Oddly, it is simply because it is a much harder problem. The basic theories for representing information are the basic theories of computation, but then you have to manipulate and transform them. Different ways of representing this make certain computations much simpler, and others much more complex. Software is equivalent to hardware, so anything that can be done in software would have to be fundamentally possible in dedicated hardware, that would find it incredibly easier to compute than the general purpose stuff. Literally everything that can be done with a computer can be done with a dedicated machine that would find it much easier than a normal computer.
And, Or, and Not gates are simply for human convenience. We find them easy to reason about, and don't need to use to many to describe/make complicated things. The computers themselves use Nand or Nor.
You might want to look at trinary, which was a perfectly good logic that can include uncertainty. It is perhaps slightly better for computers than binary, but only the Soviet Union used it, so it was a casualty of the far superior western economy.
I'm not sure what you mean by "equivalent of information theory but for computation". Though it can be applied to other things, information theory is fundamentally about computation. Asking what its equivalent for computation is is like asking what the equivalent of a car is for driving on roads.
(Not that it seems to matter, but logic circuits go far beyond or, and and not gates. There are also nand gates, xor gates, really anything you can write a truth table for you can build a gate for.)
Logic circuits haven’t produced much
Haven't they?
and seem a bit arbitrary (why only OR, AND, and NOT gates?).
You can compose any othe gate out of them. In fact you can compose any gate out of NANDs.
One problem with quantifying computation is that any problem with only a single instance can be trivially solved.
For example,
For this reason, it doesn't really make sense to talk about computation in terms of problems with only a single instance, or even a finite number of instances.
Instead, computational complexity theory focuses on how algorithms scale for arbitrary input sizes, because that's the only type of question that's actually interesting.
So can you clarify what you're asking for? And how is this related to "qualitatively understanding things like logical uncertainty"?
A quick google search brings up "computation theory". This does not seem to be as precise as information theory, we cannot talk about "n units of computation" the same way we can talk about "m bits of information". In fact there does not seem to be a generally accepted fundamental unit of computation which we can talk about.
Computational complexity theory is well-developed but only talks about scaling.
Turing machines seem to be able to "hide" a lot of the work in the number of available symbols to go on the tape or the number of available states of the pointer.
Logic circuits haven't produced much and seem a bit arbitrary (why only OR, AND, and NOT gates?).
Seems like we need a theory of computation to qualitatively understand things like logical uncertainty.