Non-dual modal operators as a basis for 4-valued accessibility relations in Hybrid logic

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Abstract

The modal operators usually associated with the notions of possibility and necessity are classically duals. This paper aims to defy that duality in a paraconsistent environment, namely in a Belnapian Hybrid logic where both propositional variables and accessibility relations are four-valued. Hybrid logic, which is an extension of Modal logic, incorporates extra machinery such as nominals – for uniquely naming states – and a satisfaction operator – so that the formula under its scope is evaluated in the state whose name the satisfaction operator indicates.

In classical Hybrid logic the semantics of negation, when it appears before compound formulas, is carried towards subformulas, meaning that eventual inconsistencies can be found at the level of nominals or propositional variables but appear unrelated to the accessibility relations. In this paper we allow inconsistencies in propositional variables and, by breaking the duality between modal operators, inconsistencies at the level of accessibility relations arise. We introduce a sound and complete tableau system and a decision procedure to check if a formula is a consequence of a set of formulas. Tableaux will be used to extract syntactic models for databases, which will then be compared using different inconsistency measures. We conclude with a discussion about bisimulation.

Introduction

The introduction of a four-valued logic in the 70s by Nuel Belnap [3] considered an algebraic structure composed of, as the name indicates, four elements {t,f,b,n}. These elements intuitively represent the notions of “true”, “false”, “both true and false” (from a classical point of view, the same as inconsistent) and “neither true nor false” (or, in a classical interpretation, incomplete). Thus Belnap's logic is not only paraconsistent, as it excludes the Principle of Explosion, but also paracomplete, as it drops also the Principle of the Excluded Middle. Moreover, these four values may be arranged according to two partial orders: the first one, t, reflects the “quality” of the information, whereas the second, k, reflects the “quantity” of information. The bilattice structure is represented in Fig. 1. Four-valued logics have been studied in the context of computer science and artificial intelligence and have been applied in areas such as symbolic model checking [8], semantics of logic programs [11] and inconsistency-tolerant systems.

In computer programs, relational structures provide a formalism to abstractly depict connections between states; a logic able to formalize these concepts is Modal logic. The notion of satisfiability in Modal logic is local, meaning that formulas are evaluated at a state in a structure. Unfortunately, there is no internal mechanism that allows us to focus on a specific state where we would like to evaluate a formula. It is possible to overcome this limitation if we add to Modal logic a new class of propositional variables, called nominals, which are true at exactly one state, and a satisfaction operator @, that acts as a jump operator. We are in the presence of a more powerful system in terms of expressivity, however still decidable and as complex as standard Modal logic K, called (Basic) Hybrid logic [5]. This extension allows us to refer to a specific state and describe what happens there: the formula @iφ holds at a state if and only if φ holds at the state named by i – actually the state of evaluation of the @-formula is not relevant, it either holds everywhere or nowhere; in particular, we are also able to specify equalities and transitions between (named) states.

Inconsistencies are generally thought of as undesirable and many argue that databases should be inconsistency-free; as such, there are tools designed to eradicate contradictions in order to keep systems consistent. Nonetheless this approach fails to use the benefits of paraconsistency and sometimes precious information is lost, as is the case when contradictions are seen as mistakes and one fails to see that their root is a fraudulent operation. Therefore, since contradictory information is everywhere and is actually the norm rather than the exception in the real world, it should be embraced, formalized and used in our favour. One possible use of paraconsistency is that it allows us to compare between different sources and choose the most reliable one based on the information we have in our hands. Observe that this is something that we naturally do in our daily lives: there are situations where we even expect divergences, something as simple as a set of different opinions about a certain subject is an almost guaranteed source of contradictions. Paraconsistent logics are flexible logical systems able to handle heterogeneous and complex data; they accommodate inconsistency in a sensible manner that treats inconsistent information as informative. Four-valued logics are in this category.

The present paper introduces a new four-valued, also known as Belnapian, Hybrid logic where the duality between modal operators is broken. We argue that this is the only way of capturing the real meaning of negation: just because it is not possible that φ, formally represented by ¬φ, it does not mean that the negation of φ is mandatory, represented by ¬φ. We interpret “positive” modal formulas (where negation does not occur immediately before the modal operator) in an almost classical fashion – the subtle difference is the use of positive relations that capture the evidence about the presence of transitions; we interpret “negative” modal formulas (where negation appears directly before the modal operator) in a distinct way and by resorting to negative relations that capture the evidence about the absence of transitions. In particular, @i¬j shall be interpreted as “there is no transition from state i to state j”, whereas @i¬j is interpreted as “all transitions from state i lead to states different from j”. Inconsistencies at the level of the accessibility relation are allowed and correspond to cases when @ij and @i¬j occur. The logic is called double-Belnapian since it assigns one of four (Belnapian) values to both propositional variables and pairs of states (the accessibility problem). We introduce a tableau system for the logic and a tableau-based procedure in order to check if a formula is a consequence of a set of formulas. The tableau construction algorithm terminates and the system is sound and complete. Another section introduces measures of inconsistency for models and databases. Finally, we talk about bisimulation and how a classical extension does not preserve satisfiability, however a slight change in the definition gives us the desired result.

Paraconsistent versions of modal logic where both the accessibility relations and the propositional variables are allowed a four-valued behaviour are not a novelty. The works of Wansing and Odintsov with BKFS logic [17] and Rivieccio and Jung with Modal bilattice logic MBL [18] are some examples of such logics. For a version of many-valued Modal logic check Fitting's work [12].

Even though proposals of paraconsistent Hybrid logics can be found in [6] and more recently in [9], the work on many-valued Hybrid logic MVHL in [16] seems to be the only version where paraconsistency is present at the level of propositional variables and the accessibility relation. The double-Belnapian Hybrid logic DBHL that we introduce in this paper is neither an extension of pre-existing paraconsistent Modal logics with Hybrid logic features, nor can it be captured by MVHL. The first distinguishing point is the fact that in the semantics for disjunction we resort to the classical notion of disjunctive syllogism. This will force a link between a disjunct and its negation since in case they both hold the other disjunct must hold as well in order to make the whole disjunction hold. Notwithstanding, that is not the main characteristic of DBHL. The main novelty here is the fact that modal operators [π] and π are not considered duals. We argue that this approach is the way to capture the meaning of negation when it appears directly before the modal operator and this is how we will obtain inconsistencies at the level of accessibility relations. If the duality was kept, the usual semantics for modal operators would make it so that saying that in a structure it is possible to make a π-transition between the state named by i and a state where p holds, i.e., the structure satisfies the formula @iπp, and that it is not possible to make such transition, i.e. ¬@iπp holds in the structure, results in an explosion created at the level of propositional variable since the latter would be equivalent to @i[π]¬p. It is clear that the focus of negation is not the transition – we want it to be. At this point we would like to mention that DBHL as appears in this paper differs from the also double-Belnapian version in [10] in the semantics for ¬[π]φ. The subtle difference is that, as the reader will have the opportunity to check, in DBHL we resort to the non-satisfiability of φ, whereas in the older version we resorted to the satisfiability of ¬φ. Satisfaction coincides for pure formulas, i.e. formulas not involving propositional variables, but has a clearly distinct behaviour in other cases.

We propose a paraconsistent and paracomplete version of Hybrid logic such that in a structure both @iπj and @i¬πj may hold or not; they will be interpreted as “there is evidence of a π-transition from the state named by i to the state named by j” and “there is evidence of the lack of a π-transition from the state named by i to the state named by j”, respectively. The latter is not compatible with the interpretation of @i[π]¬j which is that “there is evidence that all π-transitions from the state named by i terminate in a state which is not named by j”.

The structures underlying this system will incorporate two valuations in order to deal with contradictions at the level of propositional variables, V+ and V, and will, analogously, consider two families of accessibility relations, (Rπ+)πMod and (Rπ)πMod in order to deal with contradictions at the level of the accessibility relations. The semantics for nominals is the usual: each nominal holds at a unique state.

Section snippets

Double-Belnapian Hybrid Logic, DBHL

Let Lπ=Prop,Nom,Mod be a hybrid (multimodal) similarity type where Prop is a countable set of propositional variables, Nom is a countable set disjoint from Prop and Mod is a countable set of modality labels. We use p,q,r, etc. to refer to the elements in Prop. The elements in Nom are called nominals and we typically write them as i,j,k, etc. Modalities are usually represented by π,π, etc.

Definition 1

The well-formed formulas over Lπ, Form(Lπ), are defined by the following recursive definition:

φ,ψ:=i|p||¬

A tableau system for DBHL

In this section we will introduce a sound, complete and terminating tableau system for DBHL and a decision procedure that checks if a formula is a consequence of a set of formulas, called a database. In order to do it, we consider an extra-logical operator that acts on the satisfaction relation in the following sense: for a multistructure G, a state w and a formula φForm(Lπ),G,wφG,wφ and, analogously,GφGφ.

It is easy to check that Gφ if and only if it is false that wW, G,wφ if

Inconsistency measures

The idea of measuring the amount of inconsistent information in paraconsistent structures has been widely addressed in [13], [14] and [15], where a variety of different measures have been proposed. An inconsistency measure is simply a function that assigns a non-negative real value to sets of formulas. Each inconsistency measure is a strategy for analysing inconsistent information by showing how conflicting a set of formulas is. Some measures are more fine-grained than others, but in general

DB-bisimulation

Bisimulation is the fundamental notion of equivalence between models in Modal logic and extensions to Hybrid logic are not a novelty, [1]. In this section the notion of bisimulation is extended to multistructures. Curiously, the classical construction will not preserve satisfiability of formulas between bisimilar models. However, that will be the case for a posterior definition of DB-bisimulation, which performs a small, yet significant, change in the previous definition.

Definition 20

Let G=(W,(Rπ+)πMod,(Rπ

Conclusion

The paper presents a four-valued Hybrid logic where propositional variables and accessibility relations are paraconsistent and paracomplete. The major novelty about this work is the fact that the duality between modal operators is no longer valid. However, the multistructures with which we work are such that they can be described by a set of atomic formulas, a diagram, just like structures can in standard Hybrid logic. We also introduced a sound and complete tableau system, discussed

Declaration of Competing Interest

The authors certify that they have NO affiliations with or involvement in any organization or entity with any financial interest (such as honoraria: educational grants; participation in speakers' bureaus; membership, employment, consultancies, stock ownership, or other equity interest; and expert testimony or patent-licensing arrangements), or non-financial interest (such as personal or professional relationships, affiliations, knowledge or beliefs) in the subject matter or materials discussed

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