The Theory at Work

Chapter 7 puts the singularity theory to work on a number of semantic paradoxes that have intrinsic interest of their own. These include a transfinite paradox of denotation, and variations on the Liar paradox, including the Truth-Teller, Curry’s paradox, and paradoxical Liar loops. The transfinite paradox of denotation shows the need to accommodate limit ordinals. The Truth-Teller, like the Liar, exhibits semantic pathology-but, unlike the Liar, it does not produce a contradiction. The distinctive challenge of the Curry paradox is that it seems to allow us to prove any claim we like (for example, the claim that 2+2=5). Paradoxical Liar loops, such as the Open Pair paradox, extend the Liar paradox beyond single self-referential sentences. The chapter closes with the resolution of paradoxes that do not exhibit circularity yet still generate contradictions. These include novel versions of the definability paradoxes and Russell’s paradox, and Yablo’s paradox about truth.