Week 3: Logical Language

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1 Week 3: Logical Language Then you should say what you mean, the March Hare went on. I do, Alice hastily replied; at least at least I mean what I say that s the same thing, you know. Not the same thing a bit! said the Hatter. You might just as well say that I see what I eat is the same thing as I eat what I see! Lewis Carroll Symbolic Logic Before the development and widespread acceptance of symbols like +,,, =, 0, and ², not to mention variables like x, early mathematicians had to deal with statements that were very hard to manipulate. This was the case until at least the 16 th Century, when Robert Recorde introduced the equals sign. Without these conveniences, a problem might read: Twice the square of a number, plus five, is the same as eleven times the number. Could you imagine using such a statement directly to determine that the number in question must either be 1 or 5? On the other hand, we can work with the shorthand statement 2x = 11x quite easily. The symbols of arithmetic and algebra allow us to state general properties of numbers. For example, the fact that multiplication distributes over addition can be written as follows: x(y + z) = xy + xz. Imagine trying to explain this relatively simple fact, which applies to any three numbers, without using any of the shorthand symbols used here. Now, just as arithmetical and algebraic shorthand allows us to see more clearly into relationships between numbers, logical shorthand can help clarify relationships between mathematical statements. As a first example, it was noted before that most mathematical propositions can be written in the form If P 1, then P 2, with the symbols P 1 and P 2 representing two mathematical statements. A proposition of this form can be written in shorthand as follows: P 1 P 2 As we have seen (and will continue to see), it is often useful to consider the idea that a statement is not true. A proposition of the form P is not true can be written in shorthand as follows: P For example, the proposition if a positive whole number is even, then it is not odd has the following logical structure: P 1 P 2.

2 Week 3 Page 2 In mathematical statements, the words and and or often appear. A proposition of the form P 1 and P 2 can be written in shorthand as follows: P 1 P 2 A proposition of the form P 1 or P 2 can be written in shorthand as follows: P 1 P 2 For example, the proposition if a is even and b is odd, then a + b is odd has the following logical structure: (P 1 P 2 ) P 3. The proposition if n is a positive whole number, then n is even or n is odd has the following logical structure: P 1 (P 2 P 3 ). Mathematical English It is vitally important at this stage to understand the exact meaning of the logical symbols that have been introduced, and how they are connected to the English language, which is usually less precise than it should be. In most settings, mathematical statements including most of those that we will prove are written in plain English. We must be careful to understand a given statement s logical meaning before attempting to prove it. Also, it is usually considered bad form to use logical symbols in a formal proof, so in order to avoid confusing the reader (or even yourself), you must know how to express your logical thoughts clearly in English. A statement of the form P 1 P 2 can be expressed in English in the following ways, among others: P 1 and P 2 Both P 1 and P 2 P 1, and also P 2 P 1, but P 2 It may seem strange that, logically, and means the same thing as but. An example of this is Bob is old, but he has a full head of hair. Clearly this statement carries the same raw information as Bob is old and he has a full head of hair. In English, we often use but to express an and statement that is surprising or unusual in some way. Also note that when we wish to express a statement of the form P 1 P 2 in English, we usually say Neither P 1 nor P 2. A statement of the form P 1 P 2 can be expressed in English in the following ways, among others: P 1 or P 2 Either P 1 or P 2 The word or is often used in English to express a mutually exclusive situation (a dichotomy). For example, Clean your room or you re grounded does not usually signify that you might clean your room and still be grounded. However, in mathematical writing,

3 Week 3 Page 3 the word or should never carry with it any sense of exclusivity. For example, in mathematics, if we say a whole number is either even or odd, we are not automatically ruling out the possibility that it might be both. This is often a major point of confusion for students reading mathematics! Remember that we proved earlier that every positive whole number is either even or odd, and in an entirely separate proof, we saw that no positive whole number is both even and odd. A statement of the form P is called the negation 1 of P. The statement P can be expressed in the following ways, among others: Not P P is false It is not the case that P It is most common in English for a negation like P to be expressed in a way that depends on the nature of the statement P, usually in the most useful way possible. For example, if P is the statement x is even, we could think of P as x is not even, or as it is not the case that x is even, but it is probably most useful to think of P as x is odd. There are other subtleties involved with negations in English, especially when desires are being expressed. For example, one might think that the negation of I want you to go to the party tonight is I do not want you to go to the party tonight. However, in everyday English this phrase usually carries the same meaning as I want you to not go to the party tonight, which is not a correct negation of the original statement. Fortunately, since desires generally do not enter into the logic of mathematics, this level of imprecision is mostly avoided. A good way to think about a negation is that a statement and its negation cannot both be true, and they cannot both be false. A statement of the form P 1 P 2 is called a conditional 2 or an implication. This statement can be expressed in the following ways, among others: If P 1, then P 2 P 2 if P 1 P 2 when P 1 P 2 whenever P 1 P 1 implies P 2 P 2 follows from P 1 P 1 is sufficient for P 2 P 1 only if P 2 P 1 only when P 2 P 2 is necessary for P 1 The word implies has a much stronger meaning in mathematics than in everyday speech. When we say P 1 implies P 2, we mean that P 2 must be true if P 1 is. Words like implies and is sufficient do not always express the same certainty in everyday English as in mathematics. If statements in everyday English can also carry a slightly different meaning than the symbolic logic of P 1 P 2. For example, in everyday speech, the statement If you 1 The negation of P is sometimes also written as ~P. 2 The conditional P 1 P 2 is sometimes also written as P 1 P 2.

4 Week 3 Page 4 clean your room, then you can go to the movie usually also means If you do not clean your room, then you cannot go to the movie. Conditional statements in mathematics should never be assumed to carry this kind of extra meaning. For example, the statement if a number is both large and prime, then it is odd does not also mean if a number is not both large and prime, then it is not odd. There are more subtle ways that conditionals are often expressed in English. One example is the statement Humans are primates. This statement means that if x is a human, then x is a primate. It does not mean that if x is a primate, then x is a human. Another example is Only humans drive cars. This means to say that if x drives a car, then x is a human. It does not mean that every human drives a car, but rather that every driver is human. Exercise 1: Use the symbols of logic to give the logical structure of the proposition If x is a positive whole number, then it is not both even and odd. Exercise 2: Use the symbols of logic to give the logical structure of the proposition If x is not a whole number, then it is neither even nor odd. Exercise 3: Use the symbols of logic to give the logical structure of the proposition If a b is odd, then both a and b are odd. Exercise 4: Rewrite each of the following statements, without using logical symbols, in the form If, then : (a) In order for you to get a refund, it is necessary that you have your receipt. (b) To judge someone s mathematical experience, it is sufficient to ask them what the symbol stands for. (c) You are legally driving in Pennsylvania only if you are 16 years old. (d) Stephen King is fun to read if you like horror stories. Conditionals and Biconditionals For any conditional statement P 1 P 2, The contrapositive is P 2 P 1. The converse is P 2 P 1. The inverse is P 1 P 2. We have seen before that a conditional statement and its contrapositive are logically equivalent. We have also seen that a conditional statement and its converse are not logically equivalent. In other words, when a conditional statement is true, its contrapositive must be true; and when a conditional statement is false, its contrapositive must be false;

5 Week 3 Page 5 but the converse of a conditional statement may be true or false regardless of whether the original conditional is true or false. Exercise 5: Is a conditional statement logically equivalent to its inverse? Justify your answer. Exercise 6: What previous example demonstrated that a conditional statement and its converse are not logically equivalent? Exercise 7: Write the contrapositive of each of the following statements, without using logical symbols: (a) We will go if you go. (b) If a quadrilateral is square, it has four equal sides and four equal angles. (c) If a triangle has either two equal sides or two equal angles, then the triangle is isosceles. It is not uncommon in mathematics that a conditional statement and its converse might both be true. For example, if a whole number is even, then it is not odd, and if a whole number is not odd, then it is even. A more concise way to say this is A whole number is even if and only if it is not odd. The phrase if and only if comes up so frequently in mathematics that it is often shortened to the word iff. Remember that P 1 if P 2 can be written symbolically as P 2 P 1, and P 1 only if P 2 can be written as P 1 P 2. For this reason, a proposition of the form P 1 if and only if P 2 can be written in shorthand as follows: P 1 P 2 A statement of the form P 1 P 2 is called a biconditional 3. This statement can be expressed in the following ways, among others: P 1 if and only if P 2 P 1 is equivalent to P 2 P 1 iff P 2 If P 1 or P 2, then both P 1 is necessary and sufficient for P 2 Exercise 8: The most common way to prove a biconditional statement P 1 P 2 is to prove the statement P 1 P 2, and also separately prove the statement P 2 P 1. We have done this before. Find a biconditional statement we have proved in this way, and write that statement in the form if and only if. 3 The biconditional P 1 P 2 is sometimes also written as P 1 P 2 or P 1 P 2.

6 Week 3 Page 6 Laws of Logic There are a handful of mostly basic laws of logic that we will use in proving mathematical truths. One of the laws of logic that we have frequently seen is that a conditional statement and its contrapositive are always equivalent. Another law is that if we can derive a contradiction from any statement, then the statement must not be true. Note that each of these laws has expanded our options for how we might compose proofs. Another law of logic that we have not thought much about, but that is obviously true, is that the negation of the negation of a statement is the same as the original statement itself. That is, ( P) is equivalent to P. In other words, to say it is not the case that P is false is the same as saying P is true. Two very handy laws of logic that are slightly less obvious are known as DeMorgan s laws. Symbolically, they are the following: (P 1 P 2 ) is equivalent to ( P 1 ) ( P 2 ) (P 1 P 2 ) is equivalent to ( P 1 ) ( P 2 ) DeMorgan s laws are immensely useful in proving certain mathematical truths. For example, one previous exercise asked you to prove a proposition of the form If P 1, then P 2 and P 3. The contrapositive of this proposition is If P 2 and P 3 are not both true, then P 1 is not true. A simpler way to say this is If P 2 is not true or P 3 is not true, then P 1 is not true. This leads us to the following proof strategy for this particular proposition structure. Proposition: P 1 (P 2 P 3 ) Proof outline: Suppose either P 2 or P 3. If P 2, then. Therefore P 1. On the other hand, if P 3, then. Therefore P 1. In either case, P 1, so by contraposition, P 1 (P 2 P 3 ). Exercise 9: One of DeMorgan s laws can be expressed in English as follows: Saying Neither of the statements P 1 or P 2 is true is the same as saying The statements P 1 and P 2 are both false. Express the other law in English as clearly as you can. Exercise 10: Write an outline for a proof by contradiction of a proposition of the form P 1 (P 2 P 3 ). Exercise 11: Yet another law of logic is that a statement of the form Either P 1 is true, or both P 2 and P 3 are true is equivalent to the statement Either P 1 or P 2 is true, and either P 1 or P 3 is true. Or symbolically, the statement P 1 (P 2 P 3 ) is equivalent to (P 1 P 2 ) (P 1 P 3 ). Now, find a similar law about a statement of the form P 1 (P 2 P 3 ). Write the statement P 1 (P 2 P 3 ) in clear English, and a write the equivalent statement in both English and in logical symbols.

7 Week 3 Page 7 Exercise 12: Consider the mathematical statement x is a multiple of 2 and a multiple of 3 or a multiple of 5. This statement does not have a clear, definite meaning. Describe two different possible interpretations of this statement. Find a number x for which one of your interpretations is true while the other interpretation is false.