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This entry consists of the following sections. Section 1 attends closely to Chapter XI, of W. E. Johnson's Logic, Part I. Section 2 briefly discusses Johnson's use of the determinate-determinable relation elsewhere than Chapter XI of Logic, Part I, and connects this with A. N. Prior, Determinables, Determinates, and Determinants (1949). Section 3 describes a 1959 symposium between Stephan Körner and John Searle entitled On Determinables and Resemblance and examines both contributions critically. Section 4 describes the Munsell Color Solid so that color examples can be more exact. Section 5 describes and criticizes attempts to define the determinate-determinable relation by means of predicate entailment in the style of Searle. Section 6 pays more attention to this distinction and directs attention to a certain understanding of disjunctive predicate. Contrived disjunctive and conjunctive predicates are the typical cause of difficulties in attempts to define the determinate-determinable relation. Section 7 distinguishes independent predicates from non-independent predicates, and thus distinguishes disjunctive and conjunctive predicates from non-disjunctive and non-conjunctive predicates, in a way that assumes no prior classifications of determinates under a determinable. Section 8 explores a view advanced by Johnson and endorsed by many others that the things in the world, as distinguished from our descriptions and conceptions of them, are absolutely determinate. This section entertains the contrary view that nothing is absolutely determinate.
Johnson is interested in the logical differences between Red is a colour and Plato is a man. He says that the second sentence involves adjectival predication. Human is an adjective (property) predicated of Plato. Colour, on the other hand, is not an adjective (property) predicated of red (1921, p. 176).
The relation of a determinate to its determinable resembles that of an individual to a class, but differs in some important respects. For instance, taking any given determinate, there is only one determinable to which it can belong. Moreover, any one determinable is a literal summum genus not subsumable under any higher genus; and the absolute determinate is a literal infima species under which no other determinate is subsumable. (Part I, Introduction, 1921, p. xxxv)What makes red and blue and Carolina Blue all colors? Johnson denies that there is some property [some secondary adjective] that red and blue share that makes them both colors. The view that color is itself a property that red and blue share requires an explanation of what distinguishes the color red from the color blue. Explanations such as Red is the color of fire trucks and Blue is the color of my true love's eyes miss the point. They refer to mere contingent facts. An appropriate explanation should provide a necessary truth such as Triangles are three-sided polygons.
Rather than resemblance, the sharing of some property, that make red and blue colors, says Johnson, it is differences between colors.
In fact, the several colours are put into the same group and given the same name colour, not on the ground of any partial agreement, but on the ground of the special kind of difference which distinguishes one colour from another; whereas no such difference exists between a colour and a shape. (1921, p. 176)The absence of a difference or exclusion of this kind explains why red and square is not a determinate of a single determinable. Taking any given determinate, there is only one determinable to which it can belong (1921, p. xxxv). The nature of the exclusion also explains why no two determinates of the same determinable can qualify exactly the same entire spatio-temporal part of any object (1921, p. 181).
Arguments about the incompatibility of colors in the 1950s and 1960s were concerned with theories of necessity, analyticity, the a priori, and meaning. (Edwards and Pap, 1973, has a bibliography of such works, pp. 745-746.) These discussions rarely address specifically the relation between determinables and determinates. Nor do assumptions about the incompatibility of determinates often connect with the problem of understanding incompatibility in general. When the problem of incompatibility reopens there may be less emphasis on philosophy of language and logic and more emphasis on the branch of metaphysics that studies the nature of properties and the recently formed hybrid of metaphysics and science that studies the nature of color and other sensory qualities.
Consider Johnson's examples. The introduction of the term determinable in Chapter XI reads: I propose to call such terms as colour and shape determinables in relation to such terms as red and circular which will be called determinates (1921, p. 174). So circular is a determinate of the determinable shape despite the existence of a proper definition that distinguishes circles from other shapes. Different shapes are incompatible and are therefore under the same determinable. Being related by incompatibility (in the right way) appears to be necessary and sufficient for items to be determinates under a single determinable. Some determinates such as red cannot be differentiated by a traditional, conjunctive genus-species definition. Others such as square and circular can be so differentiated. Johnson's example of shape shows that a determinable-determinate relation does not require the impossibility of a conjunctive definition.
Rather than insist on a sharp contrast, Johnson attempts to subsume traditional species-genus relations under determinate-determinable relations:
We have now to point out that the increased determination of adjectival predication which leads to a narrowing of extension may consistnot in a process of conjunction of separate adjectivesbut in the process of passing from a comparatively indeterminate adjective to a comparatively more determinate adjective under the same determinable. Thus there is a genuine difference between that process of increased determination which conjunctivally introduces foreign adjectives, and that other process by which without increasing, so to speak, the number of adjectives, we define them more determinately.Johnson goes on to provide a symbolic representation of botanical classification in which there are five determinables. One of these determinables is number of cotyledons under which fall the determinates acotyledon, monocotyledon, and dicotyledon. The other determinables concern stamens, the corolla, forms of attachment, and divisibility. The determinables in this botanical example represent the summum genus plants as describable under these five heads (1921, p. 180). A determinable can have several dimensions.
In fact, the foreign adjective which appears to be added on in the conjunctive process, is really not introduced from the outside, but is itself a determinate under another determinable, present from the start, though suppressed in the explicit connotation of the genus. (Johnson, 1921, pp. 178-9)
Johnson's discussion of color variations illustrates such dimensions. Colors vary according to hue, saturation and brightness, and these variations are independent of one another. If hue, saturation, and brightness are determinables, they are not separate, since they depend on each other. There cannot be saturation without hue, for example, even though no determination of saturation requires any particular determination of hue. Johnson says that the determinable color is single, though complex, in the sense that the several constituent characters upon whose variations its variability depends are inseparable (1921, p. 183).
There is a difficulty here because the dependence patterns between the three variables are not entirely uniform. Hue and saturation cannot exist without each other, or without brightness, but degrees of brightness do not require either hue or saturation. Black and white movies and photographs and many other achromatic examples come to mind. Dimensions of quality space can vary in their dependence relations on each other.
In this case the several determinates are to be conceived as necessarily assuming a certain serial order, which develops from the idea of what may be called adjectival betweenness. The term between is used here in a familiar metaphorical sense imagined most naturally in spatial form. (1921, pp. 181-182)The three-place relation (Dabc) the difference between a and c is greater than that between a and b, however, does not by itself provide an adequate definition of between. A diagram helps to illustrate this point. Assume for the purpose of diagramming that the difference between a and c specifies a certain distance in quality space. A circle with center a and radius ac represents points in the space at distance ac from point a. Any distance between point a and any point b within this circle is less than the distance between a and c. A point b within this circle represents Dabc. As Figure 1 illustrates, Dabc is consistent with the distance between b and c being greater than the distance between a and c.
In this situation, b is not between a and c in any sense. For example, the difference (or distance) between red and yellow is greater than the distance between red and purplish red, but purplish red is not between red and yellow.
The word betwixt, which appears in the next paragraph, comes from Goodman (1951), pp. 244-253, but only the word, not Goodman's definition or intended sense. This is an occasion to remark that Goodman and Carnap (1928) develop constructions of quality order much more elaborate than Johnson's. They do not use a primitive equivalent to Dabc in their constructions. Johnson returns to questions of quality order in Part II of Logic, Chapter VII, The Different Kinds of Magnitude.
The following conjunctive definition, which overcomes this particular difficulty, is not a revision of a formulation by Johnson. It is the beginning, rather, of a brief attempt to define betweenness by means of Johnson's primitive Dabc: Let us say that b is betwixt a and c if and only if Dabc and Dcba. Orange, but not reddish purple, is betwixt red and yellow. Figure 2 adds to the circle in figure 1 another circle with the same radius with point c in the middle. A point b betwixt a and c is within the intersection of these two circles.
Betwixtness is too wide a notion to explicate betweenness. Suppose that the points in Figure 3, a specification of Figure 2, stand for the following colors:
R: a fully saturated, bright sample of red Y: a fully saturated, bright sample of yellow O1: a fully saturated, bright sample of orange O2: a less saturated, less bright sample of orange
O1 and O2 are both betwixt R and Y. But it is natural to represent O1 as right between R and Y. O2 is somewhat off to the side. A better definition of between will count O1 but not O2 as between R and Y.
Relying again on the notion of distance, one can distinguish two senses of between. (1) B is somewhere between A and C if and only if the distance between A and B plus the distance between B and C is equal to the distance between A and C. That is, B is located somewhere on the straight line (in Euclidean space) between A and B. (2) B is exactly or halfway between A and B if and only if B is somewhere between A and C and also the distance between A and B is equal to the distance between B and C. The following definition, built on Johnson's primitive, attempts to define somewhere between in sense (1):
b is somewhere between a and c if and only if b is betwixt a and c, and nothing is both betwixt a and b and betwixt b and c.When two circles with no interior points in common are tangent, the point in common is on the straight line segment between the two centers. Any point on a straight line segment between points x and y is the point in common between two circles with centers x and y that have no interior points in common. As Figure 4 illustrates, b is on a straight line segment between a and c if and only if the circle ab is tangent to the circle cb. This is the case if and only if nothing is betwixt a and b and also betwixt b and c; for any such thing has to be both an interior point of circle ab and an interior point of circle cb, and these circles have no interior points in common.
Figure 5, illustrates a point b that is betwixt a and c but is not situated like point b in Figure 4. In this case, point b is on the intersection of two circle that have interior points in common. Figure 5 shows this region as a shaded area. Anything within this shaded area is both betwixt a and b and betwixt b and c.
Although a distance between points in these diagrams can be equal, or double, or half, another distance between points, that is due to the conventions of drawing these diagrams. There has been no explication of these distance notions by means of the primitive Dabc. A definition of right between would provide a sufficient condition for the equality of the distance between a and b and the distance between b and c, but there is no attempt here to provide such a definition using only the primitive Dabc.
Johnson uses his notion of betweenness to draw two independent distinctions between quality order, interminable series in contrast with cyclic, and continuous series in contrast with discrete (1921, pp. 182-183). His use of the three-place relation the difference between a and c is greater than that between a and b undermines a point he insists on earlier that resemblance between determinates does not group them under a determinable. Johnson's three-place relation can also be expressed the similarity between a and b is greater than that between a and c. Comparisons between differences are also comparisons between likenesses or similarities. Perhaps his point can be expressed as follows: no two-place relation of resemblance groups determinates under a determinable, although a three-place relation can be useful for this purpose. Johnson's chapter ends with the pronouncement that "The practical impossibility of literally determinate characterization must be contrasted with the universally adopted postulate that the characters of things which we can only characterize more or less indeterminately, are, in actual fact, absolutely determinate" (1921, p. 185). Section 8 examines this claim.
In Part II, Johnson refers to determinables in several different contexts. One discussion is especially important for understanding Prior's later treatment of the topic. Johnson introduces the notion of a structural proposition which he compares to what Kant meant by analytic (1922, pp. 14-15). In a structural proposition, it is impossible to realise the meaning of the subject-term without implicitly conceiving it under that category (1922, p. 15).
Arthur N. Prior takes up the question of structural propositions that relate determinates to determinables in the two-part article Determinables, Determinates, and Determinants (Prior, 1949).
Since a subject's being in a certain universe or category, i.e., its being determinable in certain ways, is presupposed in every genuine characterization of it, an assertion that it is in this category, and is thus determinable, would have for its predicate something which cannot really be separated from the subject in order to be predicated of it. (Prior, 1949, p. 18)Prior's article reveals his very wide-ranging knowledge of the history of logic. The article together with Prior (1955), reflects a detailed knowledge of Johnson's entire logical system, not only the three-part Logic (1921, 1922, 1924), but also Johnson (1892). Although some of the topics in Prior (1949) have not prompted much subsequent discussion in connection with determinates and determinables, Prior puts his finger on one theme that is now central.
The problem of fitting the relation between determinates and determinables into a purely conjunctional logic might be summarily described as the problem of justifying the inference from This is red to This is coloured on the assumption that all formal inference consists in the passage from a conjunction to one of its conjuncts. (Prior, 1949, pp. 191-192)As mentioned earlier, there seems to be no conjunction of the proper kind of the form x is F and x is colored that is equivalent to x is red. Examples of improper conjunctions are:
x is red and x is colored,
x is either red or not colored, and x is colored.
Körner presents a logic of inexact concepts which reappears in his 1966 book Experience and Theory. This logic recognizes, in addition to traditional set members and non-members, intermediate or neutral set members. Two overlapping sets are related by exclusion-overlap if, by stipulating of each neutral candidate that it is either positive or negative, it is possible to end up with two overlapping sets and it is possible to end up with two sets related by exclusion (Körner, 1966, pp. 45-46. This clarifies or revises Körner, 1959, pp. 127-128). He gives blue and green as examples to illustrate exclusion overlap. Blue and green (strictly, the set of blue things and the set of green things) are not absolutely exact; they have neutral candidates. Since nothing is a member of the green set and also a member of the blue set, these sets exclude each other. When each neutral candidate is converted by stipulation either to a member or to a non-member, the two adjusted sets may still exclude each other, because no neutral candidate has been designated both green and blue, or if there is at least one formerly neutral candidate that is now both green and blue, the adjusted sets overlap.
Körner claims that determinates under the same determinable are linked, directly or indirectly, by exclusion-overlap. Red and green are not directly related by exclusion overlap, but they are presumably related indirectly by direct links between red and orange, orange and yellow, yellow and yellowish green, yellowish green and green. The concepts of red and green are therefore linked. Concepts are linked if and only if they are related by exclusion overlap or the ancestral of exclusion overlap (1959. pp. 130-131). Concepts P and Q are fully linked if and only if every species of P is linked with every species of Q (1959, p. 131). Full linkage is crucial to Körner's treatment of determinates and determinables.
Körner attempts to explain the determinate-determinable relation by means of full linkage. He asserts that full linkage is a stronger relation than mere linkage. It is difficult, however, to be convinced that this is true. If at least one species of P is linked to at least one species of Q, and all the species of P are linked to each other, as are all the species of Q, then every species of P is linked with every species of Q, and P and Q are fully linked. Körner claims that angry, a species of yellow or angry, is not linked with green. This would be an interesting and important result, if true. Körner here identifies a problem that occupies subsequent discussions. How are we to distinguish ordinary predicates such as green from disjunctive predicates such as yellow or angry?
It appears that on Körner's own definitions, green and yellow are linked, yellow is linked to yellow or angry, and yellow or angry is linked to angry, so green is linked to angry. Until explanations are forthcoming how one or more of these alleged linkages violate Körner's requirements, the main influence of his contribution is to direct attention to the problem of disjunctive predicates.
John Searle makes a fresh start. In his attempt to explicate the determinate-determinable relation, he uses the notion of predicate entailment. In the standard sense, entailment is a relation between items, such as propositions, that have truth-values. Searle extends this notion to a relation between predicates. Red entails colored because it is impossible for something to be red and not colored. This is a natural extension, and others have adopted it. Indeed, talk of predicate entailment leads easily to talk of property entailment: the property red entails the property colored.
Searle and others who follow him draw a sharp distinction between the determinate-determinable relation and the genus-species relation. (He repeats this distinction in Searle, 1967.) The definition of a species is by means of genus and differentia, which are logically independent. (Predicates F and G are independent when none of the following entailments hold: F entails G, G entails F, F entails not-G, and not-F entails G.) A determinate of a determinable cannot be defined in this way, by a conjunction of independent predicates. A traditional (although inadequate) definition Man is a rational animal passes the genus/differentia test. Rational and animal are independent terms. The attempted definition Red is a color that is red does not pass because red entails colored.
There are both historical and logical difficulties with this view.
The genus-species relation is an ancient philosophical topic. No crisp, clear definition can be consistent with everything that has been said before. Searle's confident exposition, however, contradicts some standard views. A logic text in wide use for many decades gives the following as a rule of definition:
The better the definition, the more completely will the differentia be something that can only be conceived as a modification of the genus: and the less appropriately therefore will it be called a mere attribute of the subject defined. (Joseph, 1916, p. 112)Aristotle mentions differentia that entail the genus, as in Walking animal (Topics, IV. 6) and Footed animal (Metaphysics, Z. 12). In his Commentary on Z. 12, Bostock says that the first differentia should entail the genus (Aristotle, 1994, pp. 176-184). Other philosophers have adopted Searle's proposal, so it is evidently attractive. It does not represent a consensus of earlier writers.
The nature of conjunction poses a logical problem for Searle's account of species. If two conjunctions are logically equivalent, it does not follow that the conjuncts of one are logically equivalent to the conjuncts of the other. The forthcoming example concerns conjunctive propositional functions about pure numbers. It is easy to construct parallel examples about mass, length, temperature, years of service, taxable income, and so on.
Ax: x is greater than 4 but less than 7
Bx: x is greater than 4 but less than 6.
Cx: x is greater than 5 but less than 7.
Dx: x is less than 6.
Ex: x is greater than 5.
Fx: Bx & Cx.
Gx: Dx & Ex.
Hx: Bx & Ex.
Ix: Cx & Dx.
The last four predicates, Fx, Gx, Hx, and Ix, are equivalent, so they entail the same predicates and are entailed by the same predicates. They all entail Ax, and Ax entails none of them. Their conjuncts, by design, have various entailment relations. Both conjuncts of Fx entail Ax. Neither conjunct of Gx entails Ax. One conjunct of Hx and of Ix entails Ax and the other conjunct in each case does not.Searle says that a species is a conjunction of two logically independent propertiesthe genus and the differentia (1959, p. 143). Does he mean (a) that every conjunction equivalent to the species satisfies this requirement or (b) that at least one conjunction satisfies the requirement? Stipulation (a) is too difficult to satisfy, for any species is equivalent to the conjunction of itself and the genus. Stipulation (b) is too easy to satisfy, as will be shown next. The following predicates continue the numerical example:
Jx: x is greater than 5 but less than 6.
Kx: Jx or (x is greater than 2 but less than 3).
Lx: Ax & Kx.
Jx and Lx are equivalent to each other and also to Fx, Gx, Hx, and Ix. If we consider Ax to be the genus, then Lx is a conjunction of the genus and a term Kx logically independent of the genus. One can perform a trick of the same kind with the standard example of color. Consider the following contrived genus and differentia definition of red as a species of the genus colored:
x is red =df (x is colored) & (x is red or x is not colored).
Searle's distinction between genus-species and determinate-determinable requires some principled way of excluded disjunctive predicates such as Kx and x is red or x is not colored. Explaining the determinate/determinable relation requires this anyway, whether or not accepts Searle's views about the relation of species to genus.Searle attempts to define the determinate-determinable relation and to eliminate hybrid, cross-type conjunctive and disjunctive predicates by using only the relation of predicate entailment. When a predicate A entails a predicate B, but B does not entail A, Searle says that A specifies B (1959, p. 145). A is a non-conjunctive specifier of B if and only if A specifies B and there is no pair of terms C and D such that A is equivalent to (entails and is entailed by) the conjunction of C and D, C specifies B, D does not entail B, and not-D does not entail B. Take the letters A, B, C, and D as abbreviations for some new predicates:
A: redA specifies B. There are terms C and D such that A is equivalent to (C & D), C specifies B, D does not specify B, and not-D does not specify B. So according to this definition, red is not a non-conjunctive specifier of colored, a result that is opposite to what Searle intends. He says that a necessary condition of As being a determinate of B is that A is a non-conjunctive specifier of B. Like everyone who addresses this topic, Searle takes the relation between red and colored to be a paradigm of the determinate-determinable relation.
C: colored but not green
D: red or (not colored and not prime)
Searle adds another condition with the intention of excluding disjunctive predicates such as yellow or angry. A determinate of a determinable must not only be a non-conjunctive specifier of the determinable, it must be logically related to all other non-conjunctive specifiers of the determinable. Suppose that the definition of non-conjunctive specifier can be emended somehow to allow red as a non-conjunctive specifier of colored. The new requirement of logical independence wrecks the project once again because it eliminates colored as a determinable. Consider darkish red to darkish orange. Darkish red to darkish orange and red are not logically related. Some things are both; some are neither; some are the first but not the second; some are the second but not the first. So Searle's requirements again disqualify his paradigm. (A similar objection occurs in Sanford, 1970, pp. 162-163.)
Any pair of surface color that the human eye can distinguish with respect to any color dimension, hue, brightness, or saturation (chroma), corresponds to a pair of points in The Munsell Color Solid. Estimates of the number of distiguishable colors range from two million to five million. For all practical purposes, one can regard the Munsell Color Solid as a continuum of colors. Pictures of the Munsell Color Solid, however, often depict the whole solid as contstructed out of 267 blocks of uniform determinate color (as in the color plates in Webster's Third or the Adobe Technical Guide to the Munsell Color System). These standard color names are ambiguous between determinate and determinable. For each of the 267 regions in the color solid, there is a determinate representative color, the center of gravity or centroid color. These are the colors of the Centroid Color Chips that science and commerce use to standardize color descriptions of minerals, paint, dye, ink, plastic, and so forth. Pale yellow is both the name of a determinate centroid color and a determinable color. The scientists who construct the color solid regard the determinable use as primary.
There are many pairs of easily distinguishable colors which receive in this system the same designation, while there are also many pairs that can scarcely be distinguished which receive different designations. This property is, of course, an unavoidable result of dividing the color solid into an arbitary number of blocks, one for each of the 267 designations. Analogous disadvantages result for identifying the time of events according to date; two events occurring on the same date may be separated by many hours, but on the other hand two scarcely separable midnight events may have to be assigned different dates (Kelly and Judd, 1976. p. 4).In all the forthcoming uses of the Munsell color names, they should be understood as names of determinables rather than determinates. There are distinguishable instances of pale yellow and some of these pairs are more easily distinguished than some pairs in which one color is pale yellow and the other is grayish yellow.
The bold face of the standard names contrasts in this entry with the upper case italics of invented names defined by means of the standard names. Some forthcoming definitions have the following pattern:
WEAK YELLOW: pale yellow or grayish yellow (89 or 90)
ROBUST YELLOW: strong yellow or moderate yellow or grayish yellow (84 or 87 or 90).
In the color solid, the regions corresponding to WEAK YELLOW and ROBUST YELLOW are as compact as any of the standard regions. These color names are as comprehensible as the standard names although, of course, they are more determinable. Weak yellow and robust yellow overlap because anything that is grayish yellow is both weak yellow and robust yellow. Neither entails the other.
Kelly and Judd contains thirty-one color charts (not themselves printed in color) that represent cross sections of the Color Solid. The vertical axis represents lightness (Munsell Value), and the horizontal axis represents saturation (Munsell Chroma). All colors that are not on the black-gray-white lightness axis have the same hue. Figure 6 is a reproduction of a chart for yellow (Kelly and Judd, 1976, p. 22) that represents the relation between pale yellow, grayish yellow, and other colors.
Without offering a rigorous proof, Thomason offers the opinion that the overall project of defining species-genus and determinate-determinable in terms only of entailment and negation is doomed. He proceeds to search for an abstract, structural characterization (p. 97) and finds an appropriate structure in the algebraic theory of semi-lattices. The resulting elegant theory is probably useful to theorists who develop taxonomic schemes. It does not, however, appear to help with the problems that Körner and Searle confront. What disqualifies colored or rectangular as a determinable of red? What disqualifies red and square as a determinate of colored?
Besides proposing lattice-theoretic requirements for natural kinds. Thomason recommends a principle of disjointness (D) (p. 98) for taxonomic systems that can be stated as follows:
(D) If two natural kinds a and b of a taxonomic system share at least one member, then every member of a is a member of b, or every member of b is a member of a.The two kinds red and darkish red or darkish orange fail to satisfy (D). They cannot both be natural kinds of the same taxonomic system. The algebraic theory of semi-lattices by itself provides no reason for favoring one or the other or neither as a natural kind. Many systems of classification appear not to accord with (D). In systems of biological, physical, and chemical taxonomy, its enforcement appears arbitrary. These considerations also undermine Johnson's claim that Taking any given determinate, there is only one determinable to which it can belong (Johnson, 1921, p. xxxv, quoted above in Section 1.A).
Although one may nevertheless attempt to respect principle (D), it is important to realize that not every system that respects (D) divides the color solid into natural kinds. Here is another definition in terms of Munsell classification:
BELLOW: pale yellow or deep blue (89 or 179).Something pale yellow can change to grayish yellow by continuously becoming a little bit darker and without changing at all in hue or saturation and without occupying regions in the color solid other than 89 or 90. Nothing pale yellow can change to deep blue in the same way. Either the change is discontinuous or the thing occupies many regions other than 89 and 179. The predicate pale yellow or dark blue (BELLOW) is a disjunction of two determinate predicates but does not itself correspond to a determinate. Pale yellow or grayish yellow (WEAK YELLOW) is determinable with respect to its disjuncts and is a suitable determinate of colored. Disjunctive or conjunctive syntactic forms by themselves are unreliable guides to naturalness or being a proper determinate or determinable Dean Zimmerman has also suggested improvements to Searle's treatment of determinables. He uses, in addition to the notion of predicate or property entailment, the notion of the Boolean part of a property.
F is a determinate falling under determinable G =df (1) F implies G, but G does not imply F; (2) there is no property H such that: (a) G&H implies F but (b) neither H nor not-H implies G; (3) every Boolean part of F implies G; and (4) for every property I such that I and G satisfy the preceding three clauses, F and I stand in some logical relation. (Zimmerman, 1997, p. 464)How does this definition fare for our paradigm, F (red) is a determinate falling under determinable G (colored)?
Clause (3) requires that every Boolean part of the property red implies the property of being colored. From Zimmerman's discussion (1997, pp. 462-463), it is not obvious whether red has a proper Boolean part or whether its only Boolean part is red itself. Whatever the answer, clause (3) seems to be satisfied for this example. Questions remain, however, about parts of color properties. If the predicate robust yellow stands for a property, are strong yellow, moderate yellow, and grayish yellow its Boolean parts? Or is the property of being robust yellow somehow distinct from the properties of being strong or moderate or grayish yellow? On the understanding of disjunctive predicate to be recommended below, robust yellow is disjunctive if and only if strong or moderate or grayish yellow is also disjunctive. An account of disjunctive predicates, in this sense, could be useful in identifying Boolean parts of properties. An appeal to the existence or non-existence of such parts to explain disjunctiveness, on the other hand, appears to assume what it purports to establish.
Clause (1) of the definition above is satisfied and (2), unfortunately, is unsatisfied. Consider the following property:
H: red or (not-colored and square)
G&H implies F, that is, Colored and (red or (not-colored and square)) implies red. H does not imply G, that is, red or (not-colored and square) does not imply colored. Not-H does not imply G, that is, not-red and (colored or not-square) does not imply colored. Zimmerman's formulation does not repair a difficulty in Searle's, namely, that the red-colored paradigm fails to satisfy the definition of the determinate-determinable relation. Predicate H, to be sure, is a hideous monstrosity that stands for something stapled together from Boolean parts of unrelated properties. A goal of the definitional enterprise is to distinguish ordinary, healthy predicates from such monstrosities. So we cannot assume the distinction in order to draw the distinction.
Clause (4) had difficulties of its own. Assign a new meaning to the letter F:
F: weak yellow
F satisfies clause (1). Pretend F also satisfies some revised and improved version of clause (2). Clause (3) does not stand in the way. But as one would expect, something is lurking in the wings:
I: robust yellow
I satisfies clauses (1), (2), and (3) as well as F. But I and F are logically unrelated. All the following are possible: not-F and not-I, F and not-I, I and not-F, and F and I. An example of the kind that thwarts Searle also impedes Zimmerman.
In a brisk Introduction to his early book Problems of Mind and Matter, in a section entitled Generic and Specific, John Wisdom says:
The fact is red and hard and red or hard are not universals; for strictly there are no conjunctive or disjunctive universals but only conjunctive and disjunctive facts. This is red or hard means Either this is red or this is hard. (Wisdom, 1963, p. 31)
D. M. Armstrong says something very similar about disjunctive properties (1978, p. 19-23). (Armstrong is willing to admit conjunctive properties.) A disjunction of property predicates such as red and hard is not itself a disjunctive property predicate. There is no property of being red or hard although the disjuncts of the meaningful predicate red or hard do (or might) correspond to the properties red and hard.
At this point the following distinction is relevant:
Disjunctive properties offend against the principle that a genuine property is identical in its different particulars. Suppose a has a property P but lacks Q, while b has Q but lacks P. It seems laughable to conclude from these premisses that a and b are identical to some respect. Yet both have the property, P or Q. (1978, p. 20)Something that satisfies the first disjunct of the following predicate need not be identical to or resemble in any relevant respect something that satisfies the second disjunct: More than ten million miles from Memphis or sings All of Me off key. This example is intended to be a disjunction of laughably unrelated components. Not every predicate of the form P or Q is good for a laugh in this way.
Perhaps there is no current consensus about the accepted meanings to the technical phrases disjunctive predicate and conjunctive predicate. If that is so, then the following principles are useful suggestions for fixing their meanings, 'a' stands here for 'acceptable' and 'u' stands for 'unacceptable':
(Conj-a) If predicates F and G are equivalent (necessarily apply to the same things), then F is conjunctive if and only if G is also conjunctive.On the other hand, it is useful to reject both the following principles:
(Disj-a) If predicates F and G are equivalent, then F is disjunctive if and only if G is also disjunctive.
(Conj-u) If F is equivalent to a predicate of the form K and L, then F is conjunctive.According to (Conj-u) and (Disj-u), all predicates are both conjunctive and disjunctive. Even redundant predicates such as F and F and F or F demonstrate this result. Additional qualifications can of course eliminate these particular examples. Then there will be more examples with undesirable consequences, and more qualifications to eliminate them. So long as the qualifications must be expressed in the terms of standard logical dependence and independence, the project recapitulates the efforts of Searle and Woods.
(Disj-u) If F is equivalent to a predicate of the form K or L then F is disjunctive.
According to (Conj-a) and (Disj-a), Wisdom and Armstrong can agree that disjunctive predicates do not stand for properties or universals and they can disagree about whether conjunctive predicates stand for properties or universals. Wisdom says they don't, Armstrong thinks that the arguments against disjunctive universals do not apply to conjunctive universals. Everyone should agree that if a predicate F is equivalent to a disjunction of two different property predicates, F may be disjunctive or may not. That depends on how the disjuncts are related. Similar remarks apply to conjunctive predicates.
Some earlier examples of predicates, or similar predicates, appear in the following list:
Pale yellow and grayish yellow do not differ with respect to hue or saturation. They differ in brightness only to the extent necessary to have distinct locations in the Munsell color solid. Pale yellow and grayish yellow are determinate with respect the determinable weak yellow, and weak yellow in turn is determinate with respect to yellow.
Bellow is not a determinate color with respect to the determinable color. One wants to deny that it is a color at all, even though BELLOW is equivalent to a disjunction of color predicates.
Predicates (A)and (B) contrast sharply. Pale yellow and grayish yellow are as similar as they can be while still excluding each other. Pale yellow and deep blue are about as dissimilar as two colors can be. Direct ungrounded appeals to resemblance or being in the same dimension or having to do with one another will not solve our problem. A theoretically satisfactory treatment of the determinate-determinable relation should explain these resemblances rather than be explained by them. A new technical term disjoint marks the apparent difference between (A) and (B). Pale yellow and deep blue are disjoint predicate. Pale yellow and grayish yellow are not disjoint. The next section provides a definition of disjointness.
So far as one can discriminate just by means of predicate entailment or the presence or absence of logical relations, the components of (C) and (D) are logically unrelated. But (C) is not a conjunctive predicate, and (D) is not a disjunctive predicate. Something that satisfies both disjuncts of (C), greater than 5 and less than 7 and greater than 4 or less than 6, can change continuously so as to satisfy the first but not the second, or can change continuously in the other direction along the same dimension so as to satisfy the second but not the first. Greater than 5 and less than 7 and greater than 4 and less than 6 obviously indicate overlapping intervals along the same dimension. (C) is a long-winded way of expressing greater than 4 and less than 7 which has no appearance of being disjunctive. In the same way, (D) is a long-winded way of expressing greater than 5 and less than 6 which has no appearance of being conjunctive.
So far as one can discriminate just by means of predicate entailment or the presence or absence of logical relations, the components of (E) and (F), these components are related to each other in the same way as those of (C) and (D). Yellow and angry are logically independent. A puzzle that the Korner-Searle Symposium poses is still unsolved. (E) is a disjunctive predicate. Yellow is not a determinate of the determinable yellow or angry. (F) is a conjunctive predicate. Yellow and angry is not a determinate of the determinable yellow.
Logically independent predicates can be determinates under the same determinable. Yellow and angry have an independence of a kind, indicated here by the new technical term B-independence, that has conditions in addition to those for logical independence. The next section specifies some conditions of B-independence.
Some spatial analogies about the boundaries of states and other regions composed of the states will motivate the forthcoming discussion of disjoint and B-independent predicates. Here are two disjunctive definitions:
x is in the Dakotas =df x is in North Dakota or x is in South Dakota.In each case, the disjuncts exclude each other. North Dakota and South Dakota have no points in common. Neither do North Dakota and North Carolina. Predicate entailment fails to capture a topological difference between the Dakotas and the North States. The Dakotas are a coherent, continuous region. The North States are a discontinuous region. Something can be both on the boundary of North Dakota and on the boundary of South Dakota. Nothing can be both on the boundary of North Dakota and on the boundary of North Carolina, for their boundaries are many miles apart. North Dakota and North Carolina are disjoint. North Dakota and South Dakota are not disjoint.
x is in the North States =df x is in North Dakota or x is in North Carolina.
There is a close analogy here with some the color examples in the last section. Weak yellow is a coherent, continuous region in the quality space of color. Bellow is a discontinuous region. Something can be both on the boundary of pale yellow and on the boundary of grayish yellow. Nothing can be both on the boundary of pale yellow and deep blue. This is a topological difference that predicate entailment does not represent.
Here are two more disjunctive geographical definitions. In the first definition, the disjuncts exclude each other:
x is in Dabraksa =df x is in South Dakota or x is in Nebraska.In the second definition, the disjuncts overlap; they are logically independent; neither includes the other; they do not jointly exhaust the total space:
x is in Longkota =df x is in the Dakotas or x is in Dabraska.Longkota is a coherent, continuous region. There is nothing inherently disjunctive about it. Boundary relations again indicate topological relations between the Dakotas and Dabraska that logical entailment and non-entailment do not capture. Consider something A on the boundary of South Dakota and Minnesota and Iowa. It is on the boundary of the Dakotas and on the boundary of Debraska and on the boundary of Longkota. But it is not on the boundary of the following two regions:
x is in the Dakotas but x is not in Debraska (that is, x is in North Dakota)A is not close to any boundary of North Dakota or Nebraska.
x is in Debraska but x is not in the Dakotas (that is, x is in Nebraska).
One of our main puzzle predicates, yellow or angry, presents a topological contrast. Anything A on the boundary of yellow and on the boundary of angry is also on the boundary of the following four predicates:
yellow and angryThis reflects the fact that being yellow and being angry are conceptually independent besides being logically independent. Slight changes in a that would move it from the boundary of yellow to being definitely yellow or definitely not yellow are independent of slight changes in a that would move it from the boundary of angry to being definitely angry or definitely not angry.
yellow but not angry
angry but not yellow
neither angry nor yellow
It is possible to define a color predicate analogous to Longkota. This definition uses two earlier definitions that are repeated here:
ROBUST YELLOW: strong yellow or moderate yellow or grayish yellow.The disjuncts of this last definition are logically independent. Nevertheless, SWELL YELLOW corresponds to a coherent continuous region in color space. Weak yellow and robust yellow have a topological relation that yellowness and anger do not have. There are locations in the Munsell Color Solid (for example, on the borderline of grayish yellow and on the borderline of light olive brown), where something x at this location is on the boundary of weak yellow and on the boundary of robust yellow and on the boundary of swell yellow but is not on the boundary of the following two regions:
WEAK YELLOW: pale yellow or grayish yellow
SWELL YELLOW: WEAK YELLOW or ROBUST YELLOW.
Weak yellow but not robust yellow.Although the predicates WEAK YELLOW and ROBUST YELLOW are logically independent, relations between the boundaries of their regions indicate a significant, objective connection.
Robust yellow but not weak yellow.
Although pale yellow is a highly determinate color predicate relative to yellow, it is far from being maximally determinate. A sentence from Kelly and Judd, quoted above in Section 4, is repeated here:
There are many pairs of easily distinguishable colors which receive in this system the same designation, while there are also many pairs that can scarcely be distinguished which receive different designations(Kelly and Judd, 1976. p. 4).Despite its relative specificity, pale yellow applies to samples that are visibly different with respect to color. The same goes for grayish yellow. So something can change gradually from pale yellow to grayish yellow. Is there some point along the way that is the precise boundary between these two color regions? This is a specific form of a question that divides philosophers who develop theories of vagueness. Without needing to adopt some view about the basic nature of borderline cases, one can admit the possibility of borderline cases between pale yellow and grayish yellow. It is possible that something can be on the borderline of each region. Something is a borderline case of pale yellow if it is neither definitely pale yellow nor definitely not pale yellow.
Körner also uses a logic of inexact concepts to treat yellow or angry. One of Searle's complaints about Körner is that "His definition excludes any exact concept as a possible candidate for a determinate" (Searle, 1959, p. 156). Does this objection apply to the suggestions in this section? The next section returns to the question whether there are perfectly exact concepts. Borderline cases are used in this section to locate boundaries. Suppose that more than five feet tall but less than six feet tall is perfectly exact. Anything just slightly taller than five feet or just slightly shorter than six feet is on the boundary. For the purposes of exploring relations to other predicates, we can replace an exact predicate with one that it slightly inexact. For example, amend the definition of the allegedly exact predicate by adding so far as one can tell by using a wall, a pencil, a carpenter's level, and a yardstick. The definitions as amended definitely apply to some things, definitely do not apply to others, and also have some borderline cases left over. (Following a harmless practice, this article refers both to borderline cases of predicates and to borderline cases of properties or regions.)
This project treats borderlines and boundaries as interchangeable. The following is an attempt to generalize and formalize the suggestions made above:
The B operator is used to talk about boundaries and borderline cases. BFx means x is a borderline case of F.The definition of disjoint predicates promised at the end of the section follows:
Fx and Gx are disjoint predicates if and only if Fx and Gx are exclusive predicates and For any x, if BFx, then not-BGx.Disjoint predicates do not, or in a modal version, cannot, share borderline cases. A predicate is exclusively disjunctive if and only it is equivalent to a disjunction of disjoint predicates.
A specification of a condition of B-independence was also promised at the end of Section 5. Let us say that two predicates Fx and Gx intersect if and only if there is something x such that:
B(F & G)x & B(F & not-G)x & B(not-F & G)x & (not-F & not-G)x.The boundaries of B-independent predicates not only intersect; they intersect wherever they have a point in common. Fx and Gx are B-independent only if:
For any x, if BFx and BGx, then B(F & G)x & B(F & not-G)x & B(not-F & G)) & B(not-F & not-G)x.A predicate is inclusively disjunctive only if it is equivalent to a disjunction of B-independent predicates. A predicate is conjunctive only if it is equivalent to a conjunction of B-independent predicates. Yellow or angry is inclusively disjunctive. WEAK YELLOW or ROBUST YELLOW is not inclusively disjunctive. Yellow and angry is conjunctive. WEAK YELLOW and ROBUST YELLOW is not conjunctive.
This approach appears to solve the puzzle that Körner formulated and that Searle and others attempted to solve without success. Conjunctive predicates do not correspond to determinates. Disjunctive predicates do not correspond to determinables.
The discussion above provides only a necessary condition of B-independence. Attempting to deal with Nelson Goodman's puzzle about grue and other perverse artificial predicates requires reference to more complicated relations between boundary conditions. These further conditions which are represented as both necessary and sufficient for B-independence are not spelled out here. They appear, in successive versions, in three articles by Sanford 1970, 1981 (in which the later parts are nearly incomprehensible because of over-compression and lack of diagrams), and 1994 (which has some diagrams). All three articles attempt to clarify the determinate-determinable relation by explaining the nature of disjunctive and conjunctive predicates.
A physical object is determinate in all respects, it has a perfectly precise colour, temperature, size, etc. It makes no sense to say that a physical object is light-blue in colour, but is no definite shade of light blue. (p. 59)Understanding what it is for color, temperature, size, etc., predicates to be perfectly precise helps in understanding what it is for color, temperature, size, etc., properties to be perfectly precise. A precise predicate is not vague; it is exact rather than inexact; it has no borderline cases. Precision contrasts with vagueness.
Specificity, on the other hand, contrasts with generality. Light blue is more specific than blue which is more specific than colored. The more specific a predicate, the narrower the range it covers. Is light blue more specific than smooth? The absence of an inclusion relation in either direction makes it difficult to answer this question. Attempts actually to compare the numerical results after counting all the light blue things in the world and all the smooth things can lead only to frustration and failure. This entry does not address the problem of comparing degrees of specificity of determinates under different determinables.
Specificity and exactness are independent in several ways. There can be predicates F and G such that:
F and G are not identical and are both unspecific and inexact. For example: F: about the size of a cat, G: about the size of a dog.Although specificity and precision are independent in these ways, they are also significantly connected with respect to absolute determinacy. Any absolutely specific predicate is also absolutely precise. Suppose, for example, that Armstrong blue is a predicate for an absolutely specific shade of blue. Two things that are Armstrong blue do not differ at all with respect to hue or brightness or saturation. Given a predicate F and two objects a and b such that a is a borderline case of F and b is a definite positive case of F, a and b differ along some relevant dimension. But anything that differs along any relevant dimension from something that is definitely F, when F is an absolutely specific predicate, is not F. Absolutely specific predicates cannot have borderline cases.
F is more specific and more exact than G. Example: F: pale yellow and G: yellow (in the ordinary rather inclusive sense).
F is more specific than G, and G is more exact than F. Example: F: about the size of a cat. G: has a volume not less than 50.3 cubic inches and not more than 2000.8 cubic inches. Anything F is G and not everything G is F, so F is more specific than G. But G is more exact. It requires, at the boundaries, determination to the nearest tenth of a cubic inch.
F and G are both specific and exact. Examples: F: pale yellow, G: deep blue. Neither of these color predicates is absolutely precise, but each is quite precise compared to ordinary color terms. The Munsell Color Solid is constructed with the intention that each of the 267 regions has approximately the same degree of specificity.
No predicate that can have borderline cases is absolutely specific. So if there are no absolutely precise or exact predicates, neither are there absolutely specific predicates. Johnson presumably would not question this conclusion since he says that literally determinate characterization is practically impossible. He and Armstrong claim things in the world are absolutely determinate, not the predicates we do or could use to apply to things in the world.
If things in the world are absolutely determinate, this presumably does not require that any absolute determination persists through the passage of time or space. If a cumulus cloud changes continuously in shape and size, this does not by itself preclude its having, at any one time, an absolutely determinate shape and size. The impression that clouds do not have exact boundaries, however, is probably not based entirely on their changeability.
Objects with absolutely determinate sizes have absolutely determinate boundaries. If a thing has an absolutely determinate length along some axis at a given time, then there is exactly one real number n such that its length in (say) meters is n. Things that we come across in ordinary life such as plants, animals, buildings, furniture, electronic equipment, clothing, and kitchenware do not have exact boundaries, nor do larger items such as mountains, lakes, continents, and stars.
The view that things in the world are absolutely determinate is implausible if it requires clouds, brains, and dinner plates to be absolutely determinate. But this requirement can be put aside. If the microstructure of the world is absolutely determinate, that is absolute determinacy enough. If all the atoms within you and in your vicinity have absolutely determinate properties, then the indeterminate mass and shape and volume of you, your brain, and your teeth somehow supervene on the determinate microstructure. Here is a well-known passage from David Lewis:
The reason it's vague where the outback begins is not that there's this thing, the outback, with imprecise borders; rather there are many things, with different borders, and nobody has been fool enough to try to enforce a choice of one of them as the official referent of the word outback. (Lewis, 1986, p. 212)
The view that there are many things with precise borders does not by itself refute the view that there are things with imprecise borders. Here is a kind of parody of the passage just quoted. It ends with the same point but begins with a contrary contention:
The outback is a big thing, and it is vague where it begins. The reason it has imprecise borders is that there are many things, with different precise borders, and nobody has been fool enough to enforce a choice of one of them as the official reference of the word outback.For the outback (a cloud, your brain), there are many more-or-less precise aggregates of particles such that each is about as good a candidate as there is to be identified with the outback (a cloud, your brain). This entry does not address the problem of the many, how to understand the relation between a single macro-object and many overlapping aggregates of micro-objects that more or less coincide with it.
However one resolves the problem of the many, the question of absolute determinacy becomes the question of absolute determinacy of the physical basis, the microstructure. After quoting the passage from Lewis above, Roberto Casati and Achille Varzi write:
There are plenty of objects out thereplenty of slightly distinct and yet precisely determinate aggregates of land molecules. And when we say Mount Everest or the outback, each one of a large variety of such aggregateseach with its own perfectly crisp mereotopological structurehas an equal claim to being a referent of that term. (Casati and Varzi, 1999, p. 95)And what evidence is there for this precisely determinate perfect crispness? Logic and metaphysics cannot answer this question from its own resources. Science textbooks represent particles and atoms as clouds. Textbook writers fifty years ago knew that the picture of perfect little spheres, the electrons, in elliptical orbits around a nucleus was misleading. Now the picture is simply obsolete.
An attempt to measure the precise dimensions of a polished copper cube might begin using an ordinary school supply ruler, then using a machinist's steel rule, then using a micrometer, then, starting with a low power optical microscope, using a series of increasingly powerful microscopes. At the microscopic level one can discriminate increments of length too small for a mechanical micrometer to detect. This does not produce a more precise determination of the length of the cube if nothing at this level coincides with the boundary of the cube. So the search for the exact measurements of the cube is abandoned and replaced by a hope to find absolute determinacy somewhere at the foundations, the fundamental basis, or the limit. There is a (weak) kind of non-deductive argument here.
Given a greatest degree of precision determined by the best instruments, sooner or later a more advanced technology produces instruments that are still more precise. This process of making measurement increasingly precise never ends; it asymptotically approaches absolute precision at dimensionless points of matter or spacetime or something.This vision of absolute determinacy at the limit is apparently attractive. It appears to be internally consistent. It also appears, however, to be inconsistent with physics.
Middle-sized objects do not have perfectly precise boundaries because there are microscopic objects that are neither definitely included nor definitely excluded from the object. Some larger microscopic objects lack perfectly precise boundaries for the same reason. There is no reason to believe that this process continues infinitely downward. Electron diameters are imprecise, but not because there are swarms of micro-electron-dust, each particle of which is also a swarm of something even smaller. Nor does the process stop with some basic items that really are absolutely determinate.
Instruments can measure the velocity of a tennis ball. They do not, of course, measure velocity with absolutely determinacy. They do not discriminate, say, 114.0 from 114.1 miles per hour. Given some understanding of margins of error, it is meaningful for one to say that a tennis ball was going 114 miles per hour at some temporal instant t. The notions of a limit and of convergence provide this meaning. They provide no support for believing in the possibility of a momentary tennis ball that exists neither before instant t nor after instant t but does exist precisely at instant t and travels 114 miles per hour during its instantaneous existence. If it is possible for something to have a property for an instant, it does not follow that an instantaneous thing can have that property.
The same goes for spatial points. If it is possible for something to have a property at a point, it does not follow that it is possible that something punctiform should have this property. When a region is pale yellow, we can say that any point in the region is pale yellow. But no point by itself can be pale yellow.
Johnson said that it is a universally adopted postulate that the characters of things which we can only characterise more or less indeterminately, are, in actual fact, absolutely determinate. In saying it is a postulate, Johnson does not mean we merely assume it in order to deduce its consequences. He means rather that it is both obviously true and cannot be inferred from truths that are even more obvious. But the so-called postulate seems not to be true.
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First published: April 26, 2002
Content last modified: April 26, 2002