According to a main idea of Gentzen the meanings of the logical constants are reflected by the introduction rules in his system of natural deduction. This idea is here understood as saying roughly that a closed argument ending with an introduction is valid provided that its immediate subarguments are valid and that other closed arguments are justified to the extent that they can be brought to introduction form. One main part of the paper is devoted to the exact development (...) of this notion. Another main part of the paper is concerned with a modification of this notion as it occurs in Michael Dummett’s book The Logical Basis of Metaphysics. The two notions are compared and there is a discussion of how they fare as a foundation for a theory of meaning. It is noted that Dummett’s notion has a simpler structure, but it is argued that it is less appropriate for the foundation of a theory of meaning, because the possession of a valid argument for a sentence in Dummett’s sense is not enough to be warranted to assert the sentence. (shrink)

I see the question what it is that makes an inference valid and thereby gives a proof its epistemic power as the most fundamental problem of general proof theory. It has been surprisingly neglected in logic and philosophy of mathematics with two exceptions: Gentzen’s remarks about what justifies the rules of his system of natural deduction and proposals in the intuitionistic tradition about what a proof is. They are reviewed in the paper and I discuss to what extent they succeed (...) in answering what a proof is. Gentzen’s ideas are shown to give rise to a new notion of valid argument. At the end of the paper I summarize and briefly discuss an approach to the problem that I have proposed earlier. (shrink)

The ideas of the German philosopher and mathematician Gottlob Frege lie at the root of the analytical movement in philosophy. Frege and Other Philosophers comprises all of Professor Dummett's published and previously unpublished essays on Frege, with the exception of those included in his Truth and Other Enigmas. In some of these essays he explores the relation of Frege's ideas to those of his predecessors and contemporaries. In others he considers critically some interpretations of Frege, and develops the argument (...) for a sound understanding of Frege's thought which he first delineated in The Interpretation of Frege's Philosophy. Several of the essays illustrate his contention that Frege's work remains the best starting point for the investigation of problems concerning truth, meaning, thought, and language, which are still among the most contentious issues in modern analytical philosophy. Any discussion of Frege's ideas is therefore also an exploration of fundamental and as yet unresolved issues of philosophy. (shrink)

The Justification of Deduction is the title of one of Michael Dummett’s essays. It names also an important theme in his writings to which he returned in the book The Logical Basis of Metaphysics. In the essay he distinguishes different levels of justification of increasing philosophical depth. At the third and deepest level, the focus is on explaining deduction rather than on justifying it. The task is to explain how deduction can be both legitimate and useful in giving us knowledge. (...) I suggest that it can be described as essentially being the task to say what it is that gives a deduction its epistemic force. It is a fact that deduction has such force, consisting in its capacity to provide grounds for assertions and thereby extend our knowledge, but it is a fact that has to be explained. What is it that gives a deduction this capacity? This task is more challenging than is usually assumed. Obviously, it is not the validity of its inferences, as this is usually understood, which gives a deduction its epistemic force. Truth conditional theory of meaning does not seem to have any satisfactory solution to offer, and I argue that nor have inferential theories of meaning, which take the meaning of sentences to be determined by inference rules accepted in a language. In the last part of the paper, I sketch a different approach to the problem. The main idea is here to give the concept of inference a richer content, so that to perform an inference is not only to make a speech act in which a conclusion is claimed to be supported by a number of premisses, but is in addition to operate on the grounds for the premisses with the aim of getting a ground for the conclusion. I suggest that it is thanks to such operations that deductions provide grounds for their final conclusions. (shrink)

Geach defended the thesis of relative identity according to which, for any theory there is an extension, in whose language the meaning of the identity sign does not coincide with the meaning of the identity sign in the original language. The argument fails for formalized languages, since the interpretation of a first‐order language demands that a domain be specified for the variables, and in doing this we specify the finest possible interpretation of the identity sign. Yet an analogous question (...) for natural languages is whether for any expression of restricted generality there is a definite identity‐relation associated with the range of generality. Geach's thesis fails for natural languages, too, and furthermore, it is dependent upon the failure of the thesis for formalized languages. (shrink)

Our model of time is the classical continuum of real numbers, and our model of other measurable quantities that change over time is that of functions defined on real numbers with real numbers as values. This model is not derived from reality or from our experience of it, but imposed on reality; and the fit is very imperfect. In classical mathematics, the value of a function for any real number as argument is independent of its value for any other (...) argument: the analogue is Hume's doctrine that events are loose and separate. This makes continuity in the change of any quantity a contingent law of physica, rather than a conceptual necessity. The article explores alternatives to this classical model. (shrink)

In this article the contemporary debate between realism and anti-realism in analytical philosophy is analyzed and discussed. It is claimed that the nature of the reference relation which holds between language and the world is central in this discussion which has both logical, semantical, and epistemological aspects. In a firstpart, A Tarski's (semantic) theory of truth is explained and it is shown how, amongst several theories of truth, Tarski's may be called a realist one. However, a Tarski-style semantics need not (...) lead to a realist theory of Truth and Reference, as is shown in a subsequent part. Here, H Putnam's views on these matters are discussed. It is shown how Putnam in the course of his intellectual history has changed his view from a realist to an anti-realist position. Putnam'sarguments show the importance of logical and metalogical issues in the debate. In a final part, Putnam's views are criticized and it is shown how a line of argument, analogous to Putnam's, may lead to a realist theory of meaning and reference. (shrink)

Ulrich Meyer's objections to Dummett's arguments on the time continuum fail because he takes Dummett to endorse Hume's atomistic doctrine that events are ‘loose and separate’, In fact, Dummett rejects this doctrine. He used it in his original article only to indicate that certain implications which are conceptually possible fom the point of view of the classical model of time are not actually conceptually possible.

Justificationism differs from realism about how linguistic meaning is given, and hence in its associated conception of truth, and in particular in rejecting bivalence. Empirical discourse differs from mathematical primarily in that an effective decision-procedure for an empirical statement may cease to be available at a later time. The contrast is not that empirical knowledge is derived from what is mind-dependent, namely perception, whereas mathematical knowledge is not so derived. Mathematical knowledge does not accrue simply because a proof exists: the (...) proof has to be understood and recognized to be valid. Most ordinary mathematical proofs are indirect: they supply an effective means, in principle, to construct a direct or canonical proof. An indirect justification for asserting an empirical statement does not, in general, supply a means for bringing into existence a direct justification; it merely provides a ground for supposing that a direct justification would be or have been available for someone suitably placed to make the necessary observations. But it is by what constitutes direct evidence for a statement that its meaning is given; a grasp of its meaning does not rest on an ability to survey all conceivable indirect evidence. A direct justification of an empirical statement of the kind capable of being used as a report of observation must be an actual or possible observation by a suitably placed observer. A possible observation need not be explained by means of a counterfactual: it may be taken as consisting of the appropriate physical stimuli. This way of understanding it evades all three of the untenable choices Peacocke offers the justificationist. Unlike mathematical ones, empirical statements are often justifiably asserted on probabilistic grounds. When the statement admits of a conclusive justification, probabilistic evidence will not figure as a direct justification for asserting it, and hence as determining its meaning, so no circularity is involved, as Peacocke charges. There may, however, be empirical statements that can neither be reports of observation nor admit of a conclusive justification. Such statements can be asserted only on inductive or abductive grounds; this goes to characterize their meanings. Their meanings must therefore be explained by specifying such grounds as the most direct justifications of them that can be given, although they are difeasible rather than conclusive. Peacocke appears to deny that there can be statements that transcend all possible verification. To vindicate such a view from a realist standpoint is surely very difficult: it requires a demonstration that we could not grasp what it would be for such a statement to be true. It is very obscure what a realist's conception of truth is. The principle of bivalence remains a strong mark of differentiation between a justificationist and a realist conception of truth; a clear argument why the principle should be accepted would greatly help to clarify how the realist conceives of truth. I enjoyed reading Peacocke's article, but remain as perplexed as I was before just how he does conceive of truth. (shrink)

This chapter concerns that harmony is a particular relationship between the introduction rule and the elimination rule for a given connective. The Harmony Thesis says that a connective is defective unless its associated introduction and elimination rules are in harmony. It also says that a connective is defective if the logical principles which regulate its use go beyond a pair of harmonious introduction and elimination rules. The chapter scrutinizes the most influential arguments which have been put forward for the Harmony (...) Thesis. It discusses Dummett‐Prawitz's argument for Harmony, which revealed the huge difficulties that confront the project of trying to explicate the notions of consequence and validity directly in terms of the rules which, for the Inferential Role Semantics (IRS) theorist, constitute the meanings of the connectives. The chapter concludes by discussing briefly how the failure of the Harmony Thesis affects the prospects for IRS. (shrink)

The paper examines Dummett's argument for the indefinite extensibility of the concepts set, ordinal, real number, set of natural numbers, and natural number. In particular it investigates how the indefinite extensibility of the concept set affects our understanding of the notion of real number and whether the argument to the indefinite extensibility of the reals is cogent. It claims that Dummett is right to think of the universe of sets as an indefinitely extensible domain but questions the cogency (...) of the further claim that this fact raises an issue as to what sets or real numbers there are. (shrink)

The paper examines Dummett's argument for the indefinite extensibility of the concepts set, ordinal, real number, set of natural numbers, and natural number. In particular it investigates how the indefinite extensibility of the concept set affects our understanding of the notion of real number and whether the argument to the indefinite extensibility of the reals is cogent. It claims that Dummett is right to think of the universe of sets as an indefinitely extensible domain but questions the cogency (...) of the further claim that this fact raises an issue as to what sets or real numbers there are. (shrink)

Geoffrey Hunter; VIII*—Dummett's Arguments about the Natural Numbers, Proceedings of the Aristotelian Society, Volume 80, Issue 1, 1 June 1980, Pages 115–126, h.

Dummettian anti-realism–the refusal to endorse bivalence–is generally thought to be associated with idealism This paper argues that this is only true of the position developed by early Dummett. In a later manifestation Dummettian anti-realism is better thought of as providing the logic for anti-realisms of an error theoretic kind. Early on Dummett distinguished deep from shallow arguments for giving up bivalence: deep arguments followed a strong ‘sufficiency’ reading of Frege’s context principle, and made the sentence the primary vehicle of meaning. (...) Enriched by an account of truth in terms of verifiability, deep arguments implied a form of linguistic idealism. From within a perspective that had already made ontology relative to theory, it was natural for the difference between realism and idealism to hinge on the notion of truth for sentences. Having given up the distinction between deep and shallow arguments against bivalence, Dummett, post 1990, asserts that every rejection of bivalence leads to a form of anti-realism. In the intervening years, he has also come to cast doubt on the sufficiency reading of the context principle, particularly as developed by Crispin Wright. These doubts open the space for some more simple-minded criticisms of this strong version of the context principle, developed in the paper. Once the sufficiency reading of the context principle is given up, arguments for failing to assert bivalence come to hinge on beliefs about ‘genuine’ existence. These forms of anti-realism cannot be taken to imply idealism. Rather, in many cases they will be the expression of views of an error-theoretic kind, that are quite compatible with a thorough-going rejection of idealism. (shrink)

This book presents an in-depth and critical reconstruction of Prawitz’s epistemic grounding, and discusses it within the broader field of proof-theoretic semantics. The theory of grounds is also provided with a formal framework, through which several relevant results are proved. Investigating Prawitz’s theory of grounds, this work answers one of the most fundamental questions in logic: why and how do some inferences have the epistemic power to compel us to accept their conclusion, if we have accepted their premises? Prawitz proposes (...) an innovative description of inferential acts, as applications of constructive operations on grounds for the premises, yielding a ground for the conclusion. The book is divided into three parts. In the first, the author discusses the reasons that have led Prawitz to abandon his previous semantics of valid arguments and proofs. The second part presents Prawitz’s grounding as found in his ground-theoretic papers. Finally, in the third part, a formal apparatus is developed, consisting of a class of languages whose terms are equipped with denotation functions associating them to operations and grounds, as well as of a class of systems where important properties of the terms can be proved. (shrink)

Neil Tennant and Joseph Salerno have recently attempted to rigorously formalize Michael Dummett's argument for logical revision. Surprisingly, both conclude that Dummett commits elementary logical errors, and hence fails to offer an argument that is even prima facie valid. After explicating the arguments Salerno and Tennant attribute to Dummett, I show how broader attention to Dummett's writings on the theory of meaning allows one to discern, and formalize, a valid argument for logical revision. Then, after correctly providing (...) a rigorous statement of the argument, I am able to delineate four possible anti-Dummettian responses. Following recent work by Stewart Shapiro and Crispin Wright, I conclude that progress in the anti-realist's dialectic requires greater clarity about the key modal notions used in Dummett's proof. (shrink)

A collection of all but two of the author's philosophical essays and lectures originally published or presented before August 1976.

Concerns a problem posed, but not solved, by Frege in part III of his Grundgesetze. As a preliminary to defining ‘real number’, Frege attempts to analyse the notion of a quantitative domain. He was unaware of the previous attempt of Otto Holder to do this; it is remarked how much weaker Frege's assumptions were in deriving theorems than Holder's. Frege deals with groups on which there is a right‐invariant semilinear ordering, although he does not use this terminology. He is uncertain (...) whether it can be deduced that the ordering is also left‐invariant, and proves as much as possible without invoking the assumption that it is. The independence proof, due to Dr Peter Neumann, establishes that it could not be deduced. (shrink)

Prawitz has recently developed a theory of epistemic grounding that differs in many respects from his earlier semantics of arguments and proofs. An innovative approach to inferences yields a new conception of the intertwinement of the notions of valid inference and proof. We aim at singling out three reasons that may have led Prawitz to the ground-theoretic turn, i.e.: a better order in the explanation of the relation between valid inferences and proofs; a notion of valid inference based on which (...) valid inferences and proofs are recognisable as such; a reconstruction of the deductive activity that makes inferences capable of yielding justification per se. These topics are discussed by Prawitz with reference to a very general and ancient question: why and how correct deduction has the epistemic power to compel us to accept its conclusions, provided its premises are justified? We conclude by remarking that, in spite of some improvements, the ground-theoretic approach shares with the previous one a problem of vacuous validity which, as Prawitz himself points out, blocks in both cases a satisfactory explanation of epistemic compulsion. (shrink)

This volume examines the notion of an analytic proof as a natural deduction, suggesting that the proof's value may be understood as its normal form--a concept with significant implications to proof-theoretic semantics.

One of Etchemendy's arguments against the Tarskian and model‐theoretic notion of logical truth is based on a reduction principle according to which a universally quantified sentence is true if, and only if, all of its instances are logically true. The reduction of logical truth to mere truth reveals that the concept of validity at play in Tarski and in model‐theory relies upon extra‐logical assumptions. A similar reduction had already been put forward by Prawitz, although not with focus on extra‐logical assumptions. (...) Prawitz's reduction concerns rather the possibility of understanding the Tarskian and model‐theoretic approach as implying any epistemic modality. We take into account a possible strategy that an advocate of Tarski and model‐theory may use against Etchemendy's objection, and argue that this strategy fails with respect to Prawitz's criticism. This permits us to shed light upon a difference between Prawitz's and Etchemendy's standpoints: while Etchemendy's reduction shows that the Tarskian and model‐theoretic notion of logical truth cannot be understood as an adequate formal counterpart of truth‐preservation solely by virtue of logical form, Prawitz's reduction insists on the fact that truth‐preservation solely by virtue of logical form is insufficient for epistemic modality to obtain. (shrink)

It has been common in contemporary logic and philosophy of logic to identify the validity of an inference with its conclusion being a (logical) consequence of its premisses.

Michael Dummett is a leading contemporary philosopher whose work on the logic and metaphysics of language has had a lasting influence on how these subjects are conceived and discussed. This volume contains some of the most provocative and widely discussed essays published in the last fifteen years, together with a number of unpublished or inaccessible writings. Essays included are: "What is a Theory of Meaning?," "What do I Know When I Know a Language?," "What Does the Appeal to Use Do (...) for the Theory of Meaning?," "Language and Truth," "Truth and Meaning," "Language and Communication," "The Source of the Concept of Truth," "Mood, Force, and Convention," "Frege and Husserl on Reference," "Realism," "Existence," "Does Quantification Involve Identity?," "Could there be Unicorns?," "Causal Loops," "Common Sense and Physics," "Testimony and Memory," "What is Mathematics About?," "Wittgenstein on Necessity: Some Reflections," and "Realism and Anti-Realism." Serving well as a companion to Dummett's other collections, the essays in this volume are not forbiddingly technical or specialized, and have relevance to many areas of analytic philosophy. (shrink)

In this work Dummett discusses, section by section, Frege's masterpiece The Foundations of Arithmetic and Frege's treatment of real numbers in the second volume ...

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)

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.

Dag Prawitz The Bulletin of Symbolic Logic, Volume 19, Issue 1, Page 119-122, March 2013.