"If my computations are correct and I pay the electric bill, then I will run…
2015
"If my computations are correct and I pay the electric bill, then I will run out of money. If I don't pay the electric bill, the power will be turned off. Therefore, If I don't run out of money and the power is still on then my computations are incorrect."
Convert this argument into logical notations using the variables \(𝑐,𝑏,𝑟,𝑝\) for propositions of computations, electric bills, out of money and the power respectively. (Where ¬ means NOT).
- A.
\(\text{if } (c \wedge b) \rightarrow r \text{ and } \neg b \rightarrow \neg p, \text{ then } (\neg r \wedge p) \rightarrow \neg c\) - B.
\(\text{if } (c \vee b) \rightarrow r \text{ and } \neg b \rightarrow \neg p, \text{ then } (r \wedge p) \rightarrow c\) - C.
\(\text{if } (c \wedge b) \rightarrow r \text{ and } \neg p \rightarrow \neg b, \text{ then } (\neg r \vee p) \rightarrow \neg c\) - D.
\(\text{if } (c \vee b) \rightarrow r \text{ and } \neg b \rightarrow \neg p, \text{ then } (\neg r \wedge p) \rightarrow \neg c\)
Attempted by 50 students.
Show answer & explanation
Correct answer: A
Formalization of premises: (c ∧ b) → r and ¬b → ¬p.
Premise 1: (c ∧ b) → r (If computations are correct and I pay the bill, then I will run out of money.)
Premise 2: ¬b → ¬p (If I don't pay the bill, the power will be turned off.)
Derivation:
Take the contrapositive of Premise 1: ¬r → ¬(c ∧ b), which is equivalent to ¬r → (¬c ∨ ¬b).
Take the contrapositive of Premise 2: p → b.
Assume ¬r ∧ p. From p → b and p we get b. From ¬r → (¬c ∨ ¬b) and ¬r we get (¬c ∨ ¬b). Since b is true, ¬b is false, so the disjunction yields ¬c.
Conclusion: Therefore (¬r ∧ p) → ¬c. The formalization with premises (c ∧ b) → r and ¬b → ¬p and conclusion (¬r ∧ p) → ¬c is valid and matches the natural-language argument.
A video solution is available for this question — log in and enroll to watch it.