In Bertrand Russell’s autobiography he describes his intellectual development through poignant stories. One story that has stuck with me is his recounting of his encounter with geometry as a boy. To sum it up: young Russell was looking to get to the bottom of things. When his brother showed him the axioms of Euclidean geometry and assured him that there was no truth or foundations deeper, we just have to believe them for the useful properties of Euclidean geometry to hold true, he became distressed. How could faith be at the root of truth?

I believe this proposition haunted Russell his entire life. It seems like a great contradiction: how could belief be the source of hard universal truth? We typically consider them opposite things. Saying I have a belief implies I don’t have all the information; that my rationality is in someway bounded and I cannot provide proof. Saying that something is “the truth” implies I can provide a proof; that I have enough information on the matter to declare certainty. The nature of proof is where the rub lies.

What is proof? The simplest form of logical proof can be found in syllogisms dating from the classical Greek philosophers.

All men are mortal.

Socrates is a man.

Therefore Socrates is mortal.


The idea of syllogisms is that we have some propositions we assume to be true, then we simply apply logical inference to arrive at another proposition that must be true given the assumptions. Over the thousands of years since the Greeks, logic has become much more sophisticated by the development of higher order logic such as the predicate calculus; however, we have never escaped the fact that all logical proof is bounded by assumptions by the very definition of logical proof or logic itself. Given a logical state of the world, as captured by axioms, we can use rules of inference to paint a logical picture of what is implied by those truths. Logic breaks down when we are unable to state definitively what the truths or assumptions are. Probability emerged as a way of dealing with uncertainty in our conception of the world.

Regardless of it’s limits, logic is still extremely useful. All computation can be put on a logical footing — see computers. What’s interesting to me, and what I study, is that from logic and computation we are able to implement non-logical, probabilistic, models of the world that work really well in doing tasks traditionally seen as requiring intelligence.

In sum: logic is a tool for telling us what follows from a given state of affairs. Using logic as some sort of defense usually ignores that there are beliefs underpinning the logical conclusions which cannot be logically defended by the definition of axioms.