Definition

The problem of definition enters philosophy with Socrates. In dialogue after dialogue (the Meno, the Laches, the Euthyphro, the Republic) Socrates refuses to accept particular instances in answer to questions about universals. When Meno enumerates the virtues of men, women, children, and slaves, Socrates objects: he asked for the one thing that all virtues have in common, not a list of different kinds.

This demand for the universal is simultaneously a demand for the Form. Socrates believes that genuine definition must capture what the thing really is (its essence or nature) not merely catalog instances or offer partial accounts. The definition of "figure" must cover all figures; the definition of "virtue" must cover all virtues. Anything less is not definition but enumeration.

Plato deepens the Socratic demand by connecting definition to the Forms. To define "beauty" is to state what Beauty Itself is, the separately existing universal that the many beautiful things approximate but do not exhaust. Genuine definition achieves what ordinary language only gestures toward. The Sophist, with its intricate analysis of being, not-being, and motion, explores the difficulty of defining concepts so fundamental that their definition requires a new kind of philosophical dialectic.

Plato's Socratic requirement (that definition reach the universal essence, not merely list instances) remains the standard against which all subsequent accounts are measured. Aristotle will accept the requirement but give it a different metaphysical foundation; Hobbes will reject it as too demanding; Locke will distinguish real from nominal essence to explain why it is often unattainable in practice.

Aristotle accepts the Platonic demand that definition state the essence but gives it a precise logical form. A genuine definition specifies the genus (the wider class to which the definiendum belongs) and the differentia (the characteristic that distinguishes it within that genus). "Man is a rational animal" gives the genus (animal) and the differentia (rational). The definition captures the universal essence shared by all members of the species.

Aristotle's technical discussion of definition in the Posterior Analytics and Topics addresses a series of difficult questions. Can definitions be proved, or must they be assumed as first principles? What is the relation between definition and scientific demonstration? How do we test whether a proposed definition is accurate? The Topics provides a set of common argument patterns that can be used to test proposed definitions by checking whether the genus and differentia are properly assigned and mutually consistent.

One of Aristotle's key contributions is the distinction between nominal and real definition. A nominal definition states what the name means; a real definition states what the thing is. Real definition requires knowledge of the essence, which is attainable through intellectual perception of the universal in the particular. Real definitions are not merely verbal conventions but truths about the nature of things.

Aristotle's account of definition as genus plus differentia becomes the standard framework, but it carries a tension he does not resolve: if every genus must itself be defined, the regress ends only at terms that are indefinable, yet the demand for definition was supposed to terminate at essence, not at brute verbal convention. Locke draws from this the conclusion that real definition of natural things is largely beyond our reach; Spinoza insists that genetic, causal definition can still give us genuine knowledge of essences.

Aquinas defends Aristotle's account of real definition while addressing a series of theological complications. The central problem: if real definition states the essence, and essence is known through abstraction from sensory experience, then how can we define immaterial beings (God, angels, the human soul)? These lie beyond sensory experience and therefore beyond the standard method of abstraction.

His solution employs three types of discourse about God: via positiva (affirming what God is by analogy with creatures), via negativa (denying what God is not), and via eminentiae (affirming what God possesses in a perfected degree). The divine essence exceeds all our concepts; we cannot give a real definition of God as we can of man or triangle. The best we can do is analogical predication, which gestures toward the divine reality without capturing it in a finite concept.

For material things, Aquinas maintains the Aristotelian account with a refinement: the essence abstracted from experience is not the individual thing's essence but the universal essence common to all members of the species. Definition states this universal essence. The definition "man is a rational animal" is true not just of Socrates but of humanity as such. This universality is what makes definition scientifically useful: it yields knowledge about kinds, not merely about individuals.

Aquinas's account of definition preserves the Aristotelian framework while complicating it with a problem no ancient philosopher had faced: if the most important being is also the one that defeats all definition, then the summit of inquiry is not definition but analogy and negation. Locke will secularize this limit, arguing that even ordinary natural things resist real definition because their inner constitutions are hidden from us.

Hobbes takes a radically nominalist position on definition. There are no universal essences in nature; there are only particular things and the names we give them. Definitions are not truths about the nature of things but conventions that fix the meanings of words. "Man is a rational animal" does not capture the essence of humanity; it stipulates how the word "man" is to be used in a particular discourse.

This has important consequences for science and philosophy. Hobbes agrees with Aristotle that science begins from definitions and proceeds by demonstration. But for Hobbes the definitions are not axioms about reality; they are arbitrary starting points that could in principle have been chosen differently. The conclusions of science follow necessarily from the definitions, but whether the definitions capture any real features of the world is a separate question.

Definitions can still be more or less adequate in a pragmatic sense. In geometry (the only science Hobbes takes as his model) definitions serve as the ultimate principles from which all theorems are derived. Philosophic disputes, on his account, arise from faulty or undefined terms. Clear definition is the beginning of all genuine inquiry: "the errors of definitions multiply themselves, according as the reckoning proceeds."

Hobbes's nominalism separates definition from knowledge of essences and makes it purely a matter of linguistic convention. Locke will develop this insight into his more nuanced distinction between nominal and real essence. The rationalist tradition, from Descartes to Spinoza, will resist the move to pure verbalism by insisting that some definitions at least must capture something real.

Pascal's short treatise On Geometrical Demonstration distinguishes two kinds of definitions that are ordinarily conflated. The first are "definitions of names," in which a writer assigns a chosen label to an idea or thing that has already been pointed out. These Pascal holds to be entirely free: nothing is more permissible, he says, than to give whatever name we please to a thing we have clearly pointed out. The second are "definitions of things," which would state what the thing really is. Pascal refuses these the title of definitions at all, since they are properly propositions subject to proof rather than conventions of speech. Geometry, the science Pascal takes as the model of demonstration, admits only the first kind, and its whole rigor depends on keeping the naming of terms strictly apart from the affirmation of their existence.

Yet this does not leave the geometer without standards. Pascal lays down three conditions. A definition must not contradict itself, since a name applied to an impossibility fixes nothing at all. It must not give the same name to two different things, or the subsequent reasoning will silently trade on an ambiguity. And it must not attempt to define terms that are already perfectly understood, since the attempt would obscure rather than illuminate them. Certain words belong to this last class. "Being," "time," "equal," "number": these are indefinable not because their meaning is unknown but because it is too immediate to admit of explanation by still simpler words. To undertake a definition of being, Pascal notes, is at once to use "it is" in the formula, and so to presuppose the very term one is trying to define. The work of the geometer begins at the point where the work of the dictionary must stop.

The doctrine that some terms stand as primitives is hardly Pascal's alone, but he gives it a mathematician's inflection. What comes first in the order of demonstration is not what is most abstract but what is most immediately grasped, and the task of definition is a task of settlement rather than discovery. In this he stands close to Hobbes in regarding definitions in science as acts of fixing language rather than as insights into essence, and close to Descartes and Spinoza in treating definitions as the principles from which demonstration proceeds. Where he differs from both camps is in carefully separating the question of naming from the question of what exists. Whether his sharp distinction between nominal freedom and real existence can be maintained is a question that occupies much of the later tradition, and it is treated more fully in the chapters on PRINCIPLE and REASONING.

Kant will later take up Pascal's distinction between mathematical definition, which is free to construct its own concepts, and philosophical definition, which must work out what is already obscurely contained in given concepts. The dispute over whether metaphysics can legitimately imitate the geometer's method runs from Spinoza's Ethics through the Critique of Pure Reason, and Pascal stands among its earliest and most careful witnesses.

Locke's treatment of definition in Book III of the Essay is built around a distinction that shapes the entire subsequent tradition: nominal essence versus real essence. The nominal essence of a thing is the abstract idea we form by sorting together the observable qualities that typically co-occur. "Gold" means, in our nominal definition, a substance that is yellow, heavy, malleable, fusible, and soluble in aqua regia. This definition is arbitrary (we chose these properties as defining) and conventional (others might sort differently).

Real essence, by contrast, is the inner constitution of a thing, the microscopic structure that causes its observable qualities. We assume there is such a constitution; it would explain why gold has the properties it has. But our ideas of real essences are, in practice, unavailable to us. We cannot see into the atomic structure of things; our senses give us only the macroscopic qualities.

The implication is that Aristotelian real definition (definition of real essence by genus and differentia) is largely unattainable in natural science. We can give nominal definitions that are useful for communication and practical purposes. We cannot give real definitions that state the inner constitution of natural kinds. Locke's account is both more modest and more practically adequate than Aristotle's.

Locke's distinction between nominal and real essence has continued to inform discussions in the philosophy of language and science. The question of whether science aims at real definitions, stating what things truly are, or only at nominal ones that fix how terms are used, is treated more fully in the chapters on SCIENCE and NATURE.

Spinoza proposes what he calls "genetic" or "causal" definition: a proper definition of an uncreated thing must state the proximate cause from which all the thing's properties can be derived. Definition by genus and differentia, as Aristotle proposes, tells you that a thing belongs to a class with certain properties. A causal definition tells you how the thing is generated or constituted, from which all its properties follow necessarily.

He illustrates with geometry. The definition of a circle as "the figure described by any line whose one end is fixed and the other end moves" is genetic: it specifies the construction from which all the properties of circles follow. In contrast, "a figure in which all lines drawn from the center to the circumference are equal" merely states a property. The first is a genuine definition; the second is, at best, a partial description.

This genetic ideal of definition connects to Spinoza's rationalist metaphysics. God (or Nature, or Substance) is defined as "that which is in itself and is conceived through itself." From this definition alone, by pure reasoning, Spinoza derives the entire system of his philosophy. Real definition yields a priori knowledge of essences: understanding how a thing is constituted is the same as understanding its place in the necessary order of things.

Spinoza's genetic ideal is a response to Plato's original demand that definition reach the essence and not merely list properties. Its vulnerability is the assumption that genuine definitions can be stipulated as first principles; Kant argues that in philosophy, unlike mathematics, concepts are given before definitions and cannot simply be laid down as starting points.

Hume's theory of definition is a special case of his theory of ideas. Every genuine idea is a copy of an impression (a direct sensory or emotional experience). Defining a concept means tracing it to its original impressions. If no impression can be found, the "concept" is not a genuine idea but a confused noise of words.

This provides a radical criterion of significance for philosophical concepts. "Cause" appears to mean more than constant conjunction. Philosophers claim it involves necessary connection. Hume asks: what impression does "necessary connection" come from? He concludes that it comes from the feeling of expectation in the mind, a subjective determination, not a feature of the objective world. We project the feeling onto things.

The same analysis applies to "substance" (no impression of bare substance, only of qualities), "the self" (no impression of a continuing self, only of a bundle of perceptions), and "power" (no impression of objective efficacy). Definition, on Hume's account, is an empirical project: every concept must earn its credentials by pointing to its experiential source. Philosophy is full of pseudo-concepts that, when subjected to this test, turn out to be empty words.

Hume's empiricist criterion of definition has proved both useful and disruptive as applied to the history of philosophy. It exposes genuine confusions in philosophical vocabulary. But it also threatens to discard too much: Kant argues that the concept of causation cannot be reduced to impressions without losing the necessity that makes it a genuine concept.

Kant distinguishes sharply between mathematical definition and philosophical analysis. Mathematical concepts are constructed by pure intuition (we literally draw or imagine the figure) and their definitions are both complete and certain. A circle is exactly what its construction says it is; there are no hidden properties waiting to be discovered. Mathematical definitions are therefore genuinely initial: they establish the concept rather than analyzing one already given.

Empirical concepts cannot be strictly defined. The concept "gold" can always be made more precise as our knowledge of gold increases; there is no fixed boundary. Pure philosophical concepts (the categories) can be "analyzed" (their components made explicit) but not defined in the mathematical sense, since they are not constructed by us but given to us as conditions of experience. To claim to define "substance" or "causation" in a strict sense is to claim a priori knowledge of reality that we do not have.

This has a direct bearing on the rationalist program of Spinoza and Leibniz, who tried to construct philosophy geometrically from definitions. Kant argues the attempt is misguided: philosophical knowledge does not proceed from definitions to consequences but from analysis of given concepts to their conditions. The proper method of philosophy is analysis, not construction.

Kant's distinction between mathematical construction and philosophical analysis bears on the subsequent development of logic and the philosophy of language. Frege and Russell inherit the ideal of logically perfect definitions and attempt to extend mathematical rigor to philosophy. The result, mathematical logic, addresses as a technical program what Kant had regarded as impossible for philosophy in the traditional sense.

Mill's treatment of definition in the System of Logic navigates between Hobbes's pure nominalism and Aristotle's essentialism. Against Hobbes, Mill argues that not all definitions are equally arbitrary. Definitions of "real kinds" (natural groupings of things that share many properties and respond to inquiry with more properties) are constrained by nature. "Gold" is not an arbitrary grouping; things that share gold's observable properties also share many others. The nominal definition points at a real cluster of properties that makes scientific inquiry productive.

Against Aristotle, Mill insists that even the best scientific definitions are of names, not of things. "Man is a rational animal" defines the name "man" by specifying its meaning; it does not penetrate to some inner essence lying behind the observable properties. Real essences, in Locke's sense, are the province of chemistry and physics, the inner constitutions that explain why kinds have the properties they do. Philosophical definition does not reach there.

Mill's distinction between real kinds and arbitrary groupings helps explain why scientific classification is not merely conventional. Elements in the periodic table are real kinds; "large things" is not. Scientific progress consists partly in finding the real joints in nature, groupings that are not merely convenient but that correspond to the deepest structural regularities.

Mill's treatment of definition grounds the utility of scientific classification in the structure of nature rather than in pure logic or the essences of things. The questions this account raises about natural kinds and scientific realism are treated more fully in the chapters on SCIENCE and NATURE.

James inherits from Locke the suspicion that definitions can ever reach the real essences of natural substances, and from Mill the attention to how nominal definitions latch onto the useful groupings we call real kinds. What he adds is a pragmatic account of what the selective work of definition is for. When we classify an object, we do not take in all its properties at once. We single out a few and treat those as the characters that define it, setting the rest aside as accidental. The grounds of this selection, James argues, are neither intrinsic to the object nor logically forced upon us. They are the interests, theoretical or practical, that guide the classifier at the moment of classifying.

This leads James to a striking formula: the essence of a thing is that one of its properties which is so important for our interests that in comparison with it we may neglect the rest. A substance like oil has, on his account, as many essences as it has uses. For the lamplighter oil is combustible fuel; for the cook, a medium of heat; for the engineer, a reducer of friction. Each definition is correct in its context, and no one of them discloses the oil's "real" essence. Nature indifferently submits to any and all of the divisions we wish to make among existing things, and some schemes of classification are more significant than others only by reference to the purposes that motivate them. James is careful not to deny that objective resemblances exist; he denies only that such resemblances pick out essences all by themselves.

The doctrine inverts the Aristotelian order. For Aristotle a definition captures what a thing is, and the interest of the definer follows from that. For James the interest comes first, and the definition follows from it. This is the core of what has come to be called the pragmatic theory of definition, and it connects James's treatment of classification to his doctrine of truth and his account of reasoning, where the test of any concept is always what it enables us to do. The same posture frees him from Aristotle's quarrel with Plato over whether division should proceed by genus and differentia or by some other route. If classification is always a tool and never a mirror, then the question of the single right division does not arise. Adjacent questions about natural kinds and scientific realism are treated more fully in the chapters on UNIVERSAL AND PARTICULAR and SCIENCE.

James's pragmatic account leaves open the question how the purposes that govern classification are themselves to be assessed, and whether there is a single purpose broad enough to privilege one scheme of definitions as scientifically fundamental. On the first he is comfortable with a plurality of legitimate schemes. On the second, his later work in Pragmatism and The Meaning of Truth suggests that the comparative reach and fertility of a classification is itself a matter for patient experiment rather than for a priori decision.