McTaggart recognizes two and only two fundamentally distinct kinds of Determination, which he calls "Intrinsic" and "Extrinsic" respectively. Closely connected with intrinsic determination are two other notions, of which he makes considerable use, viz., "Presupposition" and "Requirement". He regards Causation as a particular case of intrinsic determination. Closely connected with extrinsic determination are two other notions, which he calls "Manifestation" and "Organic Unity". Neither of them appears to be very important. They are rather intimately bound up with each other, and the latter cannot be explained until we have dealt with the notion of "Groups" and "Compound Particulars", which will form the subject of the next Book of this volume. I propose therefore to defer the treatment of Manifestation for the present, and to deal with it, together with Organic Unity, in Book IV. In Book III I shall first consider Intrinsic Determination; then the other two kinds of connexion which are intimately related to it, viz ., Presupposition and Requirement, and Causation. Finally, I shall devote a chapter to Extrinsic Determination.

1. Implication.

      The notion from which we start is that of the falsity of a conjunctive proposition, i.e., the notion of two propositions, p and q, not being both true. Now, of course, if p be itself false, the conjunction of p with any other proposition whatever will also be false, i.e., if p be false pq will be false, whatever q may be. Similally, if q be false pq will be false, whatever p may be. Sometimes the only known reason why pq is false is that p is known to be false or that q is known to be false. The only known reason why the conjunctive proposition that Charles I died in his bed and Julius Caesar was murdered is false is that it is known to be false that Charles I died in his bed. But this is not always the case. Sometimes a conjunctive proposition pq is false because of some relation between p and q which makes it impossible that both should be true. The fact that p and q are related by such a relation can often be seen by direct inspection by a person who does not know that p is false and does not know that q is false. We know for example, that it if false that the matter at the centre of the earth is wholly solid and wholly liquid, although we do not know that it is false that this matter is wholly solid and do not know that it is false that this matter is wholly liquid. When two propositions are so related that it is impossible for both to be true, we say that they are "mutually inconsistent". We may call a conjunction of two propositions which are inconsistent with each other "an inconsistency".

      It is, of course, quite possible that there may be cases in which, though pq is false because p and q are mutually inconsistent, we cannot see their inconsistency. In such cases we could know that pq is false only if we knew that p is false or knew that q is false. But the vitally important fact is that there are cases in which we can see that p and q are mutually inconsistent without having to know beforehand that p is false and without having to know beforehand that q is false. The importance of this fact is that, if it were not for it, there could be no legitimate inference whatever. If I know that pq is false, and I know that p is true, I know ipso facto that q is false. But this is of no use to me for inferring that q is false, if my knowledge that pq is false rests on my previous knowledge that p is false or on my previous knowledge that q is false. If my knowledge that pq is false rests on my previous knowledge that p is false, I cannot also be in the situation of knowing that p is true, and therefore cannot draw the conclusion that q is false. If, on the other hand, my knowledge that pq is false rests on my previous knowledge that q is false, I cannot use it to infer that q is false. In the first case one of my premises would contradict my only ground for believing the other. In the second case my only ground for believing one of the premises is that I already know the conclusion which I profess to be going to prove. Suppose, however, that I can see that p and q are mutually inconsistent, without needing to know that p is false and without needing to know that q is false. Then, if I know that p is true, I can legitimately infer that q is false. My argument is now neither self-contradictory nor circular. We may sum up as follows. If and only if there are inconsistencies, and we can recognize some of them by inspection, we can make legitimate inferences. And it is quite certain that there are inconsistencies, and that we can recognize some of them by inspection.

      Now there are a great many relations between propositions which make them inconsistent with each other, and can be seen to do so. We can, I think, divide them into two great classes, viz.,

1. those which depend wholly on the forms of the propositions, and
2. those which depend partly on the content of the propositions.

      The next step is to restate the above results in such a way that they do not presuppose that there really are propositions, in the unique and indefinable sense discussed in Chap. IV of the present work. We cannot, of course, say that the fact that all men are mortal is inconsistent with the fact that some immortal beings are human, for it is not a fact that some immortal beings are human. But we could say that a belief that all men are mortal and a belief that some immortal beings are human could not both concord with the facts to which they refer. If a belief of the first kind concorded with a fact, that fact would be of such a form that there could be no fact of the right form to concord with a belief of the second kind. And the same would be true if "second" and "first" be everywhere interchanged in the last sentence. There is thus no difficulty in getting rid of the reference to propositions; and, now that this is clear, we can safely go on talking in terms of "propositions" if we find it convenient to do so.

      At length we come to the notion of "Implication". The statement that "p implies q" is defined by Russell to mean the same as the statement that "either p is false or q is true". An equivalent definition, which brings it into line with our previous discussion, would be that it means the same as the statement that "p and not-q are not both true". Let us start with this latter definition. I think it is unfortunate that the word should have been used in Cambridge in this way, since this is not the way in which it is commonly used. This has led to a great deal of misunderstanding, and to tiresome verbal controversies with logicians in Oxford and other places. When "implication" is defined in this way, it follows at once that the proposition that Charles I died in his bed implies the proposition that Julius Caesar was murdered (and equally the proposition that Julius Caesar was not murdered). For, since the proposition that Charles I died in his bed is false, every conjunctive proposition which contains this as a conjunct will be false; and so, by the definition, the proposition that Charles I died in his bed will imply every other proposition, whether true or false. It also follows at once that the proposition that Julius Caesar was murdered is implied by the proposition that Charles I was beheaded (and equally by the proposition that Charles I was not beheaded). For it is true that Julius Caesar was murdered, and therefore the proposition that Julius Caesar was not murdered is false. Consequently every conjunctive proposition which contains as a conjunct the proposition that Julius Caesar was not murdered is false. And so, by the definition, the proposition that Julius Caesar was murdered will be implied by every other proposition, whether true or false.

      Now McTaggart, always objected to this use of the word "implication", and never intentionally used it in this way himself. What most people mean by "p implies q" is obviously that p and not-q are inconsistent with each other. This is commonly expressed in Cambridge by saying that p "entails" q. I propose henceforth to use the word "entails" where McTaggart uses the word "implies". With these verbal explanations everything that I have said about inconsistency can be applied, mutatis mutandis, to entailment. For example, q can be legitimately inferred from p if and only if the person who makes the inference can see that p and not-q are inconsistent without having to know beforehand that p is false and without having to know beforehand that q is true. If he had to know beforehand that p is false, his second premise, viz.. that p is true, would contradict his only ground for believing his first premise, viz., that p and not-q are not both true. If he had to know beforehand that q is true, his argument would be circular, since his only ground for believing his first premise is his pre-existing belief in the proposition which he professes to be going to infer from it. The only word of warning that is needed here is the following. Whilst "p is inconsistent with q" is equivalent to "q is inconsistent with p", "p entails q" is not equivalent to "q entails p". For "p entails q" means that p and not-q are mutually inconsistent, whilst "q entails p" means that q and not-p are mutually inconsistent.

      The statement that "p implies q" is generally written "p ⊃ q". It means (in Cambridge), as we have seen, simply that p and not-q are not both true. The statement that "p entails q" may be written "p → q". It means, as we have seen, that p and not-q are mutually inconsistent. It is evident that (p → q) → (p ⊃ q), but not conversely. There is one other point to notice. If p → q,, then p ⊃ q is a necessary proposition, i.e., if p is inconsistent with not-q, the conjunction of p and not-q is impossible. But the converse of this does not hold. The conjunction of p and not-q would be impossible if p were impossible or if q were necessary. Thus the conjunction of 2 x 2 = 5 and "Some men are immortal" is impossible, because 2 x 2 = 5 is impossible. But, so far as we know, there is no inconsistency between these two propositions, and so we cannot say that 2 x 2 = 5 entails that all men are mortal. The relations between these various notions may be summed up as follows:

(p → q) → [(p ⊃ q) is necessary] → (p ⊃ q)	}	·
(p and not-q are inconsistent) → [(p. not-q) is impossible]
→ [(p. not-q) is false]

      It remains to apply this to propositional functions, as distinct from propositions. Let φ and ψ be two characteristics. Then the statement that "φx implies ψx for all values of x" means "It is not the case that there is an x which has φ and lacks ψ". The first statement is generally written φx ⊃_x ψx, and the second is generally written ∼ (∃x) . φx . ∼ψx. This is what Russell calls "Formal Implication". Such a statement is always true if nothing has φ, and it is always true if everything has ψ. For example, since there are no frictionless fluids, there are no frictionless fluids with spherical molecules. Equally there are no frictionless fluids with non-spherical molecules. Thus the property of being a frictionless fluid formally implies the property of having non-spherical molecules, and formally implies the property of having spherical molecules. The statement that "φx entails ψx for all values of x" means that the possession of φ is inconsistent with the non-possession of ψ. This may be called "Formal Entailment". The statement that φ formally entails ψx may be written φ →_x ψx. This, of course, entails that it is impossible for any thing to have φ and lack ψ; and this entails that there is in fact nothing which has φ and lacks ψ. Thus we have the following irreversible series of entailments:

(φx →x ψx) → [(φx ⊃x ψx) is necessary] → (φx ⊃x ψx)	}	·
(φx and ∼ψx are inconsistent) → [(φx. ∼ψx) is impossible]
→ (Nothing has φ and lacks ψ)

      We can now distinguish between a "logical inconsistency" between p and not-q and an "ontological inconsistency" between them. In the first case we can say that p "logically entails" q. In the second case we can say that p "ontologically entails" q. The proposition that all men are mortal logically entails the proposition that all immortal beings are non-human. The proposition that the triangle ABC is equilateral ontologically entails the proposition that the triangle ABC is equiangular.

2. Intrinsic Determinations

      McTaggart introduces the notion of Intrinsic Determination in § 108 of the Nature of Existence. His statements about it are somewhat confused, and it will be necessary to disentangle what he probably meant from the various inconsistent things which he said.

      Intrinsic determination is said to be a relation between characteristics. "If it is true that, whenever something has the quality X, something has the quality Y, this involves that, beside the relation between the two propositions 'Something has A' and 'Something has Y', there is a relation between the qualities X and Y." It is this supposed relation between the qualities which McTaggart proposes to call "Intrinsic Determination". He adds that "the quality X will be said to determine intrinsically the quality Y whenever the proposition that something has the quality X implies the proposition that something has the quality Y". There are two remarks to be made about this at the outset. (i) The first statement quoted above makes no explicit mention of implication, whilst the second does. Probably McTaggart thought that the second statement quoted was merely a different verbal formulation of the first, but it is by no means clear that this is so. (ii) It is plain that not all entailment could be correlated with a relation between characteristics. Purely logical entailment could not be correlated with a relation between a certain characteristic and a certain other characteristic Y, since it depends wholly on the forms of propositions and in no way on their special content. If intrinsic determination is a relation between characteristics, which is connected with the relation of entailment between propositions, it must be connected with what I have called "ontological entailment" and not with what I have called "logical entailment". Now it is not at all clear at first sight what McTaggart means to convey by his definition of "intrinsic determination".

      (i) Taking the second statement which I have quoted, the ostensible meaning is clear enough. If this statement be interpreted literally, it means that φ intrinsically determines ψ if and only if the proposition "There is at least one instance of φ" entails the proposition "There is at least one instance of ψ", i.e., if and only if (∃x) . φx: → :(∃x) . ψx.

      (ii) But the first statement quoted, which constitutes McTaggart's explicit definition of "intrinsic determination", is not so clear. In that statement he says that φ will intrinsically determine ψ, if and only if, whenever something has φ, something has ψ. Now what does "whenever" mean in this connexion? Taken literally, it would mean that φ intrinsically determines ψ provided that, at any moment when something has φ, something has ψ; i.e., provided that

(∃x) . φ(x,t) : ⊃_t : (∃x) . ψ(x,t).

      (iii) When McTaggart begins to give examples he uses phrases which seem inconsistent with his definitions. Thus, at the bottom of p. 111, he says that "the occurrence of blueness intrinsically determines the occurrence of spatiality". (My italics.) And, at the top of p. 112, ho says that the quality of one person to be a husband intrinsically determines the occurrence in someone else of the quality of being a wife. (My italics, again.) Here we have a mass of verbal inconsistencies. "Intrinsic determination" was defined at the beginning of § 108 as a relation between characteristics. Yet in the very same section it is said, in the first example, to relate two occurrences, and, in the second example, to relate a quality and an occurrence of a quality. Now, for McTaggart, a quality is a characteristic and not a fact, whilst an occurrence of a quality is either a fact or an event, and is certainly not a characteristic

      I will now try to unravel this verbal tangle. I think it is certain that McTaggart means intrinsic determination to be a relation between characteristics, and not between facts. And I think that he does intend to mean by it the relation in which φ stands to ψ if and only if the proposition "There is at least one instance of φ" entails the proposition "There is at least one instance of ψ"; i.e., if and only if

(∃x) . φx : → : (∃x) . ψx.

      I believe the cause of the verbal confusions to be the following. There is another, and closely connected, relation between characteristics, which I am going to call "Conveyance", which McTaggart does not explicitly notice or name. Often he is thinking of conveyance when he is talking of intrinsic determination. This, we shall see, accounts for his statements which introduce the word "whenever". The source of the confusion is that, if φ conveys ψ, it necessarily follows that φ intrinsically determines ψ, and we hardly ever know that φ intrinsically determines ψ except by inference from knowledge that φ conveys ψ. In fact conveyance is much the more important and interesting relation of the two. I will now explain and illustrate, and try to justify, these remarks.

      I define the statement that φ "conveys" ψ to mean that, if anything has φ, it necessarily follows that that same thing has ψ; i.e., conveyance is the relation which φ has to ψ if and only if φx →_x ψx. Thus, for example, the characteristic of having shape conveys the characteristic of being extended.

      Now it follows logically from φx : →_x ψx that

(∃x) . φx : → : (∃x) . ψx.

      Let us now consider McTaggart's examples. One is that "the occurrence of blueness intrinsically determines the occurrence of spatiality". Strictly speaking, this should mean only that, if there were at least one instance of blueness, it would necessarily follow that there is at least one instance of spatiality. Now this may quite properly be expressed by saying that the "occurrence" of blueness in the universe involves the "occurrence" of spatiality in the universe. But, of course, this is much less than we are entitled to assert. What we knows is that, if anything were blue, it would necessarily follow that that very same thing would be spatial. And this is our only ground for asserting the weaker proposition that, if blueness should occur somewhere in the universe, it would necessarily follow that spatiality would occur somewhere in the universe. Thus we here assert the weaker proposition that blueness intrinsically determines spatiality only because we are entitled to assert the stronger proposition that blueness conveys spatiality.

      Now let us consider McTaggart's other example. The quality of being a husband is said to determine intrinsically the occurrence of the quality of being a wife. This again is true, on our interpretation of McTaggart's meaning. It is impossible that something should have the characteristic of being a husband and that nothing should have the characteristic of being a wife. Now here, of course, φ does not convey ψ, since it is not the case that it is impossible for any term to be a husband without itself being a wife. The difficulty is all the other way, as the Emperor Heliogabalus discovered. Nevertheless the proposition that "being a husband" intrinsically determines "being a wife" is derived from a proposition about the conveyance of one characteristic by another. It is derived from the proposition that "being a husband" conveys "having a wife". We will now proceed to generalize this example.

      If we represent the relation of being a husband by R, and its converse, the relation of being a wife, by R*. the situation is as follows. "X is a husband" will be represented by (∃Y) . XRY. "Y is a wife" will he represented by (∃X). YR*X. Then

      (i) The proposition that "being a husband" intrinsically determines "being a wife" will be represented by

(∃X, Y) . XRY : (∃Y, X) . YR*X.

      (ii) The proposition that "being a husband" conveys "having a wife" will be represented by

(∃Y) . XRY : →_x : (∃Y) . YR*X.

      (iii) The proposition (ii) entails the proposition (i), and is not entailed by it. But there is a still stronger and more definite proposition which in turn entails (ii) and is not entailed by it. This is the proposition that, if X is a husband of Y, then it necessarily follows that Y is a wife of X, whoever X and Y may be. This is symbolized by XRY →_X,Y YR*X. It is an instance of the general proposition that, if X be a dyadic relation, then, if any pair of terms are related by it in a certain order, the same pair of terms are necessarily related by its converse in the opposite order.

      It is not at all clear to me that there are any cases in which we know that φ intrinsically determines ψ except by inference from knowledge about the conveyance of one characteristic by another. As we have seen, the relation of conveyance from which we infer that φ intrinsically determines ψ may not relate φ and ψ themselves. It may relate φ to some other characteristic which stands to ψ in some special relation, such as that between "having a wife" and "being a wife". Certainly McTaggart has given no example of intrinsic determination which is known independently of knowledge of conveyance.