Is there any relation between an Argument and corresponding conditional?

Question

In my philosophy class, where we cover theory of knowledge, I leaned about Gettier problem. Gettier's counterexample to JTB is following: From A has Ford which is justified false belief, B can deduct ,using inference rule such as $\vee $ intro, A has a Ford or C is in L.A. And C is really in L.A. Then B has a justified true belief "A has a Ford or C is in L.A." So, JTB is not a necessary condition of knowledge.

But, if we have a set of false thing, then we can deduce anything from the set. So, I asked my prof that Is there any relation between counter-factual conditional and an argument which has at least one false premise. But he replied that what we concern here is argument not conditional. And we could deduct anything from the set only if at least one of the element of it is contradiction.

But I need some more explanation about it.

Answer 1

I'm not sure what your professor meant. The principle of explosion says you can infer anything from a contradiction. The proof is easy, we start with a contradiction and then validly prove some new arbitrary sentence:

1. A & ~A (Premise)
2. A (1, by conjunction elimination)
3. ~A (1, by conjunction elimination)
4. A or B (2, by disjunction introduction)
5. B (3, 4, by disjunction elimination)

But note that to be able to derive an arbitrary conclusion, it isn't enough just that one of your premises is false--you need the premises to be contradictory, i.e. necessarily false. The reason for this is that what the rules of inference do is tell us that if our assumptions are true, then so are our conclusions. Validity preserving rules of inference do not guarantee that we will not be able to infer falsehoods from false premises!

Now, in the Gettier case, somebody has a justified true belief that happens to be true only by accident, and therefore intuitively it seems that this person lacks knowledge.

I think what you are asking is something like this: If I have a justified true belief that A, then don't I also have a justified true belief in anything that follows from A, such as the sentence "A or C"? I think the answer there is: Yes. But in the Gettier cases, it is supposed to be a fact that I don't know A because my belief is just true by luck, and that would seem to be just as true of the sentence "A or C" as well. In other words, I don't know "A or C" any more than I know "A", and that's just what we'd expect. Even if we use our rules of logic perfectly correctly, if we feed garbage premises into the rules, we will still end up concluding falsehoods.