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Busaniche and Cignoli (2011) also describe "twist structures" where truth values are pairs of binary numbers representing "truth" and "falsity", and negation is swapping:$$\mathrm{\neg }(a,b)=(b,a).$$When I mentioned twist structures earlier, I restricted attention to the 3-valued case $F=(0,1),B=(1,1),T=(0,1)$, but of course $N=(0,0)$ may be thought of as "neither true nor false" and can be used as an input to the equations and circuits I showed. However, unlike with ${\mathbb{F}}_4$, there is no isomorphism between $B$ and $N.$ Here's the implication:$$(a,b)\Rightarrow (c,d):=((a\Rightarrow c)\wedge (d\Rightarrow b),a\wedge d)$$where the arrows on the rhs are regular binary conditionals.$$\begin{array}{ccccc}\Rightarrow & F& N& B& T\\ F& T& T& T& T\\ N& N& T& N& T\\ B& F& N& B& T\\ T& F& N& F& T\end{array}$$This has a left adjoint $P\otimes Q:=\mathrm{\neg }(P\Rightarrow \mathrm{\neg }Q)$. Notice that if you ignore the $N$ rows and columns, you get the relevant implication and conjunction of #RM3.

But we can form another adjoint pair! Notice that $a\Rightarrow F=\mathrm{\neg }◊a$ is the type of negation used in the paraconsistent conditional $(a\Rightarrow F)\vee b$. If we "and with the contrapositive", we get an implication ${\Rightarrow }_F$ that is exactly the same as $\Rightarrow$, except that $N{\Rightarrow }_{F}N$ is $N$ instead of $T$. It also has left adjoint $\mathrm{\neg }(a{\Rightarrow }_F\mathrm{\neg }b)$.

Perhaps this question is best left to #philosophy. What is the proper value for "if I don't know, then I don't know"? It's either $T$ or $N$. Another question is "is $N$ valid?" If it isn't valid, then Modus Ponens fails for ${\Rightarrow }_F$.