ENTAILMENT
II.—B JOHN WATLING.
1. Lewy’s dilemmas and his resolution of them.
Lewy’s dilemmas arise in this way. He produces an argument which appears to prove that statement
that the sentence ‘Es gibt keinen der zugleich ein bruder und nicht männlich ist’ expresses the
proposition that there is nobody who is a brother and is not male, entails the statement that it is
necessary that there is nobody who is a brother and is not male. Yet it seems that it cannot be true that
this entailment holds. Again, he produces an argument which seems to prove that the statement that
the proposition that Caesar is dead is materially equivalent to the proposition that Russell is a brother,
entails the statement that the proposition that Russell is a brother is materially equivalent to the
proposition that Russell is male. Yet, as before, it seems that it cannot be true that this entailment holds.
I will briefly summarize these arguments. The former goes like this: First there is an argument
to show that the statement that the sentence ‘Es gibt keinen der zugleich ein bruder und nicht männlich
ist’ expresses the proposition that there is nobody who is a brother and is not male, entails the
statement that this sentence expresses a necessary proposition. Then it is argued that the two
statements, that this sentence expresses the proposition that there is nobody who is a brother and is not
male, and that this sentence expresses a necessary proposition, together entail the statement that it is
necessary that there is nothing which is a brother and is not male. And from these two entailments the
paradoxical conclusion follows.
Similarly with the latter argument: first it is argued that the statement that the proposition that
Caesar is dead is materially equivalent to the proposition that Russell is a brother, entails the statement
that the proposition that Caesar is dead is materially equivalent to the proposition that Russell is male.
Then, it is asserted that the statements, that the proposition that Caesar is dead is materially equivalent
to the proposition that Russell is a brother, and that the proposition that Caesar is dead is materially
equivalent to the proposition that Russell is male, together entail the statement that the proposition that
Russell is a brother is materially equivalent to the proposition that Russell is male. And from these two
arguments the conclusion follows.
Using Lewy’s symbolism the arguments proceed:
A entails C, A and C entail B, therefore A entails B;
D entails F, D and F entail E, therefore D entails E.
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Lewy’s own resolution of the dilemma is this: First, he is certain that neither of the conclusions
is true. Therefore he has to explain why these arguments are not valid when they seem so; and why they
seem valid when they are not. His explanation is that the assertion that one statement entails another is
ambiguous; that there are two entailment relations, either of which may be intended by someone who
asserts that one statement entails another. One of these relations, which Lewy calls ‘necessitation’, does
hold from A to C, and from A and C to B. But it does not hold from A to B; Lewy believes that this is
possible because necessitation is not transitive. Similarly, although D necessitates F, and D and F
together necessitate E, D does not necessitate E. But the other entailment relation, which Lewy believes
is transitive, does not hold between A and C, nor between D and F. The arguments are not valid,
because that kind of entailment for which the premisses hold is not transitive, and in that kind which is
transitive the premisses do not hold. The arguments seem valid because we do not distinguish these two
kinds of entailment.
I do not accept Lewy’s resolution of the dilemmas because I do not believe there can be a kind
of entailment which is not transitive, because I find Lewy’s explanation of what necessitation is to be
unsatisfactory, and because I think his restriction of it to obtain the transitive kind of entailment is both
too strict, so that entailments which are valid become invalid, and not strict enough, so that paradoxes
similar to Lewy’s remain. I will discuss Lewy’s two kinds of entailment, and go on to say how I think
that the dilemmas are to be resolved.
2. Necessitation.
First, from the definition Lewy gives of necessitation, however statements that one proposition
is a reason for another are to be explained, it follows that necessitation is a transitive relation. Lewy
gives his definition in this way: ‘In other words, we can say, I think, that P necessitates Q if and only if,
for all propositions R, “R is a reason for P” strictly implies “R is a reason for Q”.’ Therefore if P
necessitates Q and Q necessitates S, then, for all R, that R is a reason for P strictly implies that R is a
reason for Q, and that R is a reason for Q strictly implies that R is a reason for S. Therefore, since strict
implication is transitive, it follows that, for all R, that R is a reason for P strictly implies that R is a
reason for S; that is, P necessitates S.
Secondly, from Lewy’s explanation of what it means to say that one proposition is a reason for
another it follows that no necessary or contradictory proposition can be a reason for another
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proposition, and that no proposition can be a reason for either a necessary or a contradictory
proposition. I will explain why I believe that this is so.
It is rather difficult to discuss Lewy’s definition of necessitation because, although he gives a
list of conditions which are to be sufficient for one proposition to be a reason for another, he does not
say what conditions are necessary. Not only that, but some of his sufficient conditions are stated
ambiguously. For example, one sufficient condition for R to be a reason for P is given as ‘(4) “ If R is
true, P is true.”’ I suppose that this conditional (4) is not intended to be truth functional; for, if ‘R
materially implies P’ is a sufficient condition for ‘R is a reason for P’, then it is necessarily true that a
contradiction is a reason for every statement, and every statement a reason for a necessary statement.
And, if this is so, then, if Q is a necessary proposition, no matter what proposition P is, that R is a
reason for P strictly implies that R is a reason for Q. If this is Lewy’s definition of necessitation then
every proposition necessitates a necessary proposition. But if this is so, then A necessitates B, and D
necessitates E, which Lewy denies.
Perhaps it is obvious that Lewy did not intend the conditional (4) to be truth functional; but I
wanted to establish the point in order to make it clear that his explanation of what he means by ‘is a
reason for’ rests entirely on the other conditions in his list. That is, on conditions such as ‘(1) Since R is
true, P must be true’, and ‘(5) If R were true, P would necessarily be true too.’
Now Lewy says he thinks it is a mistake to hold that it is logically impossible that any
proposition should be a reason for a contradictory proposition; but I do not see how, on Lewy’s
explanation of what it is to be a reason, it can make sense to assert that a proposition is a reason for a
contradic tion. For to assert this would be to assert that if that proposition were true then the
contradiction would be true, and how can it make sense to suppose a contradiction to be true ? A
necessarily false proposition is often defined as one which cannot be conceived true. Similar arguments
apply against assertions that a contradiction is a reason for another proposition, that a necessary
proposition is a reason for another proposition, and that a proposition is a reason for a necessary
proposition. For what sense can there be in the supposition that a proposition which cannot
conceivably be false is true ? Now if it makes no sense either to assert or deny that a proposition can be
a reason for a contradiction, then it makes no sense to assert that a proposition necessitates a
contradiction. And similarly with assertions about the necessitation of necessarily true propositions.
It might be argued, against what I have just said, that it must make sense to suppose a
contradiction true, for such suppositions are required in arguments by reductio ad absurdum: in such
arguments assertions like ‘If it were true that p, then . . .‘, where p is some contradiction, very often
occur. But I do not believe that these assertions involve conceiving of the truth of a contradiction.
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There is evidently an ambiguity of statements beginning ‘If it were . . .’ I might say ‘If the country of his
birth were China, then he would be Chinese’ without being committed to the belief that it makes sense
to suppose that Great Britain (the country of his birth) is China. I say ‘If the country of his birth were
China’ in order to imply that I do not know what country he was born in. My assertion is not about the
country of his birth, it is not about Great Britain, in the way in which the assertion that if the watch he
is wearing were clean it would keep good time, is about his watch. Perhaps in the former kind of
assertion it is more correct not to use the subjunctive, but rather to say ‘If the country of his birth is
China, then he is Chinese.’ Now it is clear that the suppositions in reductio ad absurdum arguments are
of the first kind. They are not statements about some contradictory statement, any more than ‘If the
country of his birth is China then he is Chinese ‘is about Great Britain, the country of his birth. I may
say ‘If the statement that it is raining and not raining is true, then it is true that it is raining’ without
speaking about, or conceiving the truth of the statement that it is raining and not raining; if indeed, as I
very much doubt, there is such a statement. I may consider a proposition, or what I believe to be a
proposition, whilst knowing scarcely anything about it; and when I assert that something follows from a
certain proposition, I may have in mind certain properties of this proposition, or supposed proposition,
but I may know very little else about it. I may know that if some proposition of this kind is true, then
something else follows; but I may not know if the proposition I suppose myself to be considering really
exists.
Saying, of the statement that it is raining and not raining, that if it were true then it would be
raining, may be compared to saying, of the soldier who shaves those, and only those, soldiers who do
not shave themselves, that if he were to shave himself then he would not shave himself. No one
supposes that to say this is to suppose that a certain man shaves himself; for no such man exists, or
could possibly exist.
It is quite correct to say that if it is true that it is raining and that it is not raining, then it is true
that it is raining; but it does not follow from this that the statement that it is raining and that it is not
raining entails the statement that it is raining. Now the sense in which it is true to say of the statement
that it is raining and that it is not raining, that if it is true then it is true that it is raining, is not a sense
in which the statement that it is raining and is not raining is a reason for the statement that it is raining.
To believe this would be as absurd as to believe that it follows from the fact that if the country of his
birth were China then he would be Chinese, that the statement that Great Britain is China is evidence
for the statement that he is Chinese. Therefore when Lewy gives, as a sufficient condition of P’s being a
reason for Q the condition that if P were true, Q would be, he must intend the subjunctive to express a
genuine supposition, the supposition of a different truth value, and not merely to express doubt. But it
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is impossible genuinely to suppose a contradiction true, and it is meaningless to assert, intending a
genuine supposition, that if a contradiction, or a necessary statement, were true, then some other
statement would be. Lewy has given an explanation of what it is for one proposition to be a reason for
another in which only contingent propositions can be reasons for others, and in which the only
propositions for which other propositions can be reasons are contingent propositions.
3. Entailment holds only between contingent propositions.
I do not suggest that this consequence of Lewy’s explanation of what it is for one proposition
to be a reason for another, if I am right and it is indeed a consequence, is an argument against his
explanation. In fact I believe that it is an argument in its favour, although I suppose he would not
himself agree that it is. I agree that we often speak of having reasons for, or against, necessary and
contradictory propositions, but I do not think that these assertions should be taken literally, any more
than I think that argument by reductio ad absurdum involves conceiving a contradiction to be true. But
though I cannot give an explanation of how such assertions should be interpreted, yet I think that some
sort of explanation like Lewy’s of what it is for one proposition to he a reason for another must be right,
and I also think, as Lewy does, that the notion of entailment is related to the notion of ‘being a reason
for’.
Surely if any contingent proposition entails a necessary proposition then a proposition of the
form p should entail a proposition of the form p or not-p. And if any contradiction entails a contingent
proposition, surely a proposition of the form p and not-p should entail one of the form p. But these
entailments do not hold. If I tell you that it is raining and not raining then I do not tell you that it is
raining, it does not follow from what I say that it is raining, and my statement gives you no reason at all
for believing that it is raining. There are only three reasons which anyone might have had for believing
that these entailments held. One is the principle that any proposition written as a conjunction entails
each of the propositions expressed as conjuncts. Another is that it seems necessary to accept such
entailments in order to account for the validity of reductio ad absurdum argument. And a third is that
entailment is to be identified with strict implication. Perhaps people were inclined to accept this
identification because the truth table method provided a good test for necessary statements so that it
seemed convenient to define necessary inference in terms of necessary statement. I do not believe that
any of these reasons have any force against the palpable fact that the proposition that it is raining and
not raining does not entail that it is raining.
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These arguments do not establish that it is impossible for one necessary proposition to entail
another, or for one contradiction to entail another. The fact, or what I believe to be the fact, that neither
necessary nor contradictory propositions can be reasons for other propositions, inclines me to believe
that this is impossible. Of course there is some genuine form of argument from the fact that one
proposition is necessary to the fact that another is, and if someone wishes to identify some relation of
this kind with entailment I am not sure that I can find reasons against him. But the notion of
entailment does seem to be bound up with assertion—perhaps it would be possible, though not very
enlightening, to define ‘p entails q’ as ‘p asserts everything that q asserts’—and it does seem that
propositions such as the proposition that it is both raining and not raining assert nothing, and for this
reason I do not want to agree that necessary propositions can entail one another: the proposal that
necessary statements state logical truths hardly contradicts the assertion that they say nothing. But the
decision of the question whether entailment holds between necessary propositions is not relevant to the
discussion of Lewy’s dilemmas; for the only entailments there in question are between statements of
which one at least is contingent.
If my arguments are correct, then Lewy’s paradoxes provide no reason for holding that
entailment is not transitive; for if neither necessary nor contradictory propositions can figure in
entailments, then C does not entail B, nor F entail E. It is correct that if the proposition that there is no
one who is a brother and is not male is not true, then it is not true both that the sentence S says that
there is no one who is a brother and is not male, and that S states a necessary proposition; but this is not
like saying that if it were not true that my tie is coloured, then it would not be true that my tie is green.
4. Simple Entailment.
Since I do not agree that the relation of entailment should be identified with the relation which
Lewy calls entailment, and which he says is what we intend when we say that one proposition by itself
entails another, I shall call his relation ‘simple entailment’.
Lewy argues that since A does not entail B, nor D entail E, then any relation of entailment
which is transitive must be such that in it A does not entail C, nor D entail F. The argument which
Lewy gives in setting out his dilemmas for supposing that A entails C, is that A and B entail C, and that,
since B is a necessary proposition, A alone entails C. Lewy suggests that there is a kind of entailment,
indeed the only kind of entailment which is transitive, for which arguments depending upon dropping
a necessary premiss are not always valid. He suggests that a necessary proposition may only be dropped
if it is used as a ‘rule of inference’. His definition of this kind of entailment is this: ‘I think that we can
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say that P entails Q if and only if (1) P necessitates Q, and (2) it is logically possible to shew that P
strictly implies Q without having previously shewn (or assumed) the truth of some proposition R,
which satisfies the following condition: R is different from any formal implication (in Russell’s sense) S,
such that P or a conjunct of P is a value of the antecedent of (the scope of) S, and Q or a conjunct or
disjunct of Q is a value of the consequent of (the scope of) S.’
This definition places inconvenient restrictions upon entailment. It follows, for example, that
the fact that my tie is either red or green does not simply entail the fact that my tie is coloured. The
former necessitates the latter, but it is not possible to shew that the former strictly implies the latter
without showing that being red strictly implies being coloured. Now neither the proposition that my tie
is either red or green, nor any conjunct of it, is a value of the antecedent of the generalization that if
anything is red, then it is coloured.
Indeed if it were allowed that the fact that my tie is red or green does entail that my tie is
coloured, it would be difficult to deny that the fact that the proposition that Caesar is dead is materially
equivalent to the proposition that Russell is a brother, entails the fact that the proposition that Caesar is
dead is materially equivalent to the proposition that Russell is male. For to deny this is to deny that the
fact that either Caesar is dead and Russell is a brother or Caesar is not dead and Russell is not a brother,
entails the fact that either Caesar is dead and Russell is male or Caesar is not dead and Russell is not
male. And how can this be denied if it is accepted that the fact that either my tie is red or my tie is green
entails that my tie is coloured?
But the definition of simple entailment is also unsatisfactorily weak. For even if entailment is
identified with simple entailment the following well known argument still holds:
P entails (P and Q) or (P and not-Q).
(P and Q) or (P and not-Q) entails P and (Q or not-Q).
P and (Q or not-Q) entails (Q or not-Q).
Therefore P entails (Q or not Q).
The conclusion of this argument is as paradoxical as the conclusions of Lewy’s arguments. Not only
that, but the paradox arises in the same way: because of the assumption of a necessary proposition in
the first entailment which is revealed again by the third. It differs from Lewy’s argument only in that no
necessary premiss is explicitly dropped. These entailments may be identified with what Lewy calls
entailment because the only propositions justifying them are ‘if p, then either p and q, or p and —q’; ‘if
either p and q, or p and —q, then p and (q or —q)’; and ‘if p and q, then q.’ Therefore Lewy’s restricted
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version of necessitation still allows the construction of similar arguments proving paradoxical
entailments.
The principle that any necessary proposition may be dropped from the antecedent of an
entailment without invalidating it seems to me to be an odd principle. Presumably the principle which
is true is that it is never necessary to include a necessary principle which justifies an entailment, as a
premiss of that entailment. For the denial of the principle leads to Lewis Carroll’s Achilles and the
Tortoise paradox. The principle is rather that no necessary statement need be inserted, than that any
necessary statement can be dropped.
Now I believe that, by expressing the proposition that, that Caesar is dead is materially
equivalent to that Russell is a brother, as the proposition that either Caesar is dead and Russell is a
brother or Caesar is not dead and Russell is not a brother (where ‘Caesar is not dead’ is to read ‘it is not
the case that Caesar is dead ‘), it can be made clear that D does entail F without resorting to the
principle that a necessary premiss may be dropped. For surely this proposition does entail Caesar is
dead and Russell is male or Caesar is not dead and Russell is not male. Certainly it does not simply
entail it, simple entailment was specially chosen because it did not hold between these two propositions,
but surely it entails it. Therefore this dilemma cannot be said to arise because of the use of the principle
that a necessary proposition may be dropped if it occurs in the antecedent of an entailment. I believe
that the puzzle does arise in part from the assumption of a necessary proposition, but not because one is
explicitly used as a premiss and then dropped.
I cannot find any way of showing that A entails C without using the principle that a necessary
proposition may be dropped; and I cannot, myself, decide whether A entails C or not. Therefore I
cannot be sure that what I have said about the second dilemma, holds of the first.
If Lewy’s distinction between necessitation and simple entailment fails, is there any distinction
between these arguments which prove paradoxical entailments and other arguments which prove that a
contingent proposition entails a necessary one, but which prove an apparently acceptable entailment;
for example an entailment of the form p entails (p or not-p)? I think there is, and it is this. The
arguments which establish that A entails B, that D entails E, and a proposition p entails a proposition (q
or not-q), can none of them be used to prove B from A, to prove E from D, or to prove a proposition (q
or not-q) from a proposition p. On the other hand the argument that a proposition p entails a
proposition (p or not-p) could, if correct, be used to prove a proposition (p or not-p) from a
proposition p. The former arguments cannot provide proofs because one of the steps assumes the
principle that is to be proved. However legitimate it may be to drop a necessary premiss in an
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entailment, it cannot be legitimate to assume in a proof the proposition which the proof is intended to
establish. For example, the argument
D
D entails F,
D and F entail E,
E
cannot be regarded as a proof of E from D because the step from D to F amounts to the assumption of
the necessary proposition E. Similarly the argument X above cannot be regarded as a proof of (Q or
not-Q) from P, because the step from P to (P and Q) or (P and not-Q) amounts to the assumption of
the necessary proposition (Q or not-Q). Whether the necessary proposition is assumed implicitly or
explicitly, the fault in the proof is the same.
Now the relation of ‘provable from’ is not transitive. For the first part of the argument above
might be said to provide a proof of F from D, and the second part might be said to provide a proof of E
from D and F; but the whole argument does not provide a proof of E from D. Evidently this is because
what is proved in the second part of the proof is assumed in the first part. I believe that Lewy mistakes
the fact that proof is not transitive for the fact that one sort of entailment is not transitive, and then
attempts to restrict entailment in such a way that any argument that establishes an entailment between
two propositions may also be used as a proof of the one from the other.
Any argument which establishes that one contingent proposition entails another can stand as a
proof of the one from the other, because no step from a contingent proposition to another which it
entails can amount to the assumption of a contingent proposition. Therefore, if I am right in holding
that a contingent proposition cannot entail a necessary one, nor a contradiction entail a contingent
proposition, then any argument which establishes an entailment can also serve as a proof. But there
may be arguments, and Lewy’s are examples of them, which show that if some contradiction is true
then it entails another proposition, but which do not establish entailments, and which cannot serve as
proofs.
5. How are the dilemmas to be resolved?
The second dilemma, for example, arises because, though we have what is apparently a proof
that D entails E, yet D is obviously irrelevant to E, provides no reason for believing E, and does not
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assert everything that E asserts. The solution is that the proof does not prove that D entails E, and that it
is indeed not true that D entails E. So far I agree with Lewy.
The explanation of our finding a dilemma is twofold. On the one hand we can accept the proof
because we are inclined to believe that a necessary proposition can be entailed by a contingent one;
once we have accepted this we must accept the proof as valid. The proof seems, and is, as good as the
proof that a proposition of the form p entails a proposition of the form p or not-p. On the other hand,
though we reject, correctly, that D entails E, we do so for the wrong reason and so accept, incorrectly,
that p entails p or not-p. That is, though the proofs of the two entailments seem equally good, the
former conclusion seems wrong and the latter right. Our mistake leads us to accept both invalid proofs
as valid, but to accept only one of the two incorrect conclusions as correct.
D is irrelevant to E because D is contingent and E necessary, and any proposition of the form p
is irrelevant to any proposition of the form p or not-p for the same reason. We should reject both
entailments and so we do not have to find a reason for accepting the proof of one and rejecting the
proof of the other. A proposition of the form p seems relevant to one of the form p or not-p, but
irrelevant to one of the form q or not-q, because the first entailment is an obvious consequence of
carrying over entailment rules that hold between contingent propositions to hold where one of the
propositions is necessary or contradictory, whilst the second entailment is a far less obvious
consequence of this same procedure.
That the proof, which we are inclined to accept, is such a strange proof, arises from the
peculiar character of necessary propositions. A necessary proposition may be used in arguments from
one contingent proposition to another in ways which amount to the assumption of the necessary
proposition: an example is the argument from a proposition of the form p to one of the form p and q or
p and not-q. It is this assumption of a necessary proposition which gives rise to the paradoxical nature
of the proof. But a quite unparadoxical proof, such as that of a proposition of the form p or not-p from
one of the form p, or of a proposition of the form p from one of the form p and not-p, is just as invalid.
I think that Lewy is correct in seeing the assumption of a necessary proposition as a source of his
dilemmas, but not correct in supposing that this was the source of the invalidity of his arguments.
Because of the irrelevance of D to E, and of propositions of the form p to those of the form q
or not-q, and the apparent relevance of propositions of the form p to those of the form p or not-p, e are
inclined to accept the latter entailment but not the two former. So we try to find a difference in the way
in which these propositions are supported. We see that, whilst the argument by which it is proved that
D entails E cannot be used as a proof of E from D, and the argument by which it is proved that a
proposition of the form p entails a proposition of the form q or not-q cannot be used as a proof of q or
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not-q from p—for, if used as proofs, these arguments assume what they are to prove— yet the argument
for the assertion that a proposition of the form p entails one of the form p or not-p does not suffer from
this fault; it might be taken as a proof of p or not-p from p. Therefore we conclude that no argument
establishes that p entails q unless it can be used as a proof of q from p, and so we find an excuse for
holding that a proposition of the form p does entail one of the form p or not-p, but that it does not
entail one of the form q or not-q.
In fact these arguments are all as good as one another. They are alike reductio ad absurdum
arguments. If it is false that either it is raining or it is not raining, then it is not raining; if E is false, then
so is D; if it is false that either it is raining or it is riot raining, then it is snowing. But the statement that
it is not either raining or not raining does not entail the statement that it is not raining, nor does E is
false entail D is false, nor the statement that it is not either raining or not raining entail that it is
snowing.
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