Tractable reasoning via approximation

1 Modal logic: a semantic perspective

The famous van Benthem theorem states that modal logic corresponds exactly to the fragment of first-order logic that is invariant under bisimulation. In this article we prove an exact analogue of this theorem in the framework of modal dependence logic (MDL) and team semantics. We show that Modal Team Logic (MTL) extending MDL by classical negation captures exactly the FO-definable bisimulation invariant properties of Kripke structures and teams. We also compare the expressive power of MTL to most of the variants and extensions of MDL recently studied in the area.

Multi-valued logic and Gröner bases with applications to modal logic

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.

Among non-monotonic systems of reasoning, non-monotonic modal logics, and autoepistemic logic in particular, have had considerable success. The presence of explicit modal operators allows flexibility in the embedding of other approaches. Also several theoretical results of interest have been established concerning these logics. In this paper we introduce non-monotonic modal logics based on many-valued logics, rather than on classical logic. This extends earlier work of ours on many-valued modal logics. Intended applications are to situations involving several reasoners, not just one as in the standard development.

The propositional modal logic is obtained by adding the necessity operator □ to the propositional logic. Each formula in the propositional logic is equivalent to a formula in the disjunctive normal form. In order to obtain the correspondence between the propositional modal logic and the propositional logic, we add the axiom \(\Box\varphi \leftrightarrow\Diamond\varphi \) to K and get a new system K + . Each formula in such a logic is equivalent to a formula in the disjunctive normal form, where □ k (k ≥ 0) only occurs before an atomic formula p, and \(\lnot\) only occurs before a pseudo-atomic formula of form □ k p. Maximally consistent sets of K + have a property holding in the propositional logic: a set of pseudo-atom-complete formulas uniquely determines a maximally consistent set. When a pseudo-atomic formula □ k p i (k,i ≥ 0) is corresponding to a propositional variable q ki , each formula in K + then can be corresponding to a formula in the propositional logic P + . We can also get the correspondence of models between K + and P + . Then we get correspondences of theorems and valid formulas between them. So, the soundness theorem and the completeness theorem of K + follow directly from those of P + .

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.

Recently, Baltag and van Benthem introduced a decidable logic of functional\ndependence (LFD) that extends the logic of Cylindrical Relativized Set Algebras\n(CRS) with atomic local dependence statements. Its semantics can be given in\nterms of generalised assignment models or their modal counterparts, hence the\nlogic is both a first-order and a modal logic. We show that LFD has the finite\nmodel property (FMP) using Herwig's theorem on extending partial isomorphisms,\nand prove a bisimulation invariance theorem characterizing LFD as a fragment of\nfirst-order logic.\n

Modal logic becomes action logic by adding programs as in propositional dynamic logic or the /spl mu/-calculus. Modal languages can be seen as decidable fragments of first-order logic that admit a natural bisimulation, and hence enjoy a good model theory. Recently, much stronger 'guarded fragments' of first-order logic have been identified that enjoy the same pleasant features. The latter can serve as richer action languages as well. We will develop the logic of guarded fragments as a form of process theory. In particular, moving from sequential to parallel process operations correlates with moving to first-order fragments that are close to, or perhaps just over the decidable-undecidable fence.

"This book describes the aspects of mathematical logic related to computer sciences. The materials adopted in this book are intended to attend to both the peculiarities of logical systems and the requirements of computer science."

We introduce a propositional many-valued modal logic which is an extension of the Continuous Propositional Logic to a modal system. Otherwise said, we extend the minimal modal logic to a Continuous Logic system. After introducing semantics, axioms and deduction rules, we establish some preliminary results. Then we prove the equivalence between consistency and satisfiability. As straightforward consequences, we get compactness, an approximated completeness theorem, in the vein of Continuous Logic, and a Pavelka-style completeness theorem.

The article discusses attempts to develop a conception of language in opposition to metaphysics in the of Wittgenstein and of Heidegger. Both Wittgenstein and Heidegger worked in the field of with the intention to overcome metaphysics. However, each of them understood metaphysics differently, and it resulted in different conceptions of language. In the first part, the author presents the major themes of Wittgenstein's of language, among them his formulation of metaphysics. Even if the ,,first philosophy of Wittgenstein is radically different from his ,,second philosophy, his position on metaphysics is stable: metaphysical statements are statements which violate the boundaries of sense (Sinn). In the Tractatus Wittgenstein aims at finding the limits of the world, thought and language, as well as distinguishing between sense and nonsense. The conditions for a proposition's having sense have been explored by Wittgenstein in the Tractatus: they rest on the possibility of representation or picturing. According to him, names must have a reference/meaning (Bedeutung), but they can do so only in the context of a proposition which is held together by logical form. It follows that only the factual states of affairs, which can be pictured, can be by meaningful propositions. This means that what can be said are only the propositions of natural sciences, leaving out of the realm of sense a daunting number of statements which are made and used in language. First, there are the propositions of logic. These do not represent states of affairs, and the logical constants do not stand for objects. ,,My fundamental thought is that the logical constants do not represent. That the logic of the facts cannot be represented (Tractatus logico-philosophicus 4.0312). Tautologies and contradictions, the propositions of logic, are the limits of language and thereby the limits of the world. They do not picture anything and do not, therefore, have sense, they are senseless (sinnlos). Propositions which do have sense are bipolar; they refer to the truth-conditions drawn by the propositions of logic.

(1978). Orthomodular Lattices and Quantum Physics. Mathematics Magazine: Vol. 51, No. 5, pp. 299-303.

Bisimulation is a central concept in the model theory of modal logic with extensive computational applications. It is a relation between two models in which related states have identical atomic properties and matching transition possibilities. Bisimulation captures the expressive power of propositional modal logic in the sense that bisimilar states are not distinguishable by any propositional modal formula. In recent years, fuzzy modal logic has received much attention because of the connection between uncertainty measures and fuzzy modalities. In this paper, we define the notion of fuzzy bisimulation and prove that, in a fuzzified version, it bears the same relationship to fuzzy modal logic that bisimulation bears to propositional modal logic.

Sören Halldén. On the logic of ‘better.’ Library of Theoria, no. 2. C. W. K. Gleerup, Lund, and Ejnar Munksgaard, Copenhagen, 1957, 112 pp. - Lennart Åqvist. Deontic logic based on a logic of ‘better.’Proceedings of a Colloquium on Modal and Many-valued Logics, Helsinki, 23–26 August, 1962, Acta philosophica Fennica, no. 16, Helsinki1963, pp. 285–290. - Mark Fisher. A three-valued calculus for deontic logic. Theoria (Lund), vol. 27 (1961), pp. 107–118. - Lennart Åqvist. Postulate sets and decision procedures for some systems of deontic logic. Theoria (Lund), vol. 29 (1963), pp. 154–175. - Volume 31 Issue 2