Is it possible to (knowingly, truly) believe a contradiction? (Part 1)

Let’s consider 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. I’ll work through the dialectic.

1. Is it possible to believe a contradiction?

A contradiction is a proposition or set of proposition which is “necessarily” false. It’s important to be very very careful about the formulation of “contradiction” because the standard family of notions (inconsistent, necessarily false, cannot be true, has no interpretation which makes it true, etc.) are all equivalent in a certain setting and yet can be prised apart in other settings. So, for example, in a bivalent logic, being false entails being not-true and vice versa. This isn’t true for non-bivalent logics or  logics with value overloading.

I’m going to stick with the “must be false” version because, if there’s any consensus about contradictions, it’s that they are false. (You might think they are true as well, but that’s as well. It be interesting to think about contradictions, even true contradictions, that weren’t false. But it’s also weird.)

(Assumption 1: Contradictions are false.)

All contradictions are false, but, of course, not all false proposition are contradictions. We definitely can believe all sorts of things which can be or are false. If anything is true, it true that everyone has false beliefs. So it’s possible to believe false propositions. Contradictions are false proposition, so, what’s the big deal in believing them?

(Assumption 2: We can believe false propositions.)

The usual move here is to try to connect believing with believing-as-true. Most false statements we believe have the possibility of being true. Thus, to believe something false is to be mistaken. I.e., you assign the wrong truth value to the proposition. Herein lies the problem with believing a contradiction. If we recognise that it’s a contradiction, then we must recognise that it cannot be true. Thus, presumably, that prevents it from believing it’s true (since we know its not!). Thus we can’t believe it.

(Assumption 3: We cannot believe-as-true what we know to be false.)

At this point, we have a problem in that it seems that people do believe contradictions (or at least have contradictory beliefs). However, we have an out. In the case of believing false contingent statements, the solution was our ignorance of the correct truth assignment. In the case of contradictions, perhaps we don’t know it’s a contradiction. After all, some contradictions are really hard to recognise (if Frege could miss Russell’s paradox, well, don’t feel too bad about falling into contradiction). This sometimes gets cashed out (cf. Rescher and Brandom; non-adjuctive logics generally; lots of other approaches) as we can believe inconsistent sets of propositions, but we cannot believe self-contradictions (or, perhaps, blatant self-contradictions). That is, we can have a big sea of beliefs in which we have both P and ~P but we don’t recognize that they are both in there (it’s a big sea), at least at the same time. “Of course”, the rational thing to do when the contraries come into simultaneous view is to give one up.

(This might not be easy if they follow in complicated ways from other parts of our 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 now yields the second question: Can we knowingly believe a contradiction?