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Maybe you are interested into the "algebraic" approach to logic ... See Abstract algebraic logic and Propositional Consequence Relations and Algebraic Logic with bibliography. You can see also : H.Andréka, J.D.Monk, I.Németi (eds.), Algebraic logic (1991) Dov M.Gabbay & F.Guenthner (editors), Handbook of Philosophical Logic : 2nd ed. Vol.2 (2013), ...


Note the possibility and necessity operators aren't truth functional. In other words, the so-called principle of compositionality (also known as Frege's principle) that the meaning of complex sentence depend exclusively of the meaning of its compounds does not hold. This implies that, no matter how we try, we can't fully build a truth table for those ...

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