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 âcounter

exampleâ 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 âof
exactly
the same formâ because a form of
inference which is not itself valid may have special cases
â
sub

forms
â
which
are
valid
,
+
2
the
given inference may exemplify one of these. For example âNo A is a B, therefore no B is a Câ is an
invalid form of inference (âNo horse is a dog, therefore no dog is an animalâ, which is of this form,
would take us from a truth to a falsehood);
bu
t this does
not
mean that âNo hor
se is a dog, therefore
no dog is a horseâ 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 âNo A is a B, therefore no B is an Aâ,
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
â
âdecision procedure
sâ, as they are called
â
are
available; but in others they are not. What is often done is to take a few forms of inference as
âobviouslyâ 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 âIt is not the case the case that both P and Qâ 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
Editorâs note: Prior uses a â+â for âandâ.
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
âdeductive systemsâ 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 âNo X is a Y, therefore no X is a Zâ. But why not say it exemplifies the valid sub

form
of this: âNo X is an animal, therefore no X is a dogâ? We can say this if we are prepared to c
ount
âanimalâ and âdogâ as purely formal or logical words (words like ânoâ and âisâ which are to be
left
in
when we give the form of an inference). {page 11} All logicians would in fact say that âanimalâ
and âdogâ 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 âAll Zâs are Yâs, and no X is Y, therefore no X is a Zâ, 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
â
which words are âformalâ or âlogicalâ 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. âNoâ and âisâ, are undoubtedly formal, and certain others, e.g.
âdogâ and âstoneâ
,
are undoubtedly not so. But there are borderline cases about which there is no
such agreement, e.g, in the
inference
âI am sitting down, therefore it will always be the case that I have been sitting downâ, are âwillâ
and âhas beenâ, i.e. words indicating
tense
3
{page 12} part of the logical form or not? Some
logicians say Yes, some No.
3
Editorâs 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 âAll Xs are Ysâ, âEvery X is a Yâ, âAny X
is a Yâ, âWhatever is an X is a Yâ. Are these forms to count as log
ical forms or the same? Is, for
example, âAll dogs are animals
â
of the same form as âWhatever is a square is a quadrilateralâ, 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 âAll Xs are Ysâ
and the other a premiss of the form âWhatever is an X is a Yâ 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 âAll
dogs are animalsâ and âWhatever is a squar
e is a quadrilateralâ, are both of the form âEvery X is a
Yâ. {page 14}
# Here, too, there are controversial and borderline cases. What about âOnly animals are dogsâ?
â
is
this just another way of saying âAll dogs are animalsâ, or does it exemplify a spec
ial form âOnly Ys
are Xsâ? 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 âAll Xs are Ysâ (and yet
others might define âAll Xs are Ys
â
as m
eaning the same as âOnly Ys are Xsâ). 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 âEvery X is a Yâ is a
universal
affirmative
proposition, {page 15} and if they have the
form âOnly Xs are Ysâ 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 âEvery dog is an animalâ we may validly infer âOnly animals
are dogsâ. Secondly, they may use symbolic abbreviation for their standard forms, e.g. they may
write âSome Xs are Ysâ as X:Y, and say that âX:Y,
ï
Y:Xâ, 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
â
s standard forms, if any, a particular argument in
ordinary language really fits. For example, âSome mammals eat vegetables, therefore some
vegetables eat mammalsâ might be thought to fit the form âSome Xs are Ys, therefore some Ys are
Xsâ, {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 âSome
mammals are eaters of vegetablesâ and âSome vegetables are eaters

of

mammalsâ, giving
the
inference the form âSome Xs are Ys, therefore some Zs are Wsâ, 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 âSome Xs are
Ys of Zs
, therefore some
Zs are Ys of Xsâ, but this f
orm isnât valid either (as the example shows),
and isnât a sub

variety of the valid form âSome Xs are Ys, therefore some Ys are Xsâ.