The argument you presented is not valid. It contains a logical fallacy known as "affirming the consequent." Let's break down the argument and explain why it's invalid:
"If P then Q." This statement establishes a conditional relationship between P and Q. It means that if P is true, then Q must also be true. However, it doesn't tell us anything about the truth value of P or Q independently.
"Now we know that Q is not the case." This information tells us that Q is false. In other words, we have ¬Q (not Q).
"Therefore P cannot be true." This is the conclusion drawn from the previous two statements. The argument is claiming that because Q is false, P must also be false.
The problem with this reasoning is that the truth of the premises does not guarantee the truth of the conclusion. Just because Q is false (¬Q), it doesn't mean that P is false (¬P). The argument is making an incorrect leap from "¬Q" to "¬P" based on the conditional statement "If P then Q," which is a fallacy.
To demonstrate the fallacy more clearly, let's use a specific example:
- If it is raining (P), then the ground is wet (Q).
- Now we know that the ground is not wet (¬Q).
- Therefore, it is not raining (¬P).
This argument is flawed because there are other reasons why the ground might not be wet. For instance, it could be sunny outside (¬P), and the ground is dry as a result, even though the conditional statement about rain and wet ground remains true.
To summarize, the argument is not valid, as it commits the logical fallacy of affirming the consequent. Concluding that P cannot be true solely based on the knowledge that Q is false is not logically sound.