Contents 1 Introduction 3 1.1 Deductive and Plausible Reasoning................... 3 1.1.1 Strong Syllogism......................... 3 1.1.2 Weak Syllogism.......................... 4 1.1.3 Transitivity With Strong vs. Weak Inference.......... 7 1.1.4 Building an Idealized Common Sense Machine........ 9 1
2 CONTENTS
Chapter 1 Introduction 1.1 Deductive and Plausible Reasoning Scenario A policeman on patrol hears a burglar alarm, and looks across the street to see a broken window in a jewelry store. A man in a ski mask crawls out with a bag. The policeman concludes that this man is a criminal. What s the reasoning process? 1.1.1 Strong Syllogism We can identify several different forms of inference, which can be classified into two groups, depending on how fallible they are. The strong variety are logical, deductive inferences, which are bulletproof, provided their premises are true, because the premises are absolute: if p is true, then q is always true. The most direct form of inference is modus ponens. Syllogism Type 1 (Modus Ponens) (Premise) If A is true, then B is true (e.g., if it rained, my grass is wet) (Data) A is true (it rained) (Inference) B is true (my grass is wet) 3
4 CHAPTER 1. INTRODUCTION Figure 1.1: If A is only true in situations where B is true, then we can apply modus ponens (from A to B) or modus tollens (from the negation of B to the negation of A). Given the same premise, namely that B is true whenever A is true, we can make a negative inference in the opposite direction. If B is false, then A cannot be true (since if it were, B would be true). This type of inference is called modus tollens. Syllogism Type 2 (Modus Tollens) (Premise) If A is true, then B is true (e.g., if it rained, my grass is wet) (Data) B is false (my grass is not wet) (Inference) A is not true (it must not have rained) 1.1.2 Weak Syllogism Given the premise A implies B, we can reason from A to B, or from NOT B to NOT A. A common mistake made by logic students (I ve heard this called modus morons ) is to reason from B to A, or to reason from NOT A to NOT B. It should be clear from the Venn diagram that these inferences are not logically valid: A can be false even though B is true.
1.1. DEDUCTIVE AND PLAUSIBLE REASONING 5 However, if we think of the diagram as containing possible configurations of the world, then learning that B holds rules out some of the ways that A can be false; similarly, learning that A fails rules out some of the ways that B can hold. So it seems reasonable to define the following weak syllogisms, which deal not with absolutes, but with degrees of plausibility. Syllogism Type 3 (Premise) If A is true, then B is true (e.g., if it rained, my grass is wet) (Data) B is true (my grass is wet) (Inference) A becomes more plausible (more likely that it rained) Syllogism Type 4 (Premise) If A is true, then B is true (Data) A is false (it didn t rain) (Inference) B is less plausible (less likely my grass is wet) But more or less plausible does not mean certain to be true or false. For example, what if there is a sprinkler system? But what if we don t even have the nice clean implication that A always implies B, but we have a situation more like that in Figure 1.2, where any combination is possible? Suppose we learn that B is true. This rules out both some of the ways that A could be true and some of the ways that A could be false. It s hard to say anything at all about A. But now imagine that most but not all of B overlaps with A. For example, A might indicate that there s rain and B might indicate that there s lightning. Sometimes there s lightning without rain, but usually there s rain, too. If we discover that there s lightning (B is true), then we rule out the possibility that there s lightning but no rain (A would be true), but that s relatively small as a proportion of all the ways A could happen; far more consequential is the fact that we ve ruled out the possibility of neither lightning nor rain (this makes up most of the possible worlds that do not involve rain). It seems that in this case we are licensed to say that, having observed lightning rain is now more likely (but not guaranteed).
6 CHAPTER 1. INTRODUCTION Figure 1.2: If neither event is contained inside the other, yet they overlap, then any combination is possible, and no strong syllogisms are available no matter the data. This form of reasoning is far removed from modus ponens and modus tollens: not only are we reasoning in the wrong direction, but our premise isn t even guaranteed. Still, it seems perfectly reasonable, so long as we keep our uncertainty in mind. Syllogism Type 5 (Premise) If A then B is more plausible (If there s rain, lightning becomes more likely.) (Data) B is true (There s lightning.) (Inference) A is more plausible (Rain is likely.) Questions: Which of the above characterizes the policeman s reasoning process? What s missing from the description? What if this happened all the time, and the suspects turned out to be innocent?
1.1. DEDUCTIVE AND PLAUSIBLE REASONING 7 1.1.3 Transitivity With Strong vs. Weak Inference Weak and strong syllogism behave differently when we chain inferences together, using the conclusion of one inference as the data to the next. Strong syllogisms can be joined together without any loss of certainty: the conclusion of the first strong syllogism is every bit as valid as an observation would be. For example, suppose A, B and C are propositions about a particular polygon that may or may not be true. A says the shape is a square, B says the shape is a rectangle, and C says the shape has four sides. Then we have the following transitive relationship. Strong Transitivity (Premise) If A is true then B is true (all squares are rectangles). (Premise) If B is true then C is true (all rectangles have four sides) (Data) A is true (we are given a square) (Inference) C is true (our shape has four sides) That is, since we have a square, we have a rectangle, by the first premise. Since we have a rectangle, the condition for the second premise is satisfied, which allows us to conclude that the shape has four sides. If we want to, we are licensed to chain the premises together like straws to get a new premise that says that all squares have four sides. But what if we re using a weak syllogism, such as the following? Slipping Confidence (Premise) If it s cloudy, rain is more likely. (Premise) If it rains, cancelling the baseball game is more likely. (Data) It is cloudy (Inference) (?) It is more likely that cancelling the baseball game is more likely.
8 CHAPTER 1. INTRODUCTION As we chain these weaker premises together, our certainty slips. How about this case? Weak Syllogism is not Modular (Premise) If my grass is wet, it likely rained last night. (Premise) If the sprinkler was on last night, my grass is wet. (Data) The sprinkler was on last night. (Inference) (?) It likely rained last night. Finally, compare and contrast the following intuitive, yet formally puzzling, reasoning processes. Explaining Away (Premise) If my fuel tank is empty, the gauge will read E (Premise) If the gauge battery is dead, the gauge will read E (Datum 1) The gauge reads E (Inference 1) Plausibility of empty tank goes up (Inference 2) Plausibility of dead battery also goes up (Datum 2) Battery is dead (Inference 3) Plausibility of empty tank goes back down! Here, finding a dead battery has an impact our beliefs about the fuel tank, even though these are not actually connected in any way, as can be seen in the second example. Contrast With This (Premise) If my fuel tank is empty, the gauge will read E (Premise) If the gauge battery is dead, the gauge will read E (Datum 1) Battery is dead
1.1. DEDUCTIVE AND PLAUSIBLE REASONING 9 (Inference) The gauge will read E (Inference) No effect on plausibility of empty tank! Now, when we are ignorant about the gauge, the dead battery has no impact on our beliefs about the fuel tank. There seems to be some memory about how our beliefs were formed. 1.1.4 Building an Idealized Common Sense Machine Imagine building an artificial system to do every day reasoning. Some are skeptical that machines could every really think. The mathematician von Neumann (after whom the computing architecture underlying your personal computer, the von Neumann machine is named) responded to this skepticism by saying If you will tell me precisely what it is that a machine cannot do, then I can always make a machine which will do just that!. The point was that if the human brain does something, it is physically possible, so if we know how it is done, we can make a machine do it. The modern field of Artificial Intelligence (AI) dates back to the 1950s, when scientists and engineers tried to build systems based purely on deductive reasoning. Though they had great success for artificial and highly constrained problems like chess, it was not until they started to incorporate weak inference that substantial progress began to be made in more realistic domains. Even something as seemingly simple as vision requires massive amounts of weak inference! If we want to program a computer, or build a robot, to engage in everyday reasoning of the sort described in the examples above, then it s not enough to rely on our intuitions; nor do we want to settle for fuzzy statements like rain is likely. We need to know exactly when, for example, finding a dead battery should influence our beliefs about a gas tank; and we need to know how likely it is to rain. When we are outside the realm of formal mathematics and logic, every conclusion has the possibility of being wrong, which has a cost. Even gathering more information to improve our conclusion can be costly. We need our system to be able to make intelligent decisions. Conclusion: We need a formal theory of plausible reasoning! Limitation: We will confine machine to reasoning about objective propositions, not opinions.