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The Methods of Logic

By Arthur N. Prior on NA/NA/1950

This text has been transcribed by David Jakobsen, Aalborg University, [removed] and Martin Prior

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II.
The Methods of Logic
1

A.N. Prior


{page 7} To show that a given inference is
not

valid,

we need only find a “counter
-
example” to
the form of inference which

it exemplifies, i.e. an inference of exactly the same form which takes us
from truth to falsehood. It is necessary to say “of
exactly

the same form” because a form of
inference which is not itself valid may have special cases


sub
-
forms


which
are
valid
,
+
2

the
given inference may exemplify one of these. For example “No A is a B, therefore no B is a C” is an
invalid form of inference (“No horse is a dog, therefore no dog is an animal”, which is of this form,
would take us from a truth to a falsehood);

bu
t this does
not
mean that “No hor
se is a dog, therefore
no dog is a horse” is an invalid inference, for the special sub
-
variety of the given form in which C is
the same as A, i.e. the form “No A is a B, therefore no B is an A”,
is

valid.


To show that a given inference or form of inference
is

valid is not in general so straightforward.
For a number of limited but {page 8} important branches of logic, of which we shall give an
example later, m
echanical tes
ts of validity


“decision procedure
s”, as they are called


are
available; but in others they are not. What is often done is to take a few forms of inference as
“obviously” valid
+

to show that since these are valid, so are numerous other and more complicated
forms in which the given forms
are combined or transformed in various ways. An example may be
given from the Stoic logicians. They assumed that we can safely take it for granted that, where P
and Q are any statements, the following forms of inference are valid:
-



I.

If P then Q, but i
t is not the case that Q, so it is not the case that P.


II.

It is not the case that both P and Q, but it is the case that P,
so it is not the case that Q.


Now anything of the form



III.

If both P and Q

then R, but it is not the case that R, so it is not

the case that both P and
Q


would be justified by I, of which it is {page 9} a special case. So if we have these premisses


A.

If both P and Q then R


B.

It is not the case that R


C.

It is the case that P


we

can first infer “It is not the case the case that both P and Q” from A and B by III, and then
combine this conclusion with C to obtain


D.

It is not the case that Q





1

The
text is kept

in the Prior collection at Bodleian Library, Oxford. It has been edited by Martin Prior and
David
Jakobsen.

2

Editor’s note: Prior uses a ’+’ for ’and’.

by II. So we may use the validity of I and II to establish the validity of


IV.

If both P

and Q then R, but it is not the case that R, though it is the case that P, so it is
not the case that Q.


In this way

logical principles are themselves used to build up the principles of logic into large
“deductive systems” rather like geometry or algebra.


Both in proving
+

disproving validity, it is obviously important to know what is the form of a
given inference, and

when one inference {page 10}

is of the same form as another. This task is
beset with two kinds of difficulty, one theoretical and one practical. In the first place, we have to
decide which features or parts of inferences,
+

of the statements or propositio
ns o
f which inf
erences
are made up, contribute

to their form,
+

which features or parts are non
-
formal
+

so capable of
being replaced by schematic letters. Consider
,

for example, the inference



No stone is an animal


Therefore no stone is a dog.


Is this
valid or not? It is not, if it is of exactly the same form as




No tree is a man


Therefore no tree is a plant


(which takes
us from a truth to a falsehood);

that is, if the most we can say about its form is that it is
of the form “No X is a Y, therefore no X is a Z”. But why not say it exemplifies the valid sub
-
form
of this: “No X is an animal, therefore no X is a dog”? We can say this if we are prepared to c
ount
“animal” and “dog” as purely formal or logical words (words like “no” and “is” which are to be
left
in

when we give the form of an inference). {page 11} All logicians would in fact say that “animal”
and “dog” are
not

formal words, and that the given i
nference is not valid as it stands, but looks as if
it might be because we all know that all dogs are animals, and if we state this explicitly the full
inference has the form “All Z’s are Y’s, and no X is Y, therefore no X is a Z”, which
is

valid. But
how
do we decide, in giving the form of an inference, which words to leave alone and which to
replace by letters


which words are “formal” or “logical” and which words are not? This is in fact
one of the most difficult and controverted questions in the philos
ophy of logic.
There is general
agreement that certain words, e.g. “No” and “is”, are undoubtedly formal, and certain others, e.g.
“dog” and “stone”
,

are undoubtedly not so. But there are borderline cases about which there is no
such agreement, e.g, in the

inference


“I am sitting down, therefore it will always be the case that I have been sitting down”, are “will”
and “has been”, i.e. words indicating
tense
3

{page 12} part of the logical form or not? Some
logicians say Yes, some No.




3

Editor’s note: If this text is from 1950, then this is perhaps the earliest discussion of the formal status of tense by Prior
.
It is perhaps more likely that the text, for the same reason, is later than 1950.


{page

13} Secondly, ordinary speech provides us with many alternative ways of exhibiting the
logical features of a statement or inference, e.g. we have “All Xs are Ys”, “Every X is a Y”, “Any X
is a Y”, “Whatever is an X is a Y”. Are these forms to count as log
ical forms or the same? Is, for
example, “All dogs are animals


of the same form as “Whatever is a square is a quadrilateral”, or
are they of different forms? Most logicians would agree that these are differences that can be
ignored, and that inferences wh
ich differ only in one having a premiss of the form “All Xs are Ys”
and the other a premiss of the form “Whatever is an X is a Y” are to count as having the same form.
And to save tediousness, and to develop their principles in a smoot
h

and simple way, mos
t logicians
operate with a limited selection of forms, and leave it as a special exercise to figure out which of
these forms best fit
s

a given statement in ordinary language. They might say, for example, that “All
dogs are animals” and “Whatever is a squar
e is a quadrilateral”, are both of the form “Every X is a
Y”. {page 14}

# Here, too, there are controversial and borderline cases. What about “Only animals are dogs”?


is
this just another way of saying “All dogs are animals”, or does it exemplify a spec
ial form “Only Ys
are Xs”? Some logicians find it worth while to introduce such a special form, and some do not; and
some, again, will introduce it, but will
define
it as meaning the same as “All Xs are Ys” (and yet
others might define “All Xs are Ys


as m
eaning the same as “Only Ys are Xs”). But whatever
decision particular logicians make about particular cases, they all inevitably work with a limited
number of forms, and makes
generalis
ations ab
out these forms in two main ways. In the first place,
they gi
ve their standard forms technical names; for example they may say that any proposition of
the form “Every X is a Y” is a
universal

affirmative
proposition, {page 15} and if they have the
form “Only Xs are Ys” they may call a proposition of this form an
exc
lusive

proposition
.
They can
then use these technical terms to formulate rules of inference, for example that from a universal
affirmative proposition we may validly infer the exclusive proposition in which the same terms are
used but in reverse order, e.g
. from “Every dog is an animal” we may validly infer “Only animals
are dogs”. Secondly, they may use symbolic abbreviation for their standard forms, e.g. they may
write “Some Xs are Ys” as X:Y, and say that “X:Y,


Y:X”, is a valid form of inference. By th
is
latter device, extremely complicated form
s of inference

can be set out in a compact way.


Advanced logical work is quite impossible without these schematic devices, but it is often a
tricky matter t
o decide
which of the logician

s standard forms, if any, a particular argument in
ordinary language really fits. For example, “Some mammals eat vegetables, therefore some
vegetables eat mammals” might be thought to fit the form “Some Xs are Ys, therefore some Ys are
Xs”, {page 16} and t
o be a counter
-
example to that form. But if we want to express the premiss and
conclusion of this inference in a standard way, the proper rendering of them would be “Some
mammals are eaters of vegetables” and “Some vegetables are eaters
-
of
-
mammals”, giving

the
inference the form “Some Xs are Ys, therefore some Zs are Ws”, which is not valid at all. Or if we
make use of a richer set of forms, we might regard the inference as being of

the form “Some Xs are
Ys of Zs
, therefore some
Zs are Ys of Xs”, but this f
orm isn’t valid either (as the example shows),
and isn’t a sub
-
variety of the valid form “Some Xs are Ys, therefore some Ys are Xs”.


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