H. P. Grice: Berryianum implicatum

PARADOXA BERRYANA -- Berry’s paradox Logic A paradox formulated in Principia Mathematica by Russell and Whitehead and attributed by Russell to Berry, a librarian at the Bodleian Library in Oxford. Names of integers consist of a finite sequence of syllables in English. Some of them can be named in one syllable (such as 2, 5), and others need at least two (such as 7, 14). All the names of some integers must consist of at least 19 syllables and among these there must be a least. Now the phrase The least integer not nameable in fewer than nineteen syllables expresses a finite integer. Although any name of this integer must contain at least 19 syllables, the words printed above in italics amount to a name for it and they contain only 18 syllables. This is contradictory. “A third [semantical paradox] is Berry’s, concerning the least number not specifiable in less than nineteen syllables. That number has just now been specified in eighteen syllables.” Quine, From a Logical Point of View