We present our calculus of higher-level rules, extended with propositional quantification within rules. This makes it possible to present general schemas for introduction and elimination rules for arbitrary propositional operators and to define what it means that introductions and eliminations are in harmony with each other. This definition does not presuppose any logical system, but is formulated in terms of rules themselves. We therefore speak of a foundational account of proof-theoretic harmony. With every set of introduction rules a canonical elimination (...) rule, and with every set of elimination rules a canonical introduction rule is associated in such a way that the canonical rule is in harmony with the set of rules it is associated with. An example given by Hazen and Pelletier is used to demonstrate that there are significant connectives, which are characterized by their elimination rules, and whose introduction rule is the canonical introduction rule associated with these elimination rules. Due to the availabiliy of higher-level rules and propositional quantification, the means of expression of the framework developed are sufficient to ensure that the construction of canonical elimination or introduction rules is always possible and does not lead out of this framework. (shrink)

In their Basic Logic, Sambin, Battilotti and Faggian give a foundation of logical inference rules by reference to certain reflection principles. We investigate the relationship between these principles and the principle of Definitional Reflection proposed by Hallnäs and Schroeder-Heister.

In this paper I will show the problems that are encountered when dealing with uniqueness of connectives in a bilateralist setting within the larger framework of proof-theoretic semantics and suggest a solution. Therefore, the logic 2Int is suitable, for which I introduce a sequent calculus system, displaying - just like the corresponding natural deduction system - a consequence relation for provability as well as one dual to provability. I will propose a modified characterization of uniqueness incorporating such a duality of (...) consequence relations, with which we can maintain uniqueness in a bilateralist setting. (shrink)

Modern belief revision theory is based to a large extent on partial meet contraction that was introduced in the seminal article by Carlos Alchourrón, Peter Gärdenfors, and David Makinson that appeared in 1985. In the same year, Alchourrón and Makinson published a significantly different approach to the same problem, called safe contraction. Since then, safe contraction has received much less attention than partial meet contraction. The present paper summarizes the current state of knowledge on safe contraction, provides some new results (...) and offers a list of unsolved problems that are in need of investigation. (shrink)

This collection of papers, celebrating the contributions of Swedish logician Dag Prawitz to Proof Theory, has been assembled from those presented at the Natural Deduction conference organized in Rio de Janeiro to honour his seminal research. Dag Prawitz’s work forms the basis of intuitionistic type theory and his inversion principle constitutes the foundation of most modern accounts of proof-theoretic semantics in Logic, Linguistics and Theoretical Computer Science. The range of contributions includes material on the extension of natural deduction with higher-order (...) rules, as opposed to higher-order connectives, and a paper discussing the application of natural deduction rules to dealing with equality in predicate calculus. The volume continues with a key chapter summarizing work on the extension of the Curry-Howard isomorphism, via methods of category theory that have been successfully applied to linear logic, as well as many other contributions from highly regarded authorities. With an illustrious group of contributors addressing a wealth of topics and applications, this volume is a valuable addition to the libraries of academics in the multiple disciplines whose development has been given added scope by the methodologies supplied by natural deduction. The volume is representative of the rich and varied directions that Prawitz work has inspired in the area of natural deduction. (shrink)

A comparison is given between two conditions used to define logical constants: Belnap's uniqueness and Hacking's deducibility of identicals. It is shown that, in spite of some surface similarities, there is a deep difference between them. On the one hand, deducibility of identicals turns out to be a weaker and less demanding condition than uniqueness. On the other hand, deducibility of identicals is shown to be more faithful to the inferentialist perspective, permitting definition of genuinely proof-theoretical concepts. This kind of (...) analysis is driven by exploiting the Curry–Howard correspondence. In particular, deducibility of identicals is shown to correspond to the computational property of eta expansion, which is essential in the characterization of propositional identity. (shrink)