Decidability of quantified propositional intuitionistic logic and S4 on trees

Quantified propositional intuitionistic logic is obtained from propositional intuitionistic logic by adding quantifiers \forall p, \exists p over propositions. In the context of Kripke semantics, a proposition is a subset of the worlds in a model structure which is upward closed. Kremer (1997) has shown that the quantified propositional intuitionistic logic H\pi+ based on the class of all partial orders is recursively isomorphic to full second-order logic. He raised the question of whether the logic resulting from restriction to trees is axiomatizable. It is shown that it is, in fact, decidable. The methods used can also be used to establish the decidability of modal S4 with propositional quantification on similar types of Kripke structures.Comment: v2, 9 pages, corrections and additions; v1 8 page

Representing Counterparts

This paper presents a counterpart theoretic semantics for quantified modal logic based on a fleshed out account of Lewis's notion of a 'possibility'. According to the account a possibility consists of a world and some haecceitistic information about how each possible individual gets represented de re. Following Hazen, a semantics for quantified model logic based on evaluating formulae at possibilities is developed. It is shown that this framework naturally accommodates an actuality operator, addressing recent objections to counterpart theory, and is equivalent to the more familiar Kripke semantics for quantied modal logic with an actuality operator

Ontology of sentential moods

In this paper ontological implications of the Barcan formula and its converse will be discussed at the conceptual and technical level. The thesis that will be defended is that sentential moods are not ontologically neutral since the rejection of ontological implications of Barcan formula and its converse is a condition of a possibility of the imperative mood. The paper is divided into four sections. In the first section a systematization of semantical systems of quantified modal logic is introduced for the purpose of making explicit their ontological presuppositions. In this context Jadacki's ontological difference between being and existence is discussed and analyzed within the framework of hereby proposed system of quantified modal logic. The second section discusses ontological implications of the Barcan formula and its converse within the system accommodating the difference between being and existence. The third section presents a proof of incompatibility of the Barcan formula and its converse with the use of imperatives. In the concluding section, a thesis on logical pragmatics foreclosing the dilemma between necessitism and contingentism is put forward and defended against some objections

Išvedimai tarp Buridano modalinių teiginių

In recent years modal syllogistic provided by 14th century logician John Buridan has attracted increasing attention of historians of medieval logic. The widespread use of quantified modal logic with the apparatus of possible worlds semantics in current analytic philosophy has encouraged the investigation of the relation of Buridan’s theory of modality with the modern developments of symbolic modal logic. We focus on the semantics of and the inferential relations among the propositions that underlie Buridan’s theory of modal syllogism. First, we review all inferences between propositions of necessity, possibility, contingency, and non-contingency, with or without quod est locution, that are valid in Buridan’s semantics, and offer a comprehensive diagrammatic representation that includes them all. We then ask the question if there is a way to model those results in first order modal logic. Three ways of formalizing Buridan’s propositions in quantified modal logic are considered. Comparison of inferences between the quantified formulas and Buridan’s propositions reveals that, when supplied with a suitable formalization, Buridan’s semantics of categorical statements and immediate inferences among them can be fully captured by the quantified modal system T.Straipsnyje nagrinėjama XIV a. logiko Jono Buridano modalinė logika, pastaraisiais metais sulaukianti vis didesnio viduramžių logikos tyrėjų dėmesio. Šiuolaikinėje analitinėje filosofijoje plačiai naudojama modalinė predikatų logika, paremta galimų pasaulių semantika, paskatino kelti klausimus apie Buridano modalumų teorijos ir šiuolaikinės simbolinės modalinės logikos santykį. Straipsnyje tiriama Buridano modalinės silogistikos pagrindą sudarančių teiginių semantinė interpretacija ir tarp šių teiginių galiojantys išvedimo ryšiai. Pirmiausia aptariami išvedimai tarp būtinumo, galimumo, atsitiktinumo ir neatsitiktinumo teiginių (su ir be frazės quod est), kurie yra logiškai taisyklingi pagal Buridano aprašytą semantiką. Pateikiama išsami diagraminė visų išvedimo sąryšių reprezentacija. Siekiant nustatyti, ar esama tinkamo būdo Buridano gaunamus rezultatus išreikšti predikatų logikoje su modalumais, aptariamos trys literatūroje aptinkamos Buridano teiginių formalizacijos versijos. Palyginus taisyklingus išvedimus tarp kvantifikuotų formulių ir tarp Buridano modalinių teiginių, galima tvirtinti, kad Buridano teiginių semantika ir galiojantys tiesioginiai išvedimai gali būti adekvačiai perteikiami predikatų logikos su aletiniais modalumais (sistemos T) priemonėmis

Logicism, Possibilism, and the Logic of Kantian Actualism

In this extended critical discussion of 'Kant's Modal Metaphysics' by Nicholas Stang (OUP 2016), I focus on one central issue from the first chapter of the book: Stang’s account of Kant’s doctrine that existence is not a real predicate. In §2 I outline some background. In §§3-4 I present and then elaborate on Stang’s interpretation of Kant’s view that existence is not a real predicate. For Stang, the question of whether existence is a real predicate amounts to the question: ‘could there be non-actual possibilia?’ (p.35). Kant’s view, according to Stang, is that there could not, and that the very notion of non-actual or ‘mere’ possibilia is incoherent. In §5 I take a close look at Stang’s master argument that Kant’s Leibnizian predecessors are committed to the claim that existence is a real predicate, and thus to mere possibilia. I argue that it involves substantial logical commitments that the Leibnizian could reject. I also suggest that it is danger of proving too much. In §6 I explore two closely related logical commitments that Stang’s reading implicitly imposes on Kant, namely a negative universal free logic and a quantified modal logic that invalidates the Converse Barcan Formula. I suggest that each can seem to involve Kant himself in commitment to mere possibilia

Bundled Fragments of First-Order Modal Logic: (Un)Decidability

Quantified modal logic is notorious for being undecidable, with very few known decidable fragments such as the monodic ones. For instance, even the two-variable fragment over unary predicates is undecidable. In this paper, we study a particular fragment, namely the bundled fragment, where a first-order quantifier is always followed by a modality when occurring in the formula, inspired by the proposal of [Yanjing Wang, 2017] in the context of non-standard epistemic logics of know-what, know-how, know-why, and so on. As always with quantified modal logics, it makes a significant difference whether the domain stays the same across possible worlds. In particular, we show that the predicate logic with the bundle "forall Box" alone is undecidable over constant domain interpretations, even with only monadic predicates, whereas having the "exists Box" bundle instead gives us a decidable logic. On the other hand, over increasing domain interpretations, we get decidability with both "forall Box" and "exists Box" bundles with unrestricted predicates, where we obtain tableau based procedures that run in PSPACE. We further show that the "exists Box" bundle cannot distinguish between constant domain and variable domain interpretations