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 Church’s λ conversion isn’t the same as
combinatory logic. I know what the former is (though I don’t understand it) because we have Church’s
monograph in the library. I’m 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 ‘function’ in maths
. Sometimes by ‘s
in

x’ we mean
‘λx.

sin

x
’ 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
I’ve forgotten what

[p.2]

I wrote i
n the letter! Yes
,

my
‘M’ does seem to be only a special case of your ‘T’. 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 ‘Concept Formation in Empirical Science’
3

(reviewed by me in Phil. Review
for July ‘53) is relevant. He says ‘The 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 Hilbert’s 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 Hilbert’s system and thus can function as justificatory sentence for the
definition! Hempel refers also to Peano’s ‘Les
definitions mathématiques’ in Bibliothéque du Congrés
Internationale

de Philosophie, Paris, III (1901) pp 279
-
88. (I doubt if we’ve got this in the libr
ary, though I
haven’t looked.) P
eano gives an example
4

which obviously leads to trouble


ݔ
ݕ

?

3
�
=

��

ݔ
+
3
ݕ
+
�

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.

௫
+
3
௬
+
�


is a function

of

4 argument
s
.




1

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.

2

Editors’ note:
Smart has written ‘one or two quibbles!’ but has crossed over ‘two’.

3

Editors’ 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

Editors’ note: Smart

in this letter

shortens example

with ex
.

If
(
௫
4

?

௬
3
)

is a function of 4 arguments then ‘
-

?
–
‘ is all one symbol. ‘
௫
௬
’ here has no meaning by itself and
so
[p. 4]

c
an’t mean ‘x

y’. So Peano’s

example does not seem to be a good one. The geometrical one does
seem more
instructive. The definition makes use of the expression ‘
the

straight line through P
1

and P
2
’ which
clearly does presuppose the theorem ‘here is one and only one straight line thro
ugh P
1

and P
2
’ 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 Hempel’s terminology I suppose the ‘justification’
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 can’t 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 ‘God’? In this I
argue that ‘logically necessary being’ is self
-
con
tradicting lik
e ‘round square’ simply because ‘there exists a
y’ can never be a
truth

o
f logic.

Your sentence
‘F
or what cannot be thought of as attaching
to a subject at all
cannot be thought of as attaching necessarily to a subject’
5

s
eems to me to miss
the point. For clearly ‘exists’
can be predicated of God, unicorns, lions, etc. (Even though there is a sense in which ‘it isn’t a predicate’!)

Now for the objection, I’
m not sure
that it will convince you
for you have indeed made out a
trickily
plausible
case! But I’d say that ‘being red
-
all

over and being not
-
red all over’ only accidently, so to speak,
entails non
-
exemplification.

What it primari
ly entails is [6]
‘can’t be exemplified’. But since ‘can’t be’ entails isn’t
it als
o entails isn’t
exemplified.

To

prove

exemplification won’t do, I hav
e a hunch, because ‘can’ doesn’
t entail ‘is’ in the way
‘can’t be’ entails ‘isn’t’.

The question is: Does ‘is red all over and not
-
red all over’ entail ‘can’t be exemplified’? Or does ‘can’t be
exemplified’ simply m
ean ‘
-

entails isn’t exemplified’?

But the general point remains
.

L
ogic can’t tell you what is and what isn’t but it can tell you what can be
and can’t be. But since ‘can’t’ entails ‘isn’t’ it can sometimes tell you what isn’t. But it can’t tell you what
‘is’.





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

(…)




5

Editor
s
’

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):

“
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).
”

6

Editors’ note: The scanned document ends here

and the rest of the PS is left out
.