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Monday, January 6, 2014

How to Throw Out a Garbage Can and the Theory of Logical Types

I realized I ought to throw out one of my dented old garbage cans this morning, and immediately after realized that was no simple matter.  I soon after realized that Bertrand Russell's theory of logical types provided a solution.

The problem with garbage cans is that they're recognized as containers of garbage, not garbage themselves.  This is a good thing most of the time, but not when you want to get rid of the garbage can itself.

In McLuhanesque terms, the garbage pail is a medium, and the garbage is content. Bertrand Russell addressed this distinction in a systematic way in his theory of logical types, at the beginning of the 20th century.

Indeed, Russell used his theory of logical types to try untangle one of the most intractable paradoxes in human communication: the paradox of the liar.  If someone tells you, "everything I say is a lie," do you believe that statement?  If you do - that is, if you believe that statement is true - then that means you believe that it's true that everything including that statement is a lie.   But if the statement is a lie, that means that the statement could be true, which pitches you right back into the horns of the dilemma.

Russell's solution or way out was to note that the statement "everything I say is a lie" is a statement about other statements, and therefore not one of those other statements itself, and therefore need not share the characteristics of the statements in the category.   It is a "meta" statement, or, in McLuhan's terms, it is a medium not content.   Therefore it as a meta-statement could be true - a true statement about the falsity of all the statements within its category, all the content that the statement describes.

Now, this isn't a perfect solution.  For example, if the paradox of the liar is rendered as just "this statement is lie," then the only way the theory of logical types could help us is by obliging us to consider that statement to be two statements at the same time - a meta statement or description of content, and the content, with the first being true and the second false, which is a pretty steep paradox on its own (something being both true and false at the same time).   Further, we could untangle "everything I say is a lie" just by concluding that that statement is a lie, but not then concluding that therefore everything I say is true.   If we concluded that just some things I say are lies, then the original statement could be among the statements that are true.

But as for garbage cans, Russell's theory of logical types works just fine.   The problem in how to throw out the can arises from the fact that the can is not usually itself garbage.  But when it becomes time to trash the can, all that's needed is putting a sign on the can identifying it the can as garbage - say, "please throw out this can".

I'll let you know if it works.


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