Eight distinct rules for implication in the antecedent for the sequent calculus, one of which being Gentzen’s standard rule, can be derived by successively applying a number of cuts to the logical ground sequent A → B, A ⇒ B. A naive translation into natural deduction collapses four of those rules onto the standard implication elimination rule, and the remaining four rules onto the general elimination rule. This collapse is due to the fact that the difference between a formula occurring (...) in the succedent of a premise of a sequent calculus rule and that formula instead occurring in the antecedent of the conclusion cannot be adequately expressed in the framework of natural deduction. In contrast to this, the difference between a formula occurring in the succedent of the conclusion of a sequent calculus rule and that formula instead occurring in the antecedent of a premise corresponds exactly to the distinction between the standard implication elimination rule and its general counterpart. This incongruity can be remedied by introducing a notational facility that requires of the particular premise of a rule to which it is attached to be an assumption, i.e., to prevent it from being the conclusion of another rule application. Applying this facility to implication elimination results in eight distinct rules that correspond exactly to the eight sequent calculus rules. These eight rules are presented and discussed in detail. It turns out that a natural deduction calculus (for positive implication logic) that employs the rule corresponding to the standard left implication rule of the sequent calculus as well as a rule for explicit substitution can be seen as a natural deduction style sequent calculus. (shrink)

Relevant logics are a family of non-classical logics characterized by the behavior of their implication connectives. Unlike some other non-classical logics, such as intuitionistic logic, there are multiple philosophical views motivating relevant logics. Further, different views seem to motivate different logics. In this article, we survey five major views motivating the adoption of relevant logics: Use Criterion, sufficiency, meaning containment, theory construction, and truthmaking. We highlight the philosophical differences as well as the different logics they support. We end with some (...) questions for future research. (shrink)

This open access book is a superb collection of some fifteen chapters inspired by Schroeder-Heister's groundbreaking work, written by leading experts in the field, plus an extensive autobiography and comments on the various contributions by Schroeder-Heister himself. For several decades, Peter Schroeder-Heister has been a central figure in proof-theoretic semantics, a field of study situated at the interface of logic, theoretical computer science, natural-language semantics, and the philosophy of language. -/- The chapters of which this book is composed discuss the (...) subject from a rich variety of angles, including the history of logic, the proper interpretation of logical validity, natural deduction rules, the notions of harmony and of synonymy, the structure of proofs, the logical status of equality, intentional phenomena, and the proof theory of second-order arithmetic. All chapters relate directly to questions that have driven Schroeder-Heister's own research agenda and to which he has made seminal contributions. The extensive autobiographical chapter not only provides a fascinating overview of Schroeder-Heister's career and the evolution of his academic interests but also constitutes a contribution to the recent history of logic in its own right, painting an intriguing picture of the philosophical, logical, and mathematical institutional landscape in Germany and elsewhere since the early 1970s. The papers collected in this book are illuminatingly put into a unified perspective by Schroeder-Heister's comments at the end of the book. Both graduate students and established researchers in the field will find this book an excellent resource for future work in proof-theoretic semantics and related areas. (shrink)

A complementary system for a given logic is a proof system whose theorems are exactly the formulas that are not valid according to the logic in question. This article is a contribution to the complementary proof theory of classical propositional logic. In particular, we present a complementary proof-net system, $$\textsf{CPN}$$ CPN, that is sound and complete with respect to the set of all classically invalid (one-side) sequents. We also show that cut elimination in $$\textsf{CPN}$$ CPN enjoys strong normalization along with (...) strong confluence (and, hence, uniqueness of normal forms). (shrink)

Two common forms of natural deduction proof systems are found in the Gentzen–Prawitz and Jaśkowski–Fitch systems. In this paper, I provide translations between proofs in these systems, pointing out the ways in which the translations highlight the structural rules implicit in the systems. These translations work for classical, intuitionistic, and minimal logic. I then provide translations for classical S4 proofs.

We present a multiple-assumption multiple-conclusion system for bi-intuitionistic logic. Derivations in the systems are graphs whose edges are labelled by formulas and whose nodes are labelled by rules. We show how to embed both the standard intuitionistic and dual-intuitionistic natural deduction systems into the proposed system. Soundness and completeness are established using translations with more traditional sequent calculi for bi-intuitionistic logic.

An important question for proponents of non-contractive approaches to paradox is why contraction fails. Zardini offers an answer, namely that paradoxical sentences exhibit a kind of instability. I elaborate this idea using revision theory, and I argue that while instability does motivate failures of contraction, it equally motivates failure of many principles that non-contractive theorists want to maintain.

Anderson and Belnap presented indexed Fitch-style natural deduction systems for the relevant logics R, E, and T. This work was extended by Brady to cover a range of relevant logics. In this paper I present indexed tree natural deduction systems for the Anderson–Belnap–Brady systems and show how to translate proofs in one format into proofs in the other, which establishes the adequacy of the tree systems.

The essential structure of proofs is proposed as the basis for a measure of complexity of formulas in FOL. The motivating idea was the recognition that distinct theorems can have the same derivation modulo some non essential details. Hence the difficulty in proving them is identical and so their complexity should be the same. We propose a notion of complexity of formulas capturing this property. With this purpose, we introduce the notions of schema calculus, schema derivation and description complexity of (...) a schema formula. Based on these concepts we prove general robustness results that relate the complexity of introducing a logical constructor with the complexity of the component schema formulas as well as the complexity of a schema formula across different schema calculi. (shrink)

We begin with the idea that lines of reasoning are continuous mental processes and develop a notion of continuity in proof. This requires abstracting the notion of a proof as a set of sentences ordered by provability. We can then distinguish between discrete steps of a proof and possibly continuous stages, defining indexing functions to pick these out. Proof stages can be associated with the application of continuously variable rules, connecting continuity in lines of reasoning with continuously variable reasons. Some (...) examples of continuous proofs are provided. We conclude by presenting some fundamental facts about continuous proofs, analogous to continuous structural rules and composition. We take this to be a development on its own, as well as lending support to non-finitistic constructionism. (shrink)

This paper presents a natural deduction system for orthomodular quantum logic. The system is shown to be provably equivalent to Nishimura’s quantum sequent calculus. Through the Curry–Howard isomorphism, quantum $$\lambda $$ -calculus is also introduced for which strong normalization property is established.