Write the following argument symbolically and then determine its validity:
If you are fair-skinned, you will sunburn.
If you sunburn, you will not go to the dance.
If you do not go to the dance, your parents will want to know why you didn’t go to the dance.
Your parents do not want to know why you didn’t go to the dance.
Therefore, you are not fair-skinned.
To represent the argument symbolically, we can assign variables to the statements given:
Let F represent "You are fair-skinned."
Let S represent "You will sunburn."
Let D represent "You will go to the dance."
Let P represent "Your parents will want to know why you didn't go to the dance."
The premises can then be symbolized as:
1. F → S (If you are fair-skinned, you will sunburn.)
2. S → ¬D (If you sunburn, you will not go to the dance.)
3. ¬D → P (If you do not go to the dance, your parents will want to know why you didn’t go to the dance.)
4. ¬P (Your parents do not want to know why you didn’t go to the dance.)
The conclusion can be symbolized as:
¬F (Therefore, you are not fair-skinned.)
To determine the validity of the argument, we can use the rules of logical inference. We will derive the conclusion from the premises using valid logic rules.
1. F → S (Premise 1)
2. S → ¬D (Premise 2)
3. ¬D → P (Premise 3)
4. ¬P (Premise 4)
5. F → ¬D (Hypothetical Syllogism: 1, 2)
6. ∴ ¬F (Contrapositive: 3, 4, 5)
Therefore, the argument is valid, and we can conclude that "you are not fair-skinned."