Patterns of paradox
Roy T. Cook · Journal of Symbolic Logic · 2004
We begin with a prepositional language L p containing conjunction (Λ), a class of sentence names { S α } αϵA , and a falsity predicate F . We (only) allow unrestricted infinite conjunctions, i.e., given any non-empty class of sentence names { S β } βϵB , is a well-formed formula (we will use WFF to denote the set of well-formed formulae). The language, as it stands, is unproblematic. Whether various paradoxes are produced depends on which names are assigned to which sentences. What is needed is a denotation function: For example, the L P sentence “ F ( S 1 )” (i.e., Λ { F ( S 1 )}), combined with a denotation function δ such that δ ( S 1 )“ F ( S 1 )”, provides the (or, in this context, a) Liar Paradox . To give a more interesting example, Yablo's Paradox [4] can be reconstructed within this framework. Yablo's Paradox consists of an ω-sequence of sentences { S k } kϵω where, for each n ϵ ω : Within L P an equivalent construction can be obtained using infinite conjunction in place of universal quantification - the sentence names are { S i } i ϵω and the denotation function is given by: We can express this in more familiar terms as: etc.