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)

: The open-ended character of natural languages calls for the hypothesis that humans are endowed with a recursive procedure generating sentences which are hierarchically organized. Structural relations such as c-command, expressed on hierarchical sentential representations, determine all sorts of formal and interpretive properties of sentences. The relevant computational principles are well beyond the reach of conscious introspection, so that studying such properties requires the formulation of precise formal hypotheses, and empirically testing them. This article illustrates all these aspects of linguistic (...) research through the discussion of non-coreference effects. The article argues in favor of the formal linguistic approach based on hierarchical structures, and against alternatives based on vague notions of “analogical generalization”, and/or exploiting mere linear order. In the final part, the issue of cross-linguistic invariance and variation of non-coreference effects is addressed. Keywords: Linguistic Knowledge; Morphosyntactic Properties; Unconscious Computations; Coreference; Linguistic Representations Conoscenza linguistica e computazioni inconsce Riassunto: Il carattere aperto del linguaggio naturale avvalora l’ipotesi che gli esseri umani siano dotati di una procedura ricorsiva che genera frasi gerarchicamente organizzate. Relazioni strutturali come il c-comando, espresse su rappresentazioni frasali gerarchiche, determinano tutte le proprietà formali e interpretative delle frasi. I principi computazionali rilevanti sono totalmente al di fuori della portata della coscienza introspettiva e così lo studio di tali proprietà richiede la formulazione di precise ipotesi formali e la loro verifica sperimentale. Questo articolo illustra tutti questi aspetti della ricerca linguistica, esaminando gli effetti di non-coreferenza. Si argomenta in favore dell’approccio linguistico formale basato su strutture gerarchiche e contro alternative basate su vaghe nozioni di “generalizzazione analogica” e/o che impiegano il semplice ordine lineare. Nella parte finale si affronta il tema dell’invarianza e della variazione cross-linguistica degli effetti di non-coreferenza. Parole chiave: Conoscenza linguistica; Proprietà morfosintattiche; Computazioni inconsce; Coreferenzialità; Rappresentazioni linguistiche. (shrink)

The traditional picture of logic takes it for granted that "valid arguments have a fundamental epistemic significance", but neither model theory nor traditional proof theory dealing with formal system has been able to give an account of this significance. Since valid arguments as usually understood do not in general have any epistemic significance, the problem is to explain how and why we can nevertheless use them sometimes to acquire knowledge. It is suggested that we should distinguish between arguments and acts (...) of inferences and that we have to reconsider the latter notion to arrive at the desired explanation. More precisely, the notions should be developed so that the following relationship holds: one gets in possession of a ground for a conclusion by inferring it from premisses for which one already has grounds, provided that the inference in question is valid. The paper proposes explications of the concepts of ground and deductively valid inference so that this relationship holds as a conceptual truth. Logical validity of inference is seen as a special case of deductive validity, but does not add anything as far as epistemic significance is concerned—it resides already in the deductively valid inferences. (shrink)

The concept of “necessity of thought” plays a central role in Dag Prawitz’s essay “Logical Consequence from a Constructivist Point of View” (Prawitz 2005). The theme is later developed in various articles devoted to the notion of valid inference (Prawitz, 2009, forthcoming a, forthcoming b). In section 1 I explain how the notion of necessity of thought emerges from Prawitz’s analysis of logical consequence. I try to expound Prawitz’s views concerning the necessity of thought in sections 2, 3 and 4. (...) In sections 5 and 6 I discuss some problems arising with regard to Prawitz’s views. (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)

I will be concerned with the following question: are there compelling arguments for postulating a distinction between the truth of a statement and the recognition of its truth, when truth is conceived along the lines of a suitable generalization of the intuitionistic idea that it should be characterized as the existence of a proof? I will argue that the distinction is not necessary within the conceptual framework of intuitionism by replying to two arguments to the contrary, one based on the (...) paradox of inference, the other on considerations concerning the content of a statement. (shrink)