Paper "Answered Modal Logic Catuṣkoṭi": https://github.com/advancedresearch/path_semantics/blob/master/papers-wip/answered-modal-logic-catuskoti.pdf

In the classical world of information theory, there are only bits: Yes/No, True/False, On/Off, 0/1.

This system is so convenient that it is hard for our brains to adapt to other logical systems. However, does it really matter? Can not all logical systems be reduced to bits (or propositions as they are called in Logic)?

Indeed, this correct. The challenge with what we call "Esoteric Logics", is how operations are defined, not to re-invent bits.

Answered Modal Logic was an attempt to find a logic with intrinsic path semantical properties, but was unsuccessful and replaced by Path Semantical Logic. However, recently I and Daniel Fischer revisited Answered Modal Logic to see if it has any other potential applications.

If you do not understand Answered Modal Logic, then do not feel stupid! It is a what I call a "brain-wrecker".

However, I will attempt to give a short introduction to Answered Modal Logic:

Imagine that you are asking people questions and you only write down whether they answer and erase their actual answers. What would be a natural system to encode the state of knowledge about answers?

Well, for a starter, you might think of recording a bit, telling whether people answered the question or not. However, some people might decide to not answer the question and you would have no way to distinguish people you have not asked yet from those who do not want to answer.

So, you have to add another state that encodes "no comment".

Now, you might think that this system covers all cases. Yet, when you go around asking people, somebody asks you to come back tomorrow and they will have their answer ready.

One person you ask tells you something clever, like: "I am neither answering that question nor giving you a none-answer."

Sigh, so you decide to add a 4th state for all people who promises to answer tomorrow (which promise might be broken sometimes) and people who are too clever for their own good.

This is Answered Modal Logic in a nutshell.

Now, the insane part of this esoteric logic, is that there are 2 involutions operators instead of 1, with a 3rd involution operator that extends to the corresponding 8-value logic. There are 2 families of functions instead of 1, because one involution operator maps to the other. This is so hard to reason about, that it qualifies the logic as "esoteric".

Catuṣkoṭi on the other hand... is just as incomprehensible.

1. P
2. not P
3. P and not P
4. not(P or not P)

This might seem like reducible to 2 states. However, the Catuṣkoṭi does not have a law of the excluded middle.

Now, if you are a clever mathematician, you might think that this is a kind of set, which has the states {T}, {F}, {T, F}, {}.

However, in the Dharmic traditions of Indian Logic, people were more obsessed with boundaries instead of sets.

This means, from the perspective of traditional Indian Logic, the empty set was at the same time empty and a boundary toward everything. One could say, that the empty set was used to refer to everything and nothing in one go.

So, understanding the Catuṣkoṭi is not as easy as it seems at first.

Anyway, Daniel Fischer thought there might be a way to bridge Answered Modal Logic with Catuṣkoṭi. The problem is that Answered Modal Logic has multiple involutions that are not apparent in Catuṣkoṭi. To solve this problem, we found an interpretation which helps translating between the two logics: Measurement of quantum systems.

When one measures the spin of an electron, there is answer "up" or "down", independently of whether the electron is in a superposition. So, if P is a measurement, then when no measurement is made, this preserves either the superposition or the possibility of measuring in the future, but also whether to make no measurement! Hence, a none-measurement is not not P, but P and not P.

Likewise, a negative measurement of the position of an electron, might or might not determine the position of the electron, depending on how much information is received through the negative measurement. We choose not P to represent a negative measurement.

The last case, an unknown measurement, fits the idea of a boundary from the empty set, referred to by neither making a measurement nor making a none-measurement (not(P or not P)). This idea projects toward something external, which can mean any kind of measurement. In Answered Modal Logic this corresponds to ¬□ (which is kind of like a set of the 3 other possible values).

The not operation in Catuṣkoṭi is not like the 2 involutions in Answered Modal Logic, but it still makes sense to translate between the two esoteric logics! Another thing we figured out, was that the not(...) in the 4th case is not like the other nots in Catuṣkoṭi.

Did that make sense? I hope "not"!