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This chapter consists in development of some of the key ideas of the previous chapter. It begins by showing how an arithmetical epistemology along the lines suggested in the previous chapter sits well with a structuralist conception of arithmetic and arithmetical concepts (although it is also compatible with other views). It cautions against a simplistic understanding of the envisaged grounding relationship between concepts and sensory input, arguing that the proposed account allows us to say that our arithmetical beliefs count as knowledge by the lights of Chapter 3. It shows that the account is consistent with realism as characterized in Chapter 1. The chapter spends some time discussing the crucial notion of unconceptualized sensory input, and also offers some comments on what is called here ‘ungrounded’ and ‘unfitting’ concepts.

This chapter considers property monism. It considers the requirements for the defeat of property dualism and advances an argument for the identity of dispositional and categorical property tokens which is an adaptation from a similar argument in the philosophy of the mind. The second section clarifies the nature of the monist claim. The third and fourth sections justify the claim that the argument from identity of causal role is a sound argument. The fifth section considers epistemic counterparts and the world-relative ascriptions. The sixth section examines the variable realization argument and instantiations of properties. The seventh section discusses identity conditions derived from the argument for the identity of the causal role. The eighth section presents the supplementary claims of property monism. The last two sections discuss two special cases of problematic dispositions: abstract dispositions and ungrounded dispositions.

The formal language developed in ch. 1 is modelled as a directed graph with a boundary, and the problem of assigning truth values to sentences assimilated to a boundary‐value problem. A trivalent semantics, with true, false, and ungrounded sentences results. The nature of the third truth‐value imposes constraints on the possible truth‐functional connectives in the language.

Dispositional Pluralism is more consistent with our dispositions talk than more monolithic views. Our evidence for extrinsic, ungrounded, and non-natural dispositions is comparable to that of intrinsic, grounded, and natural ones. Dispositional Pluralism has wide applicability to various philosophical issues. Secondary qualities, such as colors, can be given a dispositional account. Thinking of character traits as dispositions sheds light on the debate over Dispositionalism Situationism in moral psychology. One can give an account of gender identity as a cluster of behavioral dispositions. Finally, the potentiality of an embryo or a patient is best understood as an extrinsic dispositional property.

Some dispositions have causal bases or grounding properties. However, ungrounded dispositions do not have causal bases. Ungrounded dispositions are also known as powers, baseless dispositions, and bare dispositions. Ungrounded dispositions are not supplanted by mechanistic explanations, for even mechanistic explanations ultimately reference dispositions. While some argue that citing dispositions does not really explain anything, dispositions can in fact figure in adequate explanations. Furthermore, scientific explanations reference dispositions with no known grounds. This lends some support for the view that ungrounded dispositions are metaphysically possible. Philosophical arguments based on multiple realizability or the demand for truth-makers fail to show that ungrounded dispositions are impossible.