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?

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    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 '14 at 9:17
  • With a specific "algebraic" point of view, see : Helena Rasiowa, An algebraic approach to non-classical logics (1974). Mar 4 '14 at 10:10
  • It's my understanding that for any n =/= 2, we wouldn't call it "classical".
    – Addem
    Mar 4 '14 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 '14 at 21:15
  • @allegranza: thanks for the suggestions, they look useful. Mar 4 '14 at 21:16

Boolean algebras have complementation laws which are equivalent to the Law of the Excluded Middle. A multi-valent logic would have to abandon or modify those. One of the reasons classical logic is useful is that it expresses rules of inference and valid argumentation. If a multi-valent logic fails to do so, it has limited utility as a logic.

  • Good point about rules of inference. Intuitionistic logic is multi-valent, but Gentzen had a proof system for it. I imagine one will have to work with it explicitly to understand what this means. Mar 16 '14 at 4:39

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