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This chapter provides a taxonomy of widely endorsed conditions on a priori justification. The conditions fall into two broad categories: epistemic, which includes defeasibility, strength, and source conditions; and nonepistemic, which involves the concepts of analyticity or necessity. Two major claims are argued: (1) nonepistemic conditions are neither necessary nor sufficient for a priori justification, and (2) if a theory imposes epistemic conditions on the a priori that differ from those it imposes on the a posteriori, they must be supported by independent argument or rejected as ad hoc. Two defensible conditions on a priori justification emerge: justification by a nonexperiential source, and justification that is not defeasible by experience.

In this chapter, an analogous argument in the case of the a priori entailment view is advanced. On the one hand, the mere idea of priori entailment will not answer the argument — further material needs to be added. On the other hand, an examination of what this further material might be yields the result that either the a priori entailment view has no answer to the arguments, or else collapses into the epistemic view.

A response to a paper by Frank Jackson. Although philosophers sometimes defend certain ‘analyses’ as analytic or a priori truths, such analyses are far from obviously true and are defended inductively. Contrary to Jackson's claim, the rejection of analyticity is based on consideration of central cases.

This chapter provides a taxonomy of widely endorsed conditions on a priori justification. The conditions fall into two broad categories: epistemic, which includes defeasibility, strength, and source conditions; and nonepistemic, which involves the concepts of analyticity or necessity. Two major claims are argued: (1) nonepistemic conditions are neither necessary nor sufficient for a priori justification, and (2) if a theory imposes epistemic conditions on the a priori that differ from those it imposes on the a posteriori, they must be supported by independent argument or rejected as ad hoc. Two defensible conditions on a priori justification emerge; justification by a nonexperiential source, and justification that is not defeasible by experience.

A final problem left over from the previous chapter is that of substantive necessity. How are we to distinguish substantively necessary sentences from analytic ones? This problem arises because we have attempted to characterise analyticity in modal terms. This chapter proposes a more fine-grained ‘metaphysical picture’ of analyticity to underlie the rougher modal approximation developed in Chapter 2.

If our ability to count depends on the structure of space and time, this may explain its applicability to the things we intuit in space and time, but it will also limit its applicability to just those things: we shall, in short, be left without the ability to count anything that is not made up of spatio-temporal elements. This evidently did not bother Kant, but it has seemed implausible to many subsequent thinkers. The most notable of these was Frege, who published Die Grundlagen der Arithmetik (The Foundations of Arithmetic), a book arguing against Kant's conclusion, in 1884. Frege was the first to put forward the thesis, now known as logicism, that logic is capable of grounding mathematical truths without thereby rendering them wholly trivial.

This chapter discusses certain aspects of naive set theory, that is, set theory in which the abstraction scheme holds for all formulas of the language of set theory, and some of its implications for the philosophy of mathematics. Substitutional semantics, analyticity and paradoxicality, mathematical realism and anti-realism, and cardinality and syntax are discussed.

In the early analyses of metrics with special holonomy in dimensions 7 and 8, particularly in regards to their existence and generality, heavy use was made of the Cartan–Kähler theorem, essentially because the analyses were reduced to the study of overdetermined PDE systems whose natures were complicated by their diffeomorphism invariance. The Cartan–Kähler theory is well suited for the study of such systems and the local properties of their solutions. However, the Cartan–Kähler theory is not particularly well suited for studies of global problems for two reasons: first, it is an approach to PDE that relies entirely on the local solvability of initial value problems and, second, the Cartan–Kähler theory is only applicable in the real-analytic category. Nevertheless, when there are no other adequate methods for analyzing the local generality of such systems, the Cartan–Kähler theory is a useful tool and it has the effect of focusing attention on the initial value problem as an interesting problem in its own right. This chapter clarifies some of the existence issues involved in applying the initial value problem to the problem of constructing metrics with special holonomy. In particular, it discusses the role of the assumption of real-analyticity and presents examples to show that one cannot generally avoid such assumptions in the initial value formulations of these problems.

This chapter introduces and motivates study of the analytic/synthetic distinction, and provides a brief history of that distinction. It also contains some preliminaries which will be of use in the rest of the book.

This chapter sums up the argument of the book, and ties up a few loose ends. In particular, it considers Michael Resnik's alternative ‘pragmatic’ version of the indispensability argument, and shows how this argument, while not relying on confirmational holism, can be responded to from the fictionalist perspective developed in this book. It also considers the question of whether one should reject the existence of mathematical objects or merely remain agnostic about their existence, and argues for the stronger position which rejects the existence of such things. Finally, it considers a sense in which the view of the axioms of set theory with non‐mathematical objects as urelements as generative of a fiction can be considered to be a revival of a version of conventionalism about our mathematical hypotheses, as analytic ‘truths’ which are not in fact truths.

The final chapter examines and dissipates certain pressures that are often felt to afflict Quine or his views. These include the charge that any such view, being a view of anti‐realism about meaning, is self-refuting, as according to a well‐known argument by Paul Boghossian; and worries that the view cannot account for normativity or apparent analyticities. The book closes with a discussion of what might called the world-view of naturalism—the place of vagueness, of intuition, and of the role of regimented language.

On the book’s view, there is one entity with primitive, unexplained causal-modal attributes: God, who has natural powers. The books explains God’s non-natural powers, and in so doing explains all modal status other than God’s possibility, which it treats reductively. This chapter compares the book’s account of secular modal status to some ‘explanationist’ rivals. It emerges that its main competition comes from explanations of states of affairs’ modal status by their intrinsic content. It argues against several versions of this which involve analyticity or kindred notions. It also considers the extent to which the book’s view involves brute necessities.

What we experience of the world—for example, that pig on the lawn—does not bear logical relations to thoughts, or propositions. That pig on the lawn is not of the form of a proposition. Logical relations relate thoughts, or propositions, to one another. Quine thus correctly notes that one could, on no pain of contradiction, respond to any experience with any constellation of beliefs. He incorrectly takes this to mean that any constellation of beliefs could be consistent with one's having experienced what he did. Revision can come anywhere; if it does not, this merely reflects the ways of our people. This is a mistake, and in fact destroys the very possibility of judgement. Quine has given us no reason (even if, in fact, there is one) to think that any proposition may (not incorrectly) be held to have been falsified by experience. So goes the present argument.

Here the book turns to doing, rather than describing, Second Philosophy. In preparation for the philosophy of mathematics in Part IV, Part III takes up the question of the nature of logical truth. In this chapter, several naturalistic possibilities are surveyed and critiqued: psychologism, empiricism, conventionalism and analyticity. This sets the stage for a distinctive second-philosophical account.

Those who think philosophers appeal to intuitions when they make judgments about cases also tend to see philosophers as engaged in conceptual analysis when they appeal to cases and vice versa. The case studies considered in Chapter 8 can be used to show that both views are mistaken. Recognition of the marginal (or non-existent) role of intuitions in philosophical practice goes hand in hand with recognition of the marginal (or non-existent) role of anything reasonably labeled ‘conceptual analysis’ in philosophy. This chapter spells out some of the difficulties associated with the view that philosophy involves conceptual analysis and related notions such as conceptual modality. It criticizes possible justifications for being interested in concepts such as their alleged connections to metaphysical necessity and apriority. A number of assumptions needed to reach the conclusion that intuitions are required to engage in conceptual analysis are called into question.

Lessons Learned, Replies to Objections and Comparison to Williamson Generalizations about thought experiments are drawn, which underscore the diversity of roles they play in philosophy, and it is argued that philosophers do not implicitly rely on intuitions when they make judgments about cases. It is shown that there is no such thing as ‘the method of cases’. Replies are then presented to a number of objections. The case studies in the previous chapter support the claim that even if there are conceptual a priori truths such as ‘All vixens are female foxes’, they are not relevant to understanding the epistemology of philosophy. This argument contrasts to Williamson’s argument for the same conclusion, which rests on an argument for the claim that there are no epistemically analytic truths. Pace Williamson, the case studies provide no evidence of philosophers “psychologizing the evidence”.