Logical Paradoxes: Basics

Short: Logical paradoxes are statements or situations that seem formulated without contradiction but lead to unresolvable logical contradictions. They reveal limits and blind spots in formal systems of logic and language.

In more detail: Many paradoxes arise through self-reference, where a statement directly or indirectly refers to itself.

In detail

• Liar’s paradox: The sentence “This sentence is false” can be neither true nor false without contradicting itself.
• Russell’s paradox: The question of whether the set of all sets that don’t contain themselves contains itself led to a profound crisis in set theory.
• Sorites paradox (heap paradox): If removing a single grain of sand from a sand heap never turns it into a non-heap, exactly when does the heap stop being a heap?
• Zeno’s paradoxes: Ancient thought experiments like the paradox of Achilles and the tortoise called the very possibility of motion into question.
• Grelling–Nelson paradox: The question of whether the word “heterological” (not applying to itself) is itself heterological leads to a logical contradiction.
• Logicians’ responses: Various approaches like type theories or multi-valued logics try to formally circumvent such paradoxes.
• Practical value: Paradoxes are not mere curiosities but have sparked important foundational work in mathematics, computer science, and philosophy of language (erkenntnistheorie-wissen-und-glauben, trolley-problem-gedankenexperiment).