Question

Given the following problem: "If my engine turns over (E) and it does not start (∼S) then the distributor cap is bad (∼D) or there is no petrol in the tank (∼P) ", My engine turns over (E), it does not start (∼S), and there is petrol in my tank (P).

a. Translate these sentences into propositional logic
b. Put the logical expressions derived above into clausal form
c. Show that 'My distributor cap is bad' follows as a logical consequence of this set of sentences using resolution method.

Answer 1

To translate the given sentences into propositional logic, use letters to represent the statements and logical connectives. Put the logical expressions in clausal form by using the distributive property of logical operators. Use the resolution method to show that the conclusion follows logically from the premises.

Step-by-step explanation:

To translate the given sentences into propositional logic, we assign letters to represent the statements. Let:

E represent 'My engine turns over',

S represent 'It does not start',

D represent 'The distributor cap is bad',

P represent 'There is petrol in the tank'.

The given sentences can be translated into propositional logic as:

E ∧ ¬S ∧ P

To put the logical expressions in clausal form, we use the fact that p ∧ q ∧ r is equivalent to (p ∨ ¬q) ∧ (p ∨ ¬r). Therefore, the clausal form of the propositional logic expressions above is:

(E ∨ ¬S) ∧ (E ∨ ¬P)

To show that 'My distributor cap is bad' follows as a logical consequence of this set of sentences using resolution method, we need to negate the conclusion: ¬D. Then, we find the resolvents of the clauses (E ∨ ¬S), (E ∨ ¬P), and ¬D, until we reach a contradiction. If we obtain the empty clause, it means that the conclusion follows logically from the given premises.