Review of Symbolic Logic

ISSN: 1755-0203

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  1.  17
    A Substructural Gentzen Calculus for Orthomodular Quantum Logic.Davide Fazio, Antonio Ledda, Francesco Paoli & Gavin St John - 2023 - Review of Symbolic Logic 16 (4):1177-1198.
    We introduce a sequent system which is Gentzen algebraisable with orthomodular lattices as equivalent algebraic semantics, and therefore can be viewed as a calculus for orthomodular quantum logic. Its sequents are pairs of non-associative structures, formed via a structural connective whose algebraic interpretation is the Sasaki product on the left-hand side and its De Morgan dual on the right-hand side. It is a substructural calculus, because some of the standard structural sequent rules are restricted—by lifting all such restrictions, one recovers (...)
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  2.  5
    Interpretation, Logic and Philosophy: Jean Nicod’s Geometry in the Sensible World.Sébastien Gandon - 2023 - Review of Symbolic Logic 16 (4):1080-1109.
    Jean Nicod (1893–1924) is a French philosopher and logician who worked with Russell during the First World War. His PhD, with a preface from Russell, was published under the title La géométrie dans le monde sensible in 1924, the year of his untimely death. The book did not have the impact he deserved. In this paper, I discuss the methodological aspect of Nicod’s approach. My aim is twofold. I would first like to show that Nicod’s definition of various notions of (...)
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  3. A simple sequent system for minimally inconsisteny LP.Rea Golan - 2023 - Review of Symbolic Logic 16 (4):1296-1311.
    Minimally inconsistent LP (MiLP) is a nonmonotonic paraconsistent logic based on Graham Priest's logic of paradox (LP). Unlike LP, MiLP purports to recover, in consistent situations, all of classical reasoning. The present paper conducts a proof-theoretic analysis of MiLP. I highlight certain properties of this logic, introduce a simple sequent system for it, and establish soundness and completeness results. In addition, I show how to use my proof system in response to a criticism of this logic put forward by JC (...)
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  4.  45
    Proof Systems for Exact Entailment.Johannes Korbmacher - 2023 - Review of Symbolic Logic 16 (4):1260-1295.
    We present a series of proof systems for exact entailment (i.e. relevant truthmaker preservation from premises to conclusion) and prove soundness and completeness. Using the proof systems, we observe that exact entailment is not only hyperintensional in the sense of Cresswell but also in the sense recently proposed by Odintsov and Wansing.
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  5.  5
    Two-Sorted Frege Arithmetic is Not Conservative.Stephen Mackereth & Jeremy Avigad - 2023 - Review of Symbolic Logic 16 (4):1199-1232.
    Neo-Fregean logicists claim that Hume’s Principle (HP) may be taken as an implicit definition of cardinal number, true simply by fiat. A long-standing problem for neo-Fregean logicism is that HP is not deductively conservative over pure axiomatic second-order logic. This seems to preclude HP from being true by fiat. In this paper, we study Richard Kimberly Heck’s Two-Sorted Frege Arithmetic (2FA), a variation on HP which has been thought to be deductively conservative over second-order logic. We show that it isn’t. (...)
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  6. Towards the Inevitability of Non-Classical Probability.Giacomo Molinari - 2023 - Review of Symbolic Logic 16 (4):1053-1079.
    This paper generalises an argument for probabilism due to Lindley [9]. I extend the argument to a number of non-classical logical settings whose truth-values, seen here as ideal aims for belief, are in the set $\{0,1\}$, and where logical consequence $\models $ is given the “no-drop” characterization. First I will show that, in each of these settings, an agent’s credence can only avoid accuracy-domination if its canonical transform is a (possibly non-classical) probability function. In other words, if an agent values (...)
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  7.  9
    The Zhou Ordinal of Labelled Markov Processes Over Separable Spaces.Martín Santiago Moroni & Pedro Sánchez Terraf - 2023 - Review of Symbolic Logic 16 (4):1011-1032.
    There exist two notions of equivalence of behavior between states of a Labelled Markov Process (LMP): state bisimilarity and event bisimilarity. The first one can be considered as an appropriate generalization to continuous spaces of Larsen and Skou’s probabilistic bisimilarity, whereas the second one is characterized by a natural logic. C. Zhou expressed state bisimilarity as the greatest fixed point of an operator that there is such a process with an uncountable Zhou ordinal.
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  8.  9
    On Zardini’s Rules for Multiplicative Quantification as the Source of Contra(di)Ctions.Uwe Petersen - 2023 - Review of Symbolic Logic 16 (4):1110-1119.
    Certain instances of contraction are provable in Zardini’s system $\mathbf {IK}^\omega $ which causes triviality once a truth predicate and suitable fixed points are available.
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  9.  4
    Fractional-Valued Modal Logic.Mario Piazza, Gabriele Pulcini & Matteo Tesi - 2023 - Review of Symbolic Logic 16 (4):1033-1052.
    This paper is dedicated to extending and adapting to modal logic the approach of fractional semantics to classical logic. This is a multi-valued semantics governed by pure proof-theoretic considerations, whose truth-values are the rational numbers in the closed interval $[0,1]$. Focusing on the modal logic K, the proposed methodology relies on three key components: bilateral sequent calculus, invertibility of the logical rules, and stability (proof-invariance). We show that our semantic analysis of K affords an informational refinement with respect to the (...)
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  10.  63
    Collection Frames for Distributive Substructural Logics.Greg Restall & Shawn Standefer - 2023 - Review of Symbolic Logic 16 (4):1120-1157.
    We present a new frame semantics for positive relevant and substructural propositional logics. This frame semantics is both a generalisation of Routley–Meyer ternary frames and a simplification of them. The key innovation of this semantics is the use of a single accessibility relation to relate collections of points to points. Different logics are modeled by varying the kinds of collections used: they can be sets, multisets, lists or trees. We show that collection frames on trees are sound and complete for (...)
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  11. Hume’s Principle, Bad Company, and the Axiom of Choice.Sam Roberts & Stewart Shapiro - 2023 - Review of Symbolic Logic 16 (4):1158-1176.
    One prominent criticism of the abstractionist program is the so-called Bad Company objection. The complaint is that abstraction principles cannot in general be a legitimate way to introduce mathematical theories, since some of them are inconsistent. The most notorious example, of course, is Frege’s Basic Law V. A common response to the objection suggests that an abstraction principle can be used to legitimately introduce a mathematical theory precisely when it is stable: when it can be made true on all sufficiently (...)
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  12.  19
    Independence Proofs in Non-Classical Set Theories.Sourav Tarafder & Giorgio Venturi - 2023 - Review of Symbolic Logic 16 (4):979-1010.
    In this paper we extend to non-classical set theories the standard strategy of proving independence using Boolean-valued models. This extension is provided by means of a new technique that, combining algebras (by taking their product), is able to provide product-algebra-valued models of set theories. In this paper we also provide applications of this new technique by showing that: (1) we can import the classical independence results to non-classical set theory (as an example we prove the independence of $\mathsf {CH}$ ); (...)
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  13.  6
    Set Theory and a Model of the Mind in Psychology.Asger Törnquist & Jens Mammen - 2023 - Review of Symbolic Logic 16 (4):1233-1259.
    We investigate the mathematics of a model of the human mind which has been proposed by the psychologist Jens Mammen. Mathematical realizations of this model consists of what the first author (A.T.) has called Mammen spaces, where a Mammen space is a triple in the Baumgartner–Laver model.Finally, consequences for psychology are discussed.
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  14.  31
    Stit -logic for imagination episodes with voluntary input.Christopher Badura & Heinrich Wansing - 2023 - Review of Symbolic Logic 16 (3):813-861.
    Francesco Berto proposed a logic for imaginative episodes. The logic establishes certain (in)validities concerning episodic imagination. They are not all equally plausible as principles of episodic imagination. The logic also does not model that the initial input of an imaginative episode is deliberately chosen.Stit-imagination logic models the imagining agent’s deliberate choice of the content of their imagining. However, the logic does not model the episodic nature of imagination. The present paper combines the two logics, thereby modelling imaginative episodes with deliberately (...)
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  15.  6
    Steel’s Programme: Evidential Framework, the Core and Ultimate- L.Joan Bagaria & Claudio Ternullo - 2023 - Review of Symbolic Logic 16 (3):788-812.
    We address Steel’s Programme to identify a ‘preferred’ universe of set theory and the best axioms extending $\mathsf {ZFC}$ by using his multiverse axioms $\mathsf {MV}$ and the ‘core hypothesis’. In the first part, we examine the evidential framework for $\mathsf {MV}$, in particular the use of large cardinals and of ‘worlds’ obtained through forcing to ‘represent’ alternative extensions of $\mathsf {ZFC}$. In the second part, we address the existence and the possible features of the core of $\mathsf {MV}_T$ (where (...)
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  16. The Genealogy of ‘∨’.Landon D. C. Elkind & Richard Zach - 2023 - Review of Symbolic Logic 16 (3):862-899.
    The use of the symbol ∨for disjunction in formal logic is ubiquitous. Where did it come from? The paper details the evolution of the symbol ∨ in its historical and logical context. Some sources say that disjunction in its use as connecting propositions or formulas was introduced by Peano; others suggest that it originated as an abbreviation of the Latin word for “or,” vel. We show that the origin of the symbol ∨ for disjunction can be traced to Whitehead and (...)
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  17. Conciliatory Reasoning, Self-Defeat, and Abstract Argumentation.Aleks Https://Orcidorg Knoks - 2023 - Review of Symbolic Logic 16 (3):740-787.
    According to conciliatory views on the significance of disagreement, it’s rational for you to become less confident in your take on an issue in case your epistemic peer’s take on it is different. These views are intuitively appealing, but they also face a powerful objection: in scenarios that involve disagreements over their own correctness, conciliatory views appear to self-defeat and, thereby, issue inconsistent recommendations. This paper provides a response to this objection. Drawing on the work from defeasible logics paradigm and (...)
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  18.  34
    Gödel on Many-Valued Logic.Tim Lethen - 2023 - Review of Symbolic Logic 16 (3):655-671.
    This paper collects and presents unpublished notes of Kurt Gödel concerning the field of many-valued logic. In order to get a picture as complete as possible, both formal and philosophical notes, transcribed from the Gabelsberger shorthand system, are included.
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  19.  6
    Improving Strong Negation.Satoru Niki - 2023 - Review of Symbolic Logic 16 (3):951-977.
    Strong negation is a well-known alternative to the standard negation in intuitionistic logic. It is defined virtually by giving falsity conditions to each of the connectives. Among these, the falsity condition for implication appears to unnecessarily deviate from the standard negation. In this paper, we introduce a slight modification to strong negation, and observe its comparative advantages over the original notion. In addition, we consider the paraconsistent variants of our modification, and study their relationship with non-constructive principles and connexivity.
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  20. Szemerédi’s theorem: An exploration of impurity, explanation, and content.Patrick J. Ryan - 2023 - Review of Symbolic Logic 16 (3):700-739.
    In this paper I argue for an association between impurity and explanatory power in contemporary mathematics. This proposal is defended against the ancient and influential idea that purity and explanation go hand-in-hand (Aristotle, Bolzano) and recent suggestions that purity/impurity ascriptions and explanatory power are more or less distinct (Section 1). This is done by analyzing a central and deep result of additive number theory, Szemerédi’s theorem, and various of its proofs (Section 2). In particular, I focus upon the radically impure (...)
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  21.  23
    Subatomic Inferences: An Inferentialist Semantics for Atomics, Predicates, and Names.Kai Tanter - 2023 - Review of Symbolic Logic 16 (3):672-699.
    Inferentialism is a theory in the philosophy of language which claims that the meanings of expressions are constituted by inferential roles or relations. Instead of a traditional model-theoretic semantics, it naturally lends itself to a proof-theoretic semantics, where meaning is understood in terms of inference rules with a proof system. Most work in proof-theoretic semantics has focused on logical constants, with comparatively little work on the semantics of non-logical vocabulary. Drawing on Robert Brandom’s notion of material inference and Greg Restall’s (...)
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  22.  58
    Carnap’s Problem for Modal Logic.Denis Bonnay & Dag Westerståhl - 2023 - Review of Symbolic Logic 16 (2):578-602.
    We take Carnap’s problem to be to what extent standard consequence relations in various formal languages fix the meaning of their logical vocabulary, alone or together with additional constraints on the form of the semantics. This paper studies Carnap’s problem for basic modal logic. Setting the stage, we show that neighborhood semantics is the most general form of compositional possible worlds semantics, and proceed to ask which standard modal logics (if any) constrain the box operator to be interpreted as in (...)
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  23.  52
    The Semantic Foundations of Philosophical Analysis.Samuel Z. Elgin - 2023 - Review of Symbolic Logic 16 (2):603-623.
    I provide an analysis of sentences of the form ‘To be F is to be G’ in terms of exact truth-maker semantics—an approach that identifies the meanings of sentences with the states of the world directly responsible for their truth-values. Roughly, I argue that these sentences hold just in case that which makes something F also makes it G. This approach is hyperintensional and possesses desirable logical and modal features. In particular, these sentences are reflexive, transitive, and symmetric, and if (...)
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  24.  33
    Nonclassical Truth with Classical Strength. A Proof-Theoretic Analysis of Compositional Truth Over Hype.Martin Fischer, Carlo Nicolai & Pablo Dopico - 2023 - Review of Symbolic Logic 16 (2):425-448.
    Questions concerning the proof-theoretic strength of classical versus nonclassical theories of truth have received some attention recently. A particularly convenient case study concerns classical and nonclassical axiomatizations of fixed-point semantics. It is known that nonclassical axiomatizations in four- or three-valued logics are substantially weaker than their classical counterparts. In this paper we consider the addition of a suitable conditional to First-Degree Entailment—a logic recently studied by Hannes Leitgeb under the label HYPE. We show in particular that, by formulating the theory (...)
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  25.  42
    Self-Reference Upfront: A Study of Self-Referential Gödel Numberings.Balthasar Grabmayr & Albert Visser - 2023 - Review of Symbolic Logic 16 (2):385-424.
    In this paper we examine various requirements on the formalisation choices under which self-reference can be adequately formalised in arithmetic. In particular, we study self-referential numberings, which immediately provide a strong notion of self-reference even for expressively weak languages. The results of this paper suggest that the question whether truly self-referential reasoning can be formalised in arithmetic is more sensitive to the underlying coding apparatus than usually believed. As a case study, we show how this sensitivity affects the formal study (...)
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  26.  7
    Grzegorczyk Points and Filters in Boolean Contact Algebras.Rafał Gruszczyński & Andrzej Pietruszczak - 2023 - Review of Symbolic Logic 16 (2):509-528.
    The purpose of this paper is to compare the notion of a Grzegorczyk point introduced in [19] (and thoroughly investigated in [3, 14, 16, 18]) to the standard notions of a filter in Boolean algebras and round filter in Boolean contact algebras. In particular, we compare Grzegorczyk points to filters and ultrafilters of atomic and atomless algebras. We also prove how a certain extra axiom influences topological spaces for Grzegorczyk contact algebras. Last but not least, we do not refrain from (...)
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  27.  14
    A Correctness Proof for Al-Barakāt’s Logical Diagrams.Wilfrid Hodges - 2023 - Review of Symbolic Logic 16 (2):369-384.
    In Baghdad in the mid twelfth century Abū al-Barakāt proposes a radical new procedure for finding the conclusions of premise-pairs in syllogistic logic, and for identifying those premise-pairs that have no conclusions. The procedure makes no use of features of the standard Aristotelian apparatus, such as conversions or syllogistic figures. In place of these al-Barakāt writes out pages of diagrams consisting of labelled horizontal lines. He gives no instructions and no proof that the procedure will yield correct results. So the (...)
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  28.  12
    Russellian Definite Description Theory—a Proof Theoretic Approach.Andrzej Indrzejczak - 2023 - Review of Symbolic Logic 16 (2):624-649.
    The paper provides a proof theoretic characterization of the Russellian theory of definite descriptions (RDD) as characterized by Kalish, Montague and Mar (KMM). To this effect three sequent calculi are introduced: LKID0, LKID1 and LKID2. LKID0 is an auxiliary system which is easily shown to be equivalent to KMM. The main research is devoted to LKID1 and LKID2. The former is simpler in the sense of having smaller number of rules and, after small change, satisfies cut elimination but fails to (...)
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  29.  83
    Gödel’s Theorem and Direct Self-Reference.Saul A. Kripke - 2023 - Review of Symbolic Logic 16 (2):650-654.
    In his paper on the incompleteness theorems, Gödel seemed to say that a direct way of constructing a formula that says of itself that it is unprovable might involve a faulty circularity. In this note, it is proved that ‘direct’ self-reference can actually be used to prove his result.
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  30.  7
    Conditional Logic is Complete for Convexity in the Plane.Johannes Marti - 2023 - Review of Symbolic Logic 16 (2):529-552.
    We prove completeness of preferential conditional logic with respect to convexity over finite sets of points in the Euclidean plane. A conditional is defined to be true in a finite set of points if all extreme points of the set interpreting the antecedent satisfy the consequent. Equivalently, a conditional is true if the antecedent is contained in the convex hull of the points that satisfy both the antecedent and consequent. Our result is then that every consistent formula without nested conditionals (...)
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  31.  23
    Abstract Forms of Quantification in the Quantified Argument Calculus.Edi Pavlović & Norbert Gratzl - 2023 - Review of Symbolic Logic 16 (2):449-479.
    The Quantified argument calculus (Quarc) has received a lot of attention recently as an interesting system of quantified logic which eschews the use of variables and unrestricted quantification, but nonetheless achieves results similar to the Predicate calculus (PC) by employing quantifiers applied directly to predicates instead. Despite this noted similarity, the issue of the relationship between Quarc and PC has so far not been definitively resolved. We address this question in the present paper, and then expand upon that result. Utilizing (...)
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  32.  60
    A Note on Choice Principles in Second-Order Logic.Benjamin Siskind, Paolo Mancosu & Stewart Shapiro - 2023 - Review of Symbolic Logic 16 (2):339-350.
    Zermelo’s Theorem that the axiom of choice is equivalent to the principle that every set can be well-ordered goes through in third-order logic, but in second-order logic we run into expressivity issues. In this note, we show that in a natural extension of second-order logic weaker than third-order logic, choice still implies the well-ordering principle. Moreover, this extended second-order logic with choice is conservative over ordinary second-order logic with the well-ordering principle. We also discuss a variant choice principle, due to (...)
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  33.  12
    Probabilistic Entailment on First Order Languages and Reasoning with Inconsistencies.R. A. D. Soroush Rafiee - 2023 - Review of Symbolic Logic 16 (2):351-368.
    We investigate an approach for drawing logical inference from inconsistent premisses. The main idea in this approach is that the inconsistencies in the premisses should be interpreted as uncertainty of the information. We propose a mechanism, based on Kinght’s [14] study of inconsistency, for revising an inconsistent set of premisses to a minimally uncertain, probabilistically consistent one. We will then generalise the probabilistic entailment relation introduced in [15] for propositional languages to the first order case to draw logical inference from (...)
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  34. The Potential in Frege’s Theorem.Will Stafford - 2023 - Review of Symbolic Logic 16 (2):553-577.
    Is a logicist bound to the claim that as a matter of analytic truth there is an actual infinity of objects? If Hume’s Principle is analytic then in the standard setting the answer appears to be yes. Hodes’s work pointed to a way out by offering a modal picture in which only a potential infinity was posited. However, this project was abandoned due to apparent failures of cross-world predication. We re-explore this idea and discover that in the setting of the (...)
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  35.  46
    Rigour and Proof.Oliver Tatton-Brown - 2023 - Review of Symbolic Logic 16 (2):480-508.
    This paper puts forward a new account of rigorous mathematical proof and its epistemology. One novel feature is a focus on how the skill of reading and writing valid proofs is learnt, as a way of understanding what validity itself amounts to. The account is used to address two current questions in the literature: that of how mathematicians are so good at resolving disputes about validity, and that of whether rigorous proofs are necessarily formalizable.
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  36. Support for Geometric Pooling.Jean Baccelli & Rush T. Stewart - 2023 - Review of Symbolic Logic 16 (1):298-337.
    Supra-Bayesianism is the Bayesian response to learning the opinions of others. Probability pooling constitutes an alternative response. One natural question is whether there are cases where probability pooling gives the supra-Bayesian result. This has been called the problem of Bayes-compatibility for pooling functions. It is known that in a common prior setting, under standard assumptions, linear pooling cannot be nontrivially Bayes-compatible. We show by contrast that geometric pooling can be nontrivially Bayes-compatible. Indeed, we show that, under certain assumptions, geometric and (...)
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  37.  10
    Temporal Interpretation of Monadic Intuitionistic Quantifiers.Guram Bezhanishvili & Luca Carai - 2023 - Review of Symbolic Logic 16 (1):164-187.
    We show that monadic intuitionistic quantifiers admit the following temporal interpretation: “always in the future” (for$\forall $) and “sometime in the past” (for$\exists $). It is well known that Prior’s intuitionistic modal logic${\sf MIPC}$axiomatizes the monadic fragment of the intuitionistic predicate logic, and that${\sf MIPC}$is translated fully and faithfully into the monadic fragment${\sf MS4}$of the predicate${\sf S4}$via the Gödel translation. To realize the temporal interpretation mentioned above, we introduce a new tense extension${\sf TS4}$of${\sf S4}$and provide a full and faithful translation (...)
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  38.  31
    Quantified Modal Relevant Logics.Nicholas Ferenz - 2023 - Review of Symbolic Logic 16 (1):210-240.
    Here, I combine the semantics of Mares and Goldblatt [20] and Seki [29, 30] to develop a semantics for quantified modal relevant logics extending ${\bf B}$. The combination requires demonstrating that the Mares–Goldblatt approach is apt for quantified extensions of ${\bf B}$ and other relevant logics, but no significant bridging principles are needed. The result is a single semantic approach for quantified modal relevant logics. Within this framework, I discuss the requirements a quantified modal relevant logic must satisfy to be (...)
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  39. Operands and Instances.Peter Fritz - 2023 - Review of Symbolic Logic 16 (1):188-209.
    Can conjunctive propositions be identical without their conjuncts being identical? Can universally quantified propositions be identical without their instances being identical? On a common conception of propositions, on which they inherit the logical structure of the sentences which express them, the answer is negative both times. Here, it will be shown that such a negative answer to both questions is inconsistent, assuming a standard type-theoretic formalization of theorizing about propositions. The result is not specific to conjunction and universal quantification, but (...)
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  40.  27
    Games and Cardinalities in Inquisitive First-Order Logic.Gianluca Grilletti & Ivano Ciardelli - 2023 - Review of Symbolic Logic 16 (1):241-267.
    Inquisitive first-order logic, InqBQ, is a system which extends classical first-order logic with formulas expressing questions. From a mathematical point of view, formulas in this logic express properties of sets of relational structures. This paper makes two contributions to the study of this logic. First, we describe an Ehrenfeucht–Fraïssé game for InqBQ and show that it characterizes the distinguishing power of the logic. Second, we use the game to study cardinality quantifiers in the inquisitive setting. That is, we study what (...)
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  41.  61
    Predicativism as a Form of Potentialism.Øystein Linnebo & Stewart Shapiro - 2023 - Review of Symbolic Logic 16 (1):1-32.
    In the literature, predicativism is connected not only with the Vicious Circle Principle but also with the idea that certain totalities are inherently potential. To explain the connection between these two aspects of predicativism, we explore some approaches to predicativity within the modal framework for potentiality developed in Linnebo (2013) and Linnebo and Shapiro (2019). This puts predicativism into a more general framework and helps to sharpen some of its key theses.
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  42.  11
    The Lattice of Super-Belnap Logics.Adam Přenosil - 2023 - Review of Symbolic Logic 16 (1):114-163.
    We study the lattice of extensions of four-valued Belnap–Dunn logic, called super-Belnap logics by analogy with superintuitionistic logics. We describe the global structure of this lattice by splitting it into several subintervals, and prove some new completeness theorems for super-Belnap logics. The crucial technical tool for this purpose will be the so-called antiaxiomatic (or explosive) part operator. The antiaxiomatic (or explosive) extensions of Belnap–Dunn logic turn out to be of particular interest owing to their connection to graph theory: the lattice (...)
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  43.  75
    For Better and for Worse. Abstractionism, Good Company, and Pluralism.Andrea Sereni, Maria Paola Sforza Fogliani & Luca Zanetti - 2023 - Review of Symbolic Logic 16 (1):268-297.
    A thriving literature has developed over logical and mathematical pluralism – i.e. the views that several rival logical and mathematical theories can be equally correct. These have unfortunately grown separate; instead, they both could gain a great deal by a closer interaction. Our aim is thus to present some novel forms of abstractionist mathematical pluralism which can be modeled on parallel ways of substantiating logical pluralism (also in connection with logical anti-exceptionalism). To do this, we start by discussing the Good (...)
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  44.  7
    Rigour and Proof – Corrigendum.Oliver Tatton-Brown - 2023 - Review of Symbolic Logic 16 (1):338-338.
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