Connexive Variants of Modal Logics Over FDE

The present paper studies two approaches to the expressiveness of propositional modal logics based on first-degree entailment logic, FDE. We first consider the basic FDE-based modal logic BK and certain systems in its vicinity, and then turn to some FDE-based modal logics in a richer vocabulary, including modal bilattice logic, MBL. On the one hand, model-theoretic proofs of the definability of connectives along the lines of [McCullough, “Logical connectives for intuitionistic propositional logic”, Journal of Symbolic Logic 36, 1 (1971): 15–20. DOI: 10.2307/2271511] and [[17] Wansing, “Logical connectives for constructive modal logic”, Synthese 150, 3 (2006): 459–482. DOI: 10.1007/s11229-005-5518-5] are given for various FDE-based modal logics. On the other hand, building on [Odintsov and Wansing, “Disentangling FDE-based paraconsistent modal logics, Studia Logica 105, 6 (2017): 1221–1254. DOI: 10.1007/s11225-017-9753-9], expressibility is considered in terms of mutual faithful embeddability of one logic into another logic. A distinction is drawn between definitional equivalence, which is defined with respect to a pair of structural translations between two languages, and weak definitional equivalence, which is defined with respect to a weaker notion of translations. Moreover, the definitional equivalence of some FDE-based modal logics is proven, especially the definitional equivalence of MBL and a conservative extension of the logic BK□×BK□, which underlines the central role played by BK among FDE-based modal logics.

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 give a simple proof of a recently obtained in [12] result on the completeness of modal logics with the modality that corresponds to the intersection of accessibility relations in a Kripke model. In epistemic logic, this is the so-called distributed knowledge operator. We prove completeness for the logics in the modal languages of two types: one has the modalities □1,...,□n for the relations R1,...,Rn that satisfy a unimodal logic L, and the modality □n+1 for the intersection Rn+1=R1 ∩...∩ Rn; the other language has the modalities □i (i ∈ Σ) for the relations Ri that satisfy the logic L, and, for every nonempty subset of indices I ⊆ Σ, the modality □I for the intersection ∩i∈I Ri. While in [12] the completeness is proved only for the logics over K, KD, KT, K4, S4, and S5, here we give a "uniform" construction that enables us to obtain completeness for the logics with intersection over the 15 so-called "traditional" modal logics KΛ, for Λ ⊆ {D, T, B, 4, 5}. The proof method is based on unravelling of a frame and then taking the Horn closure of the resulting frame.

20 Modal logic for games and information

Modal logics are good candidates for a formal theory of agents. The efficiency of reasoning method in modal logics is very important, because it determines whether or not the reasoning method can be widely used in systems based on agent. In this paper, we modify the extension rule theorem proving method we presented before, and then apply it to P-logic that is translated from modal logic by functional transformation. At last, we give the proof of its soundness and completeness. ∗ Supported by National Natural Science foundation of China (Grant Nos. 60273080, 60473003); the Science and Technology Development Program of Jilin Province of China (Grant Nos. 20020306)

On the succinctness of some modal logics

In this paper we provide a semantic characterization of ground logics, a family of nonmonotonic modal logics obtained by means of a variant of the well known Mc Dermott and Doyle fixed point equation. The term ground logics carries the idea of restricting the negative introspection capabilities of the reasoning agent to the objective (i.e. non modal) part of the theory. This intuition was nicely formalized for modal logic S5 by a semantic definition based on a preference relation on Kripke models, which was obtained as the semantic counterpart to the notion of minimal knowledge initially defined by Halpern and Moses. We have then found a preference relation on Kripke models that both generalizes this notion of minimal knowledge and provides a semantic characterization for a significant subset of ground logics.KeywordsModal LogicKripke ModelGround LogicDefault LogicNonmonotonic ReasoningThese keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

We say that a Kripke model is a GL-model (Gödel and Löb model) if the accessibility relation $\prec $ is transitive and converse well-founded. We say that a Kripke model is a D-model if it is obtained by attaching infinitely many worlds $t_1, t_2, \ldots $ , and $t_\omega $ to a world $t_0$ of a GL-model so that $t_0 \succ t_1 \succ t_2 \succ \cdots \succ t_\omega $ . A non-normal modal logic $\mathbf {D}$ , which was studied by Beklemishev [3], is characterized as follows. A formula $\varphi $ is a theorem of $\mathbf {D}$ if and only if $\varphi $ is true at $t_\omega $ in any D-model. $\mathbf {D}$ is an intermediate logic between the provability logics $\mathbf {GL}$ and $\mathbf {S}$ . A Hilbert-style proof system for $\mathbf {D}$ is known, but there has been no sequent calculus. In this paper, we establish two sequent calculi for $\mathbf {D}$ , and show the cut-elimination theorem. We also introduce new Hilbert-style systems for $\mathbf {D}$ by interpreting the sequent calculi. Moreover, we show that D-models can be defined using an arbitrary limit ordinal as well as $\omega $ . Finally, we show a general result as follows. Let X and $X^+$ be arbitrary modal logics. If the relationship between semantics of X and semantics of $X^+$ is equal to that of $\mathbf {GL}$ and $\mathbf {D}$ , then $X^+$ can be axiomatized based on X in the same way as the new axiomatization of $\mathbf {D}$ based on $\mathbf {GL}$ .

<p>The abstract mathematical structures known as coalgebras are of increasing interest in computer science for their use in modelling certain types of data structures and programs. Traditional algebraic methods describe objects in terms of their construction, whilst coalgebraic methods describe objects in terms of their decomposition, or observational behaviour. The latter techniques are particularly useful for modelling infinite data structures and providing semantics for object-oriented programming languages, such as Java. There have been many different logics developed for reasoning about coalgebras of particular functors, most involving modal logic. We define a modal logic for coalgebras of polynomial functors, extending Rößiger’s logic [33], whose proof theory was limited to using finite constant sets, by adding an operator from Goldblatt [11]. From the semantics we define a canonical coalgebra that provides a natural construction of a final coalgebra for the relevant functor. We then give an infinitary axiomatization and syntactic proof relation that is sound and complete for functors constructed from countable constant sets.</p>

The formula $\triangle$A (it is noncontingent whether A) is true at a point in a Kripke model just in case all points accessible to that point agree on the truth-value of A. We can think of $\triangle$-based modal logic as a special case of what we call the general modal logic of agreement, interpreted with the aid of models supporting a ternary relation, S, say, with OA (which we write instead of $\triangle$A to emphasize the generalization involved) true at a point w just in case for all points x, y, with Swxy, x and y agree on the truth-value of A. The noncontingency interpretation is the special case in which Swxy if and only if Rwx and Rwy, where R is a traditional binary accessibility relation. Another application, related to work of Lewis and von Kutschera, allows us to think of OA as saying that A is entirely about a certain subject matter.

1 - Structures, Languages and Translations: the Structural Approach to Feature Logic

This paper considers the bi-modal logic with both $$\Box $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mo>□</mml:mo> </mml:math> and $$\Diamond $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mo>◊</mml:mo> </mml:math> arising from Kripke models with crisp accessibility whose propositions are valued over the standard Gödel algebra [0, 1]. Since this logic lacks the finite model property, we study the logic $$\textbf{GW}^\textrm{c}$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msup> <mml:mi>GW</mml:mi> <mml:mtext>c</mml:mtext> </mml:msup> </mml:math> relying on witnessed Kripke models where, for each modal formula, there is an assignment where the formula without the modality takes the same value as the modal one. We provide a cut-free sequent calculus and we exploit it to prove that $$\textbf{GW}^\textrm{c}$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msup> <mml:mi>GW</mml:mi> <mml:mtext>c</mml:mtext> </mml:msup> </mml:math> is decidable and meets the finite model property. Finally, we explore a connection between the witnessed models and the well-known bi-relational Kripke semantics.

K + T is the prepositional modal logic K with the elements of the finite set T as additional axioms.

In this article we study and solve some open problem of unification for formulas with coefficients (meta-variables) in transitive modal logics. The role of coefficients is played by propositional letters which are constants (which any unifier lets intact). We solve this problem affirmatively: we find an algorithm which constructs a finite set of the best unifiers for any unifiable formula with coefficients. This algorithm works for all transitive modal logics satisfying special general conditions. These conditions hold, in particular, for modal logics K4, S4, Grz and GL, so our results are true for these important logics. In terms of algebraic logic (or universal algebra), we solve the problem of finding solutions for equations in the free modal algebras in the signature extended by constants for free variables.

Beth's Theorem and Self-Referential Sentences