Using rules of inference
Let r be the proposition “It rains,”
let f be the proposition “It is foggy,”
let s be the proposition “The sailing race will be held,”
let l be the proposition “The life saving demonstration will go on,” and
let t be the proposition “The trophy will be awarded.”
premises (¬r ∨¬f) → (s ∧ l),
s → t, and
¬t.
We want to conclude r.
1. ¬t
2. s →t
3. ¬s ( modus tollen 1&2)
4. (¬r ∨¬f) → (s∧l)
5. (¬(s ∧l)) →¬(¬r ∨¬f) ( contrapositive)
6. (¬s∨¬l) → (r∧f)
7. ¬s∨¬l (addition from 3)
8. r ∧f (modus ponens)
9. r(conclusion)