The Barber Paradox

When Paradox just Isn’t when we consider ‘When’

When I first started studying logic in earnest I dove into the classics. I wanted to start at ground roots and work my way up to the present.

Not as far back as the Greek philosophers…well okay I did buy the Great Books of the Western World, by Encyclopaedia Brittanica, and dove into Plato and Thomas Aquinus, who, while focused on religion and God, did so through the medium of logic. I picked up the whole set of books in a wooden bookshelf for $50 at a second-hand store. It was a must-buy and still has pride of place in my lounge room.

But it was Bertrand Russell who caught my attention the most. One can almost feel the machinery of Russell’s mind working as he works out, in and through writing, what it is that he really ponders in trying to make sense of the world. I bought or downloaded onto my iPad almost everything he wrote.

I was working on software that became the Boston conceptual modelling software sold by FactEngine today, and felt a need to describe its eccentricity in logical terms in that it defies logic by treating logic as if a game.

I was wondering if I alone in the world, or if others saw things the same way. “Perhaps Russell did?”, I wondered.

Turns out that Gödel did, with his dualistic interpretation of Russell’s system of Principia Mathematica, so I bought a 1953 version of Principia Mathematica at some expense ($600, vols 1,2 & 3). But it was Russell’s Principals of Mathematics that grabbed me the most, as it contained all of Russell’s thought processes leading up to Principia (as it is called). A gem of history and logic.

It contains Russell’s thought processes in defining formal systems, and formal logic as it has come to be studied today.

To get there, Russell “discovered” paradoxes of logic that perplexed him somewhat in the aim and game of wanting to define systems free from such things.

Famous of those is The Barber Paradox.

The Barber Paradox is typically worded as:

“In a village, there is a barber who shaves all those, and only those, who do not shave themselves. The question is: Does the barber shave himself?”