The Banana Paradox

The process of mathematical reasoning is one aspect of the subject that fascinates me. But the idea of ‘logic’ could be considered a subject in itself. Logic can be thought of as the systematic study of the form of arguments. A valid argument makes use of assumptions or premises, and comes to a conclusion based on these assumptions.

But what if I told you that we can use logic to gain information about anything by looking at – well – anything? The contents of your local Walmart or Morrisons supermarket can give you minute amounts of evidence about the nature of distant galaxies – things that could not be further from your mind as you wonder down the biscuit aisle during your weekly shop, wondering whether to buy choc-chip cookies or custard creams (or both, in my case).

This idea comes in the form of what I to call the Banana Paradox. There are some different versions of this, but all are essentially the same:

(1) All ravens are black.

(2) Everything that is not black is not a raven.

(3) A banana is not black, and therefore is not a raven.

We have therefore gained evidence to support the idea that all ravens are black -by looking at a banana.

Wikipedia provides a long list of solutions to this problem. But the solution might be quite straightforward:

(4) There are far more black objects than there are ravens.

(5) There is a limited number non-black objects (call this n).

(6) One of these happens not to be a raven.

(7) If all of the non-black objects are also non-ravens, proposition (1) is true.

(8) By proving that 1 of the n non-black objects is indeed not a raven (i.e. proving that the banana is yellow), we have completed a very, very, very small fraction (1/n) of our proof that all ravens are black.

In order to prove the proposition that all ravens are black, we would have to prove that all of the n non-black objects are non-ravens.

This process is not confined only to ravens. You could replace ‘raven’ with ‘planet’ or ‘galaxy’ (with the appropriate colour) and obtain the same result – provided that the assumption of your object of choice having only one colour is a reasonable one to make.

So do not doubt the wisdom of the banana – it can give you a (very, very small) amount of evidence about almost anything you want it to.