PHIL 211
Introduction to Logic
EXAMINATIONS – 2016
TRIMESTER 2
Section A. Propositional Logic
1. Translate the following sentences into PL. At the start say what each proposition letter means in terms of an English sentence.
a. If Lola has a ball, she is happy.
b. Susan throws a ball to Lola or Susan feeds Lola, but not both.
c. Susan throws a ball to Lola unless it is raining.
d. Lola is happy only if either Susan throws a ball to her or it is not raining.
e. Susan and Lola are both happy only if Susan throws a ball to Lola. (10 Marks)
2. Evaluate the following argument using truth tables. Is it valid?
(10 Marks)
3. Do trees for the following formulas. Are they tautologies? If a formula is not a tautology, give truth values for the proposition letters that make it false.
a. ((p & ∼ (∼p ∨ q)) ⊃ r) ⊃ (p ⊃ (∼q ⊃ r))
b. (((p ⊃ q) & (q ⊃ r)) & ∼ q) ⊃ ∼r (15 Marks)
4. Prove both of the following using the natural deduction systems SD or SD+ (it is your choice whether you use the extra SD+ rules):
a. ((p ⊃ r) & (q ⊃ r)) ⊃ ((p ∨ q) ⊃ r)
b. ((p ⊃ q) & ∼r) ⊃ ((q ⊃ r) ⊃ ∼p) (15 Marks)
Section B: Quantificational Logic
5. Translate the following into QL:
a. Every dog hates some cat who hates every mouse.
b. Some cat is hated by every dog.
c. Every cat hates every mouse but loves every dog.
d. No dog hates every cat.
e. Every cat and every dog hate every mouse. (10 Marks)
6. Do an expansion of the following formula using just the names a and b: (∀y)((∃x)(Fx & Gx) ⊃ Gy). Set out a finite possible world with only a and b in which this formula is true. (10 Marks)
7. Do trees for the following formulas. Is it a tautology? If a formula is not a tautology, give a finite possible world in which it is false.
a. (∀x)(Fx ∨ Gx) ⊃ ((∀x)Fx ∨ ∼(∀x) ∼Gx)
b. (∃x)(Fx ⊃ (∀y)Fy) (10 marks)
8. Specify a finite possible world with three things in which the following formula is true:
(∀x)(∃y)xRy & (∀x)(∃y) ∼xRy
(5 Marks)