I didn't know how to ask this question before but now that I'm reading about typed lambda calculus I think I've got a better idea.
There is this answer to a question asking whether CSS is Turing complete or not (note that this question is NOT about CSS at all), which I consider is not (at least in certain way, and this "certain" is what my question below is about), even if you can simulate Rule 110 with it, which is proven to be Turing complete. My intuition behind it is that only an insignificant subset of what CSS can output can be the result of a calculation. You cannot perform any calculation on colors or sizes to be set to element's styles.
Now I found an even simpler example while reading about typed lambda calculus. This article's overview describes a lambda calculus that can use booleans. I noticed that, in addition to the untyped lambda calculus operations, there is the conditional operation (
if t1 then t2 else t3), which makes perfect sense. Without this, you wouldn't be able to do anything with booleans, and I would go ahead and say is really not Turing complete, or at least that is Turing complete but only in a trivial way.
So my question is, if this typed lambda calculus with booleans wouldn't have that conditional operator, would you say is Turing complete? If not, why not, and if yes, how would you express the fact that you cannot do any operation at all on it's main type of data which is what you'd expect from any Turing complete language in a categorical, technical way such that is clear that the Turing completeness of the language is really meaningless?
The reason why I'm giving so much importance to this is because I was recently trying to create a small Turing complete language myself and while doing it, got to a point in which my language was Turing complete but only in this trivial way (wasn't as obvious as the boolean lambda calculus without conditionals above, though). If I would have asked "Is this language Turing complete?" I would have probably got "Yes" as an answer and I would've think I'm done creating my language. But I really was not. And when asking myself why am I still working on my language, I'd like to be able to state exactly what is that my language is missing.