In my last post about generality, I tried to show how our ambition to discover ideas that are all-encompassing and eternal makes our worldview crumble, leaving us unable to think clearly even about simple issues with obvious solutions. Today, I want to discuss another instance of the same problem, in a simpler and more direct way. You can think of this essay as a prequel to “When Universality Breaks.”

What is boolean thinking

Every time someone asks you a yes/no question, you are being coerced into accepting a pattern of thought that we’ll call boolean thinking. The word “boolean” here is used in the sense of the Boolean logic, and the Boolean datatype in logic and programming — a type that admits only two values: true and false. By “boolean thinking,” I am refering to the precondition that every statement should necessarily be categorized as either true or false. This is a law in Boolen logic, known as the “law of excluded middle”).

“But every statement is either true or false,” some might object. This principle might not be entirely false, but it is also not entirely true (ba-dum-tss).

Context is key. By “context”, I mean the set of premises/postulates/axioms, which we presume in order to think. Depending on the context, a statement can be:

• Unknown or unknowable (if the context is incomplete)
• Senseless (if the question is meaningless)
• Both true and false (if the context varies)

You are probably aware of such situations, but you might still not see them as contradicting the Boolean doctrine (boolean thinking, as we shall see, is precisely that—a doctrine). It’s a mode of thought that, although not universally valid, is often useful. For instance, you can’t make plans with someone who says there’s a 40% chance they’ll go out tonight, or that the question doesn’t make sense. Thus, you might be tempted to treat all imperfections of the Boolean model as imperfections of the world — or of thinking agents themselves:

1. No statement is unknowable — somewhere out there, there must be an answer.
2. No statement is senseless — given enough effort, every statement can be interpreted.
3. A statement is both true and false only because we lack sufficient information.

People who think this way, I would say, suffer from a serious case of Boolean thinking. Fortunately, the condition is curable, provided that we understand its cause. As I mentioned, Boolean thinking always has to do with context. Generally speaking:

1. Each statement can be true in one context and false in another.
2. Every statement is senseless when presented without context.
3. Every statement is unknown when the context is incomplete.

The case against boolean logic

We’ve established that the truth or falsity of a statement depends on its context — that is, on the assumptions we take as true or false in order to justify it. Boolean thinking, boolean logic is applicable only if we agree on some universal context — a universal set of true statements on which every evaluation can rest.

Note that aside from being universal (valid for all statements) the context for boolean logic has to also be all-encompassing (relevant for every statement) i.e. the set of logical statements that form it should never, under no interpretation tell something invalid, and at the same time would let us deduce all that is valid. As I argue later, such context resembles what political philosophers call an authoritarian doctrine (although the phrase “authoritarian doctrines” is somewhat deceiving, because it isn’t the doctrines themselves that are authoritarian, but the role they play in people’s thinking patterns).

So, while boolean logic may be splendid when viewed by itself, when viewed in relation to the “real world” there is a huge issue with it, the namely that no logical context, no logical framework is strong enough to capture the things that we usually want to dissect, (the real world, if you must). Proving the claim above is a subject of a different text, for now it suffices to say that although it may not look logical or scientific, it is, however, very backed up by both logic and science. Rather than asking why this is the case, it is more appropriate to ask what makes us think the reverse, what makes us think that the real world may be captured by a boolean logical framework - I’d argue that the thought that it can be is an instance of the so called “is-ought fallacy” — the idea that something is true just because it will be good for us that it is true. But that’s a separate topic as well (see “When Universality Breaks”.

Now, we are ready to make the case against boolean logic:

**Because boolean logic overlooks the importance of context (that each proposition can be true in one context, false in another, and also neither true nor false, and senseless in another) it inspires dichotomous thinking, also known as black-and-white thinking. **

i.e. boolean logic is apt for a world where there is a single unifying and also complete framework… a universal set of axioms. But that is not our world. Our world is a place where we constantly have to compare different frameworks and, different sets of axioms, which are all incomplete (here is the place where I should reference Godel’s theorem, but I am not going to do it, as it is too cliche, (pardon my lack of diacritics))….

When we encounter something that doesn’t fit our Boolean framework, we have two options:

1. Pretend it didn’t happen (or twist our perception until it fits).
2. Declare that “the world isn’t logical” and stop trying to make sense of it.

Non-Boolean thinking, in contrast, allows multiple frameworks to coexist—without one diminishing the others.

Non-Boolean logic

Criticizing something is (nearly) pointless unless we offer an alternative and I will do my best to do just that.

Most people are aware of non-classical logics, but they tend to regard them as curiosities rather than real alternatives to traditional boolean logic. One branch of non-Boolean logic, however — intuitionistic or constructive logic — is increasingly relevant to many fields. It is, for example, the logic that is at the heart of the so called “proof assistants”. Boolean logic is also a special case of intuitionistic logic (the only difference is that it lacks the law of excluded middle).

Rather than resting on truth and falsehood, intuitionistic logic revolves around the concept of a proof. In contrast with classical logic, where a proof is primarily a process, intuitionistic logic treats a proof as an object: a construction that demonstrates the truth of a statement (you can see how this is related to programming — in intuitionistic logic, proving thing is similar to transforming some objects from one format to another).

Each proof depends on a context – a set of premises or other proofs we assume to exist. So, before evaluating any statement, intuitionistic reasoning asks: “What is the context?” i.e. “Give me the set of premises from which we are operating.”

From there, we proceed to manipulate the proofs of the premises in order to construct a proof of our statement.

You might say that this is much like “normal” logic. But there is a difference — intuitionistic logic makes us more acutely aware of the context in which we are operating. And if we start paying attention, we would observe that when trying to prove a statement, instead of the two truth values (true/false), there are actually three possibilities:

1. We might be able to construct a proof that a statement is true.
2. We might be ablet to construct a proof that it is false.
3. We might not be able to construct neither—the statement is neither true nor false.

And without the correct context, the statement migh not make sense at all.

(Yes, in a perfect world, where we know everything, we would be able to prove or disprove every statement that we can formulate, but not in this world.

And with that realization, we are free from the Boolean prison — we realize that all thinking is relative there is no a sigle truth, (nor a single falcity).

For more on intuitionistic logic, see my book Category Theory Illustrated.

Addendum: boolean thinking, authoritarianism, and propaganda

My criticism of Boolean thinking is not merely academic. The way we think about logic shapes how we think about everything — and, ultimately, how we live our lives. This is why what I call “Boolean thinking” in logic has many names elsewhere. In philosophy, it’s called Platonism. In politics, it manifests as authoritarianism.

The second point is important. You might define authoritarian ideologies in many ways, but the key thing about them is that are based on a doctrine on which all people must abide to, in their thinking — a shared “context” or set of premises. The power of authoritarian rulers arises from the way they limit the things that people can think and say: rulers define what the premises are and from then on, then you are “free” to make the conclusions yourself. To rephrase Orwell’s famous slogan:

“Who controls the premises controls the conclusions.”

Authoritarian regimes rely on propaganda, and propaganda often uses Boolean thinking: the belief that if something doesn’t follow from the “official” premises, must be necessarily false (black and white thinking). Or that if two things seem opposed, one of them must be true and the other false (false dichotomies).

To combat such propaganda techniques always remember the two rules of the contexts:

1. There are many of them.
2. No-one gets to decide which is more important.