Question

Here is an argument to show that if n2 is an odd integer, then n is an odd integer. Assume that n2 is an odd integer. Then n2 = 2k + 1 for some integer k. Let n = 2a + 1 for some integer a. This shows that n is an odd integer. There is a fallacy being used in this argument called circular reasoning.

true or false

Answer 1

If n^2 is odd, then n is odd is a true statement and does not exhibit circular reasoning. Circular reasoning, or begging the question, occurs when an argument assumes the truth of the conclusion without proper justification.

Step-by-step explanation:

The statement that if n2 is an odd integer, then n is an odd integer, is indeed true and constitutes a form of deductive reasoning. The argument works because if you assume n2 = 2k + 1, which defines an odd integer, and express n also in the form of 2a + 1, displaying the structure of an odd integer, you are not engaging in circular reasoning. Instead, you are showing that a square of an odd number must be odd. Circular reasoning occurs when the premises assume the truth of the conclusion or when the argument relies on the conclusion being true without proper justification.

For example, arguing that a book is divinely inspired and therefore the word of God, which then is used to prove the existence of God, is circular reasoning because the existence of God is assumed in order to prove itself. This type of flawed reasoning is also known as begging the question. A valid counterexample or the diagnosis of error in reasoning are important to pinpoint incorrect arguments or fallacies.

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