Question

Show that the following argument with hypotheses on lines 1–2 and conclusion on line c is

valid, by supplementing steps using the rules of inference and/or logical equivalences. Clearly label which rule you used in each step.

1. p → q
2. ∼ (q ∨ r)
c. ∼ p

Answer 1

Explanation:

`p \Rightarrow q \equiv (\\eg p \vee q)` [logical equivalence]

`\\eg (q \vee r) \equiv (\\eg q \wedge \\eg r)` [morgan laws]

if
`(\\eg q \wedge \\eg r)` is true, then
`\\eg q` is true and
`\\eg r` is too.

with
`\\eg q` true, then
`q` is false [double denial]

In the first equivalence it follows that
`\\eg p` is true [identity law]

Then it can be concluded that
`\\eg p`