Logic, Truth & the Limits of Reason

Logic Truth and the Limits of Reason · Open board

A map of the shapes an argument can take, and the exact spot where even a system built from nothing but clean rules runs out of road.

Before you start. What Is Knowledge? asked what you can know, and settled on a belief you can actually justify as true. Can Machines Think? asked whether a machine can think at all. This lesson turns the question onto the tool both of those questions were using without naming it: reasoning itself, the three different strengths it comes in, and the one thing not even a perfect version of it can do.

The idea in 30 seconds

Deduction starts from a rule and a fact and guarantees its conclusion, because it does nothing but unpack what the rule and the fact already say.
Induction builds a general rule out of a run of specific cases, and only makes that rule probable, since the very next case can still break it.
Abduction just picks whichever explanation best fits what you’re looking at, with nothing guaranteed at all.
An argument’s shape is valid when the conclusion truly follows from the premises; it is sound only when it is valid and the premises are actually true, which is why a valid argument can still be false.
In 1902 Bertrand Russell showed Gottlob Frege that Frege’s own finished, rule-perfect system produced one question it could not answer without contradicting itself.
In 1931 Kurt Gödel proved that isn’t a fixable design flaw: any system strong enough for ordinary arithmetic always contains a true statement that the system itself can never prove.

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