On Deriving Nested Calculi for Intuitionistic Logics from Semantic Systems

This paper shows how to derive nested calculi from labelled calculi for propositional intuitionistic logic and first-order intuitionistic logic with constant domains, thus connecting the general results for labelled calculi with the more refined formalism of nested sequents. The extraction of nested calculi from labelled calculi obtains via considerations pertaining to the elimination of structural rules in labelled derivations. Each aspect of the extraction process is motivated and detailed, showing that each nested calculus inherits favorable proof-theoretic properties from its associated labelled calculus

Decidability of admissibility:On a problem by friedman and its solution by rybakov

Rybakov (1984) proved that the admissible rules of IPC are decidable. We give a proof of the same theorem, using the same core idea, but couched in the many notions that have been developed in the mean time. In particular, we illustrate how the argument can be interpreted as using refinements of the notions of exactness and extendibility

Through and beyond classicality: analyticity, embeddings, infinity

Structural proof theory deals with formal representation of proofs and with the investigation of their properties. This thesis provides an analysis of various non-classical logical systems using proof-theoretic methods. The approach consists in the formulation of analytic calculi for these logics which are then used in order to study their metalogical properties. A specific attention is devoted to studying the connections between classical and non-classical reasoning. In particular, the use of analytic sequent calculi allows one to regain desirable structural properties which are lost in non-classical contexts. In this sense, proof-theoretic versions of embeddings between non-classical logics - both finitary and infinitary - prove to be a useful tool insofar as they build a bridge between different logical regions

Deductive Systems in Traditional and Modern Logic

The book provides a contemporary view on different aspects of the deductive systems in various types of logics including term logics, propositional logics, logics of refutation, non-Fregean logics, higher order logics and arithmetic

Proof-theoretic Semantics for Intuitionistic Multiplicative Linear Logic

This work is the first exploration of proof-theoretic semantics for a substructural logic. It focuses on the base-extension semantics (B-eS) for intuitionistic multiplicative linear logic (IMLL). The starting point is a review of Sandqvist’s B-eS for intuitionistic propositional logic (IPL), for which we propose an alternative treatment of conjunction that takes the form of the generalized elimination rule for the connective. The resulting semantics is shown to be sound and complete. This motivates our main contribution, a B-eS for IMLL , in which the definitions of the logical constants all take the form of their elimination rule and for which soundness and completeness are established

An Analytic Propositional Proof System on Graphs

In this paper we present a proof system that operates on graphs instead of formulas. Starting from the well-known relationship between formulas and cographs, we drop the cograph-conditions and look at arbitrary undirected) graphs. This means that we lose the tree structure of the formulas corresponding to the cographs, and we can no longer use standard proof theoretical methods that depend on that tree structure. In order to overcome this difficulty, we use a modular decomposition of graphs and some techniques from deep inference where inference rules do not rely on the main connective of a formula. For our proof system we show the admissibility of cut and a generalization of the splitting property. Finally, we show that our system is a conservative extension of multiplicative linear logic with mix, and we argue that our graphs form a notion of generalized connective

A Default-Logic Paradigm for Legal Reasoning and Factfinding

Unlike research in linguistics and artificial intelligence, legal research has not used advances in logical theory very effectively. This article uses default logic to develop a paradigm for analyzing all aspects of legal reasoning, including factfinding. The article provides a formal model that integrates legal rules and policies with the evaluation of both expert and non-expert evidence – whether the reasoning occurs in courts or administrative agencies, and whether in domestic, foreign, or international legal systems. This paradigm can standardize the representation of legal reasoning, guide empirical research into the dynamics of such reasoning, and put the representations and research results to immediate use through artificial intelligence software. This new model therefore has the potential to transform legal practice and legal education, as well as legal theory