McTaggart does not introduce the notion of "Presupposition" until § 183, Chap. XXIII, of the Nature of Existence. It is introduced there in connexion with an attempt to prove that the endless divisibility of particulars would conflict with the principle that every particular must have a sufficient description, unless certain peculiar conditions were fulfilled. Whenwe come to deal with this argument, in Chap. xx of the present work, we shall see that it can be stated without using the notion of presupposition. As presupposition is very closely connected with conveyance and intrinsic determination, and as we shall not have any need to mention it again, this seems the best place for discussing and disposing of it.

1. Presupposition.

      McTaggart's statements about presupposition are very confused and confusing; but it is quite possible to discover and to state clearly what he has in mind. I think that the verbal obscurities arise here, as they did in his treatment of intrinsic determination, from his failing to distinguish two closely connected relations. The result is that, when he uses the word "presupposition", he sometimes is thinking of one of these relations and sometimes of the other. And, as the two have different properties, he sometimes seems to be making inconsistent statements about a single relation.

      The first of these relations relates one characteristic φ to anothercharacteristic ψ. I shall express it by saying that φ "partially conveys" ψ. This relation then is a dyadic relation between characteristics, which we will call "Partial Conveyance". The second is a relation between a characteristic φ, a characteristic ψ, and a particular x. It may be expressed by the statement that "φ presupposes ψ in the instance x". Presupposition is then a triadic relation. We shall find that it can be defined in terms of partial conveyance. McTaggart, as I have said, failed to distinguish these two relations, and used the single name "Presupposition" for both of them. He also failed to notice that presupposition, in one of the senses which he has in mind, is a triadic relation, relating two characteristics and a particular. I will now explain and illustrate these notions.

      1.1. Partial Conveyance. We say that φ "partially conveys" ψ when and only when there is a class α of characteristics which obeys the following conditions.

1. α is a class of mutually exclusivecharacteristics, e.g., the various colours, or various shades of the same colour.
2. ψ is a member of α.
3. If anything has φ, then it necessarily follows that this thing has some member or other of the class α.
4. There is no member of the class α such that, if a thing has φ, it necessarily follows that this thing will have that member of α.

1. Χ, ω∈α . ⊃_x,ω : Χx →_x ∼ωx
2. ψ∈α.
3. φx →_x :. (∃Χ) : Χ∈α . Χx
4. ∼(∃Χ) :. Χ∈α : φx →_x Χx.

      The following would be an example of partial conveyance.The property of being coloured partially conveys the property of being triangular. For it is impossible to be coloured without having some shape or other, and triangularity is one of the shapes that a coloured thing may have. But it is quite possible to be coloured without being triangular, and the same is true of any shape that we choose to mention. We can call such a class as α a set of "Partial Consequents" of φ.

      The relation of a determinable, e.g., being coloured, to a determinate which falls under it, e.g., being red, is an instance of partial conveyance. But here a further condition is fulfilled, viz., that, if anything is characterised by any member of α, it necessarily follows that this thing will be characterised by φ.This further condition can hold also in cases where the two characteristics do not stand in the relation of determinable to determinate. For example, having extension partially conveys being triangular. And being triangular, or being of any other determinate shape, conveys having extension. But being triangular, being square, and so on, are not determinates under the determinable of having extension. They are determinates under the different determinable of having shape.

      1.2. Presupposition. We can now define the statement that "φ presupposes ψ in the instance x". This means simply that x is characterised by both φ and ψ, and that φ partially conveys ψ. Suppose, for example, that a certain particular x is in fact coloured and triangular. Then we should say that being coloured presupposes being triangular in the instance x.

      The meanings of our terms are now quite clear. In reading McTaggart we have only to remember that he sometimes uses the word "presupposes" when he is thinking of what we mean by "partially conveys". If he had realised that presupposition is not a two-term relation between characteristics, but is a three-term relation between two characteristics and a particular instance, he would have avoided a great deal of complication and verbal confuxion.

      We can now consider some propositions which McTaggart asserts about these relations.

      (i) He says that presupposition may be reciprocal (§ 184). This is plainly true. Suppose that x is square and has an area of one square inch. The property of being square partially conveys the property of having an area of one square inch; and, since x in fact has this area, being square presupposes, in the instance x, having an area of one square inch. But having an area of one square inch also partially conveys being square in shape; since anything that has this area must have some shape, whilst there is no shape which such a thing must have. Since x in fact is square, having an area of one square inch presupposes, in the instance x, being square in shape. Thus, in the instance x, the presupposition of these two qualities by each other is mutual.

      (ii) He says in § 184 that, if X presupposes Y or Z, and Y presupposes S or T, and Z presupposes U or V, then X will presuppose S or T or U or V. There are two comments to be made on this. (a) We must substitute for "X presupposes Y or Z" that "Y and Z are a set of partial consequents of X", and similarly throughout the proposition. (b) Even so the statement is false. We can infer from the premises that anything that had X would have S or T or U or V. And this is part of what is asserted in the conclusion. But we cannot infer, from the fact that what has X need not have Y and that what has Y need not have S, that what has X need not have S. And this is equally part of the conclusion. So McTaggart has simply committed a logical fallacy here. In the same section McTaggart says that, if X intrinsically determines Y, and Y presupposes either U or V, then X presupposes either U or V, unless X either directly or indirectly determines one of the two and thus excludes the other. The last proviso shows that he has here seen and avoided the trap into which he fell at the beginning of the paragraph. But even so his statement is not true unless we amend it to the following: "If X conveys Y, and U and V are a set of partial consequents of Y, then U and V are a set of partial consequents of X, unless X either directly or indirectly conveys one of the two and thus excludes the other".

      1.3. Total Ultimate Presupposition. We come now to the notion of a "Total Ultimate Presupposition", which McTaggart develops in § 185. It is evident that the same eharactenstic φ might presuppose in the same instance x a number of different characteristics, ψ, χ, etc. Of course each of these would have to belong to a different set of partial consequents of φ, since characteristics which belong to any one set of partial consequents of φ would, by dcfinition, be incompatible with each other. Now, if ψ and χ be two characteristics which are both presupposed by φ in a certain instance x, there are three possible kinds of relation between ψ and χ.

      (i) It might be that neither conveys the other. Suppose, for example, that φ is the property of being red, ψ the property of being round, and χ the property of being 2 square inches in area. And suppose that x is in fact red, round, and 2 square inches in area. Then, in this instance, being red presupposes being round, and also presupposes being 2 square inches in area. But a thing might be round and have any area, and it might be 2 square inches in area and have any shape.

      (ii) It might be that each conveys the other. Suppose that x is triangular, equilateral, and equiangular. Then, in this instanee, being triangular presupposes being equilateral and presupposes being equiangular. Now anything that had either equilateral or equiangular triangularity would necessarily have both.

      (iii) It might be that there is a one-sided relation of conveyance between two presupposed characteristics. Suppose that φ is the property of being coloured, that ψ is the property of being red, and that χ is the property of being scarlet. And suppose that x is in fact coloured, and red, and scarlet. Then, in this instance, being coloured presupposes being red, and it also presupposes being scarlet. But anything that was scarlet would necessarily be red, though a thing might be red without being scarlet. In this case we say that the presupposed characteristic which conveys and is not conveyed by the other presupposed charactcristic is "the more ultimate" of the two.

      It is now easy to define the notion of "Total Ultimate Presupposition". Suppose that, in a certain instance x, φ presupposes ψ₁, and ψ₂, and . . . ψ_n. Then:

1. We are to retain any of these which neither conveys nor is conveyed by any of the others.
2. Where two of them convey each other, we are to keep one and reject the other. It does not matter which of the two we keep and which we reject.
3. Where two of them stand in a relation of one-sided conveyance we are to keep the more ultimate and reject the less ultimate.

      1.31. The Principle of Total Ultimate Presuppositions. In § 196 McTaggart asserts that, if a characteristie has a presupposition at all (in a given instance), it must have a Total Ultimate Presupposition (in that instance). I have added the words in brackets, in order to take account of the fact that presupposition is a triadic relation, and that any intelligible statement in which it occurs must make mention of an instance as well as making mention of two characteristics.

      I propose to call the proposition stated above "The Principle of Total Ultimate Presuppositions". McTaggart attempts no proof of it, so it must be assumed that he regards it as self-evident.We must now consider whether it is self-evident.

      Let us first consider what would be involved if there were a case in which the Principle was not true. Suppose, if possible, that, in a certain instance x, φ had presuppositions but had no Total Ultimate Presupposition. This would mean that there is an unending series of characteristics, ψ_l, ψ₂, ..., such that

1. all of them, and also φ, characterise x,
2. φ partially conveys each of them,
3. each conveys the ones that precede it in the series, and
4. none of them conveys any one that follows it in the series.

      There are certain examples in which it looks at first sight as if this supposed state of affairs were impossible. But there are others in which it does not seem prima facie impossible. And I think that further reflexion will show that it is not certainly impossible even in cases of the first kind. I will now explain, and try to justify, these statements.

      (i) It certainly seems extremely plausible to hold that, if a particular x has a determinable characteristic, such as colour, it must have a certain absolutely determinate form of it. Suppose then that φ is the characteristic of being coloured, that ψ₁ is the characteristic of being red, that ψ₂ is the characteristic of being scarlet, and so on. Suppose that x is in fact coloured, red, scarlet, and so on. Then, it might be said, it is obvious that this series must end in a certain absolutely determinate shade, which is the colour of x. If so, the property of having this absolutely determinate shade will be the Total Ultimate Presupposition of being coloured in the instance x. There might be no objection to the series of more and more determinate shades being compact, and thus having an infinite numberof terms, like the series of ratios consisting of 1/2 and 1/1 and all the ratios between them arranged in order of magnitude. But, it will be said, it must have a last term, like the ratio 1/1 in this series; it must not be endless like the series 1, 2, . . . of the integers. I suspect that it was such series of more and more determinate specifications of a determinable which led McTaggart to think it self-evident that there must be a Total Ultimate Presupposition wherever there is a presupposition at all.

      (ii) Suppose that a certain particular x has the property of having a certain characteristic φ at some time within a certain period τ. Then it must either have φ for the whole period, or have it for less than the whole period. Suppose it has φ for less than the whole period. Then it must have φ for more than half the period or for not more than half the period. Suppose that it has φ for not more than half the period. Then it must have φ for more than one quarter of the period or for not more than one quarter of the period. Now consider the series of characteristics "having φ for less than τ", "having φ for not more than τ/2", "having φ for not more than τ/4", and so on without end. Each of them conveys all its predecessors and none of its successors. Each of them is partially conveyed by the property of "having φ at sometime within the penod τ". So, if all of them could belong to a particular x, the property of " having φ at some time within the period τ" would, in the instance x, presuppose all these other characteristics, and yet would have no Total UItimate Presupposition, since the series is plainly endless. Now, unless it were possible for all the members of such a series to belong to a particular, it would be impossible for any particular to have a characteristic at all within a period without having it for some finite time within this period. And, if this were impossiblc, all continuous change of a particular with respect to any characteristic would be impossible.

      Now this would not, of course, have worried McTaggart, who held, on other grounds that time and change are unreal. But I think that it may quite properly affect us, when we are asked to accept a certain highly abstract and unfamiliar principle as self-evident. We might admit that the of the principle does seem quite plausible at first sight, and that it does seem at first sight to be confirmed when we reflect on such a series of characteristics as being red, being scarlet, and so on. But we might add that it seems at least as plausible to hold that continuous change is metaphysically possible; and that, having now seen that the truth of the principle would be incompatible with the possibility of continuous change, we must simply suspend judgment about it.

      It may be remarked that a precisely similar argument would show that the principle is inconsistent with the possibility of there being a red band which varies continuously in shade from one end to the other. For let the band be of length x, and let s be a perfectly determinate shade of red which occurs somewhere within this band. Then the band will have a series of properties of the following kind, viz., "having the shade s throughout a length less than z", "having the shade s throughout a length not greater than x/2", "having the shade s throughout a length not greater than x/4", and so on. Each of these will be presupposed by the propertyof having the shade s somewhere within the band. Each conveys all that come before it, and is conveyed by all that come after it. And the series must be endless, if the band varies continuously in shade from one end to the other. And so, if the band is continuously shaded, the property of having the shade s somewhere within it presupposes, in the case of the band, an endless series of properties which give rise to no Total Ultimate Presupposition. Here again I would not suggest for a moment that this consequence of the Prineiple of Total Ultimate Presuppositions refutes the principle. There may be no bands which are continuously variable in shade from one end to the other. I only say that, when we see to what we are committed by accepting the principle, we may reasonably hesitate to accept it. Of course, anyone who still finds it self-evident on reflexion must still accept it and take the consequences .

      (iii) It only remains to point out that it seems quite possible to reconcile the doctrine that, if z has a colour at all, it must have a perfectly determinate shade of a certain colour, with the view that the series of more and more determinate shades under a given colour is endless, like the series of integers. Suppose that there is a compact series of perfectly determinate shades. Then to ascribe a relatively indeterminate shade to a particular might simply be to assert that its absolutely determinate shade fell somewhere or other between two shades in this compact series of absolutely determinate shades. Let us take an analogy. To say that a thing is red might be like saying that its length falls between 1/4 and 3/4 inches. To say that it is scarlet might be like saying that its length falls between 3/8 and 5/8 inches. To say that it had the shade of scarlet of a Cambridge Litt.D. gown might be like saying that its length falls between 7/18; and 9/16 inches. In general, to ascribe to it the nth relatively determinate shade in the series might be analogous to saying that its length falls between 1/2 (1 - 1/2") and 1/2 (1 + 1/2") inches. This series of ranges of length would be endless, and yet the thing would have the absolutely determinate length of 1/2 inch. Similarly, a particular x might have absolutely determinate shade of red, and yet the series of relatively determinate shades which are presupposed by its being red might be endless. So "being red" would have no Total Ultimate Presupposition in the instance x, though it would have an endless series of presuppositions in that instance.

      Now it is only a step from the last suggestion to the suggestion that the notion of relatively determinate shades is primary, and that the notion of absolutely determinate shades is definable in terms of it. May not the statement that x has a certain perfectly determinate shade of red mean that it has an endless series of relatively determinate shades of red, which converges in a certain characteristic way? The notion of absolutely determinate shade seems to me to have the same flavour of artificiality about it which attaches to the notions of points, of instants, of point-event-particles, and so on. Whitehead has suggested a method by which statements involving the latter terms can be replaced by statements about volumes, durations, etc., and their relations. I cannot help thinking that a somewhat similar method must be applied to statements in which the term "absolutely determinate shade of red", and similar terms, occur.

      When all the facts and possibilities which I have been mentioning are taken into account, I am not prepared to accept as certain that, in every instance where there is a presupposition at all, there is a TotaI Ultimate Presupposition. At any rate, it should now be plain that this principle ought not to be accepted light-heartedly and with so little diseussion as McTaggart bestows on it.

2. Requirement.

It remams to say something about the notion of "Requirement", which McTaggart introduces in connexion with Presupposition. In §184 he says that "the nature of presupposition may be expressed not unfairly by saying that X presupposes whatever it requires but does not supply". I think that the following example will make the notion of requirement quite clear. Suppose that a certain particular has the property of being a conic section. Then this conveys the disjunctive property of being either a circle or an ellipse or an hyperbola or a parabola or a pair of intersecting straight lines. It may be said then that the characteristic of being a conie section "supplies" this disjunctive property. Now suppose that the particular in question is in fact a circle. Since the property of being a conic section only partially conveys that of being a circle, we say that it "presupposes" circularity in this instance. Now, it seems to me, what the property of being a conic section has here failed to "supply" is the differentia between the generic characteristic of being a conic section and the specific characteristic of being a circle. This differentia is the characteristic of being a section perpendicular to the axis of the oone. Thus, it seems to me, the correct statement would be, not that being a conic section in this instance requires being circular, but that in this instance it requires being perpendicular to the axis of the cone. It supplies the property of being a section in some direction or other; it fails to supply the determinate direction; and yet in any particular instance the direction must be determinate. Thus what it requires in any particular instance is surely the determinate direction of the section. So I should say that being a conic section supplies being either a circle or an ellipse or an hyperbola or a parabola or two intersecting straight lines, that in this instance it presupposes being circular, and that in this instance it requires being a section perpendicular to the axis of a cone.

      On this interpretation the notion of requirement would cease to apply where there is no question of a differentia, as, for example, in the case of the determinable "being coloured " and the determinate 'being red". Suppose that a certain particular is extended, I should say that this supplies being red or blue or green or yellowor white or black. Suppose that this particular is in fact red. Then I should say that being extended in this instance prcsupposes being red. But there is nothing of which I could say that it was required in this instanee by being extended.