The Logical Simplicity of Predicates

The Logical Simplicity of PredicatesAuthor(s): Nelson GoodmanSource: The Journal of Symbolic Logic, Vol. 14, No. 1 (Mar., 1949), pp. 32-41Published by: Association for Symbolic LogicStable URL: http://www.jstor.org/stable/2268975 .

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THE JOURNAL OF SYMBOLIC LOGIC

Volume 14, Number 1, March 1949

THE LOGICAL SIMPLICITY OF PREDICATES

NELSON GOODMAN

1. The problem in general. In an earlier article,' I proposed a way of determining the relative simplicity of different sets of extralogical primitives. The calculations assumed a fully platonistic logic, committed to an indefinite hierarchy of classes, with sequences and relations defined as classes. Recently2 it has been shown that a nominalistic logic, countenancing no entities other than individuals, can be made to serve many of the purposes for which a platonistic logic had been thought necessary. The question naturally arises how we are to determine the simplicity of extralogical bases of systems founded upon a nominalistic logic. In such bases, the only extralogical predicates will be predicates (of one or more places) of individuals. The present paper offers a general method of measuring the simplicity of such bases.

We have first to consider by what standards we are to judge any proposed assignment of complexity-values. To reduce all our intuitive demands to precise principles would obviously be too difficult; but one basic demand is very roughly set forth as follows:

The assignment of complexity-values to predicates must be such that if each basis of kind A is always replaceable by some basis of kind B, while it is not the case that each basis of kind B is always replaceable by some basis of kind A, then any basis of kind B must, in the absence of other indications, have a higher complexity-value3 than any basis of kind A. The "always" indicates that the replacement must not depend on additional information that may sometimes not be available.

To illustrate the principle, every basis consisting of one one-place predicate is always replaceable by some basis consisting of one two-place predicate; for example, we may adopt as primitive the predicate "Q"4 - so explained that Q(x, y) if and only if P(x) and P(y) - and then define: "P(x) =,if Q(x, x)."5 But it is not the case that every two-place predicate can always be replaced by a one-place predicate. Accordingly, two-place predicates must have a higher value than one-place predicates.

This admittedly vague principle has to be used with the greatest caution.6

Received July 19, 1948. 1 On the simplicity of ideas, this JOURNAL, VOL. 8 (1943), pp. 107-121. 2 Steps towards a constructive nominalisnl by Nelson Goodman and W. V. Quine, this

JOURNAL, vol. 12 (1947), pp. 105-122. 3 Where the complexity-value of a basis is found by adding the values of the predicates

comprising the basis. 4 In such contexts in this paper, a capital letter is to be understood as an artificial abbre-

viation of some actual verbal predicate. The letter with quotes is thus to be taken as the name not of the letter but of the written-out predicate.

6 Here, and in two later cases, a definiens suggested to me by Prof. W. V. Quine has been adopted in place of a longer one I had originally used.

6 Indeed, the difficulties that may result from adopting this principle without all the reservations here outlined are so great that I rejected it entirely in my earlier paper. Yet it does seem to me to embody, however roughly, an intuitive demand that must be satisfied by any finally acceptable assignment of values. For that reason, I have in the present paper made what I hope is a sufficiently guarded use of the principle.

32



THE LOGICAL SIMPLICITY OF PREDICATES 33

Taken literally, it would wreak confusion upon all possible complexity calculations, since any two bases whatever are of some "same kind." Whenever we apply this principle, therefore, we must specify carefully what we mean by a "kind of basis"; and the choice of specifications has to be guided by quite inde- pendent judgments concerning what characteristics of predicates are pertinent to simplicity. At the start, we shall say that two bases are of the same kind if they contain a certain number (_ 0) of one-place predicates, a certain number of two-place predicates, and so on. Later other factors such as symmetry of predicates will have to be introduced. This progressive narrowing of kinds is the reason for the phrase "in the absence of other indications"; for example, we shall find that a two-place predicate in general must have a higher value than two one-place predicates, but that a symmetrical two-place predicate need not. But the principle stated above offers no guidance as to what "kinds" are relevant to complexity calculations; for that we must depend upon unformulated assisting intuitions.

We could hardly be content to leave the matter ultimately in this vague state. Rather, the above principle is to be used cautiously as a guiding rule in testing possible assignments of complexity-values, with the hope that this may lead to the discovery of some method of complexity measurement that will carry its own intuitive justification. In that event, we may then even be able to state the guiding rule more precisely.

Let us now see whether an assignment of the value n to n-place predicates in general will satisfy the guiding rule. One sort of case that might prove crucial is the relation between a basis consisting of one n-place predicate and a basis consisting of several predicates having a total of n-places. For example, a basis consisting of two two-place predicates and a basis consisting of one four-place predicate will each have the value 4. But if it is the case that (i) a four-place predicate cannot always be replaced by two two-place predicates while (ii) two two-place predicates can always be replaced by some four-place predicate, then a four-place predicate will have to have a higher value than two two-place predicates.

Now (i) is indeed the case. We might suppose that we could replace the four- place predicate "R" by "P" and "Q" - so explained that P(x, y) if and only if (3z)(3w)R(x, y, z, w), and Q(z, w) if and only if (3x)(3y)(Rx, y, z, w) - and then define "R(x,y,z,w) =df P(x,y).Q(z,w)." But suppose that "R(ab,cd)" and "R(eJgh)" are the only true singular sentences of "R"; then the proposed defi- nition will erroneously make "R(ab,g,h)", for example, also true. Does suppose tion (ii) also hold? The proposal here would be to replace "P" and "Q" by "R"- so explained that R(x,y,z,w) if and only if P(x,y) and Q(z,w) - and then define: "P(x,y) =df (3z)(3w)R(x,y,z,w)" and "Q(z,w) =df (3x)(3V)R(x,y,z,w)." This procedure fails if one but not both of the predicates "P" and "Q" is inapplicable-that is, such that no full sentence of it is true; for in that case "R" will also be inapplicable, and the definitions will thus make both "P" and "Q" inapplicable. Other procedures fail in other cases. It seems, then, that two two-place predicates cannot always-in particular, if we do not know whether they are applicable or not-be replaced by a four-place predicate. Therefore a basis consisting of one four-place predicate need not have a higher value than a



34 NELSON GOODMAN

basis consisting of two two-place predicates; hence there is no need on this score to assign n-place predicates a complexity-value different from n.

But while the tentative assignment of just the number of places as the complexity-value seems to satisfy our guiding rule, there are obviously many other assignments that would also satisfy it. Later (Section 4) we shall see that there are reasons for a different choice.

2. Applicable predicates. We saw that the proposed way of assigning complexity-values was saved only by the fact that some predicates may be inapplicable. Inasmuch as we usually intend to choose the primitives for any actual system from among applicable predicates, we are naturally led to inquire concerning the relative simplicity of bases stipulated to consist solely of such predicates. I shall at first consider this problem in isolation, leaving until later the question of the simplicity of such bases relative to bases for which this stipulation is not made.

Where all predicates are applicable, we can always replace a set of predicates having a total of n places by a single n-place predicate, while we cannot always replace an n-place predicate by a set of two or more predicates having a total of n places. Accordingly, an n-place predicate must have a higher complexity- value than a set of two more predicates having n places in all. For example, the value of a two-place predicate will have to be more than twice the value of a one-place predicate; the value of a four-place predicate will have to be more than twice the value of a two-place predicate, more than four times the value of a one- place predicate, and more than the sum of the values of a two-place and two one- place predicates.

How shall we assign complexity-values to meet these demands? The most obvious first suggestion is to assign to each n-place predicate the value 2n-1; so that one-place predicates will have the value 1, two-place predicates the value 3, three-place predicates the value 5, and so on. This will indeed conform to our guiding rule. But we want some further intuitive justification for this apparently arbitrary assignment. While our rule demands that the value of a four-place predicate be more than the sum of the values of a three-place and a one-place predicate, what decides that the excess shall be only 1? And is there anything in the nature of a six-place predicate to explain why it should have the value of 11 ?

Before trying to answer these questions, it might be worthwhile to look at some of the resulting values of various kinds of bases having a total of n places. The following table shows some values where the total number of places is six.

A ba8i8 COnsi8ting of: has the total value:

one 6-place predicate 11 one 5-place and one 1-place predicate 10 one 4-place and one 2-place predicate 10 two 3-place predicates 10 one 4-place and two 1-place predicates 9 one 3-place and one 2-place and one 1-place predicate 9 three 2-place predicates 9 one 3-place and 3 one-place predicates 8



THE LOGICAL SIMPLCITY OF PREDICATES 35

two 2-place and two 1-place predicates 8 one 2-place and four 1-place predicates 7 six 1-place predicates 6

Now examination of this table shows that every possible combination of two predicates having a total of six places has the value 10; that every combination of three such predicates has the value 9; every combination of four, the value 8; a combination of five, the value 7; and a combination of six, the value 6. Given the total number of places, the complexity-value diminishes as the number of predicates increases. Is this an acceptable result, and does it throw any light on the intuitive foundation of our arbitrary assignment of values?

In the first place, the result is acceptable in the sense that it is fully in accord with our guiding rule. We can in general replace a given basis by a more cohesive basis of an equal number of places, but we cannot in general replace a given basis by a more disjointed basis. Hence the more cohesive bases, in which more places are combined by predicates, should in general have higher complexity values than the more disjointed bases, in which there are more predicates and consequently fewer bonds.

Furthermore, our hitherto arbitrary assignment of complexity values can now be intuitively justified; for it can be explained as the total number of places in a basis plus the degree of what might be called its "effective cohesiveness" or "selectivity." We saw that from the two-place predicate "P" we could define two one-place predicates as follows: "Q(x) =df (3y)P(xy)", and "R(x) =df

(3y)P(y,x)". I shall speak of individuals that satisfy "Q" or that satisfy "R" as occupying respectively the first or second place of "P". A two-place predicate, besides differentiating from among all individuals those which occupy its first place and those which occupy its second place, also normally does something more: it joins each occupant of either place with some but not all the occupants of the other place. It is this selectivity which constitutes the excess of complexity of a two-place over two one-place predicates, since if two-place predicates always combined every individual that occupied the first place with every individual that occupied the second place, we could always replace a two- place by two one-place predicates. A predicate of more places is ordinarily more highly selective. A normal n-place predicate combines each occupant of each of its places with some but not all the occupants of each other place; and thus has n-1 joints. There are, of course, more than n -1 pairs of places in any pred-

7 If a predicate does not select from among all the possible combinations of occupants chosen in order from certain of its places, it will have fewer than n-I joints, as we shall see in Section 3. But predicates of this special kind are not in question at the moment, when only applicability (in addition of course to the number of places) is assumed. The degree of simplicity ascribed to a predicate with respect to certain stated characteristics is the maximum simplicity that can be inferred from those characteristics alone (or in other words-cf. Section 4-the complexity ascribed is the maximum consistent with those characteristics). Predicates which also have certain other characteristics may subse- quently be shown to be simpler. It must thus be borne in mind that many statements throughout the text are to be understood as qualified by some such phrase as "in the absence of other indications" (cf. Section 1) or "so far as determined solely by the characteristics now in question," even where no such term as "normally," "ordinarily" or "in general" is inserted as a remainder.



36 NELSON GOODMAN

icate of more than two places. But the number of joints is not the total number of such pairs, since the connections established by each pair will also be more remotely established by two or more other pairs taken together. The number of joints is rather the number of places in any minimum set of pairs of places that ties all the places of the predicate together-i.e. that provides between every two places in the predicate a route consisting entirely of steps between places paired in the set.

Now in each case in the above table it will be found that the value of a basis is the total number of places in the basis plus the total number of joints, and this holds good where more and longer predicates are involved. We have seen how and why joints may be definitionally dissolved (that is, any set of predicates having a total of n places may be defined in terms of an n-place predicate) but not in general definitionally created (that is, not every n-place predicate can be defined in terms of two or more predicates having a total of n places). Thus a basis is richer accordingly as it contains more places and more joints. Here then is the intuitive justification for assigning to n-place predicates the complexity- value 2n -1, and to a basis the sum of the thus-determined values of the predicates it contains.

3. Symmetry and other factors. When we evaluate a given predicate as proposed above, we take into account only its number of places and the fact of its applicability. Where more factors are known and considered, the complexity- value of certain predicates may be lowered. This is true, for example, where symmetry is in question. Two one-place predicates cannot always be replaced by some symmetrical two-place predicate; a symmetrical two-place predicate therefore-unlike a non-symmetrical one-has on that score no claim to a value more than twice that of a one-place predicate. More generally, a set of predicates having a total of n places cannot always be replaced by a thoroughly symmetrical8 n-place predicate. Thus we need to examine the relative simplicity of predicates that are stipulated to be symmetrical or partially symmetrical. Let us continue in the present section to confine our attention to applicable predicates.

A one-place predicate can always be replaced by a symmetrical predicate having two or more places; for example, let the one-place predicate "P" be replaced by the three-place predicate "Q" - so explained that Q(x,y,z) if and only if P(x).P(y).P(z) - and define: "P(x) =df Q(x,x,x)." But a symmetrical predicate having two or more places cannot in general be replaced by a one-place

8 Thoroughly symmetrical predicates I shall hereafter call symmetrical. A symmetrical predicate is one such that if any selection of individuals satisfies the predicate in one order then they also satisfy it in every other order. A partially symmetrical predicate is one which is symmetrical with respect to some but not all of its places; for example, the three-place predicate "P" is partially symmetrical if it is the case that (x) (y) ((3z)P(x,y,z) D (3w) P(y,x,w) 1. A predicate may be partially symmetrical with respect to scattered as well as to adjacent places. By a non-symmetrical predicate, I mean one that is not even partially symmetrical.




predicate. Thus a symmetrical predicate of more than one place must have a higher complexity-value than a one-place predicate.

A predicate of more than one place is not always replaceable by some symmetrical predicate having a greater number of places. For example, no matter how we explain a symmetrical three-place predicate in terms of an ordinary non- symmetrical two-place predicate, we shall be unable to redefine the two-place from the three-place predicate. Thus a symmetrical predicate need not on this score have a higher value than a non-symmetrical predicate of fewer places.

A symmetrical predicate, however, may always be replaced by a symmetrical predicate of more places. For example, let "Q" be so explained in terms of the symmetrical two-place predicate "P" that Q(x,y,z) if and only if P(x,y).z=x . v . P(x,y).z = y . v . P(yz).x= y; then define "P(x,y) =df Q(x,y,y)". Accordingly, a symmetrical predicate must have a higher complexity-value than any symmetrical predicate of fewer places.

The requirements so far observed may be satisfied by assigning to n-place symmetrical predicates the value n. But we have also the problem of evaluating partially symmetrical predicates; for instance, a three-place predicate that is symmetrical with respect to two of its places but not with respect to either of these and the remaining place. Such a three-place predicate can always be found to replace a basis consisting of a one-place and a symmetrical two-place predicate, but the converse does not hold. Thus the value of such a predicate must be higher than 3. Our general question now is whether some refinement of our method of counting places and joints will enable us to deal with all symmetrical and partially symmetrical predicates?

What is important about a symmetrical predicate is that the occupants of every place are the same as the occupants of any other place; or in other words, that every argument that appears in any place in some true singular sentence of the predicate also appears in each other place in some true singular sentence or other of the predicate. Thus, if "P" is a two-place symmetrical predicate and we define: "Q(x) =df (3y)P(x,y)", and "R(x) =df (3y)P(y, x)," then "Q" and "R" will be identical predicates; while if "P" is non-symmetrical, "Q" and "R" may be non-identical. In general, then, a predicate is obviously the more complex the fewer of its places have the same occupants; for a set of n different one-place predicates can in general be replaced by a predicate of n or more places only if at least n places of this predicate are such that no two have all the same occupants. Therefore, since all the places that have the same occupant as any given place belong to what I call a segment, clearly segments rather than places are to be counted in determining complexity. In an ordinary non-symmetrical predicate, each place may be a distinct segment; but a symmetrical predicate of any number of places has but one segment. A partially symmetrical n-place predicate may have any number of segments up to n -1.

A predicate may, indeed, have zero segments. If a predicate is known to apply to everything, it can be dropped without loss and thus is not more complex than a basis consisting of no extralogical predicates. Selectivity is the essential feature with places as well as with joints; for a predicate is obviously entitled



38 NELSON GOODMAN

to be counted only if it is not known to be satisfied by everything. If the two- place predicate "Q" is such that for every x there is some y such that Q(x,y), and for every y there is some x such that Q(x,y), then everything satisfies each place of "Q", and "Q" has zero segments even though it has one joint.

The number of joints is unaffected by symmetry. Although a symmetrical two-place predicate has the same occupants in its two places, it combines each of these with some but not necessarily all the others. This is just the selectivity that constitutes having a joint. Here the joint represents the excess of complexity of a two-place symmetrical over a one-place predicate. A three-place symmetrical predicate, similarly, normally has two joints. In general, symmetrical and non-symmetrical n-place predicates alike have n-1 joints.

If we now determine complexity by taking the total number of segments and joints, the value of n-place non-symmetrical predicates in general remains at 2n -1; but the value of symmetrical predicates becomes n, since they have one segment and n -1 joints. The value of partially symmetrical predicates will yary with the degree of symmetry. Take a seven-place predicate that is symmetrical with respect to its first three places, also with respect to its fourth and fifth places, also with respect to its sixth and seventh places, but non-symmetrical with respect to each two of these three sets of places. It has, then, three segments and six joints; hence a total value of 9. A thoroughly symmetrical seven-place predicate, in contrast, has the value 7; while a seven-place predicate that is symmetrical with respect to some two of its places only has the value 12- just one less than the value of a completely non-symmetrical predicate. The value of a three-place predicate is 3 if the predicate is symmetrical, 4 if it is partially symmetrical, 5 if it is non-symmetrical.

Reduction in the ratio of segments to places may result from factors other than symmetry. If the occupants of two or more places of a non-symmetrical predicate are the same, then these places are comprised within one segment, and the effect on complexity is the same as if the predicate were symmetrical with respect to these places. If, for example, "P" is such that only P(a,b,c) and P(b,c,a) and P(c,a,b), then "P" has one segment and two joints - a value of 3.

Still other factors may reduce the number of joints rather than the number of segments. Let us call a two-place predicate "self-complete" if it combines each occupant of its first place with each occupant of its second. Such a predicate has two segments but no selecting joint. Its value, therefore, is 2. It can always be replaced by two one-place predicates. An n-place non-symmetrical self- complete predicate has in general a value of n; while an n-place symmetrical self-complete predicate-which is always replaceable by a one-place predicate- has a value of 1.

It is also because self-complete predicates contain no joints that a predicate corresponding to a platonistic "ordered pair" has a value of 2. If a and b are individuals, and P(x,) if and only if (x=a).(y=b), then "P" is self-complete; it has two segments and no joints. In general, the value of a predicate corresponding to a sequence of n components is n.

The complexity-value of an individual-name such as "a" is the same as that of a one-place predicate such as "is identical with a" - and is thus 1.




Some predicates are self-complete with respect to some but not all their places. A predicate is self-complete with respect to two, not necessarily adjacent, places if it combines every occupant of each place with every occupant of the other; for example, the three-place predicate "P" is self-complete with respect to its first and third places if

(x)(y) {(g3r)(g38)(g3t)(g3w)(P(x,r,8).P(t,w,y)) D (3z)P(xzx,y) I.

A predicate is self-complete with respect to three of its places if it combines every three individuals such that each of the places in question is occupied by one of the three; and so on for more places. Now joints, of course, occur only between sets of places with respect to which a predicate is self-complete. The number of joints a predicate has is thus one less than the minimum number of separate "non- interselective" sets of this kind which together exhaust, all the places of the predicate. For example, if a five-place predicate is self-complete just with respect-to its second, third, and fifth places, its places distribute among three separate non-interselective sets. Two of these consist of one place each (first, and fourth), the other of three. The presence of the three-place set shows that two joints are missing; hence the total number of joints in the predicate is two less than n -1 - that is, 2. The total complexity-value of the predicate is of course still the number of joints plus the number of segments.

We have seen that a place is counted only if it is not satisfied by every individual, that two places are counted as two only if they are satisfied by at least some different things, and that there are said to be joints only where the predicate effects some but not all the possible combinations of the occupants of several places. Thus to measure complexity by the number of segments and joints is to measure it by degree of selectivity.

4. A general calculus of complexty. The actual number of segments and joints a basis has might be said to constitute its real complexity; but what normally concerns us is rather complexity relative to the information at hand. The effective complexity of a basis is the complexity-value as computed by taking into account only those characteristics of the basis that are revealed by that information. If, for example, we do not know whether a given primitive is symmetrical or has some one of the other characteristics that increase simplicity, then in computing the effective complexity of the primitive we proceed as if the predicate had none of those characteristics. The predicate may actually be symmetrical; and relative to further information that reveals that symmetry, the predicate will have a lower complexity-value. The effective complexity of primitives thus varies as information varies. Sometimes the effective complexity of a primitive relative to a given body of information is said to be the complexity of a given primitive "idea."

The effective complexity of a primitive predicate is, then, the maximum number of segments and joints it may have compatibly with the information at hand. Unless there is something to the contrary, an n-place predicate will thus have the value 2n -1. If the information available reduces the maximum number of segments or joints the predicate can have, the effective complexity of the predicate is reduced by just that much.



40 NELSON GOODMAN

This general principle may be applied to what was said earlier concerning the complexity of predicates when applicability is not stipulated. It is clear that no predicate, applicable or inapplicable, may have a total of more than 2n -1 segments and joints. Hence if nothing, not even applicability, is known concerning a primitive predicate, its effective complexity is 2n -1. This evaluation satisfies the guiding rule stated in Section 4 quite as well as our original proposal to assign the value n in such cases; and moreover follows from the general method of complexity measurement we have now developed. Where applicability of the predicates involved is stipulated, the value remains unchanged. If inapplica- bility is stipulated, a predicate can have neither segments nor joints, and so has zero complexity.

We saw that the guiding rule set forth in Section 1 was vague and to be used, in conjunction with other intuitions, only in the effort to discover some inde- pendently tenable method of measuring complexity. We have now formulated such a method. In terms of it we can even clarify the guiding rule somewhat, since we can now say explicitly that we regard two bases as of the same "kind" if and only if each has a certain specified number (> 0) of predicates having a maximum of no segments or joints, a certain number having a maximum of one segment and no joints, a certain number having a maximum of no segments and one joint, a certain number having a maximum of one segment and one joint, and so on.

It has been assumed, so far, that we should assign to a basis the sum of the values of its component predicates. In practice this usually works well enough. But strictly we ought first to compound into a single predicate (in the obvious way) all the primitive predicates of the basis that are stipulated to be applicable. This will not add new and superfluous joints, since the new predicate will combine every set of individuals combined by one of the original predicates with every set combined by every other such predicate. But it will make a difference in some cases because such consolidation of predicates will eliminate the separate counting of identically occupied places in different predicates. For example, if some place in "P" has just the same occupants as some place in "Q", the two places will count for two so long as they are places in different predicates; but if "P" and "Q" are replaced by "R"-so explained that R(x,y) if and only if P(x) and P(y)-the two places of R, having just the same occupants, will com- prise but one segment and so count for but one unit of complexity-value.

We may now summarize the general method for determining the effective complexity of a basis. First, consolidate all the primitive predicates that are stipulated to be applicable (and this will ordinarily mean all the predicates in the basis) into a single one. Then value all the resulting predicates, applicable or inapplicable and whatever their special characteristics may be, by determining the maximum number of segments and joints they may have compatibly with the information at hand, Finally, add these values together to determine the effective complexity of the complete basis.

5. Consequences for the complexity of classes. I shall conclude by out- lining briefly the relationship between my earlier article and the results of the present study of simplicity.




For the problem as stated in the earlier paper, the method of computing the complexity of bases for platonistic systems remains unaffected. But if the stronger guiding principle used in the present paper but rejected in the earlier paper is imposed, and if also we want to take into account the effects of assuming non-nullity (the analogue of applicability), then a thoroughgoing revision in the light of the present study will be called for. Some complications will arise from the fact that in a platonistic system a class of any type or complexity may be denoted by a simple primitive term such as "A"; and account will have to be taken of the effect of stipulating not merely that A is not null but, further, that all or just so many of its members, members of its members, and so on are not null. Also, while for nominalistic systems we get a direct quantitative compari- son of any two bases, for platonistic systems we still get only a topological com- parison whenever the primitive is of such a type and character that the partially unspecified quantity u is present in the complexity-index. However, for those who care to use platonistic primitives, this is the outline of a procedure for re- vising the method of computing their simplicity so as to comply with the more exacting demands made in this paper.

UNIVERSITY OF PENNSYLVANIA



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The Logical Simplicity of Predicates

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