Showing papers on "Paraconsistent logic published in 2008"

Constructive Negations and Paraconsistency

Abstract: Reductio ad Absurdum.- Minimal Logic. Preliminary Remarks.- Logic of Classical Refutability.- The Class of Extensions of Minimal Logic.- Adequate Algebraic Semantics for Extensions of Minimal Logic.- Negatively Equivalent Logics.- Absurdity as Unary Operator.- Strong Negation.- Semantical Study of Paraconsistent Nelson's Logic.- N4?-Lattices.- The Class of N4?-Extensions.- Conclusion.

Logical Weak Completions of Paraconsistent Logics

Abstract: Let P be an arbitrary theory and let X be any given logic. Let M be a set of atoms. We say that M is a X-stable model of P if M is a classical model of P and P∪¬M~ proves in logic X all atoms in M, this is denoted by P∪¬M~ ⊩xM. We prove that being an X-stable model is an invariant property for disjunctive programmes under a large class of logics. Two kinds of logics are mainly considered: paraconsistent logics and normal modal logics. For modal logics we use a translation proposed by Gelfond that replaces ¬a with ¬□a. As a consequence we prove that several semantics (recently introduced) for non-monotonic reasoning are equivalent for disjunctive programmes. In addition, we show that such semantics can be characterized by a fixed-point operator in terms of classical logic. We also present a simple translation of a disjunctive programme D into a normal programme N, such that the PStable model semantics of N corresponds to the stable semantics of D over the common language. We present the formal proof of this statement.

Building a Knowledge Base System for an Integration of Logic Programming and Classical Logic

Abstract: This paper presents a Knowledge Base project for FO(ID), an extension of classical logic with inductive definitions. This logic is a natural integration of classical logic and logic programming based on the view of a logic program as a definition. We discuss the relationship between inductive definitions and common sense reasoning and the strong similarities and striking differences with ASP and Abductive LP. We report on inference systems that combine state-of-the-art techniques of SAT and ASP. Experiments show that FO(ID) model expansion systems are competitive with the best ASP-solvers.

Separation Logic Contracts for a Java-Like Language with Fork/Join

Abstract: We adapt a variant of permission-accounting separation logic to a concurrent Java-like language with fork/join. To support both concurrent reads and information hiding, we combine fractional permissions with abstract predicates. As an example, we present a separation logic contract for iterators that prevents data races and concurrent modifications. Our program logic is presented in an algorithmic style: we avoid structural rules for Hoare triples and formalize logical reasoning about typed heaps by natural deduction rules and a set of sound axioms. We show that verified programs satisfy the following properties: data race freedom, absence of null-dereferences and partial correctness.

Paraconsistent Reasoning for Expressive and Tractable Description Logics

Abstract: Four-valued description logic has been proposed to reason with description logic based inconsistent knowledge bases, mainly ALC This approach has a distinct advantage that it can be implemented by invoking classical reasoners to keep the same complexity as classical semantics In this paper, we further study how to extend the four-valued semantics to more expressive description logics, such as SHIQ, and to more tractable description logics including EL++, DL-Lite, and Horn-DLs The most effort we spend defining the four-valued semantics of expressive four-valued description logics is on keeping the reduction from four-valued semantics to classical semantics as in the case of ALC; While for tractable description logics, we mainly focus on how to maintain their tractability when adopting four-valued semantics

Paraconsistent vagueness: Why not?

Abstract: The idea that the phenomenon of vagueness might be modelled by a paraconsistent logic has been little discussed in contemporary work on vagueness, just as the idea that paraconsistent logics might be fruitfully applied to the phenomenon of vagueness has been little discussed in contemporary work on paraconsistency. This is prima facie surprising given that the earliest formalisations of paraconsistent logics presented in Jaskowski and Hallden were presented as logics of vagueness. One possible explanation for this is that, despite initial advocacy by pioneers of paraconsistency, the prospects for a paraconsistent account of vagueness are so poor as to warrant little further consideration. In this paper we look at the reasons that might be offered in defence of this negative claim. As we shall show, they are far from compelling. Paraconsistent accounts of vagueness deserve further attention.

Rough Sets and 3-Valued Logics

Abstract: In the paper we explore the idea of describing Pawlak’s rough sets using three-valued logic, whereby the value t corresponds to the positive region of a set, the value f — to the negative region, and the undefined value u — to the border of the set. Due to the properties of the above regions in rough set theory, the semantics of the logic is described using a non-deterministic matrix (Nmatrix). With the strong semantics, where only the value t is treated as designated, the above logic is a “common denominator” for Kleene and Łukasiewicz 3-valued logics, which represent its two different “determinizations”. In turn, the weak semantics—where both t and u are treated as designated—represents such a “common denominator” for two major 3-valued paraconsistent logics.

Possible-translations semantics for some weak classically-based paraconsistent logics

Abstract: In many real-life applications of logic it is useful to interpret a particular sentence as true together with its negation. If we are talking about classical logic, this situation would force all o...

Paraconsistent Logic Programs with Four-Valued Rough Sets

Abstract: This paper presents a language for defining four-valued rough sets and to reason about them Our framework brings together two major fields: rough sets and paraconsistent logic programming On the one hand it provides a paraconsistent approach, based on four-valued rough sets, for integrating knowledge from different sources and reasoning in the presence of inconsistencies On the other hand, it also caters for a specific type of uncertainty that originates from the fact that an agent may perceive different objects of the universe as being indiscernible This paper extends the ideas presented in [9] Our language allows the user to define similarity relations and use the approximations induced by them in the definition of other four-valued sets A positive aspect is that it allows users to tune the level of uncertainty or the source of uncertainty that best suits applications

Merging observation and access in dynamic epistemic logic

Abstract: Logical systems have long been used to describe mathematical proof, structured computation, and linguistic meanings. In recent years, they are also coming to be used increasingly to study rational agency in its many aspects, from picking up single pieces of information to multi-agent actions of communication and goal-driven interaction generally. In particular, information flow through events of observation and communication has been studied using so-called dynamic epistemic logics of knowledge update, belief revision, and preference change. These logics use the'semantic sense'of information as ranges of possible options which get updated as new information comes in. But equally importantly, rational agents also base their actions on information from other sources, such as inference and introspection. The latter is the area of more syntax-oriented senses of logical information as something which can be'elucidated'by agents. Perhaps surprisingly, there is much less of a consensus on what this information is, and what its key mechanisms consist in, though there are many competing proposals in the logical literature. Thus, logic is really a field with many different senses of'information'. In this paper, we try to get clear on these issues by presenting one unified model of information, based on semantic ranges of possible worlds, but endowed with syntactic'access' to these worlds. This allows us to integrate external steps of'updating information'and internal steps of'elucidating information'into one system of dynamic logic. In particular, we propose two kinds of basic informational action: pure observation-based update('bare seeing') versus acts of'conscious realization'which turn implicit knowledge into explicit knowledge We show how this is a natural fit, which also provides many new research questions-many of them having to do with fitting further traditions into this picture of information-driven rational agency: including belief revision theory, situation semantics, and paraconsistent logics.

Power systems outage possibilities analysis by Paraconsistent Logic

Abstract: Para-consistent Logic is a non-classic logic whose foundations allow for the contradictions treatment without invalidating the conclusions. Several researches have been demonstrating that through algorithms named PANs - Paraconsistent Analysis Nodes is possible to establish equations determined through signals representing restrictions, risks and configurations of electric power systems networks. In this paper we presented a model with this logic where each electric power grid component is modeled to create a Net of Analysis capable to offer a risk degree for analysis and diagnosis. In that way the model offers a diagnosis and the optimal restorative strategy proposition to the electrical power after an outage. The Paraconsistent logic type used for the models is the Annotated Paraconsistent Logic (APL). The APL utilization brings certain advantages in comparison with the classic logic because allow to manipulate contradictory signals, and like this presenting a faster and reliable action to decision in some situations where the information can be vague, ambiguous or inconsistent. This method is being validated through off-line tests applied at the Eletropaulo- Eletricidade de Sao Paulo, subtransmission power system. Eletropaulo is an electric power distribution utility of Sao Paulo state, in Brazil. The results demonstrate that the Paraconsistent Logic, with their algorithms and the PANs - Para-consistent Analysis Nodes, opens a wide field for researches and developments and can be used with promising results for implementations of applied Expert Systems in analysis and electric power system re-establishment after outages to several topologic possibilities.

Formalizing common sense for scalable inconsistency-robust information integration using Direct Logic(TM) reasoning and the Actor Model

Abstract: Because contemporary large software systems are pervasively inconsistent, it is not safe to reason about them using classical logic. The goal of Direct Logic is to be a minimal fix to classical mathematical logic that meets the requirements of large-scale Internet applications (including sense making for natural language) by addressing the following issues: inconsistency robustness, contrapositive inference bug, and direct argumentation. Direct Logic makes the following contributions over previous work: * Direct Inference (no contrapositive bug for inference) * Direct Argumentation (inference directly expressed) * Inconsistency-robust deduction without artifices such as indices (labels) on propositions or restrictions on reiteration * Intuitive inferences hold including the following: * Boolean Equivalences * Reasoning by splitting for disjunctive cases * Soundness * Inconsistency-robust Proof by Contradiction Since the global state model of computation (first formalized by Turing) is inadequate to the needs of modern large-scale Internet applications the Actor Model was developed to meet this need. Using, the Actor Model, this paper proves that Logic Programming is not computationally universal in that there are computations that cannot be implemented using logical inference. Consequently the Logic Programming paradigm is strictly less general than the Procedural Embedding of Knowledge paradigm.

What is the Logic of Inference

Abstract: The topic of this paper is the question whether there is a logic which could be justly called the logic of inference It may seem that at least since Prawitz, Dummett and others demonstrated the proof-theoretical prominency of intuitionistic logic, the forthcoming answer is that it is this logic that is the obvious choice for the accolade Though there is little doubt that this choice is correct (provided that inference is construed as inherently single-conclusion and complying with the Gentzenian structural rules), I do not think that the usual justification of it is satisfactory Therefore, I will first try to clarify what exactly is meant by the question, and then sketch a conceptual framework in which it can be reasonably handled I will introduce the concept of ‘inferentially native’ logical operators (those which explicate inferential properties) and I will show that the axiomatization of these operators leads to the axiomatic system of intuitionistic logic Finally, I will discuss what modifications of this answer enter the picture when more general notions of inference are considered

Applied Logic without Psychologism

Abstract: Logic is a celebrated representation language because of its formal generality. But there are two senses in which a logic may be considered general, one that concerns a technical ability to discriminate between different types of individuals, and another that concerns constitutive norms for reasoning as such. This essay embraces the former, permutation-invariance conception of logic and rejects the latter, Fregean conception of logic. The question of how to apply logic under this pure invariantist view is addressed, and a methodology is given. The pure invariantist view is contrasted with logical pluralism, and a methodology for applied logic is demonstrated in remarks on a variety of issues concerning non-monotonic logic and non-monotonic inference, including Charles Morgan’s impossibility results for non-monotonic logic, David Makinson’s normative constraints for non-monotonic inference, and Igor Douven and Timothy Williamson’s proposed formal constraints on rational acceptance.

Logical pluralism meets logical dynamics

Abstract: Introduction Where is logic heading today? There is a general feeling that the discipline is broadening its scope and agenda beyond classical foundational issues, and maybe even a concern that, like Stephen Leacock’s famous horseman, it is ‘riding off madly in all directions’. So, what is the resultant vector? There seem to be two broad answers in circulation today. One is logical pluralism, locating the new scope of logic in charting a wide variety of reasoning styles, often marked by non-classical structural rules of inference. This is the new program that I subscribed to in my work on sub-structural logics around 1990, and it is a powerful movement today. 1 But gradually, I have changed my mind about the crux of what logic should become. I would now say that the main issue is not variety of reasoning styles and notions of consequence, but the variety of informational tasks performed by intelligent interacting agents, of which inference is only one among many, involving observation, memory, questions and answers, dialogue, or general communication. And logical systems should deal with a wide variety of these, making information-carrying events first-class citizens in their set-up. This program of logical dynamics was proposed in van Benthem 1996. The purpose of this brief paper is to contrast and compare the two approaches, drawing freely on some insights from earlier published papers. In particular, I will argue that logical dynamics sets itself the more ambitious diagnostic goal of explaining why substructural phenomena occur, by ‘deconstructing’ them into classical logic plus an explicit account of the relevant informational events. I see this as a still more challenging departure from traditional logic. Diehard mathematicians still feel at ease with logical pluralism since it is all still a ‘science of formal systems’ describing ‘inference’, while to me, inference is just one way of producing information, at best on a par, even for logic itself, with others.

Deconstructing the Laws of Logic

Abstract: I consider reasons for questioning ‘the laws of logic’ (identity, non-contradiction, excluded middle, and negation), and suggest that these laws do not accord with everyday reality. Either they are rhetorical tools rather than absolute truths, or else Plato and his successors were right to think that they identify a reality distinct from the ordinary world of experience, and also from the ultimate source of reality.

Weakening of Intuitionistic Negation for Many-valued Paraconsistent da Costa System

Abstract: In this paper we propose substructural propositional logic obtained by da Costa weakening of the intuitionistic negation. We show that the positive fragment of the da Costa system is distributive lattice logic, and we apply a kind of da Costa weakening of negation, by preserving, differently from da Costa, its fundamental properties: antitonicity, inversion, and additivity for distributive lattices. The other stronger paraconsistent logic with constructive negation is obtained by adding an axiom for multiplicative property of weak negation. After that, we define Kripke-style semantics based on possible worlds and derive from it many-valued semantics based on truth-functional valuations for these two paraconsistent logics. Finally, we demonstrate that this model-theoretic inference system is adequate—sound and complete with respect to the axiomatic da Costa-like systems for these two logics.

On the formulae-as-types correspondence for classical logic

Abstract: The Curry–Howard correspondence states the equivalence between the constructions implicit in intuitionistic logic and those described in the simplytyped lambda-calculus. It is an insight of great importance in theoretical computer science, and is fundamental in modern approaches to constructive type theory. The possibility of a similar formulae-as-types correspondence for classical logic looks to be a seminal development in this area, but whilst promising results have been achieved, there does not appear to be much agreement of what is at stake in claiming that such a correspondence exists. Consequently much work in this area suffers from several weaknesses; in particular the status of the new rules needed to describe the distinctively classical inferences is unclear. We show how to situate the formulae-as-types correspondence within the proof-theoretic account of logical semantics arising from the work of Michael Dummett andDag Prawitz, and demonstrate that the admissibility of Prawitz’s inversion principle, which we argue should be strengthened, is essential to the good behaviour of intuitionistic logic. By regarding the rules which determine the deductive strength of classical logic as structural rules, as opposed to the logical rules associated with specific logical connectives, we extend Prawitz’s inversion principle to classical propositional logic, formulated in a theory of Parigot’s lambda-mu calculus with eta expansions. We then provide a classical analogue of a subsystem of Martin-Lof’s type theory corresponding to Peano Arithmetic and show its soundness, appealing to an extension of Tait’s reducibility method. Our treatment is the first treatment of induction in classical arithmetic that truly falls under the aegis of the formulae-as-types correspondence, as it is the first that is consistent with the intensional reading of propositional equality.

Z Logic and Its Applications

Abstract: We provide an introduction to the specification language Z from a logical perspective. The possibility of presenting Z in this way is a consequence of a number of joint publications on Z logic that Henson and Reeves have co-written since 1997. We provide an informal as well as a formal introduction to Z logic and show how it may be used, and extended, to investigate issues such as equational logic, the logic of preconditions, operation and data refinement, and monotonicity.

A Semantic Analysis of a Logic for Pragmatics with Assertions, Obligations, and Causal Implication

Abstract: One of the aims of a logic for pragmatics is to provide a logical framework that formalizes reasoning about speech acts. In this paper we investigate the semantics of a fragment of the logic for pragmatics proposed by Bellin and Dalla Pozza in "A pragmatic interpretation of substructural logics" (Feferman Festschrift, ASL Lecture Notes in Logic 15, 2002). The logic deals with acts of assertion and acts of obligation, and it incorporates a rule that relates acts of obligation to acts of assertion via a notion of causal implication. As our main result we show that the logic is sound and complete with respect to a class of algebraic, Kripke, and categorical models.

5-valued Non-deterministic Semantics for The Basic Paraconsistent Logic mCi

Abstract: One of the most important paraconsistent logics is the logic mCi, which is one of the two basic logics of formal inconsistency. In this paper we present a 5-valued characteristic nondeterministic matrix for mCi. This provides a quite non-trivial example for the utility and effectiveness of the use of non-deterministic many-valued semantics.

Deontic Relevant Logic as the Logical Basis for Representing and Reasoning about Legal Knowledge in Legal Information Systems

Abstract: To represent and reason about various laws, legal rules, and precedents in legal information systems, we need a right fundamental logic system to provide us with a logical validity criterion of legal reasoning as well as a formal representation language. This position paper discusses why classical mathematical logic, its classical conservative extensions, or its non-classical alternatives are not suitable candidates for the fundamental logic, and shows that deontic relevant logic is a more hopeful candidate for the fundamental logic we need.

Paraconsistent Reasoning with Quasi-classical Semantic in $\mathcal{ALC}$

Abstract: Description logics are a family of knowledge representation formalism which descended from semantic networks. During the past decade, the important reasoning problems such as satisfiability and subsumption have been handled by tableau-like algorithms. Description logics are practical monotonic logics which, though imparting strong and conclusive reasoning mechanisms, lack the flexibility of non-monotonic reasoning mechanisms. In recent years, the study of inconsistency handling in description logics becomes more and more important. Some technologies are being applied to handle inconsistency in description logic. Quasi-classical logic, which allows the derivation of nontrivial classical inferences from inconsistent information, supports many important proof rules such as modus tollens, modus ponens, and disjunctive syllogism. In this paper, we consider the characters of $\mathcal{ALC}$ with Quasi-classical semantics and develop a sound and complete tableau algorithm for paraconsistent reasoning in $\mathcal{ALC}$.

On Constructive Linear-Time Temporal Logic

Abstract: In this paper we study a version of constructive linear-time temporal logic (LTL) with the “next” temporal operator. The logic is originally due to Davies, who has shown that the proof system of the logic corresponds to a type system for binding-time analysis via the Curry-Howard isomorphism. However, he did not investigate the logic itself in detail; he has proved only that the logic augmented with negation and classical reasoning is equivalent to (the “next” fragment of) the standard formulation of classical linear-time temporal logic. We give natural deduction and Kripke semantics for constructive LTL with conjunction and disjunction, and prove soundness and completeness. Distributivity of the “next” operator over disjunction “ (A ∨ B) ⊃ A ∨ B” is rejected from a computational viewpoint. We also give a formalization by sequent calculus and its cut-elimination procedure.

The Basic Constructive Logic for Negation-Consistency

Abstract: In this paper, consistency is understood in the standard way, i.e. as the absence of a contradiction. The basic constructive logic BKc4, which is adequate to this sense of consistency in the ternary relational semantics without a set of designated points, is defined. Then, it is shown how to define a series of logics by extending BKc4 up to minimal intuitionistic logic. All logics defined in this paper are paraconsistent logics.

Decision-making using a Paraconsistent analytic hierarchy process

Abstract: This paper presents the application of paraconsistent logic to the decision-making multi-criteria method AHP - analytic hierarchy process. The model uses pairwise comparison matrixes and multiple specialistspsila evaluations, to calculate the degree of inconsistency of each analyzed alternative. With this tool, decision-makers can acquire greater confidence in decisions based on feeling, subjective parameters or intuitions.

Common sense for concurrency and strong paraconsistency using unstratified inference and reflection

Abstract: This paper develops a strongly paraconsistent formalism called Direct Logic(TM) that incorporates the mathematics of Computer Science and allows unstratified inference and reflection using mathematical induction for almost all of classical logic to be used. Direct Logic allows mutual reflection among the mutually chock full of inconsistencies code, documentation, and use cases of large software systems thereby overcoming the limitations of the traditional Tarskian framework of stratified metatheories. Goedel first formalized and proved that it is not possible to decide all mathematical questions by inference in his first incompleteness theorem. However, the incompleteness theorem (as generalized by Rosser) relies on the assumption of consistency! This paper proves a generalization of the Goedel/Rosser incompleteness theorem: a strongly paraconsistent theory is self-provably incomplete. However, there is a further consequence: Although the semi-classical mathematical fragment of Direct Logic is evidently consistent, since the Goedelian paradoxical proposition is self-provable, every reflective strongly paraconsistent theory in Direct Logic is self-provably inconsistent! This paper also proves that Logic Programming is not computationally universal in that there are concurrent programs for which there is no equivalent in Direct Logic. Consequently the Logic Programming paradigm is strictly less general than the Procedural Embedding of Knowledge paradigm. Thus the paper defines a concurrent programming language ActorScript(TM) that is suitable for expressing massive concurrency in large software systems meta-circularly in terms of itself.