PHIL 1320 Term 2 ExamInstructions:
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)