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John Buridan on Self-Reference: Chapter Eight of Buridan's 'Sophismata', with a Translation, an Introduction, and a Philosophical Commentary

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Buridan was a 14th century French scientist and philosopher, whose idea of "impetus" anticipated Galileo's theory of inertia. George Hughes was a philosophy professor (the highest academic rank in British Commonwealth countries) at Victoria University of Wellington (New Zealand) and a world-class logician. He translated Buridan's Latin and provided a clear commentary.

In this work, Buridan attempts to solve a number of semantic paradoxes, e.g. a one-person liar "What I am saying is false"; a two-person liar where Socrates says "What Plato is saying is true" and Plato says "What Socrates is saying is false"; and other even more elaborate ones. All these have in common is a proposition referring to itself, or a set of propositions where there is some circle of internal reference, hence the title.

Before Buridan attempts to solve the paradoxes, he discusses propositions that can not be true if they exist, but can the state of affairs described is logically possible. Such propositions he describes as "not possibly true" as opposed to "logically impossible" such as "This circle is square". Buridan gave the example of a not possibly true proposition "No proposition is negative". While the facts can be as the statement asserts, if the statement exists it cannot be true because it is a negative proposition itself.

Then Buridan discusses the validity of the inferences: "All propositions are affirmative, therefore no proposition is negative" and "No proposition is negative, therefore some proposition is negative".

The point is, the usual definition of validity is "it's impossible for the premise(s) to be true and the conclusion false". Thus the first would be invalid (because the conclusion is false when it exists even though it seems to follow from the premise) and the second valid (if the premise exists it is a negative proposition, so the conclusion is true). But with such self-referential statements, this definition of validity is inadequate, because in both those cases the they are contrary to other arguments with these forms (All As are B therefore no A is non-B, No Xs are Y therefore some X is Y).

Buridan defined a valid argument as "it's impossible for the fact to be as the premise says and not as the conclusion says".

This becomes important when he presents his ingenious solution to the paradoxes.

This is not of mere academic interest. A Wellington logician, Ross Powell, studied Buridan's solutions for his Master's thesis under Prof. Hughes. Since Mr Powell also has a degree in physics, he has subsequently applied Buridan's logic to solve vexing paradoxes in quantum mechanics, e.g. the measurement problem. His articles are available on the Internet.