## Sunday, 21 October 2012

### A slight indisposition for me, a disaster for Scottish football

On Tuesday I was mildly unwell - a seasonal cold, so I spent the day in bed.  This was a disappointment because I missed  a trip to Thorpe Park with our new students (and I was genuinely sorry to miss it, despite what you are probably thinking).  On Tuesday evening the Scottish football team played Belgium - a highly fancied squad - in Brussels in a World Cup qualifying match and - another disappointment for me - they lost 2-0.

If I had not been mildly unwell on Tuesday then Scotland would have beaten Belgium.   (1)

You may think that statement is unlikely, but it is absolutely true.  At least if you are a mathematician, that is.  By the rules of propositional logic, the proposition "If p then q" is true whenever q is false.  Since the proposition "I was not mildly unwell on Tuesday" is false, then statement (1) is true.

In fact, we can strengthen it.  This statement is also true:

If I had not been mildly unwell on Tuesday, then Scotland would have beaten Belgium 5-0.    (2)

And so is this one:

If I had not been mildly unwell on Tuesday, then Scotland would have beaten Belgium 6-0.    (3)

It might appear that statements (2) and (3) contradict each other.  They don't. They are both true, and their mutual truth merely confirms that the proposition "I was not mildly unwell on Tuesday" is false.  (Who needs doctor's notes?)

What this tells us is that mathematical language sometimes does not agree with everyday language. Many people would question the truth of statement (1): in everyday language the sentence appears to have meaning.  In propositional logic if the antecedent ("I was not mildly unwell on Tuesday") is false then it has no bearing on any other proposition.  In English statement (1) implies a connection, and many people would regard the statement as meaningful (and false).  Mathematical logic suggests that this is misguided: when we move down one branch of the universal wave-function, it is meaningless to speculate on what might have happened on the branches not travelled.  It's very human to do so, and makes for good pub conversations, but ultimately we are wrong to imagine that statements like (1) are meaningful.

If only I had finished this post five seconds earlier, I would have won millions of pounds in the lottery!