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This chapter introduces ‘dialetheic’ approaches, according to which Liar sentences are both true and false, indeed both true and not true. (Some contradictions are accepted.) Dialetheism provides another way (in addition to restricting excluded middle) to retain the Intersubstitutivity Principle, but some who believe in dialetheism (such as Priest) prefer to give up the Intersubstitutivity Principle. The rationales for going one way or the other on this, the topic of gluts and gaps, and the relation to the classical glut theories of Chapter 8 are discussed.

This chapter surveys a number of dialetheic theories, pointing out some difficulties with each. The theories differ over the kind of conditionals they contain. The dialetheic theory that in some ways does best among those now known is based on the semantics of Chapters 16 and 17; however, it is somewhat unsatisfactory in not allowing reasoning by cases. But there is no clear reason to rule out the possibility of better dialetheic theories in the future.

Hypothetical imperatives are problematic: does ‘If you want to get drunk every evening, you ought to work in a bar’ entail, given ‘You want to get drunk every evening’, ‘You ought to work in a bar’? What of the inference from ‘Given that you are going to stab him, you ought to stab him with the shorter knife’, and ‘You are going to stab him’, to ‘You ought to stab him with the shorter knife’? One might cite Broome's ‘normative requirements’ supposedly governing thinking, which are not subject to such detachment. Doubts may be entertained about both the content, and the role, of these requirements. Their logic needs scrutiny, in respect of detachment and contraposition. So long as the concluding ‘ought’ is relative to the circumstance stated in the second premise, it may be allowed to follow, even if one has no reason to act upon it.

Detailed exposition and evaluation of Goodman's approach to subjunctive conditionals. The cotenability problem is shown to be an amalgam of two problems: about causal contraposition and about logical cleansing. Solutions by Parry are expounded, with help from Pollock's notion of a simple proposition.

Expounds, mainly on the basis of Adams's work, the logic of indicative conditionals—with some help from his adaptation of Venn diagrams. Failure for indicatives of or‐to‐if, contraposition, transitivity, and antecedent strengthening. Controversy over whether modus ponens holds for indicatives. Independent indicatives identified and snubbed.

This chapter argues that the difference between indicative and counterfactual conditionals traces to the overt difference in verb forms and not to any alleged covert ambiguity or context-dependence in ‘if’. ‘Would’ has a life beyond conditionals; the best hypothesis is that it is a necessity modal restricted to contextually relevant worlds. In standard counterfactual conditionals, ‘would’ scopes over ‘if’; given the invariant truth-functional semantics of ‘if’, the compositional semantics then makes counterfactual conditionals contextually restricted strict conditionals. The chapter explores the consequences of this for the logic of counterfactuals: principles such as transitivity, contraposition, and strengthening the antecedent hold, with appearances to the contrary being explained by context-shifting caused by the application of the suppositional heuristic. However, modus ponens fails because the contextual restriction may exclude the actual world.

This chapter reviews the debate regarding whether the probability of the conditional, p(If A then C), should be considered equivalent to the conditional probability, p(C|A), and considers the evidence regarding whether people, in fact, behave as though they believe this equivalence holds. It concludes with consideration of the merits of treating conditional reasoning with probabilistic reasoning, on balance.

Medieval logicians expanded Aristotle’s notation; some rules naturally come along with the expansions. Predicates are quantified, as in ‘No donkey is every animal’, and negations are sprinkled throughout sentences, so quantifier equipollences are introduced, such as ‘not some P’ is equivalent to ‘every P not’. One can now say that prefixing ‘not’ to any proposition produces its contradictory; this makes reductio proofs widely applicable. When singular terms occur, new rules are needed to let them permute with each other, with negations, and with denoting phrases. Conversion “by contraposition” is discussed; an example is converting ‘Every P is a Q’ to and from ‘Every non-Q is a non-P’; this is valid except for counterexamples involving empty terms. (‘Every chimera is an animal’ is false, but ‘Every non-animal is a non-chimera’ is true.) A set of rules of inference are given which are complete for the notation developed up to this point.