Write the following argument symbolically and then determine its validity:
1. If you are fair-skinned, you will sunburn.
2. If you sunburn, you will not go to the dance.
3. If you do not go to the dance, your parents will want to know why you didn't go the dance.
4. Therefore, you are not fair-skinned.
To write the argument symbolically, we can assign letters representing each statement:
1. Let A represent "You are fair-skinned"
2. Let B represent "You will sunburn"
3. Let C represent "You will not go to the dance"
4. Let D represent "Your parents will want to know why you didn't go to the dance"
Using these symbols, we can write the argument as follows:
1. A -> B
2. B -> C
3. C -> D
4. Therefore, ~A
The symbol '->' represents implication or "if-then" statements, "~" represents negation or "not."
Now, let's determine the validity of the argument.
One way to do this is to use logical inference, specifically the rules of inference, to determine if the conclusion follows logically from the premises.
The structure of this argument is called a chain argument, where each statement leads to the next.
The argument can be represented as a chain of implications:
A -> B -> C -> D
To prove the validity, we can assume the truth of the premises (A, B, and C) and check if the conclusion (D) must be true based on those assumptions.
Assuming A, we can infer B based on the first premise. Assuming B, we can infer C based on the second premise. And assuming C, we can infer D based on the third premise.
Therefore, if the premises are true, the conclusion must also be true. Hence, the argument is valid.