Question 1: Which rule of inference is used in each of the following arguments? Check the correct answers.

- If I work all night on this homework, then I can answer all the exercises. If I answer all the exercises, I will understand the material. Therefore, if I work all night on this homework, I will understand the material.
- Modus ponens
- Hypothetical syllogism – correct answer
- Addition
- Disjunctive syllogism
- Simplification
- Modus Tollens
- Conjunction

- Jerry is a mathematics major and a computer science major. Therefore, Jerry is a mathematics major. p = Jerry math, q = Jerry computer science. p or q is true. Therefore, p is true.
- Modus ponens
- Disjunctive syllogism
- Conjunction
- Simplification – correct answer
- Addition
- Modus Tollens
- Hypothetical Syllogism

- If it snows today, the university will close. The university is not closed today. Therefore, it did not snow. p = it snows, q = university closes. If p is true, then q is true. q is not true, therefore p is not true.
- Addition
- Modus tollens – correct answer
- Conjunction
- Hypothetical syllogism
- Simplification
- Modus ponens
- Disjunctive syllogism

- Kangaroos live in Australia and are marsupials. Therefore, kangaroos are marsupials. p = Australia, q = marsupials. p and q are true. Therefore, p is true.
- Hypothetical syllogism
- Simplification – correct answer
- Modus tollens
- Disjunctive syllogism
- Addition
- Conjunction
- Modus ponens

- If it is rainy, then the pool will be closed. It is rainy. Therefore, the pool is closed. p = rainy, q = pool closed. If p then q. p is true. Therefore, q is true.
- Conjunction
- Hypothetical syllogism
- Modus tollens
- Modus ponens – correct answer
- Simplification
- Disjunctive syllogism
- Addition

Question 2: Mike is looking for a job and Jane represents a tech company that’s hiring. Here is a part of the conversation they had.

- Mike: I read you’re looking for someone who knows Java or C++. I know Java, so I satisfy that requirement.
- Jane: How long have you been programming in Java?
- Mike: For 2 years. I’m also a guild leader in my MMO.
- Jane: I see. So you have 2 years of experience in Java.
- Mike: I also know Python.
- Jane: You know Java and Python! That’s good, but it’s still our company policy that if you want to work with us long term, you need to learn C++. We got regular training events for that.
- Mike: I am looking for long term employment. So I guess I’ll have to take that training.
- Jane: Yeah! It’s paid for by the company.
- Mike: Can we talk about work hours?
- Jane: Sure. It’s usually 40 hours per week, but every May to June, you will have to work 60 hours per week.
- Mike: Uh..
- Jane: We pay you generously though for that. It’s double pay for overtime.
- Mike: So I get paid more in May and June.
- Jane: Right.
- Mike: I heard you guys have a great work environment here.
- Jane: Absolutely. There’s free food available all day in our cafeteria. You have to try the German sausages or the vegetable lasagna they have every Tuesday.
- Mike: Well.. I’m a vegetarian, so I don’t think I’ll be trying those sausages.
- Jane: The lasagna then.
- Mike: Can we talk about benefits a bit?
- Jane: Sure. If you’re married, your wife or husband gets full health insurance.
- Mike: I don’t think we’ll be getting that benefit then.
- Jane: You’re unmarried? We have benefits for qualifying domestic partners too.
- Mike: What other benefits do you offer?
- Jane: You get to pick between having disability insurance and full life insurance. If you select disability insurance, we’ll buy you a basic life insurance policy.
- Mike: So everyone has some form of free life insurance? Nice.

Both Mike and Jane used rules of inference. Some of them are fairly obvious, others more disguised. Identify them in the order in which they were used.

- Addition – See part 1. p = Java, q = C++. p is true, therefore p or q is also true.
- Simplification – See parts 3 and 4. p = Java, q = guild leader. p and q are true, therefore p is true.
- Conjunction – See parts 5 and 6. p = Python, q = Java. p is true. q is true. Therefore, p and q are true.
- Modus Ponens – See parts 6 and 7. p = long-term, q = learn C++. If p then q. p is true. Therefore, q is true.
- Hypothetical Syllogism – See parts 10 through 13. p = May to June, q = work overtime, r = more pay. If p then q and if q then r. Therefore, if p then r.
- Disjunctive Syllogism – See parts 16 through 18. p = sausage. q = lasagna. p or q is true. If p is not true, then q is true.
- Modus Tollens – See parts 20 through 22. p = married, q = benefit. If q is false, then p is false. q is false, therefore p is false.
- Resolution – See parts 24 and 25. p = disability. q = full life insurance, r = basic life insurance. p or q and not p or r. Therefore, q or r.