A functional approach to non-monotonic logic (original) (raw)

A proof theory for Constructive Default Logic

Lecture Notes in Computer Science, 1993

We present what we call Constructive Default Logic (CDL) -a default logic in which the fixedpoint definition of extensions is replaced by a constructive definition which yield so-called constructive extensions. Selection functions are used to represent explicitly the control of the reasoning process in this default logic. It is well-known that Reiter's original default logic lacks, in general, a default proof theory. We will show that CDL does have a default proof theory, and we will also show that this is related to the fact that CDL has the existence property for constructive extensions and that it also has the semi-monotonicity property. Furthermore, we will also show that, with respect to some counter-examples that were suggested by Lukaszewicz, constructive extensions yield more intuitive conclusions than Reiter's extensions. Hence, constructive default logic does not only have heuristic advantages over Reiter's default theory from a computational point of view, but it is also more adequate with respect to knowledge representation.

Monotonic Reasoning from a Proof-Theoretic Perspective

1999

The article presents the first results we have obtained studying natural reasoning from a proof-theoretic perspective. In particular, we focus our attention on monotonicity reasoning: Inferences are made using structurally parsed sentences on which monotonic positions are displayed. The monotonicity markers are propagated through the proofs via the combined structural and logical rules for the unary operators of Multimodal Categorial Grammar (MMCG). We have chosen to work with such an expressive 'grammar logic' in order to avoid both the use of extra-logical marking devices as made in [SV91] and a too complex lexicon . With MMCG as the parser, the system is able to make the derivations fully within the logic.

Base Belief Change for Finitary Monotonic Logics

2009

We slightly improve on characterization results already in the literature for base revision. We show that consistency-based partial meet revision operators can be axiomatized for any sentential logic \(\mathcal{S}\) satisfying finitarity and monotonicity conditions (neither the deduction theorem nor supraclassicality are required to hold in \(\mathcal{S}\) ). A characterization of limiting cases of revision operators, full meet and maxichoice, is also offered. In the second part of the paper, as a particular case, we focus on the class of graded fuzzy logics and distinguish two types of bases, naturally arising in that context, exhibiting different behavior.

Strong and explicit negation in non-monotonic reasoning and logic programming

1996

Logic programs, deductive databases, and more generally non-monotonic theories, use various forms of default negation, not F, whose major distinctive feature is that not F is assumed "by defauIC, ie, it is assumed in the absence of sufficient evidence to the contrary. The meaning of "su~cient evidence" depends on the specific semantics used. For example, in Reiter's Closed World Assumption, CWA [32], not A is assumed for atomic A if A is not provable, or, equivalently, if there is a minimal model in which A is false.

New Logical Perspectives on Monotonicity

2020

Monotonicity-based inference is a fundamental notion in the logical semantics of natural language, and also in logic in general. Start- ing in generalized quantifier theory, we distinguish three senses of the notion, study their relations, and use these to connect monotonicity to logics of model change. At the end we return to natural language and consider monotonicity inference in linguistic settings with vocabulary for various forms of change. While we mostly raise issues in this paper, we do make a number of new observations backing up our distinctions.

A Unified Algebraic Framework for Non-Monotonicity

Electronic Proceedings in Theoretical Computer Science, 2019

Tremendous research effort has been dedicated over the years to thoroughly investigate non-monotonic reasoning. With the abundance of non-monotonic logical formalisms, a unified theory that enables comparing the different approaches is much called for. In this paper, we present an algebraic graded logic we refer to as Log A G capable of encompassing a wide variety of non-monotonic formalisms. We build on Lin and Shoham's argument systems first developed to formalize non-monotonic commonsense reasoning. We show how to encode argument systems as Log A G theories, and prove that Log A G captures the notion of belief spaces in argument systems. Since argument systems capture default logic, autoepistemic logic, the principle of negation as failure, and circumscription, our results show that Log A G captures the before-mentioned non-monotonic logical formalisms as well. Previous results show that Log A G subsumes possibilistic logic and any non-monotonic inference relation satisfying Makinson's rationality postulates. In this way, Log A G provides a powerful unified framework for non-monotonicity.