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This chapter is concerned with the extent to which supervaluation and revision theories can declare that their rules all preserve truth. The extent varies from one such theory to another, but none can do so entirely, and many actually declare some of their rules not to be truth-preserving. The question is raised whether this makes the theories ‘self-undermining’, and a reason is given why it doesn't. Connections to Gödel's second incompleteness theorem are drawn.

Priest has argued that his dialetheic theories declare themselves sound and in this regard are better than non-dialetheic theories. This chapter shows this to be unfounded. Indeed, dialetheic theories cannot on pain of triviality declare modus ponens to be truth-preserving, and those that accept excluded middle declare it not to be. Moreover, Priest's favored dialetheic theories say that modus ponens doesn't preserve truth even when applied to premises we accept. In this respect, they do worse than glut theories in classical logic and weakly classical theories. This however is not a problem for all dialetheic theories.

This chapter gives an initial discussion of the relation between validity and necessary truth-preservation. It argues that the so-called ‘soundness theorems’ of first order logic are misnamed, and that the soundness of first order logic is not provable (even ‘rule-circularly’) in standard set theory.

This chapter deals with classical theories that postulate truth value gluts, especially theories that accept the right-to-left half of the Tarski biconditionals. The duality with gap theories is explored, and with it the question of whether the theories are notational variants. It is shown that these theories take modus ponens not to be truth-preserving. Reasons are given why this is not as debilitating as declaring one's axioms untrue. Still, glut theories like gap theories preclude truth from having its usual role in expressing agreement and disagreement.

The Generalised Tarski Thesis (GTT) yields classical consequence when its cases are taken to be possible worlds, where possible worlds are complete and consistent with respect to negation. That (classical) precisification of ‘follows from’ is familiar and useful, however, it is not the only sense of ‘follows from’ apparent in English. Another strongly apparent sense of ‘follows from’ takes ‘from’ seriously. There is a sense of ‘follows from’ that is more restrictive than the classical sense; it demands that premises be ‘relevant’ to conclusions. That sense of ‘follows from’ is more restrictive than the classical one: it imposes constraints that go beyond truth-preservation over possible worlds. The constraints, in effect, concern the behaviour of negation. What is required is not only truth-preservation over possible worlds, but truth-preservation over cases that go beyond the constraints of worlds, beyond the constraints of completeness and consistency. The task is to specify such cases, thereby cashing out relevant consequence.

Emotions, in the sense relevant to this book, are identified as intentional states that are individuated by their content. The attribution of such states to animals is thought to be problematic. The problems are identified and resolved in this chapter. A definition of what it is for an emotion to possess moral content is supplied.

Problems in attributing beliefs to animals stem from the fact that the contents (of beliefs and desires) used in such attribution are anchored to humans. This chapter spells out a de-anchoring strategy. The result of this is that it can be appropriate to explain the behavior of an animal using contents that only humans can entertain as long as our contents track theirs. That is: (a) the truth of a belief with our content guarantees the truth of their belief, and (b) our belief and theirs share narrow content. This is important not just in the case of animals. There are good reasons for thinking that tracking begins at home. There are no stable belief contents shared by different humans or even attaching to a single human through time. Content must be de-anchored in order to make sense of ourselves as well as other animals.