Is it possible to (knowingly, truly) believe a contradiction? (Part 2)

We are considering three variants of a question:

  1. Is it possible to believe a contradiction?
  2. It is possible to knowingly believe a contradiction?
  3. It is possible to truly believe a contradiction?

Each question arises from a certain way of saying “yes” to the prior question.

In part 1, I argued that we can  know we believe a contradiction in the sense that it’s highly likely for us to have contradictory beliefs:

This “yes” answer roughly says: We can believe a contradiction if we do not know we believe it.

Even this needs some care. Given how crap we all are at thinking, we can be pretty sure we have inconsistent beliefs. So, we might well know that we believe a contradiction! But knowing that we believe some contradiction doesn’t mean we know which contradiction we believe. So maybe that’s ok!

This raise the question can we knowingly believe a contradiction in the sense of knowing which contradiction it is we believe.

2. It is possible to knowingly believe a contradiction?

If it is possible to have believed a contradiction (though we didn’t know it at the time), then it’s possible to know that we have done so. Thus, it’s possible to come to know we believe a contradiction. This isn’t all that uncommon to see!

For me, the paradigm is Frege. I’m pretty sure he believed in his axioms for arithmetic. They implied Russell’s paradox. “Arithemetic totters!” (so the story goes). He came to realise there was a contradiction in his belief set. He could even name it! Russell’s paradox (the set of all sets that do not contain themselves)!

So we can have contradictory beliefs. We can come to recognise them as contradictory. So what’s the barrier to continuing to believe them? We might even say that we know that we can do so, since we were doing so all along!

But but but! Perhaps rationality demands that when we recognise that we are believing a specific contradiction that we give up one or the other contrary (or both).  This seems to reduce to the irrationality of believing a known falsehood.

It may be irrational, but so?! While it may be irrational to believe a contradiction it doesn’t seem impossible. The basic intuition is that believing is believing as true. For a “known” false contingent belief, we can sort of “forget” that it is false, or pretend that it is true (which is easy to do). But there’s no scenario in which a contradiction is true, so how can be believe it as true? On this line, if we cannot believe it as true, we cannot actually belief it.

What, then, were we doing when we had the contradictory belief set? Well, there ignorance came in. We believed things as true which were necessarily false because we didn’t know that they were necessarily false. We didn’t put together the facts of their falseness. And this is why when we come to realise the contradiction, the story goes, we can’t believe it. There’s no place to hide from the falsity, so our belief (as true) vanishes.

(Of course, with a strong will to believe we might cloud our insight into the falsity. But that’s just either denial or imposing ignorance.)

I think we need a very strong version of “knowingly” to make this work. To wit, the belief, b,

  1. must be fully occurrent (all parts are strongly “in view”)
  2. must be unmistakably contradictory (P & ~P, please)
  3. must be believed as true

If it’s not fully occurrent we can keep rapidly shifting our focus and our believing. If it’s not unmistakably contradictory, we can fail to understand it fully. If it doesn’t have to be believed as true, well, then what’s the deal? (I can easily keep a contradiction in some buffer!)

This works only if you must believe B exclusively as true. Note that it’s important that whether we can believe B as non-exclusively true (or as false) is separate from whether B can be non-exclusively true.  That is, we can have the following possibilities:

  1. (Some variant of the law of excluded middle) B, if true, cannot be false, and vice versa
  2. B can be simultaneously true and false
  3. If we believe B, we must believe it exclusively as true (i.e., we must believe it to be not false)
  4. If we believe B, we must believe it as true (but we may also believe that it is false)
  5. Believing B is independent of our attitudes toward its truth status

Now, we clearly have loads of logics where 2 holds, so we don’t have a mathematical problem. The issue, of course, is whether these capture the behavior of believing. After all, even if 2 is the right logic for things, 3 might still be right for how we believe. For example, 5 just doesn’t seem to be true of beliefs (rather than “entertainings”).  In the worst case, it pushes the problem back a little. (I.e., even if not all believings are believing as exclusively true, what happens when we try to believe something “necessarily” false as exclusively true?)

(Note that there’s a symmetry with disbelieving tautologies.)

We can think that we shouldn’t, by and large, believe falsehoods (though many falsehoods are valuable to believe because they keep us sane, or on the straight and narrow, or are generally useful in our cognitive and affective systems) while still acknowledging that we are able to believe them. The knowingly challenge suggests that the only way we can believe false things is by not (fully?) knowing that they are false. (We can handle the case of massless ropes by treating them as useful fictions; we can “believe” false things as “true in a fictional context” or the like.)

Perhaps the heart of the problem of occurrently believing as true a known-as-such contradiction is that there is no room to articulate the truth.

Consider the proposition that my left foot is colored neon pink. This is not true at the moment, but it’s easy for me to construct a situation in which it is true (i.e., it has lots of models). The proposition that by left foot is both colored neon pink and not colored neon pink has no models. If I try to construct one (even after abandoning visualisation), I fail because it has no models. We can conceive of ordinary believing as true of falsehoods as framing them in one of their models. We can then always relativize our believe as a conditional one, If we’re in situation S, then B is true. If S is a model of B, this conditional is itself true and our knowingly believing a falsehood reduces to a believing as true of a conditional.

This move is not available for us with a contradiction. There’s no backing conditionals available to us so no way to capture what truth we are believing.

However, perhaps there’s a way out. As I wrote earlier, there are logics wherein propositions can be both true and false, which suggests that there are models which are somewhat different than classical first order models. Or more importantly, if a false proposition can also be true that suggest that there is a (perhaps non-conditional) backing truth we can appeal to to substantiate our belief as true.