Historian and Philosopher of Logic Stephen Read on the history of paradoxes, semantic paradoxes, and its direct connection to the foundations of mathematics

How have paradoxes been used throughout history? How have paradoxes influenced the study of mathematics? In what ways do paradoxes challenge knowledge and truth? These and other questions are answered by Historian and Philosopher Stephen Read.

If we hang him, he’ll have spoken the truth and so we should let him cross the bridge. But, if we let him cross the bridge he will have lied, and so we should have hung him. So, Sancho Panza, how shall we judge this case? And it takes a while for Sancho Panza to appreciate the paradox, but eventually he gives his judgement which is to hang the half of him that lied and let the half of him that spoke the truth cross the bridge.

Philosopher David Edmonds on deontological ethics, Kantian ground of human rights, and usefulness of philosophers

You could have a set of numbers, you could have a set of sets, you could have a set of sets that are members of themselves, you could perhaps have a set of sets that are not members of themselves. But, then he thought, hang on. If you had a set of sets that weren’t members of themselves, would that set be a member of itself or not?

I’ve described a number of semantic paradoxes mostly to do with truth. I’ve then shown that they are very similar to some set theoretical paradoxes at the heart of the foundations of mathematics, and actually they ramify out. There are also epistemic paradoxes which apply to concepts like knowledge as well as concepts like truth.

Professor Emeritus of History and Philosophy of Logic, University of St. Andrews in the Arché Research Centre for Logic, Language, Metaphysics and Epistemology, University of St. Andrews

Here’s one of my ways of presenting the liar paradox. Suppose the Cretan says: all Cretans are liars. This proposal is true or false? For example suppose proposal “all Cretans are liars” is a conclusion of a correct syllogism (the syllogism construct the Cretan according to the formal rules). This is deductively true statement. It does not depend on the intentions of the Cretan. And does not affect the validity of his proposals. He is a liar, but in the framework of formal procedures – telling the truth. Of course, this example is similar to the stories of the Barber and the concept of Russell. And indirectly reminds us of the program of Descartes.