How does a teacher respond to a student that says that she does not see how a compound statement consisting of two simple sentences that are false can be true? How do teachers respond to a student that says that the hypothesis is false, an a argument cannot be valid? Why are commands and questions not statements?
If a student tells me that an argument that has a false hypothesis cannot be valid, I would reply that we need to look carefully at the meaning of validity in the context of logic. In everyday speech, we tend to use “valid” to mean the same thing as “true” or “accurate.” In logic, this is not the way the term is used.
In logic, an argument is valid if the conclusion would have to be true given that the hypotheses are true. In other words, we do not ask whether the hypothesis is true. Instead, we simply assume that it is true. Assuming the hypothesis is true, we then determine whether the conclusion must also be true. Let us look at an example of this. The statements for this example are:
- All Presidents of the United States are white men.
- Barack Obama is President of the United States.
- Therefore, Barack Obama is a white man.
We know this statement is not true (at least by our standards since we tend to say that someone who is half black and half white is black). However, it is still valid. Our hypothesis is that all presidents are white men. We know that this is not true, but we do not care. If all presidents were white and if Obama is president, he would have to be a white man. Therefore, this argument is valid even though the hypothesis is not true.