This design is a little bit of fun but it also has serious purpose. It shows that the use of simple diagrams can help to clarify an idea that words have confused. They could be described as Venn diagrams except that they are so simple that they make no real use of the special qualities of John Venn’s invention, which dates from the 1880s.
It also serves to illustrate the way the vital word ‘if’ is used in mathematical thinking. ‘If is known as a connective – that is, it connects two ideas together and it is incredibly important in mathematics and in logical statements generally, not least because if means that you can often identify something quite complicated by a simply feature it displays. Take the statement ‘if A then B’, it doesn’t much matter what A and B are, they might be objects, actions, a mixture of the two or any number of weird and wonderful things. All that matters for the statement to be true is if you find that A is true, then you will always find that B is true.
So, leaving aside unfortunate accidents of birth and traffic, we might say ‘If an animal is a cat (A), then it has four legs (B).’ To most people, that probably seems quite clear and understandable. The problem is that it can be said in a number of different ways that sometimes don’t seem quite as clear. So, for instance, the statements ‘All cats have four legs’ or ‘Only four-legged animals can be cats’ or ‘If it doesn’t have four legs it’s not a cat’ have exactly the same meaning as our original ‘If…’ but somehow they don’t feel the same and, indeed, can sometimes sound quite confusing.
And then there is the sometimes vexed question of that the statement doesn’t mean. In this case, to say ‘All cats have four legs’ doesn’t mean that all animals with four legs are cats. It also doesn’t mean that ‘Some cats have four legs’, because that would mean that sometimes we have A (a cat) but not B (four legs).
So the rule about ‘if’ statements is that you can’t turn them round (with an exception we’ll look at in a moment). ‘If A then B’ doesn’t mean ‘If B then A’.
Using Venn diagrams allows us to short-circuit all the confusions introduced by language. If we represent all cats by a red circle and all four-legged creatures by a green one, then the following diagram says it all: Note that the sizes of the circles don’t, in this case, mean anything (though we could use size to indicate the relative size of the two groups if we wanted to). All that matters here is that the entire circle representing cats is contained within the circle representing four-legged animals. From the diagram, it is clear to see that all cats have four legs and that there are other four-legged animals which aren’t cats.
In the ‘Only the good die young’ print, this situation is represented by the first pair of circles, showing that what the phrase means is that the entire group of people who die young are contained within the group of the good, though the diagram suggests that there may be others in the good group who don’t die young.
The fact that you can’t turn an ‘if’ statement around is illustrated by the second pair of circles, top right. ‘Only the good die young’ expressed as ‘if’ statement is ‘If someone dies young then they were good.’ The reverse of that is ‘If someone is good then they will die young’ or ‘All of the good die young’, which is what the second pair of circles illustrates.
With those two under our belt we can turn to the special and very important case of ‘If and only if’ statements. These are the exception referred to above, where an ‘if’ statement can be turned around. ‘If and only if’ statements work equally in both directions. So ‘If and only if A then B’ means, at one and the same time both ‘If A then B’ and ‘If B then A’.
On the print, this is the circle in the middle, which is actually meant to represent two circles of equal size overlaid. As an ‘if’’ statement it means ‘If and only if someone dies young, then they were good.’ You will be the exception if you don’t find that a little confusing at first. It says two parallel things:
‘If someone dies young then they were good’ and ‘Only if someone dies young were they good’. Put another way it says that everyone who dies young was good and that everyone good will die young. The circles representing those who are good and those who die young are one and the same. And because the circles overlap exactly, we know that no-one who dies young was not good and that no-one good will escape dying young!
Fortunately for the few of us who are good, the statement is more interesting as a piece of logic than a description of reality.
The two diagrams at the bottom of the print represent the ‘if’ statements ‘If someone is good they may die young’, which is not really a formal ‘if’ statement in the sense described above because it describes a possible rather than inevitable link – and it can be reversed. The final statement illustrated is a proper ‘if’ statement of the form ‘If A then not B’ – proper because one thing inevitably leads to another and because it can’t be reversed simply – the fact that good don’t die young doesn’t mean that if you don’t die young you’re good.
So if you take anything away from this design, let it be firstly that sometimes words confuse things and a diagram (or even some numbers) can help to clarify matters. Secondly, if it must be done in words, it can be worth trying to identify the ‘if’ statement underlying things. And thirdly it’s worth pondering the incredible power of the word ‘if’. There are countless cases in the history of mathematics (and of science in general) where great strides have been made because someone realized that something very complex and previously difficult to detect gave itself away in some previously unnoticed simple way. From then on it is ‘If (simple) A then (highly complicated) B.’