Formal Indecision

Godel

A formal system in mathematics is a system which contains a set of axioms (unquestionable statements of truth), and then adds to these a set of production rules by which new true statements can be generated. The process of production can go on forever allowing a never-ending list of truths to be derived.

A famous result called Gödel’s incompleteness theorem shows that in such a system, there are some true statements that are nevertheless un-provable (undecidable) using the rules of the system. It shows that a self-consistent Continue reading

The Raven and the Shoe

Raven
The hypothesis that all ravens are black is logically equivalent to the statement that all non black things are non ravens, and this is supported by the observation of a white shoe.

This is a paraphrasing of a famous paradox due to Hempel. There is a lot of fairly impenetrable discussion on the Wikipedia page about this paradox, some of which I believe to be incorrect, and so I include a readable resolution based on a Bayesian perspective here, and I also relate this issue to our ability to know the truth Continue reading