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#RM3 is seen to have the structure of a symmetric monoidal closed category. The ordinary conjunction and disjunction are Cartesian, but we also have a non-Cartesian tensor product that is left adjoint to the relevant implication. This structure is also described as a residuated lattice.

Symmetry is optional, and we can easily define non-commutative versions of the operators, which is done for example in the Lambek calculus model of natural language, and Linear Logic (linear here means "sequential").

RM3 is robust to paradox. The usual paradoxes of the "material conditional" fall, they are paradoxes in binary logic because the conditional there is NOT "material", it is just called that. A true material conditional, like a material witness, is a relevant conditional. Paradoxes of relevance are often called "informal" fallacies because in binary logic they are tautologies. Only the relevant implication can see them.

TL,DR; there is really only one difference between binary logic and RM3, which means that there are really two differences but they are the same. Everything else works the same way, but to avoid paradox, all you need to remember is not to do either of$$\begin{array}{rcl}\varphi & \Rightarrow & \mathrm{\perp }\\ \mathrm{\top }& \Rightarrow & \varphi \end{array}$$Both are invalid. That is, reasoning from inconsistency, or reasoning towards inconsistency, are both bad. And since at the start, we equated "Both" with "Neither", we can also say reasoning from/towards "I don't know" is just as bad.