Introduction to Logic:Problems and Solutions, Exercises of Logic

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Introduction to Logic:

Problems and solutions

Contents

I Informal Logic 5

1 Theory of Argumentation 7 1.1 Lecture 1: Identification of Arguments...................... 7 1.2 Lecture 2: Non- arguments............................. 7 1.2.1 Identify arguments from the following passages............ 7 1.3 Lecture 3: Types of Arguments: Deductive vs Inductive............ 9 1.4 Lecture 4:Nature and Scope of Deductive and Inductive Arguments..... 9 1.4.1 Is the following argument best classified as deductive or inductive?. 9 1.5 Lecture 5: Truth, Validity and Soundness..................... 10 1.6 Lecture 6: Strength of Inductive arguments, Counter example method.... 10 1.6.1 Construct counter examples for the following invalid arguments... 10 1.6.2 Evaluate the following Deductive and Inductive Arguments...... 10 1.6.3 Answers................................... 11 1.7 Lecture 7: Toulmin’s Model of Argumentation.................. 12 1.7.1 Identify claim, support, warrant, rebuttal in the following arguments 12

2 Fallacies 13 2.1 Lecture 8: Identification of Formal and Informal Fallacies........... 13 2.1.1 Determine whether the fallacies committed by the following argu- ments are formal fallacies or informal fallacies.............. 13 2.2 Lecture 9: Informal Fallacies: Fallacies of relevance............... 14 2.3 Lecture 10: Fallacies of Weak Induction and Fallacies arising out of ambiguity in Language..................................... 16 2.3.1 Identify the fallacies of weak induction committed by the following arguments, giving a brief explanation for your answer. If no fallacy is committed, write no fallacy......................... 16 2.3.2 Identify the fallacies of presumption, ambiguity, and grammatical anal- ogy committed by the following arguments, giving a brief explanation for your answer. If no fallacy is committed, write no fallacy....... 17

II Traditional Logic 19

3 Syllogistic Logic 20 3.1 Lecture 11: Introduction and motivation for Syllogistic Logic......... 20 3.1.1 Name the form of each of the following categorical statements ( A , E , I , or O ). Identify the subject and predicate terms in each case. Then state the quantity (universal or particular) and quality (affirmative or negative).................................... 20

4.5.2 Symbolize and construct proofs for the following valid arguments us-

 - 4.5.1 Check Consistency of the group of statements - ing Semantic Tableaux Method - fying each step that is not a premise, using Semantic Tableaux Method: 4.5.3 Construct proofs for the following more challenging problems, justi- 

• 4.6 Lecture 20: Knights and Knaves Puzzles
  • 4.6.1 Knights and Knaves Puzzles
• 4.7 Lecture 21:Semantic Tableaux Method: Further Examples
  • 4.7.1 See Scanned copy for Lady or Tiger Puzzles
• 4.8 Lecture 22 Natural Deduction Method
• 4.9 Lecture 23: Natural Deduction: Examples - sequence in the following arguments 4.9.1 Using Natural Deduction method, show that the following logical con-
• 4.10 Lecture 24: Conjunctive and Disjunctive Normal Forms - terms: 4.10.1 Find a disjunctive normal form for each of the following propositional
• 4.11 Lecture 25 CNF, DNF and satisfiability and Validity:
• 4.12 Lecture 26 Resolution and refutation method - guments are valid or invalid 4.12.1 Using resolution refutation method, show whether the following ar-
• 4.13 Lecture 27 Resolution and refutation method: Examples
• 4.14 Lecture 28: Axiomatic Propositional Logic
• 4.15 Lecture 29:Hlbert Ackermann Axiomatic system - tem 4.15.1 Prove the following theorems using Russell-Whitehead Axiomatic Sys-
• 4.16 Lecture 30:Proofs in the PM system
  • 4.16.1 See Scanned copy for problems and solutions
• 4.17 Lecture 31: Hilbert and Ackermann System
• 4.18 Lecture 32: Characteristics of formal system PM
• 5 Predicate Logic
  • 5.1 Lecture 33: Outlines of Predicate Logic
    • 5.1.1 Symbolize the following in Predicate Logic
  • 5.2 Lecture 34: Building blocks of Predicate Logic
    • 5.2.1 Which of the following are sentences in Predicate Logic
    • 5.2.2 Answers
  • 5.3 Lecture 35: Quantifiers, freedom, bondage
    • 5.3.1 Determine free and bound variables of next formulas
  • 5.4 Lecture 36: Translation in to predicate Logic
    • 5.4.1 Translating simple syllogistic sentences
  • 5.5 Lecture 37: Semantics of Predicate Logic
  • 5.6 Lecture 38:Truth, satisfiability, validity in Predicate Logic
  • 5.7 Lecture 39: Formation Trees for wff’s in predicate Logic
  • 5.8 Lecture 40: Semantic Tableaux Method for Predicate Logic - ments are valid or invalid? 5.8.1 Using Semantic Tableaux method show whether the following argu-
  • 5.9 Lecture 41: Semantic Tableaux method: Satisfiability, Validity
    • 5.9.1 Which of the following wff in PL are valid?
    • 5.9.2 Answers for5.9.1
    • 5.9.3 Use semantic tableaux to prove the following claims
    • 5.9.4 See answers in the scanned copy
  • 5.10 Lecture 42: Natural Deduction in Predicate Logic
    • 5.10.1 See section Scanned copy for questions and Answers
  • 5.11 Lecture 43: Important theorems in First order Logic
  • 5.12 Lecture 44: Limitations of first order logic and Introduction to the course
    • 5.12.1 Some Questions
    • 5.12.2 Answers
  • 5.13 References
• IV Description and Syllabus
• 6 About the Course
  • 6.1 Description of Course:
  • 6.2 Syllabus

Chapter 1

Theory of Argumentation

1.1 Lecture 1: Identification of Arguments

1.2 Lecture 2: Non- arguments

1.2.1 Identify arguments from the following passages

1. If you want to find a good job, you should work hard. You do want to find a good job. So you should work hard.
2. Once upon a time there was a prince and a princess. They lived happily together and one day they decided to have a baby. But the baby grew up to be a nasty and cruel person and they regret it very much.
3. Cutting the interest rate will have no effect on the stock market this time round as people have been expecting a rate cut all along. This factor has already been reflected in the market.
4. For a long time, astronomers suspected that Europa, one of Jupiter’s many moons, might harbor a watery ocean beneath its ice-covered surface. They were right. Now the technique used earlier this year to demonstrate the existence of the European ocean has been employed to detect an ocean on another Jovian satellite, Ganymede, according to work announced at the recent American Geo-physical Union meeting in San Francisco. The Economist 16.12.
5. Scientific discoveries are continually debunking religious myths. Further, science pro- vides the only hope for solving the many problems faced by humankind. Hence, sci- ence provides a more accurate view of human life than does religion.
6. India’s wait for an Olympic gold medal in an individual event is finally over! Shooter Abhinav Bindra fired his way to victory in the 10-meter air rifle event on Monday, giving India her first gold medal in Beijing 2008 Olympics. Bindra’s triumph gives India her first Olympic gold medal ever in an individual event and a gold medal in any Olympic event after 28 years.
7. It is wrong for society to kill a murderer. This follows for the reason that if a murderer is wrong in killing his victim, then society is also wrong in killing the murderer. And a murderer is wrong in killing his victim.

8. Since particle like behavior and wave like behavior are the only properties that we ascribe to light, and since these properties now are recognized to belong not to light itself, but to our interaction with light,... it appears that light has no properties inde- pendent of us! To say that something has no properties is the same as saying that it does not exist. The next step in this logic is inescapable. Without us, light does not exist. [Gary Zukav, The Dancing Wu Li Masters (New York: Bantam Books, 1979), p. 95]
9. The earth is getting warmer. Why? There are many reasons, but here are two im- portant ones. First, the burning of coal, oil, and natural gas has greatly increased the carbon dioxide in the atmosphere. And carbon dioxide retains heat. Second, chlo- rofluorocarbons, which are used in air conditioners and refrigerators, have attacked the ozone layer, thus leaving the earth exposed to ultraviolet rays from the sun.
10. If inflation is receding, the government’s economic policies are sound. Inflation is indeed receding. Therefore, the government’s economic policies are sound.
11. If I were you, I would accept the fact that we are not free. In fact, you would be wise to stop treating others with the respect that real freedom would require and simply approach others as stimulus and response machines, to be manipulated for your own gain.
12. Make sure that you follow the guidelines to live a ethical life. 1. Begin each day with a prayer. 2. Work hard. 3. Love your family. 4. Make light of your troubles. 5. Follow the Golden Rule. 6. Read from the scriptures. 7. Show kindness. 8. Read worthwhile books. 9. Be clean and pure. 10. Have charity in your heart. 11. Be obedient and respectful. 12. End the day in prayer. These twelve rules, the “Quaker Dozen,” were written long ago in a family Bible. But I believe they still fit today’s prob- lems. (Adapted from Olive Ireland Theen, “Grandfather’s Quaker Dozen,” in William Nichol, ed., A New Treasury of Words to Live By, 1959)[Statement of Belief/Opinion]
13. item Although you usually cannot eliminate the personal feelings that are influencing your perceptions, you can become aware of them and try to compensate for their bias. For instance, if you are asked to evaluate a group of people, one of whom is a good friend, you should try to keep these personal feelings in mind in order to make your evaluation as accurate as possible (John Chaffee, The Thinker’s Way, 1998)

Answers

1. 1. Argument; Modus Ponens (MP).
2. 2. Non-Argument, Non Inferential, and is a chronological description of facts com- posed of statements but no premise or conclusion.
3. 3. Argument: The conclusion is that this time, cutting interest rate will have no effect on the stock market.
4. Not an argument. A Report
5. Argument; Conclusion: Science provides a more accurate view......
6. Non-Argument; Report.
7. Argument; Conclusion: it is wrong for society kill a murderer.

Answers

1. Deductive Argument; Modus Ponens.
2. Inductive: Conclusion probably follows form the premises.
3. Inductive, Weak Analogy
4. Deductive; True by definition.
5. Inductive, Strong
6. Inductive argument.
7. Inductive Argument
8. Deductive Argument.
9. Deductive Argument

1.5 Lecture 5: Truth, Validity and Soundness

1.6 Lecture 6: Strength of Inductive arguments, Counter ex-

ample method

1.6.1 Construct counter examples for the following invalid arguments

1. 1. If Ravi is a philosopher, then Ravi is wise ( A! B .)
   2. Ravi is not a philosopher. : A So, 3. Ravi is not wise. : B
2. All who seek public office are noble. Some who seek public office are not wise persons. So, some wise persons are not noble.

Answers:

1. If lemons are red, then lemons have a color.
   2. Lemons have a color. So, 3. Lemons are red.
2. All Cats are Animals. Some animals are not four-legged. Some four legged-species are not animals.

1.6.2 Evaluate the following Deductive and Inductive Arguments

1. Some professors wear glasses. Mr. Einstein wears glasses. Therefore, Mr. Einstein is a professor.
2. The overwhelming majority of mutations are not beneficial to an organisms survival. So the odds are that no mutation is going to give an organism super powers.
3. When a lighted match is slowly dunked into water, the flame is snuffed out. But gaso- line is a liquid, just like water. Therefore, when a lighted match is slowly dunked into gasoline, the flame will be snuffed out.

4. No book in English begins numbering its pages on a left-hand page. This is a book in English, therefore it will begin its numbering on a right-hand page
5. It usually takes 2-3 days for a delivery to ship from the warehouse to your door via most major shipping services. You ordered on Tuesday morning, so it is safe to assume your package will arrive Thursday or Friday.
6. All math teachers are over 7 feet tall. Mr. Damodhar. is a math teacher. Therefore, Mr. Damodhar is over 7 feet tall.
7. No one who can afford health insurance is unemployed. All politicians can afford health insurance. Therefore, no politician is unemployed.
8. Just as a football player does not become great without pain, so too with a pianist. I will bet every great football player has at one time or another torn muscles, ligaments, or tendons; many have broken something; surely all have come away from practice bruised. So you want to be great? You want to be a concert pianist one day, a virtuoso? Then I want to see you hurt! I want to see you bleed, I want to see sprained or crushed fingers!
9. Roses are red and beautiful. Einstein was a genius. Therefore, if roses are red and beautiful, Einstein was a genius.
10. If you know that you are not real, then you are not real. If you know that you are not real, then you are real. So, you cannot know that you are not real.
11. Most of the logic quizzes in the course have been very easy so far. The teacher an- nounced that the next quiz will be extremely difficult. Therefore, the next quiz will be very easy as well.

1.6.3 Answers

1. Invalid;
2. Inductive Argument, Strong, Cogent
3. Inductive Argument (analogy), Weak, Uncogent.
4. Valid, Sound
5. Inductive, Strong.
6. Valid, Unsound; the first premise is false.
7. Valid, Sound.
8. Weak analogy, Uncogent.
9. Valid, Unsound
10. Strong, Uncogent.

Chapter 2

Fallacies

2.1 Lecture 8: Identification of Formal and Informal Falla-

cies

2.1.1 Determine whether the fallacies committed by the following argu-

ments are formal fallacies or informal fallacies.

1. If Rasputin was really mad, then he deceived Czar Nicholas II. Rasputin was not really mad. Therefore, he did not deceive Czar Nicholas II.
2. Everything that runs has feet. Th e Columbia River runs very swiftly. Therefore, the Columbia River has feet.
3. All people who believe we create our own reality are people who lack social responsi- bility. All people governed by selfish motives are people who lack social responsibility. Therefore, all people who believe we create our own reality are people governed by selfish motives.
4. The ship of state is like a ship at sea. No sailor is ever allowed to protest orders from the captain. For the same reason, no citizen should ever be allowed to protest presidential policies.
5. Renowned violinist Pinchas Zukerman has said, “When it comes to vodka, Smirnoff plays second fiddle to none.” We must therefore conclude that Smirnoff is the best vodka available.
6. If the Chinese government systematically kills its unwanted orphans, then the Chinese government is immoral. The Chinese government is indeed immoral. Therefore, the Chinese government systematically kills its unwanted orphans.
7. Sarah Jessica Parker, Ben Affleck, and Julia Roberts are Democrats. Therefore, it must be the case that all Hollywood stars are Democrats.
8. Congresswoman Michele Bachmann argues in favor of drilling for oil in the Arctic Na- tional Wildlife Refuge. But consider this. Bachmann is a total moron, a complete idiot who would not recognize an oil well if she bumped into one. Clearly her arguments are ridiculous.
9. If plastic guns are sold to the public, then terrorists will carry them aboard airliners undetected. If plastic guns are sold to the public, then airline hijackings will increase.

Therefore, if terrorists carry plastic guns aboard airliners un detected, then airline hijackings will increase

Answers

1. Formal fallacy.
2. Informal fallacy.
3. Formal fallacy.
4. Informal Fallacy
5. Informal fallacy.
6. Formal fallacy.
7. Informal Fallacy
8. Informal fallacy.
9. Formal fallacy

Source: Patrick Hurley, Concise Introduction to Logic, 11th Edition pp 121

2.2 Lecture 9: Informal Fallacies: Fallacies of relevance

1. The position open in the accounting department should be given to Frank Thompson. Frank has six hungry children to feed, and his wife desperately needs an operation to save her eyesight.
2. Erica Evans, who takes orders at the local Taco Bell, argues persuasively in favor of increasing the minimum wage. But this is exactly what you would expect. Erica is paid the minimum wage, and if the minimum wage is increased, then her own salary will go up. Obviously Ericas arguments are worthless.
3. Th e school board argues that our schools are in desperate need of repair. But the real reason our students are falling behind is that they spend too much time with their computers. Becoming educated means a lot more than learning how to point and click. The school board should send a letter to the parents urging them to monitor their kids computer time.
4. Whoever thrusts a knife into another person should be arrested. But surgeons do precisely this when operating. Therefore, surgeons should be arrested.
5. You should read Irving Stones latest novel right away. It is sold over a million copies, and practically everyone in the Manhattan cocktail circuit is talking about it.
6. Friedrich Nietzsches philosophy is not worth the paper its printed on. Nietzsche was an immoral reprobate who went completely insane from syphilis before he died.
7. Surely you welcome the opportunity to join our protective organization. Th ink of all the money you will lose from broken windows, overturned trucks, and damaged merchandise in the event of your not joining.

4. Accident
5. Ad Populaum.
6. Ad Hominem , Abusive
7. Appeal to force.
8. Strawmen
9. Ignoratio Elenchi
10. Tu quoque (you, too).
11. It is not fallacy. This is Modus ponens.
12. Ad Populum.
13. Red herring.
14. Appeal to pity
15. Fallacy of Accident.

2.3 Lecture 10: Fallacies of Weak Induction and Fallacies

arising out of ambiguity in Language

2.3.1 Identify the fallacies of weak induction committed by the follow-

ing arguments, giving a brief explanation for your answer. If no

fallacy is committed, write no fallacy.

1. If a car breaks down on the freeway, a passing mechanic is not obligated to render emergency road service. For similar reasons, if a person suffers a heart attack on the street, a passing physician is not obligated to render emergency medical assistance.
2. There must be something to psychical research. Three famous physicists Oliver Lodge, James Jeans, and Arthur Stanley Eddington took it seriously.
3. The accumulation of pressure in a society is similar to the buildup of pressure in a boiler. If the pressure in a boiler increases beyond a critical point, the boiler will explode. Accordingly, if a government represses its people beyond a certain point, the people will rise up in revolt.
4. A few minutes after Governor Harrison finished his speech on television, a devastat- ing earthquake struck southern Alaska. For the safety of the people up there, it is imperative that Governor Harrison make no more speeches.
5. Lester Brown, universally respected author of the yearly State of the World report, has said that the destruction of tropical rain forests is one of the ten most serious worldwide problems. Th us, it must be the case that this is indeed a very serious problem.
6. The abstinence only policy for birth control just doesnt work. After all, it didnt work for Jamie Lynn Spears, and it didnt work for Bristol Palin, either.

7. Probably no life exists on Venus. Teams of scientists have conducted exhaustive studies of the planet’s surface and atmosphere, and no living organisms have been found.
8. We dont dare let the animal rights activists get their foot in the door. If they sell us on the idea that dogs, cats, and dolphins have rights, next it will be chickens and cows. Th at means no more chicken Kiev or prime rib. Next it will be worms and insects. Th is will lead to the decimation of our agricultural industry. Th e starvation of the human race will follow close behind
9. No one has proved conclusively that Americas nuclear power plants constitute a dan- ger to people living in their immediate vicinity. Th erefore, it is perfectly safe to con- tinue to build nuclear power plants near large metropolitan centers.
10. There are more churches in New York City than in any other city in the nation, and more crimes are committed in New York City than anywhere else. So, if we are to eliminate crime, we must abolish the churches.

Answers:

1. Weak Analogy
2. Appeal to Unqualified Authority.
3. No fallacy
4. False Cause
5. Appeal to unqualified authority.
6. Hasty Generalization.
7. No Fallacy
8. Slippery Slope
9. Appeal to Ignorance
10. False Cause.

Source: Patrick Hurley (11Ed)- pp 149–

Lecture 11: Fallacies arise out of ambiguity

Source: Patrick Hurley (11Ed)- pp 170–

2.3.2 Identify the fallacies of presumption, ambiguity, and grammati-

cal analogy committed by the following arguments, giving a brief

explanation for your answer. If no fallacy is committed, write no

fallacy.

1. Either we require forced sterilization of Third World peoples or the world population will explode and all of us will die. We certainly dont want to die, so we must require forced sterilization.
2. Every sentence in this paragraph is well written. Therefore, the paragraph is well written.

Part II

Traditional Logic

Chapter 3

Syllogistic Logic

3.1 Lecture 11: Introduction and motivation for Syllogistic

Logic

3.1.1 Name the form of each of the following categorical statements (

A , E , I , or O ). Identify the subject and predicate terms in each

case. Then state the quantity (universal or particular) and quality

(affirmative or negative).

Source: Frances Howard- Snyder (et..al), Power of Logic(4th Edition), pp 201-

1. All hungry cannibals are dangerous people.
2. No Ohioans are Texans.
3. Some diamonds are not valuable objects.
4. No green vegetables are minerals.
5. Some outlaws are heroes.
6. All equilateral triangles are geometrical figures.
7. Some poems are not sonnets.

Answers:

1. A, hungry cannibals, dangerous people, universal, affirmative
2. E, Ohioans, Texans, universal, negative
3. O, diamonds, valuable objects, particular, negative
4. E, green vegetables, minerals, universal, negative
5. I, outlaws, heroes, particular, affirmative
6. A, equilateral triangles, geometrical figures, universal, affirmative.
7. O, poems, sonnets, particular, negative