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This exam consists of three parts. Part A consists of questions concerning categorical logic.

Part B consists of questions concerning propositional logic. Part C consists of questions

concerning predicate logic. The mark for each question is indicated with the question.

FOR ALL QUESTIONS SHOW YOUR WORK. REMEMBER: WHEN YOU ARE ASKED TO EXPLAIN

SOMETHING, YOUR EXPLANATION IS REQUIRED AS PART OF THE EVALUATION. PART MARKS CAN BE

AWARDED

** Total possible high score = 200 marks**

PART A:

1. Translations. The following categorical propositions are not in standard form. Restate them in

standard form, and identify each of your reformulations as an A, E, I or O proposition. (2 marks each.

Total marks = 10.)

a) Not all Democrats support massive welfare spending.

b) Everyone who arrives by six o’clock in the morning will receive free admission.

c) It is not the case that the only things in life that are certain are death and taxes.

d) Distinguished-looking people are often tall.

e) People are not machines.

2. Assuming the Aristotelian Square of Opposition, what can be inferred about the truth or falsehood of

the remaining propositions in the following set (a) if we assume the first one (*) to be true, and (b) if we

assume the first one (*) to be false? (5 marks)

(*)Some college professors are not entertaining lecturers. (*)

All college professors are entertaining lecturers.

No college professors are entertaining lecturers.

Some college professors are entertaining lecturers.

3. Provide the converse, obverse, and contrapositive for each of the following propositions. (10 marks)

a) Some results of plastic surgery are things beyond belief.

b) Some Las Vegas casinos are not places likely to increase your wealth.

c) All microwave foods are things best left uneaten.

4. Assuming the Aristotelian interpretation, provide a Venn Diagram for the following propositions.

Ensure you clearly indicate the subject and predicate terms. (2 marks each. 10 marks total.)

a) All banana splits are healthy desserts.

b) All pigs are fantastic pets.

c) Some poets are dead people.

d) Some Olympic gold medal winners are drug cheats.

e) No dead people are people who tell tales.

5. Translate the following syllogisms into a standard form categorical syllogisms indicating their

mood and figure, and test them for validity by means of a Venn diagram. Be sure to properly label your

diagram. Assume the Boolean interpretation. Note – (c) is an enthymeme. Complete it as a valid

categorical syllogism (5 marks each, 20 marks total).

a) Some students are not female, but some males are not students; thus, some males are not

females.

b) Since no gillygongs are visible, we can conclude that no dogs are gillygongs for all dogs are

visible.

c) Sam’s Steak House must have pretty low prices. Uncle George took Aunt Tillie there for dinner

last night.

d) Some syllogisms are arguments whose validity or invalidity is not readily apparent. All syllogisms

are arguments whose validity or invalidity needs to be established. Therefore, some arguments

whose validity or invalidity needs to be established are not arguments whose validity or

invalidity is not readily apparent.

6. The following categorical syllogisms are expressed solely by their mood and figure. Use the rules of

validity provided in the reference material for this exam to discern the validity of the syllogisms. If the

syllogism is valid indicate that. If the syllogism is invalid, indicate all the rules that it violates. (2 marks

each. 10 marks total.)

a) AAA-3

b) EAE-3

c) OAO-4

d) EIO-1

e) OAO-2

7. Sorites. Rephrase this Sorites as a chain of categorical syllogisms. Indicate the mood and figure of

each syllogism (5 marks)

1. All scavengers are flesh eaters.

2. No vegetarians are flesh eaters.

3. All hoofed mammals are vegetarians.

Therefore, hoofed mammals are scavengers.

PART B:

Questions # 1 – #8 are worth 5 marks each. Ensure you provide explanation where requested. (40 marks

total.)

1. Provide an example of an ordinary language argument in the argument form modus ponens.

Provide an example of an ordinary language argument in the related formal fallacy of denying the

antecedent. Why is this fallacy so persuasive?

2. Provide a statement in symbolic form that has at least two simple statements as components and

is a contradiction. Provide a truth table for your statement that indicates its self-contradictory

nature.

3. As we know, for large arguments in our propositional logic with more than 4 simple statements

showing the validity by using a truth table is cumbersome and unwieldy. In such cases, we use the

indirect method. Briefly explain what the indirect method involves and why we are justified in

using it to determine the validity of a symbolized argument.

4. Consider the following derivation taken from a proof of validity.

n. (A v B) v C ….

n+1. ~ A ….

n+2 C n – (n+1) DS

Explain why this derivation exhibits an incorrect use of the DS rule of inference.

5. Given the following conclusion for an argument you are going to prove, what would be your first

assumption if you were using Conditional Proof (CP)? If you were to use more than one

application of CP, what would be the other assumption(s) in order of appearance?

{(A ? B) ? [(A ? C) ? A]}

6. Would the following statement qualify as a contradiction and so could then be used as the last

step in the application of an Indirect Proof (IP)? Provide a truth table to support your answer.

(~ A ? ~ B) • (A ? B)

7. Prove the following tautology using CP, IP, and any of the 18 rules of inference.

~ [(A • ~ B) • ~ (A v B)]

8. Determine whether the following pair of symbolized statements are logically equivalent. Provide a

truth table to support your answer.

~ (H • K) v ~ (K v M) (~ H v ~ K) v (~ K • ~ M)

9. Translate the following arguments symbolically using the letters indicated for the simple

propositions. Discern the validity of each of these arguments. For question (a), you must discern

the validity using a truth table. For the remaining questions you may use a truth table or the

indirect method. (5 marks each, 15 marks total.)

a. If John is a plumber, then either Sam is a bricklayer, or Tim is a carpenter. But Sam is not

a bricklayer, nor is Tim a carpenter. Therefore, John is not a plumber. (J, S, T)

b. Either I will take a trip to Europe this summer or I will save my money and get married in

September. If I do the latter, I will move to Denver. So if I don’t go to Europe this

summer I will move to Denver. (E, S, M, D)

c. If Russia intervenes in Iran, then if the United States acts to protect its interests in the

Middle East, either there will be a confrontation between Russia and the United States

or Israel will act as a surrogate for the United States. Israel will act as a surrogate if and

only if it is absolutely assured of unlimited supplies of American weapons. If the United

States meets this condition, however, the friendly Arab states will turn against the

United States; the United States cannot allow that to happen. Therefore, if Russia

intervenes in Iran and the United States acts to protect its interests in the Middle East,

there will be a confrontation between the two superpowers. (R, U, C, I, A, F)

10. Provide proof, using the rules of inference (see attached on last page), of the validity of the

following arguments. Use IP or CP if you desire. (5 marks each. 20 marks total)

a. 1. D ? C

2. ~(C • ~ S) /? D ? S

b. 1. C ? (D • M)

2. ~ M /? ~ C

c. 1. A ? H

2. (F v W) ? L /? (H ? F) ? (A ? L)

d. 1. (K v L) ? (M • N)

2. (N v O) ? (P • ~ K) /? ~ K

PART C:

1. Symbolic Translation. Translate the following ordinary language arguments into predicate logic using

the suggested notation. (10 marks each. Total = 20 marks)

a. Architects and dentists are well-paid professionals No well-paid architect eats at Burger

King, and no professional shops at Giant Tiger. Therefore, no architect eats at Burger King

or shops at Giant Tiger.

(Ax: x is an architect; Dx: x is a dentist; Wx: x is well-paid; Px is a professional; Bx: x eats at Burger King;

Sx: x shops at Giant Tiger.)

b. Some teachers are forced to take a second job. No individual who works at two jobs can be

fully alert on the job. A teacher who is not fully alert on the job will annoy students. So,

some teachers will annoy students.

(Tx: x is a teacher; Jx: x takes two jobs; Ax: x is fully alert on the job; Sx: x annoys students)

2. Translate the following valid arguments into predicate logic using the suggested symbolic notation.

Then provide a proof of validity for each argument. (10 marks each. 20 marks total.)

a. Smart people are tall. All tall people wear clothes. Thus, smart people wear clothes.

(Px, Tx, Sx)

b. Socrates is mortal. Therefore, everything is either mortal or not mortal (s: Socrates, Mx: x

is mortal.)

3. Provide proof, using the rules of inference and quantification rules (see attached on last page), of the

validity of the following valid argument. Use IP or CP if you desire. (5 marks)

1. (x) [Wx ? (Xx ? Yx)]

2. ( ?[Xx x) • (Zx • ~ Ax)]

3. (x) [(Wx ? Yx) ? (Bx ? Ax)]

? ( ?( Zx x) • ~Bx)

4. Proving Invalidity. The following arguments are invalid. Show them to be invalid using the finite

universe method. (5 marks each. Total = 10 marks)

a. 1. (x) (Dx ? ~Ex)

2. (x) (Ex ? Fx)

? (x) (Fx ? ~Dx)

b. 1. (x) (Hx ? Mx)

2. ( ?(Mxx)• Bx)

? ( ?(Hx x) • Bx)