Determine if the argument is valid or invalid. Give a reason to justify answer.

Question

If it is cold, then you need a coat.
You do not need a coat.
It is not cold.
(Points : 2)
Valid by the law of detachment
Valid by the law of contraposition
Invalid by fallacy of the converse
Invalid by fallacy of the inverse
Valid by the law of syllogism
Valid by disjunctive syllogism

Answer 1

Let
P: "it is cold"
Q: "you need a coat"

The first statement is therefore
P → Q

The second part has been written in two separate statements, ¬Q, ¬P.

If we can assume it to be related as "It is not cold, therefore you do not need a coat", or
¬Q → ¬P, then you can take one of the given choices, using the following help:

law of detachment:
(P∧Q) ∧ P => Q

law of contraposition:
(P → Q) ≡ (¬Q → ¬P)

converse of P → Q:
Q → P

inverse of P → Q:
¬P → ¬Q

law of syllogism (transitivity):
(P → Q) ∧ (Q → R) => (P → R)

disjunctive syllogism:
[P → (Q ∨ R)] ∧ ¬R => (P → Q)