A paradox arises when a set of apparently incontrovertible premises gives unacceptable or contradictory conclusions. To solve a paradox will involve either showing that there is a hidden flaw in the premises, or that the reasoning is erroneous, or that the apparently unacceptable conclusion can, in fact, be tolerated. Paradoxes are therefore important in philosophy, for until one is solved it shows that there is something about our reasonings and our concepts that we do not understand. Famous families of paradoxes include the semantic paradoxes and Zeno’s paradoxes. At the beginning of the 20th century, Russell’s paradox and other set-theoretic paradoxes led to the complete overhaul of the foundations of set theory, whilst the Sorites paradox has led to the investigation of the semantics of vagueness, and fuzzy logics. Paradoxes are indexed under their other titles: See analysis, paradox of; lottery paradox; Moore’s paradox; omnipotence, paradox of; prediction paradox; St Petersburg paradox, two-envelope paradox.