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Are there logic systems that are internally consistent that have a different makeup to the logic system that we use?

There's a nice article on intuitionist logic in the Stanford encyclopedia.The differences between it and classical logic become more profound inconnection with quantifiers such as "all" and "some". For example, inintuitionistic logic, it can be true both that not everything has someproperty and that there is nothing that does not have it! Andin the intuitionistic theory of the real numbers, we can actually findsuch a property and prove such a statement: We can prove that not everynumber is either positive, negative, or equal to zero; we can alsoprove that there is no number that is neither positive, nor negative,nor equal to zero. Think about that for a while. Of course, we can only "prove" it in the sense that it follows fromthe principles of intuitionistic analysis. Whether it is true dependsupon whether those principles are true. Since you mentioned internal consistency, perhaps I'll mention something even stranger, so-called paraconsistent logics . These are systems in which...

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On Dan's comment. The distinction between so-called weak counterexamples and strong ones is, of course, important. But it really is possible to prove, in intuitionistic analysis, the negation of the claim that every real is either negative, zero, or positive. The argument uses the so-called continuity principles for choice sequences. I don't have my copy of Dummett's Elements of Intuitionism here at home, but the argument can be found there. A short form of the argument, appealing to the uniform continuity theorem—which says that every total function on [0,1] is uniformly continuous—can be found in the Stanford Encyclopedia note on strong counterexamples . There is an important point here about the principle of bivalence, which says that every statement is either true or false. It's sometimes said that intuitionists do not, and cannot, deny the principle of bivalence but can only hold that we have no reason to affirm it. What's behind this claim is the fact that we can prove that we will not...

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