Aspects of Inductive Logic

We introduce Propositional Typicality Logic (PTL), a logic for reasoning about typicality. We do so by enriching classical propositional logic with a typicality operator of which the intuition is to capture the most typical (or normal) situations in which a given formula holds. The semantics is in terms of ranked models as studied in KLM-style preferential reasoning. This allows us to show that KLM-style rational consequence relations can be embedded in our logic. Moreover we show that we can define consequence relations on the language of PTL itself, thereby moving beyond the propositional setting. Building on the existing link between propositional rational consequence and belief revision, we show that the same correspondence holds in the case of rational consequence and belief revision defined on the language of PTL. Finally we also investigate different notions of entailment for PTL and propose two appropriate candidates. The present text is an extended version of a paper which appeared in the Proceedings of the 13th European Conference on Logics in Artificial Intelligence [Booth et al., 2012].

A propositional typicality logic for extending rational consequence

Book Information The Blackwell Guide to Philosophical Logic. Edited by Lou Goble. Blackwell Publishers. Oxford. 2001. Pp. x + 510. Paperback, £16.99.

The Blackwell Guide to Philosophical Logic

The goal of this article is to design a uniform proof-theoretical framework encompassing classical, non-monotonic and paraconsistent logic. This framework is obtained by the control sets logical device, a syntactical apparatus for controlling derivations. A basic feature of control sets is that of leaving the underlying syntax of a proof system unchanged, while affecting the very combinatorial structure of sequents and proofs. We prove the cut-elimination theorem for a version of controlled propositional classical logic, i.e. the sequent calculus for classical propositional logic to which a suitable system of control sets is applied. Finally, we outline the skeleton of a new (positive) account of non-monotonicity and paraconsistency in terms of concurrent processes.

Unifying logics via context-sensitiveness

We introduce and explore the notion of duality for entailment relations induced by preference orderings on states. We discuss the relationship between these preferential entailment relations from the perspectives of Boolean algebra, inference rules, and modal axiomatisation. Interpreting the preference relations as accessibility relations establishes modular Godel-Lob logic as a suitable modal framework for rational preferential reasoning.

Semantics for Dual Preferential Entailment

Contents Purpose 1.Introduction 1.1. We are all nonmonotonic 1.2. Recalling some features of classical consequence 1.3. Some misunderstandings and a habit to suspend 1.4. Three ways of getting more conclusions out of your premises 2. Using background assumptions 2.1. From classical consequence to pivotal assumptions 2.2. From pivotal assumptions to default assumptions 2.3. Specializations and generalizations 3. Restricting the set of valuations 3.1. From classical consequence to pivotal valuations 3.2. From pivotal valuations to default valuations 3.3. Specializations and generalizations 4. Using additional rules 4.1. From classical consequence to pivotal rules 4.2. From pivotal rules to default rules 4.3. Generalizations and variants 5. Connections with probabilistic inference 5.1. Basic concepts and axioms for probability 5.2. Probabilistic characterizations of classical consequence 5.3. Supraclassical probabilistic consequence relations 5.4. Bringing probabilistic and qualitative inference closer together Appendix: Proof of Theorem 4.3-1. Acknowledgements References 2 Purpose This overview is directed to those who have heard of nonmonotonic reasoning and would like to get a better idea of what it is all about. What are its driving ideas? In what ways does it differ from classical logic? How does it relate to probability? More generally, how does it behave and how can it be used? We will try to answer these questions as clearly as possible, without undue technicality on the one hand, nor vagueness or hand waving on the other. For the outsider, nonmonotonic logic can be a rather mysterious affair. Even from the inside, it can appear to lack unity, with multiple systems proposed by as many authors going in different directions. The few available textbooks tend to perpetuate this impression. Our main purpose is to take some of the mystery out of the subject and show that it is not as unfamiliar as may at first sight seem. In fact, it is easily accessible to anybody with a minimal background in classical propositional logic and a few basic mathematical tools-provided certain misunderstandings and a tenacious habit, which we will signal in the first sections, are avoided and put aside. As we shall show, there are monotonic logics that act as natural bridges between classical consequence and the principal kinds of nonmonotonic logic to be found in the literature. These logics, which we call paraclassical, are very simple to define and easy to study. They provide three main ways of getting more out of your premises than is allowed by strict deduction, that …

How to Go Nonmonotonic

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How to Go Nonmonotonic