Symbols, operations, and Truth tables

1. Symbols, operations, and Truth tables Adapted from Patrick J. Hurley, A Concise Introduction to Logic (Belmont: Thomson Wadsworth, 2008).

2. What is a statement? A statement is a sentence that is asserted that can be true or false.

3. Which of the following are statements? • A. Chocolate truffles are loaded with calories. • B. What is Kharoum? • C. Melatonin helps relieve jet lag. • D. No wives ever cheat on their husbands. • E. Let’s go to the movies tonight. • G. Tiger Woods plays golf. • H. Turn off the t.v. right now! • E. X is an even number. • F. Anything multiplied by 2 is an even number.

4. Symbolizing Statements • Simple statements are statements that do not have any other sentence embedded in it. It can be symbolized by giving it a letter, usually ‘P’ or ‘Q’. • Example: • P = Fast foods tend to be unhealthy. • Q = James Joyce wrote Ulyesses. • P = The number 3 is odd. • Q = Even numbers are divisible by 2.

5. Compound Statements • Compound statements are statements that contain at least one simple statement as a component. • Examples: • It is not the case that the sky is neon green. • Ice cream is cold and hot chocolate is hot. • Either we go left or we go right. • If it rains, the ground gets wet. • You will reach your goal if and only if you try.

6. Operators • The operators at work in the previous slide can be summed up this way: Operator Translation  ‘not’, ‘it is not the case that’  ‘and’, ‘also’, ‘but’ V ‘or’, ‘unless’  ‘if…then…’, ‘only if’

7. Symbolizing Compound Statements ‘It is not the case that A.’ A ‘D and C.’ D  C ‘Either P or E.’ P V E ‘If W then F.’ W F

8. Translating Operators In the following slides, we will look at examples of natural language and how they would be translated into logical symbols.

9. Negation -  All of these natural language examples of compound statements have the exact same meaning in logic: Rolex does not make cars. = R It is not the case that Rolex makes cars. = R It is false that Rolex makes cars. = R

10. Conjunction -  All of these natural language examples of compound statementshave the exact same meaning in logic: Prof. Reitz teaches Math and Prof. Park teaches Philosophy. = RP Prof. Reitz teaches Math, but Prof. Park teaches Philosophy. = RP Prof. Reitz teaches Math;however, Prof. Park teaches Philosophy=RP

11. Disjunction - V All of these natural language examples of compound statementshave the exact same meaning in logic: It is raining or it is sunny. = R V S Either it is raining or it is sunny. = R V S It is raining unless it is sunny. = R V S Unless it is raining, it is sunny. = R V S

12. Conditional -  All of these natural language examples of compound statementshave the exact same meaning in logic: If it rained, the ground got wet. = R  W The ground got wet if it rained. = R  W It rained only if the ground got wet. = R  W

13. Truth Functions As we said earlier, all statements can be true or false. But how do we determine the truth value of compound statements? There is a simple technique for determining the truth value of compound statements. It is the truth table technique. Each operator has a unique pattern of truth and falsehood.

14. Truth Tables In order to do a truth table, determine all the statements involved and symbolize (usually with a ‘P’ or ‘Q’). The way you determine the number of rows is to count the number of letters and plug into the following formula: L = 2n E.g.

15. Truth Tables Next, write out all the possible truth combinations that is possible for the given compound statement. E.g.

16. Truth Tables Then, using those truth values determine the truth value for the compound statement as a whole. E.g.

17. Determining the Truth Value for Operators Each operator comes with its own rules for how to determine the truth value of the statements they occur in. We just saw the table for the disjunction. Disjunctive statements are true when either P or Q are true, or both. They need not be true at the same time. The only time the statement is false if both statements are false. Let’s take a look at truth tables for the others operators.

18. Disjunction Translate and determine the truth value for the following disjunctive statements: • Apples are red or they are green. • Bananas are yellow or they are pink. • Strawberries are black or they are bright blue.

19. Disjunction Here are the answers: • R V G – T • Y V P – T • B V L - F

20. Negation Since negation in its simplest form can apply to a single statement, we only need two rows in the table (2 to the first power). Simply put, the negation operator has the opposite truth value of the original statement.

21. Negation Translate and determine the truth value for the following negative statements: • Bananas are not yellow. • It is not the case that bananas are purple.

22. Negation Here are the answers: 1. B – F 2. B - T

23. Conjunction Unlike disjunction, in conjunction, both P and Q must be true for the statement conjoining them to be true. For example, in ‘Bachelors are unmarried and male’, if it turned out that someone was either married or a woman or both, then this statement would be false. Only if both P and Q are true can the conjunction be true. See the next slide for the table.

24. Conjunction

25. Conjunction Translate and determine the truth value for the following conjunctive statements: • Bananas are yellow and they are green. • Bananas are yellow and they are purple. • Bananas are purple and they are pink.

26. Conjunction Here are the answers: 1. Y  B - T 2. Y  P - F 3. P I - F

27. Conditional Conditional statements are statements that follow the format “If P, then Q”, where P = antecedent, and Q = consequent. They express relationships of necessity and sufficiency. In this case, P is a sufficient condition for Q and Q is a necessary condition for P. For example, in ‘If Fido is a dog, then Fido is an animal’, Fido’s being a dog is a sufficient condition for being an animal. But being an animal is a necessary condition for being a dog.

28. Conditional In conditional statements, falsity is only possible if the antecedent is true. If the antecedent is false, then anything can follow, both meaningful and nonsensical things. For example, if I say, “If chipmunks can sing…” (which is false), then anything can follow. “If chipmunks can sing, pigs can fly.” In this case, the statement is true. If something ridiculous is possible, then so is something just as ridiculous. But if the antecedent is true, a true consequent must follow, or else the compound statement is false. See the next slide.

29. Conditional If the antecedent is true, the consequent must also be true. “If Fido is a dog, then Fido is an animal” is true, assuming that Fido is actually a dog. But “If Fido is a dog, then Fido is a plant” is false.

30. Conditional Translate and determine the truth value for the following conditional statements: • If dolphins were mammals, then they would be warm-blooded. • If dolphins were fish, they would be cold-blooded. • If dolphins were mammals, they would be cold-blooded.

31. Conditional Here are the answers: 1. M  W – T 2. F  C – T 3. M  C - F

32. Longer Compound Statements Let’s try some more complex statements. Please see the board.

33. Longer Compound Statements Try constructing truth tables for the following statements on your own or in a group. • (P  Q)  (P  Q) • (P  Q) V (Q  P) • [(C  D)  C]  D