## 2006: Volume 4

*Inexpressiveness of First-Order Fragments*

William C. Purdy12 pages. Published January 25, 2006

It is well-known that first-order logic is semi-decidable. Therefore, first-order logic is less than ideal for computational purposes (computer science, knowledge engineering). Certain fragments of first-order logic are of interest because they are decidable. But decidability is gained at the cost of expressiveness. The objective of this paper is to investigate inexpressiveness of fragments that have received much attention.

*Some non-standard interpretations of the axiomatic basis of Lesniewski’s Ontology*

Rafal Urbaniak34 pages. Published January 25, 2006

We propose an intuitive understanding of the statement: ‘an axiom (or: an axiomatic basis) determines the meaning of the only specific constant occurring in it.’ We introduce some basic semantics for functors of the category

*s*/*n*,*n*of Lesniewski’s Ontology. Using these results we prove that the popular claim that the axioms of Ontology determine the meaning of the primitive constants is false.*Natural Derivations for Priest,*An Introduction to Non-Classical Logic

Tony Roy146 pages. Published March 20, 2006

This document collects natural derivation systems for logics described in Priest,

*An Introduction to Non-Classical Logic*. It provides an alternative or supplement to the semantic tableaux of his text. Except that some chapters are collapsed, there are sections for each chapter in Priest, with an additional, final section on quantified modal logic. In each case, (i) the language is briefly described and key semantic definitions stated, (ii) the derivation system is presented with a few examples given, and (iii) soundness and completeness are proved. There should be enough detail to make the parts accessible to students would work through parallel sections of*An Introduction to Non-Classical Logic*.*Logic of Violations: A Gentzen System for Reasoning with Contrary-To-Duty Obligations*

Guido Governatori and Antonio Rotolo23 pages. Published March 30, 2006

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.

*An Analysis of Inconsistent and Incomplete Necker Cubes*

Chris Mortensen10 pages. Published March 30, 2006

This paper aims to distinguish and classify sixteen versions of the Necker cube. In particular, it is shown how to describe inconsistent and incomplete theories which correspond in a systematic way to these sixteen diagrams. Concerning two of these sixteen cubes, there is a natural intuition that there is a sense in which they inconsistent. It is seen that this intuition is vindicated by an analysis in which their corresponding theories turn out to be globally inconsistent but not locally inconsistent, while various other cubes of the sixteen are merely locally inconsistent. The Routley functor is seen to be useful in classifying the relations between these diagrams.

*Expressive Three-valued Truth Functions*

Stephen Pollard18 pages. Published May 29, 2006

The expressive truth functions of two-valued logic have all been characterized, as have the expressive unary truth functions of finitely-many-valued logic. This paper introduces some techniques for identifying expressive functions in three-valued logics.

Copyright © 2006, *Philosophy Department, University of Melbourne*.

Individual papers are copyright their authors.