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This chapter criticizes the traditional conventionalist notion of analyticity. A notion of analyticity is proposed, on which sentences widely regarded as contingent a priori are, although indeed contingent, a posteriori yet analytic. A particular candidate is considered: ‘π is the ratio (if there is a fixed such ratio) of the circumference of a circle to its diameter’.

A consequence of the Mādhyamika notion of two truths is that the normative role of truth is flattened, and thus the conventional authority of epistemic practices is undermined. This chapter offers a way out of the dismal slough of pure conventionalism by putting the notion of the two truths in dialogue with the twentieth-century philosopher Rudolph Carnap. A consideration of two truths in the context of Carnap’s pragmatism brings to surface the relationship between the semantic issue of the two truths and practical matters such as action and ethics.

This concluding chapter summarizes the discussion in the preceding chapters. The book sought an answer to the question: can we give an account of arithmetic which does not make it depend for its truth on the way the world is? And if so, what constrains the world to conform to arithmetic? All the accounts we have considered have turned out to be flawed. But what is striking is how often an account was rejected not for philosophical reasons but for technical ones. It is shown that since the problem of arithmetic participates in the larger puzzle of the relationship between thought, language, experience, and the world, we can distinguish accounts that look to each of these to supply the content we require.

This chapter clarifies and motivates the naturalist premise of the indispensability argument, which holds that we should look to our best scientific theories to discover what we have reason to believe. It distinguishes Quinean naturalism from an even more modest form of naturalism, according to which philosophers should hold back from questioning the truth of utterances made in the context of successful scientific theorizing. And it considers the debate between Carnap and Quine over ontological questions, following Quine in accepting that practical reasons to speak as if there are Fs can sometimes be viewed as providing evidence for the existence of Fs. It notes that Quine's claim that practical reasons are always evidential can only be plausible when applied to our best scientific theories, from which merely practical ways of speaking have been ironed out, hence the focus on the indispensability of mathematics to our best theories.

In his Logische Aufbau, Rudolf Carnap presented a structuralist philosophy of science. Carnap exhibited a fluctuating awareness of the difficulty besetting that programme, and of the limited options his epistemology allowed for escape. The option he finally chose has the form of solution that Newman suggested, somewhat ironically, to Russell. But Carnap attempted simultaneously to refer to experience and to claim that the notions he needed are experience-independent. The basic problem returned after some decades when Hilary Putnam put it to good use in his seminal critique of metaphysical realism. It is argued that in both cases, a dissolution of problem is possible with the introduction of indexical reference.

The mechanization and mathematization of the world-picture in modern science has a rightful claim upon us to develop views of science that do justice to its revolutionary impact on the sciences themselves. This claim is honoured in the century-long attempts to develop structuralist views of science, the recurrent defeats of such views notwithstanding. It is argued that the defeats were not inevitable. The structuralism here developed however, as a view of science rather than a view of nature, can find its proper articulation only in an empiricist setting. Essential to an empiricist structuralism is the following core construal of the slogan that all we know is structure: (I) Science represents the empirical phenomena as embeddable in certain abstract structures (theoretical models); (II) Those abstract structures are describable only up to structural isomorphism. This faces at once the Weyl's paradox introduced in Chapter 8; but the lessons concerning indexicality learned from the cases of Russell, Carnap, and Putnam show the way out of the paradox.

This chapter treats C. I. Lewis' criticism of Bertrand Russell's material implication. This chapter gives an overview of the differences between Russell's conception of logic and contemporary ones. Lewis's criticism is often taken to rest on the divergence between material implication and our intuitive conception of logical consequence. The chapter argues that a more fundamental criticism Lewis makes is internal to Russell's conception: for Russell a system of logic must not merely enable correct inferences to be made; it must state correct implications. Lewis came to hold that a primitive notion of possibility is required if the laws of logic are to capture the facts of inferential practice. But Lewis's incorporation of modality is meant to revise Principia to make its axioms properly logical. The chapter concludes the chapter by sketching some other aspects of this history, from Wittgenstein through Carnap, Quine and the full re-emergence of modal notions in analytic philosophy.

The chapter begins with the observation that one disagreement between logical empiricism and early analytic philosophy, on the one hand, and early phenomenology, on the other, concerned the relations between logic and metaphysics. This chapter seeks to open up that disagreement. The chapter first discusses Frege’s doctrine of “being” and being and elaborates the interpretation she proposed and developed in the eighties. The chapter argues that Frege’s ideas of “being” and being fall between those of Carnap and Heidegger. The chapter also discusses Carnap’s attack against metaphysics in general and against Heidegger’s metaphysics in particular. In the end, it makes a few remarks on Frege’s view of judging and being. The chapter argues that the so called veridical meaning of “is” played a central role in Frege’s conceptual notation; moreover, if we take that point seriously, we may have reason to modify the view that Frege held the thesis concerning the ambiguity of “is”.

Like Kant, Carnap holds that many traditional philosophical controversies are actually ill-posed pseudo-problems. In Carnap's eyes, for example, there's no fact about which the skeptic and the non-sceptic are disagreeing. The only real question in the vicinity concerns the best kind of linguistic framework to adopt — in particular, whether or not to adopt evidential rules strong enough to allow the existence of physical objects to be confirmed — and this is a matter of conventional, pragmatic choice, not of truth or falsity. When the Second Philosopher insists she has good evidence for the existence of ordinary objects, and even of unobservable atoms, Carnap's response echoes Kant's: for her purposes, operating inside the linguistic framework of science, she's quite right, but there is another inquiry, the Logic of Science, where other concerns (ending the pseudo-debate between the sceptic and his opponent or the scientific realist and his opponent) are addressed by other methods (rational reconstruction). Like Kant, Carnap sees two distinct levels of inquiry, the Second Philosopher only one.

This chapter returns to the compare-and-contrast approach of Part I, focusing this time on van Fraassen's constructive empiricism. The constructive empiricist holds that no evidence can support the claim that unobservable entities exist, which seems to deny the force of the accepted scientific evidence for the existence of atoms. In a move familiar from Kant, Carnap, and Putnam, van Fraassen agrees that the Second Philosopher's evidence is valid for her purposes, but posits another level of inquiry — philosophical or epistemological inquiry — where the existence of atoms can never be confirmed. As before, the Second Philosopher sees this other inquiry as unmotivated and its methods unclear, but this doesn't ally her with van Fraassen's opponents, the scientific realists, who join him in thinking the ordinary evidence needs some kind of supplementation. The chapter closes with an examination of Worrall's anti-naturalism.

This chapter explores logical semantics, that is, the structural meaning of logical expressions like connectives (it is not the case that, either or, if … then), quantifiers (there is, for all, etc.), and modalities (it is necessary that, it is possible that). It focuses on truth-theoretical semantics for formalized languages, a tradition emerging from Carnap's and Tarski's work in the first half of the last century that specifies the meaning of these expressions in terms of the truth-conditions of the sentences in which they occur. It considers Tarski-style definitions of the semantics of a given language in a stronger metalanguage, Tarski's impossibility results, and attempts to overcome them in the post-Tarskian tradition.

This chapter focuses on David Kaplan's work where formal logic plays an explicit or strongly implicit role. Topics covered include David's dissertation, which is squarely in the Frege‐Church tradition but takes into account the then‐novel ideas of Carnap on modal logic; his isolation and clarification of the idea of a standard name of something; and the paradox of Knower. It is argued that Kaplan's philosophical work often has a formal system or its semantics either in front of or in back of the discussion. There is, however, none of the flaunting of symbols that one learns early on can be used to intimidate those who are quantificationally challenged. The formalism is used for what it is good for—to crystallize ideas and to test their consequences, which is the true method of analytic philosophy.

Suppose that the world contains n events, each of which is such that, judged in the light of the totality of known rightmaking and wrongmaking properties, it would be morally wrong for anyone to allow the event in question. What is the logical probability — given only that evidence — that at least one of those n events is such that the totality of the rightmaking and wrongmaking properties, both known and unknown, of allowing that event makes that action morally wrong? Answering this question is the task of this chapter and it requires a rigorous application of inductive logic. It is shown that a Carnapian, structure-description approach entails that the probability must be less than ( 1 n + 1 ) . But the existence of such an event is logically incompatible with the existence of God, and so the probability that God exists, relative to evidence of the sort described, must be less than ( 1 n + 1 ) .

The goal of this paper is to characterize a particularly austere form of contemporary naturalism called ‘Second Philosophy’ using radical skepticism about the external world as a diagnostic instrument. It begins by presenting Second Philosophy alongside the relevant form of First Philosophy, then develops an extended compare-and-contrast with responses to skepticism from transcendental philosophy (represented by Kant) and therapeutic philosophy (represented by Kant again, and then by Carnap, Wittgenstein and Austin). The hope is that this exercise will bring Second Philosophy into sharper focus.

This chapter provides a broadly sympathetic historical account of post-Kantian empiricist approaches to mathematics and logic. It focuses primarily but on John Stuart Mill’s radical empiricism and logical positivism, but also on Rudolf Carnap and Moritz Schlick. The later work of W. V. O. Quine is also treated.