16. Universal derivation


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1 16. Universal derivation 16.1 An example: the Meno In one of Plato s dialogues, the Meno, Socrates uses questions and prompts to direct a young slave boy to see that if we want to make a square that has twice the area of a given square, then we should use the diagonal of the given square as a side in the new square. Socrates draws a square 1 foot on a side in the dirt. The young boy at first just suggests that we should double the side of a square, but Socrates shows him that this would result in a square that is four times the area of the given square; that is, a square of the size four square feet. Next, Socrates takes this 2x2 square, which has four square feet, and shows the boy how to make a square double its size. Socrates: Tell me, boy, is not this a square of four feet which I have drawn? Socrates: And now I add another square equal to the former one? Socrates: And a third, which is equal to either of them? Socrates: Suppose that we fill up the vacant corner? Boy: Very good. Socrates: Here, then, there are four equal spaces? So what Socrates has drawn at this point looks like: If he wanted to draw a square of area 2x4, he has now drawn one twice that big. But Socrates has a goal and method in drawing again the square four times the size of the original.
2 Socrates: And how many times larger is this space than this other? Boy: Four times. Socrates: But it ought to have been twice only, as you will remember. Boy: True. Socrates: And does not this line, reaching from corner to corner, bisect each of these spaces? Socrates has now drawn the following: Socrates: And are there not here four equal lines which contain this space? Boy: There are. Socrates: Look and see how much this space is. Boy: I do not understand. Socrates: Has not each interior line cut off half of the four spaces? Here, by spaces, Socrates means each of the four quarters of the diagram. Socrates: And how many spaces are there in this section? Boy: Four. Socrates: And how many in this? We presume Socrates is pointing at the half of the space that is cut by the new line. Boy: Two. Socrates: And four is how many times two? Boy: Twice. Socrates: And this space is of how many feet? Now, Socrates is referring to the entire area cut out by the four diagonals.
3 Boy: Of eight feet. Socrates: And from what line do you get this figure? Boy: From this. Socrates: That is, from the line which extends from corner to corner of the figure of four feet? Socrates: And that is the line that the learned call the diagonal. And if this is the proper name, then you, Meno's slave, are prepared to affirm that the double space is the square of the diagonal? Boy: Certainly, Socrates. It is curious that merely by questioning the slave (who would have been a child of a Greek family defeated in battle, and would have been deprived of any education), Socrates is able to get him to complete a proof. Plato takes this as a demonstration of a strange metaphysical doctrine that we once knew everything and have forgotten it, and now we just need to be helped to remember the truth. But we should note a different and interesting fact. Neither Socrates nor the slave boy ever doubts that Socrates s demonstration is true of all squares. That is, while Socrates draws squares in the dirt, the slave boy never says, Well, Socrates, you ve proved that to make a square twice as big as this square that you have drawn, I need to take the diagonal of this square as a side of my new square. But what about a square that s much smaller or larger than the one you drew here? That is in fact a very perplexing question. Why is Socrates s demonstration good for all, for any, squares? 16.2 A familiar strangeness We have saved for last the most subtle issue with reasoning with quantifiers: how shall we prove something is universally true? Consider the following argument. We will assume a first order logical language that talks about numbers, since it is sometimes easier to imagine something true of everything in our domain of discourse if we are talking about numbers. All numbers evenly divisible by eight are evenly divisible by four. All numbers evenly divisible by four are evenly divisible by two. All numbers evenly divisible by eight are evenly divisible by two. Let us assume an implicit translation key, and then we can say that the following is a translation of this argument. x(ex Fx) x(fx Tx) x(ex Tx)
4 This looks like a valid argument. But to prove it, we need some way to be able to prove a universal statement. But how could we do such a thing? There are infinitely many numbers, so surely we cannot check them all. How do we prove that something is true of all numbers, without taking an infinite amount of time and creating an infinitely long proof? The odds are that you already know how to do this, although you have never reflected on your ability. You most likely saw a proof of a universal claim far back in grade school, and without reflection concluded it was good and proper. For example, when you were first taught that the sum of the interior angles of a triangle is equivalent to two right angles, you might have seen a proof that used a single triangle as an illustration. It might have gone something like this: assume lines DE and AG are parallel, and that two other lines AC and BC cross those parallel lines, and cross each other at C. Assume also that the alternate angles for any line crossing parallel lines are equal. Assume also that a line is equivalent to two right angles, or 180 degrees. Then, in the following picture, a =a, b =b, and a +b +c=180 degrees. Thus, a+b+c=180 degrees. Most of us think about such a proof, see its reasoning, and agree with it. But if we reflect for a moment, we should see that it is quite mysterious why such a proof works. That s because, it aims to show us that the sum of the interior angles of any triangle is the same as two right angles. But there are infinitely many triangles (in fact, logicians have proved that there are more triangles than there are natural numbers!). So how can it be that this argument proves something about all of them? Furthermore, in the diagram above, there are infinitely many different sets of two parallel lines we could have used. And so on. This also touches on the case that we saw in the Meno. Socrates proves that the area of a square A twice as big as square B does not simply have sides twice as long as the sides of B. But he and the boy drew just one square in the dirt. And it won t even be properly square! How can they conclude something about every square based on their reasoning and a crude drawing? In all such cases, there is an important feature of our proof. Squares come in many sizes, triangles come in many sizes and shapes. But what interests us in such
5 proofs is all and only the properties that all triangles have, or all and only properties that all squares have. We refer to a triangle, or a square, that is abstract in a strange way: we draw inferences about, and only refer to, its properties that are shared with all the things of its kinds. We are really considering a special, generalized instance. We can call this special instance the arbitrary instance. If we prove something is true of the arbitrary triangle, then we conclude it is true of all triangles. If we prove something is true of the arbitrary square, then we conclude it is true of all squares. And so on Universal derivation To use this insight, we will introduce not an inference rule, but rather a new proof method. We will call this proof method universal derivation or, synonymously, universal proof. We need something to stand for the arbitrary instance. For a number of reasons, it is traditional to use unbound variables for this. However, to make it clear that the variable is being used in this special way, and that the wellformed formula so formed is a sentence, we will use a prime that is, the small mark ʹ to mark the variable. Let α be any variable. Our proof method thus looks like this. [αʹ ]... Φ(αʹ ) αφ(α) universal derivation Where αʹ does not appear in any open proof above the beginning of the universal derivation. The strange semantic idea of an arbitrary instance is perhaps less mysterious when we consider the actual syntactic constraints on a universal derivation. I should not be able to say anything about an arbitrary instance αʹ unless I have done universal instantiation of a universal claim. No other sentence should allow claims about αʹ. For example, you cannot perform existential instantiation to an arbitrary instance, since we required that existential instantiation be done to special indefinite names that have not appeared yet in the proof. But if we can only makes claims about αʹ using universal instantiation, then we will be asserting something about αʹ that we could have asserted about anything in our domain of discourse. Seen in this way, from the perspective of the syntax of our proof, instead of in terms of the mysterious semantics of the arbitrary instance, the universal derivation hopefully seems very intuitive. This schematic proof has a line where we indicate that we are going to use αʹ as the arbitrary object. This is not necessary, and is not part of our proof. Rather, like the explanations we write on the side, it is there to help someone understand our proof. It says, this is the beginning of a universal derivation, and αʹ stands for the arbitrary object. Since this is not actually a line in the proof, we need not number it.
6 We can now prove our example above is valid.
7 1. x(ex Fx) premise 2. x(fx Tx) premise [xʹ ] 3. Exʹ assumption for conditional derivation 4. (Exʹ Fxʹ ) universal instantiation, 1 5. Fxʹ modus ponens, 5, 4 6. (Fxʹ Txʹ ) universal instantiation, 2 7. Txʹ modus ponens, 7, 6 8. (Exʹ Txʹ ) conditional derivation, x(ex Tx) universal derivation, 38 [A section is cut here where I prove some problems from the homework. This will be replaced after Wednesday.] 16.5 Illustrating invalidity Consider the following argument: All humans are mortal. Flipper is mortal. Flipper is human. This is an invalid argument. If Flipper were a dolphin, for example, the premises would be true and the conclusion false. Because we cannot use truth tables to describe the semantics of quantifiers, we have kept the semantics of the quantifiers intuitive. A complete semantics for first order logic is called a model, and requires some set theory. This presents a difficulty: we cannot demonstrate that an argument using quantifiers is invalid without a semantics. Fortunately, there is a heuristic method that we can use that does not require developing a full model. We will develop and intuitive and partial model. We already did this above, when we described the why the argument was invalid. The idea is that we will come up with an interpretation of the argument, where we ascribe a meaning to each predicate, and a referent for each term, and where this interpretation makes the premises obviously true and the conclusion obviously false. This is not a perfect method, since it will depend upon our understanding of our interpretation, and because it requires us to demonstrate some creativity. But this method does illustrate important features of the semantics of the first order logic, and used carefully it can help us see why a particular argument is invalid. First, we will translate the argument into our first order logic. Leaving the translation key implicit, the argument has the form:
8 x(hx Mx) Md Hd What we need to do is simply find an interpretation for the predicates and the name such that the premises are obviously true and the conclusion is obviously false. (Presumably this was not the case for our original argument, or we would not be testing its validity.) It is best to do this with numbers, since there is less vagueness of the meaning of the predicates. So suppose our domain of discourse is the natural numbers. Then, we need to find an interpretation of the predicates that makes the first two lines true and the conclusion false. Here is one: Hx: x is evenly divisible by 2 Mx: x is evenly divisible by 1 d: 3 The argument would then have as premises: All numbers evenly divisible by 2 are evenly divisible by 1; and, 3 is evenly divisible by 1. These are both true. But the conclusion would be: 3 is evenly divisible by 2. This is false. Let us consider another example. Here is an invalid argument: x(fx Gx) Fa Gb We can illustrate that it is invalid by finding an interpretation that shows the premises true and the conclusion false. Our domain of discourse will be the natural numbers. We interpret the predicates and names in the following way: Fx: x is greater than 10 Gx: x is greater than 5 a: 15 b: 2 Given this interpretation, the argument translates to: Any number greater than 10 is greater than 5; 15 is greater than 10; therefore, 2 is greater than 5. The conclusion is obviously false, whereas the premises are obviously true. Some students find it strange that we would just make up meanings for our predicates and names; it seems to them that we are doing something illegitimate in our reasoning. However, as long as our interpretations of the predicates and names follow our rules, our interpretation will be acceptable. Recall the rules for predicates are that they have an arity, and that each predicate of arity n is true or false (never both, never neither) of each n things in the domain of discourse. The rule for names is that they refer
9 to only one object. If you have doubts, consider a valid argument, and try to come up with some interpretation that makes it invalid. You will find that you cannot do it, if you respect the constraints on predicates and names. Thus, an informal model used to illustrate invalidity must have three things: 1. a domain of discourse; 2. an interpretation of the predicates; and 3. an interpretation of the names. If you can find such a model that makes the premises obviously true and the conclusion obviously false, you have illustrated that the argument is invalid.
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