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Now we're going to switch gears rather abruptly and look at the two closely related topics of meaning and definitions. I especially want us to consider definitions in their philosophical context, because philosophers are constantly asking for definitions, or accounts, or analyses -- at this point it matters little what to call them -- of various philosophical concepts such as existence, causality, meaning, knowledge, goodness, justice, beauty, art, and so on. In fact, you might regard it as the essential task of philosophy, to clarify the most pervasive concepts descriptive of the universe and human life.

Now, just as arguments can be good or bad, definitions can be good or bad. So we will spend some time thinking about what makes a good definition good, and what makes a bad definition bad. But before that we should discuss a little the topic of meaning, a topic we will return to later this quarter. A definition is, after all, supposed to give us the meaning of a word; so naturally there are certain aspects of the topic of definition that cannot be understood until we understand a few things about meaning.

Let?s begin by drawing a distinction, between the extension and the intension (both spelled with an "s") of a word. This is very similar to a distinction you might already be familiar with -- between a word?s denotation and connotation. But for clarity let?s stick with "extension" and "intension."

The extension of a word is the set of things it extends to, or applies to. So the extension of the word "dog" includes all the dogs in the world: Fido, Rover, Lassie, Rex, and so on. The extension of the phrase "people in Sanger?s class this evening" includes all the people sitting in this class right now. Now, that this is a sense of the word "meaning" should be understandable. The meaning of a word is, sometimes, just the things that the word picks out, or applies to. So let?s get a nice example on the table. The phrase "Sanger?s fingers" picks out exactly eight objects (doesn?t include his thumbs, we?ll say). All right? So these objects right here are the extension of the phrase "Sanger?s fingers."

You can think of a word?s, or a phrase?s, intension as a description, or set of properties, which applies to each member of the word?s extension, and which distinguishes those things from everything not in the extension. This is a rather complicated definition, but it?s not too difficult of a concept. Here?s an example: what would be the intension of the phrase "Sanger?s fingers"? Well, two properties: first, the property of being a finger; second, the property of being possessed by Sanger. Those two properties together make up the intension of the phrase "Sanger?s fingers." Anything that has both of those properties is part of the extension of the phrase "Sanger?s fingers"; so for example as I hold up my left pointer finger, I can observe that it has the property of being a finger, and that it is possessed by Sanger; so the intention of the phrase "Sanger?s fingers" applies to my left pointer finger; and so my left pointer finger is part of the extension of the phrase "Sanger?s fingers." Moreover, anything that is not a finger, or not possessed by Sanger, is not part of the extension of the phrase "Sanger?s fingers."

I?m saying all this now simply to help you understand better the difference between a word?s extension and its intension. To help drive the point home, let me give you another example. 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, I imagine, several hundreds of millions of men. Kind of sad if you think about it. Anyway, the intension of this word isn?t nearly so large; in fact, it can be stated very 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.

All right, now remember that I said that a definition of a word is an attempt to state the meaning of the word. With this background, I think you can see that the sort of definition that we?re interested in, in philosophy, 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?re looking for. Because, 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.

Now I come to a different topic about meaning: namely, 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).

So let?s begin with ambiguity. One and the same word can have many distinct meanings; so the word is, as we say, ambiguous. We?ve just encountered some great examples of an ambigious words, the word "meaning" itself can mean "extension" or "intension." The word "unclear" can mean "ambiguous" or "vague." But let?s get a more ordinary example of an ambiguous word. Try this for size. "Bank": as a noun, this can mean either a financial institution where you put your money, or the edge of a river where you might go fishing; and it can mean other things still when used as a verb. Now, the different meanings that a word can have are its senses. So "financial institution" is one of the senses of the word "bank," and "edge of the river" is another sense. So you can see that if a word is ambiguous, then it has more than one sense. Sometimes there are some rather fine distinctions that can be drawn between different senses of the same word. For example, take the word "good"; this can mean "useful," as in "That?s a good hammer," or "exemplary," as in "She is a good student," or "moral," as in "They are good people." No doubt "good" has other senses as well.

Now, in philosophy, it is extremely important, before one starts trying to come to grips with some deep, important philosophical term, that one specify just exactly what sense of the term one is examining. For example, I might ask, "What does ?good? mean?" but I can?t even begin to answer until I have said which sense of "good" I mean. So discussions of philosophical concepts, especially in the past eighty years or so, have often begun by talking about the different senses in which a word may be used.

So much, then, for ambiguity. That?s one way in which meanings can be unclear. There is another way, which is different from ambiguity, namely vagueness. Here the stock example is "bald." Some men are definitely bald, and there?s no debating the matter; for example, Patrick Stewart (Captain Picard of Star Trek: The Next Generation). Other men are definitely not bald; for example, President Clinton. And then there are a bunch of men, you know, the sort of men who carefully comb a scanty amount of hair over their scalps, about whom we don?t know whether to say they?re bald or not. There just isn?t any clear line between being bald and not being bald. Another good example of a vague concept is the concept of a heap -- two or three grains of sand is not a heap, but a thousand is. So how many grains of sand does it take to make a heap? There isn?t any clear line.

Now, when you look at a man with thinning hair, or a small pile of sand, and you don?t know whether to call the man "bald," or the sand a "heap," then you have found a borderline case. So we can make a general principle, which in fact might work as a definition of the word "vague." Here it is: to say a term, or a concept, is vague is to say it has borderline cases; in other words, cases where there is no clear fact of the matter whether the concept applies or not. Consider those animals in Alaska that are the result of breeding Huskies and wolves: are they dogs, or aren?t they? It?s not clear: so we can say that they are borderline cases of dogs. This isn?t to say they are dogs, and it?s not to say that they aren?t dogs. It?s to say that our concept of doghood isn?t clear enough to let us rule conclusively in this case.

It turns out that vagueness is important philosophically. Suppose we want to come up with a definition of "right" in a moral sense of this term. Generally, we can say that we want a definition to cover actions that are clearly right, and exclude actions who are clearly wrong; but then what do we do with the borderline cases of actions that aren?t clearly right and aren?t clearly wrong? Surely there are such cases. What philosophers often say is that we should try to come up with a definition that is itself unclear, on just those cases that are borderline cases. Others say that we have an interest in making our definitions more precise than ordinary language, or our ordinary concepts, would by themselves allow. Well, having mentioned that issue, I?m just going to sidestep it. You should just be aware that some philosophers want their definitions to be unclear in precisely those areas in which the ordinary concept to be defined is unclear; while other philosophers want their definitions to be more precise than the ordinary concepts.

Now I have introduced the notions of ambiguity and vagueness. These are not dangerous, difficult, or skeptical notions; 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 doesn?t apply in every case. So certainly, a word that is both ambiguous and vague can have loads of meaning.

Now let?s 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?re 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. But in fact, why don?t we just define "definition" that way? --

The definition of a concept, or of (a given sense of) a word or phrase, is a description of its intention -- that is, the set of properties that characterizes all and only members of the extension of the word; the extension is all the things that the concept, word, or phrase applies to.

Now, some philosophers have some criticisms of this sort of definition of the word "definition"; or perhaps it would be better to say that some philosophers think that it is, for various reasons, impossible to give definitions of most concepts, words, and phrases. We don?t have time, or the conceptual resources, to explore these criticisms. And even if those philosophers are right, they will, most of them, still acknowledge that in philosophy we should do something like give definitions of important philosophical concepts. So for our purposes, in just getting introduced to philosophy, it will do just fine. So a definition is, to put it very briefly, a description of the intension of a word. When you have stated the intension of a word, you have defined it.

Now if you go and look at some introductory logic textbooks, you might see a list of kinds of definitions, such as "lexical" definitions, "stipulative" definitions, "ostensive" definitions, and so on. But we have little reason to consider all of those different categories of definitions; we are primarily interested in one sort, what might be called a intensional (again, spelled with an "s") definitions.

So now about intensional definitions specifically. Now I just gave you a definition of "definition." Well, really that was a definition of "intensional definition." That is the sense of the word "definition" that we defined; so you already know what an intensional definition is because I just told you. What I want to do is to give you some idea about how you might go about formulating an intensional definition. And since this is a class about philosophy we will use an example from philosophy -- in particular our example is from the field of metaphysics.

Suppose you wanted to give a definition of causality, or causation. You want to know what it means to say that the white 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? You have some rough idea of the extension of the term "causality": namely, you?re familiar with all sorts of particular cause-and-effect relations. I just listed a bunch for you. The set of all those particular cause-and-effect relations is the extension of the term "causality." 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?

Well, 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. 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? That might sound like an impossibly abstract and difficult question, that nobody can answer -- what sort of thing is causality? Who knows?

Well, here?s an answer: 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." So causality is a relationship, a relation, just as (to take another example) similarity is a relation. You know, 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 similarity is one kind of relation; and causality is another kind of relation.

So we can say that, taking causality as a species, then the genus of that species is relation. But we are not even close to being done in giving a definition of "causality," because obviously we would have to know what kind of relation causality is. There are lots of different kinds of relations out there -- similarity, being larger than, being warmer than, being next to, and so on -- and so we need to have some way to distinguish the causality relation from these other kinds of relations.

Now suppose we did come 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.

This can be clarified with another hackneyed example. Suppose we wanted to define the phrase "human being." Following the Greeks we say that human beings are members of a species. So we ask what the genus, or general category, of the species is; and the Greeks (but not us -- go with it for now) would say that the genus is animal. So the genus is animal and the species is human being. Now we ask: what are the differentia of the species, that is, the distinguishing characteristics, that is, the properties that human beings have, that other animals don?t have? Well, the Greeks said, it?s rationality. The things that humans have that other animals don?t is rationality. So rationality is the differentia of the human species, according to the Greeks; and so an old Greek like Aristotle would be prone to say, "Man is the rational animal." And he meant to be giving a definition of "man," or of "human being."

So if we?re looking for a definition of a philosophical concept like causality, we might begin like this: "Causality is a relation that ... ." The question then is what goes after the "that." Now, so as not to let you die of suspense, I?ll just give you an example of a definition of causality; it may not be a good definition, in fact I?m sure it isn?t. But it?s a plausible attempt. So here it is:

Causality is a relation that holds between events, where the first event (called the cause) precedes the second (called the effect), and where events like the first are consistently followed by events like the second.

So when we say there?s 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 that sufficient heating is always, or consistently, followed by boiling. So we can say that causality has a rather complex differentia: it?s a relation that holds between events; moreover, the first event precedes the second; and finally, events like the first are consistently followed by events like the second. So, according to this definition, those three rather complex properties together make up the differentia of the species causality.

This is, in fact, a rather crude attempt at a definition of "causality." But it should give you an example of the sort of thing that we?re looking for in a definition of a philosophical concept. The whole point of going into all this stuff about causality is to give you an example about how you might formulate an intensional definition of a philosophical concept.

All right, we now have one last topic to discuss, and that is the way that definitions can go wrong. Just as arguments can go wrong in various ways (remember, we called those fallacies), definitions too can go wrong in various ways. So I am simply going to list five ways that definitions in philosophy can be inadequate. Bear in mind I am saying this is how definitions can go wrong in philosophy. For other purposes the following definitions may be fine.

First, a definition is no good if it simply gives a one-word synonym. For example, suppose I define the word "virtue" -- an important word in ethics -- just using the word "excellence." It might be perfectly true that all virtues are excellences and all excellences are virtues, but the word "excellence" by itself is not a good definition of "virtue," in philosophy. If this isn?t just obvious to you, consider this. Suppose you are asking me for enlightenment on what virtue is; you know how the word "virtue" is used in English, but you don?t really understand the basic nature of virtue. Then I tell you: "virtue" means "excellence." Now what is stopping you from asking, "But what does ?excellence? mean?" Surely, if you?ve got a basic philosophical confusion about what "virtue" means, then you?ve also got a basic philosophical confusion about what "excellence" means. If you?re confused about one then you?re confused about the other; so it won?t do to define one simply by stating the other!

The second sort of error is closely related to the first. A definition is no good in philosophy if it uses a very near synonym in the definition. For example, suppose I define "beautiful" as "possessing aesthetic value." The words "beautiful" and "aesthetic" are very nearly the same in meaning; so if I?m deeply confused, or curious, about beauty, then I am of course going to be confused or curious about the aesthetic. And it is totally unenlightening to be told that what is beautiful possesses aesthetic value. We already knew that. The question is what general characteristics are possessed by all beautiful objects, or all objects that have aesthetic value. Here?s another example: suppose I define "good" as "possessing valuable characteristics." "Good" and "valuable" are near synonyms, so it won?t do to define "good" using the term "valuable." Again, if we?re confused about goodness, we are no doubt also confused about what?s valuable -- about what characteristics are in general valuable.

There is a general name for the first two sorts of error: in either case we may say that the definition offered is circular. A circular definition is somewhat similar to a question-begging argument: neither offers us enlightenment about the thing we wanted to be enlightened about. I think you can already see, then, that giving a good definition of philosophical terms is going to be very difficult, if we?re going to avoid giving a circular definition.

Anyway, on to the third sort of error that definitions are prone to. They can be too broad. Suppose I define "bachelor" as "unmarried male." On first glance this might look all right, but it applies to a lot of other things besides bachelors: for example, male dogs, and male babies. Those are unmarried males but they aren?t bachelors. So definitions should not be too broad, and a definition is too broad if (and you can treat this as a definition of the words "too broad") it applies to things that are not part of the extension of the word defined.

The fourth sort of error is complementary to the third: they can be too narrow. That is, they can exclude some things that they should apply to. In other words, they don?t describe some members of the word?s extension. Here?s an example of a narrow definition: "piece of furniture" means "object that is used to sit on." The trouble with this definition is that some pieces of furniture are not used to sit on; for exaple, we put objects on them (like tables) or we put our feet on them (like footstools), and so forth. So even though some pieces of furniture are objects that are used to sit on, not all furniture is used to sit on. We need a broader definition: we might add other qualifying characteristics, like "used to put feet up on" or "used to put household objects on," for example. That would make the extension of the definition bigger -- that is, the definition would apply to more things, and more of the things that we use the word "furniture" to describe.

Let me give you examples of definitions that are too broad, and too narrow, in philosophy. Suppose I defined "causality" as "relation where one thing precedes another thing." This is way too broad. All sorts of things precede other things without causing them. For example, the sinking of the Titanic preceded the Vietnam War, but the sinking of the Titanic certainly did not cause the Vietnam War. So "relation where one thing precedes another thing" applies to all sorts of relations that aren?t causal relations.

Now here?s a philosophical definition that is too narrow. "Right action" means "telling the truth." Surely, telling the truth is almost always right. But right actions very often don?t involve telling the truth at all. For example, it is right for me to pay back my debts; but when I do this I am not telling the truth; I can pay back debts, and so do something right, without saying anything at all. So a definition of "right action" as "telling the truth" is way too narrow; the definition is going to have to include a lot more than just truth-telling.

That brings us to the fifth and final way definitions can go wrong: they can go wrong by using ambiguous, obscure, or figurative language. Suppose I defined "love" as "the insensible quivering of the soul." This is absolute nonsense, and totally useless as a definition in philosophy, however nice it might be in poetry (although I doubt it). Given a definition like this one has the right to ask: but what is the insensible quivering of the soul? How would I recognize it? Is my soul insensibly quivering right now? And so on. I?m sure you get the idea. Definitions should be stated in plain, straightforward language that can be understood by the people to whom you?re giving the definition.


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