MATH 10 Module 2 Logic and Reasoning(2).pdf

Module 2 Logic and Reasoning Introduction Logic plays a very important role in mathematics – it is the foundation on which the discipline is built. Remember all the proofs you had to write to establish some theorems in algebra and...

Module 2 Logic and Reasoning Introduction Logic plays a very important role in mathematics – it is the foundation on which the discipline is built. Remember all the proofs you had to write to establish some theorems in algebra and geometry? You start with the given assumptions, use definitions and perhaps some known results, and argue logically to establish the conclusion. We consider mathematics as a language with its own symbols and “grammar”. These symbols may represent various mathematical objects like numbers, sets, or functions. The grammar will be the rules when combining these symbols. We apply logic to deduce properties of these objects and rules based on some axioms. One cannot overemphasize the importance of logic in mathematics, but logic, or logical reasoning, is just as important in our everyday life. In this era of fake news, post-truths, false advertising, we must be able to discern what is true or false. We should be able to determine if a certain argument is valid or not. Learning Outcomes After studying this module, you should be able to: 1. Determine whether a statement has truth value; 2. Negate simple and compound statements; 3. Describe the various forms of the conditional; 4. Use truth tables to determine the truth value of a statement; 5. Determine whether an argument is valid or invalid using Euler diagrams or truth tables; and 6. Illustrate deductive and inductive reasoning. 1.0 Mathematical Statements and Connectives In this section, we shall study a very basic object in mathematical logic, statements, and operations and relations on statements. These will be important when we discuss valid and invalid arguments. 1.1 Mathematical Statements A mathematical statement is a statement that can be assigned a truth value and classified as true or false, but not both. Lowercase letters, p,q,r,s,...., are used to represent mathematical statements. LOGIC AND REASONING Example 1.1 The following are mathematical statements: p:1+1=2 q:2+3=6 r : All roses are red. s : The Philippines has more than 7,100 islands. Note that p and s are true statements while q and r are false. Note that all the above statements are declarative statements. In other words, mathematical statements are declarative sentences that are either true or false, but not both. We shall not consider declarative statements whose truth value is not clear or a matter of opinion as mathematical statements. Questions, exclamations, and imperatives are not considered as mathematical statements as well, since these sentences do not have a truth value. Example 1.2 The following are NOT mathematical statements: 1) Happy Birthday! 2) Message me. 3) Can we be friends? 4) 5+1 5) x+3=0 6) Mathematics is interesting. 7) 7 is a lucky number. The first three sentences do not have a truth value and note that they are not declarative statements. The expressions “5 + 1” and “x + 3 = 0” may seem to be mathematical statement because they involve mathematical symbols, but they have no truth value, and in fact, have no meaning. However, “For all numbers x, x + 3 = 0” is a mathematical statement which is false. Statements (6) and (7) are declarative statements but are not considered as mathematical statements since they have no definite truth value. In particular, mathematics may be interesting for some people, so it could be true for these people, but false for those who find mathematics boring. In the same manner, some people might consider 7 as their lucky number and some may think of this as false. In fact, the sentence is vague since “lucky” has to be defined. However, the statement “7 is an odd number” is a true mathematical statement. EXERCISES 1.1 Determine whether the following are mathematical statements or not. 1. Math 10 is a GE course. 6. π is a special number. 2. What is your name? 7. The chairs are pink or the earth is round. 3. I am a UP student. 8. 3+4–5 4. x x) or simply 7,000 ≥ 𝑥. This is equivalent to x is at most 7,000. EXERCISES 2.1 Negate the following statements: 1. 100 is a multiple of 10. 2. x+y≤z 3. All trees are tall. 4. There are 12 months in a year. 5. Juan is at least 18 years old. 2.2 Negation of Compound Statements To negate a conjunction or disjunction, we use the following equivalences: 1. ~ ( p ∧ q ) ⇔ ~p ∨ ~q 2. ~ ( p ∨ q ) ⇔ ~p ∧ ~q The first equivalence was established using a truth table in Section 1.4 and the proof of the second rule is left as an exercise. Example 2.2 Negate the following statements: 1. The chairs are red and UP is at least 100 years old. 2. 1 + 1 < 5 or all roses are red. 3. Jose Rizal is both intelligent and nationalistic. Page 11 of 36 July 2018 LOGIC AND REASONING Answers: 1. The chairs are NOT red or UP is less than 100 years old. 2. 1+1 ≥ 5 and not all roses are red. 3. Jose Rizal is not intelligent or he is not nationalistic. EXERCISES 2.2 Negate the following statements: 1. The Philippines is an archipelago or cats are mammals. 2. 1 + 1 = 3 and 5 > 1. 3. I like watching movies and reading books. 4. p ∨ (r ∧q) 5. ( p ∨ q ) ∧ ( ~r ∨ ~s) 3 Negation of Statements with Quantifiers We can also negate statements with the quantifiers all, some, none. Let us consider the following statements with quantifiers and their negation. Example 2.3 Negate the following statements: 1. p: All roses are red. 2. q: Some roses are red. 3. r : No roses are red. Answers: 1. The negation of the statement “All roses are red. ” is simply ~ p: Not all roses are red. This is also equivalent to ~ p : Some roses are not red. 2. Note that “Some roses are red” means there are roses which are red. The negation is: there are no roses which are red, that is, ~ q: No roses are red. This is NOT equivalent to “Some roses are not red.” If some roses are red, this could also mean that some roses are not red. So the negation should be none are red. 3. The negation of “No roses are red” is ~ p: Not all roses are red. Page 12 of 36 July 2018 LOGIC AND REASONING Exercises 2.3 Negate the following statements. 1. All cats are mammals. 2. Some of the islands are not inhabited. 3. No man is an island. 4. All UP students are honest or no politician is corrupt. 5. Some violets are blue and no roses are violet. 3 The Conditional Everyday we have to make decisions, as simple as which route to take in going to school. The decisions we make normally depend on some conditions or premises or information given. For example, it it rains, then you will not walk going to school or not go to class at all. We also encounter a lot of conditional statements or “if – then’s” in advertisements, instructions, arguments, and ordinary conversations. Hence, it is important to understand the conditional and the many forms it takes and statements which are not equivalent to a given conditional. 3.1 Equivalent Forms for the Conditional The conditional p → q or "If p then q" is equivalent to the following statements: q if p. q is necessary for p. p only if q. All p are q. p is sufficient for q. Either not p or q. Example 3.1 Consider the conditional If it is a bird then it flies. The premise is the statement p: It is a bird. The conclusion is the statement q: It flies. The conditional p → q is equivalent to 1. q if p: It flies if it is a bird. 2. p only if q: It is a bird only if it flies. Page 13 of 36 July 2018 LOGIC AND REASONING 3. p is sufficient for q: Being a bird is sufficient for it to fly. 4. q is necessary for p. Flying is necessary for it to be a bird. 5. All p are q. (This is referred to as a universal statement) All birds fly. 6. Either not p or q. Either is not a bird or it flies. EXERCISES 3.1 1. Give five (5) equivalent statements for the following: a) If you care for the environment, then you should recycle. b) All animals are friendly. c) No insect is useless. 2. Using the equivalent form “~p ∨ q” of the conditional, determine the negations of p → q and verify this using a truth table. 3. State the negation of the statements in (1). 3.2 The Converse, Inverse, and Contrapositive A common mistake in restating conditionals is interchanging the premise and the conclusion. Consider the conditional p → q : If it is a bird then it flies. Which of the following statements is equivalent to p → q? q → p : If it is a bird then it flies. ~p → ~q : If it is not a bird then it does not fly. ~q → ~p : If it does not fly then it is a bird. The correct answer is ~q → ~p which is called the contrapositive of p → q. Let us use a truth table to compare the truth values of p → q, q → p, ~p → ~q, and ~q → ~p. p q p→q q→p ~p → ~q ~q → ~p T T T T T T T F F T T F Page 14 of 36 July 2018 LOGIC AND REASONING F T T F F T F F T T T T Hence, we have the equivalent statements: p → q ⇔ ~q → ~p q → p ⇔ ~p → ~q Definition : Given the conditional p → q, we call 1. ~p → ~q the contrapositive of p → ; 2. q → p the converse of p → q; 3. ~p → ~q the inverse of p → q. Remark : From the above truth table, we have seen that the conditional is equivalent to its contrapositive and not equivalent to its converse and inverse. Example 3.2 : Consider the statement: All even numbers are divisible by two. Formulate the statement as a conditional and give its converse, inverse and contrapositive. Answer: Conditional : If it is an even number, then i is divisible by two. Converse : If a number is divisible by two, then it is an even number. Inverse : If it is not an even number then it is not divisible by two. Contrapositive: If it is not divisible by two, then it is not an even number. EXERCISES 3.2 1. Which of the following statements is/are equivalent to “All Filipinos are law-abiding.” A. If you are a Filipino, then you are law-abiding. B. If you are not a Filipino, then you are not law-abiding. C. If you are not law-abiding, then you are not a Filipino. D. If you are law-abiding, then you are a Filipino. E. You are law-abiding if you are a Filipino. F. You are law-abiding only if you are a Filipino. Page 15 of 36 July 2018 LOGIC AND REASONING 2. Which of the following statements is/are NOT equivalent to “No gorilla is playful.” A. If you are playful, then you are a gorilla. B. If you are a gorilla, then you are not playful. C. Either you are playful or you are a gorilla. D. If you are not a gorilla, then you are playful E. If you are playful then you are not a gorilla. F. You are not playful only if you are a gorilla. 3. Give the contrapositive, inverse, and converse of the statements in Exercise 3.1. 4. ACTIVITY: Give five (5) examples of conditional statements coming from advertisements (TV, radio, magazines or newspapers). 4 Valid and Invalid Arguments Charles Dodgson, author of Alice’s Adventures in Wonderland and better known as Lewis Caroll, was not only a famous writer. He was also a mathematician and logician. He created numerous amusing puzzles with absurd implications and nonsensical statements to train people on logical reasoning. The following are puzzles by Lewis Carrol: Puzzle 1: All babies are illogical. Nobody is despised who can manage a crocodile. Illogical persons are despised. Puzzle 2: No kitten that loves fish is unteachable. No kitten without a tail will play with a gorilla. Kittens with whiskers always love fish. No teachable kitten has green eyes. No kittens have tails unless they have whiskers. The objective in each puzzle is to draw a conclusion based on all the premises given. Logical reasoning is important in puzzle and problem solving, as it provides a systematic way to come up with a solution. We are also regularly inundated by arguments or reasoning which may seem nonsensical or puzzling. We also encounter phrases like : God is love. Love is blind. Therefore, God is blind. This seems logical or valid, but is it? Hence, logical reasoning should be applied when we discern the validity or soundness of arguments. In this section, we discuss valid and invalid arguments, common forms or valid arguments and some examples of fallacies we could encounter in everyday life. Page 16 of 36 July 2018 LOGIC AND REASONING 4.1 Euler Diagrams An Euler diagram represents statements, the way Venn diagrams represent sets. For example, consider the statement p: Jose is a UP student. The statement involves inclusion in a set, that is, Jose is an element of the set of UP students. We can use a circle to represent the set of UP students and if we let x represent Jose, the Euler diagram of statement p is given by : UP students x We can also represent statements with the quantifiers all, some, and none. Example 4.1. Draw an Euler diagram for the following statements: 1. All cats are mammals. 2. Some dogs are hairy. 3. No even number is an odd number. Answers: 1. All cats are mammals. This statement involves two sets: set of cats and set of mammals, represented by circles. Since all cats are mammals, this means the set of cats is contained in the set of mammals. The diagram is given by: mammals cats The conditional "If it is a cat then it is a mammal" is equivalent to "All cats are mammals" so it is represented by the same diagram. 2. Some dogs are hairy. This statement involves the set of dogs and the set of hairy things (which may include animals). The quantifier "some" signifies that there may be dogs which are not hairy and there may be hairy things which are not Page 17 of 36 July 2018 LOGIC AND REASONING dogs. This is represented by two intersecting sets and the region where they intersect contains the set of dogs which are hairy. dogs hairy things hairy dogs Hairy dogs 3. No even number is an odd number. We consider the sets of even numbers and odd numbers. The quantifier "none" means the two sets must not intersect, otherwise, there will be even numbers which are odd. even odd numbers numbers In general, we have the following Euler diagrams for statements with quantifiers all, some, or none involving members of two sets A and B: All A are B. Some A are B. A B A B Page 18 of 36 July 2018 LOGIC AND REASONING No A is B. A B EXERCISES 4.1 Draw an Euler Diagram for the following statements: 1. Juan is a Math 10 student. 2. Maria is not a Filipino. 3. All actors are artists. 4. Some scientists are actors. 5. No professor is infallible. 6. Ducks are yellow. 7. If you are a Filipino, then you are honest. 4.2 Valid and Invalid Arguments An argument consists of premises, say p1, p2,..., pn, and a conclusion q and consider the conjunction p1∧p2∧... ∧pn = p. The argument p→q is valid if the premises are assumed to be true, then the conclusion must also hold true. That is, the statement p→q is an implication. We can use Euler diagrams to determine the validity of an argument. We construct a diagram which represents the premises. The argument is valid if the conclusion is satisfied by the Euler diagram representing all premises. Note that the premises are assumed to be true although the statements may not be true in the strict sense. Example 4.2. Consider the argument: All dogs are hairy. Cotton is a dog. Therefore, Cotton is hairy. To determine if it is valid, we draw an Euler diagram for the premises. The first premise is represented by two sets, the set of dogs is inside the set of hairy things: Page 19 of 36 July 2018 LOGIC AND REASONING hairy things dogs and the second premise, we have Cotton (represented by x) inside the set of dogs: dogs x The premises, considered together, are represented by the following diagram: \ hairy things dogs x Since x is in the set of the smaller set (dogs), it follows that s is also in the set of hairy things. This means the conclusion "Cotton is a dog" follows naturally from the premises so the argument is valid. To show an argument is invalid, it suffices to exhibit an Euler diagram satisfying all the premises but not the given conclusion. Example 4.3. Determine the validity of the argument All dogs are hairy. My pet Cotton is hairy. Therefore, Cotton is a dog. Answer: The first premise is the same as the first premise in Example 4.2 so the Euler Diagram is the same. On the other hand, the second premise “Cotton is hairy” hairy things represented by the diagram on the right (x represents Cotton). x Page 20 of 36 July 2018 LOGIC AND REASONING Observe that the diagram below satisfies the two premises: hairy things dogs x However, the conclusion “Cotton is a dog” is contradicted in the diagram. This means that being hairy does not automatically imply being a dog. One may be hairy without being a dog. Hence, the argument is INVALID. Remark: One may also show that an argument is invalid by exhibiting two different diagrams representing the premises. In the above example, we have the two diagrams: hairy things hairy things dogs dogs x x which both satisfy the two premises given but give two possible conclusions. In the diagram on the left, Cotton (x) is a dog and also hairy, but in the second diagram, Cotton is hairy but is not a dog. If an argument is valid, there should only be one possible conclusion. Let us look at more examples. Example 4.4. Determine the validity of the following arguments: 1. All dogs are hairy. My pet Donut is not hairy. Therefore, Donut is not a dog. 2. All cats are mammals. My pet Donut is not a cat. Therefore, Donut is not a mammal. 3. All cats are mammals. All mammals are animals. Therefore, all cats are animals. 4. All parrots are birds. Some birds are colorful. Therefore, some parrots are colorful. Page 21 of 36 July 2018 LOGIC AND REASONING 5. All parrots are birds. Some parrots fly. Therefore, some birds fly. 6. All parrots are birds. All cats are mammals. No bird is a mammal. Therefore, no parrot is a cat. Solutions: 1. This argument is VALID. The two premises are represented by the following diagram, with D representing Donut. hairy things D dogs Since Donut is not in the set of hairy things, it follows that Donut is also not in the set of dogs. Therefore, the conclusion is implied by the premises. Take note that the premise given “All dogs are hairy” is not really a true statement since some dogs are not hairy, but we evaluate the validity of the argument and NOT the truth value of the conclusion. 2. The argument is INVALID. Although Donut is not a cat, it does not automatically follow that Donut is not a mammal. This can be illustrated with a diagram: mammals cats D which satisfies both premises but not the conclusion. 3. The two premises are represented by: All cats are mammals. All mammals are animals. animals mammals Page 22 of 36 July 2018 mammals cats LOGIC AND REASONING The Euler diagram representing both premises is given by: A A - animals M M - mammals C C - cats The diagram shows that any element of the set of cats is automatically an element of the set of animals. Hence, the argument is VALID. 4. We have the two Euler diagrams for each premise All parrots (P) are birds (B). Some birds (B) are colorful (C). B B C P If both premises are considered, we have the Euler diagram B P C which satisfies the two premises but not the conclusion. That is, it does not follow that some parrots fly. Hence, the argument is invalid. 5. The following diagram represents the two premises: All parrots (P) are birds (B). Some parrots (P) fly (F). Page 23 of 36 July 2018 LOGIC AND REASONING B P F Since (F) intersects (P), it automatically intersects (B). That is, the parrots which fly are the birds which fly. Hence, the argument is valid. 6. The premises are represented by the diagram All parrots are birds. All cats are mammals. No bird is a mammal. Therefore, no parrot is a cat. All parrots (P) are birds (B). All cats (C) are mammals (M). B M P C The diagrams, taken together, satisfy the third premise as well, that no bird is a mammal. As can be seen, the conclusion that no parrot is a cat follows from the three premises. Therefore, the argument is valid. Remark: As mentioned in the first example, the argument may be valid even if the premises are not universally true. That is, the argument is valid but not “sound”. In fact, we can have premises that may be meaningless, but the conclusion (which is also meaningless) can still follow logically if the premises are assumed to be true. Let us consider the following argument: Example 4.5. All booms (B) are zooms (Z). All feeps (F) are meeps (M). No zoom is a meep. Therefore, no boom is a feep. The argument is valid. We have following diagram representing the premises: Z M Page 24 of 36 July 2018 B F LOGIC AND REASONING Note that since no zeep is a meep, this will ensure that no boom is a feep. This argument is actually in the same form as the argument given in Example 6 above. EXERCISES 4.2 1. Determine if the following arguments are valid or invalid. Justify your answers by drawing a diagram. a. All Filipinos enjoy singing. Juan is a Filipino. Therefore, Juan enjoys singing. b. Some physicists are poets. Einstein is a physicist. Therefore, Einstein is a poet. c. All lions are animals. Some lions have manes. Therefore, some animals have manes. d. All parrots are birds. Some birds are colorful. Therefore, some parrots are colorful. e. All booms (B) are zooms (Z). All feeps (F) are meeps (M). No boom is a feep. Therefore, no zoom is a meep. 2. Consider the following premises : Every sane person can do logic. No cat can do logic. Those who are not sane cannot serve in the government. Which of the following is not a valid conclusion? a) Cats cannot serve in the government. b) No cat is sane. c) Logical persons cannot serve in the government. d) Those who serve in the government are not logical. 3. Consider the following premises: All physicists are scientists. Some scientists are artists. All mathematicians are artists. Some physicists are mathematicians. No scientist is illogical. Which is a valid conclusion? a) No physicist is illogical. b) Some mathematicians are illogical. c) Some physicists are artists. 4. Find a valid conclusion for the following Lewis Carrol puzzles (HINT: Express the given statements in conditional form): a. Babies are illogical. Nobody is despised who can manage a crocodile. Illogical persons are despised. Page 25 of 36 July 2018 LOGIC AND REASONING b. No ducks waltz. No officers ever decline to waltz. All my poultry are ducks. c. No kitten that loves fish is unteachable. No kitten without a tail will play with a gorilla. Kittens with whiskers always love fish. No teachable kitten has green eyes. No kittens have tails unless they have whiskers. 4.3 Valid Argument Forms Let us look at the valid arguments given in previous examples: 1. All dogs are hairy. Cotton is a dog. Therefore, Cotton is hairy. 2. All dogs are hairy. My pet Donut is not hairy. Therefore, Donut is not a dog. 3. All cats are mammals. All mammals are animals. Therefore, all cats are animals. Arguments of the above form are always valid and satisfy valid argument forms called modus ponens, modus tollens, and syllogism, respectively. To define the form of these arguments, recall that the conditional p → q is equivalent to the universal statement “All p are q.” So we have can restate the above as All p are q. p→q All dogs are hairy. If it is a dog, then it is hairy. All cats are mammals. If it is a cat, then it is a mammal. All mammals are animals If it is a mammal, then it is an animal. We have the following general forms: Modus ponens If it is a dog then it is hairy. p→q Page 26 of 36 July 2018 LOGIC AND REASONING Cotton is a dog. p Therefore, Cotton is hairy Therefore q Modus tollens If it is a dog then it is hairy. p→q My pet Donut is not hairy. ~q Therefore, Cotton is a dog. Therefore ~p Syllogism If it is a cat, then it is a mammal. p→q If it is a mammal then it is an animal. q→r Therefore, if it is a cat, then it is an Therefore p → r animal. We can use truth tables to show the validity of these arguments. In particular, we find the truth value of the conditional whose premise is the conjunction of the premises and the conclusion is the given conclusion in the argument, that is, 1. (Modus ponens) [(p →q) ∧ p] → q 2. (Modus tollens) [(p → q) ∧ ~q] → ~p. 3. (Syllogism) [(p →q) ∧ (q → r) ] → (p → r) If the statement is true for each of the four cases in the truth table, then the argument is valid. In this case, we call the statement a tautology and the conditionals given are in fact, implications. We have the following truth tables: 1. Modus ponens p q p→q (p → q ) ∧ p [(p →q) ∧ p] → q Page 27 of 36 July 2018 LOGIC AND REASONING T T T T T T F F F T F T T F T F F T F T 2. Modus Tollens p q p→q (p → q) ∧ ~ q [(p → q) ∧ ~q] → ~ p T T T T T T F F F T F T T F T F F T F T 3. Syllogism (Note we have three statements p, q, r.) (p → q) ∧ [(p → q) ∧ (q→r)] p q r p→q q→r p→r (q→r) → (p→r) T T T T T T T T T T F T F F F T T F T F T F T T T F F F T F F T F T T T T T T T F T F T F F T T F F T F T F T T F F F F T F T T Example 4.6. Verify that the following arguments are valid and determine if they are of the given valid argument forms (modus ponens, modus tollens, or syllogism). Page 28 of 36 July 2018 LOGIC AND REASONING 1. All UP students need to enroll in Math 10. Maria is a UP student. Therefore, Maria has enroll in Math 10. 2. If I sing, then it rains. It is not raining. Therefore, I did not sing. 3. If you love cats, then you love animals. If you love animals, then you should be kind to animals. Therefore, if you love cats, you should be kind to animals. Answers: The given arguments are valid and this can be shown using Euler Diagrams (left as an exercise). Arguments 1, 2, 3 are examples of a modus ponens, modus tollens, and syllogism, respectively. EXERCISES 4.3 1. Verify that the following arguments are valid and classify according to the type of valid argument form. a) If it rains, I will sleep. I did not sleep. Therefore, it did not rain. b) If it’s a reptile, then it’s cold-blooded. Barney is not cold-blooded. Therefore, Barney is not a reptile. c) ~All scientists are hardworking. If you are hardworking, then you contribute to our country’s economic growth. Therefore, all scientists contribute to our country’s economic growth. 2. Using Euler Diagrams, establish the validity of the modus pones, modus tollens and syllogism. 4.4 Fallacies Consider the two invalid arguments previously discussed: 1. All dogs are hairy. My pet Cotton is hairy. Therefore, Cotton is a dog. 2. All cats are mammals. My pet Donut is not a cat. Therefore, Donut is not a mammal. These are examples of two invalid argument forms, the fallacy of the converse and the fallacy of the inverse, respectively. In symbols, these are of the form: Fallacy of the Converse Fallacy of the Inverse p→q p→q Page 29 of 36 July 2018 LOGIC AND REASONING q ~p Therefore p Therefore ~ q Let us look at the truth tables for these two arguments. We consider the two conditionals {[(p → q) ∧ q] → p} and { [(p → q) ∧ ~p] → ~q}. 1. (Fallacy of the Converse) [(p → q) ∧ q] → p p q p→q (p → q ) ∧ q [(p →q) ∧ q] → p T T T T T T F F F T F T T T F F F T F T 2. (Fallacy of the Inverse) [(p → q) ∧ ~p] → ~q p q p→q (p → q) ∧ ~p [(p → q) ∧ ~p] → ~q T T T F T T F F F T F T T T F F F T F T Observe that unlike the valid argument forms, the arguments given above are not tautologies since they are not true for each of the four cases. They are called fallacies. The argument {[(p →q) ∧ q] → p} is false when p is true and q is false, and {[(p → q) ∧ ~p] → ~q} is false when p is false and q is true. Example 4.6. Verify that the following arguments are invalid and determine whether the argument is an example of the fallacy of the converse or the fallacy of the inverse. 1. If you can add, then you can subtract. You cannot add. Therefore, you cannot subtract. Page 30 of 36 July 2018 LOGIC AND REASONING 2. All UP students are diligent. Maria is diligent. Therefore, Maria is a UP student. 3. If I sing, then it rains. If it does not rain, then birds sing. Birds are not singing. Therefore, I am singing. Answers: The above statements can be shown to be invalid using Euler diagrams. Arguments 1 and 3 are fallacies of the converse while argument 1 is an example of a fallacy of the inverse. In argument 3, the premise “Birds are not singing” implies that it is raining (by the contrapositive of the second premise). If it rains, no conclusion may be drawn since the given premise is “If I sing, then it rains.” Other fallacies We also have fallacies other than the forms given above. In fact, we probably encounter these fallacies more frequently: 1. Ad Hominem. The argument is based on the character of the opponent instead of the argument itself. This may also involve insulting the opponent to make opponent’s argument seem false. Example: Maria wears leather shoes, so she cannot be a vegetarian. 2. Ad Populum. This fallacy occurs when an argument is assumed to be valid since many people believe in it. Example: Surveys indicate that 55% of the population believe that use of contraceptives is dangerous. Therefore, contraceptives should be banned. 3. Appeal to Authority. The argument is claimed to be valid because a famous or famous person, who is not an expert in the pertinent field, supports it. Example: Brand X is the best toothpaste since our president uses it. 4. False Cause. This fallacious argument correlates two events, even if unrelated. Example: Every time I go to sleep, the sun goes down. Therefore my going to sleep causes the sun to set. 5. Hasty Generalization. In this fallacy, a generalization is made based on a few examples supporting the claim. Example: I don’t excel in tennis. I also don’t excel in volleyball. Therefore, I don’t excel in sports. Exercises 4.4 1. Verify that the following arguments are invalid and determine whether the argument is an example of the fallacy of the converse or the fallacy of the Page 31 of 36 July 2018 LOGIC AND REASONING inverse: a. All birds have feathers. I am not a bird. Therefore, I have no feathers. b. If you are smart, then you are successful. You are successful. Therefore you are smart. 2. Give your own example of the fallacies discussed. 3. Give at least three (3) other fallacies aside from the forms discussed in this section and give an example for each. You may refer to these sources: a. https://www.youtube.com/watch?v=0RyVj2FPGyg b. https://www.youtube.com/watch?v=VDGp04CfM4M c. http://www.don-lindsay-archive.org/skeptic/arguments.html d. https://www.huffingtonpost.com/mario-livio/logical- fallacies_b_1932906.html 4. ACTIVITY : Give five (5) examples of fallacies found in advertisements or newspaper/magazine articles and explain why these are fallacies and how they are used to mislead. 4 Inductive and Deductive Reasoning Problem solving is done not only in formal mathematics courses or limited to mathematicians and scientists. Almost any profession or discipline requires problem solving. In addition, if you love solving sudoku puzzles, rubiks cube, logic, or pattern problems, these recreational activities are also forms of problem solving which basically involves finding a conclusion or answer from known facts. Correct reasoning is important when we solve problems We discuss here two processes of reasoning, deductive and inductive, and use these to solve some problems. Deductive reasoning (or logical deduction) is the process of reasoning from a general statement to a specific instance. We have seen this in the valid argument forms given in the previous sections. The classic example of a deductive argument: All men are mortal. Socrates is a man. Therefore, Socrates is mortal. is the primary form of deductive reasoning, which is the valid argument form modus ponens. The statement “ All men are mortal” is a general statement and the conclusion is deduced from this and the specific instance that Socrates is a man. The modus tollens and syllogism are also forms of deductive reasoning. Inductive reasoning is the process of reasoning from specific instances to a general statement. The following is an example of an inductive argument: Page 32 of 36 July 2018 LOGIC AND REASONING 2 is an even number. 12 is an even number. 22 is an even number. Therefore, all numbers ending in 2 are even. Although the conclusion drawn in this example is true, this is not always the case when inductive reasoning is employed. Look at numbers of the form 2p- 1, where p is a prime number (p = 2, 5, 7, …). We have 22 - 1 = 4 – 1 = 3 23 - 1 = 8 – 1 = 7 25 - 1 = 32 – 1 = 31 Note that 3, 7, 31 are prime numbers. By inductive reasoning, we can conclude that 2p- 1 is a prime number when p is a prime number. However, this is false. The number 211- 1 = 2048 – 1 = 2047 is not prime since it is the product of 23 and 89. (Trivia: Prime numbers which can be expressed in the form 2p- 1 where p is a prime number are called Mersenne primes, named after the French scholar Marin Mersenne who took interest in prime numbers of this form. The search for Mersenne primes continues and the largest known as of December 2017 is 277,232,917 – 1.) Hence, When a general statement is concluded from specific examples using inductive reasoning, this still has to be formally established or proved using kown facts, in which case, deductive reasoning is necessary. Exercises 5.1 1. Determine whether the following arguments use inductive or deductive reasoning. a. All Filipinos are nationalistic. Jose Rizal is a Filipino. Hence, Jose Rizal is nationalistic. b. Math 10 is easy. Math 20 is easy. Therefore, all math courses are easy. c. In a mystery case, it is known that Jose did it or Maria did it. Maria did not do it. Therefore, Jose did it. d. Ducks do not waltz. I can waltz. Therefore, I am not a duck. 2. Activity: Circles and Chords. Given a circle, if you mark n points on the circle, what is the maximum number of chords (a chord is line joining two points on a circle) that can be drawn? What is the maximum number of regions formed? Investigate the case for n = 1, 2, 3, 4, 5, 6. Can you Page 33 of 36 July 2018 LOGIC AND REASONING generalize the results? Let us illustrate, for n = 2, 3 and do the case for 1, 4, 5, 6 Number of points (marked x) Max. no. of Max. no. of chords regions n=2 x 1 2 x n=3 x x 3 4 x Example 5.1. Place the numbers 1, 2, 3, 4, 5, 6 in the circles in the figure below, such that each side totals 12. Solution: The problem involves writing the numbers 1, 2, 3, 4, 5, 6 such that if you add the numbers on one side of the triangle formed by the singles, the sum should be 12. This is a problem where guess and test may be employed. However, if you just guess and write the numbers randomly, solving it might take some time until you “guess” the correct solution. Since the sum is 12, you cannot place the smaller numbers 1, 2, 3 on the same side. Moreover, to get a sum of 12, you will need exactly two of the higher numbers 4, 5, 6, but not all three, on one side. Hence, 4, 5, 6 will be on the vertices of the triangle and 1, 2, 3 may now be placed in the appropriate circles. The solution is. 4 3 2 Page 34 of 36 July 2018 LOGIC AND REASONING 5 1 6 Example 5.2. Find the next number in the sequence 1, 1, 2, 3, 5, 8, ?. Solution: This problem, and in general, finding the next number in a sequence, involves pattern finding. In this case, you could have observed that 1 + 1 = 2, 1 + 2 = 3, 2 + 3 = 5, 3 + 5 = 8. So the next number is obtained by adding the two previous numbers. So the answer is 13. This sequence is called the Fibonacci sequence. Exercises 5.2 Solve the following problems and explain how you arrived at your solution using Polya’s process. 1. Find the next two (2) numbers, x and y, in the given sequences: a. 1, 4, 9, 16, 25, x, y b. 3, 9, 15, 21, 27, x, y c. 3, 15, 75, x, y d. 1, 6, 15, 28, 45, x, y e. 2, 6, 22, 56, 114, x, y 2. Three school children Junie, Glory, and Mickey are sitting side by side. Junie always tells the truth, Glory sometimes tells the truth and Mickey never tells the truth. The child on the left says “Junie is in the middle”. The child in the middle says “I’m Glory”, and the child on the right says “Mickey is in the middle”. Determine the seating arrangement of the three. 3. In a certain jungle, there are three tribes: Tribe T, Tribe L, Tribe X. Members of Tribe T always tell the truth, Tribe L members never tell the truth and Tribe X members sometimes tell the truth and they sometimes lie. If you meet one tribe member and he tells you, “I always lie,” which tribe does he come from? If you meet another tribe member and he says, “I sometime lie,” which tribe can he come from? 4. Jose must take a cat, a mouse, and a sack of rice across a river with his boat. The boat to be used can only accommodate Juan and either the cat, mouse or the sack of rice. However, if left together, the cat will eat the mouse. Also, if the mouse is left alone with the rice, it will eat the rice. The cat does not eat rice. The mouse and rice are safe when Jose is present. What is the minimum number of times Jose needs to cross the river so he could get everything across? 5. Four children, Amy, Susie, Tessie, and Eddie are lined up according to height, each holding a balloon. The child in front (the shortest) is holding Page 35 of 36 July 2018 LOGIC AND REASONING neither a red nor blue balloon. Susie is holding a red balloon. Tessie sees exactly two balloons in front of her. The child holding the blue balloon is right in front of the child with the yellow balloon. Amy is in front of Tessie. One child is holding a white balloon. Determine the arrangement of the four children (from shortest to tallest) and the color of the balloon they are holding. References: 1. Mathematical Ideas, C.D. Miller, V. Heeren, J. Hornsby, C. Heeren, Pearson, 2015. 2. Mathematics in Life, Society, and the World, H. Parks, G. Musser, R. Burton,Hornsby W. Siebler, Prentice Hall, 1977. 3. Mathematics: A Human Endeavor, H.R. Jacobs, W.H. Freeman and Co., 1977. 4. An Introduction to Mathematical Reasoning, P. Eccles, Cambridge University Press, 1998. 5. Sources from the web: a. https://owlcation.com/humanities/Logical-Fallacies-Logical-Fallacies- and-How-They-Are-Used b. http://www.math.hawaii.edu/~hile/math100/logice.htm c. https://www.huffingtonpost.com/mario-livio/logical- fallacies_b_1932906.html d. http://www.fallacydetective.com/articles/categoriescategory/fallacies/ Page 36 of 36 July 2018