Showing papers on "Normal modal logic published in 2018"

A logic of goal-directed knowing how

Abstract: In this paper, we propose a decidable single-agent modal logic for reasoning about goal-directed “knowing how”, based on ideas from linguistics, philosophy, modal logic, and automated planning in AI. We first define a modal language to express “I know how to guarantee \(\varphi \) given \(\psi \)” with a semantics based not on standard epistemic models but on labeled transition systems that represent the agent’s knowledge of his own abilities. The semantics is inspired by conformant planning in AI. A sound and complete proof system is given to capture valid reasoning patterns, which highlights the compositional nature of “knowing how”. The logical language is further extended to handle knowing how to achieve a goal while maintaining other conditions.

A modal separation logic for resource dynamics

Abstract: The logic of Bunched implications (BI), and its boolean version (Boolean BI), are logics that allow us to express properties on resources and to provide logical frameworks for the so-called separation logics. In this paper we study a new modal separation logic that extends Boolean BI with two kinds of modalities, in order to deal with resources having dynamic properties (which depend on the current state of a system) and also to capture some resource evolutions or transformations. We show how we can model concurrent processes manipulating resources, and we provide a sound and complete tableau calculus, with a counter-model extraction method, for proving properties expressed in this logic.

Gamma graph calculi for modal logics

Abstract: We describe Peirce’s 1903 system of modal gamma graphs, its transformation rules of inference, and the interpretation of the broken-cut modal operator. We show that Peirce proposed the normality rule in his gamma system. We then show how various normal modal logics arise from Peirce’s assumptions concerning the broken-cut notation. By developing an algebraic semantics we establish the completeness of fifteen modal logics of gamma graphs. We show that, besides logical necessity and possibility, Peirce proposed an epistemic interpretation of the broken-cut modality, and that he was led to analyze constructions of knowledge in the style of epistemic logic.

Expressivity in chain-based modal logics

Abstract: We investigate the expressivity of many-valued modal logics based on an algebraic structure with a complete linearly ordered lattice reduct. Necessary and sufficient algebraic conditions for admitting a suitable Hennessy–Milner property are established for classes of image-finite and (appropriately defined) modally saturated models. Full characterizations are obtained for many-valued modal logics based on complete BL-chains that are finite or have the real unit interval [0, 1] as a lattice reduct, including Łukasiewicz, Godel, and product modal logics.

A Normal Modal Logic for Trust in the Sincerity

Abstract: In the field of multi-agent systems, as some agents may be not reliable or honest, a particular attention is paid to the notion of trust. There are two main approaches for trust: trust assessment and trust reasoning. Trust assessment is often realized with fuzzy logic and reputation systems which aggregate testimonies -- individual agents' assessments -- to evaluate the agents' global reliability. In the domain of trust reasoning, a large set of works focus also on trust in the reliability as for instance Liau's BIT modal logic where trusting a statement means the truster can believe it. However, very few works focus on trust in the sincerity of a statement -- meaning the truster can believe the trustee believes it. Consequently, we propose in this article a modal logic to reason about an agent's trust in the sincerity towards a statement formulated by another agent. We firstly introduce a new modality of trust in the sincerity and then we prove that our system is sound and complete. Finally, we extend our notion of individual trust about the sincerity to shared trust and we show that it behaves like a KD system.

Stable modal logics

Abstract: Stable logics are modal logics characterized by a class of frames closed under relation preserving images. These logics admit all filtrations. Since many basic modal systems such as K4 and S4 are not stable, we introduce the more general concept of an M-stable logic, where M is an arbitrary normal modal logic that admits some filtration. Of course, M can be chosen to be K4 or S4. We give several characterizations of M-stable logics. We prove that there are continuum many S4-stable logics and continuum many K4-stable logics between K4 and S4. We axiomatize K4-stable and S4-stable logics by means of stable formulas and discuss the connection between S4-stable logics and stable superintuitionistic logics. We conclude the article with many examples (and nonexamples) of stable, K4-stable, and S4-stable logics and provide their axiomatization in terms of stable rules and formulas.

Proving Structural Properties of Sequent Systems in Rewriting Logic

Abstract: General and effective methods are required for providing good automation strategies to prove properties of sequent systems. Structural properties such as admissibility, invertibility, and permutability of rules are crucial in proof theory, and they can be used for proving other key properties such as cut-elimination. However, finding proofs for these properties requires inductive reasoning over the provability relation, which is often quite elaborated, exponentially exhaustive, and error prone. This paper aims at developing automatic techniques for proving structural properties of sequent systems. The proposed techniques are presented in the rewriting logic metalogical framework, and use rewrite- and narrowing-based reasoning. They have been fully mechanized in Maude and achieve a great degree of automation when used on several sequent systems, including intuitionistic and classical logics, linear logic, and normal modal logics.

Rosser provability and normal modal logics.

Abstract: In this paper, we investigate Rosser provability predicates whose provability logics are normal modal logics. First, we prove that there exists a Rosser provability predicate whose provability logic is exactly the normal modal logic ${\sf KD}$. Secondly, we introduce a new normal modal logic ${\sf KDR}$ which is a proper extension of ${\sf KD}$, and prove that there exists a Rosser provability predicate whose provability logic includes ${\sf KDR}$.

On the expressive power of first-order modal logic with two-dimensional operators

Abstract: Many authors have noted that there are types of English modal sentences cannot be formalized in the language of basic first-order modal logic. Some widely discussed examples include “There could have been things other than there actually are” and “Everyone who is actually rich could have been poor.” In response to this lack of expressive power, many authors have discussed extensions of first-order modal logic with two-dimensional operators. But claims about the relative expressive power of these extensions are often justified only by example rather than by rigorous proof. In this paper, we provide proofs of many of these claims and present a more complete picture of the expressive landscape for such languages.

Reverse Public Announcement Operators on Expanded Models

Abstract: Past public announcement operators have been defined in Hoshi and Yap (Synthese 169(2):259–281, 2009) and Yap (Dynamic logic montreal, 2007), to describe an agent’s knowledge before an announcement occurs. These operators rely on branching-time structures that do not mirror the traditional, relativization-based semantics of public announcement logic (PAL), and favor a historical reading of past announcements. In this paper, we introduce reverse public announcement operators that are interpreted on expanded models. Our model expansion adds accessibility links from an epistemic model $$\mathcal {M}$$ to a filtrated submodel of the canonical model for $$\mathbf K _g$$ . Here $$\mathbf K _g$$ is the minimal normal modal logic together with $$\mathbf S5 $$ axioms for the universal operator U. This yields a highly general pre-announcement version of $$\mathcal {M}$$ that makes our operators potentially useful for studying non-standard interpretations of rescinded announcements in PAL. Indeed, we find that our reverse announcement operators cannot be represented by product update, and that they have an intimate connection with the knowledge forgetting of Zhang and Zhou (Artif Intell J 173(16–17):1525–1537, 2009). We show that the logic resulting from adding reverse announcements to PAL is sound and complete.

Proof Theory for Functional Modal Logic

Abstract: We present some proof-theoretic results for the normal modal logic whose characteristic axiom is $$\mathord {\sim }\mathord {\Box }A\equiv \mathord {\Box }\mathord {\sim }A$$ . We present a sequent system for this logic and a hypersequent system for its first-order form and show that these are equivalent to Hilbert-style axiomatizations. We show that the question of validity for these logics reduces to that of classical tautologyhood and first-order logical truth, respectively. We close by proving equivalences with a Fitch-style proof system for revision theory.

On the Modal Logic of the Non-orthogonality Relation Between Quantum States

Abstract: It is well known that the non-orthogonality relation between the (pure) states of a quantum system is reflexive and symmetric, and the modal logic $$\mathbf {KTB}$$ is sound and complete with respect to the class of sets each equipped with a reflexive and symmetric binary relation. In this paper, we consider two properties of the non-orthogonality relation: Separation and Superposition. We find sound and complete modal axiomatizations for the classes of sets each equipped with a reflexive and symmetric relation that satisfies each one of these two properties and both, respectively. We also show that the modal logics involved are decidable.

Actualism and modal semantics

Abstract: According to actualism, modal reality is constructed out of valuations (combinations of truth values for all propositions). According to possibilism, modal reality consists in a set of possible worlds, conceived as independent objects that assign truth values to propositions. According to possibilism, accounts of modal reality can intelligibly disagree with each other even if they agree on which valuations are contained in modal reality. According to actualism, these disagreements (possibilist disagreements) are completely unintelligible. An essentially actualist semantics for modal propositional logic specifies which sets of valuations are compatible with the meanings of the truth-functional connectives and modal operators without drawing on formal resources that would enable us to represent possibilist disagreements. The paper discusses the availability of an essentially actualist semantics for modal propositional logic. I argue that the standard Kripkean semantics is not essentially actualist and that other extant approaches also fail to provide a satisfactory essentially actualist semantics. I end by describing an essentialist actualist semantics for modal propositional logic.

Complete Additivity and Modal Incompleteness

Abstract: In this paper, we tell a story about incompleteness in modal logic. The story weaves together a paper of van Benthem, `Syntactic aspects of modal incompleteness theorems,' and a longstanding open question: whether every normal modal logic can be characterized by a class of completely additive modal algebras, or as we call them, V-BAOs. Using a first-order reformulation of the property of complete additivity, we prove that the modal logic that starred in van Benthem's paper resolves the open question in the negative. In addition, for the case of bimodal logic, we show that there is a naturally occurring logic that is incomplete with respect to V-BAOs, namely the provability logic GLB. We also show that even logics that are unsound with respect to such algebras do not have to be more complex than the classical propositional calculus. On the other hand, we observe that it is undecidable whether a syntactically defined logic is V-complete. After these results, we generalize the Blok Dichotomy to degrees of V-incompleteness. In the end, we return to van Benthem's theme of syntactic aspects of modal incompleteness.