Russell's Two Legacies, plus addendum

 Russell’s Two Legacies

With diamond addendum

 

Bertrand Russell has two legacies: his Paradox and his Principia. His Paradox is concisely stated, wittily illustrated with a quaint tale, and logically devastating. He asked: consider the set of all sets that do not contain themselves. Does it contain itself, or not? The answer is: yes as much as no!

          Russell illustrated his Paradox with a quaint tale about a village’s  barber. The barber shaves all those, and only those, who do not shave themselves. Does the barber shave himself? Yes as much as no!

          You can tell similar tales. For instance, that village is watched by a watchman, who watched all those, and only those, who do not watch themselves. Who watches the watchman? The watchman watches himself as much as he does not.

          The village jester laughs at all those, and only those, who do not laugh at themselves. Does the jester laugh at the jester? The answer to this question is, yes as much as no.

          Distantly related to all of these is Gödel’s Paradox: “ ‘is not provable when it quotes itself’ is not provable when it quotes itself.” That statement uses coding to call itself unprovable. This yields a paradox unless proof does not equal truth.

Russell’s Principia is an attempt to suppress Russell’s Paradox, and win the day for the logicist program. It is long, dense, unreadable by most, and its mission was defeated by Gödel’s Paradox. Russell tried to ban self-reference, but self-reference is inherent in arithmetic, given coding.

Russell’s Principia is long, dull, obscure, unreadable, specialized, and failed. Russell’s Paradox is short, sweet, clear, witty, adaptable, and effective.

I think that Russell’s Paradox will long outlive his Principia, which will be remembered only for taking hundreds of pages to prove that 1+1=2. I see in this an object lesson in pragmatic memetics:

Keep It Simple, Stupid.

 

          Diamond Addendum:

          I resolve the paradoxes with “diamond logic”, in which truth values have two components, each of which can be true or false. That makes four values in all, including the paradox values ‘true but false’ and ‘false but true’.

          I illustrate this with a new tale from Russell’s Village. I say that the village has two barbers, who between them shave all those, and only those, who do not shave themselves; and they shave each other, but not themselves.

          Likewise, the village has two watchmen, who watch each other but not themselves; and two jesters, who laugh at each other but not themselves.