Modal Logic

Edited by Matteo Pascucci (Slovak Academy of Sciences, Central European University)
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  1. Modal Logic: The System S5.Gabriel Andrus - manuscript
    A brief overview of the system S5 in modal logic as defined by Brian F. Chellas, author of "Modal Logic: An Introduction." The history and usage of modal logic are given mention, along with some applications. Very much a draft. Written for PhileInSophia on July 5, 2021.
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  2. The Logic of Logical Necessity.Andrew Bacon & Kit Fine - manuscript
    Prior to Kripke's seminal work on the semantics of modal logic, McKinsey offered an alternative interpretation of the necessity operator, inspired by the Bolzano-Tarski notion of logical truth. According to this interpretation, `it is necessary that A' is true just in case every sentence with the same logical form as A is true. In our paper, we investigate this interpretation of the modal operator, resolving some technical questions, and relating it to the logical interpretation of modality and some views in (...)
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  3. Logics of Formal Inconsistency Enriched with Replacement: An Algebraic and Modal Account.Walter Carnielli, Marcelo E. Coniglio & David Fuenmayor - manuscript
    One of the most expected properties of a logical system is that it can be algebraizable, in the sense that an algebraic counterpart of the deductive machinery could be found. Since the inception of da Costa's paraconsistent calculi, an algebraic equivalent for such systems have been searched. It is known that these systems are non self-extensional (i.e., they do not satisfy the replacement property). More than this, they are not algebraizable in the sense of Blok-Pigozzi. The same negative results hold (...)
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  4. Cathoristic Logic.Richard Evans - manuscript
    Cathoristic logic is a multi-modal logic where negation is replaced by a novel operator allowing the expression of incompatible sentences. We present the syntax and semantics of the logic including complete proof rules, and establish a number of results such as compactness, a semantic characterisa- tion of elementary equivalence, the existence of a quadratic-time decision pro- cedure, and Brandom’s incompatibility semantics property. We demonstrate the usefulness of the logic as a language for knowledge representation.
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  5. We Belong Together? A Plea for Modesty in Modal Plural Logic.Simon Hewitt - manuscript
    It is often assumed that pluralities are rigid, in the sense of having all and only their actual members necessarily. This assumption is operative in standard approaches to modal plural logic. I argue that a sceptical approach towards the assumption is warranted.
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  6. A Modal Logic for Gödelian Intuition.Hasen Khudairi - manuscript
    This essay aims to provide a modal logic for rational intuition. Similarly to treatments of the property of knowledge in epistemic logic, I argue that rational intuition can be codified by a modal operator governed by the axioms of a dynamic provability logic, which embeds GL within the modal $\mu$-calculus. Via correspondence results between modal logic and the bisimulation-invariant fragment of second-order logic, a precise translation can then be provided between the notion of 'intuition-of', i.e., the cognitive phenomenal properties of (...)
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  7. How Gödelian Ontological Arguments Fail.Matthew Parker - manuscript
    Ontological arguments like those of Gödel (1995) and Pruss (2009; 2012) rely on premises that initially seem plausible, but on closer scrutiny are not. The premises have modal import that is required for the arguments but is not immediately grasped on inspection, and which ultimately undermines the simpler logical intuitions that make the premises seem plausible. Furthermore, the notion of necessity that they involve goes unspecified, and yet must go beyond standard varieties of logical necessity. This leaves us little reason (...)
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  8. Modal Epistemology.Juhani Yli-Vakkuri & John Hawthorne - manuscript
    Some central epistemological notions are expressed by sentential operators O that entail the possibility of knowledge in the sense that 'Op' entails 'It is possible to know that p'. We call these modal-epistemological notions. Using apriority and being in a position to know as case studies, we argue that the logics of modal epistemological notions are extremely weak. In particular, their logics are not normal and do not include any closure principles.
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  9. Chemical Possibility and Modal Semantics.Mark Sharlow - 2007
    This paper is a study of a distinctively chemical notion of possibility. This is the notion of possibility that occurs in chemical discourses when chemists speak of the possibility or impossibility of achieving a given result through chemical means. This notion pertains to the possibility of processes, not of compounds, so it differs from the kind of chemical possibility mentioned in Wittgenstein's Philosophical Investigations or the kinds discussed in the literature on Putnam's Twin Earth argument. I argue that this process-oriented (...)
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  10. A Theory of Necessities.Andrew Bacon & Jin Zeng - forthcoming - Journal of Philosophical Logic:1-49.
    We develop a theory of necessity operators within a version of higher-order logic that is neutral about how fine-grained reality is. The theory is axiomatized in terms of the primitive of *being a necessity*, and we show how the central notions in the philosophy of modality can be recovered from it. Various questions are formulated and settled within the framework, including questions about the ordering of necessities under strength, the existence of broadest necessities satisfying various logical conditions, and questions about (...)
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  11. Meta-Inferences and Supervaluationism.Luca Incurvati & Julian J. Schlöder - forthcoming - Journal of Philosophical Logic:1-34.
    Many classically valid meta-inferences fail in a standard supervaluationist framework. This allegedly prevents supervaluationism from offering an account of good deductive reasoning. We provide a proof system for supervaluationist logic which includes supervaluationistically acceptable versions of the classical meta-inferences. The proof system emerges naturally by thinking of truth as licensing assertion, falsity as licensing negative assertion and lack of truth-value as licensing rejection and weak assertion. Moreover, the proof system respects well-known criteria for the admissibility of inference rules. Thus, supervaluationists (...)
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  12. Epistemic Multilateral Logic.Luca Incurvati & Julian J. Schlöder - forthcoming - Review of Symbolic Logic:1-44.
    We present epistemic multilateral logic, a general logical framework for reasoning involving epistemic modality. Standard bilateral systems use propositional formulae marked with signs for assertion and rejection. Epistemic multilateral logic extends standard bilateral systems with a sign for the speech act of weak assertion (Incurvati and Schlöder 2019) and an operator for epistemic modality. We prove that epistemic multilateral logic is sound and complete with respect to the modal logic S5 modulo an appropriate translation. The logical framework developed provides the (...)
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  13. The Logic of Sequence Frames.Fabio Lampert - forthcoming - Review of Symbolic Logic:1-44.
    This paper investigates and develops generalizations of two-dimensional modal logics to any finite dimension. These logics are natural extensions of multidimensional systems known from the literature on logics for a priori knowledge. We prove a completeness theorem for propositional n-dimensional modal logics and show them to be decidable by means of a systematic tableau construction.
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  14. Dynamic Consequence for Soft Information.Olivier Roy & Ole Thomassen Hjortland - forthcoming - Journal of Logic and Computation.
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  15. Intuitionism and the Modal Logic of Vagueness.Susanne Bobzien & Ian Rumfitt - 2020 - Journal of Philosophical Logic 49 (2):221-248.
    Intuitionistic logic provides an elegant solution to the Sorites Paradox. Its acceptance has been hampered by two factors. First, the lack of an accepted semantics for languages containing vague terms has led even philosophers sympathetic to intuitionism to complain that no explanation has been given of why intuitionistic logic is the correct logic for such languages. Second, switching from classical to intuitionistic logic, while it may help with the Sorites, does not appear to offer any advantages when dealing with the (...)
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  16. On the Relation Between Modality and Tense.Fabrice Correia & Sven Rosenkranz - 2020 - Inquiry: An Interdisciplinary Journal of Philosophy 63 (6):586-604.
    ABSTRACT We critically review two extant paradigms for understanding the systematic interaction between modality and tense, as well as their respective modifications designed to do justice to the contingency of time’s structure and composition. We show that on either type of theory, as well as their respective modifications, some principles prove logically valid whose truth might sensibly be questioned on metaphysical grounds. These considerations lead us to devise a more general logical framework that allows accommodation of those metaphysical views that (...)
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  17. An Axiomatic Approach to Theodicy Via Formal Applied Systems.Gesiel B. Da Silva - 2020 - Dissertation, University of Campinas
    Edward Nieznański developed two logical systems in order to deal with a version of the problem of evil associated with two formulations of religious determinism. The aim of this research was to revisit these systems, providing them with a more appropriate formalization. The new resulting systems, namely, N1 and N2, were reformulated in first-order modal logic; they retain much of their original basic structures, but some additional results were obtained. Furthermore, our research found that an underlying minimal set of axioms (...)
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  18. Formal Theodicy: Religious Determinism and the Logical Problem of Evil.Gesiel B. Da Silva & Fábio Bertato - 2020 - Edukacja Filozoficzna 70:93-119.
    Edward Nieznański developed two logical systems to deal with the problem of evil and to refute religious determinism. However, when formalized in first-order modal logic, two axioms of each system contradict one another, revealing that there is an underlying minimal set of axioms enough to settle the questions. In this article, we develop this minimal system, called N3, which is based on Nieznański’s contribution. The purpose of N3 is to solve the logical problem of evil through the defeat of a (...)
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  19. Diamonds Are Forever.Cian Dorr & Jeremy Goodman - 2020 - Noûs 54 (3):632-665.
    We defend the thesis that every necessarily true proposition is always true. Since not every proposition that is always true is necessarily true, our thesis is at odds with theories of modality and time, such as those of Kit Fine and David Kaplan, which posit a fundamental symmetry between modal and tense operators. According to such theories, just as it is a contingent matter what is true at a given time, it is likewise a temporary matter what is true at (...)
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  20. Sketch of a Proof-Theoretic Semantics for Necessity.Nils Kürbis - 2020 - In Nicola Olivetti, Rineke Verbrugge & Sara Negri (eds.), Advances in Modal Logic 13. Booklet of Short Papers. Helsinki: pp. 37-43.
    This paper considers proof-theoretic semantics for necessity within Dummett's and Prawitz's framework. Inspired by a system of Pfenning's and Davies's, the language of intuitionist logic is extended by a higher order operator which captures a notion of validity. A notion of relative necessary is defined in terms of it, which expresses a necessary connection between the assumptions and the conclusion of a deduction.
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  21. Might Generics.Brian Rabern - 2020 - Snippets 39:8-9.
    How do generics interact with modals? This note offers one observation about an interaction with 'might' that presents a challenge for standard theories.
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  22. Two Kinds of Logical Impossibility.Alexander Sandgren & Koji Tanaka - 2020 - Noûs 54 (4):795-806.
    In this paper, we argue that a distinction ought to be drawn between two ways in which a given world might be logically impossible. First, a world w might be impossible because the laws that hold at w are different from those that hold at some other world (say the actual world). Second, a world w might be impossible because the laws of logic that hold in some world (say the actual world) are violated at w. We develop a novel (...)
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  23. A Decidable Multi-Agent Logic for Reasoning About Actions, Instruments, and Norms.Kees van Berkel, Tim Lyon & Francesco Olivieri - 2020 - In Mehdi Dastani, Huimin Dong & Leon van der Torre (eds.), Logic and Argumentation. pp. 219 - 241.
    We formally introduce a novel, yet ubiquitous, category of norms: norms of instrumentality. Norms of this category describe which actions are obligatory, or prohibited, as instruments for certain purposes. We propose the Logic of Agency and Norms (LAN) that enables reasoning about actions, instrumentality, and normative principles in a multi-agent setting. Leveraging LAN , we formalize norms of instrumentality and compare them to two prevalent norm categories: norms to be and norms to do. Last, we pose principles relating the three (...)
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  24. A Strange Kind of Power: Vetter on the Formal Adequacy of Dispositionalism.David Yates - 2020 - Philosophical Inquiries 8 (1):97-116.
    According to dispositionalism about modality, a proposition <p> is possible just in case something has, or some things have, a power or disposition for its truth; and <p> is necessary just in case nothing has a power for its falsity. But are there enough powers to go around? In Yates (2015) I argued that in the case of mathematical truths such as <2+2=4>, nothing has the power to bring about their falsity or their truth, which means they come out both (...)
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  25. 'On a Supposed Puzzle Concerning Modality and Existence'.Thomas Atkinson, Daniel J. Hill & Stephen K. McLeod - 2019 - Organon F: Medzinárodný Časopis Pre Analytickú Filozofiu 26 (3):446-473.
    Kit Fine has proposed a new solution to what he calls ‘a familiar puzzle’ concerning modality and existence. The puzzle concerns the argument from the alleged truths ‘It is necessary that Socrates is a man’ and ‘It is possible that Socrates does not exist’ to the apparent falsehood ‘It is possible that Socrates is a man and does not exist’. We discuss in detail Fine’s setting up of the ‘puzzle’ and his rejection, with which we concur, of two mooted solutions (...)
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  26. Negation on the Australian Plan.Francesco Berto & Greg Restall - 2019 - Journal of Philosophical Logic 48 (6):1119-1144.
    We present and defend the Australian Plan semantics for negation. This is a comprehensive account, suitable for a variety of different logics. It is based on two ideas. The first is that negation is an exclusion-expressing device: we utter negations to express incompatibilities. The second is that, because incompatibility is modal, negation is a modal operator as well. It can, then, be modelled as a quantifier over points in frames, restricted by accessibility relations representing compatibilities and incompatibilities between such points. (...)
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  27. Two-Sided Trees for Sentential Logic, Predicate Logic, and Sentential Modal Logic.Jesse Fitts & David Beisecker - 2019 - Teaching Philosophy 42 (1):41-56.
    This paper will present two contributions to teaching introductory logic. The first contribution is an alternative tree proof method that differs from the traditional one-sided tree method. The second contribution combines this tree system with an index system to produce a user-friendly tree method for sentential modal logic.
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  28. Remark on Al-Fārābī's Missing Modal Logic and its Effect on Ibn Sīnā.Wilfrid Hodges - 2019 - Eshare: An Iranian Journal of Philosophy 1 (3):39-73.
    We reconstruct as much as we can the part of al-Fārābī's treatment of modal logic that is missing from the surviving pages of his Long Commentary on the Prior Analytics. We use as a basis the quotations from this work in Ibn Sīnā, Ibn Rushd and Maimonides, together with relevant material from al-Fārābī's other writings. We present a case that al-Fārābī's treatment of the dictum de omni had a decisive effect on the development and presentation of Ibn Sīnā's modal logic. (...)
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  29. Cut-Free Calculi and Relational Semantics for Temporal STIT Logics.Tim Lyon & Kees van Berkel - 2019 - In Francesco Calimeri, Nicola Leone & Marco Manna (eds.), Logics in Artificial Intelligence. Springer International Publishing. pp. 803 - 819.
    We present cut-free labelled sequent calculi for a central formalism in logics of agency: STIT logics with temporal operators. These include sequent systems for Ldm , Tstit and Xstit. All calculi presented possess essential structural properties such as contraction- and cut-admissibility. The labelled calculi G3Ldm and G3Tstit are shown sound and complete relative to irreflexive temporal frames. Additionally, we extend current results by showing that also Xstit can be characterized through relational frames, omitting the use of BT+AC frames.
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  30. Automating Agential Reasoning: Proof-Calculi and Syntactic Decidability for STIT Logics.Tim Lyon & Kees van Berkel - 2019 - In M. Baldoni, M. Dastani, B. Liao, Y. Sakurai & R. Zalila Wenkstern (eds.), PRIMA 2019: Principles and Practice of Multi-Agent Systems. 93413 Cham, Germany: Springer. pp. 202-218.
    This work provides proof-search algorithms and automated counter-model extraction for a class of STIT logics. With this, we answer an open problem concerning syntactic decision procedures and cut-free calculi for STIT logics. A new class of cut-free complete labelled sequent calculi G3LdmL^m_n, for multi-agent STIT with at most n-many choices, is introduced. We refine the calculi G3LdmL^m_n through the use of propagation rules and demonstrate the admissibility of their structural rules, resulting in auxiliary calculi Ldm^m_nL. In the single-agent case, we (...)
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  31. Disappearing Diamonds: Fitch-Like Results in Bimodal Logic.Weng Kin San - 2019 - Journal of Philosophical Logic 48 (6):1003-1016.
    Augment the propositional language with two modal operators: □ and ■. Define ⧫ to be the dual of ■, i.e. ⧫=¬■¬. Whenever (X) is of the form φ → ψ, let (X⧫) be φ→⧫ψ . (X⧫) can be thought of as the modally qualified counterpart of (X)—for instance, under the metaphysical interpretation of ⧫, where (X) says φ implies ψ, (X⧫) says φ implies possibly ψ. This paper shows that for various interesting instances of (X), fairly weak assumptions suffice for (...)
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  32. Contingent Existence and the Reduction of Modality to Essence.Trevor Teitel - 2019 - Mind 128 (509):39-68.
    This paper first argues that we can bring out a tension between the following three popular doctrines: (i) the canonical reduction of metaphysical modality to essence, due to Fine, (ii) contingentism, which says that possibly something could have failed to be something, and (iii) the doctrine that metaphysical modality obeys the modal logic S5. After presenting two such arguments (one from the theorems of S4 and another from the theorems of B), I turn to exploring various conclusions we might draw (...)
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  33. A Neutral Temporal Deontic STIT Logic.Kees van Berkel & Tim Lyon - 2019 - In P. Blackburn, E. Lorini & M. Guo (eds.), Logic, Rationality, and Interaction. Berlin, Heidelberg: pp. 340-354.
    In this work we answer a long standing request for temporal embeddings of deontic STIT logics by introducing the multi-agent STIT logic TDS . The logic is based upon atemporal utilitarian STIT logic. Yet, the logic presented here will be neutral: instead of committing ourselves to utilitarian theories, we prove the logic TDS sound and complete with respect to relational frames not employing any utilitarian function. We demonstrate how these neutral frames can be transformed into utilitarian temporal frames, while preserving (...)
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  34. Boxes and Diamonds: An Open Introduction to Modal Logic.Richard Zach - 2019 - Open Logic Project.
    A textbook for modal and other intensional logics based on the Open Logic Project. It covers normal modal logics, relational semantics, axiomatic and tableaux proof systems, intuitionistic logic, and counterfactual conditionals.
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  35. The Broadest Necessity.Andrew Bacon - 2018 - Journal of Philosophical Logic 47 (5):733-783.
    In this paper the logic of broad necessity is explored. Definitions of what it means for one modality to be broader than another are formulated, and it is proven, in the context of higher-order logic, that there is a broadest necessity, settling one of the central questions of this investigation. It is shown, moreover, that it is possible to give a reductive analysis of this necessity in extensional language. This relates more generally to a conjecture that it is not possible (...)
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  36. The Logical Contingency of Identity.Hanoch Ben-Yami - 2018 - European Journal of Analytic Philosophy 14 (2):5-10.
    I show that intuitive and logical considerations do not justify introducing Leibniz’s Law of the Indiscernibility of Identicals in more than a limited form, as applying to atomic formulas. Once this is accepted, it follows that Leibniz’s Law generalises to all formulas of the first-order Predicate Calculus but not to modal formulas. Among other things, identity turns out to be logically contingent.
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  37. From Display to Labelled Proofs for Tense Logics.Agata Ciabattoni, Tim Lyon & Revantha Ramanayake - 2018 - In Anil Nerode & Sergei Artemov (eds.), Logical Foundations of Computer Science. Springer International Publishing. pp. 120 - 139.
    We introduce an effective translation from proofs in the display calculus to proofs in the labelled calculus in the context of tense logics. We identify the labelled calculus proofs in the image of this translation as those built from labelled sequents whose underlying directed graph possesses certain properties. For the basic normal tense logic Kt, the image is shown to be the set of all proofs in the labelled calculus G3Kt.
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  38. On the Expressive Power of First-Order Modal Logic with Two-Dimensional Operators.Alexander Kocurek - 2018 - Synthese 195 (10):4373-4417.
    Many authors have noted that there are types of English modal sentences cannot be formalized in the language of basic first-order modal logic. Some widely discussed examples include “There could have been things other than there actually are” and “Everyone who is actually rich could have been poor.” In response to this lack of expressive power, many authors have discussed extensions of first-order modal logic with two-dimensional operators. But claims about the relative expressive power of these extensions are often justified (...)
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  39. Actuality and the a Priori.Fabio Lampert - 2018 - Philosophical Studies 175 (3):809-830.
    We consider a natural-language sentence that cannot be formally represented in a first-order language for epistemic two-dimensional semantics. We also prove this claim in the “Appendix” section. It turns out, however, that the most natural ways to repair the expressive inadequacy of the first-order language render moot the original philosophical motivation of formalizing a priori knowability as necessity along the diagonal.
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  40. Gestalt Shifts in the Liar Or Why KT4M Is the Logic of Semantic Modalities.Susanne Bobzien - 2017 - In Bradley Armour-Garb (ed.), Reflections on the Liar. Oxford University. pp. 71-113.
    ABSTRACT: This chapter offers a revenge-free solution to the liar paradox (at the centre of which is the notion of Gestalt shift) and presents a formal representation of truth in, or for, a natural language like English, which proposes to show both why -- and how -- truth is coherent and how it appears to be incoherent, while preserving classical logic and most principles that some philosophers have taken to be central to the concept of truth and our use of (...)
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  41. Contingent Existence and Iterated Modality.Cian Dorr - 2017 - Analysis 77 (1):155-165.
    A discussion of a view, defended by Robert Adams and Boris Kment, according to which contingent existence requires rejecting many standard principles of propositional modal logic involving iterated modal operators.
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  42. Modal Logic S4 as a Paraconsistent Logic with a Topological Semantics.Marcelo E. Coniglio & Leonardo Prieto-Sanabria - 2017 - In Carlos Caleiro, Francisco Dionisio, Paula Gouveia, Paulo Mateus & João Rasga (eds.), Logic and Computation: Essays in Honour of Amilcar Sernadas. London, UK: College Publications. pp. 171-196.
    In this paper the propositional logic LTop is introduced, as an extension of classical propositional logic by adding a paraconsistent negation. This logic has a very natural interpretation in terms of topological models. The logic LTop is nothing more than an alternative presentation of modal logic S4, but in the language of a paraconsistent logic. Moreover, LTop is a logic of formal inconsistency in which the consistency and inconsistency operators have a nice topological interpretation. This constitutes a new proof of (...)
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  43. Weak Rejection.Luca Incurvati & Julian J. Schlöder - 2017 - Australasian Journal of Philosophy 95 (4):741-760.
    ABSTRACTLinguistic evidence supports the claim that certain, weak rejections are less specific than assertions. On the basis of this evidence, it has been argued that rejected sentences cannot be premisses and conclusions in inferences. We give examples of inferences with weakly rejected sentences as premisses and conclusions. We then propose a logic of weak rejection which accounts for the relevant phenomena and is motivated by principles of coherence in dialogue. We give a semantics for which this logic is sound and (...)
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  44. On the Concept of a Notational Variant.Alexander W. Kocurek - 2017 - In Alexandru Baltag, Jeremy Seligman & Tomoyuki Yamada (eds.), Logic, Rationality, and Interaction (LORI 2017, Sapporo, Japan). pp. 284-298.
    In the study of modal and nonclassical logics, translations have frequently been employed as a way of measuring the inferential capabilities of a logic. It is sometimes claimed that two logics are “notational variants” if they are translationally equivalent. However, we will show that this cannot be quite right, since first-order logic and propositional logic are translationally equivalent. Others have claimed that for two logics to be notational variants, they must at least be compositionally intertranslatable. The definition of compositionality these (...)
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  45. Natural Deduction for Diagonal Operators.Fabio Lampert - 2017 - In Maria Zack & Dirk Schlimm (eds.), Research in History and Philosophy of Mathematics: The CSHPM 2016 Annual Meeting in Calgary, Alberta. Cham: Birkhäuser. pp. 39-51.
    We present a sound and complete Fitch-style natural deduction system for an S5 modal logic containing an actuality operator, a diagonal necessity operator, and a diagonal possibility operator. The logic is two-dimensional, where we evaluate sentences with respect to both an actual world (first dimension) and a world of evaluation (second dimension). The diagonal necessity operator behaves as a quantifier over every point on the diagonal between actual worlds and worlds of evaluation, while the diagonal possibility quantifies over some point (...)
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  46. A Purely Recombinatorial Puzzle.Fritz Peter - 2017 - Noûs 51 (3):547-564.
    A new puzzle of modal recombination is presented which relies purely on resources of first-order modal logic. It shows that naive recombinatorial reasoning, which has previously been shown to be inconsistent with various assumptions concerning propositions, sets and classes, leads to inconsistency by itself. The context sensitivity of modal expressions is suggested as the source of the puzzle, and it is argued that it gives us reason to reconsider the assumption that the notion of metaphysical necessity is in good standing.
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  47. Necessity, a Leibnizian Thesis, and a Dialogical Semantics.Mohammad Shafiei - 2017 - South American Journal of Logic 3 (1):1-23.
    In this paper, an interpretation of "necessity", inspired by a Leibnizian idea and based on the method of dialogical logic, is introduced. The semantic rules corresponding to such an account of necessity are developed, and then some peculiarities, and some potential advantages, of the introduced dialogical explanation, in comparison with the customary explanation offered by the possible worlds semantics, are briefly discussed.
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  48. Post Completeness in Congruential Modal Logics.Peter Fritz - 2016 - In Lev Beklemishev, Stéphane Demri & András Máté (eds.), Advances in Modal Logic, Volume 11. College Publications. pp. 288-301.
    Well-known results due to David Makinson show that there are exactly two Post complete normal modal logics, that in both of them, the modal operator is truth-functional, and that every consistent normal modal logic can be extended to at least one of them. Lloyd Humberstone has recently shown that a natural analog of this result in congruential modal logics fails, by showing that not every congruential modal logic can be extended to one in which the modal operator is truth-functional. As (...)
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  49. Appendix to Juhani Yli-Vakkuri’s ‘Epistemicism and Modality’.Peter Fritz - 2016 - Canadian Journal of Philosophy 46 (4-5):836-838.
    A formal result is proved which is used in Juhani Yli-Vakkuri’s ‘Epistemicism and Modality’ to argue that certain two-dimensional possible world models are inadequate for a language with operators for ‘necessarily’, ‘actually’, and ‘definitely’.
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  50. Introduzione alle Logiche Modali.Marcello Frixione, Samuele Iaquinto & Massimiliano Vignolo - 2016 - Roma-Bari: Laterza.
    La logica modale è nata per studiare i ragionamenti su ciò che è possibile e ciò che è necessario. Negli ultimi decenni, a partire dal lavoro di logici e filosofi quali Rudolf Carnap, Saul Kripke e David Lewis, la sua applicazione è stata progressivamente estesa ad altri ambiti, quali il ragionamento sul tempo, sulla conoscenza e sui sistemi di norme. Queste ricerche hanno condotto a un complesso e intrigante dialogo con alcune fondamentali branche della filosofia: la metafisica, l’epistemologia, la filosofia (...)
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