Question

Determine if statement (3) follows from statements (1) and (2) by the Law of Detachment, the Law of Syllogism, or is invalid.

(1) If you buy shoes, then you won’t be able to afford a Mother’s Day gift.
(2) You could not afford a Mother’s Day gift.
(3) You bought shoes.

Answer 1

The argument provided by the student commits the logical fallacy of affirming the consequent and is invalid. The lack of affordably for a Mother's Day gift does not necessarily imply that shoes were bought, as there could be multiple reasons for not affording the gift.

Step-by-step explanation:

The student has asked to determine the validity of a logical argument using given statements. We are provided with:

1. If you buy shoes, then you won’t be able to afford a Mother’s Day gift.
2. You could not afford a Mother’s Day gift.
3. You bought shoes.

The provided argument is an example of the logical fallacy known as affirming the consequent. This fallacy occurs when one assumes that the presence of the consequent (in this case, not being able to afford a Mother's Day gift) necessarily confirms the preceding antecedent (buying shoes), which is not logically sound. Therefore, statement (3) does not necessarily follow from statements (1) and (2) and is thus invalid. The mistake lies in assuming that the only way one could not afford a Mother's Day gift is by purchasing shoes. There could be many reasons for this lack of affordability, and the fact that it happened does not specifically imply that shoes were purchased.

When evaluating logical arguments, it is essential to differentiate between necessary and sufficient conditions. Statement (1) can be read as “Buying shoes is a sufficient condition for not being able to afford a Mother's Day gift,” but it is not a necessary condition. Thus, while buying shoes would lead to not being able to afford the gift, not being able to afford the gift does not necessarily mean that shoes were purchased.