Logic

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

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The kind of logic that most mathematicians assume in their work is known as classical logic. Classical logic accepts the Law of the Excluded Middle, which states that for every statement P, "P or not-P" is true. Some logicians and mathematicians (though not many) work within systems of reasoning that do not assert this Law. Most constructive logics fall into this category, in particular intuitionism does. For a little more on intuitionism see Question 139. These days, there isn't much of a debate within the mathematical community about which is "the correct logic". There has been considerable debate about this within philosophy. Especially pertinent here are the writings of the Oxford philosopher Michael Dummett.

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 contradictions—statements of the form p & ¬ p—can be true.Of course, such statements are also false. But paraconsistent systemslack the classical principle sometimes called "explosion", whichpermits one to infer anything whatsoever from a contradiction. Suchsystems are of interest in connection with paradoxes. Consider, forexample, the sentence "This very sentence is not true". If it is true,then it's not, and if it's not true, then it is. That is usually takento be a problem, since the sentence can't be both true and false. Soone wants to know where the reasoning that seems to lead to thatconclusion went astray. But paraconsistent logics allow one to say:What problem? That sentence is both true and false! It's an interesting option.

I have a minor quibble with one of Richard's statements. He says that an intuitionist "can prove that not every number is either positive, negative, or equal to zero." I don't think an intuitionist would claim to be able to prove that. Rather, an intuitionist would say that he is unable to prove that every number is either positive, negative, or equal to zero. It's a small point, but there is a difference between being unable to prove that something is true and being able to prove that it is false.

For more on this, see the entry in the Stanford Encyclopedia of Philosophy on Weak Counterexamples. Notice that conclusion 2 in that entry is "we cannot now assert that every real number is either positive, negative, or equal to zero," which is different from saying "we can now assert that not every real number is either positive, negative, or equal to zero."

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 be able to find a statement P and show that it is neither true nor false. That is to say, we can prove that there is no statement that is neither true nor false. But that does not, by itself, show that it is incoherent to hold that not every statement is either true or false. The two claims are intuitionistically consistent.

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