Logical Deduction using the Local Computation Framework

We generalize two well-known model-theoretic characterization theorems from propositional modal logic to first-order modal logic (FML, for short). We first study FML-definable frames and give a version of the Goldblatt–Thomason theorem for this logic. The advantage of this result, compared with the original Goldblatt–Thomason theorem, is that it does not need the condition of ultrafilter reflection and uses only closure under bounded morphic images, generated subframes and disjoint unions. We then investigate Lindstrom type theorems for first-order modal logic. We show that FML has the maximal expressive power among the logics extending FML which satisfy compactness, bisimulation invariance and the Tarski union property.

We define a natural notion of standard translation for the formulas of conditional logic which is analogous to the standard translation of modal formulas into the first-order logic. We briefly show that this translation works (modulo a lightweight first-order encoding of the conditional models) for the minimal classical conditional logic $\mathsf {CK}$ introduced by Brian Chellas in [3]; however, the main result of the article is that a classically equivalent reformulation of these notions (i.e., of standard translation plus theory of conditional models) also faithfully embeds the basic Nelsonian conditional logic $\mathsf {N4CK}$ , introduced in [11] into $\mathsf {QN4}$ , the paraconsistent variant of Nelson’s first-order logic of strong negation. Thus $\mathsf {N4CK}$ is the logic induced by the Nelsonian reading of the classical Chellas semantics of conditionals and can, therefore, be considered a faithful analogue of $\mathsf {CK}$ on the non-classical basis provided by the propositional fragment of $\mathsf {QN4}$ . Moreover, the methods used to prove our main result can be easily adapted to the case of modal logic, which makes it possible to improve an older result [10, Proposition 7] by S. Odintsov and H. Wansing about the standard translation embedding of the Nelsonian modal logic $\mathsf {FSK}^d$ into $\mathsf {QN4}$ .

As an extension of the traditional modal logic, the fuzzy first-order modal logic is discussed in this paper. A description of fuzzy first-order modal logic based on constant domain semantics is given, and a formal system of fuzzy reasoning based on the semantic information of models of first-order modal logic is established. It is also introduced in this paper the notion of the satisfiability of the reasoning system and some properties associated with the satisfiability are proved.

Many authors have noted that a number of English modal sentences cannot be formalized into standard first-order modal logic. Some widely discussed examples include “There could have been things other than there actually are” and “Everyone who’s actually rich could have been poor.” In response, many authors have introduced an “actually” operator @ into the language of first-order modal logic. It is occasionally noted that some of the example sentences still cannot be formalized with @ if one allows only actualist quantifiers, and embedded versions of these example sentences cannot be formalized even with possibilist quantifiers and @. The typical justification for these claims is to observe that none of the most plausible candidate formalizations succeed. In this paper, we prove these inexpressibility results by using a modular notion of bisimulation for first-order modal logic with “actually” and other operators. In doing so, we will explain in what ways these results do or do not generalize to more expressive modal languages.

A well established technique toward developing the proof theory of a Hilbert-style modal logic is to introduce a Gentzen-style equivalent (a Gentzenisation), then develop the proof theory of the latter, and finally transfer the metatheoretical results to the original logic (e.g., [1, 6, 8, 18, 10, 12]). In the first-order modal case, on one hand we know that the Gentzenisation of the straightforward first-order extension of GL, the logic QGL, admits no cut elimination (if the rule is included as primitive; or, if not included, then the rule is not admissible [1]). On the other hand the (cut-free) Gentzenisations of the first-order modal logics M3 and ML3 of [10, 12] do have cut as an admissible rule. The syntactic cut admissibility proof given in [18] for the Gentzenisation of the propositional provability logic GL is extremely complex, and it was the basis of the proofs of cut admissibility of the Gentzenisations of M3 and ML3, where the presence of quantifiers and quantifier rules added to the complexity and length of the proof. A recent proof of cut admissibility in a cut-free Gentzenisation of GL is given in [5] and is quite short and easy to read. We adapt it here to revisit the proofs for the cases of M3 and ML3, resulting to similarly short and easy to read proofs, only slightly complicated by the presence of quantification and its relevant rules.

Modal logics for qualitative possibility theory

Kripke transformed both the model theory of modal logic and the understanding of metaphysical modality. In particular, he analysed the model-theoretic conditions for the Barcan formula and its converse to hold. However, the relation between the model theory of modal logic and the correctness of formulas on their metaphysical readings is not straightforward. To mediate between them, a notion of metaphysical universality is defined such that an intended model structure (on the metaphysical reading) should validate all and only metaphysically universal formulas. Technical problems for defining intended model structures arise in both propositional and first-order modal logic; there may be ‘too many’ worlds and ‘too many’ individuals. Given that these problems can be overcome, a special reason is explained why contingentism but not necessitism forbids an intended model structure for quantified modal logic. But even granted necessitism, it is controversial which first-order modal formulas are metaphysically universal.

"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."

Introducing possibilistic logic in ILP for dealing with exceptions

Modal Logics for Qualitative Possibility and Beliefs

First-order modal logic, in the usual formulations, is not sufficiently expressive, and as a consequence problems like Frege's morning star/evening star puzzle arise. The introduction of predicate abstraction machinery provides a natural extension in which such difficulties can be addressed. But this machinery can also be thought of as part of a move to a full higher-order modal logic. In this paper we present a sketch of just such a higher-order modal logic: its formal semantics, and a proof procedure using tableaus. Naturally the tableau rules are not complete, but they are with respect to a Henkinization of the true semantics. We demonstrate the use of the tableau rules by proving one of the theorems involved in Godel's ontological argument, one of the rare instances in the literature where higher-order modal constructs have appeared. A fuller treatment of the material presented here is in preparation

21 Modal logic and philosophy

Controlled Query Evaluation (CQE) protects confidential information, stored in an information system. It prevents harmful inferences due to a user's knowledge and reasoning. In this article we extend CQE to incomplete first-order databases, a data model which suits a broader range of applications than a previously studied propositional incomplete data model. Because of the complexity of the underlying implication problem, which describes the user's reasoning, the representation of the user's knowledge is the main obstacle to effective inference control. For knowledge representation, we introduce first-order modal logic to CQE. Especially, we deal with knowledge about a restricted data model in first-order logic. The restricted data model considered gives rise to a new problem: if the user is aware of the data model, his reasoning must be modeled appropriately. In the analysis of this “reasoning” model we consider both confidentiality and availability. Finally we show, how the considered data model can be reduced to the propositional case and analyze confidentiality properties of the resulting implementation.

Automatic reasoning tools play an important role when developing provably correct software. Both main approaches, program verification and program synthesis employ automated reasoning tools, more specifically, automated theorem provers. Besides classical logic, non-classical logics are particularly relevant in this field. This chapter presents calculi to automate theorem proving in classical and some important non-classical logics, namely first-order intuitionistic and first-order modal logics. These calculi are based on the connection method, which permits a goal-oriented and, hence, a more efficient proof search. The connection calculi for these non-classical logics extend the calculus for classical logic in an elegant and uniform way by adding so-called prefixes to atomic formulae. The leanCoP theorem prover is a very compact PROLOG implementation of the connection calculus for classical logics. We present details of the implementation and describe some basic techniques to improve its efficiency. leanCoP is adapted to non-classical logics by integrating a prefix unification algorithm, which depends on the specific logic. This results in leading theorem provers for the aforementioned non-classical logics.

We consider a general format for sequent rules for not necessarily normal modal logics based on classical or intuitionistic propositional logic and provide relatively simple local conditions ensuring cut elimination for such rule sets. The rule format encompasses e.g. rules for the boolean connectives and transitive modal logics such as S4 or its constructive version. We also adapt the method of constructing suitable rule sets by saturation to the intuitionistic setting and provide a criterium for translating axioms for intuitionistic modal logics into sequent rules. Examples include constructive modal logics and conditional logic $\mathbb{VA}$ .