Question

12) Write the argument below in symbolic form. If the argument is valid, prove it. If the argument is not valid, give a counterexample:

If I watch football, then I don't do mathematics.
If I do mathematics, then I watch hockey. If I don't watch hockey, then I watch football​

Answer 1

The student's argument is written in symbolic form and analyzed for validity. Upon examination, the argument does not have a standard valid form, such as disjunctive syllogism, and a counterexample demonstrates that the premises do not necessitate the conclusion, proving the argument to be invalid.

Step-by-step explanation:

To write the argument in symbolic form, we can use the following symbols for the statements:

• Let W = I watch football
• Let M = I do mathematics
• Let H = I watch hockey

The argument in symbolic form is:

1. W → ¬M (If I watch football, then I don't do mathematics.)
2. M → H (If I do mathematics, then I watch hockey.)
3. ¬H → W (If I don't watch hockey, then I watch football.)

To test if the argument is valid, let's analyze its structure using deductive reasoning. We have three conditionals which form a chain argument. However, from the premises provided, the conclusion is not explicitly stated. A disjunctive syllogism or other standard forms don't apply directly here, hence we must look for the logical flow and determine validity.

To find a counterexample, assume the premises are true but the conclusion is false: it's possible that I don't do mathematics (not-M), therefore I must watch hockey (H) according to the second premise M → H, which contradicts the third premise (¬H → W).

Our example shows that the premises can be true, but they don't necessitate the conclusion in a valid manner. Therefore, the argument is invalid because the conclusion does not logically follow from the premises.

Answer 2

The argument can be written in symbolic form, and it is valid.

Step-by-step explanation:

To put the argument into symbolic form, we can assign variables to the statements:

1. If I watch football, then I don't do mathematics: F → ¬M
2. If I do mathematics, then I watch hockey: M → H
3. If I don't watch hockey, then I watch football: ¬H → F

This argument can be written symbolically as:

• F → ¬M
• M → H
• ¬H → F

To prove the argument is valid, we need to show that the conclusion follows logically from the premises. By using the rules of logical deduction, we can show that if the premises are true, then the conclusion must also be true. In this case, the argument is valid.