9. TWO KINDS OF LOGIC Let's begin today by looking at one of the simplest examples of how logic can help us to see the formal relations between words and the truth value of the proposition they compose. A proposition's truth value, as I explained in the last session, is quite distinct from its actual, material truth. It refers to the truth or falsity a proposition would have under any given set of conditions. Thus, we can find the truth value of a proposition without knowing anything at all about its actual content, provided we know what kind of proposition it is. One way of doing this is to construct what is called a "truth table" for the proposition. Let's take as an example the proposition: "If you read the textbook, then you will do well on the final exam." (Actually, I wish I didn't have to give you a written exam--or any grade, for that matter--since it will be nearly impossible to judge how much real philosophy you have learned just by administering a conventional test. The purpose of this course is not to teach you about philosophy--that could be tested quite easily--but to teach you to philosophize. Nevertheless, the university pays me, and they require me to give you a "grade", so let's use logic to remind ourselves of one good way to make that grade a high one.) The first step in constructing a truth table is to reduce the proposition in question to its simplest logical form. In this case we can replace "you read the textbook" with p and "you will do well on the final exam" with q, giving us the proposition "If p, then q". This can be expressed entirely in symbols as "pq", where the arrow means "implies" (which is equivalent to saying "If ..., then ..."). The second step is to substitute for each variable all the possible combinations of "T" ("is true") and "F" ("is false"), and determine for each combination whether the resulting proposition would be true or false. The appropriate letter (T or F) is then written in the far right column, as in Figure 9.1a. If it is true that "you do the readings", then, as the truth table tells us, my proposition will be true only if you pass the exam. If, on the contrary, the p statement is false, then the proposition will be logically true regardless of the truth or falsity of q. The reason for this rather surprising result can be seen more clearly if we convert the proposition into its equivalent "Either ..., or ..." proposition. If p really does imply q, then either q is true or p is false. Since p's truth implies q's truth, the falsity of q would imply the falsity of p. That means "pq" is equivalent to "- p / q" (i.e., "Either -p, or q"). If we now construct a new truth table, as in Figure 9.1b, we find that the second row is the only one in which both choices are false; each of the other three propositions has at least one true option, so that the overall proposition can be judged to be true. (Note that the first column in Figure 9.1a contains the opposite values as those in Figure 9.1b, because the former is a function of p, while the latter is a function of -p.) (a) "If ..., then ..." (b) "Either ..., or ..." [Figure 9.1: Two Truth Tables] Being aware of the truth value of various types of proposition can help us to avoid being fooled by arguments that attempt to prove something by presupposing a false p. Since the whole proposition is formally true regardless of the truth or falsity of q, we can use such an argument to "prove" the truth of something which is actually false. For example, if I want to make myself appear to be your favorite teacher, I could argue: "If you are the Governor of Hong Kong, then I am your favorite teacher." Assuming you are not the Governor in disguise, this proposition is true whether or not I am actually your favorite teacher! For it is the same as saying "Either you are not the Governor of Hong Kong, or I am your favorite teacher." So when faced with a proposition in which p is false, always be sure to remember that it is a fallacy to assume from the truth value of such a proposition that q on its own is actually true. Now let's imagine that you do the recommended readings for this class, but you end up failing the exam anyway. If the proposition whose form we analyzed in Figure 9.1a is actually true, then you could come and argue from logic alone that you should pass the class. For example, you could remind me that the proposition I stated at the beginning of today's class implies another proposition: "Either you did not do the readings, or you passed the exam." The truth value of this proposition, as shown in Figure 9.1b, requires that in order for it to be true, at least one of its two parts must be true. So if p ("you did not do the readings") is false, and if my original proposition is true, then as the truth table shows, q ("you passed the exam") must be true. So don't ever say logic is too abstract to have any practical value! Such tables can, in principle, be constructed for any proposition, though they would become quite cumbersome for propositions with many discrete components. For our purposes, however, these simple examples will suffice. In the next few lectures we will meet some patterns that are rather similar to those used in such truth tables. But for the rest of today's lecture I want to focus on what I believe is the single most important distinction in logic: namely, the distinction between "analysis" and "synthesis". Rather than giving a universal definition of these two terms, I will explain how they can be applied to make three key distinctions: namely, distinctions between methods of argumentation, types of proposition, and kinds of logic. The distinction between analytic and synthetic methods of argumentation is more commonly known as the distinction between "deduction" and "induction". A deduction is an argument that starts by positing two or more propositions, called "premises", which are presupposed to be true. A conclusion is then drawn which is supposed to follow necessarily from the premises. The archetypes of all deductions are those with three steps, called "syllogisms". And the most common one of all is the "categorical" syllogism Socrates used to convince his friends not to worry about his impending death, since death is inevitable. It looks like this: All human beings are mortal. Socrates is a human being. \ Socrates is mortal. (The symbol "\" stands for "therefore".) In this case the first proposition (or "major premise") puts forward a universal assumption; the second proposition (or "minor premise") puts forward a particular test case; and the third proposition, of course, draws a necessary ("categorical") conclusion, which is called the "inference". The only way the conclusion could prove to be false would be if one of the two premises is false, unless of course the formal relations between the terms in these propositions are in some way fallacious. A good way to test whether or not the terms in a deduction contain a fallacy is to convert the propositions into a set of corresponding logical symbols. In the example given above, which is commonly known as "universal implication" (because of the use of the word "all"), the words are typically converted into symbols such as the following: All h's are m. S is an h. \ S is m. As far as formal logic is concerned, the validity of this syllogism remains exactly the same, whether "h" refers to humans or horses, whether "m" refers to mortality or morality, and whether "S" refers to Socrates or Saddam Hussein! But remember: proving an argument to be valid still leaves open the question of whether the premises are actually true. (The remaining words, "all", "are", etc., can also be converted into symbols--but I don't want to scare you away from logic at this early stage!) Another important aid for anyone who wishes to explore the formal structure of any deductive argument was provided more than two thousand years ago, by Aristotle, the founder of formal logic. He developed a virtually complete system of all the possible forms of deductive argument. Until about a hundred years ago, this system was regarded by virtually all philosophers as giving an unsurpassable account of all the basic propositions of formal logic. This, without a doubt, gives Aristotle the honor of introducing the single most universally recognized and longest-lasting contribution ever made to philosophy. However, for our purposes it is not necessary to learn all the details of Aristotle's system, especially since his ideas have been superseded in numerous respects during this century. What is more significant here is that deduction is not the only respectable form of philosophical argument. For this analytic method is complemented by an equally significant synthetic method. This method, called induction, requires us to begin by appealing to various material facts which, taken together, point to the desired conclusion. In other words, unlike the necessity governing a true deduction, induction always involves some guesswork. Or, to borrow from Kant's terminology, we could say deduction remains entirely within the realm of concepts, while induction requires an appeal to intuition as well. Perhaps an example will help to illuminate this difference. Let's say we want to prove that the proposition "The sun always rises in the east" is true. In order to deduce the truth of this statement, we would need to find at least two true assumptions which together necessitate such a conclusion. For example, we might choose the following: All planets revolve around a star in such a way that the star always appears to rise on the planet's eastern horizon. The earth is a planet and the sun is a star. \ The sun always rises in the east. In order to arrive at this same conclusion by induction, however, we would need to argue in something like the following way: My father says the first day he ever saw the sun rise, it rose in the east. My mother says the sun rose in the east on the day I was born. The first day I remember seeing the sun rise, it rose in the east. Last week I woke up early and saw the sun rising in the east. Yesterday I did the same thing. I have never heard anyone say they have seen the sun rise in the north, south, or west. \ The sun always rises in the east. In the third part of this course we will raise the question as to whether we are ever capable of reaching a necessary truth by induction. But for now I am only trying to show you the difference between it and deduction. The terms "analysis" and "synthesis", as labels for distinguishing deductive and inductive methods of argumentation, are at least as old as Euclid. In his Elements, Euclid made it abundantly clear that these two methods should not be seen as mutually exclusive, but as complementary. His method was to demonstrate the correctness of each of his geometrical theorems by first using an analytic (deductive) method of argumentation, and then supporting its conclusion with synthetic (inductive) reasoning. Following his lead, we can picture the opposing directions of these two methods as follows: (a) Deduction (b) Induction [Figure 9.2: Two Methods of Argumentation] Whereas the actual process of constructing a deduction (as opposed to its written form) starts by thinking up a conclusion, which is then proved by searching for two or more true assumptions upon which it is based, the process of induction starts by collecting innumerable bits of evidence, and using them as the basis to draw a conclusion. As I mentioned earlier, the terms "analytic" and "synthetic" have been used in several quite different ways by philosophers. For a long time Euclid's way of using the terms, to refer to two methods of argumentation, was the commonly accepted usage. But Kant proposed quite a new way of using the same terms, by suggesting that they can refer to two distinct types of proposition. According to Kant, an analytic proposition is one in which the subject is "contained in" the predicate, whereas in a synthetic proposition the subject goes "outside" the predicate. Thus, for example, "Red is a color" is analytic, because the concept "red" is already included as one of the constituents of the concept "color". Likewise, "This chalk is white" is synthetic, because you would not know that this thing I'm holding in my hand is chalk if I merely told you it is white. Using these two examples, we can picture Kant's initial description of this distinction in terms of the two maps shown in Figure 9.3. Kant also gave several other, more rigorous, guidelines for determining whether a proposition is analytic or synthetic. The truth of an analytic proposition can always be known through logic alone; so, if the meanings of the words are already known, the proposition is not informative. An analytic proposition is self-explanatory. All I have to do is say "white" and any of you who understand the meaning of this word will already know I'm talking about a color. So like the conclusion of a good deduction, the truth of an analytic proposition is purely conceptual, and therefore, necessary. The truth of a synthetic proposition, by contrast, requires an appeal to something more than mere concepts. Like an inductive argument, an appeal will be made to some intuition--i.e., to some factual state of affairs. As a result, synthetic propositions are always informative, and the truth of their conclusions is contingent on a given state of affairs continuing to exist. When I tell you this piece of chalk hidden in the palm of my hand is white, the truth of my statement depends on whether or not I have somehow fooled you by slipping it into my pocket, or by replacing it with a blue piece of chalk, etc. (a) "Yellow is a color." (b) "This chalk is white." [Figure 9.3: Analytic and Synthetic Propositions] I hope you will try out a few of your own sample propositions to test your grasp of this distinction between analytic and synthetic propositions. Some philosophers nowadays think there are so many propositions which are difficult to pin down as either analytic or synthetic that the whole distinction is worthless. However, I believe such "grey areas" cause problems only when we forget to look at the context of a proposition, or when we forget to apply each of Kant's guidelines with sufficient care. In any case, this is not an issue we can resolve in an introductory course such as this. Instead, I'll just mention here that Kant combined this distinction between analytic and synthetic propositions (or "judgments", as he also referred to them) with another distinction, between "a priori" and "a posteriori" kinds of knowledge. "A priori" refers to something which can be known to be true without appealing to experience; by contrast, something is "a posteriori" if a demonstration of its truth requires such an appeal to experience. This gives rise to four possible kinds of knowledge, two of which are non-controversial: analytic a priori knowledge is simply logical knowledge, and synthetic a posteriori knowledge is simply empirical knowledge. Kant believed there is no analytic a posteriori knowledge; I contend, however, that this category actually defines a very important, though often neglected, type of knowledge. Although I won't defend this contention here (but see my "A Priori Knowledge in Perspective", and Kant's System of Perspectives, pp.129-140), I will simply tell you that all our hypothetical beliefs about the world can be described in this way. The synthetic a priori class of knowledge occupied most of Kant's attention; for he argued that all transcendental knowledge is of this type. This is why he said the question "How are synthetic judgments a priori possible?" is the central question of all Critical philosophy. Although we won't have time to discuss the intricacies of these different logical classifications, it is important to see their interrelationships, as the following map makes clear: [Figure 9.4: Four Perspectives on Knowledge] I would like to end today by introducing a third way of using the terms "analytic" and "synthetic". You won't find this use of these terms in any logic textbook, as far as I know. But I think it makes a useful addition to the way they have been used in the past. I believe it is very helpful to use these terms to distinguish between two distinct kinds of logic. "Analytic logic" is the entire body of logic based on the principles of reasoning set forth by Aristotle. The most basic of all these principles is what is often called the "law of contradiction". However, for reasons which will become clear later, I suggest we call it the "law of noncontradiction", especially since this principle tells us how we can avoid contradicting ourselves. Aristotle states this law in Categories by saying that a thing cannot both be and not be in the same respect at the same time. In other words, it is impossible for a thing to be both black and white, both "A" and "-A", etc. The simplest symbolic expression of this law is: "A is not -A" or "A=-A" It would be difficult to overstate the profound influence this law has had on the past two thousand three hundred years of philosophy. For upon it is based nearly all of the arguments western philosophers have put forward. Moreover, we would not be able to communicate with each other without assuming that when we use a word, we want those who hear us to think of the thing to which that word refers, and not its opposite! Deduction and analytic propositions are, I believe, two aspects of analytic logic. In both cases, as we have seen, they are paired with complementary, synthetic functions: induction and synthetic propositions. This raises a very important question: Is there also a complementary form of logic itself, equal and opposite to analytic logic, out of which these (and other) non-analytic logical functions arise? And if so, is there a law governing this alternative logic? I'd like you all to think through these two questions on your own between now and the next time we meet. Then I will begin the next lecture by offering my own answer to these questions. QUESTIONS FOR FURTHER THOUGHT 1. Could an argument be valid without being a deduction or an induction? 2. Could there be a proposition which is both analytic and synthetic? 3. Could there be a proposition which is neither analytic nor synthetic? 4. How could knowledge be both analytic and a posteriori? RECOMMENDED READINGS 1. T.L. Heath, Introduction to The Thirteen Books of Euclid's Elements (Cambridge: Cambridge University Press, 1956), section 8, "Analysis and Synthesis", pp.137-140. 2. Immanuel Kant, Critique of Pure Reason, tr. Norman Kemp Smith (London: Macmillan, 1929), "Introduction", pp.41-62. 3. Stephen Palmquist, "A Priori Knowledge in Perspective: (II) Naming, Necessity, and the Analytic A Posteriori", The Review of Metaphysics 41.2 (December 1987), pp.255-282. 4. D.W. Hamlyn, "Analytic and Synthetic Statements", Paul Edwards (ed.), The Encyclopedia of Philosophy, vol. 1, (London: Collier Macmillan Publishers, 1967), pp.105-109.