The Nachlass of A.N. Prior
a joint project between
Department of Information Studies - University of Copenhagen
and
Department of Communication and Psychology - Aalborg University

Many-valued Logics

By Arthur N. Prior on 02/05/1957

This text has been transcribed by Jørgen Albretsen, University of Copenhagen, [removed]

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-
item?record_id=168375 (starts at 15:17)

{717}

May 2 1957

THE LISTENER

Many
-
valued Logics

The last

of three talks on 'The Logic Game' by A.N. PRIOR

1

2

Of this alone is Deity bereft,

To make undone whatever hath been done.

So wrote the Greek poet Agathon, and his words were quoted with approval by Aristotle in his
Ethics
. I
had something to say in my l
ast talk
3

about the analogies that may be drawn between the logic of
necessity and possibility on the one hand and the logic of past, present and future on the other; but what
these words of the poet point to is not just an analogy but a positive connectio
n between these two
things. What is necessary

par excellence
, in the sense of being beyond any possibility of alteration, is
the past, 'whatever hath been done'. It is only with respect to the future that different possibilities are
really open; with respe
ct to the past we can say that this or that is 'possible' only in the sense that we do
not know whether it was this or that that happened.

No one has yet produced a rigorous and satisfactory formal presenta
tion of these connections between
time and necess
ity; but the person who has come nearest to it is the late Professor Lukasiewicz,
4

and to
have made a start on this is perhaps his most far
-
reaching achievement as a logician. What he put
forward with this sort of problem in view was what he called a three
-
valued logic, that is a logic with
three truth
-
values. This word 'truth
-
value' sounds alarmingly mathematical, but if is just a technical
phrase designed to close up a rather obvious gap in the English language. When we ask whether
something is heavy or l
ight, we would be said to be asking about the weight of the object. Again, when
we ask whether something is cheap or expensive, we would be said to be asking about its cost. Then



1

Editors


Note: This text has been transcribed and edited by Jørgen Albretsen. The text is kept in the Bodleian Library,
Oxford, Box 5. The page numbers in the original text have been put in curly brackets. In the Virtual Lab for Prior Studies,
http://research.prior.aau.dk

the text is stored under VL id 74.

2

Editors’ Note: With respect to the audio recording: Max Cresswell has with the kind help of Nga Taonga Sound & Vision
-

The New Zealand Archive of Film, Television a
nd Sound, localised what very probably seems to be th
e only existing
recording of Arthur Norman Prior
's voice.

PLEASE NOTE: only part of the recording is preserved. See footnote 5.

http://www.ngataonga.or
g.nz/collections/catalogue/catalogue
-
item?record_id=168375 (starts at 15:17)

3

In the original text, a footnote marked with a * at this point reads: “Printed in The Listener of April 25”. It refers to the

talk
Symbolism and Analogy: The second of three tal
ks on ‘The Logic Game’ by A.N. Prior
, here found in VL

id
73. VL

id
72
contains
The Necessary and the Possible: The first of three talks on ‘The Logic Game’ by A.N. Prior
, The Listener, April 18
1957.

4

Throughout the text, this spelling is used, not the
correct: Łukasiewicz (starting with the 'L with stroke').

again, when we ask whether a man is tall or short, we would be said to be ask
ing about his height. Now
suppose we ask whether a statement is false or true


what are we asking about then? We have no
ordinary word for it, so the logicians quite sensibly have coined one


they say we are asking about the
statement's 'truth
-
value'. Th
e truth
-
value of a statement is its truth or falsehood; at least, we ordinarily
assume that those are the oniy two possibilities. Once you know this, it is obvious what a three
-
valued
logic is; it is a logic in which allowance is made for statements being
neither true nor false but some
third thing of the same sort. And Lukasiewicz' view, when he first elaborated this system, seems to
have been that statements about what has already come to pass, and what is bound to come to pass, are
either true or false;
but if is some still
-
future matter the outcome of which depends on chance or free
choice, and the issue is not yet settled, then there are not as yet any facts one way or the other with
which statements about this matter could agree or disagree, so we may
say that such statements are
neither true nor false but neuter, to give the third truth
-

value a name.

To explain one of the main results which Lukasiewicz achieve by means of these neuter statements I
must introduce another {718} technical term which is n
ot as alarming as it sounds, the term truth
-
function. In ordinary two
-
valucd logic we would say that a statement of the form 'p and q' is a truth
-
function of its two parts, meaning by this that whether the statement as a whole is true or false depends
enti
rely on whether its parts are true or false. For example, if the separate statements 'grass is green' and
'the sky is blue' are both of them true, then the compound statement 'grass is green and the sky is blue'
is automatically true too; and if either of
the two separate parts is false, the compound is automatically
false. We could set this out if we wanted to in a sort of multiplication table


in fact it will be exactly
like a multiplication table for the numbers 1 and 0. A truth and a truth (that is a t
ruth joined to a truth
by 'and'), is a truth, just as 1 X 1 = 1; a truth and a falsehood is a falsehood, just as 1 X 0 = 0; a
falsehood and a truth is a falsehood, just as 0 X 1 =0; and a falsehood and a falsehood is a falsehood,
just as 0 X 0 = 0. Again,
whether the statement 'grass is not green' is true or false depends entirely on
whether the simple 'grass is green' is true or false. If 'grass is green' is true, 'grass is not green' is
automatically false, and if 'grass is green' is false, 'grass is not
green' is automatically true. So here we
have another truth
-
function, and once again we could express its properties by means of a table, this
time like the table for 1
-
x where x is 0 or 1. 'Not' applied to a truth gives a falsehood, just as 1
-
1=0, and
'no
t' applied to a falsehood gives a truth, just as 1
-
0=1.

Even the chief of all the logician's special words, the word 'if', has a sense in which it is truth
-
functional, though it is not often used in this sense in common speech. You can see what an advantag
e
it is to be dealing just with truth
-
functions. Suppose you have a long formula, with lots of p's and q's in
it that can stand for any statements at all, and you want to know whether this formula expresses a
logical law, that is, you want to know whether
it is true regardless of what the p's and q's stand for. If
there is nothing in it but truth
-
functions, all you need to do is consider all the possible ways in which
truth and falsehood can be distributed among your p's and q's, and work out by your tables

what the
whole thing comes to in each case; if it always comes out true, what you have is a law, and if it does
not, it is not.

Expression of a Law

Let us take a simple example


the formula 'not both
-
p and not
-
p'. I have not given you a symbol for
'both'

, it is actually K, so this formula works out as NKpNp. Suppose, first, that p is true. Then 'not p',
by the table for 'not' will be false, and 'both p and not p' will combine a true and a false statement; and
this, by the table for 'and', will be false.
And since 'both p and not p' is false, 'not both p and not p' is
true, by the table for 'not'. All that is on the supposition that p is true; but if you suppose p false the
tables will give you the same result by a different route; so the thing is true eit
her way


that is, this
formula NKpNp expresses a logical law.

This method of verifying and falsifying formulae by calculation on the basis of truth
-
value tables goes
back to C.S. Pierce, in a paper of 1885, and it is now, almost everywhere, one of the fir
st things that
students of logic learn. It came to this country, I think, through Wittgenstein, who started this
unfortunate practice of describing any
thing verifiable by truth
-
value calculation as a 'tautology'.

There seem to be branches of logic in whic
h truth
-
value calculation is not enough, and the logic of
necessity and possibility is one of them. For it would seem that whether 'necessarily p' and 'possibly p'
are true or false, does not depend solely on whether the simple p is true or false. It does,

of course,
depend on it up to a point; if the plain p is true, we know that 'possibly p' is also true, and if the plain p
is false, we know that 'necessarily p' is also false. But if p is true, that is not enough to decide whether
'necessarily p' is true


some truths, we would ordinarily say, are necessary truths while others are
merely contingent, that is, although they happen to be true they might very well not have been. And if p
is false, that is not enough to decide whether 'possibly p' is false


so
me falsehoods, certainly, are also
impossibilities but not all of them.

What Lukasiewicz argued was that it only seems impossible to represent the forms 'necessary p' and
'possibly p' as truth
-
functions because we assume that the only truth
-
values are trut
h and falsehood; if
we admit neuter statements, it is a different story. Whether the forms 'necessarily p' and 'possibly p' are
true or false is indeed not fully determined by whether the plain p is true or false; but whether
'necessarily p' and 'possibly
p' are true, false or neuter is fully determined by whether the plain p is true,
false or neuter.
5

It works like this: 'necessarily p' is automatically true if the plain p true, and false if p is
false or neuter; and 'possibly p' is true if p is true or ne
uter, and false if p is false. This does not sound
right; but it is not so bad if you remember the relation between necessity and time with which I started,
and then remember what the three truth
-
values are. A statement is true only if it is a correct acco
unt
either of something which has already come to pass, and so cannot any longer not have come to pass,
or of something which has yet to come to pass but is already bound to do so. So only what is necessary
for one reason or the other is really true either

because it has already happened and cannot be altered or
because it is bound to happen. And a statement is false only if it is an incorrect account either of
something which has already come to pass, and therefore cannot now be as the statement said, or o
f
something which has yet to come to pass but is bound to do so other
wise than as the statement says. So
only what is impossible for one reason or the other is really false.

The True and the Necessary

This does not mean that whatever is possible is true,
for although the possible comprises nothing that is
false, it does comprise, beside what is true, what is not as yet either false or true. The fact that the true



5

[Start of sound recording]
. Prior makes several additions and alterations wh
en reading the article text, to explain details
further and surely to make it fit better to a talk over the radio.

and the necessary are co
-
extensive does not mean that the man who says 'My horse is bound to w
in' is
never worse off than the man who says simply 'My horse will win'; for if the possibility of his horse's
winning and the possibility of its not winning arc both still open, the man who says 'My horse is bound
to win' has said something definitely fal
se, while the man who just says 'My horse will win' has indeed
said something which is not true, but it is not false either.

When looked at in this way, this account of the necessary and the possible is attractive; the real
difficulties of three
-
valued log
ic arise when we consider words like 'and'. On ordinary two
-
valued
assump
tions, no logical word is more obviously and flatly truth
-
functional than 'and' but it is difficult to
preserve this truth
-
functional character of 'and' in a three
-
valued logic; that

is, it is difficult to maintain
that whether the compound 'p and q' is true, false or neuter depends solely on whether its two parts are
true, false or neuter. For when the two parts are both of them neuter, we want to say in some cases that
the combinati
on of them is neuter also, and in other cases definitely false. Suppose the two parts are
'John will wear a blue tie' and 'James will wear a red handkerchief' and both James and John have still
to make up their minds on these matters; we are in this case h
appy enough about saying that the
compound statement 'John will wear a blue tie
and

James will wear a red handkerchief ' is neuter in
truth
-
value just as its parts are. But suppose the two parts are 'John will wear a blue tie' and 'John will
not wear a blu
e tie'. If John has not yet decided whether to wear a blue tie or not, we again have a pair
of neuter propositions, but the combination 'John both will and will not wear a blue tie' doesn't seem at
all neuter


most of us would want to rule that out as pla
in false, and we would have no qualms in this
case about equating the false with the impossible.

The Three
-
valued 'And'

Lukasiewicz took this dilemma squarely by the horns, and ruled that the three
-
valued 'and' is truth
-
functional, and that the result of c
om
bining two neuter statements by means of 'and' is always another
neuter statement, even if it happens to be of the form 'p and not
-
p'. In this way he did give a clear and
rigorous formulation to a logical system allowing for a third truth
-
value; but one

cannot help feeling
that he only did it by putting at least one poor word to something like forced labour, and I do not think
we have yet heard the end of this story.

Lukasiewicz went on from the study of three
-
valued logic to the elaboration of similar s
ystems
involving not merely one but an indefinite number of truth
-
values in between the usual two; though in
the closing years of his life he was for some reason strongly attracted to the view that the number of
truth
-
values is precisely four. This last de
velopment is not one in which I am personally able to follow
him;
6

my own view is that the future of the subject lies in the direction of the system with an infinity of
values. It also seems to me that the notion of a truth
-
value needs to be broadened. Wit
h some sorts of
logical calculation, what is important about a statement is not its simple truth or falsity, but when it is
true, or in what possible circumstances it would be true, and it is sometimes useful to describe as a
'truth
-
value' the property of
being true until five past six last night and never true {719} thereafter. In
this area, as in others, the logician needs to have a nose for analogies; and what is important to him is
the set of alternative possibilities we must reckon with when performing

a given sort of calculation





6

One major omission by Prior in his radio talk, though, at this point, where the part of the text about precisely four truth
values is left
out.

that is what decides whether a system is two
-
valued or many
-
valued, in the most recent use of these
terms.

I will finish with a little piece of history. As long ago as 1885, in the paper in which the idea of
calculating with
truth
-
values was first sketched, C.S. Peirce wrote this: 'According to ordinary logic, a
proposition is either true or false, and no further distinction is recognised. This is the descriptive
conception, as the geometers say; the metric conception would be

that every proposition is more or less
false, and that the question is one of amount'. Truth, on such a view, is zero falsehood, and it might
have been better to talk about 'falsehood
-
values' instead of 'truth
-
values'. But Peirce only threw this out
in pa
ssing as a possible line of development; it was left to Lukasiewicz and his school to make
something of it.

-
Third Programme



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