The semantical framework for the positive view of this book is one in which entailment is understood primarily in terms of theory closure. This chapter outlines both the history of the notion, beginning with Alfred Tarski’s theory of closure operators, and the relationship between closure operators and the entailment connective. At the end of the chapter, it is shown how closure operators can be used to model a simple logic, Graham Priest’s logic N4.

In investigating Being and the Nothing, Heidegger finds himself committed to inconsistent theses. The rational response to this conundrum would seem to be revising his commitments, but Heidegger explores a different option: retaining the inconsistent theses and rejecting the principle of noncontradiction instead. While such a stance would be anathema to most philosophers, it appears to align him with dialetheism, or the view that there can be true contradictions, whose most prominent advocate is Graham Priest. This chapter compares Heidegger’s questioning of the principle of noncontradiction and Priest’s arguments for revising traditional logic. It finds that, despite the superficial resemblance in their aims, Heidegger and Priest have fundamentally opposed conceptions of the status of logic vis-à-vis metaphysics. Heidegger’s conception of logic is actually much closer to Wittgenstein’s, especially as interpreted by Stanley Cavell. Both Wittgenstein and Heidegger stress that "attunements" enable us to communicate with one another, and they regard logic as a distillation of the structures within which we make sense. This comparison allows us to understand the importance of anxiety within Heidegger’s thought: According to him, anxiety is an experience that so alters our attunements and their associated logic that we can intelligibly philosophize about Being.

Following Graham Priest, I argue in this chapter that the dialetheic (or true-contradictory) violation of this law with respect to recursion and infinity is called for by logical consistency itself. However, I suggest that the resulting nihilistic hypostasisation of emptiness may arbitrarily construe the dialetheic as a dogmatic gesture at the margins of the Principle of Non-Contradiction, rather than, as for Nicholas of Cusa, an apophatic gesture which may indicate an unknown plenitude rather than an enthralling absence.