In this investigation we explore a general strategy for constructing modal theories where the modal notion is conceived as a predicate. The idea of this strategy is to develop modal theories over axiomatic theories of truth. In this first paper of our two part investigation we develop the general strategy and then apply it to the axiomatic theory of truth Friedman-Sheard. We thereby obtain the theory Modal Friedman-Sheard. The theory Modal Friedman-Sheard is then discussed from three different perspectives. First, we (...) show that Modal Friedman-Sheard preserves theoremhood modulo translation with respect to modal operator logic. Second, we turn to semantic aspects and develop a modal semantics for the newly developed theory. Third, we investigate whether the modal predicate of Modal Friedman-Sheard can be understood along the lines of a proposal of Kripke, namely as a truth predicate modified by a modal operator. (shrink)

I argue that the ‘aoristic’ operators, which are intended to describe the logic of vagueness, do not form a standard modal logic. I redefine the operators so that they do form a standard modal logic, provide a semantics of that logic, and argue that the logic is not as strong as standardly claimed.

Michael Dummett argues, against Saul Kripke, that there could have been unicorns. He then claims that this possibility shows that the logic of metaphysical modality is not S5, and, in particular, that the B axiom is false. Dummett’s argument against B, however, is invalid. I show that although there are number of ways to repair Dummett’s argument against B, each requires a controversial metaphysical or semantic commitment, and that, regardless of this, the case against B is undermotivated. Dummett’s case is (...) still of interest, however, as if his assumptions are correct, S5 has to go, with the natural culprit being S4. (shrink)

This paper examines causal theories of reference with respect to how plausible an account they give of non-physical natural kind terms such as ‘gene’ as well as of the truth of the associated theoretical claims. I first show that reference fixism for ‘gene’ fails. By this, I mean the claim that the reference of ‘gene’ was stable over longer historical periods, for example, since the classical period of transmission genetics. Second, I show that the theory of partial reference does not (...) do justice to some widely held realist intuitions about classical genetics. This result is at loggerheads with the explicit goals usually associated with partial theories of reference, which is to defend a realist semantics for scientific terms. Thirdly, I show that, contrary to received wisdom and perhaps contrary to physics and chemistry, neither reference fixism nor partial reference are necessary in order to hold on to scientific realism about biology. I pinpoint the reasons for this in the nature of biological kinds, which do not even remotely resemble natural kinds (i.e., Lockean real essences) as traditionally conceived. (shrink)

In 'Fiction and Fictionalism', Mark Sainsbury has recently dubbed “Selection Problem” a serious trouble for Meinongian object theories. Typically, Meinongianism has been phrased as a kind of realism on nonexistent objects : these are mind-independent things, not mental simulacra, having the properties they have independently from the activity of any cognitive agent. But how can one single out an object we have no causal acquaintance with, and which is devoid of spatiotemporal location, picking it out from a pre-determined, mind-independent set (...) ? In this paper, I set out a line of response by distinguishing different ways in which a thing may not exist. I show that the selection problem (a) does not arise for past, currently nonexistent objects ; (b) may not arise also for future existents (provided one massages naïve intuitions a bit) ; and (c) even for mere possibilia ; but (d) is a real snag for purely fictional objects, such as Holmes or Gandalf. As for (d), I propose a solution that forces Meinongianism to introduce a kind of ontological dependence of purely fictional nonexistents upon existents. The strategy complicates the intuitively simple, naïve Meinongian framework a bit, but looks quite promising. (shrink)

It is widely held that if an object a is identical (or non-identical) to an object b, then it is necessary that a is identical (non-identical) to b. This view is supported an argument from Leibniz's Law and a popular conception of de re modality. On the other hand, there are good reasons to allow for contingent identity. Various alternative accounts of de re modality have been developed to achieve this kind of generality, and to explain what is wrong with (...) the argument from Leibniz's Law. (shrink)

Tarski apresentou sua definição de operador de consequência com a intenção de expor as concepções fundamentais da consequência lógica. Um espaço de Tarski é um par ordenado determinado por um conjunto não vazio e um operador de consequência sobre este conjunto. Esta estrutura matemática caracteriza um espaço quase topológico. Este artigo mostra uma visão algébrica dos espaços de Tarski e introduz uma lógica proposicional modal que interpreta o seu operador modal nos conjuntos fechados de algum espaço de Tarski. DOI:10.5007/1808-1711.2010v14n1p47.

We present an embedding of quantified multimodal logics into simple type theory and prove its soundness and completeness. A correspondence between QKπ models for quantified multimodal logics and Henkin models is established and exploited. Our embedding supports the application of off-the-shelf higher-order theorem provers for reasoning within and about quantified multimodal logics. Moreover, it provides a starting point for further logic embeddings and their combinations in simple type theory.

The paper is a brief survey of the most important semantic constructions founded on the concept of possible world. It is impossible to capture in one short paper the whole variety of the problems connected with manifold applications of possible worlds. Hence, after a brief explanation of some philosophical matters I take a look at possible worlds from rather technical standpoint of logic and focus on the applications in formal semantics. In particular, I would like to focus on the fruitful (...) marriage of possible world semantics and algebra and its evolution leading to very general construction of Wójcicki called referential semantics and some of its refinements. The presentation is informal and sketchy; the main purpose is to put in one place a short, and readable I hope, description of the most important constructions and to point out the main sources of these solutions. (shrink)

This chapter investigates the computational consequences of a broadly Gricean view of language use as intentional activity. In this view, dialogue rests on coordinated reasoning about communicative intentions. The speaker produces each utterance by formulating a suitable communicative intention. The hearer understands it by recognizing the communicative intention behind it. When this coordination is successful, interlocutors succeed in considering the same intentions— that is, the same representations of utterance meaning—as the dialogue proceeds. In this paper, I emphasize that these intentions (...) can be formalized; we can provide abstract but systematic representations that spell out what a speaker is trying to do with an utterance. Such representations describe utterances simultaneously as the product of our knowledge of grammar and as actions chosen for a reason. In particular, they must characterize the speaker’s utterance in grammatical terms, provide the links to the context that the grammar requires, and so arrive at a contribution that the speaker aims to achieve. Because I have implemented this formalism, we can regard it as a possible analysis of conversational processes at the level of computational theory. Nevertheless, this analysis leaves open what the nature of the biological computation involved in inference to intentions is, and what regularities in language use support this computation. (shrink)

This paper addresses questions of universality related to ontological engineering, namely aims at substantiating (negative) answers to the following three basic questions: (i) Is there a ‘universal ontology’?, (ii) Is there a ‘universal formal ontology language’?, and (iii) Is there a universally applicable ‘mode of reasoning’ for formal ontologies? To support our answers in a principled way, we present a general framework for the design of formal ontologies resting on two main principles: firstly, we endorse Rudolf Carnap’s principle of logical (...) tolerance by giving central stage to the concept of logical heterogeneity, i.e. the use of a plurality of logical languages within one ontology design. Secondly, to structure and combine heterogeneous ontologies in a semantically well-founded way, we base our work on abstract model theory in the form of institutional semantics, as forcefully put forward by Joseph Goguen and Rod Burstall. In particular, we employ the structuring mechanisms of the heterogeneous algebraic specification language HetCasl for defining a general concept of heterogeneous, distributed, highly modular and structured ontologies, called hyperontologies. Moreover, we distinguish, on a structural and semantic level, several different kinds of combining and aligning heterogeneous ontologies, namely integration, connection, and refinement. We show how the notion of heterogeneous refinement can be used to provide both a general notion of sub-ontology as well as a notion of heterogeneous equivalence of ontologies, and finally sketch how different modes of reasoning over ontologies are related to these different structuring aspects. (shrink)

Variabilism is the view that proper names (like pronouns) are semantically represented as variables. Referential names, like referential pronouns, are assigned their referents by a contextual variable assignment (Kaplan 1989). The reference parameter (like the world of evaluation) may also be shifted by operators in the representation language. Indeed verbs that create hyperintensional contexts, like ‘think’, are treated as operators that simultaneously shift the world and assignment parameters. By contrast, metaphysical modal operators shift the world of assessment only. Names, being (...) variables, refer rigidly in the latter merely intensional contexts, but may vary their reference in hyperintensional contexts. This conforms to the intuition that the content of attitude ascriptions encapsulates referential uncertainty. Furthermore, names in hyperintensional contexts are ambiguous between de re* and de dicto* interpretations. This fact is used to account for asymmetric mistaken identity attributions (for example, Biron thinks Katherine is Rosaline, but he doesn’t think Rosaline is Katherine). -/- The variable theory compares favourably with its alternatives, including Millianism and descriptivism. Millians cannot account for the behaviour of names in hyperintensional contexts, while descriptivists cannot generate a necessary contrast between intensional and hyperintensional contexts. No other theory can capture the facts pertaining to the existentially bound use of names. (shrink)

In Kantian philosophy, the term “condition of possibility” is central, but carries the following ambiguity. According to one reading, “condition of possibility” merely means “necessary condition”. However, it is demonstrated that a deeper interpretation of the term “possibility” proves to be more fruitful. This reading allows us to reconstruct an important background assumption of Kant: Every condition of the possibility of experience holds necessarily, provided that experience is possible. Or more generally: All conditions of the possibility hold necessarily as long (...) as the conditioned is actually possible. -/- This conclusion is understood through the axiomatic system S5. Therefore I argue by referencing the schematism chapter that Kant’s notion of modal terms can be formalized through S5. (shrink)

L.E.J. Brouwer famously took the subject’s intuition of time to be foundational and from there ventured to build up mathematics. Despite being largely critical of formal methods, Brouwer valued axiomatic systems for their use in both communication and memory. Through the Dutch Mathematical Society, Gerrit Mannoury posed a challenge in 1927 to provide an axiomatization of intuitionistic arithmetic. Arend Heyting’s 1928 axiomatization was chosen as the winner and has since enjoyed the status of being the _de facto_ formalization of intuitionistic (...) arithmetic. We argue that axiomatizations of intuitionistic arithmetic ought to make explicit the role of the subject’s activity in the intuitionistic arithmetical process. While Heyting Arithmetic is useful when we want to contrast constructed objects with platonistic ones, Heyting Arithmetic omits the contribution of the subject and thus falls short as a response to Mannoury’s challenge. We offer our own solution, Doxastic Heyting Arithmetic, or \({\textsf{DHA}}\), which we contend axiomatizes Brouwerian intuitionistic arithmetic. (shrink)

In this paper, the predicate counterparts, defined both axiomatically and semantically by means of Kripke frames, of the modal propositional logics $\textbf {GL}$, $\textbf {Grz}$, $\textbf {wGrz}$ and their extensions are considered. It is proved that the set of semantical consequences on Kripke frames of every logic between $\textbf {QwGrz}$ and $\textbf {QGL.3}$ or between $\textbf {QwGrz}$ and $\textbf {QGrz.3}$ is $\Pi ^1_1$-hard even in languages with three (sometimes, two) individual variables, two (sometimes, one) unary predicate letters, and a single (...) proposition letter. As a corollary, it is proved that infinite families of modal predicate axiomatic systems, based on the classical first-order logic and the modal propositional logics $\textbf {GL}$, $\textbf {Grz}$, $\textbf {wGrz}$ are not Kripke complete. Both $\Pi ^1_1$-hardness and Kripke incompleteness results of the paper do not depend on whether the logics contain the Barcan formula. (shrink)

The article deals with infinitary modal logic. We first discuss the difficulties related to the development of a satisfactory proof theory and then we show how to overcome these problems by introducing a labelled sequent calculus which is sound and complete with respect to Kripke semantics. We establish the structural properties of the system, namely admissibility of the structural rules and of the cut rule. Finally, we show how to embed common knowledge in the infinitary calculus and we discuss first-order (...) extensions of infinitary modal logic. (shrink)

dolce, the first top-level (foundational) ontology to be axiomatized, has remained stable for twenty years and today is broadly used in a variety of domains. dolce is inspired by cognitive and linguistic considerations and aims to model a commonsense view of reality, like the one human beings exploit in everyday life in areas as diverse as socio-technical systems, manufacturing, financial transactions and cultural heritage. dolce clearly lists the ontological choices it is based upon, relies on philosophical principles, is richly formalized, (...) and is built according to well-established ontological methodologies, e.g. OntoClean. Because of these features, it has inspired most of the existing top-level ontologies and has been used to develop or improve standards and public domain resources (e.g. CIDOC CRM, DBpedia and WordNet). Being a foundational ontology, dolce is not directly concerned with domain knowledge. Its purpose is to provide the general categories and relations needed to give a coherent view of reality, to integrate domain knowledge, and to mediate across domains. In these 20 years dolce has shown that applied ontologies can be stable and that interoperability across reference and domain ontologies is a reality. This paper briefly introduces the ontology and shows how to use it on a few modeling cases. (shrink)

The Unified Foundational Ontology (UFO) was developed over the last two decades by consistently putting together theories from areas such as formal ontology in philosophy, cognitive science, linguistics, and philosophical logics. It comprises a number of micro-theories addressing fundamental conceptual modeling notions, including entity types and relationship types. The aim of this paper is to summarize the current state of UFO, presenting a formalization of the ontology, along with the analysis of a number of cases to illustrate the application of (...) UFO and facilitate its comparison with other foundational ontologies in this special issue. (The cases originate from the First FOUST Workshop – the Foundational Stance, an international forum dedicated to Foundational Ontology research.). (shrink)

This paper presents rules in sequent calculus for a binary quantifier I to formalise definite descriptions: Ix[F, G] means ‘The F is G’. The rules are suitable to be added to a system of positive free logic. The paper extends the proof of a cut elimination theorem for this system by Indrzejczak by proving the cases for the rules of I. There are also brief comparisons of the present approach to the more common one that formalises definite descriptions with a (...) term forming operator. In the final section rules for I for negative free and classical logic are also mentioned. (shrink)

Some philosophers, like Kripke, Williamson, Hawthorne, and Turri, have offered examples of claims that are allegedly contingent and a priori justifiable. If any of these examples is genuine, this would upend the traditional epistemological classification on which (a) all and only a priori justifiable claims are necessary and (b) all and only a posteriori ones are contingent. I argue here that these examples are not genuine. This conclusion is not new, but the strategy pursued here is to formalize these muchdiscussed (...) examples in symbolic logics. Once formalized, a perspicuous representation of their logical form will bring into sharp relief that these examples are not both contingent and a priori. Two takeaways are (1) that the traditional epistemological classification remains plausible and (2) that one’s proposed examples of contingent a priori claims should be supported by a formalization in one’s preferred background symbolic logic. (shrink)

In this paper, I will defend a new compositional semantics for single-domain free logic. This semantics makes use of a distinction between the semantic value of a singular term and its semantic referent. The semantic value of a singular term is conceived of as a set that either contains the semantic referent or no element at all. The semantic referent is the object that the term designates. Before I will introduce this new semantics for single-domain predicate and an S5-type modal (...) logic in detail, I will present five related problems of compositionality of the standard semantics for these logical systems. After that I will show in detail how my new proposed alternative semantics can be used to solve nearly all of the outlined five problems. (shrink)

The grounding relation is routinely characterized by means of logical postulates. The aim of this paper is twofold. First, I show that a subset of those postulates is incompatible with a minimal characterization of metaphysical modality. Then I consider a number of ways for reconciling ground with modality. The simplest and most elegant solution consists in adopting serious actualism, which is best captured within a first-order modal language with predicate abstraction governed by negative free logic. I also explore a number (...) of alternative strategies by revising the ground-theoretic postulates, while keeping the modal ones fixed. As I argue, each of those strategies is either unviable, highly contentious, or insufficiently motivated. (shrink)

Kripke (1980) hypothesizes a link between rigidity and scope: a singular term is rigid over a space S of possibilities just in case it is scopeless with respect to modals that quantify over S. Kripke’s hypothesis works well when we consider the interaction of singular terms with metaphysical modals, but runs into trouble when we consider the interaction of singular terms with epistemic modals. After describing the trouble in detail, and considering one non-solution to it, I develop a novel version (...) of dynamic semantics that resolves the problem. (shrink)

This essay introduces a puzzle about the interaction between quantifiers and epistemic modals. The puzzle motivates the idea that whether an object satisfies an epistemically modalized predicate depends on the mode of presentation of the domain of quantification. I compare two ways of implementing this idea, one using counterpart theory, the other using Aloni's 'conceptual covers' theory, and then provides some evidence in favor of the former.

In this paper, I develop a new set of doxastic logical systems and I show how they can be used to solve several well-known problems in doxastic logic, for example the so-called problem of logical omniscience. According to this puzzle, the notions of knowledge and belief that are used in ordinary epistemic and doxastic symbolic systems are too idealised. Hence, those systems cannot be used to model ordinary human or human-like agents' beliefs. At best, they can describe idealised individuals. The (...) systems in this paper can be used to symbolise not only the doxastic states of perfectly rational individuals, but also the beliefs of finite humans. Proof-theoretically, I will use a tableau technique. Every system is combined with predicate logic with necessary identity and ‘possibilist’ quantifiers and modal logic with two kinds of modal operators for relative and absolute necessity. The semantics is a possible world semantics. Finally, I prove that every tableau system in the paper is sound and complete with respect to its semantics. (shrink)

Open futurism is the indeterministic position according to which the future is ‘open’, i.e., there is now no fact of the matter as to what future contingent events will actually obtain. Many open futurists hold a branching conception of time, in which a variety of possible futures exist. This paper introduces two challenges to branching-time open futurism, which are similar in spirit to a challenge posed by Fine to tense realism. The paper argues that, to address the new challenges, open (...) futurists must adopt an objective, non-perspectival notion of actuality and subscribe to an A-theoretic, dynamic conception of reality. Moreover, given a natural understanding of “actual future”, it is perfectly sensible for open futurists to hold that a unique, objectively actual future exists, contrary to a common assumption in the current debate. The paper also contends that recognising the existence of a unique actual future helps open futurists to avoid potential misconceptions. (shrink)

We present in this note a plea for Barcan formula. This view connects Barcan formula with a modal principle that expresses the -Introduction rule of first-order logic.

Name der Zeitschrift: Deutsche Zeitschrift für Philosophie Jahrgang: 65 Heft: 4 Seiten: 657-688.

We show that basic hybridization makes it possible to give straightforward Henkin-style completeness proofs even when the modal logic being hybridized is higher-order. The key ideas are to add nominals as expressions of type t, and to extend to arbitrary types the way we interpret \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$@_i$\end{document} in propositional and first-order hybrid logic. This means: interpret \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$@_i\alpha _a$\end{document}, where \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} (...) \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$\alpha _a$\end{document} is an expression of any type \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$a$\end{document}, as an expression of type \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$a$\end{document} that rigidly returns the value that \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$\alpha_a$\end{document} receives at the i-world. The axiomatization and completeness proofs are generalizations of those found in propositional and first-order hybrid logic, and we automatically obtain a wide range of completeness results for stronger logics and languages. Our approach is deliberately low-tech. We don’t, for example, make use of Montague’s intensional type s, or Fitting-style intensional models; we build, as simply as we can, hybrid logicover Henkin’s logic. (shrink)

A consequence relation \ is monotonic iff for premise sets \ and conclusion \, if \, \, then \; and non-monotonic if this fails in some instance. More plainly, a consequence relation is monotonic when whatever is entailed by a premise set remains entailed by any of its supersets. From the High Middle Ages through the Early Modern period, consequence in theology is assumed to be monotonic. Concomitantly, to the degree the argument formulated by Anselm at Proslogion 2–4 is taken (...) up by later commentators, it is accepted or rejected in accordance with a monotonic notion of consequence. Examining Anselm’s use of parallelism in the Proslogion, I show Anselm embeds his famous argument in Proslogion 2–4 in a non-monotonic context. The results here presented challenge some deeply ingrained ideas governing the historiography of the long twelfth century, particularly concerning how the theology of the later eleventh through the twelfth century relates to the scholasticism of the thirteenth. (shrink)

The literature on quantum logic emphasizes that the algebraic structures involved with orthodox quantum mechanics are non distributive. In this paper we develop a particular algebraic structure, the quasi-lattice ( $${\mathfrak{I}}$$ -lattice), which can be modeled by an algebraic structure built in quasi-set theory $${\mathfrak{Q}}$$. This structure is non distributive and involve indiscernible elements. Thus we show that in taking into account indiscernibility as a primitive concept, the quasi-lattice that ‘naturally’ arises is non distributive.

The contribution of this paper lies with providing a systematically specified and intuitive interpretation pattern and delineating a class of relational structures and models providing a natural interpretation of logical operators on an underlying propositional calculus of Positive Lattice Logic and subsequently proving a generic completeness theorem for the related class of logics, sometimes collectively referred to as Generalized Galois Logics.

In this paper, we propose a decidable single-agent modal logic for reasoning about goal-directed “knowing how”, based on ideas from linguistics, philosophy, modal logic, and automated planning in AI. We first define a modal language to express “I know how to guarantee \ given \” with a semantics based not on standard epistemic models but on labeled transition systems that represent the agent’s knowledge of his own abilities. The semantics is inspired by conformant planning in AI. A sound and complete (...) proof system is given to capture valid reasoning patterns, which highlights the compositional nature of “knowing how”. The logical language is further extended to handle knowing how to achieve a goal while maintaining other conditions. (shrink)

In this paper we examine Prior's reconstruction of Master Argument [4] in some modal-tense logic. This logic consists of a purely tense part and Diodorean definitions of modal alethic operators. Next we study this tense logic in the pure tense language. It is the logic $K_t 4$ plus a new axiom $:p \wedge Gp \supset PGp'$. This formula was used by Prior in his original analysis of Master Argument. is usually added as an extra axiom to an axiomatization of the (...) logic of linear time. In that case the set of moments is a total order and must be left-discrete without the least moment. However, the logic of Master Argument does not require linear time. We show what properties of the set of moments are exactly forced by in the reconstruction of Prior. We make also some philosophical remarks on the analyzed reconstruction. (shrink)

While standard first-order modal logic is quite powerful, it cannot express even very simple sentences like “I could have been taller than I actually am” or “Everyone could have been smarter than they actually are”. These are examples of cross-world predication, whereby objects in one world are related to objects in another world. Extending first-order modal logic to allow for cross-world predication in a motivated way has proven to be notoriously difficult. In this paper, I argue that the standard accounts (...) of cross-world predication all leave something to be desired. I then propose an account of cross-world predication based on quantified hybrid logic and show how it overcomes the limitations of these previous accounts. I will conclude by discussing various philosophical consequences and applications of such an account. (shrink)

In this paper, we discuss formal reconstructions of Anselm’s ontological argument. We first present a number of requirements that any successful reconstruction should meet. We then offer a detailed preparatory study of the basic concepts involved in Anselm’s argument. Next, we present our own reconstructions—one in modal logic and one in classical logic—and compare them with each other and with existing reconstructions from the reviewed literature. Finally, we try to show why and how one can gain a better understanding of (...) Anselm’s argument by using modern formal logic. In particular, we try to explain why formal reconstructions of the argument, despite its apparent simplicity, tend to become quite involved. (shrink)

This paper introduces an axiomatisation for equational hybrid logic based on previous axiomatizations and natural deduction systems for propositional and first-order hybrid logic. Its soundness and completeness is discussed. This work is part of a broader research project on the development a general proof calculus for hybrid logics.

Can we find propositions that cannot rationally be denied in any possible world without assuming the existence of that same proposition, and so involving ourselves in a contradiction? In other words, can we find transworld propositions needing no further foundation or justification? Basically, three differing positions can be imagined: firstly, a relativist position, according to which ultimately founded propositions are impossible; secondly, a meta-relativist position, according to which ultimately founded propositions are possible but unnecessary; and thirdly, an absolute position, according (...) to which such propositions are necessary. In this short essay I show that under the premise of modal logic S5 with constant domain there are ultimately founded propositions and that their existence is even necessary, and I will give some reasons for the superiority of S5 over other logics. (shrink)

This article opposes a view widely accepted in studies concerning the history of modal logic, according to which (i) the approach of C. I. Lewis towards constructing modern modal logic was purely syntactical (i.e. limited to the construction of axiomatic systems S1-S5 of propositional modal logic), and (ii) the notion of a possible world was incorporated into modern logic and philosophy mainly by authors such as Rudolf Carnap and Saul Kripke. The article presents Lewis' definition of a possible world, and (...) his formulation of the truth-conditions of statements containing strict implication as their main connective in terms of possible worlds. The main question of the article is whether it is possible to consider Lewis' work in this area as an early stage of the development of possible world semantics, and if so, in what sense? The article concludes by answering affirmatively, due to soundness and completeness proofs with respect to S5 using Lewis' semantics. (shrink)

We describe a general logical framework, Justification Logic, for reasoning about epistemic justification. Justification Logic is based on classical propositional logic augmented by justification assertions t: F that read t is a justification for F. Justification Logic absorbs basic principles originating from both mainstream epistemology and the mathematical theory of proofs. It contributes to the studies of the well-known Justified True Belief vs. Knowledge problem. We state a general Correspondence Theorem showing that behind each epistemic modal logic, there is a (...) robust system of justifications. This renders a new, evidence-based foundation for epistemic logic. As a case study, we offer a resolution of the GoldmanRed Barns in Justification Logic. Furthermore, we formalize the well-known Gettier example and reveal hidden assumptions and redundancies in Gettier’s reasoning. (shrink)

This is a companion paper to Braüner where a natural deduction system for propositional hybrid logic is given. In the present paper we generalize the system to the first-order case. Our natural deduction system for first-order hybrid logic can be extended with additional inference rules corresponding to conditions on the accessibility relations and the quantifier domains expressed by so-called geometric theories. We prove soundness and completeness and we prove a normalisation theorem. Moreover, we give an axiom system first-order hybrid logic.

Despite their controversial ontological status, the discussion on arbitrary objects has been reignited in recent years. According to the supporting views, they present interesting and unique qualities. Among those, two define their nature: their assuming of values, and the way in which they present properties. Leon Horsten has advanced a particular view on arbitrary objects which thoroughly describes the earlier, arguing they assume values according to a sui generis modality, which he calls afthairetic. In this paper, we offer a general (...) method for defining the minimal system of this modality for any given first-order theory, and possible extensions of it that incorporate further aspects of Horsten’s account. The minimal system presents an unconventional inference rule, which deals with two different notions of derivability. For this reason and the failure of the Necessitation rule, in its full generality, the resulting system is non-normal. Then, we provide conditional soundness and completeness results for the minimal system and its extensions. (shrink)

The Thousands of Problems for Theorem Provers (TPTP) World is a well-established infrastructure that supports research, development and deployment of automated theorem proving systems. This paper provides an overview of the logic languages of the TPTP World, from classical first-order form (FOF), through typed FOF, up to typed higher-order form, and beyond to non-classical forms. The logic languages are described in a non-technical way and are illustrated with examples using the TPTP language.

Recently, interest has surged in similarity-based epistemologies of possibility. However, it has been pointed out that the notion of ‘relevant similarity’ is not properly developed in this literature. In this paper, I look at the research done in the field of analogical reasoning, where we find that one of the most promising ways of capturing relevance in similarity reasoning is by relying on the predictive analogy similarity relation. This takes relevant similarity to be based on shared properties that have structural (...) relations to the property of interest. I argue that if we base our epistemology of possibility on similarity reasoning on the predictive analogy similarity relation, we require prior knowledge of the specifics of these structural relations. I discuss a number of possible responses to this on behalf of the similarity theorists given their methodological approach to the epistemology of modality more generally. They could either opt for making explicit the metaphysics underlying these structural relations, in which case they need to spell out how we can come to know these relations. Or they could opt for developing a theory that explains why we do not need to have explicit knowledge of these structural relations; for example by suggesting that we make use of epistemic shortcuts. (shrink)

In this paper, I present and motivate a modal set theory consistent with the idea that there is only one size of infinity.