Philosophers discuss the meaning, function, and possibility of offering definitions. It is typical (e.g., in college logic texts) to distinguish a number of different kinds of definition: dictionary or lexical definition[?], intensional definition[?], extensional definition[?], ostensive definition[?], stipulative definition[?], operational definition and others.
Philosophers are constantly asking for definitions, or accounts[?], or analyses[?], of various philosophical concepts such as existence, causality, meaning, knowledge, goodness, justice, beauty, art, and so on. Thus they hope to clarify in some manner the most pervasive concepts descriptive of the universe and human life.
Just as arguments can be good or bad, definitions can be good or bad. A definition is supposed to give us the meaning of a word; there are certain aspects of the topic of definition that cannot be understood until one reviews a few features of meaning. We will therefore review the topics of extension, intension, ambiguity, and vagueness.
Begin with the distinction between the extension and the intension (both spelled with an "s") of a word. This is very similar to a familiar distinction--between a word's denotation and connotation. Take the word 'bachelor'. The extension of this word is all and only the bachelors in the world. The extension of this word would include several hundreds of millions of men. The intension of this word can be stated relatively briefly, because it includes just two properties: the property of being a man, and the property of being unmarried. So all bachelors are unmarried men, and only bachelors are unmarried men.
The sort of definition that philosophers are interested in, insofar as they are interested in definitions at all, is one that identifies the intension, not the extension of the word. An excellent definition of the word 'bachelor' is 'unmarried man'. A less-than-excellent definition of the same word would be a list of names of all of the men in the world who are bachelors. Aside from being practically impossible, such a list is just not what we are looking for. After all, what we are interested in is a description of what all those things we call 'bachelors' have in common, which distinguishes them from all non-bachelors. A lst of all bachelors would give us no means to determine whether any new human is a bachelor or not.
Notice now that there are two different ways in which the meanings of words can be unclear. Words can be unclear in the sense of being ambiguous, or in the sense of being vague (or, it so happens, in both senses). Most words are, in fact, both ambiguous and vague. This is not a skeptical or even a controversial claim; to say that many, or perhaps even most, words are both ambiguous and vague is not to say that they have no meaning. It is to say, first, that many individual words many distinct senses; and, second, that those senses are often, in ordinary language, not so precise as to be able to allow us to rule that the word does or does not apply in every case. So certainly, a word that is both ambiguous and vague can have a rich fund of meaning.
With this background, it will be easier to discuss definitions. Suppose we have decided on some word, or concept associated with the word, to define. Suppose also that we have identified which sense of the word we are interested in, and we have noted clear cases, some unclear cases, and some borderline cases of the application of the word. So we ask: how can this word be defined? We already know that we want a description of the intension of the word: that is, we want an account of the set of properties that characterizes all and only members of the extension. In that case, it seems the following is a servicable account of the meaning of '(intensional) definition':
The above is a rough definition of 'intensional definition'. How might one go about formulating an intensional definition? Consider an example from philosophy--in particular our example is from the field of metaphysics.
Suppose we want to give a definition of causality, or causation. We want to know what it means to say that the cue ball causes the eight ball to roll into pocket, or that heat causes water to boil, or that the Moon's gravity causes the Earth's tides, or that a hard blow to the arm causes a bruise. What does all this talk of 'causes' mean? We all have some rough idea of the extension of the term 'causality': we are familiar with all sorts of particular cause-and-effect relations. The set of all those particular cause-and-effect relations is the extension of the term 'causality' (or of 'causal relation'). But what properties do all these particular cause-and-effect relations have in common? We say heat causes boiling, and punching causes bruising; so what do these two relations have in common, that we can both call them causal relations?
We can begin by taking a clue from the ancient Greeks, who treated concepts to be defined as species, or a specific category, of a genus, or a general category. Beginning with Socrates, and codified by Aristotle, the ancient Greeks sought so-called genus-species definitions[?]. So we begin by asking: what is the genus of causality, the general category into which it falls? In other words, what sort of thing is causality?
Causality is a kind of relationship, or simply relation for short. We do, after all, speak of causal relations between things. The causality relation is the relation that holds between what we call a 'cause' and what we call an 'effect'. Another example of a relation is similarity. Suppose we say Jack and Jill are similar in appearance; then we say there is a certain relationship, the relation of similarity between them.
So we can say that, taking causality as a species, then the genus of that species is relation. Obviously we would have to know what kind of relation causality is.
Suppose we came up with some properties that allowed us to distinguish causality from all those other relations. They would be the distinguishing properties of causation. Those distinguishing properties would be called, by the ancient Greeks, the differentia of the species. The differentia of a species are the properties that the species has, and that other members of the genus do not have. So the differentia of a species are the distinguishing characteristics of the species. If we discover any, we can formulate a genus-differentia definition.
So if we are looking for a definition of a philosophical concept like causality, we might begin like this: "Causality is a relation that ... ." Here is an example:
(Something like this definition was offered by the Scottish philosopher David Hume.)
So when we say there is a causal relation between heat and water boiling, we say: the heating came before the boiling, and whenever water is heated sufficiently, then it boils. So sufficient heating is always, or consistently, followed by boiling. In this case, we can say that causality has a rather complex differentia: it holds between events; moreover, the first event precedes the second; and finally, events like the first are consistently followed by events like the second. According to this definition, those three rather complex properties together are the differentia of the species causality.
This is only a crude attempt at a definition of 'causality'. But it does provide an example of the sort of thing that we are looking for in a definition of a philosophical concept.