First principles of logic

Rethinking truth and knowledge

It is hard to think in the most abstract terms possible, forget everything we know, and think really hard on what is foundationally true, not true as a consequence of something else, particularly when the area you are thinking about is thinking itself.

In this post I will explain what is in my opinion the most fundamental fact of logic that can be used to understand the rest of it.

The basics

The first thing to establish is that which nobody can question: the meaning of true and false. Astonishingly, some people do question the meaning of those concepts (for example the infamous Sam Harris podcast with Jordan Peterson: What is True?), but unsurprisingly this exercise leads nowhere. No, these concepts just are.

1. True (⊤): that which is the case.

2. False (⊥): that which is not the case.

That’s it.

The third wheel

My contention is that there is a third concept that anchors the rest of logic that is not considered enough.

First it has to be understood that there is no single defined method of representing logic. The most common ones are propositional logic and predicate logic, but these don’t address sufficiently well the issue I’m trying to tackle here. For that we will need a less common one: modal logic.

The basic notions from modal logic we are going to need are that of necessity and possibility. Something is necessarily true (□) if it has to be true, or simply put: if it’s true. On the other hand something is possibly true (◇) if it may be true, or: if we haven’t determined that it’s true or not.

At first glance we haven’t discovered anything consequential, but we haven’t considered negation yet. In classical logic the negation of true (¬⊤) is false (⊥), however, in modal logic the negation of true (¬□⊤) is possibly false (◇¬⊤).

I can’t state strongly enough how unbelievably crucial this notion is.

We know if a person is wearing a wedding ring, that person isn’t single. If being single is p, then we can say it’s necessarily the case that person isn’t single as: □¬p. However, what if a person is not wearing a wedding ring? We can say the opposite: it’s not necessarily the case that person isn’t single: ¬□¬p. In other words: it’s possible that person is single: ◇p. Possibly the case is not necessarily the case.

At first this sounds like a convoluted way of stating the obvious, however, experiments like the Wason selection task shows humans get these sort of intuitions wrong the vast majority of the time. So it is worth pondering this notion of possibility, and what it means in relation to truth.

So our new basics are

1. True (□⊤): that which we know is true.

2. False (□⊥): that which we know is false.

3. Unknown (◇⊤, ◇⊥): that which we don’t know if it’s true or false.

Embracing the unknown

Humans have a natural fear of the unknown, however, anyone that has taken an expedition into epistemology knows uncertainty is our friend. It is the known that haunts us without we knowing it.

The greatest enemy of knowledge is not ignorance, it is the illusion of knowledge.

We have to let go our desire to know things, and accept that there are many things we don’t know, and in fact: many we cannot know.

Once we embrace this powerful ally, we can begin to understand that it is not to be relegated to a few rare circumstances, it is in fact there to help us in each and every question of life.

The unknown is the default.

A rational person treats each and every statement as unknown, it is called the default position. Nothing is assumed to be true, or false, and even when there’s ample evidence that something is the case, the possibility that it isn’t is considered to be always present.

Any time there’s a dispute about the truth of X, a rational person assumes the default position. The person that claims X is true has the burden of proof, but so does the person that claims X is false. Until that burden of proof is met either way, the rational skeptic remains neutral: in the unknown territory.

So when a skeptic rejects the truth of X, he is not implying that X is false: there’s no dichotomy (remember there’s a third option). Rejecting X is stating that X is not necessarily true (¬□X), which is not the same as stating that X is necessarily false (□¬X).

In many debates there will be certainty bullies that will try to lock you to a position. If somebody claims Trump wants to become an autocrat, and you are not convinced of that claim, they will immediately push you to prove the opposite. Resist that urge to prove anything. You don’t have to prove that X is necessarily false. If they made the claim that X is true, they have the burden of proof.

It’s OK to say “I don’t know”.

Bolting the negation

Something so simple in logic is so hard for human minds, so it’s worth to try to nail it down.

When O. J. Simpson was acquitted, did the jury find him innocent? If guilty is □p, then the opposite is ¬□p, which we saw before is the same as ◇¬p, in other words: possibly not guilty. But possibly not guilty is not the same as innocent.

There is a reason the justice system in USA declares the accused as “not guilty”, never as “innocent”. This reason eludes most people, but it has everything to do with the lack of certainty. Using the wedding ring as an analogy, the fact that Simpson was not found with a ring doesn’t mean he is necessarily single: he may be single, he may not, the only thing that was stated is that he wasn’t found married.

So no: Simpson was never found innocent.

Let’s do the reverse. If Simpson is possibly not guilty (◇¬p), the opposite (¬◇¬p) is necessarily guilty (□p). This matches. On the other hand, if Simpson is innocent (□¬p), the opposite (¬□¬p) is possibly guilty (◇p).

So we have two relations:

• The negation of necessarily innocent is possibly guilty

• The negation of possibly innocent is necessarily guilty

It’s not so simple as true is the negation of false, and vice versa.

The binary truth

The knee-jerk reaction of most people when dealing with the uncertainty of guilt or innocence is to say: “No! O. J. Simpson is either guilty or he is not!”. This is of course the case, but one thing is the truth, and another thing is our knowledge of the truth.

Consider this example. Right now it’s either before midday or after midday in my location. Do you believe it’s after midday? If your answer is “no”, the certainty bully would say: “Aha! So you believe it’s before midday!”. This is flawed reasoning.

We saw in the last section what is the correct reasoning. If you don’t know if it’s after midday, that means you think it’s possible that is before midday, but not certain. If you don’t know if it’s before midday, that means you think it’s possible that it’s after midday, but not certain.

So it’s perfectly consistent with logic to say “I don’t believe X is true”, and “I don’t believe X is false”.

The lack of belief is misunderstood. An atheist doesn’t believe that gods don’t exist, he lacks the belief that gods exist.

Similarly, the absence of evidence is not the same as evidence of absence.

Yes, the truth is out there, lacking the belief in it doesn’t imply denying its existence, it’s just accepting our proper relationship to it, or more precisely: our lack of proper relationship.

Conclusion

Logic is difficult for humans. It’s much better for us to accept we are bad at it, and tread carefully. It’s OK to accept our limitations. Skepticism is not about discerning truth: it’s about not discerning falsehoods. Rationality is not about being good at logic: it’s about not being bad at it.