The Principle of Explosion

Consider the following argument:

1. A tomato is a fruit.


2. A tomato is not a fruit.


3. Therefore David Beckham has a pimple on his bottom.

Contrary to appearances, it’s perfectly valid. Given that a tomato both is and is not a fruit, we can validly infer that David Beckham has a pimple on his bottom. In fact, given contradictory premises we can validly infer just about anything. For example, the following argument is also perfectly valid:

1. There are nine planets in the solar system.


2. There are not nine planets in the solar system.


3. Therefore I am a great philosopher

As is this one:

1. The sky is blue.


2. The sky is not blue.


3. Therefore Ringo was the most creative Beatle.

Ex falso quodlibet

Logicians have long appreciated that contradictory premises entail any conclusion. In medieval times this startling but valid rule of inference was known as ex falso quodlibet. Nowadays it is often labelled ‘the principle of explosion’.

Doubtlessly, many readers will protest that whatever Aristotle or medieval logicians might say, ex falso quodlibet is a load of bunk. Tomatoes are one thing. David Beckham’s bottom is another thing entirely. So how can we possibly argue from one to the other? Let’s find out.

Some background

Before demonstrating the validity of the principle of explosion we need to do a bit of spadework. First, consider the statement ‘John is sad or Alan is hungry’. This is comprised of two clauses linked by an ‘or’. In classical logic statements like this are true provided that one or both of the constituent clauses is true. This means that if ‘John is sad’ is true then ‘John is sad or Alan is hungry’ is also true.

Next, consider the statement ‘Mary is sleepy or Sarah is noisy’. If this statement is true but ‘Mary is sleepy’ is false then clearly ‘Sarah is noisy’ must be true.

The principle of explosion

Armed with this information, let’s re-examine the David Beckham argument. We’ll begin with the premises:

1. A tomato is a fruit


2. A tomato is not a fruit

Now, if ‘a tomato is a fruit’ is true then ‘a tomato is a fruit or David Beckham has a pimple on his bottom’ is also true. So, now we can say:

1. A tomato is a fruit


2. A tomato is not a fruit


3. A tomato is a fruit or David Beckham has a pimple on his bottom (from 1)

But if it’s true that ‘a tomato is a fruit or David Beckham has a pimple on his bottom’ and also true that ‘a tomato is not a fruit’ then it must be true that ‘David Beckham has a pimple on his bottom’. So:

1. A tomato is a fruit


2. A tomato is not a fruit


3. A tomato is a fruit or David Beckham has a pimple on his bottom (from 1)


4. David Beckham has a pimple on his bottom (from 2 and 3)

To make the logical structure of the argument clearer, let’s employ some shorthand. Let ‘P’ stand for the proposition ‘a tomato is a fruit’ and let ‘Q’ stand for the proposition ‘David Beckham has a pimple on his bottom’. Thus we have:

1. P is true


2. P is false


3. P is true or Q is true (from 1)


4. Q is true (from 2 and 3)

Clearly, ‘P’ and ‘Q’ could stand for any propositions we care to devise. Thus we have a general proof that anything can be inferred from contradictory premises.

I am the Pope

According to one old story, the English mathematician and philosopher A N Whitehead was once challenged to demonstrate that one can prove anything using contradictory premises. ‘Starting from the premise that four equals three, prove that you are the Pope,’ he was asked.

‘Easy!’ Whitehead replied. ‘Four equals three; subtract two from each side; then two equals one. It is commonly known that the Pope and I are two people. Therefore the Pope and I are one.’ (Some versions of the story substitute J. M. E. McTaggart, G. H. Hardy or Bertrand Russell for Whitehead. But never mind. It’s a great story anyway.)

Valid, but not sound

The principle of explosion is a perfectly good rule of inference according to classical logic. You really can derive anything from contradictory premises. However, we have to remember that ex falso quodlibet arguments, though valid, are not sound. Classical logicians hold that contradictions are never true, which means that one of the premises must always be false. This means we can never use the principle of explosion to actually prove anything.