In this chapter, negation and disjunction are integrated into the semantics developed in Chapters 7 and 8. Here, the semantics of negation is given in terms of an incompatibility relation between theories. A corresponding incompatibility relation is added to the formal language, and a more intuitive and conceptually satisfying set of rules for negation are added to the natural deduction system.

This chapter generalizes the ideas given in the previous chapter. It sets out the notion of a model based on a set of theories. One of these theories is the logic itself. It is, so to speak, the correct theory of theories. It correctly states the principles under which all the theories (including itself) are closed. But each theory has associated with it a closure operator. Some of these operators get the principles of theory closure quite wrong in the sense that they do not apply correctly to every theory in the model. The interaction between these closure operators can be altered in various ways, giving rise to different logical systems. The resulting formal semantics can be represented in the manner of Kit Fine’s “Models for Entailment”.

I consider a range of objections to the identity of negative actions and positive events, which have appeared in the recent literature. Most of these objections are appeals to Leibniz’s Law: they attempt to show that a certain negative action can’t be identical to a certain positive event, because these entities have different properties (e.g. different spatiotemporal locations, modal profiles, or causal roles). I show that these objections rely on confusions (e.g. between the things we do and our particular doings of them, or between action sentences and nominals which purport to denote particular doings). Thus, we can (and in the light of the previous chapters, we should) identify negative actions with positive events.

This chapter meets several of the objections that have been offered against truthmaker theory being a worthy metaphysical enterprise. It counters the charges that truthmaker theory is maximalist or nothing, and that it requires up-front contentious ontological posits. It also rebuts challenges from Timothy Williamson and Kit Fine that truthmaker theory is an inferior if not pointless endeavor. It identifies and argues against “truthmaker deflationists” who believe that the point of truthmaker theory can be accomplished without resort to any ontological investigation whatsoever. Finally, it considers challenges from those who might take the relation of grounding to be a competing and superior metaphysical notion.