Question

construct arguments determine a valid conclusion from the given statements, if possible. then state whether your conclusion was determined using the law of detachment or the law of syllogism. if no valid conclusion can be determined, select no valid conclusion. justify your argument. given: if you are an athlete, then you enjoy sports. if you are competitive, then you enjoy sports. select choice select choice

Explanation:

Step1: Identify the logical statements

Let \( p \): "you are an athlete", \( q \): "you enjoy sports", \( r \): "you are competitive". The given statements are \( p \to q \) and \( r \to q \).

Step2: Analyze possible logical laws

The Law of Detachment requires a statement \( p \) (or \( r \)) to be true to conclude \( q \). The Law of Syllogism requires a common term between the consequent of one and the antecedent of another (e.g., \( p \to q \) and \( q \to r \) to get \( p \to r \)). Here, both have consequent \( q \), no common term for syllogism, and no given \( p \) or \( r \) is true for detachment.

Answer:

No valid conclusion can be determined. The two conditional statements (\( \text{athlete} \to \text{enjoy sports} \) and \( \text{competitive} \to \text{enjoy sports} \)) share the same consequent but different antecedents. The Law of Detachment needs a true antecedent to infer the consequent, and the Law of Syllogism needs a common term between the consequent of one and the antecedent of the other (which is not present here), so no valid conclusion is possible.