Relations are ways in which several entities stand to each other. They usually connect distinct entities but some associate an entity with itself. The adicity of a relation is the number of entities it connects. The direction of a relation is the order in which the elements are related to each other. The converse of a relation carries the same information and has the opposite direction, like the contrast between "two is less than five" and "five is greater than two". Both relations and properties express features in reality with a key difference being that relations apply to several entities while properties belong to a single entity.
Many types of relations are discussed in the academic literature. Internal relations, like resemblance, depend only on the monadic properties of the relata. They contrast with external relations, like spatial relations, which express characteristics that go beyond what their relata are like. Formal relations, like identity, involve abstract and topic-neutral ideas while material relations, like loving, have concrete and substantial contents. Logical relations are relations between propositions while causal relations connect concrete events. Symmetric, transitive, and reflexive relations are distinguished by their structural features.
Metaphysical difficulties like the question of where relations are located lie at the center of discussions of their ontological status. Eliminativism is the thesis that relations are mental abstractions that are not a part of external reality. A less radical position is reductionism, which claims that relations can be explained in terms of other entities, like monadic properties, and are not a substantial addition to reality. According to realists, relations have a mind-independent existence. A strong form of realism is relationalism, which states that all of reality is relational at its most basic level. Historically, eliminativism and reductionism were the dominant views. This only changed toward the end of the 19th century, when various developments in the fields of mathematics, logic, and science prompted a more realist outlook.
Definition and essential features A relation is a manner in which multiple entities stand to each other. It is a connection or association between entities and can be understood as a feature characterizing these entities as a whole. Many relations hold between distinct entities. For example, the first-born sibling stands in the relation of being older than to their other siblings. But an entity can also stand in a relation to itself. For instance, every entity stands in the relation of identity to itself. Relations can hold between diverse entities, including objects, people, and concepts. If a relation holds between entities then the relation together with the entities constitutes a fact or state of affairs.
The word "relationship" is often used as a synonym. The entities related to each other are called the relata. The term "relation" comes from the Latin terms and , which mean reference or towardness.
In mathematics and logic, relations are defined as set-theoretic structures. For example, the relation less than is defined as the set of all ordered pairs in which the first element is less than the second element. This set includes pairs like [1,2], [1,3], and [2,17]. Mathematical functions are a special type of relation in which one or several elements are uniquely associated with exactly one other element.
Relations have various characteristic features, like the number of relata they have and the direction in which they connect them. They are closely associated with properties and share several aspects with them.
Adicity The adicity of a relation is the number of places or relata it has. The terms "arity" and "degree" are used as synonyms. For instance, the relation being larger than has an adicity of two since it involves two entities: a smaller entity and a larger entity. Another example is the relation of being adjacent to. Relations with an adicity of two are called dyadic or binary. Triadic or ternary relations have an adicity of three, like the relation of giving, which involves a giver, a receiver, and a given object. The relation of being between is also triadic since it requires two entities on the sides and one in the middle, as in "5 is between 2 and 23".
Unigrade relations are relations that have a fixed adicity: they always apply to the same number of entities. They contrast with multigrade relations, for which the number of their relata varies from one occasion to another. It is controversial whether there are genuine multigrade relations. Some theorists, like David M. Armstrong, argue that the adicity of a relation is an essential feature of it. According to this view, there are no multigrade relations on the fundamental level of reality. However, this view is not generally accepted and some cases of multigrade relations have been suggested. Putative examples include causal relations (which may relate several causes to several effects), logical consequence (which relates several premises to a conclusion they support), and expressions like being the tallest among (for which the adicity depends on the size of the comparison group). Multigrade predicates are also common in everyday language. For instance, the predicate "lift" is dyadic in the sentence "John is lifting a table" and triadic in the sentence "John and Mary are lifting a table". Other examples are predicates that have optional arguments, as in the sentences "John is eating a cake" and "John is eating".
Direction and converse The direction of a relation is the order in which the elements are related to each other. For instance, if Abelard loves Eloise then the relation of loving goes from Abelard to Eloise. If Eloise loves Abelard then the direction goes in the opposite direction. Both of these facts have the same constituents. They are only distinguished by their direction. Only non-symmetric relations have a direction since the order of the relata does not make a difference for symmetric relations.
The converse of a non-symmetric relation is a second relation that always accompanies the first relation but has a different order of elements. For example, the converse of above is below. This means that whenever x is above y then y is below x. The same is the case for relations like coming before and coming after as well as being a parent of and being a child of. Binary relations have exactly one converse while tertiary and higher-degree relations have several converses.
A relation and its converse carry the same information. For this reason, it is controversial whether they should be considered as two distinct relations instead of seeing them as the same relation. This difficulty has prompted some philosophers to conceptualize the directional aspect of non-symmetric relations differently or to deny the existence of converse relations. According to positionalism, relations do not have a direction but have different unique positions that are filled by their relata. For instance, the relation love has two positions: one for the lover and one for the beloved. This view explains how relata can play different roles in a relation without implying that the relation has a direction or a converse.
Contrast to properties Relations are usually contrasted with properties. Properties are held by a single entity and express what this entity is like. Relations connect several entities and are features that apply to them as a whole. A closely associated contrast is that properties belong to entities or inhere in them while relations are not found in the relata but stand between them.
However, there are also many parallels between properties and relations and there is no general agreement on a strict dichotomy between the two. Both are often used to describe and explain repeating patterns in the world and many of the ontological distinctions applied to properties also affect relations. For example, both properties and relations may be understood as universals that are instantiated by individuals at a specific place and time. Like properties, relations are either determinable or determinate. Determinable relations are not specified, like the relation of being distant from. Determinate relations are fully specified, like the relation of being exactly one meter distant from.
It is possible to conceptualize properties and propositions as special types of relations. According to this view, the difference between these phenomena only concerns how many entities they apply to: regular relations are polyadic and thus apply to several entities; properties are monadic relations and only apply to a single entity; propositions are relations with a degree of zero and do not apply to any entities.
The opposite perspective is also possible: to conceptualize relations as polyadic or relational properties. It is usually accepted that relational properties accompany relations: if x stands in a relation to y, then x has the relational property of bearing a relation to y. For example, since Antony was married to Cleopatra, he had the relational property of being married to Cleopatra. However, it is not generally accepted that relations are correctly understood as or can be reduced to relational properties. Historically, properties have received significantly more attention from metaphysicists than relations.
Types Various types of relations are distinguished in the academic literature based on their ontological status, the domains they apply to, and the structures they form.
Internal and external An influential distinction differentiates between internal and external relations. A relation is internal if it only depends on what the relata are like: it is determined by the characteristics or the nature of the relata alone. External relations are not fixed this way and carry characteristics that go beyond the intrinsic features of their relata. Mathematical relations between numbers are examples of internal relations. For instance, the number six stands in the relation of being greater than to the number five. This relation is internal because it is essential to the numbers six and five that six is greater than five. Other traditional examples of internal relations are resemblance and difference. Spatial relations are normally understood as external relations, like the relation of a book to the table it is lying on. The same is true for temporal and causal relations.
However, the precise characterization of the distinction between internal and external relations is disputed and there are various incompatible ways to define them. According to G. E. Moore, a relation is internal if it follows from the existence of its relata that the relation also exists. This means that the relation is essential to the relata and the relata cannot exist without this relation. For Moore, external relations are different since they could fail to obtain even if its relata exist. Another definition is defended by philosophers like Armstrong, who hold that a relation is internal if it is necessitated by the properties or the intrinsic features of the relata. David K. Lewis provides a slightly different formulation by claiming that internal relations supervene on the intrinsic properties of their elements. Some philosophers talk of ideal relations to refer to relations that solely depend on the qualities of the related terms, in contrast to real relations, for which this is not the case.