Imaginary propositional logic Has a field of logic been explored, where the conventional form of propositional logic is extended in such way, that the statements (whose truth values are evaluated) can have an additional imaginary part – much in the same way as the complex set adds imaginary numbers?
The ”imaginary propositional logic” I'm imagining here would have a new unary operator (let's use ”@” here as an example) with the following rules:


*

*@A = @A (imaginary statement A being equal to imaginary statement A is true)

*@@A = not A (imaginary imaginary statement A becomes not A in the same way as 1j * 1j becomes -1)

 A: We can arrange for just that property to hold; the smallest example is the four-valued boolean algebra.
Representing its elements as pairs of binary truth values (with the logical operators operating on each slot independently), then we can define


*

*$@(TT) = TF$

*$@(TF) = FF$

*$@(FF) = FT$

*$@(FT) = TT$


This is of questionable use, since it doesn't appear to have any reasonable interaction with other logical connectives. I imagine that will be the case in general.

There's a simple recipe for finding examples. In any boolean algebra (except for the one-element algebra) you can partition all of the truth values into ordered pairs of the form $( P, \neg P )$. Then:


*

*Partition these into pairs of pairs $((P,\neg P), (Q, \neg Q))$ in any fashion you like.

*For each such pair of pairs, define $@P = Q$, $Q = \neg P$, $@\neg P = \neg Q$, and $@\neg Q = P$.


The number of truth values in every finite boolean algebra is a multiple of four (except for the 1 and 2 element boolean algebras), so this is always possible.
A: There is a direct treatment of this question in the work of Louis H. Kauffman in a 30 year old article called Self Reference and Recursive Forms. Kauffman explores self-reference and recursion in systems such as G. Spencer Brown's Laws of Form. He goes on to explore how imaginary numbers are analogous to self-referential systems and demonstrates the analogy in logic to complex numbers. 
A pure imaginary logical value can be understood as an oscillating value between True and False. But the negation of an oscillating truth value is a new oscillating value that is 180 degrees out of phase with the original value. Kauffman has some diagrams that make this clear. Imagine a two axis diagram. The horizontal axis extends between real True on the right to real False on the left. The vertical axis extends from imaginary True on the top to imaginary False on the bottom. 
We can then represent real True and False with the ordered pairs [T, T] and [F,F] respectively. Likewise, we can represent imaginary True and False as [F,T] and [T,F] respectively. In this 2-D space, an operator @ (the notation similar to one Kauffman used to represent the "square root of not") rotates our value vector by 90 degrees. It is defined as follows: @[A,B] = [~B,A]. This is analogous to rotation of a complex number vector in a phasor diagram by 90 degrees or by multiplying by i. Subsequent operation of @, i.e. @@, is equivalent to ~, the logical negation. With this formalism, it is possible to derive operation tables for logical OR, logical AND, and Logical IMPLICATION. 
The elegance of this approach is its relationship to the representation of complex numbers and with the intuition of an imaginary logic value representing an oscillating logical state. In this sense, then, an imaginary logical value is both True and False in a timeless realm (where classical logic lives), but resolves into an oscillation in abstract time between True and False.
