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In common natural languages, there are two interpretations of the word "or".

Can you construct a formal logic based on the excluding notion of "or", such that from a contradictory ($A$ and $\mathbb{not}(A)$ is true simultaneously) it doesn't follow, that all formulas are true?

That logic doesn't have to be very strong, but should still look like something which can be used to compute intuitive conclusion rules from some axioms.

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I don't know what you mean by "constructing a logic from [...]or". The rest seems like you might be interested in paraconsistent logic. – Michael Greinecker Jul 23 '12 at 13:20
@MichaelGreinecker: I just assumed here that both or's have some representation by a chain of other symbols, just like $\Longrightarrow$ is "not or". What I mean is to that this version of or as primitive symbol and avoid the axiom $A\Longrightarrow A\ or\ B$ (Disjunction introduction). – NikolajK Jul 23 '12 at 13:23
You can still construct or out of and, not and xor, so I don't see how you would be "avoiding" that axiom, you're just making it more complicated to state. – Robert Mastragostino Jul 23 '12 at 13:26
@RobertMastragostino: (edit: I mean MichaelGreinecker) I guess the "Tradeoff" section in the link you gave answers my question, thx. – NikolajK Jul 23 '12 at 13:28
Possibly relevant: Can the principle of explosion be removed from constructive logic?. My conclusion in that thread was that I should go back and reread Graham Priest's book In Contradiction. – MJD Jul 23 '12 at 13:44
up vote 2 down vote accepted

Graham Priest's book In Contradiction: A Study of the Transconsistent is fascinating, very readable, and discusses this exact question in exhaustive detail.

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