Simplified Kripke-Style Semantics for Some Normal Modal Logics

We investigate model theoretic characterisations of the expressive power of modal logics in terms of bisimulation invariance. The paradigmatic result of this kind is van Benthem's theorem, which says that a first-order formula is invariant under bisimulation if and only if it is equivalent to a formula of basic modal logic. The present investigation primarily concerns ramifications for specific classes of structures. We study in particular model classes defined through conditions on the underlying frames, with a focus on frame classes that play a major role in modal correspondence theory and often correspond to typical application domains of modal logics. Classical model theoretic arguments do not apply to many of the most interesting classes -for instance, rooted connected frames, well-founded frames, finite rooted connected frames, finite transitive frames, finite equivalence frames - as these are not elementary. Instead we develop and extend the game-based analysis (first-order Ehrenfeucht-Fraisse versus bisimulation games) over such classes and provide bisimulation preserving model constructions within these classes.

Some results concerning the lattice of normal modal logics A(K) (see for instance [B1, B2]) show the extreme complexity of this structure and the consequent impossibility of a complete description of it. On the other hand, many other results which appear in the literature give descriptions of significative parts of A(K) (see [B 3, Ma, Ra]). In our paper we concentrate the attention on the lattice of the extensions of the logics KAltn, originally introduced in [-Se]. We recall that KAIt. is the normal modal propositional logic characterized by the axiom

A new deduction system for deciding validity for the minimal decidable normal modal logic K is presented in this article. Modal logics could be very helpful in modelling dynamic and reactive systems such as bio-inspired systems and process algebras. In fact, recently the Connectionist Modal Logics has been presented, which combines the strengths of modal logics and neural networks. Thus, modal logic K is the basis for these approaches. Soundness, completeness and the fact that the system itself is a decision procedure are proved in this article. The main advantages of this approach are: first, the system is deterministic, i.e. it generates one proof tree for a given formula; second, the system is a validity-checker, hence it generates a proof of a formula (if such exists); and third, the language of deduction and the language of a logic coincide. Some of these advantages are compared with other classical approaches.

We introduce KBE, a modal epistemic logic for reasoning about Knowledge, Belief and Estimation, three attitudes involved in an agent's decision-making process. In our logic, Knowledge and Belief are captured by S4.2, a modal logic holding a distinguished position among the epistemic logics investigated in AI and Philosophy. The Estimation operator of KBE is a kind of generalized 'many' or 'most' quantifier, whose origins go back to the work of J. Burgess and A. Herzig, but its model-theoretic incarnation ('weak filters') has been introduced by K. Schlechta and V. Jauregui. We work with complete weak filters ('weak ultrafilters') as we are interested in situations where an estimation can be always reached. The axiomatization of KBE comprises 'bridge' axioms which reflect the intuitive relationship of 'estimation' to 'knowledge' and 'belief', several introspective properties are shown to hold and it comes out that believing i¾? can be equivalently defined in KBE as 'estimating that i¾? is known', an interesting fact and an indication of the intuitive correctness of the introduced estimation operator. The model theory of KBE comprises a class of frames combining relational Kripke frames with Scott-Montague semantics, in which neighborhoods are collections of 'large' sets of possible worlds. Soundness and completeness is mentioned and a tableaux proof procedure is sketched.

In the logical context, ignorance is traditionally defined recurring to epistemic logic. In particular, ignorance is essentially interpreted as “lack of knowledge”. This received view has - as we point out - some problems, in particular we will highlight how it does not allow to express a type of content-theoretic ignorance, i.e. an ignorance of φ that stems from an unfamiliarity with its meaning. Contrarily to this trend, in this paper, we introduce and investigate a modal logic having a primitive epistemic operator I, modeling ignorance. Our modal logic is essentially constructed on the modal logics based on weak Kleene three-valued logic introduced by Segerberg (Theoria, 33(1):53–71, 1997). Such non-classical propositional basis allows to define a Kripke-style semantics with the following, very intuitive, interpretation: a formula φ is ignored by an agent if φ is neither true nor false in every world accessible to the agent. As a consequence of this choice, we obtain a type of content-theoretic notion of ignorance, which is essentially different from the traditional approach. We dub it severe ignorance. We axiomatize, prove completeness and decidability for the logic of reflexive (three-valued) Kripke frames, which we find the most suitable candidate for our novel proposal and, finally, compare our approach with the most traditional one.

Kripke frames are generalized to finite-chain graded frames. The minimal finite-chain graded modal logic is shown to be sound and complete with respect to the class of all finite-chain graded frames. Finite-chain algebras are defined for giving algebraic semantics for this modal logic. A Jonsson-Tarski style representation theorem for finite-chain algebras is proved. This new kind of multimodal logic differs from both classical normal modal logic and graded modal logic. Some results for extensions of the minimal finite-chain graded modal logic are also obtained.

Decidability of order-based modal logics

Focused proofs are sequent calculus proofs that group inference rules into alternating positive and negative phases. These phases can then be used to define macro-level inference rules Gentzen’s original and tiny introduction and structural rules. We show here that the inference rules of labeled proof systems for modal logics can similarly be described as pairs of such phases within the LKF focused proof system for first-order classical logic. We consider the system $${ G3K} $$ of Negri for the modal logic $$K$$ and define a translation from labeled modal formulas into first-order polarized formulas and show a strict correspondence between derivations in the two systems, i.e., each rule application in $${ G3K} $$ corresponds to a bipole—a pair of a positive and a negative phases—in $${ LKF} $$ . Since geometric axioms (when properly polarized) induce bipoles, this strong correspondence holds for all modal logics whose Kripke frames are characterized by geometric properties. We extend these results to present a focused labeled proof system for this same class of modal logics and show its soundness and completeness. The resulting proof system allows one to define a rich set of normal forms of modal logic proofs.

Coalgebraic semantics of modal logics: An overview

Completeness and Correspondence in the First and Second Order Semantics for Modal Logic

We characterize the modal logics of elementary classes of Kripke frames as precisely those modal logics that are axiomatized by modal axioms synthesized in a certain effective way from sentences of hybrid logic. These are pure positive hybrid sentences with arbitrary existential and relativized universal quantification over nominals. The proof has three steps. The first step is to use the known result that the modal logic of any elementary class of Kripke frames is also the modal logic of the closure of this class under disjoint unions, generated subframes, bounded morphic images, and ultraroots. This latter class can be defined by the first-order sentences of a special syntactic form (called pseudo-equations by Goldblatt) that are valid in the former class. The second step is to translate these pseudo-equations into equivalent quasi-positive hybrid sentences. In the third and main step, we show that any quasi-positive sentence S generates an infinite set of modal formulas called approximants, which together axiomatize a canonical modal logic that is sound and complete for the class of frames validating S. The proof is analogous to standard proofs of Sahlqvist's theorem. It generalizes to sets of quasi-positive sentences. The main result now follows.

This paper studies finitary modal logics as specification languages for Set-coalgebras (coalgebras on the category of sets) using Stone duality. It is well-known that Set-coalgebras are not semantically adequate for finitary modal logics in the sense that bisimilarity does not in general coincide with logical equivalence. Stone-coalgebras (coalgebras over the category of Stone spaces), on the other hand, do provide an adequate semantics for finitary modal logics. This leads us to study the relationship of finitary modal logics and Set-coalgebras by uncovering the relationship between Set-coalgebras and Stone-coalgebras. This builds on a long tradition in modal logic, where one studies canonical extensions of modal algebras and ultrafilter extensions of Kripke frames to account for finitary logics. Our main contributions are the generalisations of two classical theorems in modal logic to coalgebras, namely the Jónsson-Tarski theorem giving a set-theoretic representation for each modal algebra and the bisimulation-somewhere-else theorem stating that two states of a coalgebra have the same (finitary modal) theory iff they are bisimilar (or behaviourally equivalent) in the ultrafilter extension of the coalgebra.

Propositional modal provability logics like G and Grz have arithmetical interpretations where □φ can be read as ‘formula φ is provable in Peano Arithmetic’. These logics are decidable but are characterized by classes of Kripke frames which are not first‐order definable. By abstracting the aspects common to their characteristic axioms we define the notion of a formula generation map F(P) in one propositional variable. We then focus our attention on the properly displayable subset of all (first‐order definable) Sahlqvist modal logics. For any logic L from this subset, we consider the (provability) logic LF obtained by the addition of an axiom based upon a formula generation map F(P) so that LF = L + F(P). The class of such logics includes G and Grz. By appropriately modifying the right introduction rules for □, we give (not necessarily cut‐free) display calculi for every such logic. We define the pseudo‐displayable subset of these logics as those whose display calculi enjoy cut‐elimination for sequents of the form ⊤ ⊢ φ for any formula φ. We then show that for any provability logic LF having a conservative tense extension, there is a map f on formulae such that LF is pseudo‐displayable if and only if f maps theorems of LF to theorems of the underlying logic L and vice versa. By using a standard renaming technique we can guarantee that there is a polynomial‐time translation from LF into L. All proofs are purely syntactic and show the versatility of display calculi since similar results using traditional Gentzen calculi are not possible for as broad a range of logics and require further conditions. Our maps generalize previously known maps from G into K4. An application of our results gives an O(n.log n)3) translation from the (‘second order’) provability logic Grz into a decidable subset of first‐order logic. Since each of our logics L is a Sahlqvist logic, it is first‐order definable, and hence each L has a translation into first‐order logic. Our results therefore show that all pseudo‐displayable logics LF are ‘essentially first‐order’ even though their characteristic axiom may not be first‐order definable.

We consider the relationship between the algorithmic properties of the validity problem for a modal or superintuitionistic propositional logic and the size of the smallest Kripke countermodels for non-theorems of the logic. We establish the existence, for every degree of unsolvability, of a propositional logic whose validity problem belongs to the degree and whose every non-theorem is refuted on a Kripke frame that validates the logic and has the size linear in the length of the non-theorem. Such logics are obtained among the normal extensions of the propositional modal logics $\textbf {KTB}$, $\textbf {GL}$ and $\textbf {Grz}$ as well as in the lattice of superintuitionistic propositional logics. This shows that the computational complexity of a modal or superintuitionistic propositional logic is, in general, not related to the size of the countermodels for its non-theorems.

The paper presents a study of abstract argumentation theory from the point of view of modal logic. The key thesis upon which the paper builds is that argumentation frameworks can be studied as Kripke frames. This simple observation allows us to import a number of techniques and results from modal logic to argumentation theory, and opens up new interesting avenues for further research. The paper gives a glimpse of the sort of techniques that can be imported, discussing complete calculi for argumentation, adequate model-checking and bisimulation games, and sketches an agenda for future research at the interface of modal logic and argumentation theory.