Determine if the argument is valid or invalid. Give a reason to justify answer.
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
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)
To determine if the argument is valid or invalid, we need to analyze the structure of the argument and see if it follows a valid logical pattern.
The argument can be represented as follows:
1. If it is cold, then you need a coat. (Premise)
2. You do not need a coat. (Premise)
3. It is not cold. (Premise)
To determine the validity, we need to see if the conclusion necessarily follows from the premises.
In this case, the argument is invalid.
If it were valid, the conclusion would be "It is not cold," but this conclusion is not necessarily true based on the given premises. The premises do not support the conclusion, as they only provide information about needing a coat if it is cold and not needing a coat. The argument does not establish a direct relationship between the statement "It is not cold" and the other premises.
Therefore, the correct answer is: Invalid by fallacy of the converse.
The argument is valid by the law of contraposition.
The given argument states that if it is cold, then you need a coat. The contrapositive of this statement is if you do not need a coat, then it is not cold.
In the argument, it is stated that you do not need a coat. Therefore, according to the contrapositive, it implies that it is not cold.
Since the conclusion of the argument aligns with the contrapositive statement, the argument is valid.