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Here’s an idea many philosophers and logicians have about the function of logic in our cognitive life, our inquiries and debates. It isn’t a player. Rather, it’s an umpire, a neutral arbitrator between opposing theories, imposing some basic rules on all sides in a dispute. The picture is that logic has no substantive content, for otherwise the correctness of that content could itself be debated, which would impugn the neutrality of logic. One way to develop this idea is by saying that logic supplies no information of its own, because the point of information is to rule out possibilities, whereas logic only rules out inconsistencies, which are not genuine possibilities. On this view, logic in itself is totally uninformative, although it may help us extract and handle non-logical information from other sources.



The idea that logic is uninformative strikes me as deeply mistaken, and I’m going to explain why. But it may not seem crazy when one looks at elementary examples of the cognitive value of logic, such as when we extend our knowledge by deducing logical consequences of what we already know. If you know that either Mary or Mark did the murder (only they had access to the crime scene at the right time), and then Mary produces a rock-solid alibi, so you know she didn’t do it, you can deduce that Mark did it. Logic also helps us recognize our mistakes, when our beliefs turn out to contain inconsistencies. If I believe that no politicians are honest, and that John is a politician, and that he is honest, at least one of those three beliefs must be false, although logic doesn’t tell me which one.

Leif Parsons

The power of logic becomes increasingly clear when we chain together such elementary steps into longer and longer chains of reasoning, and the idea of logic as uninformative becomes correspondingly less and less plausible. Mathematics provides the most striking examples, since all its theorems are ultimately derived from a few simple axioms by chains of logical reasoning, some of them hundreds of pages long, even though mathematicians usually don’t bother to analyze their proofs into the most elementary steps.

For instance, Fermat’s Last Theorem was finally proved by Andrew Wiles and others after it had tortured mathematicians as an unsolved problem for more than three centuries. Exactly which mathematical axioms are indispensable for the proof is only gradually becoming clear, but for present purposes what matters is that together the accepted axioms suffice. One thing the proof showed is that it is a truth of pure logic that those axioms imply Fermat’s Last Theorem. If logic is uninformative, shouldn’t it be uninformative to be told that the accepted axioms of mathematics imply Fermat’s Last Theorem? But it wasn’t uninformative; it was one of the most exciting discoveries in decades. If the idea of information as ruling out possibilities can’t handle the informativeness of logic, that is a problem for that idea of information, not for the informativeness of logic.

Related More From The Stone Read previous contributions to this series.

The conception of logic as a neutral umpire of debate also fails to withstand scrutiny, for similar reasons. Principles of logic can themselves be debated, and often are, just like principles of any other science. For example, one principle of standard logic is the law of excluded middle, which says that something either is the case, or it isn’t. Either it’s raining, or it’s not. Many philosophers and others have rejected the law of excluded middle, on various grounds. Some think it fails in borderline cases, for instance when very few drops of rain are falling, and avoid it by adopting fuzzy logic. Others think the law fails when applied to future contingencies, such as whether you will be in the same job this time next year. On the other side, many philosophers — including me – argue that the law withstands these challenges. Whichever side is right, logical theories are players in these debates, not neutral umpires.

Another debate in which logical theories are players concerns the ban on contradictions. Most logicians accept the ban but some, known as dialetheists, reject it. They treat some paradoxes as black holes in logical space, where even contradictions are true (and false).

A different dispute in logic concerns “quantum logic.” Standard logic includes the “distributive” law, by which a statement of the form “X and either Y or Z” is equivalent to the corresponding statement of the form “Either X and Y or X and Z.” On one highly controversial view of the phenomenon of complementarity in quantum mechanics, it involves counterexamples to the distributive law: for example, since we can’t simultaneously observe both which way a particle is moving and where it is, the particle may be moving left and either in a given region or not, without either moving left and being in that region or moving left and not being in that region. Although that idea hasn’t done what its advocates originally hoped to solve the puzzles of quantum mechanics, it is yet another case where logical theories were players, not neutral umpires.

As it happens, I think that standard logic can resist all these challenges. The point is that each of them has been seriously proposed by (a minority of) expert logicians, and rationally debated. Although attempts were made to reinterpret the debates as misunderstandings in which the two sides spoke different languages, those attempts underestimated the capacity of our language to function as a forum for debate in which profound theoretical disagreements can be expressed. Logic is just not a controversy-free zone. If we restricted it to uncontroversial principles, nothing would be left. As in the rest of science, no principle is above challenge. That does not imply that nothing is known. The fact that you know something does not mean that nobody else is allowed to challenge it.

Of course, we’d be in trouble if we could never agree on anything in logic. Fortunately, we can secure enough agreement in logic for most purposes, but nothing in the nature of logic guarantees those agreements. Perhaps the methodological privilege of logic is not that its principles are so weak, but that they are so strong. They are formulated at such a high level of generality that, typically, if they crash, they crash so badly that we easily notice, because the counterexamples to them are simple. If we want to identify what is genuinely distinctive of logic, we should stop overlooking its close similarities to the rest of science.

Read previous posts by Timothy Williamson.

Timothy Williamson is the Wykeham Professor of Logic at Oxford University, a Fellow of the British Academy and a Foreign Honorary Member of the American Academy of Arts and Sciences. He has been a visiting professor at M.I.T. and Princeton. His books include “Knowledge and its Limits” (2000) and “The Philosophy of Philosophy” (2007) and, most recently, “Modal Logic as Metaphysics,” which will be published next year.