Relations in Philosophy

A relation is a way in which two or more things stand to one another. Being taller than, being the parent of, being between, being west of, loving, causing, and resembling are all relations. Ordinary thought is full of them: almost every interesting fact about the world concerns how things are connected rather than what each thing is like taken alone. Yet relations have long puzzled philosophers. Where is the relation "west of" when Glasgow is west of Edinburgh? Is it in Glasgow, in Edinburgh, somewhere between them, or nowhere at all? Can a relation be reduced to properties of the things it relates? Does a relation need yet another relation to tie it to its relata, and if so, does that start an infinite regress? Is reality fundamentally made of things, or of relations?

This article explains the basic concepts philosophers use to talk about relations, surveys the main types of relation, describes the major positions on whether relations really exist, and traces the history of the topic from Aristotle to contemporary metaphysics and philosophy of physics. It ends with common misconceptions and frequently asked questions.

What a relation is

Relations and relata

When a relation holds between some things, those things are called its relata (singular relatum). In "Abelard loves Eloise," the relation is loving and the relata are Abelard and Eloise. When a relation holds between its relata, philosophers often say that the relation and the relata together make up a fact or state of affairs: the fact that Abelard loves Eloise. The word "relation" comes from the Latin relatio, connected to the verb referre, conveying the idea of reference or "towardness": one thing is related to another by bearing on it.

Usually relations connect distinct things. But some relations can link a thing to itself. Everything is identical to itself, and a person can admire themselves.

Relations and properties

Relations are often contrasted with properties. A property, such as being red or having a mass of two kilograms, belongs to a single thing and characterizes what that thing is like. A relation holds between several things. Both are often treated as universals, features that can be shared by many different things: many objects can be red, and many pairs of people can be siblings.

Properties and relations share other features. Both can come in more and less specific versions. Being distant from is a general, or determinable, relation; being exactly one meter distant from is a determinate version of it. Some philosophers treat properties as simply one-place relations, and some even treat whole propositions as zero-place relations. Others go the other way and treat relations as special relational properties, such as the property of being taller than Ben Vane.

Historically, metaphysicians have given far more attention to properties than to relations. One theme of this article is that this neglect has been costly, because many of the hardest problems about properties take a sharper form for relations.

Relations in logic and mathematics

In logic and mathematics, relations are given a precise set-theoretic definition. A two-place relation on a set is simply a set of ordered pairs. The "less than" relation on whole numbers is the set of all pairs (x, y) such that x is smaller than y, including (1, 2), (1, 3), and (2, 17). Three-place relations are sets of ordered triples, and so on.

This definition is enormously useful but deliberately thin. It identifies a relation with the list of things it holds between, its extension. Philosophers are often interested in a richer notion: what it is in the world that makes 1 less than 2, or Abelard love Eloise. Two different relations could in principle hold between exactly the same pairs; the set-theoretic definition cannot tell them apart.

Key features of relations

Adicity: how many places

The adicity of a relation, also called its arity or degree, is the number of relata it takes.

A relation with a fixed adicity is called unigrade. A relation that can take different numbers of relata on different occasions is called multigrade. Proposed examples include being the tallest among, which can hold among three people or thirty; causation, since an event may have one cause or many; and logical consequence, since a conclusion may follow from one premise or several. The term "multigrade" in this sense is credited to Henry Leonard and Nelson Goodman (1940). Whether there really are any multigrade relations at a fundamental level is controversial. The Australian philosopher David M. Armstrong held that adicity is essential to a relation and that fundamental reality contains no multigrade relations.

Direction, order, and converses

Some relations seem to have a direction: they run from one relatum to another. In "Abelard loves Eloise," the love runs from Abelard to Eloise; "Eloise loves Abelard" says something different, and the two can come apart. Only non-symmetric relations show this kind of direction.

Each binary relation has a converse: a relation that holds between the same things in the reverse order. Above has the converse below; parent of has the converse child of; taller than has the converse shorter than.

This raises a puzzle. When Abelard loves Eloise, Eloise is loved by Abelard. Are these two facts or one? If one, then the relation loves and its converse is loved by seem to produce the same fact, which suggests they are not really distinct relations. If two, then the world seems to contain twice as many facts as necessary. Peter Geach (1957) argued that a relation and its converse cannot be separated. The Stanford Encyclopedia of Philosophy presents this as an inconsistent set of three attractive principles:

• Converses: every non-symmetric relation has a distinct converse.
• Identity: the fact that arises when a relation applies to some things is identical to the fact that arises when its converse applies to them in reverse order.
• Uniqueness: no single fact arises from more than one relation.

All three cannot be true together. The traditional "directional" view, associated with Bertrand Russell's The Principles of Mathematics (1903), holds that non-symmetric relations run from one relatum to another, and it faces pressure from this triad.

Two main alternatives have been developed:

• Positionalism holds that relations have distinct argument places, or positions, that relata fill, without any intrinsic direction. On this view, loving has a "lover" position and a "beloved" position. Abelard loves Eloise when Abelard occupies the lover position and Eloise the beloved position. The converse is not a separate relation; it is the same relation described differently. Versions have been developed by Hector-Neri Castañeda, Armstrong, Francesco Orilia, and Maureen Donnelly. Critics object that positionalism has trouble with symmetric relations and with multigrade relations.
• Anti-positionalism, proposed by Kit Fine in his paper "Neutral Relations" (2000), denies that relations have built-in positions. Instead, it explains the different ways a relation can apply to things by how states can be transformed into one another by substituting relata. Fine described relations as entities "capable of giving rise to a diversity of completions."

A further option, defended by Fraser MacBride (2014), treats the fact that a relation relates, and the way it applies, as primitive features that cannot be analyzed further. The SEP entry on relations concludes that there may be several distinct "problems of order" rather than one, and that different theories address different ones.

Logical structure of binary relations

Logicians classify binary relations by structural features. These are useful in philosophy as well as mathematics.

Some combinations have special names. An equivalence relation is reflexive, symmetric, and transitive; equality and "having the same birthday" are examples. A strict partial order is irreflexive, asymmetric (or antisymmetric), and transitive; "less than" is an example.

The question of whether non-symmetric relations exist is itself philosophically contested. Russell defended them vigorously in 1903. More recently, Armstrong (1997), Cian Dorr (2004), and Peter Simons (2010) have argued, in different ways, that at the fundamental level only symmetric relations exist, and that apparently non-symmetric ones can be understood in other terms. MacBride (2015) has responded to these arguments.

Types of relations

Internal and external relations

One of the most important distinctions in the philosophy of relations is between internal and external relations. The rough idea is that internal relations are fixed by what their relata are like in themselves, while external relations add something beyond that.

Examples help. If one thing is red and another is red, they resemble each other in color. Given their colors, the resemblance is guaranteed; nothing more is needed. Resemblance looks internal. Likewise, given the numbers 6 and 5, it is fixed that 6 is greater than 5. By contrast, two red balls could be one meter apart or a thousand kilometers apart without either ball being any different in itself. Spatial distance looks external. So do temporal relations and, traditionally, causal relations.

The distinction was given its modern form by the English philosopher G. E. Moore in a 1919 paper, "External and Internal Relations." But philosophers have defined it in several incompatible ways.

These definitions can disagree about cases. Having the same shape is internal on Armstrong's definition, because it is necessitated by the shapes of the two objects. On Moore's definition it is external, because the objects could have existed with different shapes, so the relation does not follow merely from their existence.

The American philosopher David Lewis, in On the Plurality of Worlds (1986), offered an influential refinement. On his view, an internal relation is one that supervenes on the intrinsic natures of its relata taken separately: any two things with the same intrinsic natures as these would stand in the same internal relations. An external relation does not supervene on the separate natures of the relata, but does supervene on the intrinsic nature of the composite they form together. Distance is his central example.

Formal and material relations

A second distinction, discussed by Kevin Mulligan (1998) in terms of "thin" and "thick" relations, separates:

• Formal (thin) relations, which are abstract and topic-neutral: they can hold between things of any kind. Identity, difference, resemblance, set membership, and entailment are examples. Formal relations are generally considered internal.
• Material (thick) relations, which have concrete content specific to certain kinds of things. Colliding, smiling at, loving, killing, and giving are examples.

Mulligan argued that thin relations such as greater than remain needed even when a relation like taller than appears to reduce to the heights of the relata, since something must explain why those heights suffice to make one taller.

Logical and causal relations

Logical relations hold between propositions or statements: one proposition's truth depends on another's. Entailment, contradiction, and contrariness are examples. These are typically known a priori and are studied by logic.

Causal relations hold between concrete events or states: the spark caused the fire. They are studied empirically and have traditionally been regarded as external and contingent, governed by the laws of nature. An alternative tradition grounds causation in the powers or dispositions of objects; if those powers are intrinsic, causal relations may turn out to be internal after all.

The word "because" can express either kind. "The argument is valid because the conclusion follows from the premises" expresses a logical relation; "the window broke because the ball hit it" expresses a causal one.

Spatial and temporal relations

Spatial relations include being above, below, within, or a certain distance from. Temporal relations include occurring before, after, or at the same time as. Both generally hold only between concrete things or events, and both are usually classed as external.

The status of spatial and temporal relations is at the heart of a long debate in the philosophy of physics. Substantivalism holds that space (or spacetime) is a real entity in its own right, a container in which objects are located. Relationalism about spacetime holds that there is nothing to space and time over and above the spatial and temporal relations among objects and events. The debate was famously conducted in the correspondence between Gottfried Wilhelm Leibniz and Samuel Clarke (1715 to 1716), with Clarke defending Isaac Newton's substantivalist view and Leibniz arguing that space is merely an order of coexisting things. Classical physics treated space and time as separate; relativity theory treats them as a unified spacetime whose structure is affected by mass and energy, and the debate has continued in that new setting.

Do relations exist? The main positions

Why relations are puzzling

Relations differ from ordinary objects, sometimes called substances, because they depend on their relata: there is no "taller than" without things to be taller. They differ from properties because they belong to several things at once and cannot easily be located in just one of them.

This gives rise to the location problem. J. M. E. McTaggart, writing around 1920, used the example of Glasgow being west of Edinburgh. The sentence is true, but where is the relation? Not in Glasgow alone, nor in Edinburgh alone. Possible answers include that relations are somehow divided between their relata, that they occupy the space between them, or that they are abstract entities with no location at all. None is obviously satisfactory.

Relations also connect to the ancient problem of the one and the many: how can reality be both a plurality of distinct things and a unified whole? Relations seem to be what binds the many into one, which is why theories of relations have so often been tied to theories of reality as a whole.

Realism

Realists about relations hold that relations exist independently of minds. Strong realists hold that relations are part of fundamental reality, alongside or even prior to the things they relate. Weaker realists accept that relations exist but treat them as derived or emergent, built out of more basic features.

Armstrong held a form of realism on which relations are universals instantiated in spacetime, and on which science, rather than philosophy alone, determines which relations exist. His books Universals and Scientific Realism (1978) and A World of States of Affairs (1997) are central texts.

Eliminativism

Eliminativists about relations deny that relations exist at all. On some versions, relations are abstractions produced by the mind when it compares things: the world contains things with properties, and we create relations by noticing how they compare. Absolute idealists in the nineteenth century often rejected genuine relations for a deeper reason: they held that reality is a single whole, with only one ultimate subject, so there are no genuinely distinct things for relations to connect.

Reductionism

Reductionists accept that relational statements can be true but hold that relations are not an addition to reality: they can be fully explained by non-relational features, usually properties of the relata.

The standard example, used in the Wikipedia and SEP treatments, is two Scottish mountains. Ben Vorlich is taller than Ben Vane. Ben Vorlich has a height of 3,094 feet and Ben Vane a height of 3,002 feet. Given those two heights, nothing more is needed for the first to be taller than the second. The relation seems to come "for free." Armstrong described internal relations of this kind as "no addition of being."

Reductionism is widely accepted for internal relations. It is much more controversial for external relations such as distance, which do not seem to follow from the intrinsic properties of their relata.

A subtle point concerns the difference between truthmaking and ontological commitment. Perhaps the two heights are what make "Ben Vorlich is taller than Ben Vane" true. It does not follow that the sentence carries no commitment to a relation. Critics argue that a thin relation of being greater than is still needed to explain why those two heights make the comparison true.

Relationalism

Relationalism, in the broad metaphysical sense, is a strong form of realism holding that reality is relational at its most fundamental level. Instead of things that happen to be related, there are relations, and things are either derivative or nothing more than positions in a structure. This contrasts with the "substance" view that dominated much of the history of philosophy, on which independent things come first and relations second.

Positions compared

Bradley's regress

The argument

The most famous argument against relations was given by the British idealist F. H. Bradley in Appearance and Reality (1893). Bradley argued that relations cannot be real because they lead to a vicious infinite regress.

The argument can be stated as a dilemma. Take a relation R said to hold between A and B. Either R is nothing to A and B, or it is something to them.

• If R is nothing to A and B, then it cannot relate them. A, B, and R would just be three separate items.
• If R is something to A and B, then R must itself be related to A and to B. But that requires further relations, R1 linking R to A and R2 linking R to B. Those in turn must be related to what they relate, requiring yet more relations, and so on without end.

Either way, Bradley concluded, relations cannot do what they are supposed to do. Years later he described the attempt to make relations work as "an endless renewal of our failure." His positive conclusion was monism: reality is a single, undivided whole, and the appearance of many related things is just that, an appearance.

Responses

Philosophers have answered Bradley in several ways.

• Facts or states of affairs: Some, including Herbert Hochberg, Armstrong, and Keith Hossack, propose that what unifies relata and a relation is not another relation but the fact or state of affairs they compose. The fact that A is R to B is a basic entity, not something assembled with glue.
• Tropes: Others, including Anna-Sofia Maurin and Peter Simons, propose that relations are particular instances, or tropes, that are essentially dependent on their relata, so that they need nothing further to tie them in.
• Rejecting the premise: Others deny that relations need to be related to their relata at all. A relation simply is the kind of thing that relates; asking what relates it to its relata is a confusion.
• Pragmatic rejection: Bertrand Russell, in a 1924 essay, rejected Bradley's argument because it cannot be squared with scientific discourse, which seems committed to external relations. The SEP compares this to W. V. O. Quine's later indispensability argument: if our best science needs relations, we should accept them.

MacBride has objected that the facts and tropes responses both presuppose that relations can relate, so they do not so much answer Bradley as assume his question away. Bradley's regress remains a live topic in contemporary metaphysics.

History

Ancient and medieval thought

Aristotle included relation (Greek pros ti, "toward something") among his ten categories, the basic kinds of being. But he treated relations as the least substantial of the categories: dependent on substances and on other accidents such as quantity and quality. Later Neoplatonists, including Plotinus and Porphyry, largely followed Aristotle's framework while emphasizing that relations are nonetheless real.

Medieval scholastic philosophers developed a range of views.

These debates were partly driven by theology. The Christian doctrine of the Trinity was articulated in terms of relations among the divine persons, which pressed scholastic thinkers to say what relations are.

Early modern philosophy

Thomas Hobbes took a nominalist line: relations are nothing over and above the related things; they exist only in our comparison of them.

Gottfried Wilhelm Leibniz famously denied that relations are fundamental. One reason was the location problem: a relation, he held, would have to be an accident in two subjects at once, with "one leg in one, and the other in the other," which is impossible. In his metaphysics of monads, the ultimate constituents of reality are simple substances that are "windowless" and do not interact. Relational truths are grounded in the monadic properties of the relata. For example, the truth that Adam is the father of Cain is grounded in Adam's having the property of being Cain's father and Cain's having the property of being Adam's child, together with the harmony God has established among all monads.

Immanuel Kant included relation among the four headings of his table of categories, the basic concepts through which experience is structured, alongside quantity, quality, and modality. He held that the world as we experience it is constituted largely by relations.

G. W. F. Hegel argued that the nature and value of things arise from their relations to other things and ultimately to the whole.

The logic of relations

A decisive shift came in logic. Aristotelian syllogistic logic, which dominated for two thousand years, was built on subject-predicate statements such as "All men are mortal." It had great difficulty expressing relational reasoning, such as the inference from "every horse is an animal" to "every head of a horse is a head of an animal."

In the mid-nineteenth century, Augustus De Morgan developed a logic of relations, including the idea of composing relations: combining brother of with father of, for example, yields uncle of (a brother of one's father). Charles Sanders Peirce extended this work considerably, developing an algebra of relations and later a graphical logic. Peirce compared relations to chemical molecules, with relata bonding to relations in the way atoms bond according to their valency. He also argued that three-place relations cannot in general be reduced to two-place ones.

Gottlob Frege, in his Begriffsschrift (1879), created modern predicate logic, in which relations are handled as naturally as properties. For Frege, relations are "unsaturated" or incomplete: they have open places that must be filled by objects to yield a complete thought. This logical revolution made relations impossible to ignore.

Bradley, Moore, and Russell

At the end of the nineteenth century, British philosophy was dominated by absolute idealism, of which Bradley was a leading figure. His monism implied that plurality is an illusion and that no genuine relations exist.

G. E. Moore and Bertrand Russell led the revolt against idealism around 1900, and the reality of relations was at the center of it. Moore appealed to common sense and developed the internal/external distinction. Russell argued in The Principles of Mathematics (1903) that relational reasoning, especially in mathematics, cannot be understood without accepting relations.

Russell's key argument concerned asymmetric relations such as greater than or before. He considered two ways of reducing relations:

• The monadistic view reduces "A is greater than B" to properties of A and B separately. But, Russell argued, if A has magnitude a and B magnitude b, we still need a relation between a and b (one is greater than the other) to complete the account. The relation has not been eliminated.
• The monistic view, associated with idealism, treats "A is greater than B" as a property of the whole composed of A and B. But the whole composed of A and B is the same whole as the one composed of B and A, so this view cannot distinguish "A is greater than B" from "B is greater than A."

Russell concluded that asymmetric relations are irreducible and that external relations are real. He later argued that external relations are required to give an accurate scientific description of the external world. This argument played a central role in the birth of analytic philosophy.

The early Ludwig Wittgenstein, in the Tractatus Logico-Philosophicus (1921), took a different line: in a state of affairs, objects hang together directly, like links in a chain, without a separate relational entity holding them together.

Twentieth-century and contemporary metaphysics

In the second half of the twentieth century, debates about relations were reshaped by Armstrong's theory of universals and by David Lewis's metaphysics.

Lewis proposed a thesis he called Humean supervenience: roughly, that everything about the world supervenes on the arrangement of local intrinsic properties at spacetime points, together with spatiotemporal relations among those points. On this picture, spatiotemporal relations are the only fundamental external relations; everything else is fixed by local matters of particular fact. Lewis acknowledged that physics might turn out to refute the thesis, but he defended it as a working hypothesis about what the world could be like.

Relations and physics

Entanglement

Quantum mechanics has been one of the strongest sources of pressure on views that try to reduce relations to intrinsic properties. In a quantum entangled state, two particles can be described together by a single state that cannot be broken down into separate states for each particle. Paul Teller argued in 1986 that entanglement commits physics to "relational holism": relations that do not supervene on the intrinsic properties of the relata. If this is right, Humean supervenience fails. Michael Esfeld has gone further, suggesting that at the fundamental level there may be no intrinsic properties at all.

Other physics

The SEP entry on relations notes several other ways in which physics seems to undercut a picture of the world as made of independent bearers of intrinsic properties. In classical mechanics, quantities such as velocity are represented by vectors and tensors whose definition involves relations among points (as argued by Jeremy Butterfield). In relativity theory, mass-energy and momentum depend on the structure of spacetime. In quantum field theory, quantities such as mass and charge depend on the scale at which a system is described. Each finding suggests that what looked like intrinsic properties may be partly relational.

Ontic structural realism

The most radical response is ontic structural realism, developed by James Ladyman, Steven French, Don Ross, and others. In works including Every Thing Must Go (2007) by Ladyman and Ross, they argue that there are, at the fundamental level, no objects in the ordinary sense: there is only structure, a network of relations. What we call objects are features of the structure. Critics, including Stathis Psillos, object that relations without relata make no sense: a relation must relate something.

Similarly radical ideas have appeared elsewhere. The philosopher Randall Dipert proposed that the world is a mathematical graph whose concrete entities are subgraphs. In physics itself, Carlo Rovelli's relational interpretation of quantum mechanics, first proposed in 1996, holds that the properties of a physical system are defined only relative to other systems.

Relations elsewhere in philosophy

The concept of relations is central in many other areas.

• Philosophy of perception: Relationalist theories of color hold that colors are not monadic properties of surfaces but relations between objects, perceivers, and viewing conditions. This would explain why the same surface can look different under different lighting or to different perceivers.
• Philosophy of mind: Some theories of perception, sometimes called naive realism or relationalism about experience, hold that perceptual experience is fundamentally a relation between a perceiver and the perceived object, rather than an inner representation.
• Social theory: Relational sociology studies society through the interactions and relations among people, treating relations, rather than individuals or social structures alone, as the primary units of analysis.
• Ethics: Relational approaches emphasize that moral obligations arise from relationships, such as those between parent and child or between friends.

A worked hypothetical

The following example is hypothetical and is meant only to show how the positions above would treat one simple case.

Two identical twins, Ana and Ben, stand in a room. Ana is two meters to the left of Ben, and the twins resemble each other closely. Consider three true statements: "Ana resembles Ben," "Ana is two meters from Ben," and "Ana is the sister of Ben."

• An eliminativist says that the room contains Ana and Ben, each with their own properties, and that resemblance, distance, and sisterhood are ways a mind compares them. Nothing else is there.
• A reductionist says that resemblance is fixed by the twins' intrinsic features: give two people those faces and heights and they resemble each other. So resemblance is internal and "no addition of being." Distance is harder: Ana and Ben could have exactly the same intrinsic features while standing ten meters apart. Many reductionists therefore treat distance as one of a small number of external relations that must be accepted as basic, as Lewis did.
• A realist accepts all three relations as features of the world, perhaps holding that the internal ones are cheap and the external ones genuine additions.
• A relationalist might go further and say that what it is to be Ana is partly a matter of her place in a network of relations, spatial, biological, and social.

Sisterhood is an instructive extra case. It depends on the twins' shared parents, a fact about history rather than about their present intrinsic features. Whether it counts as internal or external therefore depends on which definition one uses, and on whether facts about origins are treated as part of a thing's nature.

Common misconceptions

"Relations are just properties with extra steps"

Relations can sometimes be modeled as properties: one can talk about the property of being taller than Ben Vane. But Russell's arguments about asymmetric relations show that eliminating relations in favor of properties is far from straightforward. The relational property "being taller than Ben Vane" itself mentions a relation.

"The mathematical definition settles the philosophical question"

Identifying a relation with a set of ordered pairs is a precise and useful convention. But it uses order, an inherently relational notion, in its definition, and it cannot distinguish two relations that happen to hold between exactly the same things. It describes which things are related, not what relating is.

"If a relation is internal, it does not exist"

Reductionists hold that internal relations are "no addition of being," but this is a claim about fundamentality, not existence. It remains true that Ben Vorlich is taller than Ben Vane; the question is whether the world contains anything beyond the two mountains and their heights that makes it true.

"Bradley's regress was refuted long ago"

Russell and others rejected Bradley's conclusion, and few philosophers today accept his monism. But there is no consensus on where exactly his argument goes wrong, and the question of what unifies a relation with its relata is still actively debated.

"Relationalism means everything is subjective"

Saying that something is relational, such as a color depending on perceivers and lighting, does not make it subjective or arbitrary. A relation can hold objectively. Being west of Edinburgh is a relational fact about Glasgow, and it is not a matter of opinion.

Frequently asked questions

What is the difference between a relation and a relationship?

In everyday English, "relationship" often refers to a personal connection between people. In philosophy, "relation" is a general term for any way two or more things stand to each other, including logical, spatial, causal, and mathematical relations. A personal relationship is one kind of relation.

Why does it matter whether relations are reducible?

It matters for what one thinks reality is made of at the most basic level. If all relations reduce to properties, then a complete inventory of things and their properties would describe the world completely. If some relations are irreducible, as entanglement may suggest, then such an inventory would leave something out: the world would be more than the sum of its parts.

Is causation an internal or an external relation?

Traditionally, causation has been regarded as external: the same two events could have existed without one causing the other, and what connects them is a matter of the laws of nature. Philosophers who ground causation in the intrinsic powers of objects argue it may instead be internal. The question is unresolved.

What is a multigrade relation?

It is a relation that can take different numbers of relata on different occasions. "Being the tallest among" can hold among two people or a hundred. Whether such relations are genuinely fundamental or can always be analyzed in terms of fixed-adicity relations is debated.

What did Leibniz think about relations?

Leibniz held that relations are not fundamental features of reality. They are grounded in the properties of individual substances, combined with a mental act of comparison. He also argued that space is not a container but an order of relations among coexisting things.

What is the location problem?

It is the question of where a relation is. If Glasgow is west of Edinburgh, the relation does not seem to be in either city alone, and it is not clear what it would mean for it to be "between" them. The problem pushes some philosophers toward eliminativism and others toward treating relations as non-spatial.

How do relations matter in science?

Physics describes the world largely through relations: distances, forces between bodies, spacetime structure, and the correlations of entangled systems. Some philosophers take this as evidence that relations are fundamental, perhaps more fundamental than objects themselves.

Summary

Relations are the ways things stand to one another. Philosophers classify them by how many relata they take, by logical features such as symmetry and transitivity, and by whether they are internal or external, formal or material, logical or causal, spatial or temporal. For most of history, relations were treated as second-class citizens, dependent on substances and properties. Leibniz tried to reduce them away; Bradley argued they lead to an infinite regress and concluded that reality is a single whole. The development of the logic of relations by De Morgan, Peirce, and Frege, and Russell's arguments about asymmetric relations, put relations at the center of analytic philosophy. Contemporary debates concern the problem of order and converses, whether non-symmetric relations exist, how to answer Bradley, and whether physics, especially quantum entanglement, shows that some relations are irreducible or even that the world is fundamentally a structure of relations.

Sources

• Relation (philosophy), Wikipedia)
• Relations, Stanford Encyclopedia of Philosophy
• Bradley's regress, Wikipedia
• Binary relation, Wikipedia
• Leibniz-Clarke correspondence, Wikipedia
• Structural Realism, Stanford Encyclopedia of Philosophy
• Relational quantum mechanics, Wikipedia