Logic, Explainability and the Future of Understanding—Stephen Wolfram Writings

In symbolic logic, one introduces symbols like p and q to stand for statements (or “propositions”) like “this is an interesting essay”. Then one has certain “rules of logic”, like that, for any p and any q , NOT (p  AND  q ) is the same as (NOT  p ) OR  (NOT  q ).

But where do these “rules of logic” come from? Well, logic is a formal system. And, like Euclid’s geometry, it can be built on axioms. But what are the axioms? We might start with things like p  AND  q = q  AND  p , or NOT NOT  p = p . But how many axioms does one need? And how simple can they be?

It was a nagging question for a long time. But at 8:31pm on Saturday, January 29, 2000, out on my computer screen popped a single axiom. I had already shown there couldn’t be anything simpler, but I soon established that this one little axiom was enough to generate all of logic:

But how did I know it was correct? Well, because I had a computer prove it. And here’s the proof, as I printed it in 4-point type in A New Kind of Science (and it’s now available in the Wolfram Data Repository):

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