Lativity and interstitiality

[Patrik Eklund]

Dear all,

Some of you may recall how I have used the word "lativity" in the 
context of from-to constructions in logic. For instance, at the 
beginning of logic constructions we start with the signature and from 
there we can come to terms, and from there we come to sentences, and so 
on. Important in doing so is that we always close the door behind us. 
For instance, once we have terms, we fix that "object of all terms" when 
we proceed to construct sentences based on terms, i.e., we cannot find a 
new term and say "Oh! Here's a new one. Let's throw it in to whatever we 
already had as terms.".

Maybe "lativity" is not the best word for that from-to and closing the 
door behind you, but that is the word I have used.

The interstice in that from-to lativity is also important. From 
signature to term, there is the actual interstitial construction of 
terms. This can be done with a set-theoretic construction, with a mix of 
set theory and category theory, and with a purely categorical 
construction of terms, also in the many-sorted case, and even over any 
monoidal closed category apart from the category of sets and functions. 
Of course, in the monoidal cat case, "terms" are not what they usually 
look like, and often we simply have the "object of all terms". However, 
there are some categories, like the Goguen category for "fuzzy sets", 
either over a distributive lattice, or over quantales. In that 
particular Goguen case, terms are "many-valued" terms so that even 
operators are many-valued. So, in a simple example, if the result of 2+2 
is uncertain, it may not be just about the uncertainty of 2 but indeed 
also about the uncertainty of +. This "terms over Goguen's category" can 
also be done for lambda terms, where the underlying signature can be 
arranged so that the function type constructor is an operator (on a 
certain level) and even application can be arranged as an operator. 
Application can then be "uncertain". I would be curious to see what 
Wolfram's trees would look like with such terms.

What I try to say is that whenever we aim at being very constructive in 
those interstices we may open up views of things we may not have seen 
before while dealing with those interstices in more intuitive ways.

Similarly, I'm not sure "interstitiality" is the best word for this 
particular thing. I am perhaps inspired by the so called interstitial 
cells of Cajal (ICC) e.g. in the gastrointestinal tract. They are 
myenteric and they also have a certain pace making role in the 
intestines. The neuron has provided inspiration for neural networks, so 
maybe those ICCs could have some relations in that "lativity" and 
interstitiality" of logic and logical constructions from-to.

The above is more of gut feelings than computations or mathematics above 
my spine, and I didn't check what ChatGPT would say about this.

Best,

Patrik