## A Discovery about Basic Logic

Logic is a foundation for many things. But what are the foundations of logic itself?

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:

✕
((p·q)·r)·(p·((p·r)·p))==r |

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):