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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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    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.