A tableau proof theory for CWPL

We prove that the positive fragment of first-order intuitionistic logic in the language with two variables and a single monadic predicate letter, without constants and equality, is undecidable. This holds true regardless of whether we consider semantics with expanding or constant domains. We then generalise this result to intervals [QBL, QKC] and [QBL, QFL], where QKC is the logic of the weak law of the excluded middle and QBL and QFL are first-order counterparts of Visser's basic and formal logics, respectively. We also show that, for most "natural" first-order modal logics, the two-variable fragment with a single monadic predicate letter, without constants and equality, is undecidable, regardless of whether we consider semantics with expanding or constant domains. These include all sublogics of QKTB, QGL, and QGrz -- among them, QK, QT, QKB, QD, QK4, and QS4.

9 First-order modal logic

In a seminal paper, Lin and Reiter introduced the notion of progression for basic action theories in the situation calculus. Earlier works by Moore, Scherl and Levesque extended the situation calculus to account for knowledge. In this paper, we study progression of knowledge in the situation calculus. We first adapt the concept of bisimulation from modal logic and extend Lin and Reiter's notion of progression to accommodate knowledge. We show that for physical actions, progression of knowledge reduces to forgetting predicates in first-order modal logic. We identify a class of first-order modal formulas for which forgetting an atom is definable in first-order modal logic. This class of formulas goes beyond formulas without quantifyingin. We also identify a simple case where forgetting a predicate reduces to forgetting a finite number of atoms. Thus we are able to show that for local-effect physical actions, when the initial KB is a formula in this class, progression of knowledge is definable in first-order modal logic. Finally, we extend our results to the multi-agent case.

In this paper, we present a knowledge representation system based on a first-order modal logic and discuss the deductive inference mechanism of this system. A possible-world model, which is used in discussing the semantics of a modal logic, can be regarded as structured knowledge, and modal operators can be used to describe various kinds of properties on a possible-world model. In this paper, we introduce a new concept, "viewpoints of modalities", in order to describe the knowledge structure effectively and compactly. We also show that schema formulas available in this framework are useful for the description of metaknowledge such as property inheritance. Therefore, a modal logic is suitable for representing both structured knowledge and metaknowledge. We construct a knowledge representation system based on a subset of a first-order modal logic, and give a complete deductive inference rule which is as effective as SLD resolution.

We present a syntactic abstraction method to reason about first-order modal logics by using theorem provers for standard first-order logic and for propositional modal logic.

Some sentences of natural languages, such as John might have been taller than Mary is, ascribe relations to objects each of which is associated with a possible world. In this example, Mary is associated with the actual world whereas John is associated with a possible world that can be distinct from the actual one. This phenomenon – the phenomenon of crossworld predication – cannot be reflected by means of standard modal logic. Recently, I elaborated a nonstandard logic accommodating crossworld predication; I shall call it CWPL (a logic for crossworld predication). CWPL is a first-order modal logic with equality; its formal language contains individual constants and lambda-operator. Semantically, CWPL is based on crossworld interpretation of predicates that assigns extensions to each n-ary predicate with respect to n-tuples of possible worlds rather than single possible worlds. Models for CWPL have varying domains. In CWPL’s semantics, truth values of formulae are relativized, inter alia, to partial functions from variables to possible worlds – this enables us to employ crossworld interpretation of predicates when evaluating formulae. In my 2023 paper, I presented syntax and semantics of CWPL; in the present paper, I elaborate a tableau proof theory for a slightly simplified version of CWPL that I shall call CWPL0. CWPL0 is simpler than CWPL in following respects: it is a logic without equality, its formal language does not contain individual constants, and it is based on models with constant domain. In the present paper, a tableau system for CWPL0 is presented and its completeness with respect to the crossworld semantics is established. The presented tableau system is based on tableau proof theory for standard modal logic proposed by Fitting and Mendelsohn, but their proof theory is modified so as to be adequate to crossworld semantics. I also show that the simplifications of CWPL mentioned above are not essential in the sense that the presented proof theory for CWPL0 can be transformed into a proof theory for CWPL without loss of completeness.

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.

First order modal logic (𝖥𝖮𝖬𝖫) is built by extending First Order Logic (𝖥𝖮) with modal operators. A typical formula is of the form \(\forall x \exists y \Box P(x,y)\) . Not only is 𝖥𝖮𝖬𝖫 undecidable, even simple fragments like that of restriction to unary predicate symbols, guarded fragment and two variable fragment, which are all decidable for 𝖥𝖮 become undecidable for 𝖥𝖮𝖬𝖫. In this paper we study Term Modal logic (𝖳𝖬𝖫) which allows modal operators to be indexed by terms. A typical formula is of the form \(\forall x \exists y~\Box _x P(x,y)\) . There is a close correspondence between 𝖳𝖬𝖫 and 𝖥𝖮𝖬𝖫 and we explore this relationship in detail in the paper. In contrast to 𝖥𝖮𝖬𝖫, we show that the two variable fragment (without constants, equality) of 𝖳𝖬𝖫 is decidable. Further, we prove that adding a single constant makes the two variable fragment of 𝖳𝖬𝖫 undecidable. On the other hand, when equality is added to the logic, it loses the finite model property.

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.

This paper responds to criticism of the Kripkean account of logical truth in first-order modal logic. The criticism, largely ignored in the literature, claims that when the box and diamond are interpreted as the logical modality operators, the Kripkean account is extensionally incorrect because it fails to reflect the fact that all sentences stating truths about what is logically possible are themselves logically necessary. I defend the Kripkean account by arguing that some true sentences about logical possibility are not logically necessary.

I consider the first-order modal logic which counts as valid those sentences which are true on every interpretation of the non-logical constants. Based on the assumptions that it is necessary what individuals there are and that it is necessary which propositions are necessary, Timothy Williamson has tentatively suggested an argument for the claim that this logic is determined by a possible world structure consisting of an infinite set of individuals and an infinite set of worlds. He notes that only the cardinalities of these sets matters, and that not all pairs of infinite sets determine the same logic. I use so-called two-cardinal theorems from model theory to investigate the space of logics and consequence relations determined by pairs of infinite sets, and show how to eliminate the assumption that worlds are individuals from Williamson's argument.

Standpoint extensions of KR formalisms have been recently introduced to incorporate multi-perspective modelling and reasoning capabilities. In such modal extensions, the integration of conceptual modelling and perspective annotations can be more or less tight, with monodic standpoint extensions striking a good balance as they enable advanced modelling while preserving good reasoning complexities. We consider the extension of C² – the counting two-variable fragment of first-order logic – by monodic standpoints. At the core of our treatise is a polytime translation of formulae in said formalism into standpoint-free C², requiring elaborate model-theoretic arguments. By virtue of this translation, the NEXPTIME-complete complexity of checking satisfiability in C² carries over to our formalism. As our formalism subsumes monodic S5 over C², our result also significantly advances the state of the art in research on first-order modal logics. As a practical consequence, the very expressive description logics ?ℋ?ℐ?ℬs and ?ℛ?ℐ?ℬs which subsume the popular W3C-standardized OWL 1 and OWL 2 ontology languages, are shown to allow for monodic standpoint extensions without any increase of standard reasoning complexity. We prove that NEXPTIME-hardness already occurs in much less expressive DLs as long as they feature both nominals and monodic standpoints. We also show that, with inverses, functionality, and nominals present, minimally lifting the monodicity restriction leads to undecidability.

Neighborhood-Sheaf Semantics for First-Order Modal Logic

Two of the most active research areas in automated deduction in modal logic are the use of translation methods to reduce its derivability problem to that of classical logic and the extension of existing automated reasoning techniques, developed initially for the propositional case, to first-order modal logics. This paper addresses both issues by extending the translation method for propositional modal logics known as □ -as-Pow (read “box-as-powerset”) to a widely used class of first-order modal logics, namely, the class of locally quantified modal logics. To do this, we prove a more general result that allows us to separate (classical) first-order from modal (propositional) reasoning. Our translation can be seen as an example application of this result, in both definition and proof of adequateness.KeywordsModal LogicPropositional VariableRelation SymbolConstant SymbolModal FormulaThese 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.

This article describes the relationship between network search space, search strategies and search logic with logical language. The logicalproperty of complex network search algorithm is determined by the initial state (the search state) and the search strategy of complex network space, which can be defined logically. The article also presents the defi- nition of complex network search states set and its relationship with first-order modal predicate logic. All possible net- work search state sets in the search strategy can be described with first-order modal predicate logic. Analysis of net work search state properties lays a good foundation for the intelligentization of complex network search processes in the future.