The contributions to this volume represent a broad range of aspects of proof-theoretic semantics. Some do so in the narrower, and some in the wider sense of the term. Some deal with issues I have been concerned with directly, and some tackle further problems. All of them open interesting new perspectives and develop the field in different directions. I will briefly comment on the significance of each contribution here.

In this autobiographical sketch, which is followed by a bibliography of my writings, I try to relate my intellectual development to problems, ideas and results in proof-theoretic semantics on which I have worked and to which I have contributed.

The initial premise of this paper is that the structure of a proof is inherent in the definition of the proof. Side conditions to deal with the discharging of assumptions means that this does not hold for systems of natural deduction, where proofs are given by monotone inductive definitions. We discuss the idea of using higher order definitions and the notion of a functional closure as a foundation to avoid these problems. In order to focus on structural issues we introduce (...) a more abstract perspective, where a structural proof theory becomes part of a more general theory of functional closures. A notion of proof equations is discussed as a structural classifier and we compare the Russell and Ekman paradoxes to illustrate this. (shrink)

In the present paper, the Fregean conception of proof-theoretic semantics that I developed elsewhere will be revised so as to better reflect the different roles played by open and closed derivations. I will argue that such a conception can deliver a semantic analysis of languages containing paradoxical expressions provided some of its basic tenets are liberalized. In particular, the notion of function underlying the Brouwer–Heyting–Kolmogorov explanation of implication should be understood as admitting functions to be partial. As argued in previous (...) work, the correctness of an inference rule should not be defined in terms of transmission of provability, but rather should be grounded on weaker principles, which I show that are motivated by consideration about the content of the inversion principle. In the conclusion, I briefly address the issue of compositionality, arguing that the violation of compositionality induced by paradoxes are no worse than others that are regarded, at least by Dummett, as wholly unproblematic. (shrink)

Developing early results of Prawitz, Tennant proposed a criterion for an expression to count as a paradox in the framework of Gentzen’s natural deduction: paradoxical expressions give rise to non-normalizing derivations. Two distinct kinds of cases, going back to Crabbé and Tennant, show that the criterion overgenerates, that is, there are derivations which are intuitively non-paradoxical but which fail to normalize. Tennant’s proposed solution consists in reformulating natural deduction elimination rules in general form. Developing intuitions of Ekman we show that (...) the adoption of general rules has the consequence of hiding redundancies within derivations. Once reductions to get rid of the hidden redundancies are devised, it is clear that the adoption of general elimination rules offers no remedy to the overgeneration of the Prawitz–Tennant analysis. In this way, we indirectly provide further support for a solution to one of the two overgeneration cases developed in previous work. (shrink)

A proof-theoretic test for paradoxicality was famously proposed by Tennant: a paradox must yield a closed derivation of absurdity with no normal form. Drawing on the remark that all derivations of a given proposition can be transformed into derivations in normal form of a logically equivalent proposition, we investigate the possibility of paradoxes in normal form. We compare paradoxes à la Tennant and paradoxes in normal form from the viewpoint of the computational interpretation of proofs and from the viewpoint of (...) proof-theoretic semantics. (shrink)

It has been argued that reduction procedures are closely connected to the question about identity of proofs and that accepting certain reductions would lead to a trivialization of identity of proofs in the sense that every derivation of the same conclusion would have to be identified. In this paper it will be shown that the question, which reductions we accept in our system, is not only important if we see them as generating a theory of proof identity but is also (...) decisive for the more general question whether a proof has meaningful content. There are certain reductions which would not only force us to identify proofs of different arbitrary formulas but which would render derivations in a system allowing them meaningless. To exclude such cases, a minimal criterion is proposed which reductions have to fulfill to be acceptable. (shrink)

Dag Prawitz (1971) put forward the idea that an admissible reduction process does not affect the identity of proofs represented by derivations in natural deduction. The idea relies on his conjecture that two derivations represent the same proof if and only if they are equivalent in the sense that they are reflexive, transitive and symmetric closure of the immediate reducibility relation. Schroeder-Heister and Tranchini (2017) accept Prawitz’s conjecture and propose the triviality test as the criterion for admissible reductions. In the (...) present paper, we will consider two main troubles of the triviality test. The first is the obscurity of a method of evaluating admissible reductions. The second is the circularity problem that the triviality test already assumes the set of admissible reduction procedures. For the solution of the problems, we will propose the spoiler test which immunes the problems of the triviality test and has the role of the criterion for admissible reductions. At last, we shall cover a plausible problem of the spoiler test that can be caused by Crabbé’s case. (shrink)

In this dissertation, we shall investigate whether Tennant's criterion for paradoxicality(TCP) can be a correct criterion for genuine paradoxes and whether the requirement of a normal derivation(RND) can be a proof-theoretic solution to the paradoxes. Tennant’s criterion has two types of counterexamples. The one is a case which raises the problem of overgeneration that TCP makes a paradoxical derivation non-paradoxical. The other is one which generates the problem of undergeneration that TCP renders a non-paradoxical derivation paradoxical. Chapter 2 deals with (...) the problem of undergeneration and Chapter 3 concerns the problem of overgeneration. Chapter 2 discusses that Tenant’s diagnosis of the counterexample which applies CR−rule and causes the undergeneration problem is not correct and presents a solution to the problem of undergeneration. Chapter 3 argues that Tennant’s diagnosis of the counterexample raising the overgeneration problem is wrong and provides a solution to the problem. Finally, Chapter 4 addresses what should be explicated in order for RND to be a proof-theoretic solution to the paradoxes. (shrink)

In his paper, “On paradox without self-reference”, Neil Tennant proposed the conjecture for self-referential paradoxes that any derivation formalizing self-referential paradoxes only generates a looping reduction sequence. According to him, the derivation of the Liar paradox in natural deduction initiates a looping reduction sequence and the derivation of the Yablo's paradox generates a spiral reduction. The present paper proposes the counterexample to Tennant's conjecture for self-referential paradoxes. We shall show that there is a derivation of the Liar paradox which generates (...) a spiraling reduction procedure. Since the Liar paradox is a self-referential paradox, the result is a counterexample to his conjecture. Tennant has believed that classical reductio has no essential role to formalize paradoxes. As our counterexample applies the rule of classical reductio, he may reject the counterexample. In this sense, it will be briefly argued that classical reductio and his rules for the liar sentence share some inferential role. If classical reductio should not be used in paradoxical reasoning, neither should be his rules for the liar sentence. (shrink)