Question

Determine if this argument is valid. If so, indicate the law(s) of logic used.

a. If a figure has three sides, then it is a triangle.
b. If a figure is a triangle, then the sum of the interior angles is 180°
c. Therefore, if the sum of the interior angles is not 180 degrees, then the figure does not have three sides.

Answer 1

The student's argument is valid and relies on the logic of a disjunctive syllogism, particularly modus tollens. The structure ensures that if the premises are true, the conclusion must necessarily be true, which matches the properties described in the argument about a figure being a triangle.

Step-by-step explanation:

The argument presented can be analyzed for validity using logical laws, particularly the structure known as disjunctive syllogism.

Lets translate the original statements into a logical form:

1. If a figure has three sides (P), then it is a triangle (Q).
2. If a figure is a triangle (Q), then the sum of the interior angles is 180° (R).
3. If the sum of the interior angles is not 180° (not R), then the figure is not a triangle (~Q).
4. Therefore, the figure does not have three sides (~P).

The pattern here is a modus tollens, which is a valid argument form. The first premise (P > Q) and the second premise (Q > R) combine to establish that P is sufficient for R. We then have not R, which allows us to deduce not P by modus tollens. So, this argument is valid because if the premises are true, the conclusion necessarily follows; the sum of interior angles of a figure being not 180° means that the figure cannot be a triangle and therefore must not have three sides.