Topics Logic

Question This is a question about pure logic. There are two theries: Theory A and Theory B. Theory A assumes AssumptionA. Theory B assume AssumptionB. The two assumptions are mutually exclusive: if AssumpionA then not AssumptionB and vice versa. I believe that a philosophical result is that Theory A and Theory B cannot prove anything about each other. All you can do is preface each result with the assumption. For example, if Theory A proves X and Theory B proves Y, then we can say "If AssumptionA, then X" and "If AssumptionB then Y". Who first proved this? Where is it documented? Eugene

Accepted:
August 18, 2016

Comments

I'm going to step through

I'm going to step through this carefully to make sure I follow.

We have two theories: A and B. Theory A has an assumption: A and theory B has an assumption B. And A and B are mutually exclusive—can't both be true.

Let's pause. To say that a theory has an assumption means that if the theory is true, then the assumption is true. It doesn't mean that if the assumption is true, then the theory is true. A silly example: the special theory of relativity assumes that objects can move in space. But from the assumption that objects can move in space, the special theory of relativity doesn't follow; you need a lot more than that. Otherwise, the "assumption" would be the real theory.

You ask if it's true that neither theory can prove anything about the other. If I understand the question aright, it's not true. For one thing, trivially, if we take A as a premise, then by your own description, it follows that B is false. That seems like a case of proving something about B

But let's consider something a little more interesting. Suppose that theories A and B are about the same general phenomena, and have many assumptions in common. They differ, however, in that one makes an assumption X and the other makes the assumption that X is not true. Although I don't have an example handy (it's late in the afternoon), I'm pretty sure there are many examples. (Okay, here's one: many-worlds quantum mechanics assumes that quantum theory is complete as it stands; Bohmian quantum mechanics assumes that it needs to be supplemented with the so-called "quantum potential," at least on one version.) Suppose I like theory A. I'm aware, however, that theories A and B have an important assumption P in common. I prove that from assumption P, a certain interesting conclusion Q follows for theory A. But since B shares assumption P, Q also follows for theory B. So I've proved something about theory B from within A. And this might be important. It might be that even though Q is consistent with B, Q seems much more "natural" in the context of theory A.

Sorry that I've left that awfully abstract, but it's a style of argument that we see a lot in philosophy and, I think, also in theoretical disputes in science. A proponent of a theory shows that her theory has a certain consequence that seems natural, reasonable, whatever in that context. She point out that since her theory and her rivals theory share the crucial assumptions, the consequence also holds for her rival's theory. But the consequence is not welcome from her rival's point of view. In fact, proving it may have been difficult, and it may be something her rival assumed or hoped didn't apply to his theory. That's potentially a strike against the rival's position, though the devil is in the details.

The overall point: it's a matter of basic logic that if a set of premises P1, P2, P3... imply a conclusion Q, then they still imply Q even if we add other premises. Who proved that? No one in particular. It's part of the very concept of logical implication. Since rival theories A and B can share assumptions, some conclusions about B can be proved from within A. That is: some things that follow from A also follow from B exactly because of those shared assumptions.

Whether this proves to be interesting depends on how A and B are related. But I'm pretty sure that no philosopher or logician would disagree with what I've said, even though there's no particular documentary source for this point. It's just a consequence of the fact that to say P implies Q is to say that it's inconsistent to hold P and reject Q.