Higher-Order Modal Logic—A Sketch

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.

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.

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.

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.

Many writers have commented on the revival of philosophical theology that began in analytic circles in the 1960s. Those of us who received our training in the 50s were reacting to two things. First, our predecessors’ preoccupation with the question of religious discourse’s meaningfulness. While not necessarily disagreeing with Basil Mitchell’s, say, or John Hick’s responses to Antony Flew’s charge of ‘death by a thousand qualifications’, most of us lost interest in the debate. Second, a conviction that Hume’s and Kant’s vaunted critiques of natural theology didn’t withstand careful scrutiny. On the positive side, developments in modal logic, probability theory, and so on, offered tools for introducing a new clarity and rigour to traditional disputes. Alvin Plantinga’s work on the ontological argument, or Richard Swinburne’s use of Bayesian techniques to formulate his cumulative argument for God’s existence, are paradigmatic examples.

The Local Computation Framework has been used to improve the efficiency of computation in various uncertainty formalisms. This paper shows how it can be applied to logical deduction in first-order logic, modal and conditional logics, circumscription and possibilistic logic.

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.

1 Modal logic: a semantic perspective

In this paper we generalize the existing tableau methods for modal logics. First of all, while usual modal tableaux are based on trees, our basic structures are rooted directed acyclic graphs (RDAG). This allows natural tableau rules for some modal logics that are difficult to capture in the usual way (such as those having an accessibility relation that is dense or confluent). Second, tableau rules rewrite patterns, which are (schemas of) parts of a RDAG. A particular case of these rules are the single-step rules recently proposed by Massacci. This allows in particular tableau rule presentations for K5, KD5, K45, KD45, and S5 that respect the subformula property. Third, we divide modal tableau rules into propagation rules and structural rules. Structural rules construct new edges and nodes (without adding formulas to nodes), while propagation rules add formulas to nodes. This distinction allows to prove completeness in a modular way.

In this paper a technique is presented which provides us with a means to develop resolution-based calculi for (first-order) modal and temporal logics. The approach is based on three parts: A special translation technique from modal and temporal logic formulae into classical predicate logic, a certain kind of saturation technique which is to be applied to given background theories, and an extraction of either suitable “simpler” background theories or logic-specific inference rules. The former is interesting in case existing classical logic theorem provers are to be utilized; the latter gains importance if one is prepared to extend theorem provers that are already at hand.

This paper deals with automated deduction for classical and partial logics, especially for the three-valued logic L3, which has been introduced, for example, in the study of natural language semantics. Based on ideas from a Plaisted's Gentzen style system for classical two-valued logic, we present a new tree-structured proof procedure (TMPR) together with a new completeness proof using proof transformation techniques and some improvements including the generation and use of lemmata. TMPR extends SLD-resolution with a Prolog-style backward chaining to full first-order logic by a controlled use of case analysis. This is done without having to extend negative goals needed, for example, for model elimination. A classification of TMPR, model elimination and related calculi in a common tableau framework is given. Thereafter, we present our extension of the TMPR proof procedure to L3 and show its soundness and completeness. As a side result, a TMPR proof system for the four-valued logic L4 is given. Finally, the restriction of TMPR to ‘L3-Horn clauses’ is considered, and, additionally, an idea for similarly extending model elimination and related systems to L3 (and L4) is illustrated.

If the words in Spinoza’s Ethics are considered as symbols, then certain words in the definitions of the Ethics can be replaced with symbols from set theory and we can reexamine Spinoza’s first definitions within a logical–mathematical frame. The authors believe that, some aspects of Spinoza’s work can be explained and illustrated through mathematics. A semantic relation between the definitions of the philosopher and set theory is presented. It is explained each chosen symbol. The ontological argument is developed through modal logic. And finally, we present some conclusions drawn from this work.

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.

We introduce here the study of generalnonmonotonic rule systems. These deal with situations where a conclusion is drawn from a system of beliefsS (and seen to be inS), basedboth on some being inS and on some restraints not being inS. In the monotone systems of traditional logic there are no restraints, conclusions are drawn solely based on premises being inS. Nonmonotonic rule systems capture the essential syntactic, semantic, and algorithmic features of many nonmonotone systems such as default logic, negation as failure, truth maintenance, autoepistemic logic, and also important combinatorial questions from mathematics such as the marriage problem. This reveals semantics and syntax and proof procedures and algorithms for computing belief sets in many cases where none were previously available and entirely uniformly. In particular, we introduce and study deductively closed sets, extensions and weak extensions. Semantics of nonmonotonic rule systems is studied in part II of this paper and extensions to predicate classical, intuitionistic, and modal logics are left to a later paper.