Validity in Simple Partial Logic

Mathematical Logic Quarterly, 1983

2012

Specifications of programs frequently involve operators and functions that are not defined over all of their (syntactic) domains. Proofs about specifications–and those to discharge proof obligations that arise in justifying steps of design–must be based on formal rules. Since classical logic deals only with defined values, some extra thought is required. There are several ways of handling terms that can fail to denote a value—this paper provides a semantically based comparison of three of the best known approaches.

Science of Computer Programming, 1999

Partiality abounds in specifications and programs. We present a three-valued typed logic for reasoning equationally about programming in the presence of partial functions. The logic in essence is a combination of the equational logic E and typed LPF. Of course, there are already many logics in which some classical theorems acquire the status of neither-true-nor-false. What is distinctive here is that we preserve the equational reasoning style of E, as well as most of its main theorems. The principal losses among the theorems are the law of the excluded middle, the anti-symmetry of implication, a small complication in the trading law for existential quantification, and the requirement to show delinedness when using instantiation. The main loss among proof methods is proof by mutual implication; we present some new proof strategies that make up for this loss. Some proofs are longer than in E, but the heuristics commonly used in the proof methodology of E remain valid. We present a Hilbert-style axiomatisation of the logic in which modus ponens and generalisation are the only inference rules. The axiomatisation is easily modified to yield a classical axiomatisation of E itself. We suggest that the logic may be readily extended to a many-valued logic, and that this will have its uses.

Journal of Philosophical Logic, 1993

Partial logic may be seen as a three-valued logic whose third truth value can be soundly interpreted as "undefined" or "neither true nor false" (see [4]). It is well known that three valued partial logic-or partial information-can be modelled within classical modal logic (see [3]). However, for many purposes it seems more convenient to allow not only partiality, but also incoherence in logic, dealivgg with a four-valued logic whose non-classical truth values should be interpreted as "undefined" and "both true and false", respectively. The interest of this interpretation was pointed out by J. M. Dunn in [5] and [6]. Some more motivations for such a logic can be found in [2], [9], or in Barwise and Perry's Situation Semantics (see especially [I] and [10]). This paper aims to prove that this four-valued partial and incoherent logic can also be reduced to classical modal logic, more precisely, to the modal system $4, and that when attention is restricted to the three-valued case, the reducing modal logic becomes just $4 plus McKinsey's axiom, i.e., the system often known as $4. l.

2010

Abstract The Logic of Partial Functions (LPF) is used to reason about propositions that include terms that can fail to denote values. This paper provides semantics for LPF. A Structural Operational Semantics (SOS) provides an intuitive introduction; this is followed by a denotational semantics where the space of denotations is relations which provide an intuitive model of undefined terms. Finally, we illustrate how the denotational semantics can be used as a basis for proofs about propositions that include terms that can fail to denote.

2011

After a very brief introduction to the general subject of Knowledge Representation and Reasoning with Logic Programs we analyse the syntactic structure of a logic program and how it can influence the semantics. We outline the important properties of a 2-valued semantics for Normal Logic Programs, proceed to define the new Minimal Hypotheses semantics with those properties and explore how it can be used to benefit some knowledge representation and reasoning mechanisms. The main original contributions of this work, whose connections will be detailed in the sequel, are: • The Layering for generic graphs which we then apply to NLPs yielding the Rule Layering and Atom Layering-a generalization of the stratification notion; • The Full shifting transformation of Disjunctive Logic Programs into (highly nonstratified) NLPs; • The Layer Support-a generalization of the classical notion of support; • The Brave Relevance and Brave Cautious Monotony properties of a 2-valued semantics; • The notions of Relevant Partial Knowledge Answer to a Query and Locally Consistent Relevant Partial Knowledge Answer to a Query; • The Layer-Decomposable Semantics family-the family of semantics that reflect the above mentioned Layerings; • The Approved Models argumentation approach to semantics; • The Minimal Hypotheses 2-valued semantics for NLP-a member of the Layer-Decomposable Semantics family rooted on a minimization of positive hypotheses assumption approach; • The definition and implementation of the Answer Completion mechanism in XSB Prolog-an essential component to ensure XSB's WAM full compliance with the Well-Founded Semantics; • The definition of the Inspection Points mechanism for Abductive Logic Programs; ix x • An implementation of the Inspection Points workings within the Abdual system [21] We recommend reading the chapters in this thesis in the sequence they appear. However, if the reader is not interested in all the subjects, or is more keen on some topics rather than others, we provide alternative reading paths as shown below. 1-2-3-4-5-6-7-8-9-12 Definition of the Layer-Decomposable Semantics family and the Minimal Hypotheses semantics (1 and 2 are optional) 3-6-7-8-10-11-12 All main contributions-assumes the reader is familiarized with logic programming topics 3-4-5-10-11-12 Focus on abductive reasoning and applications

Information Processing Letters, 1995

2012

Abstract It has been pointed out by a number of authors that partial terms (ie terms that can fail to denote a value) arise frequently in the specification and development of programs. Furthermore, earlier papers describe and argue for the use of a nonclassical logic (the" Logic of Partial Functions") to facilitate sound and convenient reasoning about such terms. This paper addresses some of the issues that arise in trying to provide (semi-) decision procedures-such as resolution-for such a logic.

Notre Dame Journal of Formal Logic

We develop a bottom-up approach to truth-value semantics for classical logic of partial terms based on equality, and apply it to prove the conservativity of the addition of partial description and selection functions, independently of any strictness assumption.

Journal of Logic and Computation, 2016

The use of hybrid logics allows the description of relational structures, at the same time that allows establishing accessibility relations between states and, furthermore, nominating and making mention to what happens at specific states. However, the information we collect is subject to inconsistencies, namely, the search for different information sources can lead us to pick up contradictions. Nowadays, by having so many means of dissemination available, that happens frequently. The aim of this work is to develop tools capable of dealing with contradictory information that can be described as hybrid logics' formulas. To build models, to compare inconsistency in different databases, and to see the applicability of this method in day-today life are the basis for the development of this dissertation.