Showing papers in "Journal of Applied Non-Classical Logics in 2008"
TL;DR: A relational possible worlds semantics as well as sound and complete display sequent calculi for the logics under consideration are presented.
Abstract: In this paper, a family of paraconsistent propositional logics with constructive negation, constructive implication, and constructive co-implication is introduced. Although some fragments of these ...
67 citations
TL;DR: It is proved that G'3 can define the same class of functions as Lukasiewicz 3 valued logic, and it is identified some normal forms for this logic.
Abstract: We present a Hilbert-style axiomatization of a recently introduced logic, called G'3 G'3 is based on a 3-valued semantics. We prove a soundness and completeness theorem. The replacement theorem holds in G'3. As it has already been shown in previous work, G'3 can express some non-monotonic semantics. We prove that G'3can define the same class of functions as Lukasiewicz 3 valued logic. Moreover, we identify some normal forms for this logic.
38 citations
TL;DR: This paper develops several extensions of SQEMA where that syntactic condition is replaced by a semantic one, viz. downward monotonicity, and proves correctness for a large class of modal formulae containing an extension of the Sahlqvist formULae, defined by replacing polarity with monotonic.
Abstract: In (Conradie et al., 2006a) we introduced the algorithm SQEMA for computing first-order equivalents and proving canonicity of modal formulae, and thus established a very general correspondence and canonical completeness result. SQEMA is based on transformation rules, the most important of which employs a modal version of a result by Ackermann that enables elimination of an existentially quantified predicate variable in a formula, provided a certain negative polarity condition on that variable is satisfied. In this paper we develop several extensions of SQEMA where that syntactic condition is replaced by a semantic one, viz. downward monotonicity. For the first, and most general, extension SemSQEMA we prove correctness for a large class of modal formulae containing an extension of the Sahlqvist formulae, defined by replacing polarity with monotonicity. By employing a special modal version of Lyndon's monotonicity theorem and imposing additional requirements on the Ackermann rule we obtain restricted versio...
32 citations
TL;DR: In many real-life applications of logic it is useful to interpret a particular sentence as true together with its negation as mentioned in this paper, and this situation would force all of us to assume that the negation is true.
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...
24 citations
TL;DR: For the first time the non-Archimedean valued probability logic is constructed on the base of BL∀∞, which is built as an ω-order extension of the logic BL∄∞.
Abstract: In this paper the non-Archimedean multiple-validity is proposed for basic fuzzy logic BL∀∞ that is built as an ω-order extension of the logic BL∀. Probabilities are defined on the class of fuzzy subsets and, as a result, for the first time the non-Archimedean valued probability logic is constructed on the base of BL∀∞.
23 citations
TL;DR: In the paper, complexity of intuitionistic propositional logic and its natural fragments such as implicative fragment, finite-variable fragments, and some others are considered.
Abstract: In the paper we consider complexity of intuitionistic propositional logic and its natural fragments such as implicative fragment, finite-variable fragments, and some others. Most facts we mention here are known and obtained by logicians from different countries and in different time since 1920s; we present these results together to see the whole picture.
17 citations
TL;DR: This paper introduces apartness algebras and apartness frames intended to be abstract counterparts to the apartness spaces of (Bridges et al., 2003), and proves a discrete duality for them.
Abstract: Apartness spaces were introduced as a constructive counterpart to proximity spaces which, in turn, aimed to model the concept of nearness of sets in a metric or topological environment. In this pap...
15 citations
TL;DR: This paper gives a detailed description of CondLean, a theorem prover for some standard conditional logics, and introduces a goal-directed proof search mechanism, derived from the above mentioned sequent calculi based on the notion of uniform proofs.
Abstract: In this paper we focus on theorem proving for conditional logics. First, we give a detailed description of CondLean, a theorem prover for some standard conditional logics. CondLean is a SICStus Prolog implementation of some labeled sequent calculi for conditional logics recently introduced. It is inspired to the so called “lean” methodology, even if it does not fit this style in a rigorous manner. CondLean also comprises a graphical interface written in Java. Furthermore, we introduce a goal-directed proof search mechanism, derived from the above mentioned sequent calculi based on the notion of uniform proofs. Finally, we describe GOALDUCK, a simple SICStus Prolog implementation of the goal-directed calculus mentioned here above. Both the programs CondLean and GOALDUCK, together with their source code, are available for free download at.
14 citations
TL;DR: In this article, the authors show that a family of compatible functions on commutative residuated lattices is locally affine complete, i.e. neither S1 is definable as a polynomial in the language of S2 nor S2 in that enriched with S1.
Abstract: Let L be a commutative residuated lattice and let f : Lk → L a function. We give a necessary and sufficient condition for f to be compatible with respect to every congruence on L. We use this characterization of compatible functions in order to prove that the variety of commutative residuated lattices is locally affine complete. Then, we find conditions on a not necessarily polynomial function P(x, y) in L that imply that the function x ↦ min{y є L | P(x, y) ⪯ y} is compatible when defined. In particular, Pn(x, y) = yn → x, for natural number n, defines a family, Sn, of compatible functions on some commutative residuated lattices. We show through examples that S1> and S2, defined respectively from P1 and P2, are independent as operations over this variety; i.e. neither S1 is definable as a polynomial in the language of L enriched with S2 nor S2 in that enriched with S1.
12 citations
TL;DR: The autoreferential Kripke-style semantics for this class of modal algebras is based on the set of possible worlds equal to the complete lattice of algebraic truth-values, while all intermediate possible worlds represent the different levels of paraconsistent logics.
Abstract: In this paper we consider the class of truth-functional modal many-valued logics with the complete lattice of truth-values. The conjunction and disjunction logic operators correspond to the meet an...
12 citations
TL;DR: This analysis yields a characterization of invariance and safety under bisimulation as natural conditions for logical operations in modal and dynamic logics, and some new transfer results between first-order logic and modal logic.
Abstract: Consider any logical system, what is its natural repertoire of logical operations? This question has been raised in particular for first-order logic and its extensions with generalized quantifiers, and various characterizations in terms of semantic invariance have been proposed. In this paper, our main concern is with modal and dynamic logics. Drawing on previous work on invariance for first-order operations, we find an abstract connection between the kind of logical operations a system uses and the kind of invariance conditions the system respects. This analysis yields (a) a characterization of invariance and safety under bisimulation as natural conditions for logical operations in modal and dynamic logics, and (b) some new transfer results between first-order logic and modal logic.
TL;DR: In this paper, a clausal logic allowing to handle term-graphs is defined, and a complete (w.r.t validity) calculus for these fragments is proposed.
Abstract: A clausal logic allowing to handle term-graphs is defined. Term-graphs are a generalization of terms (in the usual sense) possibly containing shared subterms and cycles. The satisfiability problem for this logic is shown to be undecidable (not even semi-decidable), but some fragments are identified for which it is semi-decidable. A complete (w.r.t validity) calculus for these fragments is proposed. Some simple examples give a taste of this calculus at work.
TL;DR: It is shown that the serializability problem for FOTLT(n) formulae is decidable.
Abstract: We define the logic FOTLT(n), to be a monadic monodic fragment of first-order linear temporal logic, with 2n propositions representing the read and write steps of n two-step concurrent database transactions and a time-dependent predicate representing queries giving the sets of data items accessed by those read and write steps at given points in time. The models of FOTLT(n) contain interleaved sequences of the steps of infinitely many occurrences of the n transactions accessing unlimited data over time. A property of serializability is specified for FOTLT(n) formulae. We show that the serializability problem for FOTLT(n) formulae is decidable.
TL;DR: It is shown that various non-classical connectives “hidden” in the classical logic in the form of G˛s with ˛ —a classical connective, and s—a propositional variable belong to the 4-valued logic of Lukasiewicz.
Abstract: Objects of consideration are various non-classical connectives “hidden” in the classical logic in the form of G˛s with ˛ —a classical connective, and s—a propositional variable One of them is negation, which is defined as G ⇒ s; another is necessity, which is defined as G ∧ s The new operations are axiomatized and it is shown that they belong to the 4-valued logic of Lukasiewicz A 2-point Kripke semantics is built leading directly to the 4-valued logical tables
TL;DR: It is shown that the extended languages have the same expressive power as the first-order language over the class of all relational structures of equivalence relations in local agreement by providing appropriate translation of formulae.
Abstract: In this paper we investigate several extensions of the first order-language with finitely many binary relations. The most interesting of the studied extensions appears to be the monadic second-order one. We show that the extended languages have the same expressive power as the first-order language over the class of all relational structures of equivalence relations in local agreement by providing appropriate translation of formulae. The decidability of the considered extensions over the above mentioned class of structures is also shown.
TL;DR: The technique proposed subsumes many other results, including the Ackermann's lemma and various forms of fixpoint approaches when the “ordering” relations are interpreted as implication and reveals the common principle behind these approaches.
Abstract: In the paper we present a technique for eliminating quantifiers of arbitrary order, in particular of first-order. Such a uniform treatment of the elimination problem has been problematic up to now, ...
TL;DR: It is proved that although all these logics built up from B+ have the characteristic paradoxes of consistency, they lack the K rule (and so, the K axioms).
Abstract: The logic B+ is Routley and Meyer's basic positive logic. We show how to introduce a minimal intuitionistic negation and an intuitionistic negation in B+. The two types of negation are introduced in a wide spectrum of relevance logics built up from B+. It is proved that although all these logics have the characteristic paradoxes of consistency, they lack the K rule (and so, the K axioms).
TL;DR: A first-order characterization of indiscernibility and complementarity is obtained through a duality result between hyper arrow structures and certain structures of relational type characterized by first- order conditions.
Abstract: In this paper, we study indiscernibility relations and complementarity relations in hyper arrow structures. A first-order characterization of indiscernibility and complementarity is obtained through a duality result between hyper arrow structures and certain structures of relational type characterized by first-order conditions. A modal analysis of indiscernibility and complementarity is performed through a modal logic which modalities correspond to indiscernibility relations and complementarity relations in hyper arrow structures.
TL;DR: The main result provides a decision algorithm for T MAZ (so, it is proved that TMAZ is decidable), which also solves the satisfiability problem and considers the admissibility problem for inference rules in T MAz, and shows that this problem is decailable.
Abstract: The paper deals with a temporal multi-agent logic T MAZ , which imitates taking of decisions based on agents' access to knowledge by their interaction. The interaction is modeled by possible communication channels between agents in special temporal Kripke/Hintikka-like models. The logic T MAZ distinguishes local and global decisions-making. T MAZ is based on temporal Kripke/Hintikka models with agents' accessibility relations defined on states of all possible time clusters C(i) (where indexes i range over all integer numbers Z). The main result provides a decision algorithm for T MAZ (so, we prove that T MAZ is decidable). This algorithm also solves the satisfiability problem. In the final part of the paper, we consider the admissibility problem for inference rules in T MAZ , and show that this problem is decidable for T MAZ as well.
TL;DR: It is argued that, for certain artificial agents employing a modal operator TK, the diachronic Dutch book argument for Bayesian updating is on firm ground.
Abstract: This paper employs epistemic logic to investigate the philosophical foundations of Bayesian updating in belief revision. By Bayesian updating, we understand the tenet that an agent's degrees of belief--assumed to be encoded as a probability distribution--should be revised by conditionalization on the agent's total knowledge up to that time. A familiar argument, based on the construction of a diachronic Dutch book, purports to show that Bayesian updating is the only rational belief-revision policy. We investigate the conditions under which the premises of this argument might be satisfied. Specifically, we consider the case of an artificial agent whose language (of thought) features a modal operator TK, where TK P has the interpretation "My total knowledge is P". We show that every proposition of the form TK P is epistemically categorical: it determines, for every proposition Q in the agent's language, whether the agent knows that Q. We argue that, for certain artificial agents employing such a language, the diachronic Dutch book argument for Bayesian updating is on firm ground.