Randolph Diagrams

This is what a genius looks like.

There is something aesthetic and elegant about Randolph diagrams, unfortunately they aren’t commonly used. I found out about them when reading Embodiments of Mind by the glorious and bearded Warren McCulloch.

The Original Proposal:
In the book he refers to them as “Venn functions” and they are briefly explained as being derived from Venn’s diagrams for sets but in McCulloch’s case they were used to express logical statements. If you draw a Venn diagram of two circles intersecting you are left with four spaces ( a/b, a&b, b/a, U ), adding a jot into a space to denote truth or leaving it blank for false gives you the 16 possible logic combinations. They are great examples of the isomorphism between logic and set theory:

He used these as tools to help teach logic to neurologists, psychiatrists and psychologists. Later he developed them into a probablistic logic which he applied to John vonn Neumann‘s logical neuron nets. Which I will discuss in the next post.

Randolph’s Diagrams:

The truth values for three statements.

McCulloch does mention that they could be used to apply more than two statements but doesn’t show how, later John F. Randolph developes the system as an alternative visualisation of set relations neatly coping with more than two sets (something Venn diagrams begin to struggle with after five). For each additional statement/set a new line is introduced in each quadrant. Four statements would be a large cross with four smaller crosses, one in each quadrant.

Wikipedia has an example of the tautological proof for the logical argument, modus ponens, which can be found here, but I thought it would be good to show how three values are handled – so we’ll use syllogism, as in “Socrates is a man, all men are mortal, therefore Socrates is mortal” being reduced in it’s logical form to tautology:

((A implies B) and (B implies C)) implies (A implies C)


6 thoughts on “Randolph Diagrams

  1. Hi!

    how do i read the diagrams?
    for a venn diagram, one closed figure is a set, the other one is another set, call those a and b, and the intersection is the a intersection b … etc…

    i understand, that a dot indicates a truth. a dot above a line indicates: a true, and below it means (not a) is true, same with the other line. so a line in the position left (of four positions up left right down) would mean ((a true) and ((not b) true))… is it really so?

    • Hmm, hopefully this picture will help, just whipped it up so there may be spelling mistakes!

      I’m not a sure what “so a line in the position left (of four positions up left right down) would mean ((a true) and ((not b) true))… ” means but that image shows how the lines relate to each other.

      • Thanks for making and sharing this picture. It nice to be able to relate the Venn Diagram and Randolph Diagram visually.

    • I’m afraid I can’t remember, it was an embedded *.mov file in a stand alone website somewhere out there. It was a bit of a deep google dive that got me to it so I wanted to put it on Youtube to be more accessible.

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