Suppose my argument has a single premise and a single conclusion; if this single premise is always false, do I regard the argument as valid still? (Because technically, an argument can only be invalid if it can happen that premises are true and its conclusion is still false, right?) Eg Premise: John is married and John is single. Conclusion: John has children
(Not sure if that was a good example :)) )
Yes, an argument can be true even if the premise(s) is false. Validity is only about the structure of the argument, the form, which is why it's called formal logic. Having a false premise is irrelevant to the formal validity, and is instead an informal error.
As Mauro pointed out in his comment on your question, all conclusions can be shown to follow from a contradiction, so the example you gave is indeed a valid argument, although I don't think the example you gave is really that relevant to your real question. A more relevant example to your actual question would be an argument like this. "All bachelors are married. John is a bachelor. Therefore John is married." So just assume that john actually is a bachelor for this example. This is a completely valid argument, at least in terms of formal logic. Note, however, that even though it is a valid argument, it gives a false conclusion because it has a false premise (that all bachelors are married), and that premise will always be false.
Note also that it COULD give a true conclusion, if we had also made the other premise (that john is a bachelor) false, meaning that john really isn't a bachelor. Then the same argument would give the same conclusion, that john is married, but now the conclusion would be true, because the premise that john is a bachelor is false.
Basically it's just that validity determines whether that form of argument works and is completely consistent. If you take a false premise, then your knowledge about whether you'll get a true conclusion goes out the window (depending on how many premises are false and how many are true, etc).
In classical logic, yes.
Here is a proof by contradiction (therefore mildly unconvincing) that John, in your example, has children:
Let's assume that John do not have children and find a contradiction. Well, from the axioms, John is married and is single, which is a contradiction. Therefore the assumption is wrong, and John have children.
Note that you can also prove that John do not have children, and that he is also not married and not single.
