Recursive Structures

Have you ever wanted to articulate the distinction between verbs like 'want' and 'know' based on the idempotence of 'knowing that you know X', but lack thereof in 'liking to like X'? Why doesn't this simple nomenclature exist?

A recursively applicable operator is one where the output type is the same as the input. Thus, both the verbs 'like' and 'know' are recursively applicable, because, for both, both the output and input type are ideas.

Some recursively applicable operators have strict idempotence. Knowing X is the same as knowing that you know X, which is the same as knowing that you know that you know X. Both infinite metacognitive awareness is implied from the bottom-up (knowing X implies all the rest), and infinite sub knowledge is implied from the top-down (knowing that you know X implies that you know X, etc.).

This will get confusing fast unless we introduce some shorthand. Let us now refer to knowing as the operator K. Rephrasing the previous paragraph in these terms, ∀n, K(Kⁿ(X)) = Kⁿ(X) ∧ Kⁿ(K(X)) = Kⁿ(K(K(X))). I believe that this is somewhat contested in philosophy, but in everday parlance, I think it is trivially true. In the future, let us refer to collapsing in the first way as top-down, and the second way as bottom-up.

Such operators with this strict idempotence can be said to be recursively collapsible (RC). We can always write them in their simplest form with a single operator. When discussing linguistics, the semantic field indicated by the first operator remains unchanged with subsequent uses.

Well, what are some other structures? Let us now consider liking, L. Well, it is certainly collapsible in the top-down direction, but what about bottom up? Does L(X) => L(L(X))? I don't think so.

The presence of the top-down collapse but not bottom-up is interesting. Let us refer to operators which collapse in one direction as partially collapsible (PC). In this case, it is top-down PC. L(L(X)) contains strictly more information than L(X).

This observation implies something subtle about the semantics of the operator. In the use of every operator, there is an implicit subject and object. Upon recursive application of an operator, the prior subject becomes the object. In an operator that is RC, the implicit subject and implicit object are identical. In an operator that is top-down PC, such as L, the subject is slightly broader than the object. The person doing the liking is subtly separated from what is liked.

Let us now consider an RC operator, K. There is NO implicit separation between the knower and the knowledge.

One well founded criticism of this is that language is not properly defined. However, this is precisely the point of this essay! Language evolves through usage, leading to weird structures such as these, based on what is most useful to convey. By cataloguing the structures we can tame this evolution, perhaps uncovering deeper connections as to WHY certain operators have some recursive properties, and what that may say about human minds, and the society that shapes them!

Consider Berridge's Incentive Salience Theory: He separates Liking, Wanting and Learning as three separate cognitive conditions, with some neurobiological basis. There is emerging neurobiological basis for this distinction: liking roughly corresponds to hedonic hotspots, wanting to dopaminergic craving, learning as rule based cortical representations.

Perhaps, even before we had this neurobiological grounding, we could have observed the recursive structures of verbal operators like wanting (W), liking (L) and learning (Le). Learning in this case is according to Berridge's definition, meaning something akin to 'learning to like'.

W is not RC. This implies the evolutionary necessity of this distinction, which perhaps implies two separate cognitive states being denoted by W(X) and W(W(X)). Perhaps similarities can be observed between the meaning implied by W(W(X)) and L(X), and L(L(X)) and Le(X).

Let us now consider a bottom-up PC operator. Unfortunately, the example I have in mind is not easy to capture: to understand that you need (UN). UN(X) => UN(UN(X)), but UN(UN(X)) =/= UN(X). A simple example: understanding that you need to understand that you need to do the dishes does not imply that you actually understand that you need to do the dishes! The second often semantically implies more processing than the first! Note that although this example seems contrived when put in English, it's meaning is not, and quite commonplace.

Keep this notion of processing in mind- we will come back to it. For now, let's recentre on the subject object distinction. As we said before, top-down PC implies an implicit subject that subsumes the object. Perhaps this implies a degree of metacognitive awareness of the object. In the case of wanting, it is implied that the self is much greater than the want.

What about bottom-up PC? Well, the subject is smaller than the object. Perhaps this implies some degree of LACK of metacognitive awareness of the object. Understanding that you need to understand that you need to do the dishes implies that you are cognitively stretching beyond yourself, exploring something that is not encoded within yourself.

This implies a lack of psychological integration between the object being discussed and the subject discussing it. Think of when such bottom-up PC operators would be most common: when discussing things that we SHOULD do. Perhaps things that we have an internal conflict about.

This can be extended much further. I'm sure there are many more insights that can be garnered from this theory. I hope you enjoyed!