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One Variable Relevant Logics are S5ish Journal of Philosophical Logic Pub Date : 2024-03-22 Nicholas Ferenz
Here I show that the one-variable fragment of several first-order relevant logics corresponds to certain S5ish extensions of the underlying propositional relevant logic. In particular, given a fairly standard translation between modal and one-variable languages and a permuting propositional relevant logic L, a formula \(\mathcal {A}\) of the one-variable fragment is a theorem of LQ (QL) iff its translation
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A General Schema for Bilateral Proof Rules Journal of Philosophical Logic Pub Date : 2024-03-02 Ryan Simonelli
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Naïve Truth and the Evidential Conditional Journal of Philosophical Logic Pub Date : 2024-03-01 Andrea Iacona, Lorenzo Rossi
This paper develops the idea that valid arguments are equivalent to true conditionals by combining Kripke’s theory of truth with the evidential account of conditionals offered by Crupi and Iacona. As will be shown, in a first-order language that contains a naïve truth predicate and a suitable conditional, one can define a validity predicate in accordance with the thesis that the inference from a conjunction
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Common Ground in Non-face-to-face Communication: In Sensu Diviso or In Sensu Composito Journal of Philosophical Logic Pub Date : 2024-02-26 Merel Semeijn
Traditional definitions of common ground in terms of iterative de re attitudes do not apply to conversations where at least one conversational participant is not acquainted with the other(s). I propose and compare two potential refinements of traditional definitions based on Abelard’s distinction between generality in sensu composito and in sensu diviso.
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A Hyperintensional Logic of Non-prime Evidence Journal of Philosophical Logic Pub Date : 2024-02-26 Pietro Vigiani
We present a logic of evidence that reduces agents’ epistemic idealisations by combining classical propositional logic with substructural modal logic for formulas in the scope of epistemic modalities. To this aim, we provide a neighborhood semantics of evidence, which provides a modal extension of Fine’s semantics for relevant propositional logic. Possible worlds semantics for classical propositional
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Questions & Indexicality Journal of Philosophical Logic Pub Date : 2024-02-22 Thom van Gessel
The truth conditions of sentences with indexicals like ‘I’ and ‘here’ cannot be given directly, but only relative to a context of utterance. Something similar applies to questions: depending on the semantic framework, they are given truth conditions relative to an actual world, or support conditions instead of truth conditions. Two-dimensional semantics can capture the meaning of indexicals and shed
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Grzegorczyk and Whitehead Points: The Story Continues Journal of Philosophical Logic Pub Date : 2024-02-16 Rafał Gruszczyński, Santiago Jockwich Martinez
The paper is devoted to the analysis of two seminal definitions of points within the region-based framework: one by Whitehead (1929) and the other by Grzegorczyk (Synthese, 12(2-3), 228-235 1960). Relying on the work of Biacino & Gerla (Notre Dame Journal of Formal Logic, 37(3), 431-439 1996), we improve their results, solve some open problems concerning the mutual relationship between Whitehead and
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First-Degree Entailment and Truthmaker Functions Journal of Philosophical Logic Pub Date : 2024-02-10 Roderick Batchelor
We define a concept of truthmaker function, and prove the functional completeness, w.r.t. truthmaker functions in this sense, of a set of four-valued functions corresponding to standard connectives of the system of relevance logic known as First-Degree Entailment or Belnap–Dunn logic.
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An Algebraic View of the Mares-Goldblatt Semantics Journal of Philosophical Logic Pub Date : 2024-01-26 Andrew Tedder
An algebraic characterisation is given of the Mares-Goldblatt semantics for quantified extensions of relevant and modal logics. Some features of this more general semantic framework are investigated, and the relations to some recent work in algebraic semantics for quantified extensions of non-classical logics are considered.
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Truthlikeness and the Number of Planets Journal of Philosophical Logic Pub Date : 2024-01-25 Theo A. F. Kuipers
Examples of hypotheses about the number of planets are frequently used to introduce the topic of (actual) truthlikeness but never analyzed in detail. In this paper we first deal with the truthlikeness of singular quantity hypotheses, with reference to several ‘the number of planets’ examples, such as ‘The number of planets is 10 versus 10 billion (instead of 8).’ For the relevant ratio scale of quantities
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A Semantic Framework for the Impure Logic of Ground Journal of Philosophical Logic Pub Date : 2024-01-25 Louis deRosset
There is a curious bifurcation in the literature on ground and its logic. On the one hand, there has been a great deal of work that presumes that logical complexity invariably yields grounding. So, for instance, it is widely presumed that any fact stated by a true conjunction is grounded in those stated by its conjuncts, that any fact stated by a true disjunction is grounded in that stated by any of
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Jeffrey Conditionalization Permits Undermining Journal of Philosophical Logic Pub Date : 2024-01-22 Marc Lange
It has frequently been argued recently that Jeffrey Conditionalization (JC) does not permit undermining. For JC to be inapplicable in cases where the evidence could be undermined would severely compromise JC’s range. However, this paper contends that the argument fails to show that JC cannot accommodate undermining. This response turns on using the proper partition to capture the direct impact of our
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Supervaluationism, Modal Logic, and Weakly Classical Logic Journal of Philosophical Logic Pub Date : 2024-01-17 Joshua Schechter
A consequence relation is strongly classical if it has all the theorems and entailments of classical logic as well as the usual meta-rules (such as Conditional Proof). A consequence relation is weakly classical if it has all the theorems and entailments of classical logic but lacks the usual meta-rules. The most familiar example of a weakly classical consequence relation comes from a simple supervaluational
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A Dynamic Logic of Data-Informed Knowledge Journal of Philosophical Logic Pub Date : 2024-01-16 Kaya Deuser, Junli Jiang, Pavel Naumov, Wenxuan Zhang
With agents relying more and more on information from central servers rather than their own sensors, knowledge becomes property not of a specific agent but of the data that the agents can access. The article proposes a dynamic logic of data-informed knowledge that describes an interplay between three modalities and one relation capturing the properties of this form of knowledge. The main technical
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8 Valued Non-Deterministic Semantics for Modal Logics Journal of Philosophical Logic Pub Date : 2024-01-10 Pawel Pawlowski, Daniel Skurt
The aim of this paper is to study a particular family of non-deterministic semantics for modal logics that has eight truth-values. These eight-valued semantics can be traced back to Omori and Skurt (2016), where a particular member of this family was used to characterize the normal modal logic K. The truth-values in these semantics convey information about a proposition’s truth/falsity, whether the
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Boolean Valued Models, Boolean Valuations, and Löwenheim-Skolem Theorems Journal of Philosophical Logic Pub Date : 2023-12-23 Xinhe Wu
Boolean-valued models for first-order languages generalize two-valued models, in that the value range is allowed to be any complete Boolean algebra instead of just the Boolean algebra 2. Boolean-valued models are interesting in multiple aspects: philosophical, logical, and mathematical. The primary goal of this paper is to extend a number of critical model-theoretic notions and to generalize a number
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Tarski’s Theory of the Formal Correctness of Definitions Journal of Philosophical Logic Pub Date : 2023-12-20
Abstract In his 1933 monograph on the concept of truth, Alfred Tarski claimed that his definition of truth satisfied “the usual conditions of methodological correctness”, which in a 1935 article he identified as consistency and back-translatability. Following the rules of defining for an axiomatized theory was supposed to ensure satisfaction of the two conditions. But Tarski neither explained the two
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Ab Esse ad Posse Non Valet Consequentia Journal of Philosophical Logic Pub Date : 2023-12-11 Daniel Dohrn
While knowledge of mere possibilities is difficult to understand, knowledge of possibilities that are actual seems unproblematic (as far as we know the actual world). The principle that what is actual is possible has been near-universally accepted. After summarizing some sporadic dissent, I present a proposal for how the validity of the principle might be restricted. While the principle certainly holds
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The Implicative Conditional Journal of Philosophical Logic Pub Date : 2023-11-27 Eric Raidl, Gilberto Gomes
This paper investigates the implicative conditional, a connective intended to describe the logical behavior of an empirically defined class of natural language conditionals, also named implicative conditionals, which excludes concessive and some other conditionals. The implicative conditional strengthens the strict conditional with the possibility of the antecedent and of the contradictory of the consequent
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Reasoning about Dependence, Preference and Coalitional Power Journal of Philosophical Logic Pub Date : 2023-11-28 Qian Chen, Chenwei Shi, Yiyan Wang
This paper presents a logic of preference and functional dependence (LPFD) and its hybrid extension (HLPFD), both of whose sound and strongly complete axiomatization are provided. The decidability of LPFD is also proved. The application of LPFD and HLPFD to modelling cooperative games in strategic form is explored. The resulted framework provides a unified view on Nash equilibrium, Pareto optimality
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Mathematical Modality: An Investigation in Higher-order Logic Journal of Philosophical Logic Pub Date : 2023-11-28 Andrew Bacon
An increasing amount of contemporary philosophy of mathematics posits, and theorizes in terms of special kinds of mathematical modality. The goal of this paper is to bring recent work on higher-order metaphysics to bear on the investigation of these modalities. The main focus of the paper will be views that posit mathematical contingency or indeterminacy about statements that concern the ‘width’ of
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A Step Towards Absolute Versions of Metamathematical Results Journal of Philosophical Logic Pub Date : 2023-11-29 Balthasar Grabmayr
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A Sound and Complete Tableaux Calculus for Reichenbach’s Quantum Mechanics Logic Journal of Philosophical Logic Pub Date : 2023-11-18 Pablo Caballero, Pablo Valencia
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The Liar Paradox and “Meaningless” Revenge Journal of Philosophical Logic Pub Date : 2023-11-16 Jared Warren
A historically popular response to the liar paradox (“this sentence is false”) is to say that the liar sentence is meaningless (or semantically defective, or malfunctions, or…). Unfortunately, like all other supposed solutions to the liar, this approach faces a revenge challenge. Consider the revenge liar sentence, “this sentence is either meaningless or false”. If it is true, then it is either meaningless
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Modal Information Logics: Axiomatizations and Decidability Journal of Philosophical Logic Pub Date : 2023-11-03 Søren Brinck Knudstorp
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Subject-Matter and Intensional Operators II: Applications to the Theory of Topic-Sensitive Intentional Modals Journal of Philosophical Logic Pub Date : 2023-10-26 Thomas Macaulay Ferguson
In frameworks in which topic-theoretic considerations—e.g., tracking subject-matter or topic—are given equal importance with veridical considerations, assigning topics to formulae in a satisfactory way is of critical importance. While intuitions are more-or-less solid for extensional formulae in a propositional language, arriving at a compelling account of the subject-matter of intensional formulae
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Axiomatizing Rumsfeld Ignorance Journal of Philosophical Logic Pub Date : 2023-10-24 Jie Fan
In a recent paper, Kit Fine presents some striking results concerning the logical properties of (first-order) ignorance, second-order ignorance and Rumsfeld ignorance. However, Rumsfeld ignorance is definable in terms of ignorance, which makes some existing results and the axiomatization problem trivial. A main reason is that the accessibility relations for the implicit knowledge operator contained
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Truth Meets Vagueness. Unifying the Semantic and the Soritical Paradoxes Journal of Philosophical Logic Pub Date : 2023-10-09 Riccardo Bruni, Lorenzo Rossi
Semantic and soritical paradoxes display remarkable family resemblances. For one thing, several non-classical logics have been independently applied to both kinds of paradoxes. For another, revenge paradoxes and higher-order vagueness—among the most serious problems targeting solutions to semantic and soritical paradoxes—exhibit a rather similar dynamics. Some authors have taken these facts to suggest
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Logic-Sensitivity and Bitstring Semantics in the Square of Opposition Journal of Philosophical Logic Pub Date : 2023-10-06 Lorenz Demey, Stef Frijters
This paper explores the interplay between logic-sensitivity and bitstring semantics in the square of opposition. Bitstring semantics is a combinatorial technique for representing the formulas that appear in a logical diagram, while logic-sensitivity entails that such a diagram may depend, not only on the formulas involved, but also on the logic with respect to which they are interpreted. These two
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Logical Multilateralism Journal of Philosophical Logic Pub Date : 2023-09-25 Heinrich Wansing, Sara Ayhan
In this paper we will consider the existing notions of bilateralism in the context of proof-theoretic semantics and propose, based on our understanding of bilateralism, an extension to logical multilateralism. This approach differs from what has been proposed under this name before in that we do not consider multiple speech acts as the core of such a theory but rather multiple consequence relations
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Abduction as Deductive Saturation: a Proof-Theoretic Inquiry Journal of Philosophical Logic Pub Date : 2023-09-15 Mario Piazza, Gabriele Pulcini, Andrea Sabatini
Abductive reasoning involves finding the missing premise of an “unsaturated” deductive inference, thereby selecting a possible explanans for a conclusion based on a set of previously accepted premises. In this paper, we explore abductive reasoning from a structural proof-theory perspective. We present a hybrid sequent calculus for classical propositional logic that uses sequents and antisequents to
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A Minimal Probability Space for Conditionals Journal of Philosophical Logic Pub Date : 2023-09-14 Anna Wójtowicz, Krzysztof Wójtowicz
One of central problems in the theory of conditionals is the construction of a probability space, where conditionals can be interpreted as events and assigned probabilities. The problem has been given a technical formulation by van Fraassen (23), who also discussed in great detail the solution in the form of Stalnaker Bernoulli spaces. These spaces are very complex – they have the cardinality of the
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A Basis for AGM Revision in Bayesian Probability Revision Journal of Philosophical Logic Pub Date : 2023-08-30 Sven Ove Hansson
In standard Bayesian probability revision, the adoption of full beliefs (propositions with probability 1) is irreversible. Once an agent has full belief in a proposition, no subsequent revision can remove that belief. This is an unrealistic feature, and it also makes probability revision incompatible with belief change theory, which focuses on how the set of full beliefs is modified through both additions
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A Logical Analysis of Instrumentality Judgments: Means-End Relations in the Context of Experience and Expectations Journal of Philosophical Logic Pub Date : 2023-08-10 Kees van Berkel, Tim S. Lyon, Matteo Pascucci
This article proposes the use of temporal logic for an analysis of instrumentality inspired by the work of G.H. von Wright. The first part of the article contains the philosophical foundations. We discuss von Wright’s general theory of agency and his account of instrumentality. Moreover, we propose several refinements to this framework via rigorous definitions of the core notions involved. In the second
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The Value of the One Value: Exactly True Logic revisited Journal of Philosophical Logic Pub Date : 2023-08-07 Andreas Kapsner, Umberto Rivieccio
In this paper we re-assess the philosophical foundation of Exactly True Logic (\(\mathcal {ET\!L}\)), a competing variant of First Degree Entailment (\(\mathcal {FDE}\)). In order to do this, we first rebut an argument against it. As the argument appears in an interview with Nuel Belnap himself, one of the fathers of \(\mathcal {FDE}\), we believe its provenance to be such that it needs to be taken
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Quantum Epistemology and Constructivism Journal of Philosophical Logic Pub Date : 2023-08-05 Patrick Fraser, Nuriya Nurgalieva, Lídia del Rio
Constructivist epistemology posits that all truths are knowable. One might ask to what extent constructivism is compatible with naturalized epistemology and knowledge obtained from inference-making using successful scientific theories. If quantum theory correctly describes the structure of the physical world, and if quantum theoretic inferences about which measurement outcomes will be observed with
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Internal Categoricity, Truth and Determinacy Journal of Philosophical Logic Pub Date : 2023-08-03 Martin Fischer, Matteo Zicchetti
This paper focuses on the categoricity of arithmetic and determinacy of arithmetical truth. Several ‘internal’ categoricity results have been discussed in the recent literature. Against the background of the philosophical position called internalism, we propose and investigate truth-theoretic versions of internal categoricity based on a primitive truth predicate. We argue for the compatibility of a
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On the Provable Contradictions of the Connexive Logics C and C3 Journal of Philosophical Logic Pub Date : 2023-07-29 Satoru Niki, Heinrich Wansing
Despite the tendency to be otherwise, some non-classical logics are known to validate formulas that are invalid in classical logic. A subclass of such systems even possesses pairs of a formula and its negation as theorems, without becoming trivial. How should these provable contradictions be understood? The present paper aims to shed light on aspects of this phenomenon by taking as samples the constructive
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Communication Pattern Logic: Epistemic and Topological Views Journal of Philosophical Logic Pub Date : 2023-07-28 Armando Castañeda, Hans van Ditmarsch, David A. Rosenblueth, Diego A. Velázquez
We propose communication pattern logic. A communication pattern describes how processes or agents inform each other, independently of the information content. The full-information protocol in distributed computing is the special case wherein all agents inform each other. We study this protocol in distributed computing models where communication might fail: an agent is certain about the messages it
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The Logic of Lexical Connectives Journal of Philosophical Logic Pub Date : 2023-07-21 Giorgio Sbardolini
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From Epistemic Norms to Logical Rules: Epistemic Models for Logical Expressivists Journal of Philosophical Logic Pub Date : 2023-07-10 Niklas Dahl
In this paper I construct a system of semantics for classical and intuitionistic propositional logic based on epistemic norms governing belief expansion. Working in the AGM-framework of belief change, I give a generalisation of Gärdenfors’ notion of belief systems which can be defined without reference to a logical consequence operator by using a version of the Ramsey Test. These belief expansion systems
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Intuitionistic Mereology II: Overlap and Disjointness Journal of Philosophical Logic Pub Date : 2023-05-22 Paolo Maffezioli, Achille C. Varzi
This paper extends the axiomatic treatment of intuitionistic mereology introduced in Maffezioli and Varzi (Synthese, 198(S18), 4277–4302 2021) by examining the behavior of constructive notions of overlap and disjointness. We consider both (i) various ways of defining such notions in terms of other intuitionistic mereological primitives, and (ii) the possibility of treating them as mereological primitives
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Solutions to the Knower Paradox in the Light of Haack’s Criteria Journal of Philosophical Logic Pub Date : 2023-04-17 Mirjam de Vos, Rineke Verbrugge, Barteld Kooi
The knower paradox states that the statement ‘We know that this statement is false’ leads to inconsistency. This article presents a fresh look at this paradox and some well-known solutions from the literature. Paul Égré discusses three possible solutions that modal provability logic provides for the paradox by surveying and comparing three different provability interpretations of modality, originally
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Identity and Harmony and Modality Journal of Philosophical Logic Pub Date : 2023-04-15 Julian J. Schlöder
Stephen Read presented harmonious inference rules for identity in classical predicate logic. I demonstrate here how this approach can be generalised to a setting where predicate logic has been extended with epistemic modals. In such a setting, identity has two uses. A rigid one, where the identity of two referents is preserved under epistemic possibility, and a non-rigid one where two identical referents
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A Logical Modeling of Severe Ignorance Journal of Philosophical Logic Pub Date : 2023-04-11 S. Bonzio, V. Fano, P. Graziani, M. Pra Baldi
In the logical context, ignorance is traditionally defined recurring to epistemic logic. In particular, ignorance is essentially interpreted as “lack of knowledge”. This received view has - as we point out - some problems, in particular we will highlight how it does not allow to express a type of content-theoretic ignorance, i.e. an ignorance of φ that stems from an unfamiliarity with its meaning.
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The Logic of Action and Control Journal of Philosophical Logic Pub Date : 2023-04-05 Leona Mollica
In this paper I propose and motivate a logic of the interdefined concepts of making true and control, understood as intensional propositional operators to be indexed to an agent. While bearing a resemblance to earlier logics in the tradition, the motivations, semantics, and object language theory differ on crucial points. Applying this logic to widespread formal theories of agency, I use it as a framework
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Modus Ponens and the Logic of Decision Journal of Philosophical Logic Pub Date : 2023-03-17 Nate Charlow
This article presents and discusses a prima facie counterexample to modus ponens. To appropriately theorize the case, I argue for conceptualizing the notions of logical consequence and logical commitment in “normative” terms, so that logical commitment does not attach to the premises of a spurious dominance argument—and, more generally, does not attach to unreasonable decision states, or to syntactically
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Varieties of Self-Reference in Metamathematics Journal of Philosophical Logic Pub Date : 2023-03-14 Balthasar Grabmayr, Volker Halbach, Lingyuan Ye
This paper investigates the conditions under which diagonal sentences can be taken to constitute paradigmatic cases of self-reference. We put forward well-motivated constraints on the diagonal operator and the coding apparatus which separate paradigmatic self-referential sentences, for instance obtained via Gödel’s diagonalization method, from accidental diagonal sentences. In particular, we show that
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Reflective Mereology Journal of Philosophical Logic Pub Date : 2023-02-16 Bokai Yao
I propose a new theory of mereology based on a mereological reflection principle. Reflective mereology has natural fusion principles but also refutes certain principles of classical mereology such as Universal Fusion and Fusion Uniqueness. Moreover, reflective mereology avoids Uzquiano’s cardinality problem–the problem that classical mereology tends to clash with set theory when they both quantify
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Is, Ought, and Cut Journal of Philosophical Logic Pub Date : 2023-02-14 Norbert Gratzl, Edi Pavlović
In this paper we use proof-theoretic methods, specifically sequent calculi, admissibility of cut within them and the resultant subformula property, to examine a range of philosophically-motivated deontic logics. We show that for all of those logics it is a (meta)theorem that the Special Hume Thesis holds, namely that no purely normative conclusion follows non-trivially from purely descriptive premises
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Consistent Theories in Inconsistent Logics Journal of Philosophical Logic Pub Date : 2023-02-07 Franci Mangraviti, Andrew Tedder
The relationship between logics with sets of theorems including contradictions (“inconsistent logics”) and theories closed under such logics is investigated. It is noted that if we take “theories” to be defined in terms of deductive closure understood in a way somewhat different from the standard, Tarskian, one, inconsistent logics can have consistent theories. That is, we can find some sets of formulas
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That’s It! Hyperintensional Total Logic Journal of Philosophical Logic Pub Date : 2023-02-02 Stephan Krämer
Call a truth complete with respect to a subject matter if it entails every truth about that subject matter. One attractive way to formulate a complete truth is to state all the relevant positive truths, and then add: and that’s it. When the subject matters under consideration are non-contingent, a non-trivial conception of completeness must invoke a hyperintensional conception of entailment, and of
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Wright’s Strict Finitistic Logic in the Classical Metatheory: The Propositional Case Journal of Philosophical Logic Pub Date : 2023-01-21 Takahiro Yamada
Crispin Wright in his 1982 paper argues for strict finitism, a constructive standpoint that is more restrictive than intuitionism. In its appendix, he proposes models of strict finitistic arithmetic. They are tree-like structures, formed in his strict finitistic metatheory, of equations between numerals on which concrete arithmetical sentences are evaluated. As a first step towards classical formalisation
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The Logic of Framing Effects Journal of Philosophical Logic Pub Date : 2023-01-17 Francesco Berto, Aybüke Özgün
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A Class of Implicative Expansions of Belnap-Dunn Logic in which Boolean Negation is Definable Journal of Philosophical Logic Pub Date : 2023-01-09 Gemma Robles, José M. Méndez
Belnap and Dunn’s well-known 4-valued logic FDE is an interesting and useful non-classical logic. FDE is defined by using conjunction, disjunction and negation as the sole propositional connectives. Then the question of expanding FDE with an implication connective is of course of great interest. In this sense, some implicative expansions of FDE have been proposed in the literature, among which Brady’s
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Editorial Introduction: Substructural Logics and Metainferences Journal of Philosophical Logic Pub Date : 2023-01-04 Eduardo Barrio, Paul Égré
The concept of substructural logic was originally introduced in relation to limitations of Gentzen’s structural rules of Contraction, Weakening and Exchange. Recent years have witnessed the development of substructural logics also challenging the Tarskian properties of Reflexivity and Transitivity of logical consequence. In this introduction we explain this recent development and two aspects in which
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Valueless Measures on Pointless Spaces Journal of Philosophical Logic Pub Date : 2022-12-12 Tamar Lando
On our ordinary representations of space, space is composed of indivisible, dimensionless points; extended regions are understood as infinite sets of points. Region-based theories of space reverse this atomistic picture, by taking as primitive several relations on extended regions, and recovering points as higher-order abstractions from regions. Over the years, such theories have focused almost exclusively
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Conservative Translations Revisited Journal of Philosophical Logic Pub Date : 2022-12-01 J. Ramos, J. Rasga, C. Sernadas
We provide sufficient conditions for the existence of a conservative translation from a consequence system to another one. We analyze the problem in many settings, namely when the consequence systems are generated by a deductive calculus or by a logic system including both proof-theoretic and model-theoretic components. We also discuss reflection of several metaproperties with the objective of showing
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Elementary Belief Revision Operators Journal of Philosophical Logic Pub Date : 2022-12-01 Jake Chandler, Richard Booth
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A Simple Logic of Concepts Journal of Philosophical Logic Pub Date : 2022-11-22 Thomas F. Icard, Lawrence S. Moss
In Pietroski (2018) a simple representation language called SMPL is introduced, construed as a hypothesis about core conceptual structure. The present work is a study of this system from a logical perspective. In addition to establishing a completeness result and a complexity characterization for reasoning in the system, we also pinpoint its expressive limits, in particular showing that the fourth