relation a two-or-more-place property (e.g., loves or between), or the extension of such a property. In set theory, a relation is any set of ordered pairs (or triplets, etc., but these are reducible to pairs). For simplicity, the formal exposition here uses the language of set theory, although an intensional (property-theoretic) view is later assumed.
The terms of a relation R are the members of the pairs constituting R, the items that R relates. The collection D of all first terms of pairs in R is the domain of R; any collection with D as a subcollection may also be so called. Similarly, the second terms of these pairs make up (or are a subcollection of) the range (counterdomain or converse domain) of R. One usually works within a set U such that R is a subset of the Cartesian product U$U (the set of all ordered pairs on U). Relations can be: (1) reflexive (or exhibit reflexivity): for all a, aRa. That is, a reflexive relation is one that, like identity, each thing bears to itself. Examples: a weighs as much as b; or the universal relation, i.e., the relation R such that for all a and b, aRb. (2) symmetrical (or exhibit symmetry): for all a and b, aRb P bRa. In a symmetrical relation, the order of the terms is reversible. Examples: a is a sibling of b; a and b have a common divisor. Also symmetrical is the null relation, under which no object is related to anything. (3) transitive (or exhibit transitivity): for all a, b, and c, (aRb & bRc) P aRc. Transitive relations carry across a middle term. Examples: a is less than b; a is an ancestor of b. Thus, if a is less than b and b is less than c, a is less than c: less than has carried across the middle term, b. (4) antisymmetrical: for all a and b, (aRb & bRa) P a % b. (5) trichotomous, connected, or total (trichotomy): for all a and b, aRb 7 bRa 7 a % b. (6) asymmetrical: aRb & bRa holds for no a and b. (7) functional: for all a, b, and c, (aRb & aRc) P b % c. In a functional relation (which may also be called a function), each first term uniquely determines a second term. R is non-reflexive if it is not reflexive, i.e., if the condition (1) fails for at least one object a. R is non-symmetric if (2) fails for at least one pair of objects (a, b). Analogously for non-transitive. R is irreflexive (aliorelative) if (1) holds for no object a and intransitive if (3) holds for no objects a, b, and c. Thus understands is non-reflexive since some things do not understand themselves, but not irreflexive, since some things do; loves is nonsymmetric but not asymmetrical; and being a cousin of is non-transitive but not intransitive, as being mother of is. (1)–(3) define an equivalence relation (e.g., the identity relation among numbers or the relation of being the same age as among people). A class of objects bearing an equivalence relation R to each other is an equivalence class under R. (1), (3), and (4) define a partial order; (3), (5), and (6) a linear order. Similar properties define other important classifications, such as lattice and Boolean algebra. The converse of a relation R is the set of all pairs (b, a) such that aRb; the complement of R is the set of all pairs (a, b) such that –aRb (i.e. aRb does not hold).
A more complex example will show the power of a relational vocabulary. The ancestral of R is the set of all (a, b) such that either aRb or there are finitely many cI, c2, c3, . . . , cn such that aRcI and c1Rc2 and c2Rc3 an. . . and cnRb. Frege introduced the ancestral in his theory of number: the natural numbers are exactly those objects bearing the ancestral of the successor-of relation to zero. Equivalently, they are the intersection of all sets that contain zero and are closed under the successor relation. (This is formalizable in second-order logic.) Frege’s idea has many applications. E.g., assume a set U, relation R on U, and property F. An element a of U is hereditarily F (with respect to R) if a is F and any object b which bears the ancestral of R to a is also F. Hence F is here said to be a hereditary property, and the set a is hereditarily finite (with respect to the membership relation) if a is finite, its members are, as are the members of its members, etc. The hereditarily finite sets (or the sets hereditarily of cardinality ‹ k for any inaccessible k) are an important subuniverse of the universe of sets.
Philosophical discussions of relations typically involve relations as special cases of properties (or sets). Thus nominalists and Platonists disagree over the reality of relations, since they disagree about properties in general. Similarly, one important connection is to formal semantics, where relations are customarily taken as the denotations of (relational) predicates. Disputes about the notion of essence are also pertinent. One says that a bears an internal relation, R, to b provided a’s standing in R to b is an essential property of a; otherwise a bears an external relation to b. If the essential–accidental distinction is accepted, then a thing’s essential properties will seem to include certain of its relations to other things, so that we must admit internal relations. Consider a point in space, which has no identity apart from its place in a certain system. Similarly for a number. Or consider my hand, which would perhaps not be the same object if it had not developed as part of my body. If it is true that I could not have had other parents – that possible persons similar to me but with distinct parents would not really be me – then I, too, am internally related to other things, namely my parents. Similar arguments would generate numerous internal relations for organisms, artifacts, and natural objects in general. Internal relations will also seem to exist among properties and relations themselves. Roundness is essentially a kind of shape, and the relation larger than is essentially the converse of the relation smaller than. In like usage, a relation between a and b is intrinsic if it depends just on how a and b are; extrinsic if they have it in virtue of their relation to other things. Thus, higher-than intrinsically relates the Alps to the Appalachians. That I prefer viewing the former to the latter establishes an extrinsic relation between the mountain ranges. Note that this distinction is obscure (as is internal-external). One could argue that the Alps are higher than the Appalachians only in virtue of the relation of each to something further, such as space, light rays, or measuring rods. Another issue specific to the theory of relations is whether relations are real, given that properties do exist. That is, someone might reject nominalism only to the extent of admitting one-place properties. Although such doctrines have some historical importance (in, e.g., Plato and Bradley), they have disappeared. Since relations are indispensable to modern logic and semantics, their inferiority to one-place properties can no longer be seriously entertained. Hence relations now have little independent significance in philosophy. See also ESSENTIALISM , IDENTITY, META- PHYSICS , POSSIBLE WORLDS , SET THEORY , SPAC. S.J.W.