Complexity function and complexity of validity of modal and superintuitionistic propositional logics

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 + .

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.

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.

We study a framework for the specification of architecture styles as families of architectures involving a common set of types of components and coordination mechanisms. The framework combines two logics: 1 interaction logics for the specification of architectures as generic coordination schemes involving a configuration of interactions between typed components; 2 configuration logics for the specification of architecture styles as sets of interaction configurations. The presented results build on previous work on architecture modelling in BIP. We show how propositional interaction logic can be extended into a corresponding configuration logic by adding new operators on sets of interaction configurations. We provide a complete axiomatisation of the propositional configuration logic, as well as a decision procedure for checking that an architecture satisfies given logical specifications. To allow genericity of specifications, we study first-order and second-order extensions of the propositional logic. We provide examples illustrating the application of the results to the characterization of architecture styles. Finally, we provide an experimental evaluation using the Maude rewriting system to implement the decision procedure for the propositional logic.

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.

"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."

For a novice this book is a mathematically-oriented introduction to modal logic, the discipline within mathematical logic studying mathematical models of reasoning which involve various kinds of modal operators - `like it is necessary' in philosophy, `it is believed' in cognitive science, `it is provable' in mathematics and `it is true after executing a program' in computer science. It is an advanced text which starts with very fundamental concepts and gradually proceeds to the front line of current research, introducing in full details the modern semantical and algebraic apparatus and covering practically all classical results in the field. It contains both numerous exercises and open problems, and presupposes only minimal knowledge in mathematics. A specialist can use the book as a source of references. For the first time results and methods of many directions in propositional modal logic - from completeness and duality to algorithmic problems - are collected and systematically presented in one volume. Unlike other books, modal logic is treated here as a uniform theory rather than a collection of a few particular systems. It is the only book presenting the theory of superintuitionistic logics.

In multi-agent systems, an agent generally forms her belief based on information from multiple sources, such as messages from other agents or perception of the external environment. While modal epistemic logic has been a standard formalism for reasoning about agent's belief, it lacks the expressive power for tracking information sources. Justification logic (JL) provides the missing expressivity by using justification terms to keep track of the belief formation process. However, because JL does not make a clear distinction between potential and actual evidence, the interpretation of justification formulas in JL turns out to be ambiguous. In this paper, we present a justification-based multi-source reasoning formalism built upon second-order propositional modal logic. Our framework not only inherits the source-tracking advantage of JL but also allows the distinction between the actual observation and simply potential admissibility of evidence.

This book, written by one of the most distinguished of contemporary philosophers of mathematics, is a fully rewritten and updated successor to the author's earlier The Unprovability of Consistency (1979). Its subject is the relation between provability and modal logic, a branch of logic invented by Aristotle but much disparaged by philosophers and virtually ignored by mathematicians. Here it receives its first scientific application since its invention. Modal logic is concerned with the notions of necessity and possibility. What George Boolos does is to show how the concepts, techniques, and methods of modal logic shed brilliant light on the most important logical discovery of the twentieth century: the incompleteness theorems of Kurt Godel and the 'self-referential' sentences constructed in their proof. The book explores the effects of reinterpreting the notions of necessity and possibility to mean provability and consistency.

1 Modal logic: a semantic perspective

Substitution Frege and extended Frege proof systems in non-classical logics

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.

This handbook provides a systematic survey of the central areas of philosophical logic. It is divided into four volumes, each devoted to a major sub-field within the disciplines. The articles should be of value to general philosophers, linguists, logicians, mathematicians and computer scientists. Volume 1, Elements of Classical Logic, deals with the background to what has come to be considered the standard formulation of predicate logic - both as far as its semantics and proof theory are concerned. The central chapter on predicate logic is followed by chapters outlining various alternative, but essentially equivalent ways of constructing the semantics for first-order logic as well as its proof theory. In addition, this volume contains a discussion of higher-order extensions of first-order logic and a compendium of the algorithmic and decision-theoretic prerequisites in the study of logical systems. Volume 2, Extensions of Classical Logic, surveys the most significant extensions of predicate logic and their applications to various philosophical fields of inquiry. The 12 chapters in this volume together provide a succinct introduction to a variety of intensional frameworks, a discussion of the most well-known logical systems, as well as an overview of major applications and of the open problems in the respective fields. Volume 3, Alternatives to Classical Logic, consists of a series of surveys of some of the alternatives to the basic assumptions of classical logic. These include many-valued logic, partial logic, free logic, relevance and entailment logics, dialogue logic, quantum logic and intuitionism. Volume 4, Topics in the Philosophy of Language, presents a panorama of the applications of logical tools and methods in the formal analysis of natural language. Since a number of developments in philosophical logic were originally stimulated by concern arising in the semantic analysis of natural language discourse, the chapters in this volume provide some criteria of evaluation of the applications of work in philosophical logic. In revealing both the adequacies and inadequacies of logical investigations in the semantic structures of natural discourse, these chapters also point the way to future developments in philosophical logic in general and thus close again the circle of inquiry relating logic and language.

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

There has been a significant interest in modal logics with intersection, prominent examples including epistemic and doxastic logics with distributed knowledge, propositional dynamic logic with intersection, and description logics with concept intersection. Completeness proofs for such logics tend to be complicated, in particular on model classes such as S5 used, e.g., in standard epistemic logic, mainly due to the undefinability of intersection of modalities in standard modal logic. A standard proof method for the S5 case uses an “unraveling-folding” technique to achieve a treelike model to deal with the problem of undefinability. This method, however, is not easily adapted to other logics, due to its reliance on S5 in a number of steps. In this paper we demonstrate a simpler and more general proof technique by building a treelike canonical model directly, which avoids the complications in the processes of unraveling and folding. We illustrate the technique by showing completeness of the normal modal logics K, D, T, B, S4 and S5 extended with intersection modalities. Furthermore, these treelike canonical models are compatible with Fischer-Ladner-style closures, and we combine the methods to show the completeness of the mentioned logics further extended with transitive closure of union modalities known from PDL or epistemic logic. Some of these completeness results are new.