Patterns of paradox

Extract

We begin with a prepositional language L_p containing conjunction (Λ), a class of sentence names {Sα}_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(S₁)” (i.e., Λ{F(S₁)}), combined with a denotation function δ such that δ(S₁)“F(S₁)”, 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}_iϵω and the denotation function is given by:

We can express this in more familiar terms as:

etc.