Russell's quote above is extremely incisive. Modern mathematics is indeed made up of statements of the form statement P implies statement Q. That is, we have ‘If statement P is true, then statement Q is true also.’ Usually, however, this structure is hidden, mainly to make mathematics more comprehensible – it would be hard to read if we always wrote it that way.

The second part of Russell's quote is also true but a lot more subtle. One could argue that the statement ‘The Moon is made of cheese implies the Moon is a tasty snack’ is true because if the Moon was cheese, then it would be tasty. The point is that the statement makes sense and is true yet it has nothing to say on whether the Moon really is made of cheese or whether it really is a tasty snack. All it says is that if it is cheesy, then it is tasty. It is worth bearing this example in mind as we proceed.

Instead of Russell's P and Q we will, in general, use A and B to denote our statements.

‘If …, then …’ statements

Most mathematical statements are of the form

‘If statement A is true, then statement B is true.’

They may be heavily disguised but when you break them down, that is what you will find. This type of statement is called an implication.