Philosophy assignment 10 questions

PHIL 110; Spring 2020

Final Exam

This is an “open book” exam, in the sense that as you complete the exam you may look at your notes, the

textbook, my lecture slides … or any other resource that you find helpful.

The exam is designed to be taken in three hours, but you may complete it more quickly or more slowly if

you prefer.

If you find one of the questions unclear, you may email me (at tmdonald@sfu.ca) to ask for clarification.

Otherwise, you should complete this exam without help from anyone, and without collaborating.

You should upload your answers to Canvas as a pdf:

• You could write your answers by hand, and then scan your answers to pdf.

• Alternatively, you could type your answers. For help with this, please refer to the document “How

to Type Your Answers to the PHIL 110 Final”, on Canvas.

Question One (5 points)

Which of the following three inferences are valid? (There is no need to explain your answers.)

(a) Premise: If Jon forgot to turn the oven on, then the dinner has been ruined.

Premise: The dinner has been ruined.

Conclusion: Jon forgot to turn the oven on. NOT VALID

(b) Premise: When he retired, Jon moved from Vancouver to Jamaica.

Conclusion: Jon likes sunny weather. NOT VALID

(c) Premise: Jon is either in Vancouver or in Jamaica.

Premise: Jon is not in Vancouver.

Conclusion: Jon is in Jamaica. VALID

Question Two (5 points)

Consider the following argument:

If moral laws are made by people, then moral relativism is true. But moral relativism is

not true. Therefore, moral laws are not made by people. Moral laws are either made by

people, or by God. Therefore, moral laws are made by God.

(a) What is the conclusion of this argument?

MORAL LAWS ARE MADE BY GOD.

(b) Identify two premises of this argument.

FOR FULL MARKS, STATE ANY OF TWO OF THE FOLLOWING:

• IF MORAL LAWS ARE MADE BY PEOPLE, THEN MORAL RELATIVISM IS TRUE.

• MORAL RELATIVISM IS NOT TRUE.

• MORAL LAWS ARE EITHER MADE BY PEOPLE, OR BY GOD.

(c) Identify two inference rules that are used in the argument. MT, DS

Question Three (5 points)

Briefly explain the distinction between strict and loose generalizations, giving examples. What is a

counterexample? Can a loose universal generalization be refuted by a single counterexample?

When you assert a strict universal generalization, you say that something is true in every single case

without exception. When you assert a loose universal generalization, you say that something is normally

true, or usually true, or typically true, or true in most cases.

For example:

STRICT: Every single triangle without exception has three sides.

LOOSE: Dogs typically have four legs.

A strict universal generalization can be refuted a single example – a “counterexample”. For example, if

someone says that every bird can fly, and intends this as a strict generalization, you can refute them by

showing them a single penguin.

A loose universal generalization cannot be refuted by just one counterexample.

Question Four (9 points)

Consider the following argument:

Premise One: It’s not true that Ashni and Ben were both cooking. (A & B)

Premise Two: Ashni was cooking. A

Premise Three: If Ashni was cooking and Ben wasn’t, then the meal was delicious. ((A & B) → D)

Conclusion: The meal was delicious. D

Symbolize the argument, using the following abbreviations:

A: Ashni was cooking.

B: Ben was cooking.

D: The meal was delicious.

Is the argument valid? Justify your answer in detail.

The argument is valid. You can justify this either by giving a proof or by using a truth table. I’ll do it both

ways:

1. (A & B) Prem

2. A Prem

3. ((A & B) → D) Prem

4. B 1, 2 CS

5. (A & B) 2, 4 Conj

6. D 3, 5, MP

A B D B (A & B) (A & B) (A & B) ((A & B) → D) T T T F T F F T T T F F T F F T T F T T F T T T T F F T F T T F F T T F F T F T F T F F F T F T F F T T F T F T F F F T F T F T

Question Five (2 points)

Krishna has been asked to prove the following statement:

For any natural number n, (n2 + 3n + 10) is even.

Krishna responds by using a computer to check that there are no counterexamples to the statement

below 1,000,000,000.

Has Krishna proved the statement? Briefly explain your answer.

No, Krishna has not proved the statement. Krishna has shown that there are no counterexamples to the

statement below one billion, but he has not shown that there are no counterexamples above one billion.

Question Six (9 points)

Exactly one of these two inferences is valid. Give a natural deduction proof of the valid inference.

(1) Premise: ∀x(Ax → Bx)

Premise: ∃x Ax

Conclusion: ∃x Bx

(2) Premise: ∀x(Ax → Bx)

Premise: ∃x Bx

Conclusion: ∃x Ax

(1) is valid:

1. ∀x(Ax → Bx) Premise

2. ∃x Ax Premise

3. Ai 2, EI

4. (Ai → Bi) 1, UI

5. Bi 3, 4 MP

6. ∃x Bx 5, EG

Question Seven (2 points)

Consider the following mathematical proof:

Claim There do not exist whole numbers a and b such that ( 𝑎

𝑏 )

3

= 2.

Proof Suppose for contradiction that there do exist whole numbers a and b,

where ( 𝑎

𝑏 )

3

= 2.

Then by “cancelling down” the fraction, we can find numbers c and d which are not both

even, where ( 𝑐

𝑑 )

3

= 2.

Then c 3 = 2d 3, so c 3 is even. Now we know that the cube of an odd number must always

be odd. So c is even. So for some number k, c = 2k.

Now since c 3 = 2d 3 and c = 2k, we have (2k)3 = 2d 3. Thus 8k 3 = 2d 3, and so 4k 3 = d 3.

This implies that d 3 is even, and so d is even.

But now we’ve shown that both c and d are even – which contradicts our initial

statement that c and d are not both even. Thus we have arrived at a contradiction and the

proof is complete.

Identify one inference rule that is used in this proof.

The RA rule is used – note the giveaway phrase “suppose for contradiction”.

Question Eight (5 points)

Using the following symbols, symbolize the statements listed below.

Universe of Discourse: the people at a party

a Ashni

b Ben

c Chiara

Lxy x loves y.

Sx x is a singer.

(1) Ashni loves Ben, but Ben doesn’t love her back.

(Lab & Lba)

(2) Ben loves Chiara and nobody else.

(Lbc & ∀x(Lbx → x = c)

(3) There are at least two people who love Ben.

∃x ∃y((Lxb & Lyb) & x ≠ y)

(4) Everyone who Ashni loves, Ben loves too.

∀x(Lax → Lbx)

(5) Ashni loves someone, and that person loves everyone who loves Ben.

∃x(Lax & ∀y(Lyb → Lxy))

Question Nine (6 points)

In each part of this question, there are two statements. You should choose one statement to be the

premise, and the other to be the conclusion, in such a way that the resulting argument is valid. (You do

not need to explain your answers). I use the same symbols as in Question Eight.

(a) ((Sa → Sb) & (Sb → Sc)) PREMISE

(Sa → Sc) CONCLUSION

(b) ∃x (Sx  Lxc) CONCLUSION

∃x (Sx & Lxc) PREMISE

(c) ∃x ∀y Lxy PREMISE

∀y ∃x Lxy CONCLUSION

Question Ten (10 points)

“Every statement is either true or false.” Do you agree or disagree? Explain your answer.

SEE LECTURE 14 FOR MY ANSWER.