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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](http://en.wikipedia.org/wiki/Paraconsistent_logic).2012-07-23
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    @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).2012-07-23
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    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.2012-07-23
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    @RobertMastragostino: (edit: I mean MichaelGreinecker) I guess the "Tradeoff" section in the link you gave answers my question, thx.2012-07-23
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    @NickKidman I believe that was Mr. Greinecker's link, if we're giving credit where credit is due. I'm no logician.2012-07-23
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    Possibly relevant: [Can the principle of explosion be removed from constructive logic?](http://math.stackexchange.com/questions/148210/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*.2012-07-23
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    @MarkDominus: Haha, it's funny how your comment to username *sclv* is basically a copy of the discussion here.2012-07-23
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    @Nick That is funny. Definitely check out the Priest book.2012-07-23
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    This may be more of probability and statistics, but Simpson's paradox comes to mind. The reason I'm thinking of this is that to describe it in a general context, it states that adding another variable to the expression can reverse a relation. Thus we could possible reverse a contradiction or false statement with the addition of another variable in the equation, which would be like a more thorough study of the logic, adding a new observation. I hope that this makes sense. Here's Wikipedia's description: http://en.wikipedia.org/wiki/Simpson's_paradox2012-07-23

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