LEG - Law of Excluded Gap

“Nel mezzo del cammin di nostra vita
mi ritrovai per una selva oscura,
ché la diritta via era smarrita..”

A recurring problem when you talk with what I would call “first-order religious devotees” is that they have a system of thought that explains everything. This is particularly a problem for themselves because it makes them intellectually lazy. They have a world view that is complete and they interpret/react using this framework. Symmetrically, the same phenomenon can be observed on hard core scientists/atheists. They are both representatives of people believing in the Law of Excluded Gap (LEG). LEGs supporters are people who know the path to knowledge. Even when they admit they don’t know, they deeply believe they know how they could know. And if someone comes with an alternative path to knowledge they immediately discard both the messenger and the message.

The foot in the door

To borrow from a popular sentence those days, I would dare to say that I am not a psychologist but it appears to me that, in addition to the laziness hypothesis, the fear of the foot in the door is at play in LEGs supporters. The fear is that if your system of belief cannot provide you tools to explain everything then it crumbles into rubble (like a vampire in the light of the sun). Examples abound in the different camps (there are more than the two I mentioned). For instance, very quickly, some striking examples:

1. Creationism is an extreme example in which proponent of a literal interpretation of the Bible cannot sort out scientifically established facts other than saying it is a test of his faith. Therefore comforting his worldviews.
2. The scientific approach to what constitutes a healthy diet is not impressive to say the least. “Official recommendations” change on a regular basis, and you can find studies (even official policies) for almost anything you can expect: meat is good, meat is bad, fasting is good, fasting is bad, veganism will make you a superhuman, veganism is going to destroy your health… On the other hand, all religions and cultures incorporate dietary recommendations. Those recommendations are often downplayed as superstitious by hardcore scientists who nonetheless have trouble demonstrating that their approach is better in any way.
3. The unfolding controversy (can we call it a discussion ?) in France, and internationally, on the chloroquine-azithromycin with relation to the COVID19 crisis, shows two opposite sides that only share their unconditional condemnation of the other side. One side advocates a strict obedience to double-blind protocol and doesn’t want to hear anything else, while on the other side very optimistic declarations are made “the epidemics is over, the solution is here, etc.”
4. A less scientific/religious opposition, between two competing LEG’s faction, is also witnessed in the great divide in the US politics. The polarization of the political debate is such that almost any subject is instantly analyzed through a partisan lens and no phenomenon, no matter how trivial, is linked to low scale politics. What I want to focus on here is not really the fierce opposition, but the fact that it leads their supporter to erase any trace of skepticism.

I could pile up examples, but the idea is clear at this point. The common point of these examples is that the influence of LEG is greatly enhanced when there are opposite factions. The fear of loosing ground pushes the paranoid factions to ignore their ignorance, which is seen as a foot in the door from their foes. So they erase any gap in their worldviews.

Halting problem and semi-decidability

The bad news for LEG supporters is that deep mathematical and logical results from the past century demonstrate that gaps in knowledge are not only unavoidable, but much more common than one may think. The famous Gödel’s incompleteness results and one of its avatar, the halting problem of Turing machines, are often misunderstood. One implication, in the field of logic, is that provability and negation do not commute. It has led to numerous work among which are the intuitionistic logics in which there is no law of the excluded middle (which basically says that a theorem is true or false and no other choice is possible). From an epistemic point of view, the halting problem is more à propos. The crux of the idea is that there is no generic method to know whether or not a given program is going to stop. When a computer program runs you cannot know for sure if it is going to end or not: you don’t know and there are no other ways to do than to wait. The halting problem is told to be semi-decidable because if the program stops, one day you will know but if it doesn’t you will wait for ever. This is linked to the law of the excluded middle and the incompleteness of logic by this remark: imagine that your program is trying to prove a theorem. While your program is running you don’t know if the theorem is true or wrong. It turns out that there are true theorems for which there is no proof. In fact this is exactly what Gödel incompleteness means. For such theorems your program will never find a proof and thus compute forever. You will remain in a state in which you don’t know if the theorem is true or false. Another way to put it is that “not knowing” is not equivalent to “knowing that no”, which is the equivalent of the non commutativity between proof and negation in the logical world.

Semi-decidability is very tricky because intuitively, when we think, we presuppose that the law of the excluded middle always holds. This translates into the Law of Excluded Gap in knowledge.

reflexivity do not always lead to paradox

Skepticism, can, and must apply to itself

What can be concluded from these remarks? The first point is that there is no universal knowledge or way to build knowledge. It doesn’t mean that anything is right (that would be a false symmetrical reasoning that doesn’t hold as we just saw) and that knowledge is impossible. It leads to the second point: you must apply your skepticism everywhere, even to the skepticism itself, because sometimes things are true and proved. Skepticism is one of those rare moral imperatives that do not lead to paradox when applied on itself. When you spot gaps in your knowledge it is a call to effort, imagination, work and humility. If you don’t spot gaps, then you are missing something important. And the last sentence is not an opinion, it is a technical fact.