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Consider for example a predicate ‘troop’ that applies to all truths, but perhaps not only to truths. Consider moreover a troop Liar: • This sentence is not troop. Will this sentence be troop or not? If the sentence is not troop, it must be true, since that is precisely what it says. But if it is not troop, it must lie outside of that category of ‘troop’ specified as including all truths. It must therefore be false. Contradiction. We therefore have a reductio argument against the supposition that the sentence above is not troop.

THE WIDER LESSONS OF REFLEXIVITY We have outlined conceptual structures within a range of traditional paradoxes explicitly enough to graph the dynamics of paradoxical oscillation. Implicative dynamics diagrams also allow us a glimpse of the formal extent of oscillation phenomena, characterized structurally, both within and beyond its instantiation in set-theoretical and semantic paradox. Our aim in the chapters that follow is to carry these first lessons of reflexivity farther. In the next chapter we turn to instances of related structure at the core of major 20th century limitative theorems in logic and computation.

His second achievement was to show a limitative theorem corresponding to Gödel’s: that there are some things that nonetheless could never be computed by such a machine. THE LIMITATIVE THEOREMS 47 We began in chapter 1 with categories of things that stand in certain relations to themselves: sets that take themselves as members, for example, or words that apply to themselves. We considered sentences that take themselves as subjects. In the Gödel result we have formulae within axiomatic systems which by Gödel numbering have a complex interpretational relationship to themselves.

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