In Chapter 8, I deal with the threat that the present account strips arithmetical knowledge of all the important characteristics traditionally associated with it: apriority, objectivity, necessity and universality. I argue that apriority can be saved in the strong sense of arithmetical knowledge being contextually a priori in the context set by our cognitive and physical capacities. Objectivity can be saved in the sense of maximal inter-subjectivity, while necessity can be saved in the sense of arithmetical theorems being true in all possible worlds where cognitive agents with proto-arithmetical abilities have developed. Finally, universality of arithmetical truths is saved through arithmetic being universally applicable and shared by all members of cultures that develop arithmetic based on proto-arithmetical abilities.

Moral rationalists have claimed a priori status for moral principles, including the commonsense principles described in Chapter 4. Intuitionists – prominently including Ross – have even claimed self-evidence for such principles. How can this claim be justified? Central to the case is the idea that normative properties are a priori grounded in certain non-normative natural properties. This chapter explains such grounding. In doing so, it distinguishes two kinds of normativity: a kind belonging to a priori grounds of obligation, e.g. promising – normativity in upshot – and another belonging to propositions, such as moral judgments, that employ normative concepts: this is normativity in content. The chapter shows how the commonsense principles control critical discourse in the constitutive ways appropriate to entrenched a priori generalizations. It also shows how their apriority squares with the empiricality of singular moral judgments and how, even where obligations conflict, singular judgments of overall obligation may be justified and known.