Why I Don’t Believe The Law of the Excluded Middle

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The Law of the Excluded Middle says that there are only two “truth-values:” namely True and False. It also says that X and not-X is false, while X or not-X is true. If we accept LoEM already, then LoEM is either true or false. However, there are already two reasons I don’t like doing that:

  1. It introduces the concept of “actually true” and “actually false” independent of whether or not we’ve chosen to believe something.

  2. It’s self-referential. So if I believe it, that means I have to believe it really hard. Weakly believing it is a lot like not believing it at all, which I find to be unfair.

Because it is asking to me to believe it completely if I believe it at all, I feel more comfortable choosing to consider it “false” on my terms, which is merely that it gave me no other choice because it defined itself to be false when it is only believed weakly.

It’s a bit like associating the interval [0,1) with False and {1} with True. This is because if I have some collection of propositions (say {A_i}), then if I take the “and” of them all together to make A, then A is false if at least one of the A_i’s is false. Thus, not only is there no sense of a “partially true” proposition, even if I could measure the proportion of true pieces of a proposition, any proportion besides exactly 1 is considered exactly 0. If you’ve given me a collection of statements, and all you told me is that one of those statements is wrong, but not which one, I would still feel as though you’d given me something useful.

Conversely, if you gave me a collection of statements, and you told me that all you knew about them was that the “or” of them all together was true, I’d probably feel like you could have given me a bunch of junk.

But if we think in “calculus” mode, then a huge statement made of lots of regular-sized statements concatenated together with “and” can be identified with a regular-sized statement that is implicitly constructed out of a potentially infinite number of infinitesimal-sized concepts, and thus could be factored into a huge statement made of regular-sized statements concatenated with “and.” Thus if only one if these infinitesimal-sized concepts is false, the whole thing is false.

We know that “Newtonian physics” is technically not true. Or do we? All we know is that it has been superseded by several more advanced, more accurate-within-their-own-domains, but mutually-incompatible theories. Which means those are technically not true either. But we continue to use all of them, including Newtonian physics. How is that possible?

This means that in LoEM, the “false” is made enormously (infinitely, in the limit) powerful, compared to the “true.” But this doesn’t seem to match how “false” apparently works in reality. This is basically why I doubt the LoEM.

You know what’s really great about doubting the LoEM? It’s that even if you want to assume it’s true, you can, and you can see what happens. Furthermore, I do trust that what we know about what happens is indeed what happens when you assume the LoEM, and therefore, that we don’t really face that much danger when we want to entertain counterfactuals. If you want absolute certainty (in the form of consistency), apparently you have to make do with not knowing that you have it.

People who believed in the LoEM wanted to know things with absolute certainty. They also were worried about what would happen if you believed in things that were inconsistent. Let’s say you take a bunch of axioms. Now, suppose you could “prove false” with it (that means prove a contradiction). This is equivalent to one of your statements being the negation of another. Since your theory is all of your axioms (and derived propositions) “and”-ed together, your entire theory is false.

I assume that “false” means something meaningful, as well as that it’s generally a bad thing when you believe something that is false. If “false” meant in reality what it means in LoEM, it would be infinitely bad to believe something false. One bad belief spoils the entire batch!

An example of something bad happening that could come from believing in a contradiction is called the “Principle of Explosion.” This is what people refer to when they say, “From a contradiction, anything follows.” How it works feels like a cheat: Assume you believe any proposition P as well as its negation, not-P. Now, suppose you have any proposition Q. Because you believe P, you also believe Q or P. Does that seem like it should get us very much?

But we also believe not-P. Therefore, because Q or P, and also that not-P, therefore Q! Does that seem valid to you? It doesn’t to me!

Those steps above constitute a fully general proof for “Q” which is an arbitrary statement. So I’ve already proven “everything” just from doing that. But a proof is supposed to do the work of convincing me that something is true. That doesn’t feel like a proof of anything, let alone a proof of everything. So assuming a contradiction does not allow us to feel like we’ve proven whatever we want to be true.

That’s because if I wanted to prove something arbitrary from a contradiction, the proof would need to take the same form as described above, which already does not feel convincing. So I strongly feel that from a contradiction, anything does not follow.

People use the Principle of Explosion rhetorically during arguments in the following way: Arguer A makes a “big claim” that Arguer B finds preposterous, but hard to refute. Arguer B spends time debating Arguer A, trying to find any pair of claims by the latter that seem to contradict. Once Arguer B finds something (which might not be hard to do once pressing A for lengthy explanations), B claims “from a contradiction, anything can be proven! Which is how you arrived at (big claim).”

But while A might indeed be wrong about (big claim), that is not how A arrived at it. A did not merely take a proof similar to the one I gave above, consider that valid, then swap out Q for (big claim), and feel triumphant. He did not do that because no one could do so and hope to feel confident in any arbitrary claim they thought they could prove that way. That isn’t how people reason, so we can’t claim that people arrive at faulty conclusions that way.

I forgot to mention: I am not even quite sure what it means to believe P and not-P simultaneously. Perhaps it is possible in some way, but what it seems to mean in this context is that I can arbitrarily swap P for not-P whenever I feel like it. If I can do that, then I can use that as a rule to “get to” anywhere I want, presumably.

But what if I was permitted to kind of believe P and not-P, and also kind of not believe it? That would mean that Q or P is not necessarily true (nor necessarily false), nor is not-P necessarily true (nor necessarily false). So that means I don’t have a proof of Q (which I didn’t want anyway).

Given that not-P in this case is the LoEM, it looks to me like kind of believing it and kind of not believing it prevents the “explosion” from happening.

Here’s my goal, stated in very hand-wavy, fuzzy, intuitionistic terms:

I want to be able to entertain a diverse array of theories that may not only contradict each other, they may also be internally inconsistent as well. But I want to do this in a way that I can “tease out” the consequences of the theories and their inconsistencies by pointing to what would need to be changed to make one compatible (or more compatible) with another.

It seems like it should be possible for reasons I mentioned earlier: Physicists seem to be granted the luxury of holding mutually-incompatible theories in their heads, while also getting to know where and how they don’t agree with each other or reality. For example, if I want to believe in time travel, then I have to believe in closed-timelike-curves. If I want to believe in those, while also getting to make one if it didn’t exist before, I would need to have negative mass, apparently. It may not be the case that negative mass exists or can exist, but it is at least nice that we get to know what would have to be true in order to make our proposition true.

If I wanted to partially believe in LoEM, then I would see that if I wanted to prove that time travel is not possible, I could use that above reasoning to conclude that because negative mass isn’t possible, time travel is not.

If I wanted to partially disbelieve in LoEM, than I could also partially disbelieve that we had disproven the possibility of time travel. Maybe we could “tease out” the consequences of negative mass, to see what else would need to be for it to exist.

Here’s a question to conclude this piece with: Is Löb’s theorem still true even if we don’t accept LoEM? (My guess is yes.)