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COS3761 FORMAL LOGIC III — COMPREHENSIVE
PRACTICE EXAM 2026/2027 COMPLETE (100) CURRENT
TESTING QUESTIONS AND CORRECT ANSWERS WITH
DETAILED RATIONALES.
COS3761
Prepare for the COS3761 Practice Exam with a focused study resource designed to
reinforce key concepts in Formal Logic III. It supports review of propositional logic,
predicate logic, natural deduction, semantics, satisfiability, and basic modal logic.
Use the material to strengthen logical reasoning, practice applying formal rules, and
identify areas that may require additional review before the exam. This resource is
best suited for University of South Africa (UNISA) COS3761 Formal Logic III students
preparing for their examinations.
MULTIPLE CHOICE.
SECTION 1: PROPOSITIONAL LOGIC — TRANSLATION (Questions 1-10)
1. Let P = "The house has seven bathrooms", Q = "The house is used as a
guesthouse". Which of the following correctly translates "The house is
used as a guesthouse only if it has seven bathrooms"?
a) P → Q
b) Q → P
c) P ∧ Q
d) P ∨ Q
Answer: b) Q → P
Rationale: "Only if" introduces a necessary condition. "Q only if P" means Q →
P. The house being used as a guesthouse (Q) implies that it has seven
bathrooms (P).
2. Let S = "Susan saves money", B = "Susan buys a house". Which
proposition correctly translates "Susan saves money and buys a house"?
a) S ∨ B
b) S → B
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c) S ∧ B
d) S ↔ B
Answer: c) S ∧ B
Rationale: The conjunction "and" is represented by ∧. Both statements must
be true.
3. Let R = "It is raining", C = "The ground is wet". Which formula translates
"If it is raining, then the ground is wet"?
a) R ∧ C
b) R → C
c) C → R
d) R ↔ C
Answer: b) R → C
Rationale: "If ... then ..." is represented by material implication (→).
4. Let M = "Mary studies", P = "Mary passes the exam". Which formula
translates "Mary passes the exam only if she studies"?
a) P → M
b) M → P
c) P ∧ M
d) P ∨ M
Answer: a) P → M
Rationale: "P only if M" means P → M. Passing implies studying.
5. Let J = "John likes the house", H = "Susan buys a house". Which formula
translates "John likes the house if Susan buys it"?
a) H → J
b) J → H
c) H ∧ J
d) H ∨ J
Answer: a) H → J
Rationale: "If Susan buys a house, then John likes it" is H → J.
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6. Let A = "The alarm is on", B = "The door is locked". Which formula
translates "The alarm is on and the door is locked"?
a) A → B
b) A ∨ B
c) A ∧ B
d) A ↔ B
Answer: c) A ∧ B
Rationale: "And" is conjunction (∧). Both conditions must hold.
7. Let P = "Peter likes the house", S = "Susan buys a house". Which formula
translates "Peter likes the house if and only if Susan buys it"?
a) P → S
b) S → P
c) P ↔ S
d) P ∨ S
Answer: c) P ↔ S
Rationale: "If and only if" is the biconditional (↔), meaning P implies S and S
implies P.
8. Let T = "The train is late", M = "The meeting is canceled". Which formula
translates "The meeting is canceled if the train is late"?
a) T → M
b) M → T
c) T ∧ M
d) T ∨ M
Answer: a) T → M
Rationale: "If the train is late, then the meeting is canceled" is T → M.
9. Let C = "The car starts", K = "The key is in the ignition". Which formula
translates "The car starts only if the key is in the ignition"?
a) C → K
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b) K → C
c) C ∧ K
d) C ∨ K
Answer: a) C → K
Rationale: "C only if K" means C → K.
10. Let S = "The sun is shining", W = "The flowers are wilting". Which
formula translates "Either the sun is shining or the flowers are wilting"?
a) S → W
b) S ∧ W
c) S ∨ W
d) S ↔ W
Answer: c) S ∨ W
Rationale: "Either ... or ..." is disjunction (∨).
SECTION 2: PROPOSITIONAL LOGIC — TRUTH TABLES & VALIDITY
(Questions 11-15)
11. How many rows are in the truth table for a formula with 3 propositional
variables?
a) 4
b) 6
c) 8
d) 16
Answer: c) 8
Rationale: The number of rows in a truth table is 2ⁿ, where n is the number of
distinct propositional variables. For 3 variables, 2³ = 8 rows.
12. A formula is a tautology if:
a) It is true in at least one row of the truth table
b) It is false in at least one row of the truth table
c) It is true in all rows of the truth table
d) It has the same truth value in all rows
COS3761 FORMAL LOGIC III — COMPREHENSIVE
PRACTICE EXAM 2026/2027 COMPLETE (100) CURRENT
TESTING QUESTIONS AND CORRECT ANSWERS WITH
DETAILED RATIONALES.
COS3761
Prepare for the COS3761 Practice Exam with a focused study resource designed to
reinforce key concepts in Formal Logic III. It supports review of propositional logic,
predicate logic, natural deduction, semantics, satisfiability, and basic modal logic.
Use the material to strengthen logical reasoning, practice applying formal rules, and
identify areas that may require additional review before the exam. This resource is
best suited for University of South Africa (UNISA) COS3761 Formal Logic III students
preparing for their examinations.
MULTIPLE CHOICE.
SECTION 1: PROPOSITIONAL LOGIC — TRANSLATION (Questions 1-10)
1. Let P = "The house has seven bathrooms", Q = "The house is used as a
guesthouse". Which of the following correctly translates "The house is
used as a guesthouse only if it has seven bathrooms"?
a) P → Q
b) Q → P
c) P ∧ Q
d) P ∨ Q
Answer: b) Q → P
Rationale: "Only if" introduces a necessary condition. "Q only if P" means Q →
P. The house being used as a guesthouse (Q) implies that it has seven
bathrooms (P).
2. Let S = "Susan saves money", B = "Susan buys a house". Which
proposition correctly translates "Susan saves money and buys a house"?
a) S ∨ B
b) S → B
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c) S ∧ B
d) S ↔ B
Answer: c) S ∧ B
Rationale: The conjunction "and" is represented by ∧. Both statements must
be true.
3. Let R = "It is raining", C = "The ground is wet". Which formula translates
"If it is raining, then the ground is wet"?
a) R ∧ C
b) R → C
c) C → R
d) R ↔ C
Answer: b) R → C
Rationale: "If ... then ..." is represented by material implication (→).
4. Let M = "Mary studies", P = "Mary passes the exam". Which formula
translates "Mary passes the exam only if she studies"?
a) P → M
b) M → P
c) P ∧ M
d) P ∨ M
Answer: a) P → M
Rationale: "P only if M" means P → M. Passing implies studying.
5. Let J = "John likes the house", H = "Susan buys a house". Which formula
translates "John likes the house if Susan buys it"?
a) H → J
b) J → H
c) H ∧ J
d) H ∨ J
Answer: a) H → J
Rationale: "If Susan buys a house, then John likes it" is H → J.
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6. Let A = "The alarm is on", B = "The door is locked". Which formula
translates "The alarm is on and the door is locked"?
a) A → B
b) A ∨ B
c) A ∧ B
d) A ↔ B
Answer: c) A ∧ B
Rationale: "And" is conjunction (∧). Both conditions must hold.
7. Let P = "Peter likes the house", S = "Susan buys a house". Which formula
translates "Peter likes the house if and only if Susan buys it"?
a) P → S
b) S → P
c) P ↔ S
d) P ∨ S
Answer: c) P ↔ S
Rationale: "If and only if" is the biconditional (↔), meaning P implies S and S
implies P.
8. Let T = "The train is late", M = "The meeting is canceled". Which formula
translates "The meeting is canceled if the train is late"?
a) T → M
b) M → T
c) T ∧ M
d) T ∨ M
Answer: a) T → M
Rationale: "If the train is late, then the meeting is canceled" is T → M.
9. Let C = "The car starts", K = "The key is in the ignition". Which formula
translates "The car starts only if the key is in the ignition"?
a) C → K
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b) K → C
c) C ∧ K
d) C ∨ K
Answer: a) C → K
Rationale: "C only if K" means C → K.
10. Let S = "The sun is shining", W = "The flowers are wilting". Which
formula translates "Either the sun is shining or the flowers are wilting"?
a) S → W
b) S ∧ W
c) S ∨ W
d) S ↔ W
Answer: c) S ∨ W
Rationale: "Either ... or ..." is disjunction (∨).
SECTION 2: PROPOSITIONAL LOGIC — TRUTH TABLES & VALIDITY
(Questions 11-15)
11. How many rows are in the truth table for a formula with 3 propositional
variables?
a) 4
b) 6
c) 8
d) 16
Answer: c) 8
Rationale: The number of rows in a truth table is 2ⁿ, where n is the number of
distinct propositional variables. For 3 variables, 2³ = 8 rows.
12. A formula is a tautology if:
a) It is true in at least one row of the truth table
b) It is false in at least one row of the truth table
c) It is true in all rows of the truth table
d) It has the same truth value in all rows
