Question

Which of the following truth assignments demonstrates that the argument below is not valid?

P: It is raining.
Q: The streets are wet.
R: The sprinklers are on.
S: The dog is dry.
a) P: False, Q: False, R: False, S: True
b) P: True, Q: False, R: True, S: True
c) P: True, Q: True, R: False, S: False
d) P: False, Q: True, R: True, S: True

Answer 1

Option d) demonstrates that the argument is not valid, as it provides a scenario with the truth assignment where it is not raining (P is False), but the streets are wet (Q is True), which breaks the usual causal relationship assumed by the argument.

Step-by-step explanation:

The question revolves around assessing the validity of an argument by examining truth assignments for different propositions. To demonstrate that an argument is not valid, we need to show that it is possible for the premises to be true while the conclusion is false. Looking at the propositions given:

• P: It is raining.
• Q: The streets are wet.
• R: The sprinklers are on.
• S: The dog is dry.

For a valid argument, typically, if 'P' then 'Q' would be true. However, to showcase invalidity, we need to find an example where 'P' is true and 'Q' is false, or 'P' is false and 'Q' is true. Here are the truth assignments:

1. P: False, Q: False, R: False, S: True
2. P: True, Q: False, R: True, S: True
3. P: True, Q: True, R: False, S: False
4. P: False, Q: True, R: True, S: True

We're looking for a scenario where the usual causal relationship between P and Q is broken (i.e., it is raining (P), but the streets are not wet (Q), or it is not raining (P), but the streets are wet (Q)). Option d) provides a scenario where it is not raining (P is False), but the streets are wet (Q is True), potentially due to the sprinklers being on (R is True). This option shows that the conclusion 'If it is raining, then the streets are wet' can be false even if the streets are wet, thereby demonstrating that the argument is not valid.