Generally, when talking about the Trinaryfield the parts worth mentioning are those not within the Binaryfield. As such, this area is known as the Deep Trinaryfield to avoid confusion with the full Trinaryfield.
It is difficult to even imagine what kind of objects and concepts exist within the Deep Trinaryfield, in which all objects are neither x nor not x for any x within the Binaryfield. As such, very little is known about the contents of the Deep Trinaryfield. However, there are logics that model things similar to logic within the Trinaryfield, which are called many-valued logics.
What is known is that the Deep Trinaryfield actually contains more than the Binaryfield, as if objects can exist in more states there will be more possibilities for different objects. Any y within the Binaryfield can only be in 2 states in relation to x (either x or not x), meanwhile in the Deep Trinaryfield y can be in 3 states in relation to x. (x, not x or neither.) As such, there are more possibilities for objects within the Deep Trinaryfield compared to the Binaryfield. However, nothing more about these objects is known.
n-aryfields are a way of extending the base definitions of Binaryfield and Trinaryfield to far higher levels. Each value of n represents a different collection of objects and concepts, as n corresponds to the number of states y can be in relation to x. (y and x are two different random objects.)
If n is 1, y can only be x, meaning the Unary field contains only one object. If n is 2, y can be either x or not x, forming the Binaryfield. If n is 3, y can be either x, not x or neither, forming the Trinaryfield. If n is 4, y can be either x, not x, neither or none of the others. This continues up the chain for every number imaginable. The n-aryfield in which n is 4 is known as the Quaternaryfield.
This can progress all the way up to ω, at which point y can be in a countably infinite amount of states in relation to x. (Known as an Apeironaryfield. If there is an uncountable amount of states y can be in it is known as a Circularyfield.) Further beyond infinite numbers, you could hypothetically use post-infinites such as E_0 (the smallest number larger than itself), for which the number of states y can be in is greater than the number of states y can be in.
Another way to imagine these successive n-aryfields is expressing every single one as some form of 'x and not x' by using the previous n-aryfield as x. The Trinaryfield can be expressed as everything that either is or isn't within the Binaryfield. Furthermore, the Quaternaryfield can be expressed as everything that either is or isn't within the Trinaryfield. This can be used for every n-aryfield, though once transfinite and infinite numbers are reached, this definition is not particularly helpful and displays very little information about what the field actually contains. Nevertheless, it is still technically accurate.
Size of n-aryfields
Successive n-aryfields follow the same rule outlined previously that the more states any object can be in, the more possibilites for different objects there are. Because of this, within each n-aryfield the previous n-aryfield contains the minority of objects. This means most of the Quaternaryfield is within the Deep Quaternaryfield, most of the Quinaryfield is within the Deep Quinaryfield, and so on. Thus, each n-aryfield is exponentially larger than the previous by an unknown amount.
All n-aryfields, regardless of what is used for n, are likely within the Schemafield, though some disagree with this interpretation.