[p.2]
I wrote i
n the letter! Yes
,
my
âMâ does seem to be only a special case of your âTâ. To revert to it
CpMq =
Df
NCpq
By putting p = False it
i
s obvious the def. leads to trouble.
Now what rule of de
finition have we broken here? It
ought
to be possible to formulate it. What
Hempel says on pp 18

19 of his âConcept Formation in Empirical Scienceâ
3
(reviewed by me in Phil. Review
for July â53) is relevant. He says âThe introduction of certain kinds of nominal definitions into a given
theoret
ical system is permissible only on condition that an appropriate nondefinitional sentence, which might
be called its
justificatory sentence
, has been previously established. Thus, e.g., in Hilbertâs axiomatization of
Euclidean geometry, the line segment de
termined by two points, P
1
and P
2
, is defined, in effect, as the class
of points between P
1
and P
2
in the
straight line through
[p. 3]
P
1
and P
2
. This definition evidently presupposes
that through any two points there exists exactly one straight line; and
it is permissible only because this
presupposition can be proved in Hilbertâs system and thus can function as justificatory sentence for the
definition! Hempel refers also to Peanoâs âLes
definitions mathÃ©matiquesâ in BibliothÃ©que du CongrÃ©s
Internationale
de Philosophie, Paris, III (1901) pp 279

88. (I doubt if weâve got this in the libr
ary, though I
havenât looked.) P
eano gives an example
4
which obviously leads to trouble
Ý
Ý
?
3
�
=
��
Ý
+
3
Ý
+
�
So
1
2
?
2
3
=
3
5
and
2
4
?
2
3
=
4
7
But
1
2
=
2
4
so
3
5
=
4
7
!
What has gone wrong seems to be this.
à¯«
+
3
à¯¬
+
�
is a function
of
4 argument
s
.
1
Editor
âs note: The letter
is in the Prior archive box 3 at the Bodleian Library in Oxford and
has been transcribed
and
commented
by David Jakobsen and Simon Graf.
2
Editorsâ note:
Smart has written âone or two quibbles!â but has crossed over âtwoâ.
3
Editorsâ note: Smart refers to Foundations of the Unity of Science. Volume II, no. 7:
Fundamentals of Concept
Formation in Empirical Science
by Carl G. Hempel; International Encyclopedi
a of Unified Science, I and II.
4
Editorsâ note: Smart
in this letter
shortens example
with ex
.
If
(
à¯«
4
?
à¯¬
3
)
is a function of 4 arguments then â

?
â
â is all one symbol. â
à¯«
à¯¬
â here has no meaning by itself and
so
[p. 4]
c
anât mean âx
ï¸
yâ. So Peanoâs
example does not seem to be a good one. The geometrical one does
seem more
instructive. The definition makes use of the expression â
the
straight line through P
1
and P
2
â which
clearly does presuppose the theorem âhere is one and only one straight line thro
ugh P
1
and P
2
â which is not
universally true in certain sorts of non

Eucl. geometry. But reverting to
CpMq =
Df
NCpq.
If p is false the Truth value of Cpq is independent of what q is. So it is easy to see why the definition of
CpMq =
Df
NCpq leads to
trou
ble in this case. To use Hempelâs terminology I suppose the âjustificationâ
theorem we wo
uld need is that Cpq and NCpr ca
n always be given the sam
e truth values by suitable choice
of q for given r. And this theorem is not forthcoming.
But when is a justif
ication theor
em
needed?
[5] I canât formulate any general rules for telling as this. Can you?
Thanks for the necessary existence thing. Did I ever send you my lecture on the existence of âGodâ? In this I
argue that âlogically necessary beingâ is self

con
tradicting lik
e âround squareâ simply because âthere exists a
yâ can never be a
truth
o
f logic.
Your sentence
âF
or what cannot be thought of as attaching
to a subject at all
cannot be thought of as attaching necessarily to a subjectâ
5
s
eems to me to miss
the point. For clearly âexistsâ
can be predicated of God, unicorns, lions, etc. (Even though there is a sense in which âit isnât a predicateâ!)
Now for the objection, Iâ
m not sure
that it will convince you
for you have indeed made out a
trickily
plausible
case! But Iâd say that âbeing red

all
over and being not

red all overâ only accidently, so to speak,
entails non

exemplification.
What it primari
ly entails is [6]
âcanât be exemplifiedâ. But since âcanât beâ entails isnât
it als
o entails isnât
exemplified.
To
prove
exemplification wonât do, I hav
e a hunch, because âcanâ doesnâ
t entail âisâ in the way
âcanât beâ entails âisnâtâ.
The question is: Does âis red all over and not

red all overâ entail âcanât be exemplifiedâ? Or does âcanât be
exemplifiedâ simply m
ean â

entails isnât exemplifiedâ?
But the general point remains
.
L
ogic canât tell you what is and what isnât but it can tell you what can be
and canât be. But since âcanâtâ entails âisnâtâ it can sometimes tell you what isnât. But it canât tell you what
âisâ.
Yours
Jack
PS. Here is a pleasant puzzle I got from one of the mathematicians here. An aviator flying over darkest
Africa is told that if he has to
6
(â¦)
5
Editor
s
â
note:
The
quotation
can be found near
verbatim
in
Prior, A.N.,
Is Necessary Existence Possible?
P
hilosophy
and Pheno
menological Research Vol
15, No. 4
(
1955):
â
For what is not rightly thought of as attaching to (being
predicable of) a subject at all, cannot be thought of as attaching to a subject necessarily (or, of course, contingently).
â
6
Editorsâ note: The scanned document ends here
and the rest of the PS is left out
.