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This chapter examines the possible relevance of philosopher Nāgārjuna's work on contradictions arising at the limits of thought. It states that Nāgārjuna has endorsed the possibility of true contradictions which confirms that he is indeed a rational thinker. It provides an interpretation of how Nāgārjuna may view logic or the limits of thought. This chapter demonstrates that Nāgārjuna's two linked limit paradoxes satisfy a schema common to a number of well-known paradoxes in Western philosophy.

This chapter examines the role of language theory and unity of thought on the contradictions in the limits of thought. It is natural for a theory of language to have implications about what can and what cannot be expressed, and modern theories of language always seem to render important things beyond the limit of expression. This chapter analyses philosopher Gottlob Frege's postulates on the existence of concepts, and suggests that his claims about them are beyond the limit of the expressible. It also evaluates Ludwig Wittgenstein's notion of form, and suggests that his account for unity of thought is beyond the expressible.

This chapter discusses the contradiction in the limits of iteration, one of the four limits of thought. This type of limit arises when there is some operation that is applied over and over again as far as possible. The most notable example is the mathematical (ordinal) infinite. This chapter analyses Aristotle's view on the infinite and his perception of contradiction at the limit of the iterable. It also examines mathematician and philosopher Gottfried Leibniz's cosmological argument for the existence of God.

This chapter discusses the contradiction in the limits of conception, one of the four limits of thought. Though it is common enough to suppose that there are things beyond conception, it is difficult to do so without conceiving them in some sense, hence the contradiction. The chapter analyses this contradiction on the basis of the works and views of the philosopher Anselm on Canterbury, whose concern is God, and of Bishop Berkeley, whose concern is idealism. Berkeley's idealism prevented him from realizing the contradiction, though it has been shown that idealism is no escape from contradiction.

This chapter examines the relevance of theories on sets and classes to the contradictions in the limits of thought. It analyses possible solutions to the set-theoretic paradoxes based on Ernst Friedrich Ferdinand Zermelo's Zermelo-Fraenkel set theory and John von Neumann's notion of a proper class. It shows that, like other contemporary solutions to inclosure contradictions, Zermelo's and von Neumann's are not adequate even in the limited domain for which they were designed. This chapter suggests that the only uniform approach to all these paradoxes is the dialetheic one, and that the limits of thought which are the inclosures are truly contradictory.

This chapter sums up the key findings of this book on the limits of thought. It has traced the notions of the four limits of thought and analysed them through the works of Western philosophers, from its origins in ancient Greece to the present. Many of these philosophers whose works explored the locus of paradox and contradiction have unwillingly ended up in self-contradiction. The contradictions at the limits of thought are all captured by the notion of an inclosure and the historical figure with the clearest insight into this phenomenon was philosopher Georg Wilhelm Friedrich Hegel.

This introductory chapter explains the coverage of the book, which is about the relationship between the limits of thoughts and contradiction. It suggests that the limits of thought, which provide the boundaries beyond which certain conceptual processes cannot go, are dialetheic or subject to true contradictions. The contradiction is simply to the effect that the conceptual processes in question do cross the boundaries. The book provides case studies on the works of various philosophers concerning the limits of thought and contradiction.

This chapter discusses the contradiction in the limits of cognition, one of the four limits of thought. Cognition is about relationships that arise between agents and the world that they cognise, such as knowledge, truth, and rational belief. It determines the contradictions at the limit of cognition with a thesis called the Cognition Scheme. It also analyses the issue of relativism and scepticism through the views and works of philosophers Protagoras and Sextus Empiricus.

This chapter examines philosopher Immanuel Kant's Theory of Categories and how it may avoid the limits of thought. It analyses Kant's Critique of Pure Reason in order to chart the logical geography of the most general features of thought and spell out their limits. This chapter explores every possible way to avoid contradiction that is acceptable to Kant, but none of them is proved successful. This suggests that the theory of categories does not apply to noumena.

This chapter examines the views of philosophers Ludwig Wittgenstein and Jacques Derrida on the theory of meaning. Both philosophers reject accounts of meaning based on semantic correlates but they accept the consequent indeterminacy of sense and the relationship between consciousness and meaning. Wittgenstein and Derrida share another similarity in that their views all trespass over the limits of the expressible, and so deliver contradictions at the limits of thought.

This chapter examines how the limits of expression or limits of the inexpressible, one of the four limits of thought, arose in pre-Kantian philosophy. It explains three reasons for assuming that there are features of the world that transcend language. Good examples of these features are the ones that might be thought to constitute ultimate reality, such as the fundamental matter or God. This chapter analyses these two things based on Aristotle's account of prime matter and Nicholas Cusanus' account of God. It also discusses Cratylus' view of the flux of meaning.

This chapter examines the relevance of mathematician and philosopher George Cantor's views on infinity to the contradictions in the limits of thought. Cantor's work has clarified with an unprecedented rigour that contradictions do occur and he has narrowed down the field of possible solutions to a very small surveyable class. His notion of diagonalization has provided the cornerstone for an adequate understanding of boundary-transcendence. However, it is shown that Cantor's theory of the transfinite is not successful in removing the paradoxes of the infinite, it merely relocates them.

This chapter examines the relevance of mathematician Frank Ramsey's essay The Foundations of Mathematics to the contradictions in the limits of thought. Ramsey has provided the most acute early critical commentary on Bertrand Russell's ideas on paradox and he suggested that the paradoxes of self-reference are of two kinds. This chapter analyses several modern solutions to the paradoxes of Ramsey's Group B and finds that they have the same general failings as Ramsey's own solution.

This chapter discusses the contradictions in the limit of thoughts in theories which postulate that semantic correlates must be avoided in giving an account for meaning. It examines the views and works of philosophers Willard Van Orman Quine and Donald Davidson on the theory of meaning. The findings reveal that the Quine/Davidson theory gives rise to contradictions at the limits of expression because of the indeterminacy of sense and reference, and in an attempt to avoid contradictions at the limits of cognition.

This chapter examines the relevance of philosopher Georg Wilhelm Friedrich Hegel's infinities to contradictions in the limits of thought. It highlights Hegel's own positive account of the nature of infinity and his criticism on the basis of the distinction between phenomena and noumena. Hegel suggests that the infinities of Immanuel Kant's antimonies are contradictory and that generated infinities are truly contradictory. He also observes that certain kinds of limits behave in a contradictory fashion and he creates a contradictory category for them.

Hume's Treatise has two distinct goals: first, to reveal the limitations of human knowledge by developing an empirical psychology, and second, to decide metaphysical questions. Many commentators take these to be in conflict, and so emphasize one or the other. This chapter suggests a way to unify them: by discovering the limits of human thought, Hume can declare certain metaphysical positions not just unknowable, or true for all we can tell, but positively nonsensical. The psychological project is thus in service of the metaphysical.

This chapter traces the development of Wittgenstein's philosophy of psychology. The Tractatus is little concerned with the subject. One can but glean what his meta-psychological commitments were from occasional asides and correspondence. They concern meaning something by words and sentences, thought and thinking, and intentionality. These commitments were presupposed by the idea that there is a semantic connection between words and world, forged by mental acts of meaning. With the collapse of the Tractatus, Wittgenstein realized that the concepts of meaning and intending, of understanding and interpreting, of thinking and imagining, had to be subjected to philosophical scrutiny. This guided his preoccupation with psychological concepts in the 1930s. The range of interest widened with the development of his private language arguments. After completion of the Investigations in 1946/7, Wittgenstein's philosophical writings are focused largely on questions in philosophy of psychology, of which a synopsis is given.