Part 2 Module 1 Logic

1. Part 2 Module 1Logic In Part 2 Modules 1 through 5 we will study logic. The logical rules, patterns and formulas that we will uncover over the next few lectures are the (usually unstated) principles that form the foundations for critical reasoning. Critical reasoning ability is usually identified (by prospective employers, for example) as the single most important skill to be possessed by a college graduate, irrespective of one’s major.

2. Part 2 Module 1Statements, quantifiers, negations A statement or proposition is a declarative sentence that has truth value. To say that a sentence has truth value means that, when we hear or read the sentence, it makes sense to ask whether the sentence is true or false.

3. Examples of statements Today is Saturday. False Today I have math class. True 1 + 1 = 3 False

4. Examples of statements Some cats have fleas. This is an example of a quantified or categorical statement. Words like “all,” “some,” and “none” are called quantifiers. In logic, “some” means “at least one.” Unlike in everyday usage, in logic, “some” does not necessarily indicate plural.

5. Quantified statements Some cats have fleas. True All lawyers are dishonest. False In this course, a sentence that sounds like an opinion will be treated as an acceptable statement. In such a case we will pretend, for the sake of discussion, that a subjective, value-laden term like “dishonest” has been precisely defined.

6. Sentences that aren’t statements Not every sentence is a statement. What’s your sign? Questions are not statements. It doesn’t make sense to ask whether a question is true or false. PAY ATTENTION! Commands are not statements. It doesn’t make sense to ask whether a command is true or false.

7. Sentences that aren’t statements This statement is false. The previous sentence is a paradox. It is neither true nor false, so it isn’t a statement.

8. Names for statements We will tend to use lower case letters, like p, q, r, and so on, as names for statements.

9. Names for statements p: Today is Saturday. q: Today I have math class. r: 1 + 1 = 3 s: Some cats have fleas. u: All lawyers are dishonest.

10. Negations Let p be any statement. The negation of p, denoted ~p is another statement that is logically opposite to p. This means that ~p will always be opposite in truth value to p.

11. Negations For any statement p: In any situation that makes p a true statement, ~p will be false. In any situation that makes p a false statement, ~p will be true.

12. Negations of simple statements For each of the statements that were named at the beginning of this discussion, write the negation.

13. Negations of simple statements p: Today is Saturday. (F) ~p: Today is not Saturday. (T) q: Today I have math class. (T) ~q: Today I don’t have math class. (F) r: 1 + 1 = 3 (F) ~r: 1 + 1  3 (T)

14. Negations There is a very strong relationship between any statement p and its negation ~p: It is impossible to conceive of a situation where a statement and its negation will have the same truth value.

15. Negations of quantified statements Select the correct negation of “Some cats have fleas.” A. All cats have fleas. B. Some cats don’t have fleas. C. No cats have fleas. D. Some fleas have cats.

16. Negations of quantified statements The correct negation of “Some cats have fleas” is “No cats have fleas.” Fact: if a statement has the form “Some A are B”, its negation will have the form “No A are B.” One way to verify this fact is by diagramming. This was demonstrated in class.

17. Negations of quantified statements Select the correct negation of “All lawyers are dishonest.” A. All lawyers are honest. B. Some lawyers are honest. C. No lawyers are dishonest. D. Some lawyers are dishonest.

18. Negations of quantified statements The correct negation of “All lawyers are dishonest” is “Some lawyers are honest.” Fact: If a statement has the form “All A are B” then its negation will have the form “Some A are not B.” Again, one way to verify this fact is by diagramming. This was demonstrated in class.

19. Negations, alternative phrasing We have seen that the correct negation of “All lawyers are dishonest” is “Some lawyers are honest.” However, there are many other (non-preferred) ways to correctly state the negation of “All lawyers are dishonest.”

20. Negations, alternative phrasing Each of the following statements is a correct negation of “All lawyers are dishonest.” “It is not the case that all lawyers are dishonest.” “It is not true that all lawyers are dishonest.” (In the style of Borat:) “All lawyers are dishonest… NOT!”

21. Exercise - Negations of quantified statements Select the negation of “No beetles fight battles.” A. All beetles fight battles. B. Some beetles fight battles. C. Some beetles don’t fight battles. D. No beetles swing paddles.

22. Solution - Negations of quantified statements Select the negation of “No beetles fight battles.” In Part 2 Module 1 we have seen two rules for negations of quantified statements. StatementNegation “Some A are B.” “No A are B.” “All A are B.” “Some A are not B.” Each of these two rules works in both directions, and the verb doesn’t necessarily have to be “are.” The first rule means that if a statement has the form “None do…”, then its negation will have the form “Some do…” So, the negation of “No beetles fight battles” is “Some beetles fight battles.”

23. Exercise - Negations of quantified statements Select the negation of “Some poodles don’t leap puddles.” A. Some poodles leap puddles. B. No poodles leap puddles. C. All poodles leap puddles. D. None of these.

24. Solution - Negations of quantified statements Select the negation of “Some poodles don’t leap puddles.” In Part 2 Module 1 we have seen two rules for negations of quantified statements. StatementNegation “Some A are B.” “No A are B.” “All A are B.” “Some A are not B.” Each of these two rules works in both directions, and the verb doesn’t necessarily have to be “are.” Each of these two rules works in both directions. The second rules says that if a statement has the form “Some don’t…”, then its negation will have the form “All do…” So, the negation of “Some poodles don’t leap puddles” is “All poodles leap puddles.”

25. Compound statements A compound statement is formed by joining two or more simpler statements, using special connecting words or structures such as “and,” “or,” or “if…then.” 1 + 1 = 2 or 4 < 3 is an example of a compound statement. All lawyers are dishonest and some cats have fleas is another example of a compound statement.

26. Logical connectives Words or phrases such as “and,” “or,” or “if…then,” used to form compound statements, are called logical connectives.

27. The Conjunction Let p, q be any statements. Their conjunction is the compound statement having the form “p and q.” This is denoted p  q In order for a conjunction to be true, both terms must be true.

28. The Disjunction Let p, q be any statements. Their disjunction is the compound statement having the form “p or q.” This is denoted p  q In order for a disjunction to be true, at least one of the two terms must be true. A disjunction is false only in the case where both terms are false.

29. Symbolic statements Suppose p represents the statement “I have a dime,” and q represents the statement “I have a nickel.” The symbolic statement ~p  q corresponds to “I don’t have a dime, or I have a nickel.” The symbolic statement p  ~q corresponds to “I have a dime and I don’t have a nickel.” This last statement can also be read as “I have a dime but I don’t have a nickel.”

30. Exercise Suppose p represents a true statement, while q, r represent false statements. Find the truth value of (~r  p)  ~(~q  r) True B. False

31. Solution Suppose p represents a true statement, while q, r represent false statements. Find the truth value of (~r  p)  ~(~q  r) First, replace each variable with its specified value: (~F  T)  ~(~F  F) Next, evaluate simple terms that are negated: (T T)  ~(T F) Next, evaluate compound statements within parentheses: T  ~(T) T  F Finish the evaluation F

32. Exercise: Symbolic statements Suppose p, q represent false statements, while r represents a true statement. Find the truth value of ~[ q  (~p  r) ] True B. False

33. Solution Suppose p, q represent false statements, while r represents a true statement. Find the truth value of ~[ q  (~p  r) ] ~[ F  (~F  T) ] ~[ F  (T  T) ] ~[ F  T ] ~[ T ] F The correct answer is B. False

34. Exercise Suppose p, q represent false statements, while r represents a true statement. Find the truth value of ~[ ~r  (p  ~q) ] True B. False

35. Solution Suppose p, q represent false statements, while r represents a true statement. Find the truth value of ~[ ~r  (p  ~q) ] ~[ ~T  (F  ~F) ] ~[ F  (F  T) ] ~[ F  T ] ~[ F ] T

36. Exercise: Evaluate a symbolic statement Suppose p represents a false statement and q represents a false statement. Find the truth value of ~ (~p  q) A. True B. False

37. The conjunction and the disjunction: summary Conjunction Disjunction A and B A or B A  B A  B A  B is trueA  B is false only in the case only in the case where both terms where both terms are true. are false.

38. Solution Suppose p represents a false statement and q represents a false statement. Find the truth value of ~ (~p  q) ~ (~F  F) ~ (T  F) ~ (F) T The correct answer is “A. True”

39. Truth tables A truth table is a device that allows us to analyze and compare compound logic statements. Consider, for example, the symbolic statement p  ~q. Whether this statement turns out to be true or false will depend upon whether p is true or false, whether q is true or false, and the way the “” and “~” operators work. A truth table will show all the possibilities.

40. Truth tables As an introduction to constructing and filling in truth tables, we will make a truth table for the statement p  q and a truth table for the statement p  q.

41. Truth tables

42. Truth tables EXAMPLE 2.1.8 #1 Make a truth table for the statement p  ~q

43. Truth table for p  ~q In order to make a truth table for the statement p  ~q, we need to set up a “skeleton” two variable truth table, and include a column for ~q prior to the column for p  ~q. We fill in the column for p  ~q by referring to the p column, the ~q column, and applying the rule for the “or” connective.

44. Truth tables • Referring to the truth table shown below, be aware that the values in the rightmost column may not be correct. Insert intermediate columns as needed (“????”), fill in the truth table, and decide whether the rightmost column is correctly filled in as shown. • Yes, the rightmost column is correctly filled in. • No, the values in the rightmost column are not all correct.

45. Truth table for q  ~(p  q) To make a truth table for the statement q  ~(p  q), we set up the skeleton for a two-variable truth table. In order to fill in a column for the statement q  ~(p  q), the truth table must first have a column for ~(p  q). In order to fill in the column for ~(p  q), the truth table must first have a column for p  q.

46. A tautology The truth table column for the statement q~(pq) shows only “true.” This means that it is never possible for that statement to be false. The statement is always true, due to its logical structure. A statement that can never be false is called a tautology.

47. Tautologies A tautology isa statement that can never be false, due to its logical structure. To decide whether a symbolic statement is a tautology, make a truth table having a column for that statement. If the truth table column shows only “Trues” and no “Falses,” then the statement is a tautology. Otherwise, the statement is not a tautology.

48. Exercise: Tautologies Decide whether this symbolic statement is a tautology: (p  ~q)  (~p  q) A. Yes, this statement is a tautology. B. No, this statement isn’t a tautology.

49. Negation of a compound statement Select the correct negation of “I’m a lumberjack and I’m okay.” A. I’m a lumberjack and I’m not okay. B. I’m not a lumberjack and I’m not okay. C. I’m not a lumberjack or I’m not okay. D. None of these.

50. Solution Select the correct negation of “I’m a lumberjack and I’m okay.” One way to deal with this question is by making a truth table. Let p be the statement “I’m a lumberjack” and let q be the statement “I’m okay.” We want to find a statement that is the negation (that is, the opposite) of pq. Choice A is the statement p~q, choice B is the statement ~p~q, and choice C is the statement ~p~q.