Compatibility between modal operators in distributive modal logic

Defeasible logic is a non-monotonic formalism that deals with incomplete and conflicting information. Modal logic deals with necessity and possibility, exhibiting defeasibility; thus, it is possible to combine defeasible logic with modal operators. This paper reports on the extension of the DR-DEVICE defeasible reasoner with modal and deontic logic operators. The aim is a practical defeasible reasoner that will take advantage of the expressiveness of modal logics and the flexibility to define diverse agent types and behaviors.

We use a modified version of E.Beth’s concept of implicit definitions to show that all the usual logical operators as well as the first and second order quantifiers are implicitly defined—and for essentially the same reason that involves an account of the logical operators using a concept of filter conditions. An “inferential” proposal is then suggested for a Gentzen-like account as a necessary condition for the familiar logical operators. We then explore the question of whether our proposal can also be taken as a sufficient condition. To this end, we discuss whether other operators, like a truth operator, the counterfactual conditional, the identity, and the modal operators are also logical operators. The paper closes with a brief discussion of what is called the robustness of the logical operators: What happens to the logical operators when there is a shift from one logical structure to another which extends it, and what happens when there is a shift from one structure to one in which it is homomorphically embedded.

Between modal logic and closure operators for topological spaces as well as for formal languages there exists a strong relation. Modal logic allows to define classes of formal languages in several ways.

In the last few years, two topes-theoretic approaches to modal logic have been developed simultaneously but independently: one started by Reyes [9] and further developed in Lavendhomme, Lucas and Reyes [7], and the other due to Ghilardi and Meloni [2]. At first sight, these approaches appear quite unrelated and the formal systems to which they lead are different. It may come, therefore, as a surprise that both are particular cases of a more general one which we shall describe in detail in this paper. There is a fundamental difference between modal operators in a topos such as 0 (necessity) and other logical operators such as -, (negation): while -, is functorial in the sense that it commutes with pull-backs (and hence it defines a map -,: n n), this is not so for O. Indeed, the only operator 0: n n such that Op P and oT = T is the identity. In Reyes [9], this difficulty is circumvented by considering a topos over a base topos and restricting the domain of application of modal operators to predicates of objects only, but keeping functoriality with respect to maps. Ghilardi and Meloni [2] on the other hand, restrict the domain of application as in Reyes [9] but relax functoriality to lax functoriality for constant maps. This feature of the approach of Ghilardi and Meloni [2] will be kept in our work. Indeed, it has to be kept in any context which aims to generalize their approach. In a sequel to this paper, we shall show that by imposing a restriction on the topos itself, we may define lax modal operators on predicates of all objects of the topos.

We study a number of modal operators over intuitionistic logic defined as compositions of principal modal operator of logic in consideration with intuitionistic negation. We adopt an approach due to K. Dosen and M. Božic (see [1] and [3]), who introduced four systems of intuitionistic modal logics — HK , HK♦, HK ′ and HK♦′ — each dealing with one of the modal operators of necessity, possibility, un-necessity and impossibility, respectively, as a framework for investigating intuitionistic modal logics. By a composition we understand simply a sequence consisting of occurrences of one of four types of modal operators and intuitionistic negation and we mainly focus on basic compositions, that is compositions of the form ¬δ, δ¬ or ¬δ¬, where δ is the principal modal operator of the logic HKδ. It is well known that such compositions over classical logic can be regarded as natural definitions of four types of modal operators above by means of each other. We investigate which basic compositions yield modal operators of the same type over intuitionistic logic as over classical logic, which is a natural question, as there is no dualities between four types of modal operators over intuitionistic logic. It turns out that five basic compositions behave classically in that sense and for those five compositions we obtain theirs respective axiomatizations. Moreover, we show that logic KC is the smallest superintuitionistic logic over which all twelve basic compositions behave classically. We also investigate the so called Heyting-Kripke logicN∗ [2], which can be obtained by adding to intuitionistic non-modal base modal operator ∼, satisfying exactly the properties of negative operators ′ and ♦′ in logics HK ′ and HK♦′, respectively. We study some basic properties of N∗ such as finite model property, decidability and constructive properties. We then axiomatize two compositions, that is ¬ ∼ and ∼∼, in N∗ as necessity operators. We show that the former composition has an infinite axiomatization and the latter one is axiomatized in the form of logic, modal operator in which combines properties of positive operators and ♦ in logics HK and HK♦, respectively.

There are various operators and operations in intuitionistic fuzzy set theory. The roles of these operators and operations are very significant as they show a deeper interconnection between the two ordinary modal logic operators. It can be well noted that modal operators can change intuitionistic fuzzy sets into fuzzy sets easily. Considering all of these, we establish several equalities on intuitionistic fuzzy sets.

In this article, we present a modal logic system that allows representing relationships between sets or classes of individuals defined by a specific property. We introduce two modal operators, [a] and <a>, which are used respectively to express “for all A” and “there exists an A”. Both the syntax and semantics of the system have two levels that avoid the nesting of the modal operator. The semantics is based on a variant of Kripke semantics, where the modal operators are indexed over propositional logic formulas (“pre-formulas” in the paper). Furthermore, we present a set of axioms and rules that govern the system and we prove that the logic is correct and complete with respect to Kripke models. In the final section of the article, we discuss potential future work. We consider the possibility of combining our operator with other modalities, such as necessity or knowledge. Additionally, as an example of the utility of our modal operator, we briefly analyze a conveniently adapted Barcan formula within the framework of our system. In summary, we propose combining our modal operator with other ones as a simpler, more compact, albeit less expressive way to address quantified modal logic.

In this 1992 book, Professor Koslow advances an account of the basic concepts of logic. A central feature of the theory is that it does not require the elements of logic to be based on a formal language. Rather, it uses a general notion of implication as a way of organizing the formal results of various systems of logic in a simple, but insightful way. The study has four parts. In the first two parts the various sources of the general concept of an implication structure and its forms are illustrated and explained. Part 3 defines the various logical operations and systematically explores their properties. A generalized account of extensionality and dual implication is given, and the extensionality of each of the operators, as well as the relation of negation and its dual, are given substantial treatment because of the novel results they yield. Part 4 considers modal operators and studies their interaction with logical operators. By obtaining the usual results without the usual assumptions this new approach allows one to give a very simple account of modal logic minus the excess baggage of possible world semantics.

In this 1992 book, Professor Koslow advances an account of the basic concepts of logic. A central feature of the theory is that it does not require the elements of logic to be based on a formal language. Rather, it uses a general notion of implication as a way of organizing the formal results of various systems of logic in a simple, but insightful way. The study has four parts. In the first two parts the various sources of the general concept of an implication structure and its forms are illustrated and explained. Part 3 defines the various logical operations and systematically explores their properties. A generalized account of extensionality and dual implication is given, and the extensionality of each of the operators, as well as the relation of negation and its dual, are given substantial treatment because of the novel results they yield. Part 4 considers modal operators and studies their interaction with logical operators. By obtaining the usual results without the usual assumptions this new approach allows one to give a very simple account of modal logic minus the excess baggage of possible world semantics.

On a structuralist account of logic, the logical operators, as well as modal operators are defined by the specific ways that they interact with respect to implication. As a consequence, the same logical operator (conjunction, negation etc.) can appear to be very different with a variation in the implication relation of a structure. We illustrate this idea by showing that certain operators that are usually regarded as extra-logical concepts (Tarskian algebraic operations on theories, mereological sum, products and negates of individuals, intuitionistic operations on mathematical problems, epistemic operations on certain belief states) are simply the logical operators that are deployed in different implication structures. That makes certain logical notions more omnipresent than one would think.

A natural deduction system of quantified modal logic (S5) with an actuality operator and quantifiers (ranging, at every world, over domain of actual world) is described and proved to be complete. Its motivation and relation to other systems are discussed. / The language Predicates. One logical predicate: E ! , exists. Individual constants if you want, though for simplicity I'll ignore them (constants thought of as formalizing names or other rigid designators ought to behave like free variables). Individual (free variables): u,υ9... . Individual bound variables: x9y,... (I follow conventions of Thomason [16] here). Truth functional connectives: &, v, D, ~ . Modal operators: D (necessity), 0 (possibility), O (actuality). Ordinary quantifiers: V, 3. Actuality quantifiers: V°, 3°. The usual formation rules (bound variables never occurring free). 2 Semantics A model is a quadruple M = (W,@9D9I} where Wis a set (of worlds), @ E W (@ is the actual world), D is a function assigning to each wEWa (not necessarily nonempty) set as its domain, and /is an interpretation function assigning to each Λ-adic predicate a function assigning to each w G Wa set of ^-tuples drawn from \Jv(ΞWD(υ), with condition that [/(E!)] (w) = D(w). Note that, corresponding to various intuitive readings of predicates of formal language, and to various metaphysical positions, we might want to impose further conditions on interpretation function; these will often validate extensions of logic described below. An assignment for m is a partial function from individual parameters into Received April 4, 1989; revised June 26, 1989 ACTUALITY AND QUANTIFICATION 499 \Jw(EWD(w) (partial function in order to avoid validating OE!u, which is not derivable in system described. If you want motivation, think of some parameters as formalizing names from fiction.) We define in first instance: Truth in a model, at a world, on an assignment. Truth in a model at a world is Truth at that world in that model on every assignment. Truth in a model is Truth at actual world in model. Validity is Truth in every model; validity of an argument is validity of its associated conditional. Base clause of recursion: An atomic formula, F(uu... ,un), where F is an Az-adic predicate, is True(M, w,α) if and only if (i) a(uχ),... ,a(un) are all defined, and (ii) £ [I(F)](w). Recursion clauses: For truth functional compounds: Standard. For modal and actuality operators: ΏA is True(M, w,a) iff A is True(M, w\a) for every w' G W9 <)A is True(M, w,α) iff 4 is True(M, w',α) for at least one w' E W, OA is True(M,w,α) iffA is True(M,@,α). For ordinary quantifiers: VxA (x/u) is True(M, w,a) iff A is True(M, w,β) for every assignment β such that (i) β(v) = a(v) for every parameter v Φ u, (ii) β(u) is defined, and (iii) β(u)ED(w). 3xA (x/u) is True(M, w,α) iff A is True(M, w,β) for at least one assignment β such that (i) β(v) = a(v) for every parameter v Φ u, (ii) β(u) is defined, and (iii) β(u)EΌ(w). For actuality quantifiers: V°xA(x/u) is True(M, w,α) iff A is True(M, w9β) for every assignment β such that (i) β(v) = a(υ) for every parameter v Φ u, (ii) β(u) is defined, and (iii) jS(iι)eD(@). l°xA (x/u) is True(M, w,α) iff A is True(M,w,β) for at least one assignment β such that

In this paper we show some logical presumptions for the contradiction-form to really mean contradiction. We first give an introductory note that the same argument-form of Russell paradox could be interpreted to derive a contradiction (as Russell did) and to derive some positive non-contradictory results (such as Gödel's lemma on incompleteness and Cantor's lemma on cardinality), depending on the context. This surprisingly suggests that a logical argument of a contradiction itself is rather independent of interpreting it as contradiction or non-contradiction. In the main section (Section 2) we investigate further in the hidden logical assumptions underlying a usual derivation of contradiction (such as the last step from the Russell argument to conclude a contradiction). We show the logical form of contradiction does not always mean a contradiction in a deep structure level of logic. We use the linear logical analysis for this claim. Linear logic, in the author's opinion, provides fundamental logical structures of the traditional logics (such as classical logic and intuitionistic logic). The each traditional logical connectives split into two different kinds of connective, corresponding to the fundamental distinction, parallel or choice, of the fundamental level of logic; more precisely the connectives related to parallel-assertings and the connectives related to choice-assertions. We claim that (1) the law of contradiction is indisputable for the parallel-connectives, but (2) the law of contradiction is not justified for the choice-connectives. (In fact, the law of contradiction has the same meaning as other Aristotlean laws (the indentity, the excluded middle) from the view point of the duality principle in linear logic, and the disputability of the law of contradiction is exactly the same as the disputability of the law of excluded middle, in the linear logic level.) Here, although (1) admits the law of contradiction, the meaning of contradiction is quite diferrent and, in the author opinion, more basic than the traditional sense of contradiction. (2) tells us that the disputability of the law of contradiction for the choice-connectives is equivalent to the disputability of the law of excluded middle. However, this disputability is more basic than the traditional logicist-intuitionist issue on the excluded middle, since admitting the traditional law of excluded middle (from the classical or logicist viewpoint) is compatible with this disputability of the excluded middle (and equivalently the law of contradiction) with respect to the choice-connectives of the linear logic. Then, the traditional logics (classical and intuitionistic logics) are perfectly constituted from this fundamental level of logic by the use of reconstructibity or re-presentation operator, (which is the linear-logical modal operators). With the use of modal operator the originally splitted two groups of logical connectives merge into a single group, which makes the traditional logical connectives. (The use of slightly different modalities results in the difference between the traditional classical logic and the traditional intuitionistic logic.) With the use of modal operator, the contradiction-form becomes to get the traditional sense of contradiction. This situation shows that the traditional sense of contradiction presumes re-presentation or reconstruction of the inference-resources, which is now explicit by the use of linear logical modal operator(s), and which also makes possible the denotational or objectivity interpretation of logical language. The merge of the two different aspects (the parallel-connectives and the choice-connectives) into one, by the presence of the modal operators, also eliminates the original conflict (on the indisputability of the law of contradiction in the parallel-connectives side and the disputability of that in the choice-connectives side.)

Chang algebras as algebraic models for Chang's modal logics [1] are defined. The main result of the paper is a representation theorem for these algebras.

Tense operators in fuzzy logic

Defeasible logic is a non-monotonic formalism that deals with incomplete and conflicting information, whereas modal logic deals with the concepts of necessity and possibility. These types of logics play a significant role in the emerging Semantic Web, which enriches the available Web information with meaning, leading to better cooperation between end-users and applications. Defeasible and modal logics, in general, and, particularly, deontic logic provide means for modeling agent communities, where each agent is characterized by its cognitive profile and normative system, as well as policies, which define privacy requirements, access permissions, and individual rights. Toward this direction, this article discusses the extension of DR-DEVICE, a Semantic Web-aware defeasible reasoner, with a mechanism for expressing modal logic operators, while testing the implementation via deontic logic operators, concerned with obligations, permissions, and related concepts. The motivation behind this work is to develop a practical defeasible reasoner for the Semantic Web that takes advantage of the expressive power offered by modal logics, accompanied by the flexibility to define diverse agent behaviours. A further incentive is to study the various motivational notions of deontic logic and discuss the cognitive state of agents, as well as the interactions among them.