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This chapter presents the view that the supervenience of the experiential on the non-experiential is irreducibly a posteriori — the a posteriori entailment view. The main problem for this view emerges when we notice that Kripke, the philosopher who did most to make it prominent, also considered and rejected it. The lesson of Kripke’s discussion on this point is that the mere idea of a posteriori entailment does not solve the problem of experience, and therefore that a proponent of the a posteriori entailment view is obliged to add further material. On the other hand, an examination of what this further material might be yields the result that either the a posteriori entailment view has no answer to the arguments, or else collapses into the epistemic view.

This chapter argues against the hedged principles laid out in Chapter 6. It contends that the best explanation of the possibility of practical wisdom entails that morality can be codified. It gives three main arguments. The first argument contends that generalism as a regulative ideal better preserves a broad scope for practical wisdom than its particularist rivals. In pressing this argument, it is shown how generalism is better situated to accommodate an important role for certain kinds of fiction and narrative in moral thinking. The second argument contends that generalism as a regulative ideal is better situated to distinguish the a priori from the a posteriori elements of moral knowledge than its particularist rivals. The third argument simply appeals to pre-theoretical intuitions about the codifiability of morality.

This chapter characterizes essentialism, and distinguishes essentialism about individuals from essentialism about natural kinds, arguing that the first does, but the second need not, involve de re (as opposed to de dicto) modality. It is argued that given certain plausible assumptions, essentialism about individuals involves a commitment to necessary a posteriori truth, although the commitment is not inevitable. The implications of the concepts of possible worlds and identity across possible worlds (‘transworld identity’) are discussed.

The distinction between a priori knowledge and a posteriori knowledge has come under attack in the recent literature by Philip Kitcher, John Hawthorne, C. S. Jenkins, and Timothy Williamson. Evaluating the attacks requires answering two questions. First, have they hit their target? Second, are they compelling? My goal is to argue that the attacks fail because they miss their target. Since the attacks are directed at a particular concept or distinction, they must accurately locate the target concept or distinction. Accurately locating the target concept or distinction requires correctly articulating that concept or distinction. The attacks miss their target because they fail to correctly articulate the target concept or distinction. I go on to present a different challenge to the a priori–a posteriori distinction. This challenge is not directed at the coherence or significance of the distinction. Its target is the traditional view that all knowledge (or justified belief) is either a priori or a posteriori.

This chapter argues that the correct internalist account of perceptual experience involves four key claims, that sensory contents are qualia, that experiences of qualia are constituted in a necessary a posteriori manner by internal physical states, that these states have the modal structural architecture pioneered for color experience by Hardin, and that intentionalism is the correct account of the semantic contribution of these experiences. It considers contrary externalist arguments, including arguments rooted in the claim of Williamson and Harman that knowledge is a basic mental state, in Kant's Refutation of Idealism, and in disjunctivism.

Saul Kripke claims that there are necessary a posteriori truths and contingent a priori truths. These claims challenge the traditional Kantian view: (K) All knowledge of necessary truths is a priori, and all a priori knowledge is of necessary truths. Kripke's claims continue to be resisted, which indicates that the Kantian view remains attractive. My goal is to identify the most plausible principles linking the epistemic and the modal. My strategy for identifying the principles is to investigate two related questions. Are there compelling general supporting arguments for (K)? Are there decisive counterexamples to (K)? My investigation uncovers two intuitively plausible principles that are not open to decisive counterexamples but that enjoy no compelling independent support.

W. V. Quine's famous objection to essentialism has provoked two responses. The unfriendly response is that it conflates the de re and the de dicto. The friendly response does not dispute the point about conflation but argues that, behind Quine's argument, lies an important epistemological problem for essentialism. This chapter argues that the problem can be solved by looking at the issue through the lens of possible worlds approaches to modal questions. This serves as a preamble to addressing the corresponding issue for the necessary a posteriori. There is an important epistemological problem for the necessary a posteriori but the possible worlds approach to modal questions offers a solution to it. The solution turns out to be version of two-dimensionalism about the necessary a posteriori.

This chapter discusses Saul Kripke’s treatment of the necessary a posteriori and concomitant distinction between epistemic and metaphysical possibility. It extracts the enduring lessons of his treatment of these matters and disentangles them from errors and confusions that mar some of his most important discussions. It argues that there are two Kripkean routes to the necessary a posteriori—one correct and philosophically far-reaching; the other incorrect, philosophically misleading, and the source of damaging errors that persist to this day. It connects two false principles involved in the second, unsuccessful, route to the necessary a posteriori with the plausible and potentially correct idea that believing a singular proposition that o is F always involves also believing a richer more descriptively informative proposition in which some further property plays a role in the agent’s thoughts about o. It explains why this idea will not save the failed second route to the necessary a posteriori and suggests that it may help reconcile Kripke’s insights with the lessons of Frege’s puzzle.

This chapter discusses Kant’s relation to Rationalism and Empiricism, his attempt to reduce his agenda to the issue of synthetic a priori judgments, and the overall of plan and structure of the Critique of Pure Reason.

This chapter examines the phase error problem and also shows, via a dispersion analysis, that higher order methods can significantly improve phase accuracy. Once a solution is computed it is desirable to assess the accuracy of the solution to determine how to refine the mesh. The next section of the chapter presents a residual based a posteriori error analysis that shows how both the error in the curl of the solution and the divergence needs to be assessed. The final section concerns absorbing boundary conditions, which are often used in preference to the ‘exact’ techniques in Chapters 10-12 to ease the implementation burden. The standard Silver-Muller condition, infinite elements, and the justly popular Perfectly Matched Layer (PML) of Berenger are discussed.

Philosophers have traditionally believed that there is a close connection between the categories of a priori propositions and necessary propositions. One widely held thesis about the nature of this connection is that all a priori knowledge is of necessary propositions and that all necessary propositions are knowable a priori. Saul Kripke has recently argued that this traditional account is mistaken. In “Identity and Necessity” he argues that there are necessary a posteriori propositions, while in “Naming and Necessity” he argues, in addition to this, that there are contingent a priori propositions. The primary concern of this paper is to examine Kripke's arguments in order to determine whether he has succeeded in calling the traditional account into question.

McGinn defends a causal criterion for distinguishing a priori from a posteriori knowledge. In the case of a posteriori knowledge, the subject matter of a knower's ground for believing a proposition is the cause of that knower's coming to believe that proposition. In the case of a priori knowledge, it is not the case that the subject matter of the knower's ground for believing a proposition is the cause of that knower's coming to believe that proposition. In this essay's first section, McGinn argues that a causal condition for knowledge is no part of an analysis of knowledge tout court, but merely part of a naturalized account of a posteriori knowledge. McGinn then argues for the extensional adequacy of his account of a priori knowledge for the cases of mathematical, logical, and analytic knowledge. He concludes by considering the connections between the a priori and the a posteriori, on the one hand, and the modal notions of necessity and contingency, on the other.

This chapter argues that, although a distinction between a priori and a posteriori knowledge (or justification) can be drawn, it is a superficial one, of little theoretical significance. The point is not that the distinction has borderline cases, for virtually all useful distinctions have such cases. Rather, it is argued by means of an example, the differences even between a clear case of a priori knowledge and a clear case of a posteriori knowledge may be superficial ones. In both cases, experience plays a role that is more than purely enabling but less than strictly evidential. It is also argued that the cases at issue are not special, but typical of a wide range of others, including knowledge of axioms of set theory and of elementary logical truths. Attempts by Quine and others to make all knowledge a posteriori (‘empirical’) are repudiated. The chapter ends with a call for a new framework to be developed for analysing the epistemology of cognitive uses of the imagination.

Chapter 6 describes the idea of a posteriori error estimation and adaptive mesh refinement. The main points are the description of the adaptive finite element algorithm, its convergence and a sketch of the optimal complexity of this algorithm with respect to an approximation class of solutions. As in the last chapter the relations between the strong and weak form of the equations are the key for deriving upper and lower a posteriori error estimates. Quantities of interest are estimated.

This chapter focuses on philosophical issues in knowledge. Tradition insists that knowledge falls into two broad classes: a priori knowledge and a posteriori knowledge. These categories are conceived as exclusive and exhaustive: no piece of knowledge is both a priori and a posteriori, and any piece of knowledge is one or the other. One can characterize a posteriori knowledge as knowledge acquired by means of the senses (“by experience”) and a priori knowledge as knowledge not so acquired, but rather acquired “by reason alone” or “intuitively.” The chapter then addresses the proof or evidence of the existence of an external world, and looks deeper into what knowledge is and whether knowledge implies truth.

Although the much-debated question of whether the necessary connection from P to Q is knowable a priori or only a posteriori is important, a different one also needs attention: is the necessity logico-conceptual, or does it depend on the necessity of identity? Reinforcing points made in the last chapter, it is argued that a posteriori physicalism as often expressed is not physicalistic; when modified to become so, it requires the l-c entailment thesis – which renders the alleged identities superfluous. Neither a priori or a posteriori physicalism is attacked directly, but it is important to consider how the latter, as usually understood, compares with the redescriptive version as an expression of physicalism, and whether any of the arguments for it threaten redescriptive physicalism. The conclusion is that identity-based a posteriori physicalism is at best problematic.

It is argued that the propositions semantically expressed by attitude ascriptions report relations to the Russellian propositions expressed by their complement clauses, and hence that pairs of ascriptions that differ only in the substitution of linguistically simple, coreferential names, or indexicals are semantically equivalent. It is explained how ascriptions that are equivalent in this way may nevertheless be used by speakers to assert and convey propositions with different truth values. The end result is a theory in which a Millian account of the semantic contents of simple names and indexicals, plus a Russellian treatment of propositional attitude ascriptions, is reconciled with Fregean intuitions about the information asserted and conveyed by utterances of both simple sentences and attitude ascriptions. The chapter concludes with an examination of the consequences of this theory for Kripke's doctrines about identity sentences and the necessary a posteriori.

The distinction between a priori knowledge and a posteriori knowledge has come under attack in the recent literature by Philip Kitcher, John Hawthorne, C. S. Jenkins, and Timothy Williamson. Evaluating the attacks requires answering two questions. First, have they hit their target? Second, are they compelling? My goal is to show that the attacks fail because they miss their target. They miss their target because they fail to correctly articulate the distinction that they attack. I begin by identifying three factors that obscure the concept of a priori knowledge and maintain that the failure to take these factors into account leads to three common errors in attempts to articulate that concept. I go on to argue that the attacks of Kitcher, Hawthorne, Jenkins, and Williamson turn on these errors or variants of them.