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.

By a valid inference is here understood an inference that succeeds in its aim to justify its conclusion given that its premisses are already justified. For an inference to be valid it is thus not enough that the sentence asserted in the conclusion is a logical consequence of the sentences asserted in the premisses. A proof is understood as a succession of valid inferences that is closed (i.e. all its assumptions are discharged and all its free variables are bound by (...) some of the inferences). On the other hand, in mathematics the justification of an assertion has traditionally been taken to consist in a proof of it, which gives rise to an explanatory circle.In an attempt to avoid this circle, I have in previous works tried to explain the notion of ground for an assertion without referring to valid inferences. However, it turns out that the resulting notion of ground has unwanted consequences like the notion of proof in the so-called BHK-interpretation of intuitionistic logic. For instance, one is guaranteed a ground or a proof of an implication A → B as soon as there is in fact no ground or proof of A, regardless of whether one really knows this fact, and hence without having a real justification of the assertion of the implication.In the absence of a non-circular explanation of the notions of valid inference and proof, I suggest that we should try to formulate a number of principles of how these two epistemic notions depend on each other. A more careful explanation of the validity of an inference results in what I call a heuristic idea about the validity of inference, which is used as a guide to formulate precise principles. One such principle generalizes Gentzen’s idea that his introduction rules define the meaning of the logical constants. Another principle generalizes a previously noted property of Gentzen’s elimination rules that I have called that of satisfying the inversion principle. It results in a notion that I call non-creativity, which implies validity. (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.

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)

Reviewed Works:Gaisi Takeuti, Proof Theory.Georg Kreisel, Proof Theory: Some Personal Recollections.Wolfram Pohlers, Contributions of the Schutte School in Munich to Proof Theory.Stephen G. Simpson, Subsystems of $\mathbf{Z}_2$ and Reverse Mathematics.Solomon Feferman, Proof Theory: A Personal Report.

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)

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)

We may try to explain proofs as chains of valid inference, but the concept of validity needed in such an explanation cannot be the traditional one. For an inference to be legitimate in a proof it must have sufficient epistemic power, so that the proof really justifies its final conclusion. However, the epistemic concepts used to account for this power are in their turn usually explained in terms of the concept of proof. To get out of this circle we may (...) consider an idea within intuitionism about what it is to justify the assertion of a proposition. It depends on Heyting’s view of the meaning of a proposition, but does not presuppose the concept of inference or of proof as chains of inferences. I discuss this idea and what is required in order to use it for an adequate notion of valid inference. (shrink)

What is the appropriate notion of truth for sentences whose meanings are understood in epistemic terms such as proof or ground for an assertion? It seems that the truth of such sentences has to be identified with the existence of proofs or grounds, and the main issue is whether this existence is to be understood in a temporal sense as meaning that we have actually found a proof or a ground, or if it could be taken in an abstract, tenseless (...) sense. Would the latter alternative amount to realism with respect to proofs or grounds in a way that would be contrary to the supposedly anti-realistic standpoint underlying the epistemic understanding of linguistic expressions? Before discussing this question, I shall consider reasons for construing linguistic meaning epistemically and relations between such reasons and reasons for taking an anti-realist point of view towards the discourse in question. (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)

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

WeingartnerPaul and LeebHans-Peter, eds, Kreisel’s Interests: On the Foundations of Logic and Mathematics. Tributes; 41. London: College Publications, 2020. Pp. viii + 171. ISBN: 978-1-84890-330-2.

This collection of 38 papers gives a cross-section of ongoing research in philosophy of science and philosophical logic. The papers, written by active researchers in the field and published here for the first time, are drawn from around 650 papers that were contributed to the 9th International Congress of Logic, Methodology and Philosophy of Science in Uppsala, Sweden, 1991. Some of the speakers whose contributions attracted special interest were invited to contribute their papers to this volume. A few papers appear (...) here more or less as they were presented at the Congress, whereas others are expansions or elaborations of the talks given. There is one section with five papers on philosophical logic. The other papers deal with many different aspects of philosophy of science, including general methodological questions, problems of probability, induction and decision theory, and ethics of science and technology, as well as foundational problems about particular sciences. Five special sections are concerned with logic, mathematics and computer science, the physical sciences, the biological sciences, cognitive science, and linguistics, respectively. Finally, there is one section on the history of logic, methodology and philosophy of science. The book will be of interest to philosophers of science and logicians, as well as to all researchers interested in the foundations of their disciplines. (shrink)