Tertium Datur

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In logic, there’s a law called tertium non datur.  The English name of the law is “the law of the excluded middle” (literally: no third (option) is given).  The law states that any statement is either true or false.  There’s no in-between.

The next 4 paragraphs go into some fairly technical logics; if you think that sounds super-boring, and instead want to skip to my insightful observations from a more psychological point of view, skip down to the paragraph that begins with “This all sounds highly abstract.”

Actually, tertium non datur is not a law.  It’s an axiom.  That means it cannot and doesn’t need to be proven true.  It is just considered so obvious there’s not need to prove it.  Most modern logic is based on tertium non datur along with a couple handfuls of other axioms at the level of “if A implies B and A holds, then B must hold” (that one is called modus ponens).

Logic where we assume tertium non datur has two consequences: if something is not false, it is true, and in order to prove something, you just have to prove that the opposite leads to a contradiction.  The first is really a reformulation of tertium non datur.  Tertium non datur says “A or not A,” where A can be any statement.  The statement that “if something is not false, it is true” can be formulated as “not not A implies A.”  This sounds very reasonable, but an alternative formulation is reductio ad absurdum or the rule that if something leads to a contradiction (the absurd), it is true.  This can be formulated as “(not A implies false) implies A.” (This is seen as “A implies false = not A,” so “not A implies false” is the same as “not not A,” which implies A according to the previous rule.)

It is entirely possible to build up a complete and meaningful logic without assuming tertium non datur. The reason one would do that is because it is a terrible proof rule, which is counter-intuitive to most and makes it very easy to make mistakes.  You have to be 100% certain you didn’t make a mistake in your derivation of the absurd, and you have to be extremely careful in what you conclude as you will typically make many assumptions, and have to be careful in which of those you declare to have lead to the contradiction.

Logic without tertium non datur is called intuitionistic or constructive logic.  The reason for this is that reductio ad absurdum is non-constructive.  It doesn’t start at known truths and argues towards its goal, but rather starts by assuming its goal is false, and tries to reach an absurdity.  Intuitionistic logic instead starts from known facts, and only allows constructive proofs.  Intuitionistic logic can be thought of as having three classifications: (constructively) provable true, constructively provable false, and neither.  There is a classical proof that there are infinitely many prime numbers, which assumes there is only a finite number and constructs a number by multiplying all of these and adding one.  Now, either this new number is prime or it is a product of two numbers, where one of them can be chosen as a prime not on the list.  The proof is elegant and can be taught in some detail to high-schoolers.  It is not constructive, though, and it provides no way to construct prime numbers.  In fact, while we know there’s infinitely many primes, we don’t know how to efficiently construct prime numbers (which is why most common forms of encryption works).

This all sounds highly abstract, but if we combine this with the logical fallacy false dichotomy, and  Kahneman and Tversky’s notion of substitution, many arguments actually rely on tertium non datur.  For example, one person says that, “we cannot accept Syrian refugees because of the risk of letting in terrorists” while another says, “we have to help Syrian refugees because they are not all terrorists.”  This is a highly uncomfortable discussion, so a systems thought will be do apply substitution: apply a simple reduction to the statements so they are easier to grasp and refute.  Instead of “risk of letting in terrorists” substitute “they are all terrorists” and instead of “they are not all terrorists” substitute “none are terrorists.”  Now, apply reductio ad absurdum to these; for the first, if we can find even one non-terrorist, we have disproven the statement: they cannot all be terrorists if we have one who isn’t, so they are not all terrorists.  For the second, just find one terrorist: they cannot all not be terrorists if we have one example.  The reason for doing so is we have separated the world into “them” and “us.”  Not “them, the refugees/terrorist” and “us, the non-refugees/non-terrorists,” but “those who believe the risk outweighs the need for helping” and “those that believe the need for helping outweighs the risk.”  We have embraced the false dichotomy that if they are not with us, they are against us.   There is only “A” and “not A,” and byt proving “them” wrong, we have proven ourselves right.

Without even thinking, the two persons split the world into them and us, it substitutes the opponent’s argument to a simpler form and automatically refutes it using a simple argument that only works by assuming tertium non datur.  Without thinking about it, we refutes the other’s claim.  Not only that, but we do that using an argument that doesn’t even refute the concern the actual argument posed by the other.  You are not going to convince somebody who is afraid of terrorists by pulling out any number of refugees and proving them to not be terrorists.  Nor are you going to convince somebody concerned with helping that this is not the right thing to do of the risk of terrorism by pointing to actual examples of terrorists.  This effect is only stressed by the fact that we tend to either wildly over-estimate small risks or entirely neglect them.  to the person afraid of terrorism, the probability of becoming a victim is increased manyfold to the point of it becoming a real risk that they might become victims.  To the one who just wants to help, the risk is not only negligible, it is neglected.  The risk is perceived as actually zero.

Not only do we apply reductio ad absurdum incorrectly because of our substitution, we even apply it in a situation where it doesn’t apply due to the false dichotomy.  It is perfectly possible to live in a world where “some refugees are terrorists” and “not all refugees are terrorists.”  All it requires is at least two refugees, and the world would be a much nicer place if that was the entire population of refugees.

The good news is that while we automatically make the substitution and refute the other person’s argument, we can actively engage in a more productive train of thought by forcing ourselves.  The thesis of Kahneman’s excellent Thinking Fast and Slow is that our brain contains two systems, a fast systems which makes simplifications to quickly get answers and a slow one which makes more accurate answers.  The fast system introduces the dichotomy, does the substitution and refutes the argument you disagree with.

We can force ourselves to invoke the slower, more reliable system.  One way is to consciously think that there is a third, “tertium datur.”  I don’t actually not know Latin, so I just removed the word that looked the most like a negation in tertium non datur and uncritically went with assuming that this means “there is a third (option)” in best fast system style.  For example, are you “pro life” or “pro choice?”  or is it possible for the two exist in the same world?

I strongly believe that if we try not to refuting to other’s arguments we can have much more productive discussions.  The refugee crisis is not a small problem with easy answers.  By being honest that there might be a risk that some terrorists will sneak in with the refugees we can make better decisions.  By admitting that there will also be people who are not refugees but just immigrants who want to do better in a country with better opportunities can we have a more honest discussion.  At the same time we also have to consider whether we are willing to let a million refugees die just because we’re a bit a afraid of tanned people.

This of course nicely ignores that “they might be terrorists” might be shorthand for “I don’t like dark people.”  I explicitly ignore that here as I believe it has at least some ground in naive realism caused by the out-group homogeneity cognitive bias and the illusion of asymmetrical insight.  But I think this is a topic for another post, where I also have ample examples of why dismissing Trump’s voters as misogynists and racists is disingenuous.

If we don’t just outright dismiss people, we might be able to have a better discussion.  A discussion doesn’t necessarily have to end with one winner and one winner and one loser.  More importantly, it doesn’t even need to end in a Solomonian compromise (though that would be a solution to the pro-life/pro-choice discussion, I guess).  The otherwise terrible book The Third Alternative proposes the very interesting thesis that it is possible to come up with a solution that is not just a compromise, but a hitherto unthought of alternative.  By dismissing tertium non datur, we might be able to break out of the false dichotomy and come up with an altogether better solution.  For most questions, it’s not as simple as that, of course, but if we don’t even try we are sure not to find the better third alternative, and instead have to live with a compromise that leaves all sides unhappy or overruling one side and leaving that side very unhappy.

 

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