The relations between
*
v
' and
*
or
'
(or
*
either ... or
. . .*) are, on the whole, less intimate than those between
* . ?
and * and *, but less distant than those between
* D
' and 'if.
Let us speak of a statement made by coupling two clauses by
*
or
'
as an alternative statement
; and let us speak of the first
and second aUernatesof such a statement, on analogywith our talk
of the antecedent and consequent of a hypothetical statement.
At a bus-stop, someone might say :
*
Either we catch this bus
or we shall have to walk all the way home '. He might equally
well have said
'
If we don't catch this bus, we shall have to walk
all the way home *. It will be seen that the antecedent of the
hypothetical statement he might have made is the negation of
the first alternate of the alternative statement he did make.
Obviously, we should not regard our catching the bus as a
sufficient condition of the
'
truth
'
of either statement ;
if it
turns out that the bus we caught was not the last one, we
should say that the man who had made the statement had been
wrong. The truth of one of the alternates is no more a sufficient
condition of the truth of the alternative statement than the
falsity of the antecedent is a sufficient condition of the truth of
the hypothetical statement. And since 'p"Dpyq' (and,
equally,
*
q"3p v q ') is a law of the truth-functional system,
this fact sufficiently shows a difference between at least one
standard use of
*
or
' and the meaning given to
*
v '. Now in
all, or almost all, the cases where we are prepared to say something of the form
'
p or q ', we are also prepared to say something
of the form 4
if not-p, then q \ And this fact may us to
exaggerate the difference between
4
v
'
and
*
or \ to think that,
since in some cases, the fulfilment of one alternate Is not a
sufficient condition of the truth of the alternative statement of
which It is an alternate, the fulfilment of one alternate is
a sufficient condition of the truth of an alternative statement.
And this is certainly an exaggeration. If someone says ;
4 Either
it was John or it was Robert but I couldn't tell which
*
,
we are
satisfied of the truth of the alternative statement if either of the
alternates turns out to be true; and we say that the speaker
was wrong only if neither turns out to be true. Here we seem
to have a puzzle ; for we seem to be saying that
*
Either it was
John or it was Robert
'
entails
4
If it wasn't John, it was Robert
*
and, at the same time, that
*
It was John
'
entails the former,
but not the latter. What we are suffering from here is perhaps
a crudity in our notion of entailraent, a difficulty In applying
this too undifferentiated concept
to the facts of speech ; or, if we
prefer it, an ambiguity in the notion of a sufficient condition.
The statement that It was John entails the statement that It was
either John or Robert in the sense thai it confirms
it ; when It
turns out to have been John, the man who said that either It
was John or it was Robert is shown to have been right. But the
first statement does not entail the second in the sense that the
step
*
It was John, so It was either John or Robert
*
is a logically
proper step (unless the person saying this means by It simply
that the alternative statement made previously was correct,
i.e., 'it was one of the two '). For the alternative statement
carries the implication of the speaker's uncertainty
as to which
of the two it was, and this implication
is inconsistent with the
assertion that it was John. So in this sense of
*
sufficient condition ', the statement that it was John is no more a sufficient
condition of (no more entails) the statement that it was either
John or Robert than it is a sufficient condition of (entails) the
statement that if it wasn't John, it was Robert. The further
resemblance, which we have already noticed, between the alternative statement and the hypothetical statement, is that whatever knowledge or experience
renders it reasonable to assert the
alternative statement, also renders it reasonable to make the
statement that (under the condition that it wasn't John) it was
Robert. But we are less happy about saying
that the hypo-
2 TRUTH-FUNCTIONS [CH. 3
thetical statement is confirmed by the discovery that it was John,
than we are about saying that the alternative statement is
confirmed by this discovery. For we are inclined to say
that
the question of confirmation of the hypothetical statement (as
opposed to the question of its reasonableness or acceptability)
arises only if the condition (that
it wasn't John) turns out
to be fulfilled. This shows an asymmetry,
as regards confirmation, though not as regards acceptability, between
4
if not p,
then q
'
and *
if not qy then p
' which is not mirrored in the forms
*
either p or q
' and
*
either q or p '. This asymmetry is ignored
in the rule that
*
if not p, then q
' and
*
if not q,
then p ', are
logically equivalent/
for this rule regards acceptability rather
than confirmation. And rightly. For we may often discuss
the l truth
'
of a subjunctive conditional, where the possibility of
confirmation is suggested by the form of words employed to be
not envisaged.
It is a not unrelated difference between
*
if
'
sentences and
4
or
'
sentences that whereas, whenever we use one of the latter,
we should also be prepared
to use one of the former, the converse
does not hold. The cases in which it does not generally hold are
those of subjunctive conditionals. There is no
*
or
'
sentence
which would serve as a paraphrase
of
4 If the Germans had
invaded England in 1940, they would have won the war \ as
this sentence would most commonly be used. And this is connected with the fact that
c either . . . or . . .' is associated
with situations involving
choice or decision.
4 Either of these
roads leads to Oxford
'
does not mean the same as
'
Either this
road leads to Oxford or that road does
*
;
but both confront us
with the necessity of making a choice.
This brings us to a feature of
*
or
'
which, unlike those so far
discussed, is commonly mentioned in discussion of its relation
to
*
v
'
; the fact, namely, that in certain verbal contexts,
4
either . . . or . . .' plainly carries the implication
* and not
both . . . and . . .', whereas in other contexts, it does not.
These are sometimes spoken of as, respectively, the exclusive and
inclusive senses of
*
or
'
; and, plainly,
if we are to identify
4
v
*
with either, it must be the latter. The reason why, unlike
others, this feature of the ordinary use of
4
or
'
is commonly
mentioned, is that the difference can readily be accommodated
1 Cf. footnote to p. 86.
FT. in] TRUTH-FUNCTIONAL CONSTANTS 3
In the symbolism of the truth-functional system : It is the
difference between
*
(p y q) . ~
(p . q)
*
(exclusive sense) and
6
p v q
'
(inclusive sense).
* Or *, like
* and \ is commonly used to join words and phrases
as well as clauses. The
4
mutuality difficulties
?
attending the
general expansion of
4
x and y are/
5 into
*
x is /and y is/' do
not attend the expansion of
4
x or y isf into
c
r Is/or y is/
? . (This is not to say that the expansion can always correctly be
made.)
We may call
4
v
*
the disjunctive sign and, being warned
against taking the reading too seriously, may read it as
*
or.'
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