# Are there n-valent classical propositional logics?

Classical propositional logic is truth-functional, that is the truth of propositions are determined by the assignment of truth-values taken from {false,true} to the atomic propositions. And it is this that gives the semantics for the logic.

But noting that {false, true} is actually a boolean algebra, and in fact the smallest one, can we generalise to a B-valent classical propositional logics by taking values from some boolean algebra B, which need not be finite?

It seems to me that the usual completeness and soundness theorems should hold in this context. Is that right?

• Basically I suggest you three books; an old one : JBarkley Rosser & Atwell Rufus Turquette, Many-Valued Logics (1952), a new one : Grzegorz Malinowski, Many-Valued Logics (1993), and the encyclopedic : Dov Gabbay (editor), Handbook of the History of Logic, vol 8 : The Many Valued and Nonmonotonic Turn in Logic. Mar 4, 2014 at 9:17
• With a specific "algebraic" point of view, see : Helena Rasiowa, An algebraic approach to non-classical logics (1974). Mar 4, 2014 at 10:10
• It's my understanding that for any n =/= 2, we wouldn't call it "classical". Mar 4, 2014 at 19:42
• @addem: no, I wouldn't either. We're simply formally increasing the number of truth-values but whereas with the classical 2-valued logic we have a correspondance with reality, in that we can say it exists or that it does not, there isn't an obvious interpretation with reality that is at hand. But that doesn't mean that the extension isn't interesting. Mar 4, 2014 at 21:15
• @allegranza: thanks for the suggestions, they look useful. Mar 4, 2014 at 21:16