Understanding Logic: Necessary Truths and Valid Arguments - Prof. Georges Rey, Exams of Introduction to Philosophy

Download Understanding Logic: Necessary Truths and Valid Arguments - Prof. Georges Rey and more Exams Introduction to Philosophy in PDF only on Docsity! 3. A Little Logic 3.1 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. 3. 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: 3.2A 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. 3.2B 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). 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. 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., it's impossible for those premises to be true and that conclusion (or any conclusion, for that matter) also to be false. So argument G. is in fact valid -as is any argument with contradictory premises (in the light of this discussion, you should think about whether the "arguments" in sets A and B on p17 -where we take the fourth sentence as the conclusion, the rest as premises- are valid or not). This is one among many reasons for avoiding contradictions: if you argue from a contradiction, then you can prove anything! This last fact 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 (cf. Sober's "Surprise Principle" at S:pp29-31). 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. Bear with me; you'll find it's well worth it. 3.5 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. DO NOT USE EXAMPLES OF SOCIAL INTERACTION. They will mislead you in ways irrelevant for the purely logical purposes for which we need them. 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 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: If p then q can be paraphrased as q if p and as p only if q but If p then q cannot be paraphrased as If q then p or as q only if p (Just to alert you to the dangers of thinking about conditionals involved in social interaction, 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! 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 are something like rules of cogent conversation -e.g. just talk about what's relevant- and it's these that are being violated in this example.) 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 If p then q we may paraphrase it as either: p is (a) sufficient (condition) for q 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