Question

A set of hypothesis and a conclusion are given. Use the valid argument forms to deduce the conclusion from the hypothesis.

(i) p v q,

(ii) q ->r,

(iii)(p^s) ->t,

(iv) not r,

(v) (not q) -> (u ^ s)

(vi) t

Answer 1

Hypothesis:

1) p V q

2) q->r

3) (p s) -> t

4) ~r

5) ~q -> (u s)

Conclusion:

t

Proof:

q->r can be written as ~r->~q [Contrapositive]

Therefore we get 6) ~r->~q.

From 4) and 6) we get:-

7) ~q [Modus Ponens]

From 7) and 5) we get:-

8) u s [Modus Ponens]

From 8) we get:-

9) s [Simplification]

From 1) and 7) we get:-

10) p [Disjunctive Syllogism]

From 9) and 10) we get:-

11) p s [Conjunction]

From 11) and 3) we get:-

t [Modus Ponens]

Hence, Proved.