Logic: Concise Edition, CH 7A: Propositional Logic

"conjunction" logical operator

dot (·) "and; also; moreover"

"disjunction" logical operator

wedge (v) "or; unless"

Translate the following compound statements: 1. It is not the case that drinking hot coffee reduces sweating. 2. Hamlet is a tragedy and Kung Fu Panda is a comedy. 3. Either we reduce carbon emissions or global warming will get worse. 4. If the IRS processed my return, then I should get my refund this week. 5. You will graduate if and only if you meet all university requirements.

1. ∼ D 2. H ⋅ K 3. C ν G 4. I ⊃ R 5. G ≡ U

Exercise 7A part II Determine whether a sufficient condition exists in the following statements: 1. If Ed is a bachelor, then Ed is an adult male 2. If Ed is an adult male, then Ed is a bachelor. 3. If there is oxygen in the room, then there is a fire in the room. 4. If there is a fire in the room, then there is oxygen in the room. 5. If this is the month of June, then this month has exactly 30 days. 6. If this month has exactly 30 days, then this is the m onth of June. 7. If I live in the White House, then I am the president of the United States. 8. If I am the president of the United States, then I live in the White House. 9. If I have exactly 100 pennies, then I have at least the equivalent of $1. 10. If I have at least the equivalent of $1, then I have exactly 100 pennies. 11. If I am over 21 years of age, then I am over 10 years of age. 12. If I am over 10 years of age, then I am over 21 years of age. 13. If I am eating a banana, then I am eating a fruit. 14. If I am eating a fruit, then I am eating a banana. 15. If I hurt a human, then I hurt a mammal. 16. If I hurt a mammal, then I hurt a human.

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Exercises 7A part I: Translate the following statements into symbolic form by using logical operators and uppercase letters 1. Either it will rain tommorow or it will be sunny. 2. The food in that restauraunt stinks, and the portions are too small. 3. Your ice is not cold. 4. IF my stock portfolio is weak, then I am losing money. 5. My car does not look great, but it gets great gas mileage. 6. If you feel great, then you look great. 7. My test score was high or I was mistaken. 8. You passed the exam only if your got a C. 9. Either candy or tobacco is bad for your teeth. 10. Bill is cold and Mary is late. 11. Today is Monday or today is Tuesday. 12. He is not a U.S. senator. 13. Toothpaste is good for your teeth, but tobacco is not. 14. Driving too fast is hazardous to your health; also driving without buckling up. 15. Pizza contains all the basic food groups if, and only if, you get it w ith anchovies. 16. Lava lamps are distracting, while music in the background is soothing. 17. M y room could use a good cleaning, but I am too lazy to do anything about it. 18. You must get a passing grade on the next exam; otherwise you will fail. 19. If Carly agrees to do a job, then she will make sure it is done right. 20. It is not true that Titanic is the highest grossing film of all time. 21. I will leave a big tip only if the dinner is excellent. 22. Your paper was turned in late; however, I am willing to grant you an extension. 23. Unless you stop eating too much pepperoni, you will develop a stomach ulcer. 24. Only if your paper was turned in late, I will deduct a letter grade. 25. It is false that Grover Cleveland was the greatest U.S. president. 26. She is happy with her box of candy; however, she would have preferred a new car. 27. Only if my car has a turbocharger, it is fast. 28. Citizen Kane did not w in the Academy Award for best picture, but it is still the greatest movie ever made. 29. Barbara is going to lose her football bet and Johnny will get a night at the ballet. 30. M y father is wise only if he is honest. 31. Either my stock portfolio is strong or I am losing money. 32. If I am lazy, then my room is not clean. 33. If driving too fast is hazardous to your health, then so is driving w ithout buckling up. 34. M y father is wise and he is honest. 35. M y stock portfolio is weak only if I am losing money. 36. There are not too many circus acts in Las Vegas. 37. Only if my room could use a good cleaning, I am too lazy to do anything about it. 38. W atching circus acts is hazardous to your health and so is falling into deep holes. 39. If my father is wise, then he is honest. 40. M y car is fast, if it has a turbocharger. 41. If it rains tomorrow, then I will not have to water my plants. 42. Reading is relaxing and thinking is productive. 43. Cats and dogs make great pets. 44. The decathlon is a difficult Olympic event. 45. M y car is old, but it is still reliable. 46. Only if you are registered can you vote. 47. Either coffee or tea contains caffeine. 48. Today is M onday unless today is Tuesday. 49. Sally got a promotion, and either Louis asks for a raise or he looks for another job. 50. Either September does not have 31 days, or if July has 31 days, then so does August. 51. Both slot machines and table gaming do not take credit cards. 52. Either the United States or France has a large m ilitary presence in Europe given that both Russia and Switzerland are not part of the NATO alliance. 53. Mary does not own a motorcycle; however, if she passes the motorcycle driver's test, then either she will buy her own motorcycle or she will use Tom's. 54. If stock prices fall this year, then if unemployment rises this year, then the housing market and m anufacturing jobs will suffer dire consequences. 55. It is not the case that both illiteracy and racism are genetically determined, but both can be reduced by education. 56. If the hum an population rises past eight billion, then our species will require more food, and if other anim al species become extinct, then natural resources may become depleted. Moreover, survival may become more difficult and competition for scarce resources may become more violent. 57. Prison populations will continue to grow and longer prison sentences will be imposed only if new laws are created and profiling is not stopped; but if punishm ent is seen as retribution, then punishm ent cannot work as a deterrence. 58. If cars and factories continue to pollute the air, then either the oceans will rise or climate change will put some life forms in jeopardy; nevertheless, we can protect future generations if, and only if, we implement sound scientific advice and curb global conspicuous consumption. 59. My university has many good instructors and resources, but if I don't take advantage of all the university has to offer, then I will have wasted both my tim e and my parents' money. 60. If I get a degree and find a good job, then I can save for my retirement if, and only if, the world economy does not have a meltdown and natural disasters do not wreck our infrastructure.

Answers in key

Exercise 7A part III Determine whether a necessary condition exists in the following statements: 1. If Ed is not an adult male, then Ed is not a bachelor 2. If Ed is a not a bachelor, then Ed is not an adult male. 3. If there is not a fire in the room, then there is not oxygen in the room. 4. If there is not oxygen in the room, then there is not a fire in the room. 5. If this month does not have exactly 30 days, then this is not the m onth of June. 6. If this is not the month of June, then this month does not have exactly 30 days. 7. If I am not the president of the United States, then I do not live in the White House. 8. If I do not live in the White House, then I am not the president of the United States. 9. If I do not have at least the equivalent of $1, then I do not have exactly 100 pennies. 10. If I do not have exactly 100 pennies, then I do not have at least the equivalent of $1. 11. If I am not over 10 years of age, then I am not over 21 years of age. 12. If I am not over 21 years of age, then I am not over 10 years of age. 13. If I am not eating a fruit, then I am not eating a banana. 14. If I am not eating a banana, then I am not eating a fruit. 15. If I do not hurt a mammal, then I do not hurt a human. 16. If I do not hurt a human, then I do not hurt a mammal.

Check answer key

inclusive disjunction

a compound statement in which both disjuncts can be true at the same time example: Either July or August has 31 days

exclusive disjunction

a compound statement in which both disjuncts cannot be true at the same time example: Today is Monday or today is Wednesday

simple statement

a statement that does not have any other statement or logical operator as a component examples: • Harrisburg is the capital of Pennsylvania. • Wednesday is hump day. • Grilled hamburgers taste delicious. • Detective novels make great movies.

compound statement

a statement that has a least one simple statement and at least one logical operator as components examples: 1. It is not the case that drinking hot coffee reduces sweating. 2. Hamlet is a tragedy and Kung Fu Panda is a comedy. 3. Either we reduce carbon emissions or global warming will get worse. 4. If the IRS processed my return, then I should get my refund this week. 5. You will graduate if and only if you meet all university requirements. translated: 1. It is not the case that D. 2. H and K. 3. Either C or G. 4. If I, then R. 5. G if and only if U

"conditional" logical operator

horseshoe (⊃) "if....then...; only if"

logical operators

special symbols that are used to translate ordinary language statements

propositional logic

the basic components in propositional logic are statements

"negation" logical operator

tilde (~) "it is not the case that"

"biconditional" logical operator

triple bar (≡) "if and only if"

necessary condition

when one thing is essential, mandatory, or required in order for another thing to be realized example: If you live in New Jersey, then you live in the United States Living in the United States is necessary to live in New Jersey. If you do not live in the United States, you can not live in New Jersey

sufficient condition

whenever one event ensures that another event will happen example: If you live in New Jersey, then you live in the United States Living in New Jersey is sufficient for living in the United States