Gangadean and (Really) Easy Knowledge.

Note: My apologies to my readers for a slightly more technical post. I think (or at least hope) that it still manages to be accessible, but I'm certainly not the best judge of that. At any rate, I think it's a pretty nifty argument that was at least fun to think through.

On Gangadean's view, God could not fail to exist. That is to say, God must necessarily exist. This means that the proposition, "God exists" is a necessary truth (just like "2 + 2 = 4"). Now it's a matter of the conventions of logic that any proposition (whatsoever) entails a necessary truth. This means that an argument that has as its conclusion, a necessary truth, will be valid no matter what the premises end up being. Remember, an argument is valid by definition, if and only if, provided that the premises are true, the conclusion couldn't be false (note Gangadean agrees with this definition of validity).

The thing is whenever you are dealing with a necessary truth, it simply can't ever be false, period. So that means if you make an inference from any proposition whatsoever, to a necessary truth, the premise will entail the conclusion which means the inference will be valid by way of the definition of validity. Again an inference is valid whenever if the premises are true the conclusion is also true. But when you have a conclusion that can't possibly be false (i.e., a necessary truth), that definition is satisfied no matter what the premise (even a false premise).

Now if we have an argument or inference (an argument is just a set of inferences) that not only fits this pattern but also contains only true premises, you will have not only a valid argument, but a sound one as well. A sound argument is valid and has only true premises.

Now all of this might sound innocuous at first, but it get's problematic in a hurry. In fact, it would appear that for Gangadean and his kin, a person that believes that God exists on the basis of any true premises whatsoever, counts as knowing (with certainty) that God exists. This is based on the considerations we have just covered in conjunction with Gangadean's views about knowledge. He thinks that knowledge of God's existence (for instance) consists in truly believing that God exists on the basis of a sound argument with "God exists" as the conclusion. In other words, by Gangadean's lights a person that believes that God exists on the basis of the following argument, knows (certainly) that God exists.

(A) 

1) There are no married bachelor's
2) Therefore, God exists. 

As I mentioned, as a matter of logic, the premise entails the conclusion in (A) because any proposition whatsoever entails a necessary truth (and we're granting Gangadean that 2) expresses a necessary truth). Further, since the only premise of this argument is true (i.e., it's also a necessary truth), it turns out that this is also a sound argument. So strictly speaking, someone that actually believes that God exists in virtue of this argument knows (with certainty) that God exists according to Gangadean's theory of knowledge. After all, such a person couldn't possibly be wrong in believing a necessary truth and they are certainly justified in believing that God exists because they believe it on the basis of a logically valid and sound argument.

Admittedly, we'd be hard pressed to think that someone might be convinced by the argument just considered. The premise doesn't seem to have anything to do with the truth of the conclusion (despite the entailment). Still, the above is a proof--a demonstrative one. It isn't enough to say that it won't be very convincing for some people (or even many people) since Gangadean thinks that proving something comes apart from convincing someone of it, and it is the former that matters to knowledge not the latter. But this is a really odd result. It just seems bad, intuitively. More importantly, it also causes serious problems for Gangadean's view that the vast majority of believers outside his kin (both in the past and now) don't actually know that God exists. After all, provided that they believe that God exists on the basis of any true premise or set of premises, they will believe that God exists via a deductive proof and thus by definition know that God exists.

So perhaps Gangadean needs to rethink his views about what it takes to know something or at least what it means for something to count as a proof and what it means to believe something on the basis of a proof. Here's one more example that might be a little closer to home.

(B) 

1) It seems to me that God exists.
2) Therefore, God exists. 

Again the argument is valid insofar as 'God exists' is a necessary truth (which Gangadean is certainly committed to). Moreover, as long as the premise is true, the argument becomes sound. And the premise will be true just in case, it actually seems to me that God exists. So if I happen to be a subject for which it seems to me that God exists (I'm told there are a lot of people that this is true of), and God actually exists, then I have a sound proof on my hands in (B).

In other words, Gangadean's views about what is necessary and sufficient for knowledge faces a problem in it's present form. He says you know that God exists if and only if

(i) You believe that God exists.
(ii) You are justified (maximally) in believing that God exists.
(iii) And God actually exists.

But all three of these conditions is met by the person I am describing, from the Gangadeanian point of view. For ease of exposition, say that it seems to me that God exists and that I believe that God exists.

Now since Gangadean already accepts that God exists necessarily, he is going to grant me (iii)--after all truth is truth--it isn't like it's true for Gangadean and not for me. But this means conditions (i) and (iii) are satisfied as it concerns my belief that God exists. The proposition that God exists is true, and I believe it, ex hypothesi. Finally, if I hear or rehearse argument (B) and it is on that basis that I become convinced that God exists (and moreover, I can show it), then condition (ii) is fulfilled as well. After all, I will be maximally justified in believing something on the basis of a sound argument. This means on Gangadean's theory of knowledge, I (as the epistemic subject) know (with certainty) that God exists! But this seems really bad and I wonder how the Gangadeanian should respond.

It isn't enough for the Gangadeanian to say that there's something funny about my examples. I fully grant that this is a really bad result. The trick is for the Gangadeanian to explain specifically what goes awry here. It seems to follow straightfowardly from the theory of knowledge that Gangadean provides in conjunction with the claim that any proposition entails a necessary truth. He might try and change the very definition of entailment or argument validity so as to exclude such cases, but that move will be ad hoc (used simply as a means to save Gangadean's theory of knowledge from counterexample), unless there is some independent reason for these revisions.

So here is my challenge to the Gangadeanian. Do at least one of the following.

1) Show that arguments (A) and (B) do not actually follow from Gangdean's theory of knowledge (i.e., show that a person that believes that God exists on the basis of either argument doesn't count as having a justified true belief). But don't just assert it, prove it.

2) Independently motivate a revision to the logician's notion of entailments regarding necessary truths.

3) Argue how contrary to initial appearances, arguments (A) and (B) aren't so bad as ways of knowing that God exists.

The sharp reader will realize that this sort of "knowledge" by deductive proof of necessary truths will generalize. Plug in any necessary truth for the conclusion (e.g., any theorem of mathematics no matter how complicated) and any true premise whatsoever and you can come up with a sound argument which seems to make knowledge come too easily.