A second-in-a-row Dictionary of Mathematical Eponymy post about Boolean logic today – and another example of a Very Neat Diagram.

You’ve seen - at least, I hope you’ve seen - Venn diagrams. Beastly things. I would chuck them out the window if I could, they just don’t sit nicely with the way I visualise sets, and I have to mentally translate back and forth between my mental images and the circles.

In the 1960s, at the University of Arkansas, John F Randolph came up with another way to show logical relations. Rather than using a pair of circles for a two-event scenario, he used a grid. One side of the grid represented A and not-A; the other B and not-B. If the combination of events represented by a cell gave a ‘true’ value under the operation in question, the cell would have a dot in it - otherwise, it wouldn’t.

For example, suppose we’re interested in the XOR function, which is true if exactly one of the two events happens. This isn’t exactly how Randolph set up the diagram1, but it’s close enough:

A | A' | |
---|---|---|

B | * | |

B' | * |

Nicely, Randolph diagrams can extend naturally to events with more than two possible states, and to more than two events - putting smaller Randolph diagrams inside the cells gives you a simple way to envisage functions of four events - and you can nest them (in principle) as deeply as you like.

The reason Randolph put his diagrams on the diagonal wasn’t aesthetic, it was practical: it meant that putting in place a rule that “above the line means the event occurs, below means it doesn’t” saves a fair bit of labelling - although I’d still like to see the lines labelled!

They’re better than Venn diagrams. Any questions?

They’re also clearer (to me, at least) than truth tables, and make simplifying complicated logical expressions - and proving Boolean identities - much more intuitive.

They apply just as nicely to set theory - it’s almost as if Boolean logic and set theory were somehow linked!

It’s a bit hard to track down details of Randolph’s life, but I’ve picked up a few: he graduated from the University of Michigan in 1928, and after Arkansas became a professor at the University of Rochester until his retirement in 1968. He died in 1988.

He appears to have kept a journal while on a train-and-cycling holiday in Europe in 1938, published as *Traveling Between The Lines*.

I’d love to know more about Randolph! If you know anything, do pass it on.

- Randolph’s diagrams were at a diagonal and not, so far as I’ve seen, labelled - but I’ve straightened them up and labelled them for clarity and typesetting reasons. [↩]