Logic of Violations: A Gentzen System for Reasoningwith Contrary-To-Duty Obligations

In this paper we present a Gentzen system for reasoning with contrary-to-duty obligations. The intuition behind the system is that a contrary-to-duty is a special kind of normative exception. The logical machinery to formalise this idea is taken from substructural logics and it is based on the definition of a new non-classical connective capturing the notion of reparational obligation. Then the system is tested against well-known contrary-to-duty paradoxes.

Simple Gentzenizations for the normal formulae of contraction-less logics

In [1], we established Gentzenizations for a good range of relevant logics with distribution, but, in the process, we added inversion rules, which involved extra structural connectives, and also added the sentential constant t. Instead of eliminating them, we used conservative extension results to relate them back to the original logics. In [4], we eliminated the inversion rules and t and established a much simpler Gentzenization for the weak sentential relevant logic DW, and also for its quantificational extension DWQ, but a restriction to normal formulae (defined below) was required to enable these results to be proved. This method was quite general and hope was expressed about extending it to other relevant logics.In this paper, we develop an innovative method, which makes essential use of this restriction to normality, to establish two simple Gentzenizations for the normal formulae of the slightly weaker logic B, and then extend the method to other sentential contraction-less logics. To obtain the first of these Gentzenizations, for the logics B and DW, we remove the two branching rules (F&) and (T∨), together with the structural connective ‘,’, to simplify the elimination of the inversion rules and t. We then eliminate the rules (T&) and (F∨), thus reducing the Gentzen system to one containing only ˜ and → and their four associated rules, and reduce the remaining types of structures to four simple finite types. Subsequently, we re-introduce (T&) and (F∨), and also (F&) and (T∨), to obtain the second Gentzenization, which contains ‘,’ but no structural rules.

On the Algebraization of Some Gentzen Systems1

In this paper we study the algebraization of two Gentzen systems, both of them generating the implication-less fragment of the intuitionistic propositional calculus. We prove that they are algebraizable, the variety of pseudocomplemented distributive lattices being an equivalent algebraic semantics for them, in the sense that their Gentzen deduction and the equational deduction over this variety are interpretable in one another, these interpretations being essentially inverse to one another. As a consequence, the consistent deductive systems that satisfy the properties of Conjunction, Disjunction and Pseudo-Reductio ad Absurdum are described by giving apropriate Gentzen systems for them. All these Gentzen systems are algebraizable, the subvarieties of the variety of pseudocomplemented distributive lattices being their equivalent algebraic semantics respectively. Finally we give a Gentzen system for the conjunction and disjunction fragment of the classical propositional calculus, prove that the variety of distributive lattices is an equivalent algebraic semantics for it and give a Gentzen system, weaker than the latter, the variety of lattices being an equivalent algebraic semantics for it.