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CONCEPTUAL DISTANCE AND ALGEBRAS OF CONCEPTS Rev. Symb. Log. (IF 0.6) Pub Date : 2024-02-22 MOHAMED KHALED, GERGELY SZÉKELY
We show that the conceptual distance between any two theories of first-order logic is the same as the generator distance between their Lindenbaum–Tarski algebras of concepts. As a consequence of this, we show that, for any two arbitrary mathematical structures, the generator distance between their meaning algebras (also known as cylindric set algebras) is the same as the conceptual distance between
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BEYOND LINGUISTIC INTERPRETATION IN THEORY COMPARISON Rev. Symb. Log. (IF 0.6) Pub Date : 2023-12-21 TOBY MEADOWS
This paper assembles a unifying framework encompassing a wide variety of mathematical instruments used to compare different theories. The main theme will be the idea that theory comparison techniques are most easily grasped and organized through the lens of category theory. The paper develops a table of different equivalence relations between theories and then answers many of the questions about how
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PARACONSISTENT AND PARACOMPLETE ZERMELO–FRAENKEL SET THEORY Rev. Symb. Log. (IF 0.6) Pub Date : 2023-12-13 YURII KHOMSKII, HRAFN VALTÝR ODDSSON
We present a novel treatment of set theory in a four-valued paraconsistent and paracomplete logic, i.e., a logic in which propositions can be both true and false, and neither true nor false. Our approach is a significant departure from previous research in paraconsistent set theory, which has almost exclusively been motivated by a desire to avoid Russell’s paradox and fulfil naive comprehension. Instead
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LOGICS FROM ULTRAFILTERS Rev. Symb. Log. (IF 0.6) Pub Date : 2023-11-28 DANIELE MUNDICI
Ultrafilters play a significant role in model theory to characterize logics having various compactness and interpolation properties. They also provide a general method to construct extensions of first-order logic having these properties. A main result of this paper is that every class $\Omega $ of uniform ultrafilters generates a $\Delta $-closed logic ${\mathcal {L}}_\Omega $. ${\mathcal {L}}_\Omega
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ONTOLOGICAL PURITY FOR FORMAL PROOFS Rev. Symb. Log. (IF 0.6) Pub Date : 2023-11-13 ROBIN MARTINOT
Purity is known as an ideal of proof that restricts a proof to notions belonging to the ‘content’ of the theorem. In this paper, our main interest is to develop a conception of purity for formal (natural deduction) proofs. We develop two new notions of purity: one based on an ontological notion of the content of a theorem, and one based on the notions of surrogate ontological content and structural
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NON-FACTIVE KOLMOGOROV CONDITIONALIZATION Rev. Symb. Log. (IF 0.6) Pub Date : 2023-10-31 MICHAEL RESCORLA
Kolmogorov conditionalization is a strategy for updating credences based on propositions that have initial probability 0. I explore the connection between Kolmogorov conditionalization and Dutch books. Previous discussions of the connection rely crucially upon a factivity assumption: they assume that the agent updates credences based on true propositions. The factivity assumption discounts cases of
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AN ALGEBRAIC PROOF OF COMPLETENESS FOR MONADIC FUZZY PREDICATE LOGIC Rev. Symb. Log. (IF 0.6) Pub Date : 2023-10-18 JUNTAO WANG, HONGWEI WU, PENGFEI HE, YANHONG SHE
Monoidal t-norm based logic $\mathbf {MTL}$ is the weakest t-norm based residuated fuzzy logic, which is a $[0,1]$-valued propositional logical system having a t-norm and its residuum as truth function for conjunction and implication. Monadic fuzzy predicate logic $\mathbf {mMTL\forall }$ that consists of the formulas with unary predicates and just one object variable, is the monadic fragment of fuzzy
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ALGEBRAIC SEMANTICS FOR RELATIVE TRUTH, AWARENESS, AND POSSIBILITY Rev. Symb. Log. (IF 0.6) Pub Date : 2023-09-28 EVAN PIERMONT
This paper puts forth a class of algebraic structures, relativized Boolean algebras (RBAs), that provide semantics for propositional logic in which truth/validity is only defined relative to a local domain. In particular, the join of an event and its complement need not be the top element. Nonetheless, behavior is locally governed by the laws of propositional logic. By further endowing these structures
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CONNEXIVE IMPLICATIONS IN SUBSTRUCTURAL LOGICS Rev. Symb. Log. (IF 0.6) Pub Date : 2023-07-24 DAVIDE FAZIO, GAVIN ST. JOHN
This paper is devoted to the investigation of term-definable connexive implications in substructural logics with exchange and, on the semantical perspective, in sub-varieties of commutative residuated lattices (FL${}_{\scriptsize\mbox{e}}$-algebras). In particular, we inquire into sufficient and necessary conditions under which generalizations of the connexive implication-like operation defined in
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‘A REMARKABLE ARTIFICE’: LAPLACE, POISSON AND MATHEMATICAL PURITY Rev. Symb. Log. (IF 0.6) Pub Date : 2023-07-24 BRAM PEL
In the early nineteenth century, a series of articles by Laplace and Poisson discussed the importance of ‘directness’ in mathematical methodology. In this thesis, we argue that their conception of a ‘direct’ proof is similar to the more widely contemplated notion of a ‘pure’ proof. More rigorous definitions of mathematical purity were proposed in recent publications by Arana and Detlefsen, as well
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TESTING DEFINITIONAL EQUIVALENCE OF THEORIES VIA AUTOMORPHISM GROUPS Rev. Symb. Log. (IF 0.6) Pub Date : 2023-07-17 HAJNAL ANDRÉKA, JUDIT MADARÁSZ, ISTVÁN NÉMETI, GERGELY SZÉKELY
Two first-order logic theories are definitionally equivalent if and only if there is a bijection between their model classes that preserves isomorphisms and ultraproducts (Theorem 2). This is a variant of a prior theorem of van Benthem and Pearce. In Example 2, uncountably many pairs of definitionally inequivalent theories are given such that their model categories are concretely isomorphic via bijections
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SUBJECT-MATTER AND INTENSIONAL OPERATORS III: STATE-SENSITIVE SUBJECT-MATTER AND TOPIC SUFFICIENCY Rev. Symb. Log. (IF 0.6) Pub Date : 2023-07-12 THOMAS MACAULAY FERGUSON
Logical frameworks that are sensitive to features of sentences’ subject-matter—like Berto’s topic-sensitive intentional modals (TSIMs)—demand a maximally faithful model of the topics of sentences. This is an especially difficult task in the case in which topics are assigned to intensional formulae. In two previous papers, a framework was developed whose model of intensional subject-matter could accommodate
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A NATURAL DEDUCTION SYSTEM FOR ORTHOMODULAR LOGIC Rev. Symb. Log. (IF 0.6) Pub Date : 2023-07-10 ANDRE KORNELL
Orthomodular logic is a weakening of quantum logic in the sense of Birkhoff and von Neumann. Orthomodular logic is shown to be a nonlinear noncommutative logic. Sequents are given a physically motivated semantics that is consistent with exactly one semantics for propositional formulas that use negation, conjunction, and implication. In particular, implication must be interpreted as the Sasaki arrow
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A SIMPLIFIED PROOF OF THE EPSILON THEOREMS Rev. Symb. Log. (IF 0.6) Pub Date : 2023-07-10 STEFAN HETZL
We formulate Hilbert’s epsilon calculus in the context of expansion proofs. This leads to a simplified proof of the epsilon theorems by disposing of the need for prenexification, Skolemisation, and their respective inverse transformations. We observe that the natural notion of cut in the epsilon calculus is associative.
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RELEVANT CONSEQUENCE RELATIONS: AN INVITATION Rev. Symb. Log. (IF 0.6) Pub Date : 2023-06-30 GUILLERMO BADIA, LIBOR BĚHOUNEK, PETR CINTULA, ANDREW TEDDER
We generalize the notion of consequence relation standard in abstract treatments of logic to accommodate intuitions of relevance. The guiding idea follows the use criterion, according to which in order for some premises to have some conclusion(s) as consequence(s), the premises must each be used in some way to obtain the conclusion(s). This relevance intuition turns out to require not just a failure
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IS CANTOR’S THEOREM A DIALETHEIA? VARIATIONS ON A PARACONSISTENT APPROACH TO CANTOR’S THEOREM Rev. Symb. Log. (IF 0.6) Pub Date : 2023-06-29 UWE PETERSEN
The present note was prompted by Weber’s approach to proving Cantor’s theorem, i.e., the claim that the cardinality of the power set of a set is always greater than that of the set itself. While I do not contest that his proof succeeds, my point is that he neglects the possibility that by similar methods it can be shown also that no non-empty set satisfies Cantor’s theorem. In this paper unrestricted
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ANALYTICITY AND SYNTHETICITY IN TYPE THEORY REVISITED Rev. Symb. Log. (IF 0.6) Pub Date : 2023-06-27 BRUNO BENTZEN
I discuss problems with Martin-Löf’s distinction between analytic and synthetic judgments in constructive type theory and propose a revision of his views. I maintain that a judgment is analytic when its correctness follows exclusively from the evaluation of the expressions occurring in it. I argue that Martin-Löf’s claim that all judgments of the forms $a : A$ and $a = b : A$ are analytic is unfounded
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FINITE AXIOMATIZABILITY OF TRANSITIVE MODAL LOGICS OF FINITE DEPTH AND WIDTH WITH RESPECT TO PROPER-SUCCESSOR-EQUIVALENCE Rev. Symb. Log. (IF 0.6) Pub Date : 2023-06-23 YAN ZHANG, MING XU
This paper proves the finite axiomatizability of transitive modal logics of finite depth and finite width w.r.t. proper-successor-equivalence. The frame condition of the latter requires, in a rooted transitive frame, a finite upper bound of cardinality for antichains of points with different sets of proper successors. The result generalizes Rybakov’s result of the finite axiomatizability of extensions
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UNPRINCIPLED Rev. Symb. Log. (IF 0.6) Pub Date : 2023-06-13 GORDON BELOT
It is widely thought that chance should be understood in reductionist terms: claims about chance should be understood as claims that certain patterns of events are instantiated. There are many possible reductionist theories of chance, differing as to which possible pattern of events they take to be chance-making. It is also widely taken to be a norm of rationality that credence should defer to chance:
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FIRST-ORDER FRIENDLINESS Rev. Symb. Log. (IF 0.6) Pub Date : 2023-06-07 GUILLERMO BADIA, DAVID MAKINSON
In this note we study a counterpart in predicate logic of the notion of logical friendliness, introduced into propositional logic in [15]. The result is a new consequence relation for predicate languages with equality using first-order models. While compactness, interpolation and axiomatizability fail dramatically, several other properties are preserved from the propositional case. Divergence is diminished
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NON-CONTRACTIVE LOGICS, PARADOXES, AND MULTIPLICATIVE QUANTIFIERS Rev. Symb. Log. (IF 0.6) Pub Date : 2023-06-05 CARLO NICOLAI, MARIO PIAZZA, MATTEO TESI
The paper investigates from a proof-theoretic perspective various non-contractive logical systems, which circumvent logical and semantic paradoxes. Until recently, such systems only displayed additive quantifiers (Grišin and Cantini). Systems with multiplicative quantifiers were proposed in the 2010s (Zardini), but they turned out to be inconsistent with the naive rules for truth or comprehension.
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VARIETIES OF CLASS-THEORETIC POTENTIALISM Rev. Symb. Log. (IF 0.6) Pub Date : 2023-05-22 NEIL BARTON, KAMERYN J. WILLIAMS
We explain and explore class-theoretic potentialism—the view that one can always individuate more classes over a set-theoretic universe. We examine some motivations for class-theoretic potentialism, before proving some results concerning the relevant potentialist systems (in particular exhibiting failures of the $\mathsf {.2}$ and $\mathsf {.3}$ axioms). We then discuss the significance of these results
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ON THREE-VALUED PRESENTATIONS OF CLASSICAL LOGIC Rev. Symb. Log. (IF 0.6) Pub Date : 2023-05-11 BRUNO DA RÉ, DAMIAN SZMUC, EMMANUEL CHEMLA, PAUL ÉGRÉ
Given a three-valued definition of validity, which choice of three-valued truth tables for the connectives can ensure that the resulting logic coincides exactly with classical logic? We give an answer to this question for the five monotonic consequence relations $st$ , $ss$ , $tt$ , $ss\cap tt$ , and $ts$ , when the connectives are negation, conjunction, and disjunction. For $ts$ and $ss\cap tt$ the
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WHAT MODEL COMPANIONSHIP CAN SAY ABOUT THE CONTINUUM PROBLEM Rev. Symb. Log. (IF 0.6) Pub Date : 2023-04-25 GIORGIO VENTURI, MATTEO VIALE
We present recent results on the model companions of set theory, placing them in the context of a current debate in the philosophy of mathematics. We start by describing the dependence of the notion of model companionship on the signature, and then we analyze this dependence in the specific case of set theory. We argue that the most natural model companions of set theory describe (as the signature
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CANONICITY IN POWER AND MODAL LOGICS OF FINITE ACHRONAL WIDTH Rev. Symb. Log. (IF 0.6) Pub Date : 2023-03-22 ROBERT GOLDBLATT, IAN HODKINSON
We develop a method for showing that various modal logics that are valid in their countably generated canonical Kripke frames must also be valid in their uncountably generated ones. This is applied to many systems, including the logics of finite width, and a broader class of multimodal logics of ‘finite achronal width’ that are introduced here.
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FIRST-ORDER RELEVANT REASONERS IN CLASSICAL WORLDS Rev. Symb. Log. (IF 0.6) Pub Date : 2023-03-21 NICHOLAS FERENZ
Sedlár and Vigiani [18] have developed an approach to propositional epistemic logics wherein (i) an agent’s beliefs are closed under relevant implication and (ii) the agent is located in a classical possible world (i.e., the non-modal fragment is classical). Here I construct first-order extensions of these logics using the non-Tarskian interpretation of the quantifiers introduced by Mares and Goldblatt
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AN ALGORITHMIC IMPOSSIBLE-WORLDS MODEL OF BELIEF AND KNOWLEDGE Rev. Symb. Log. (IF 0.6) Pub Date : 2023-03-13 ZEYNEP SOYSAL
In this paper, I develop an algorithmic impossible-worlds model of belief and knowledge that provides a middle ground between models that entail that everyone is logically omniscient and those that are compatible with even the most egregious kinds of logical incompetence. In outline, the model entails that an agent believes (knows) $\phi $ just in case she can easily (and correctly) compute that $\phi
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THE ZHOU ORDINAL OF LABELLED MARKOV PROCESSES OVER SEPARABLE SPACES Rev. Symb. Log. (IF 0.6) Pub Date : 2023-02-27 MARTÍN SANTIAGO MORONI, PEDRO SÁNCHEZ TERRAF
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
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AXIOMS FOR TYPE-FREE SUBJECTIVE PROBABILITY Rev. Symb. Log. (IF 0.6) Pub Date : 2023-02-27 CEZARY CIEŚLIŃSKI, LEON HORSTEN, HANNES LEITGEB
We formulate and explore two basic axiomatic systems of type-free subjective probability. One of them explicates a notion of finitely additive probability. The other explicates a concept of infinitely additive probability. It is argued that the first of these systems is a suitable background theory for formally investigating controversial principles about type-free subjective probability.
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NECESSARY AND SUFFICIENT CONDITIONS FOR DOMINATION RESULTS FOR PROPER SCORING RULES Rev. Symb. Log. (IF 0.6) Pub Date : 2023-01-16 ALEXANDER R. PRUSS
Scoring rules measure the deviation between a forecast, which assigns degrees of confidence to various events, and reality. Strictly proper scoring rules have the property that for any forecast, the mathematical expectation of the score of a forecast p by the lights of p is strictly better than the mathematical expectation of any other forecast q by the lights of p. Forecasts need not satisfy the axioms
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DISJUNCTION AND EXISTENCE PROPERTIES IN MODAL ARITHMETIC Rev. Symb. Log. (IF 0.6) Pub Date : 2022-12-23 TAISHI KURAHASHI, MOTOKI OKUDA
We systematically study several versions of the disjunction and the existence properties in modal arithmetic. First, we newly introduce three classes $\mathrm {B}$ , $\Delta (\mathrm {B})$ , and $\Sigma (\mathrm {B})$ of formulas of modal arithmetic and study basic properties of them. Then, we prove several implications between the properties. In particular, among other things, we prove that for any
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NATURAL KIND SEMANTICS FOR A CLASSICAL ESSENTIALIST THEORY OF KINDS Rev. Symb. Log. (IF 0.6) Pub Date : 2022-12-05 JAVIER BELASTEGUI
The aim of this paper is to provide a complete Natural Kind Semantics for an Essentialist Theory of Kinds. The theory is formulated in two-sorted first order monadic modal logic with identity. The natural kind semantics is based on Rudolf Willes Theory of Concept Lattices. The semantics is then used to explain several consequences of the theory, including results about the specificity (species–genus)
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GENERALIZED PARTIAL MEET AND KERNEL CONTRACTIONS Rev. Symb. Log. (IF 0.6) Pub Date : 2022-11-24 MARCO GARAPA, MAURÍCIO D. L. REIS
Two of the most well-known belief contraction operators are partial meet contractions (PMCs) and kernel contractions (KCs). In this paper we propose two new classes of contraction operators, namely the class of generalized partial meet contractions (GPMC) and the class of generalized kernel contractions (GKC), which strictly contain the classes of PMCs and of KCs, respectively. We identify some extra
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THE LOGIC OF HYPERLOGIC. PART B: EXTENSIONS AND RESTRICTIONS Rev. Symb. Log. (IF 0.6) Pub Date : 2022-10-12 ALEXANDER W. KOCUREK
This is the second part of a two-part series on the logic of hyperlogic, a formal system for regimenting metalogical claims in the object language (even within embedded environments). Part A provided a minimal logic for hyperlogic that is sound and complete over the class of all models. In this part, we extend these completeness results to stronger logics that are sound and complete over restricted
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ON SHAVRUKOV’S NON-ISOMORPHISM THEOREM FOR DIAGONALIZABLE ALGEBRAS Rev. Symb. Log. (IF 0.6) Pub Date : 2022-09-12 EVGENY A. KOLMAKOV
We prove a strengthened version of Shavrukov’s result on the non-isomorphism of diagonalizable algebras of two $\Sigma _1$ -sound theories, based on the improvements previously found by Adamsson. We then obtain several corollaries to the strengthened result by applying it to various pairs of theories and obtain new non-isomorphism examples. In particular, we show that there are no surjective homomorphisms
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CRAIG INTERPOLATION THEOREM FAILS IN BI-INTUITIONISTIC PREDICATE LOGIC Rev. Symb. Log. (IF 0.6) Pub Date : 2022-08-12 GRIGORY K. OLKHOVIKOV, GUILLERMO BADIA
In this article we show that bi-intuitionistic predicate logic lacks the Craig Interpolation Property. We proceed by adapting the counterexample given by Mints, Olkhovikov and Urquhart for intuitionistic predicate logic with constant domains [13]. More precisely, we show that there is a valid implication $\phi \rightarrow \psi $ with no interpolant. Importantly, this result does not contradict the
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ON ZARDINI’S RULES FOR MULTIPLICATIVE QUANTIFICATION AS THE SOURCE OF CONTRA(DI)CTIONS Rev. Symb. Log. (IF 0.6) Pub Date : 2022-07-19 UWE PETERSEN
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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COLLECTION FRAMES FOR DISTRIBUTIVE SUBSTRUCTURAL LOGICS Rev. Symb. Log. (IF 0.6) Pub Date : 2022-07-13 GREG RESTALL, SHAWN STANDEFER
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
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PROBABILISTIC ENTAILMENT ON FIRST ORDER LANGUAGES AND REASONING WITH INCONSISTENCIES Rev. Symb. Log. (IF 0.6) Pub Date : 2022-07-07 SOROUSH RAFIEE RAD
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
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COUNTING TO INFINITY: GRADED MODAL LOGIC WITH AN INFINITY DIAMOND Rev. Symb. Log. (IF 0.6) Pub Date : 2022-07-07 IGNACIO BELLAS ACOSTA, YDE VENEMA
We extend the languages of both basic and graded modal logic with the infinity diamond, a modality that expresses the existence of infinitely many successors having a certain property. In both cases we define a natural notion of bisimilarity for the resulting formalisms, that we dub $\mathtt {ML}^{\infty }$ and $\mathtt {GML}^{\infty }$ , respectively. We then characterise these logics as the bisimulation-invariant
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STRONG HOMOMORPHISMS, CATEGORY THEORY, AND SEMANTIC PARADOX Rev. Symb. Log. (IF 0.6) Pub Date : 2022-05-30 JONATHAN WOLFGRAM, ROY T. COOK
In this essay we introduce a new tool for studying the patterns of sentential reference within the framework introduced in [2] and known as the language of paradox $\mathcal {L}_{\mathsf {P}}$ : strong $\mathcal {L}_{\mathsf {P}}$ -homomorphisms. In particular, we show that (i) strong $\mathcal {L}_{\mathsf {P}}$ -homomorphisms between $\mathcal {L}_{\mathsf {P}}$ constructions preserve paradoxicality
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IS CAUSAL REASONING HARDER THAN PROBABILISTIC REASONING? Rev. Symb. Log. (IF 0.6) Pub Date : 2022-05-18 MILAN MOSSÉ, DULIGUR IBELING, THOMAS ICARD
Many tasks in statistical and causal inference can be construed as problems of entailment in a suitable formal language. We ask whether those problems are more difficult, from a computational perspective, for causal probabilistic languages than for pure probabilistic (or “associational”) languages. Despite several senses in which causal reasoning is indeed more complex—both expressively and inferentially—we
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PROOF SYSTEMS FOR EXACT ENTAILMENT Rev. Symb. Log. (IF 0.6) Pub Date : 2022-05-11 JOHANNES KORBMACHER
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 hyperintensional not only in the sense of Cresswell, but also in the sense recently proposed by Odintsov and Wansing.
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THE LOGIC OF HYPERLOGIC. PART A: FOUNDATIONS Rev. Symb. Log. (IF 0.6) Pub Date : 2022-04-27 ALEXANDER W. KOCUREK
Hyperlogic is a hyperintensional system designed to regiment metalogical claims (e.g., “Intuitionistic logic is correct” or “The law of excluded middle holds”) into the object language, including within embedded environments such as attitude reports and counterfactuals. This paper is the first of a two-part series exploring the logic of hyperlogic. This part presents a minimal logic of hyperlogic and
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WHAT IS A RESTRICTIVE THEORY? Rev. Symb. Log. (IF 0.6) Pub Date : 2022-04-25 TOBY MEADOWS
In providing a good foundation for mathematics, set theorists often aim to develop the strongest theories possible and avoid those theories that place undue restrictions on the capacity to possess strength. For example, adding a measurable cardinal to $ZFC$ is thought to give a stronger theory than adding $V=L$ and the latter is thought to be more restrictive than the former. The two main proponents
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TWO-SORTED FREGE ARITHMETIC IS NOT CONSERVATIVE Rev. Symb. Log. (IF 0.6) Pub Date : 2022-04-18 STEPHEN MACKERETH, JEREMY AVIGAD
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),
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HUME’S PRINCIPLE, BAD COMPANY, AND THE AXIOM OF CHOICE Rev. Symb. Log. (IF 0.6) Pub Date : 2022-04-08 SAM ROBERTS, STEWART SHAPIRO
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
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TABULARITY AND POST-COMPLETENESS IN TENSE LOGIC Rev. Symb. Log. (IF 0.6) Pub Date : 2022-04-07 QIAN CHEN, MINGHUI MA
A new characterization of tabularity in tense logic is established, namely, a tense logic L is tabular if and only if $\mathsf {tab}_n^T\in L$ for some $n\geq 1$. Two characterization theorems for the Post-completeness in tabular tense logics are given. Furthermore, a characterization of the Post-completeness in the lattice of all tense logics is established. Post numbers of some tense logics are shown
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TAMING THE ‘ELSEWHERE’: ON EXPRESSIVITY OF TOPOLOGICAL LANGUAGES Rev. Symb. Log. (IF 0.6) Pub Date : 2022-03-28 DAVID FERNÁNDEZ-DUQUE
In topological modal logic, it is well known that the Cantor derivative is more expressive than the topological closure, and the ‘elsewhere’, or ‘difference’, operator is more expressive than the ‘somewhere’ operator. In 2014, Kudinov and Shehtman asked whether the combination of closure and elsewhere becomes strictly more expressive when adding the Cantor derivative. In this paper we give an affirmative
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Modal Quantifiers, Potential Infinity, and Yablo sequences Rev. Symb. Log. (IF 0.6) Pub Date : 2022-03-17 Michał Tomasz Godziszewski,Rafał Urbaniak
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SET THEORY AND A MODEL OF THE MIND IN PSYCHOLOGY Rev. Symb. Log. (IF 0.6) Pub Date : 2022-03-14 ASGER TÖRNQUIST, JENS MAMMEN
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 $(U,\mathcal S,\mathcal C)$, where U is a non-empty set (“the universe”), $\mathcal S$ is a perfect Hausdorff topology on U, and $\mathcal C\subseteq
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A SIMPLE SEQUENT SYSTEM FOR MINIMALLY INCONSISTENT LP Rev. Symb. Log. (IF 0.6) Pub Date : 2022-02-23 REA GOLAN
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
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TOWARDS THE INEVITABILITY OF NON-CLASSICAL PROBABILITY Rev. Symb. Log. (IF 0.6) Pub Date : 2022-02-21 GIACOMO MOLINARI
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
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, , AND REINHARDT’S PROGRAM Rev. Symb. Log. (IF 0.6) Pub Date : 2022-02-08 LUCA CASTALDO, JOHANNES STERN
In “Some Remarks on Extending and Interpreting Theories with a Partial Truth Predicate”, Reinhardt [21] famously proposed an instrumentalist interpretation of the truth theory Kripke–Feferman ( $\mathrm {KF}$ ) in analogy to Hilbert’s program. Reinhardt suggested to view $\mathrm {KF}$ as a tool for generating “the significant part of $\mathrm {KF}$ ”, that is, as a tool for deriving sentences of the
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Plural Ancestral Logic as the Logic of Arithmetic Rev. Symb. Log. (IF 0.6) Pub Date : 2022-02-07 Oliver Tatton-Brown
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A SUBSTRUCTURAL GENTZEN CALCULUS FOR ORTHOMODULAR QUANTUM LOGIC Rev. Symb. Log. (IF 0.6) Pub Date : 2022-01-27 DAVIDE FAZIO, ANTONIO LEDDA, FRANCESCO PAOLI, GAVIN ST. JOHN
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
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MEREOLOGICAL BIMODAL LOGICS Rev. Symb. Log. (IF 0.6) Pub Date : 2022-01-27 DAZHU LI, YANJING WANG
In this paper, using a propositional modal language extended with the window modality, we capture the first-order properties of various mereological theories. In this setting, $\Box \varphi $ reads all the parts (of the current object) are $\varphi $ , interpreted on the models with a whole-part binary relation under various constraints. We show that all the usual mereological theories can be captured
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RAMSIFICATION AND SEMANTIC INDETERMINACY Rev. Symb. Log. (IF 0.6) Pub Date : 2022-01-04 HANNES LEITGEB
Is it possible to maintain classical logic, stay close to classical semantics, and yet accept that language might be semantically indeterminate? The article gives an affirmative answer by Ramsifying classical semantics, which yields a new semantic theory that remains much closer to classical semantics than supervaluationism but which at the same time avoids the problematic classical presupposition
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MODES OF CONVERGENCE TO THE TRUTH: STEPS TOWARD A BETTER EPISTEMOLOGY OF INDUCTION Rev. Symb. Log. (IF 0.6) Pub Date : 2022-01-03 HANTI LIN
Evaluative studies of inductive inferences have been pursued extensively with mathematical rigor in many disciplines, such as statistics, econometrics, computer science, and formal epistemology. Attempts have been made in those disciplines to justify many different kinds of inductive inferences, to varying extents. But somehow those disciplines have said almost nothing to justify a most familiar kind
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THE GENEALOGY OF ‘’ Rev. Symb. Log. (IF 0.6) Pub Date : 2022-01-03 LANDON D. C. ELKIND, RICHARD ZACH
The use of the symbol $\mathbin {\boldsymbol {\vee }}$ for disjunction in formal logic is ubiquitous. Where did it come from? The paper details the evolution of the symbol $\mathbin {\boldsymbol {\vee }}$ 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