This is a long-awaited new edition of one of the best known Oxford Logic Guides. The book gives an introduction to intuitionistic mathematics, leading the reader gently through the fundamental mathematical and philosophical concepts. The treatment of various topics, for example Brouwer's proof of the Bar Theorem, valuation systems, and the completeness of intuitionistic first-order logic, have been completely revised.

This highly acclaimed book is a major contribution to the philosophy of language as well as a systematic interpretation of Frege, indisputably the father of ...

Knowledge and its Limits presents a systematic new conception of knowledge as a kind of mental stage sensitive to the knower's environment. It makes a major contribution to the debate between externalist and internalist philosophies of mind, and breaks radically with the epistemological tradition of analyzing knowledge in terms of true belief. The theory casts new light on such philosophical problems as scepticism, evidence, probability and assertion, realism and anti-realism, and the limits of what can be known. The arguments are (...) illustrated by rigorous models based on epistemic logic and probability theory. The result is a new way of doing epistemology and a notable contribution to the philosophy of mind. (shrink)

Such a conception, says Dummett, will form "a base camp for an assault on the metaphysical peaks: I have no greater ambition in this book than to set up a base ...

The linchpin of Williamson (2000)'s radically externalist epistemological program is an argument for the claim that no non-trivial condition is luminous—that no non-trivial condition is such that whenever it obtains, one is in a position to know that it obtains. I argue that Williamson's anti-luminosity argument succeeds only if one assumes that, even in the limit of ideal reflection, the obtaining of the condition in question and one's beliefs about that condition can be radically disjoint from one another. However, no (...) self-respecting defender of the luminosity of the mental would ever make such an assumption. Thus Williamson can only secure his controversial claims in epistemology by taking for granted certain equally controversial claims in the philosophy of mind. What emerges is that the best bet for defending an internalist epistemology against Williamson's attack is to take there to be a tight, intimate connection between (to take one example) our experiences and our beliefs upon reflection about the obtaining of those experiences, or between (to take another example) the rationality of our beliefs and our beliefs upon reflection about the rationality of those beliefs. (shrink)

The main question addressed in this chapter is how to analyze the modal ingredient in the concept of logical consequence or logical validity of an inference, here expressed by saying that the truth of the conclusion of a logically valid inference should follow by necessity of thought from the truth of the premisses. It is claimed that this modal ingredient is not taken care of by Tarski’s requirement, later developed in model theory, that for all interpretations of the non-logical terms (...) of the sentences, the conclusion is true if the premisses are. Instead we must bring in reasoning or the notion of proof in the analysis, and it is shown how one may develop a notion of proof or validity of an argument based on the constructive meaning of the logical constants. It is proposed that from a constructive point of view, a sentence is a consequence of a set of premisses if and only if there is a proof in the sense explained of the sentence from the set of premisses. Knowledge of such a proof compels us to hold the conclusion true given that we hold the premisses true. (shrink)

Suppose we want to take seriously the neoverificationist idea that an intuitionistic theory of meaning can be generalized in such a way as to be applicable not only to mathematical but also to empirical sentences. The paper explores some consequences of this attitude and takes some steps towards the realization of this program. The general idea is to develop a meaning theory, and consequently a formal semantics, based on the idea that knowing the meaning of a sentence is tantamount to (...) having a criterion for establishing what is a justification for it. Section 1 motivates a requirement of epistemic transparency imposed onto justifications conceived as mental states. In Section 2, the formal notion of justification for an atomic formula is defined, in terms of the notion of cognitive state. In Section 3, the definition is extended to logically complex formulas. In Section 4, the notion of truth-ground is introduced and is used to give a definition of logical validity. (shrink)