Chapter 2. A Little Logic

Chapter 2. A Little Logic 2.1 An Intuitive Distinction: Logic can be regarded as the study of certain specific properties and relations among sentences. Those properties and relations are ultimately specified by logic itself, but you can begin to appreciate what logicians have in mind by contrasting the following two groups of sentences: A. 1. Plato had twelve children. 2. Plato didn't have twelve children. 3. It rained in Paris on 10 Sept 1992 4. It rained somewhere on 10 Sept 1992. B. 1. Either Plato had twelve children or he didn't. 2. It's not the case both that Plato had twelve children and that he didn't. 3. If it rained in Paris on 109/1992, then it rained in Paris on 10/92. 4. If it rained in Paris on 10/992, then it rained somewhere on 10/9/92. Before proceeding, it is worth trying to formulate for yourself the difference between the two groups, adding new examples to each. This shouldn't be hard. The intuitions that inform these judgments seem to be available to almost all human beings. One way of thinking about logic is as the study of these and related intuitions, and as therefore a kind of self-study (whether this is ultimately the best way to think about is a controversial issue on which we needn't take a stand here). It is important to appreciate throughout the study that the authority of logic need not be thought of as external to you: always ask yourself, after you have understood some problematic claim, whether you yourself don't really agree with it. If, after careful reflection, you find that you don't, then that is a reason for doubting that the claim expresses a genuine logical principle (which is not to say, however, that you needn't know it for the exam). But, in general, you should find that an initially problematic claim, once you understand it, is one that you do ultimately accept, and that your temptations to think otherwise are based upon confusions that you yourself will recognize as confusions. Returning to groups A and B: one thing that seems to distinguish them is how they stand with respect to truth. And here we need appeal to nothing more than the notion of truth captured by the Redundancy Theory sketched above. Given this understanding, let's now consider which of the sentences of groups A and B are true, which false. Page 1 of 15

2.2 Modal Notions There are several different ways (or "modes") in which things can be true: they can be contingently, possibly or necessarily true (these have come to be called "modal" notions, and are studied by "modal logic"). It will be useful to examine the sentences in the light of these distinctions: 2.2A Contingency: With respect to group A one might be unsure: one might say, "It all depends upon Plato's domestic life or the weather in Paris in 1992." When the truth of a sentence "depends" upon whether the world is one way or another, we say that it is a "contingent" sentence. A little more precisely, a sentence is contingent if and only if it is possible for it to be true and it is possible for it to be false (although, of course, not at the same time). All of the sentences in set A are clearly contingent. 2.2B Possibilities and Necessities: Not so the sentences in group B. Try to imagine or conceive or describe a world in which any of them are false. Ruling out (as we always will here) any ambiguity or vagueness that might attach to the words, you should find it pretty difficult. You might even experience a peculiar mental pain in trying to do so (attend to this pain, and become familiar with it. It is sometimes a good guide to logical distinctions). We could say of each of the sentences in B that it isimpossible for it to be false; unlike the sentences of group A, they are each necessarily true, or true under all possible circumstances; as the philosopher Gottfried Leibniz (1646-1716) put it, they are "true in all possible worlds." It would be nice to be able to say what this word "possible" means here. Unfortunately, this is none too easy. For now, take it to be the very broadest notion of possibility you can coherently imagine. Start from the narrow notion of practicalpossibility and work out: it is practically possible --i.e. possible in existing practice-- for most of us to sit cross-legged. However, it is not practically possible for many of us to sit in a "full lotus" (cross-legged, with all the toes of each foot resting on the opposite thigh). For many people at the present time, this is practically impossible. However, despite being practically impossible, sitting full lotus is for most of us still physically possible, i.e. compatible with the laws of physics. With a little yoga, most of us can learn to do it. And so it goes for many things: what is practically impossible at one time is practically possible at another, so long as it's physically possible. Some things, though, are physically impossible: e.g. (according to Einstein) accelerating beyond the speed of light. This is incompatible with the laws of physics. Now, unlike what's practically possible, what's physically possible doesn't change: physical laws, although they describe change, do not themselves change (even if peoples' beliefs about them do). Page 2 of 15

But for all that we still seem able to conceive still a further range of possibility, that of the logically possible: even though it is physically impossible to accelerate beyond the speed of light, it still seems a coherent, imaginable possibility. The world would probably be a very different world than the one it is, but it would still seem a world, a possible way things could be. At any rate, it wouldn't seem as bad off as "a world" (?) in which the same thing at the same time both did and did not accelerate beyond the speed of light, or in which people were and were not mortal. These latter mark the limits of all possibility: they seem incomprehensible, unimaginable. They are logically impossible. So, for the time being, something is (logically) possible if and only if it is coherently imaginable; and a sentence is necessarily true if and only if it is not logically possible for it to be false. What seems to distinguish group A sentences from those in group B is that those in B, but not those in A, are necessarily true. 2.3 Arguments Related to the difference between sets A and B is a difference between different kinds of arguments. Abstracting from the heat and anger usually involved in arguments, let's use the word 'argument for any set of sentences one of which is designated as the conclusion: the rest are premises or intervening steps. Thus, we could, if we liked, designate the third sentence in each of A and B as conclusions, and the rest as premises, in which case both group A and group B would be arguments --just quite peculiar ones. More interesting ones are the following (conclusions are standardly indicated by such words as 'so, 'therefore, 'it follows that): C. l. Every time any woman from Kent has tried to sit lotus in the past, she has failed and developed a cramp. 2. Meg is a woman from Kent. So: 3. Next time Meg tries to sit lotus, she will fail and develop a cramp. D. 1. Every woman is mortal. 2. Meg is a woman. So: 3. Meg is mortal. E. 1. Every woman from Kent is mortal. 2. Meg is a woman from Kent. 3. All mortals are despondent. So: 4. Someone is despondent. Page 3 of 15

Just as we may intuitively distinguish groups A and B, so can we distinguish C from D and E and other groups like them. One thing one might say is that there is something "non-compelling" about C, while utterly "compelling" about D. Suppose Meg herself provided arguments C and D on behalf of her despair about sitting lotus and her despair about being mortal: what might one say in reply? It would seem natural in the first but not the second case to encourage her to try a little harder. "After all," we might argue, "just because Kentish women in the past have tried to sit lotus but failed doesn't mean that you might not succeed." But, if every women from Kent, past present or future is in fact mortal, and Meg is such a woman, then it is hopeless: she must be mortal as well. And, if, furthermore, all who are mortal are despondent, then, since Meg is mortal, she must be despondent; and so someone is despondent. As in the task of distinguishing groups A and B, it is important to realize that youare the judge here: it is you who should feel compelled in moving from premises to conclusion in D and E, but not in C. And the source of the compulsion seems similar in both kinds of cases: just as the B sentences seem different from the A sentences in being necessarily true, it would seem to be necessary that the conclusions of D and E be true if the premises are. 2.4 (Deductive) Validity Notice that this feeling of being bound by the argument doesn't depend upon the truth or falsity of the individual sentences taken by themselves. I have no idea (and doubt whether you do) whether Meg is from Kent or not, much less whether all women from Kent are poor or have trouble sitting lotus. But being unclear about the truth or falsity of the sentences individually doesn't prevent one from considering patterns of truth or falsity over the argument as a whole. In particular, using our previous notion of logical possibility to capture this feeling of "compulsion," we can say the following: In D and E, but not in C, if the premises are true then the conclusion must be true; or, equivalently: it is impossible for the premises to be true and the conclusion at the same time to be false. What I've just said is so very important that it needs to be repeated in bold. It will be our definition of a valid (deductive) argument: (V) A Valid (Deductive) Argument is one in which it is impossible for its premises to be true and its conclusion false. Philosophical usage deviates a little from ordinary talk here: where ordinarily 'valid' is sometimes used as a variant of 'true', or perhaps for a "good" argument, in philosophy it is used only in the way indicated by (V), for deductively valid arguments --hence the `deductive' is redundant. A bit more jargon that's perhaps closer to ordinary usage: The premises of a valid argument imply, or entail, its conclusion. Page 4 of 15

The conclusion of a valid argument is implied, or entailed, by its premises. Note that (V) does not say that a valid argument is one in which both the premises and the conclusion happen to be true. After all, consider the following: F. 1. The earth is round. 2. 2+2=4 so: 3. Columbus sailed in 1492. All these three sentences are true, but the argument isn't valid, since it's still possible for the premises to be true and the conclusion false (just suppose the premises are true and Columbus sailed a year later). What (V) says is that in a valid argument it is impossible for both those things to happen together: true premises and false conclusion. When this is so, when it is impossible for the premises to be true and conclusion false, we speak of an argument being deductively valid, or just valid. The premises entail or imply the conclusion; the conclusion follows from or is implied by the premises. All these phrases are meant to capture that "compulsion" that seems to take us from premises to conclusion in D and E but not in C. A good test of whether you have fully grasped the notion of valid argument as it is defined in (V) is to consider, however, the following limited, but rather surprising claim about the premises considered by themselves: Consider for example: If the premises are contradictory, then the argument is valid. G. 1. The earth is round. 2. The earth is not round. So: 3. Pigs can fly. Now ask yourself carefully: is it possible for premises 1 and 2 of this argument to be true and at the same time the conclusion, 3, to be false? Think about it. What you should notice is that there is just no way for both the premises ever to be true: it's not possible for these to be true at all. Well, if it's impossible for those premises to be true, then, by golly, it's certainly impossible for those premises to be true and for pigs not to fly! I.e., if it's impossible for those premises to be true, then it's impossible for both the following to obtain: those premises are true, and that conclusion (or any conclusion, for that matter) false. So argument G. is in fact valid --as is any argument with contradictory premises This is one among many reasons for avoiding contradictions: if you argue from a contradiction, then you can prove anything! Lest you think this somewhat surprising fact about contradictions is simply due to our stipulated definition of `validity', it is important to notice that one can arrive at the very same fact by intuitive logical reasoning. Instead of arguing from the definition of `valid', we can proceed Page 5 of 15

with the above argument G by the following steps (I'll explain the "rules" in a moment): Page 6 of 15

G': 1. The earth is round. premise 2. The earth is not round. premise 3. Either the earth is round or pigs can fly. by 1 and "Addition Rule" 4. Pigs can fly by 2 and 3, and "Dilemma Rule" (This argument was first noted by the medieval Scottish philosopher, Duns Scotus.) The "addition rule" says that is a sentence, p, is true, then it follows that the sentence "p or q" is also true, where `p or q' is the inclusive `or' that means "At least one of p or q is true" (If "The earth is round is true, then at least one of `The earth is round' or `Pigs can fly' must be true). The "Dilemma Rule" (more commonly called "Disjunctive Syllogism") says that if a dilemma of the form "p or q" is true, and one of either p or q is also false, then the other one must be true ("If the coin is either in my left or my right hand, and it's not in my left, it must be in my right"). In the next section I'll address what might be a lingering dissatisfaction with this argument as well. This fact aboujt contradictions implying anything also highlights another important fact about valid arguments: just because an argument is valid is no reason, by itself, to believe the conclusion. It would only be a reason to believe the conclusion if you also happened to be convinced of the premises, and the fact that an argument is valid is no reason in itself to accept the premises (indeed, in valid argument G, there's presumably no way you're going to be convinced of its contradictory premises). Acceptance of the conclusion is at best conditional: if the premises are true, then the conclusion must be true. But of course that's a big `if': the premises may not be true; and so one may be free to ignore the conclusion. If the premises of a valid argument are, though, in fact true, then the argument is additionally a sound one. That is: (S) A sound argument is a valid argument whose premises are true. Obviously, sound arguments are really nice: they do provide you with a true conclusion. However, it would be a mistake to confine ourselves only to sound arguments. Often we want to know whether an argument is valid whether or not we know the truth of the premises; indeed, we may often be trying to find out the truth of the premises by finding out what those premises entail, as when a scientist makes a prediction on the basis of a specific theory. Speaking of conditionals, though, we've actually been using them quite a bit throughout our discussion, sometimes in ways that you may not entirely have understood. I remember when I was an undergraduate being puzzled by them, and particularly by the funny phase philosophers (but also mathematicians and lawyers) like to use, 'if and only if'. Well, it turns out that conditionals are, indeed, peculiar, in many ways that become increasingly apparent the more one studies them. In order to give you some handle on them, and on the diverse jargon with which they are often expressed, they deserve a section all their own. But in order to get a handle on the oddities that will merge in that section it will be useful to make a distinction between the rules of logic and the rules of conversation. Bear with me; you'll find all of this is well worth it, and will bear usefully upon issues in epistemology at quite a surprising number of places. Page 7 of 15

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2.5 Implication vs. Implicature (Logic vs. Conversation) One thing that can make much of the discussion of logic seem a little weird is that some of it would seem pretty odd in ordinary conversation. In the Duns Scotus argument, G', that we just discussed, someone might understandably balk at the rule of "addition" that enable us to introduce an arbitrarily new sentence into our reasoning, as when we moved from (1) to (3). But it's important to notice that the oddity doesn't seem to be relevant to the issue of the relative truth of (1) and (3): that is, despite its oddity, you've got to admit that there's simply no denying that there's still no way for (1) to be true and (6) false! Philosophers and logicians have concluded that the oddity is therefore due not to a failure of a rule of logic, but rather of some rule of ordinary conversation. Unlike purely logical reasonings, conversations are complicated social interactions having their special rules of polite-ness and efficiency. And this leads to an important distinction between what is implied by (or the implications of) a set of sentences, and what is "conversationally implicated" by someone's utterance of them on a particular occasion --what are the "implicatures" of those sentences on that occasion. Very often when we converse, we intend to convey all sorts of information that is not strictly speaking contained in or implied by the sentences we utter. For example, if I ask you at the dinnertable, "Could you pass the salt?", I'm not really asking a question about your capabilities (which is what the question is literally asking, but which I usually check out before asking), but rather am simply requesting in a slightly roundabout way that you pass me the salt. Similarly, suppose I am asked how much money I have in my pocket, and suppose that in fact I have three dollars; do I speak falsely if I say "I have one dollar"? No. I simply speak misleadingly. It seems to be something like a tacit convention of normal conversation that when asked such questions, a speaker should provide the maximum relevant information. Although I don't speak falsely by saying 'one' when I know full well I have three, I violate that convention and so mislead the speaker. But notice this is a convention of appropriate behavior in a conversation, not a rule about the truth of the sentence `I have one dollar.' After all, anyone who has three dollars has one dollar! Consequently, logicians regard the apparent implication --that that's all the money I have-- as merely an implicature, not a genuine logical implication of the sentence. More generally, rules concerned with the appropriate use of language are regarded as rules of pragmatics; rules concerned with the literal truth of sentences, and the validity of arguments, as rules of semantics. In this book we will be concerned centrally only with rules of the latter sort. (This distinction has been the source of immense controversy between traditional "analytic" philosophers, and followers of the later Wittgenstein and J.L. Austin, who want to stress how issues of meaning cannot be divorced from issues of "ordinary use." See Grice (1975) for rich discussion --and a theory of "conversational implicature.") Page 9 of 15

In sum: p implies q, or q is an implication of p, iff it's impossible for p to be true and q false. p (merely) implicates q, or q is a (mere) implicature of p, iff p doesn't imply q, but q is presumed to be true as a convention of ordinary conversation. Thus, my saying in answer to the question "How much money do you have?", "I have one dollar" doesn't imply, but ordinarily merely implicates "I don't have more than one dollar." And "Could you pass the salt?" implicates but does not imply "You should pass me the salt." We'll see that it will often be extremely important in philosophy to ignore pragmatic matters and "cancel" implicatures, so tayt we can concentrate purely on the issues of truth. For example, in thinking about the surprising fact that inconsistent premises imply anything, what's odd and "silly" about such arguments is not that they fail to be valid, but, rather, that they fail to be conversationally relevant ("What," someone might ask, "does the claim that pigs can fly have to do with the roundness of the earth??"). If this is true, then logicians can acknowledge this oddity, cancel the implicature that says that the added sentence is normally relevant, and accept the rules of "addition" and "negative dilemma" that on reflection we can see that we have every reason to accept. 2.6 Conditionals Many claims that we make about the world are conditional, i.e. of the form If p then q, where p and q are any English sentences (don't worry about the corner quotes; they're there to indicate the generality of the treatment). Such claims are so frequent and important in logic and philosophy that it is worth getting accustomed to various features and paraphrases of them. But a provisional cautionary note: in beginning to think about conditionals, it's best to stick to geographical examples, such as 'If someone is in Boulder, then they're in Colorado'. They turn out to reveal more of the essential nature of conditionals than do the much more complex ones involved in social interaction (e.g. 'If you buy the tickets, I'll go to the movies'), which it turns out involve a lot of niceties about etiquette. Indeed, for the time being, IN THINKING ABOUT CONDITIONALS. IN A PINCH, USE ONLY GEOGRAPHICAL EXAMPLES. DO NOT USE EXAMPLES OF SOCIAL INTERACTION; They will mislead you in ways irrelevant for the purely logical purposes for which we need them (see 2.5 for the reasons why). In a conditional of the form "If p then q," the sentence that goes in for 'p' is called the antecedent, the one that goes in for 'q', the consequent. It's important to keep track of this terminology, since English has an irritating way of permitting paraphrases of conditionals so that which is antecedent and which is consequent can get slightly obscured. For example, English also Page 10 of 15

permits us to switch the order in which antecedent and consequent are written: If p then q can also be written q if p. antecedent consequent consequent antecedent Thus, 'If I'm in Boulder, I'm in Colorado' can also be expressed by, 'I'm in Colorado, if I'm in Boulder'. Despite now occurring second in the sentence, 'I'm in Boulder' is still the antecedent, and 'I'm in Colorado' is still the consequent. This should all be pretty obvious so far. Things get utterly non-obvious when one adds the little word 'only' to the stew. When I was an undergraduate I was completely baffled to notice that the very thought that is expressed by a sentence of the form If p then q can often be expressed as p only if q. Thus, 'If I'm in Boulder, then I'm in Colorado can be paraphrased 'I'm in Boulder only if I'm in Colorado. What I (still) find puzzling about this paraphrase is that the 'if' has switched places: it's now attached to the consequent, not to the antecedent! (Think of other examples so that this particular paraphrase becomes vivid to you.) Notice that this switch cannot be made without the word 'only': saying something of the form If p then q is entirely different from saying something of the converse form If q then p: saying, for example, the sentence, 'If I'm in Boulder, then I'm in Colorado is entirely different from saying the converse, 'If I'm in Colorado, then I'm in Boulder. The first is true, the second false. So switching the 'if' by itself from the antecedent to the consequent of a conditional is no way of paraphrasing that conditional. However, if the 'if is prefixed with the word 'only it turns out to be perfectly all right! Indeed, it's best to think of the expression 'only if as a single expression (a little like 'kicked the bucket or 'down in the mouth). In other words: If p then q can also be written p only if q. antecedent consequent antecedent consequent!! This is so confusing, let me summarize what I've said so far: but If p then q can be paraphrased as q if p and as p only if q If p then q CANNOT be paraphrased as If q then p or as q only if p Don't put away the book yet. Still another way of talking about conditionals is in terms of "necessary" and "sufficient" conditions. If we have a conditional of the form we may paraphrase it as either: If p then q p is (a) sufficient (condition) for q Page 11 of 15

or as: q is (a) necessary (condition) for p It really all depends upon what aspect of the relationship between p and q you want to stress. Thus, given the truth of 'If I'm living in Boulder, then I'm living in Colorado, we can also say, "Living in Boulder is a sufficient condition for living in Colorado" (or "Living in Boulder is sufficient for living in Colorado") and "Living in Colorado is a necessary condition for living in Boulder" (or "Living in Colorado is necessary for living in Boulder"). All of this gives rise to the following table of equivalences which you should be sure you master in order to avoid later confusion: LEFT RIGHT If p then q; conversely: If q then p q if p; conversely: p if q p only if q; conversely: q only if p p is sufficient for q; conversely: q is sufficient for p q is necessary for p; conversely: p is necessary for q The above all mean the same. The above all mean the same. BUT THESE ON THE LEFT DO NOT MEAN THE SAME AS THESE ON THE RIGHT!! All expressions in on the LEFT are paraphrases of one another; all expressions on the RIGHT are also paraphrases of one another; but all the expressions on the left are the converses and so are not paraphrases of the expressions on the right; i.e. none on the left mean the same as any on the right. You will do well to master this little table by giving yourself lots of (again) geographical examples. The dangers of thinking about conditionals involved in social interaction provide another nice example of the problem of confusing implication with implicature. Consider the following exchange between Jim and Sue: Jim: I'll go to this movie only if you buy tickets. Sue: OK. [She buys the tickets] Here they are; let's go in! Jim: Hey, I said I'd go only if you buy the tickets, and in logic that means no more than if I'll go, then you buy the tickets; it says nothing about the converse, if you buy the tickets, then I'll go. Indeed, I never had any intention of going. See ya! Page 12 of 15

Clearly, Jim is being a cad. Although he is strictly correct about the logic of `only if', as everyone knows, there's a lot more to social interaction than logic alone. There is presumably a rule of cogent conversation that says that in laying down the conditions for going to the movies, he should lay down all the remaining necessary conditions that in the context would be sufficient. If this pragmatic, conversational rule is followed, then, of course, the `only if' now implicates (even though it doesn't imply) "if you buy, then I'll go." Alas, we're still not finished. There's also the "bi-conditional": Sometimes we want to express both a conditional and its converse: not only if p then q, but also if q then p (e.g. "If I'm in the Capital, I'm in Washington, and, moreover, if I'm in Washington, I'm in the Capital"). For this purpose we have the biconditional, p if and only if q, which is a conjunction of two conditionals, each of which is the converse of the other. p if and only if q is equivalent to If p then q, and if q then p. This can be seen by recalling some of the above paraphrases. Since and If p then q is equivalent to p only if q, If q then p is equivalent to p if q, it follows that the expression is itself equivalent to If p then q, and if q then p p only if q, and p if q. Merely reversing the order of this conjunction, we get: p if q, and p only if q which can be condensed to: p if and only if q. In logic and philosophy, this is further abbreviated as: p iff q. (Be sure you fully understand this little "derivation": it's the sort of thing I like people to know in their heart of hearts --and, consequently, on exams!) Suppose now that I wanted to say that some condition(s) p were each individually necessary and together sufficient for q. Well, I could say that. For example, I could say that the following conditions are individually necessary and jointly sufficient for x being a bachelor: (1) x Page 13 of 15

is male, (2) x is an adult, (3) x has never been married. But, reverting to the language of conditionals, I could also say: "x is bachelor if x is a male adult who has never been married" (this expresses that the conditions are jointly sufficient) and "x is a bachelor only if x is a male adult who has never been married" (this expresses the fact that the conditions are individually necessary). Or, more briefly: "x is a bachelor iff x is an unmarried adult male." In general, p is necessary and sufficient for q means p iff q. O.K., that's enough (I'm sure you'll agree) about (bi-)conditionals (for now). One thing (bi-)conditionals are useful for is for expressing definitions --indeed, you may recall that we used the bi-conditional in expressing the Redundancy Definition of Truth, (T) The sentence 'p' is true if and only if p. You might want now to review this definition in the light of this discussion of conditionals, providing yourself various paraphrases of it (and using different examples of 'p'). But, now about definitions: 2.7 Definitions Definitions are frequently expressed by citing conditions that are individually necessary and jointly sufficient, and so expressible in the above way. That is, a definition of some term T is supposed to provide conditions that something must or needs to have in order for T to apply to it, as well as conditions that would be enough for T to apply. For example, a definition of 'bachelor cites conditions that are necessary --being male, being an adult-- such that, taken all together, those conditions are sufficient --being an adult, never married male. Notice that the words here are modal: definitions ordinarily are supposed to apply not only to actual cases, but to all possible cases. Even if it turned out that for some reason women ceased to exist, and so all the unmarried adults happened to be male, still we would want to insist that the definition include a clause about excluding women, since women were still possible beings that might need to be discussed, but still would not be bachelors. In view of the paraphrases of these words discussed above, many definitions can therefore be expressed using the 'iff' idiom. Thus, we might say that x is a bachelor iff x is an unmarried, adult male. Often in an intellectual discipline, it's convenient to introduce new terms as abbreviations of complex combinations of old ones. If the term is really not intended to bear any important relation to a previous term, this is called a stipulative definition. For example, if I was doing genealogy, I might be interested in the category of uncles who never married. And so I might introduce the term,`schmuncle', defining it stipulatively: Someone is a schmuncle iff he is an uncle and a bachelor Oftentimes, though, the new term will be intended to sharpen or capture something important in Page 14 of 15

the prior use of a term. In this case, even though the word might be provided what looks like a stipulated meaning, the relation to its earlier use gives us a reason to think of it as an "explication" of that earlier use, and in this case we speak of an explicative definition (or explication) of the common word. You've probably encountered this sort of definition in geometry class, where, for example, a circle is defined as: Something is a circle iff it is a locus of co-planar points equidistant from a given point. Another example might be the chemist's definition of 'water as 'H2O, which might be regarded as an excellent explicative definition. Many of the definitions (or "analyses") we seek in philosophical discussions are of this sort. They may be informed both by a sense of ordinary use, and a sense of the important function of the word, the purpose it serves particularly in some explanatory enterprise. Proposed explicative definitions of a term can be too wide, too narrow, or both. A proposed definition is too wide iff it applies to more things than the definiendum (the term being defined) applies to; it is too narrow iff it applies to fewer things than the definiendum. If we proposed defining 'bachelor as 'unmarried male, this would be too wide, since it would include (as ordinary use presumably does not) infant boys; if we proposed defining it as 'unmarried adult male with green eyes' this would be too narrow, since (again, by ordinary use) there are blue-eyed bachelors. If we proposed defining it as "unmarried males with green eyes" it would be both too wide and too narrow, wrongly including the infants and excluding the blue-eyed. What we want, of course, is a definition that is neither too narrow not too wide, but just right. Unfortunately, it turns out that, aside from a legal and mathematical examples like `bachelor' and maybe the surprising example of `true' these are much harder to come by than one might initially think. We'll see in chapter 3 ( 3.1) that definitions of `knowledge' have turned out to be extraordinarily difficult, with one generation of philosophers after another proposing plausible definitions, only to be refuted some ingenious counterexamples of the next (I think I mentioned in class the (semifacetious) idiom "to chisholm away at a definition," after the late American philosopher, Roderick Chisholm, who would constantly qualify his earlier definitions of some term with ever more complex clauses to cover counterexamples his students would produce). 2.8 Non-Deductive Arguments Next week; stay tuned! Page 15 of 15