A Paraconsistent Higher Order Logic

Classical logic predicts that everything (thus nothing useful at all) follows from inconsistency. A paraconsistent logic is a logic where an inconsistency does not lead to such an explosion, and since in practice consistency is difficult to achieve there are many potential applications of paraconsistent logics in knowledge-based systems, logical semantics of natural language, etc. Higher order logics have the advantages of being expressive and with several automated theorem provers available. Also the type system can be helpful. We present a concise description of a paraconsistent higher order logic with countable infinite indeterminacy, where each basic formula can get its own indeterminate truth value (or as we prefer: truth code). The meaning of the logical operators is new and rather different from traditional many-valued logics as well as from logics based on bilattices. The adequacy of the logic is examined by a case study in the domain of medicine. Thus we try to build a bridge between the HOL and MVL communities. A sequent calculus is proposed based on recent work by Muskens.Comment: Originally in the proceedings of PCL 2002, editors Hendrik Decker, Joergen Villadsen, Toshiharu Waragai (http://floc02.diku.dk/PCL/). Correcte

Multi-dimensional Type Theory: Rules, Categories, and Combinators for Syntax and Semantics

We investigate the possibility of modelling the syntax and semantics of natural language by constraints, or rules, imposed by the multi-dimensional type theory Nabla. The only multiplicity we explicitly consider is two, namely one dimension for the syntax and one dimension for the semantics, but the general perspective is important. For example, issues of pragmatics could be handled as additional dimensions. One of the main problems addressed is the rather complicated repertoire of operations that exists besides the notion of categories in traditional Montague grammar. For the syntax we use a categorial grammar along the lines of Lambek. For the semantics we use so-called lexical and logical combinators inspired by work in natural logic. Nabla provides a concise interpretation and a sequent calculus as the basis for implementations.Comment: 20 page

New Formalized Results on the Meta-Theory of a Paraconsistent Logic

Classical logics are explosive, meaning that everything follows from a contradiction. Paraconsistent logics are logics that are not explosive. This paper presents the meta-theory of a paraconsistent infinite-valued logic, in particular new results showing that while the question of validity for a given formula can be reduced to a consideration of only finitely many truth values, this does not mean that the logic collapses to a finite-valued logic. All definitions and theorems are formalized in the Isabelle/HOL proof assistant

Recent Work in Relevant Logic

This paper surveys important work done in relevant logic in the past 10 years

Handbook of the First World Congress on Logic and Religion

This is the handbook of abstracts of the 1st World Congress on Logic and Religion, which took place in João Pessoa, Brazil, April 01-05, 2015

Time-Situated Metacognitive Agency and Other Aspects of Commonsense Reasoning

Much research in commonsense reasoning (CSR) involves use of external representations of an agent's reasoning, based on compelling features of classical logic. However, these advantages come with severe costs, including: omniscience, consistency, static semantics, frozen deadlines, lack of self-knowledge, and lack of expressive power to represent the reasoning of others. Active logic was developed to address many of these, but work to date still leaves serious gaps. The present work focuses on major extensions of active logic to deal with self-knowledge, and their implementation into a newly-developed automated reasoner for commonsense active logic. Dealing with self-knowledge has been designed and implemented in the reasoner via a new treatment of quotation as a form of nesting. More sophisticated varieties of nesting, particularly quasi-quotation mechanisms, have also been developed to extend the basic form of quotation. Active logic and the reasoner are applied to classical issues in CSR, including a treatment of one agent having the knowledge and inferential mechanisms to reason about another's time-situated reasoning