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This chapter defines a paradox, roughly and superficially, as an assembly of apparently reasonable considerations that engender conflicting inclinations about what to believe, and hence a form of cognitive tension. By way of illustration, it examines various philosophical paradoxes — concerning knowledge, motion, freedom, evidence, non-existence, truth, normative facts, and vagueness. In light of these examples, the relative merits of three competing theories of the nature and meta-philosophical import of paradox are discussed: one suggested by Stephen Schiffer, which blames paradoxes on ‘glitch’ concepts; a second diagnosis, due to Wittgenstein, which stresses irrationally stubborn, scientistic over-generalization; and a third — pinned on Quine — for which the causes of paradox are conflicts amongst our fundamental habits of reasoning.

This essay discusses the programme called ‘therapeutic Bayesianism’ from three abstract points of view: substantial, foundational, and meta-philosophical. It illustrates treatments of the ‘raven’ paradox and the puzzle of diverse data, and defends the propriety of certain idealizations. It criticises a meta-philosophical perspective that does not properly distinguish science from the philosophy of science, and overvalues the use of symbolic apparatus.

This chapter lays out the meta-philosophical perspective from which the subsequent inquiry will take place, namely, that of the Second Philosopher. Beginning in her characteristic way, from within the practice, the Second Philosopher surveys a range of set-theoretic examples, historical and contemporary. Though the methods employed differ from her more familiar empirical approach, she judges that they are rational means toward the ends these various set-theoretic practices are out to achieve. Assuming for the moment that the historical and continuing inter-relations of pure mathematics with natural science are enough to warrant the Second Philosopher in regarding set theory as a body of truths, she concludes that its methods are reliable guides to those truths. The challenge, then, is to explain what sort of activity set theory is, what kind of things set are and how we know them, in such a way as to respect those methods. This means avoiding any variety of the familiar Robust Realism, which requires that apparently rational, reliable set-theoretic methods be supplemented or even rejected on the basis of an extra-mathematical metaphysics.

This chapter makes a meta-philosophical claim that Lokāyata, considered as a strong this-worldly rational approach including diverse philosophical trends, can influence one’s approach to Indian philosophy as a whole. In this light, it takes note of some of the recent Indian philosophers and thinkers who have tried to reassess and reconstruct Indian philosophy by keeping a Lokāyatic or Lokāyata-like approach at the centre. Hence the chapter refers to the following recent thinkers/philosophers: (a) the analytical philosopher Rajendra Prasad, who secularized the theory of the four puruṣārthas, (b) Debiprasad Chattopadhyaya, who turned the traditional hierarchical order among the Indian philosophical schools upside down, (c) S. S. Barlingay, who reinterpreted Advaita–Vedānta, Sāṅkhya, Jainism, and so on, in materialistic terms, and (d) B. R. Ambedkar, who reconstructed Buddhism and synthesized it with Lokāyata. The chapter concludes the work by underlining the need for a Lokāyatic turn in Indian philosophy.

This chapter defines a paradox, roughly and superficially, as an assembly of apparently reasonable considerations that engender conflicting inclinations about what to believe, and hence a form of cognitive tension. By way of illustration, it examines various philosophical paradoxes—concerning knowledge, motion, freedom, evidence, non-existence, truth, normative facts, and vagueness. In light of these examples, the relative merits of three competing theories of the nature and meta-philosophical import of paradox are discussed: one suggested by Schiffer, which blames paradoxes on “glitch” concepts; a second diagnosis, due to Wittgenstein, which stresses irrationally stubborn, scientistic over-generalization; and a third—pinned on Quine—in which the causes of paradox are conflicts amongst our fundamental habits of reasoning.

This chapter discusses problems that arise from the assumption that there is a final, ultimately true theory of the world and considers a variety of arguments (connected with the coherence theory of truth, semantic contextualism, and the denial of absolutely general quantification) against it. From this it follows that if there cannot be an ultimately true theory then it also cannot be ultimately true that the world has a non-wellfounded structure. This leaves us with a problem, for it appears as if the theory of the non-existence of the real world we defend in the book as a whole cannot be a final theory either. The chapter closes with a discussion of this final problem, together with some reflections on what the implications of the denial of ultimately true theories are for the ontological or philosophical enterprise more generally.

Chapter 8 presents solutions to all three versions of The Benacerraf Dilemma—namely, The Original Benacerraf Dilemma, The Extended Benacerraf Dilemma, and The Generalized Benacerraf Dilemma—and develops a new structuralist account of the nature of mathematics and logic (Kantian Structuralism) alongside a new rational-intuition-based account of the nature of mathematical and logical a priori knowledge (Kantian Intuitionism), comparing and contrasting the two differing accounts with Charles Parsons’s theory. The chapter then moves on to work out a positive or anti-skeptical, innatist, rational-intuition-based solution to The Extended Benacerraf Dilemma. Following this, it then goes on to generalize that solution to The Generalized Benacerraf Dilemma. These solutions and accounts, in turn, presuppose and indirectly vindicate a Kant-inspired rational-intuition-based approach to meta-philosophy.