矛盾の研究

Chapter I

The Liar's Paradox

*Consider the statement: "This sentence is false." If the sentence is true, then what it says must be the case -- but it says it is false, so it must be false. If the sentence is false, then what it says is not the case -- but it says it is false, so it must be true.

*The paradox strikes at the foundation of bivalent logic: the assumption that every proposition must be either true or false. The liar's sentence appears to be neither, or perhaps both -- a contradiction that has haunted logicians for over two millennia.

"A man says that he is lying. Is what he says true or false?"

-- Eubulides of Miletus

Modern approaches to the liar's paradox include Tarski's hierarchy of languages, Kripke's fixed-point semantics, and paraconsistent logics that permit true contradictions. Each solution resolves the paradox by modifying the logical framework, suggesting that the paradox is not merely a puzzle but a fundamental challenge to our understanding of truth.

Chapter II

Russell's Antinomy

*Consider the set of all sets that do not contain themselves. Does this set contain itself? If it does, then by definition it should not. If it does not, then by definition it should. Bertrand Russell's 1901 discovery of this paradox shattered the foundations of naive set theory and triggered a crisis in the philosophy of mathematics.

The antinomy reveals that unrestricted comprehension -- the ability to form a set from any property -- leads inevitably to contradiction. Zermelo-Fraenkel set theory resolves this by restricting which collections can be called "sets," but the deeper lesson remains: not every coherent-sounding definition produces a coherent object.

Chapter III

Zeno's Dichotomy

*To traverse any distance, you must first cross half that distance. But to cross that half, you must first cross half of it. And so on, infinitely. Therefore, motion requires completing infinitely many tasks in finite time -- which seems impossible. Yet we move.

The resolution came two millennia later with the development of calculus and the concept of convergent infinite series. The sum 1/2 + 1/4 + 1/8 + ... = 1. The infinite tasks are completed because each takes proportionally less time. Yet one might ask: does this mathematical resolution truly explain how motion is possible, or does it merely describe it?