Mathematical Logic Module 2-2 Math to the Modern World

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1. MATHEMATICAL LOGIC Prepared by: Abel E. Sadji MODULE 2

2. MODULE OBJECTIVES At the end of this module, challenge yourself to: illustrate and symbolize propositions; distinguish between simple and compound propositions; determine the truth values of propositions; illustrate the different forms of conditional propositions; illustrate different types of tautologies and fallacies; determine the validity of categorical syllogisms; establish the validity and falsity of real-life arguments using logical propositions, syllogisms, and fallacies; and determine the validity of an argument.

3. LESSONS IN THIS MODULE 01 Truth Tables, Equivalent Statements, and Tautologies 02 Logic Statement and Quantifiers

4. Logic Statement and Quantifiers

5. LOGICAL STATEMENTS AND QUANTIFIERS A proposition (or statement ) is a declarative sentence which is either true or false, but not both. The truth value of the propositions is the truth and falsity of the proposition.

6. Proposition DIRECTION: Determine which of the following are proposition and not a proposition. Manila is the capital of the Philippines. What day is it? Help me, please. He is handsome. Not a Proposition Not a Proposition Proposition

7. LOGICAL STATEMENTS AND QUANTIFIERS A propositional variable is a variable which is used to represent a proposition. A formal propositional variable written using propositional logic notation, , , and are used to represent propositions.  

8. LOGICAL STATEMENTS AND QUANTIFIERS Logical connectives are used to combine simple propositions which are referred as compound propositions. A compound proposition is a proposition composed of two or more simple propositions connected by logical connectives "and," "or," "if then," "not," and "if and only if”. A proposition which is not compound is said to be simple (also called atomic ).

9. OPERATIONS ON PROPOSITIONS There are three main logical connectives such as conjunction, disjunction, and negation. The following are briefly discussed in this section. Note that T refers to true proposition and F refers to false proposition.

10. OPERATIONS ON PROPOSITIONS CONJUNCTION The conjunction of the proposition p and q is the compound proposition "p and q." Symbolically, p q, where is the symbol for "and." If p is true and q is true, then p q is true; otherwise, p q is false. Meaning, the conjunction of two propositions is true only if each proposition is true.   p q p q T T F F T F T F T F F F p q T T F F T F T F T F F F

11. OPERATIONS ON PROPOSITIONS Common Words Associated with Conjunction p and q p but q p also q p in addition q p moreover q p q p q T T F F T F T F T F F F p q T T F F T F T F T F F F

12. EXAMPLE 2+6=9 and men are mammal. p q p q T T F F T F T F T F F F p q T T F F T F T F T F F F p: 2+6=9 q: men are mammal. Since "2 + 6 = 9", is a false proposition and the proposition "man is a mammal" is true, the conjunction of the compound proposition is false.

13. EXAMPLE Manny Pacquiao is a boxing champion and Gloria Macapagal Arroyo is the first female Philippine President. p q p q T T F F T F T F T F F F p q T T F F T F T F T F F F p: Manny Pacquiao is a boxing champion q: Gloria Macapagal Arroyo is the first female Philippine President. In the proposition "Manny Pacquiao is a boxing champion" is true while the proposition "Gloria Macapagal Arroyo is the first female Philippine President" is false therefore the conjunction of the compound proposition is false.

14. EXAMPLE Abraham Lincoln is a former US President and the Philippine Senate is composed of 24 senators. p q p q T T F F T F T F T F F F p q T T F F T F T F T F F F p: Abraham Lincoln is a former US President q: Philippine Senate is composed of 24 senators. Since both the propositions "Abraham Lincoln is a former US Philippine President" and "Philippine Senate is composed of 24 senators" are both true, thus the conjunction of the compound proposition is true.

15. OPERATIONS ON PROPOSITIONS DISJUNCTION The disjunction of the proposition p, q is the compound proposition "p or q." Symbolically, p q, where is the symbol for "or". If p is true or q is true or if both p and q are true, then p q is true; otherwise, p q is false. Meaning, the disjunction of two propositions is false only if each proposition is false.   p q p∨q T T F F T F T F T T T F

16. OPERATIONS ON PROPOSITIONS Common Words Associated with Disjunction p or q p q p∨q T T F F T F T F T T T F

17. EXAMPLE 2+6=9 or Manny Pacquiao is a boxing champion p q p∨q T T F F T F T F T T T F p: 2+6=9 q: Manny Pacquiao is a boxing champion. Note that the proposition "2 + 6 = 9" is false while the proposition "Manny Pacquiao is a boxing champion" is true; hence the disjunction of the compound proposition is true.

18. EXAMPLE Philippine Senate is composed of 24 senators or Gloria Macapagal Arroyo is the first female Philippine President. p q p∨q T T F F T F T F T T T F p: Philippine Senate is composed of 24 senators q: Gloria Macapagal Arroyo is the first female Philippine President. Since proposition "Philippine Senate is composed of 24 senators" is true and the proposition "Gloria Macapagal Arroyo is the first female Philippine President" is false, therefore the disjunction of the compound proposition is true

19. EXAMPLE Abraham Lincoln is a former US President or man is a mammal. p q p∨q T T F F T F T F T T T F p: Abraham Lincoln is a former US President q: man is a mammal Given that both propositions "Abraham Lincoln is a former US President" and "man is a mammal" are both true, thus the disjunction of the compound proposition is true.

20. OPERATIONS ON PROPOSITIONS NEGATION The negation of the proposition p is denoted by-p, where is the symbol for "not." If p is true, ∼p is false. Meaning, the truth value of the negation of a proposition is always the reverse of the truth value of the original proposition. p ∼p T F F T

21. OPERATIONS ON PROPOSITIONS Common Words Associated with Negation not p It is false that p... It is not the case that p... p ∼p T F F T

22. EXAMPLE The following are propositions for p, find the corresponding ∼p. 3+5=8. Sofia is a girl. Achaiah is not here. ANSWER: 3+5 8. Sofia is not a girl. Achaiah is here.   p ∼p T F F T

23. OPERATIONS ON PROPOSITIONS CONDITIONAL The conditional (or implication) of the proposition p and q is the compound proposition "if p then q. Symbolically, p → q, where is the symbol for "if then." p is called hypothesis (or antecedent or premise ) and q is called conclusion (or consequent or consequence ). The conditional proposition p → q is false only when p is true and q is false; otherwise, p → q is true. Meaning p → q states that a true proposition cannot imply a false proposition. p q p → q T T F F T F T F T F T T

24. OPERATIONS ON PROPOSITIONS Common Words Associated with Conditional If p, then q. p implies q. p only if q. p therefore q. p is stronger than q. p is sufficient condition for q. p q p → q T T F F T F T F T F T T

25. OPERATIONS ON PROPOSITIONS Common Words Associated with Conditional q if p. q follows p. q whenever p. q is weaker than p. q is a necessary condition for p. p q p → q T T F F T F T F T F T T

26. EXAMPLE If vinegar is sweet, then sugar is sour. p: If vinegar is sweet q: sugar is sour Since the propositions "vinegar is sweet" and the "sugar is sour" are both false, therefore the conditional of the compound proposition is true. p q p → q T T F F T F T F T F T T

27. EXAMPLE 2+5=7 is a sufficient condition for 5+6=1. p: 2+5=7 q: 5+6=1. Note that "2+5=7 " is true and "5+6=1" is false, thus the conditional of the compound proposition is false. p q p → q T T F F T F T F T F T T

28. OPERATIONS ON PROPOSITIONS BICONDITIONAL The biconditional of the proposition p and q is the compound proposition "p if and only if q". Symbolically, p q, where is the symbol for "if and only if." If p and q are true or both false, then p q is true; if p and q have opposite truth values, then p q is false.   p q p↔q T T F F T F T F T F F T

29. OPERATIONS ON PROPOSITIONS Common Words Associated with Biconditional p if and only if q. (p iff q) p is equivalent to q. p is necessary and sufficient for q p q p↔q T T F F T F T F T F F T

30. EXAMPLE 2+8=10 if and only if 6-3=3. p: 2+8=10 q: 6-3=3 Since the statements "2+8=10" and the "6-3=3" are both true, therefore the conditional of the compound proposition is true. p q p↔q T T F F T F T F T F F T

31. EXAMPLE Manila is the capital of the Philippines is equivalent to fish live in the moon. p: Manila is the capital of the Philippines q: fish live in the moon Note that "Manila is the capital of the Philippines" is true proposition while "fish live in the moon" is false, thus the conditional of the compound proposition is false. p q p↔q T T F F T F T F T F F T

32. EXAMPLE 8-2=5 is a necessary and sufficient for 4+2 = 7. p: 8-2=5 q: 4+2 = 7 Given that "8 -2 = 5" and "4 + 2 = 7" are both false, thus the conditional of the compound proposition is true. p q p↔q T T F F T F T F T F F T

33. EXAMPLE

34. Constructing Truth Tables

35. TRUTH TABLES This section shows the construction of compound propositions through truth tables which referred as standard table form. Let us construct the truth table for each of the following proposition:  

36. SOLUTION T T F F T F T F F F T T F T F T F T T T T T F F T F T F F F T T F T F T F T T T  

37. SOLUTION T T T T F F F F T T F F T T F F T F T F T F T F T T T T F F F F T T F F T T F F T F T F T F T F  

38. EQUIVALENT STATEMENTS Two propositions are said to be logically equivalent (or equivalent) if they have the same truth value for every row of the truth table, that is is a tautology. Symbolically, EXAMPLE: Show that the following are equivalent.  

39. SOLUTION   T T T T F F F F T T F F T T F F T F T F T F T F T T T F T T T F T T T F F F F F T F T F F F F F T T F F F F F F T T T F F F F F T T T T F F F F T T F F T T F F T F T F T F T F T T T F T T T F T T T F F F F F T F T F F F F F T T F F F F F F T T T F F F F F

40. SOLUTION   T T F F T F T F T T F F T F T F

41. TAUTOLOGIES There are three important classes of compound statements namely tautology, contradiction, and contingency. TAUTOLOGY . It is a compound statement that is true for all possible combinations of the truth values of the propositional variables also called logically true. CONTRADICTION . It is a compound statement that is false for all possible combinations of the truth values of the propositional variables also called logically false or absurdity . CONTINGENCY . It is a compound statement that either be true or false, depending on the truth values of the propositional variables are neither tautology nor a contradiction.

42. FALLACIES Logical Fallacies refers to faulty reasoning in logic of an argument. It is advantageous to know logical fallacies in order to avoid them in an argument. There are different types of fallacies that we might use to present our position. The following are the list of common types of fallacies with their corresponding examples.

43. FALLACIES Appeal to Authority (or Argumentum Ad Verecundiam ). It is an argument that occurs when we accept or reject a claim merely because of the sources or authorities who made their positions on a certain argument. Example 1: The government should not impose death penalty. Many respected people, such as the former Secretary of Justice, have publicly stated her opposition to it. Example 2: Floyd Mayweather signs autographs with Parker pen, so evidently Parker pen is the most reliable pen on the market.

44. FALLACIES Appeal to Force (or Argumentum Ad Baculum ). It is an argument which attempts to establish a conclusion by threat or intimidation. Example 1: You will support my idea and tell the others that I am right; because if you don’t, I will do everything for you to lose your job. Example 2: If you don't believe in God, you won't go to heaven.

45. FALLACIES Appeal to Ignorance (or Argumentum Ex Silentio ). It is an argument supporting a claim merely because there is no proof that it's wrong. Example 1: Since time people have been trying to prove that God exists. But no one has yet been able to prove it. Therefore, God does not exist. Example 2: If you can't say that there aren't Martians living in Mars, so it's safe for me to accept there are

46. FALLACIES Appeal to Pity (or Argumentum Ad Misericordiam ). It is an argument that involves an irrelevant or highly exaggerated appeal to pity to get people to accept a conclusion by making them feel sorry for someone. Example 1: Mark has worked hard on his research project, and he will be depressed if he fails. For these reasons, you must give him a passing grade. Example 2: The city engineer is a vital part of this city. If he is sent to prison, the city and his family will suffer. Therefore, you must find in your heart to forgive him.

47. FALLACIES Appeal to the People (or Argumentum Ad Populum ). It is an argument that the opinion of the majority is always valid. Example 1: Most Filipino like soda. Therefore, soda is good. Example 2: Everyone I know is voting for Juan dela Cruz, so he's probably the best choice for mayor.

48. FALLACIES Argumentum Ad Hominem (Latin for " to the man "). It is an attack on the character of a person of his opinions or arguments. It is a tactic used by an adversary when they do not have a logical counter-argument. Example 1: Don't listen to Peter's assertions on instruction, he's a simpleton. Example 2: You can't believe that Presidential candidate is going to lower taxes. He's a liar.

49. FALLACIES Circular Argument (or Petitio Principii ). If a premise of an argument presupposes the truth of its conclusions; meaning, the argument takes for granted what it's supposed to prove. Example 1: Senator Chiz Escudero is a good communicator because he speaks effectively. Example 2: God exists because the Holy Bible says so. The Holy Bible is true. Therefore, God exists.

50. FALLACIES Equivocation . It is an argument used in two or more different senses/meanings within a single argument. Example 1: Giving financial support to charity is the right thing to do. So, charities have the right to our finances. Example 2: Some real numbers less than any number. Therefore, some real numbers are less than itself.

51. FALLACIES Fallacy of Division. A reasoning which assumes that the characteristic of a group is also the characteristic of each individual in the group. Example 1: University of the Philippines is the best university in the country. Therefore, every student from UP is better than any other university in the country. Example 2: Your family is crazy. That means that you are crazy, too.

52. FALLACIES False Dilemma. It is an argument which implies one or two outcomes is inevitable and both have negative consequences, but actually there could be more choices possible. Example 1: If you don't vote for this candidate, you must be antichrist. Example 2: You either broke the glass door, or you did not. Which is it?

53. FALLACIES Hasty Generalization. It is an argument that a general conclusion on a certain condition is always true based on insufficient or biased evidence. Example 1: A MacBook broke after a month, so there must be something wrong in the manufacture of MacBook. Example 2: My cousin said that mathematics subjects were hard, and the one I'm enrolled in is hard, too. All mathematics classes must be hard.

54. FALLACIES Red Herring. It is an argument which introduces a topic related to the subject at hand. It is diversionary tactic to avoid key issues, often way of avoiding opposing argument rather than addressing them. Example 1: Some politicians may be corrupt, but there are corrupt police, corrupt lawyers, and even corrupt leaders of the church. There are also many honest police officers. Therefore, let's put corrupt politicians in perspective. Example 2: I know I forget to clean the toilet yesterday. But nothing I do pleases you

55. FALLACIES Slippery Slope (or snowball/domino theory ). It is an argument which claims a sort of chain reaction, usually ending in some extreme and after ludicrous will happen, but there's really not enough evidence for such assumption. Example 1: If high school students are given 15 minutes rather than 5 minutes break between classes, they'll just start skipping classes. Example 2: If I fail Algebra, I won't be able to graduate. If I don't graduate, I probably won't be able to get a good job, and may very well end up like a beggar.

56. FALLACIES Strawman Fallacy. It is an argument that misrepresents position of the opponent in an extreme or exaggerated form or attacking the weaker and irrelevant portion of an argument in order to make it appear weaker than it actually is. The objective is to refute the misrepresentation of the position, and conclude that the real position has been refuted. Example 1: Congressman who does not support the proposed national minimum wage increase hates the poor. Example 2: A mandatory helmet law for motorcycle drivers could never be enforced. You can't issue tickets to dead people.