By Keith Douglas

My retelling of the classic story of Hilbert’s Hotel had one more point beyond “Think about infinity!” It was about how dangerous it is to think one understands something by the vividness and life-like nature of the description, and about how dangerous it is to regard “how something appears” as the be all and end all of something important. We claim to have learned this lesson with people, but there are schools of philosophy and areas of the discipline (like the debate over qualia) where it seems that’s all many people do — appeal to appearances.

Was anyone thinking they’d like to spend their vacation in such a wonderful place? I admit I am a better philosopher than a story-teller, so perhaps not!

Following a Contradiction

Moving on, let’s spell out why in classical logic everything follows from a contradiction. Take a contradiction, A&~A. Then use conjunction elimination and get A. Apply disjunction introduction and get AvB. Apply conjunction elimination and get ~A. Apply disjunctive syllogism and get B.

This strikes people as a bit weird sometimes. In fact, there are at least two ways to avoid this weirdness.

One: Deny that one can apply conjunction elimination to the contradiction either of the times. This view is attributed to Aristotle, though I am not sure that’s correct. But note the difficulty in stating the rule in the terms that we are used to in classical logic. One needs a rule that is sensitive to several of the sub-formulas. This can lead to so-called connexive logic. (Not connective.)

Two: Deny that disjunction introduction is a valid rule. This is adopted by the so-called “relevant” or “relevance” logics of our day, often associated with the logicians Belnap and Anderson (and in a way to the very challenging G. Priest).

Do you think either of these approaches solves the problem? Is there even a problem? What would be necessary to convince you they are right? Remember that you are talking about how one argues for a logic, which, among other things, is itself used to argue. Logic changed historically — how did that happen? Can one rationally reconstruct that process? Finally, if one decides logic has to change, how does one rewrite all of current explicit reasoning to use the new logic (e.g., in mathematics)?