Hybrid journal (It can contain Open Access articles) ISSN (Print) 1572-8730 - ISSN (Online) 0039-3215 Published by Springer-Verlag[2468 journals]
QUARC+and+Classical+Logic&rft.title=Studia+Logica&rft.issn=1572-8730&rft.date=2025&rft.volume=">QUARC and Classical Logic
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Abstract: I show that Hanoch Ben-Yami’s so-called QUantified ARgument Calculus (QUARC) can be extended to what I call QUARC $$^{+}$$ which I show to be intertranslatable with a version of first-order logic in which unary predicates are non-empty. Given this result, I show that QUARC $$^{+}$$ is complete, propose an axiomatization of QUARC, and discuss the resulting expressive limitation of QUARC. PubDate: 2025-04-01
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Abstract: In this paper, we examine several lesser-known properties of da Costa’s calculi $$C_n$$, where $$1 \leqslant n < \omega $$. We demonstrate that the pair of formulas: $${\sim }(\alpha \wedge {\sim }\alpha )$$ and $$\alpha \wedge {\sim }\alpha $$, is not the sole pair from which any conclusion can be derived. We provide a proof, using $$C_1$$ as an example, that a form of ’reduction of negation’ holds true. Additionally, we propose several alternative axiomatizations for $$C_1$$, which lead to the definition of its weakenings and the introduction of da Costa-like hierarchies of paraconsistent calculi axiomatized over the positive fragment of intuitionistic propositional logic. PubDate: 2025-03-22
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Abstract: We remark that forcing in fiber bundles of structures of first order languages is not a compatible semantics with the pullback (of fiber bundles). Motivated by a combination of epistemology and geometry, we describe a semantics which behaves well with respect to the pullback. This new semantics uses parallel transport in its definition and allows to introduce two different types of extensions for the formulas: vertical and horizontal extensions. PubDate: 2025-03-04
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Abstract: Quasi MV-algebras are a generalization of MV-algebras and they are motivated by the investigation of the structure of quantum logical gates. In the first part, we present relationships between ideals, weak ideals, congruences, and perfectness within MV-algebras and quasi MV-algebras, respectively. To achieve this goal, we provide a comprehensive characterization of congruence relations of a quasi MV-algebra $${\mathcal {A}}$$ concerning the congruence relations of its MV-algebra of regular elements of $${\mathcal {A}}$$, along with specific equivalence relations concerning the complement of the set of regular elements. In the second part, we concentrate on perfect quasi MV-algebras. We present their representation by symmetric quasi $$\ell $$-groups, a special kind of quasi $$\ell $$-groups. Moreover, we establish a categorical equivalence of the category of perfect quasi MV-algebras, the category of n-perfect quasi MV-algebras, and the category of symmetric quasi $$\ell $$-groups. PubDate: 2025-02-25
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Abstract: Fractional semantics provides a multi-valued interpretation of a variety of logics, governed by purely proof-theoretic principles. This approach employs a method of systematic decomposition of formulas through a well-disciplined sequent calculus, assigning a fractional value that measures the “quantity of identity” (intuitively, “quantity of truth”) within a sequent. A key consequence of this framework is the breakdown of the traditional symmetry between truth and contradiction. In this paper, we explore the ramifications of this novel perspective on classical logic. Specifically, we (i) introduce an alternative paraconsistent consequence relation, and (ii) show how the gradual character of contradictions induces a corresponding characterization of tautologies, thereby obtaining a full-fledged informational refinement of classical logic. PubDate: 2025-02-13
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Abstract: In his 2016 article On All Strong Kleene Generalizations of Classical Logic, Stefan Wintein provides a detailed and comprehensive semantic and tableau-based analysis of the consequence relations that can be defined over the four-valued Belnap–Dunn semantics. These include familiar consequence relations like $$\textsf{FDE}$$, which takes $$\{t, b\}$$ as the set of designated values, but also much less familiar relations that don’t follow the designated-value strategy (i.e. defining logical consequence as preservation of a set of values). It turns out that many of the interesting features of these relations are made evident at the level of metainferences and, although Wintein discusses some aspects of metainferences, his work does not provide a systematic way to decide on the validity of metainferences for these logics. In this paper, we extend Wintein’s tableaux from inferences to metainferences. The paper ends with a discussion about different ways in which we can understand the notion of metainference validity. PubDate: 2025-02-13
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Abstract: We study interpolation properties for Shavrukov’s bimodal logic $$\textbf{GR}$$ of usual and Rosser provability predicates. For this purpose, we introduce a new sublogic $$\textbf{GR}^\circ $$ of $$\textbf{GR}$$ and its relational semantics. Based on our new semantics, we prove that $$\textbf{GR}^\circ $$ and $$\textbf{GR}$$ enjoy Lyndon interpolation property and uniform interpolation property. As a consequence of our proofs, we obtain the completeness and the finite frame property of $$\textbf{GR}^\circ $$ and $$\textbf{GR}$$ with respect to our new semantics. PubDate: 2025-02-05
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Abstract: Proof-theoretic semantics (P-tS) is the approach to meaning in logic based on proof (as opposed to truth). There are two major approaches to P-tS: proof-theoretic validity (P-tV) and base-extension semantics (B-eS). The former is a semantics of arguments, and the latter is a semantics of logical constants. This paper demonstrates that the B-eS for intuitionistic propositional logic (IPL) encapsulates the declarative content of a version of P-tV based on the elimination rules. This explicates how the B-eS for IPL works, and shows the completeness of this version of P-tV. PubDate: 2025-01-09
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Abstract: In the original Ulam-Rényi game with m lies/errors, Player I chooses a secret number $${\bar{x}}$$ in a finite search space S, and Player II must guess $${\bar{x}}$$ by adaptively asking Player I a minimum number of binary questions. Up to m answers may be mendacious/erroneous or may be distorted before reaching Player II. In his monograph “Fault-Tolerant Search Algorithms. Reliable Computation with Unreliable Information”, F. Cicalese provides a comprehensive account of many models of the game and their applications in error-correcting coding with noiseless feedback. Since for $$m>0$$ lies the game is not called off by contradictory answers, and repeated answers to the same question are more informative than single answers, the underlying logic of the game with m lies is not boolean. Indeed, the logic of Rényi-Ulam games is Łukasiewicz infinite-valued logic Ł$$_\infty $$. In this paper we consider Ulam-Rényi games with variable numbers of lies over infinite search spaces. We characterize the MV-algebras and the unital Specker $$\ell $$-groups given by the logic of these games. PubDate: 2025-01-06
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Abstract: Proof-theoretic semantics (P-tS) is an innovative approach to grounding logical meaning in terms of proofs rather than traditional truth-conditional semantics. The point is not that one provides a proof system, but rather that one articulates meaning in terms of proofs and provability. To elucidate this paradigm shift, we commence with an introduction that contrasts the fundamental tenets of P-tS with the more prevalent model-theoretic approach to semantics. The contribution of this paper is a P-tS for a substructural logic, intuitionistic multiplicative linear logic (IMLL). Specifically, we meticulously examine and refine the established P-tS for intuitionistic propositional logic. Subsequently, we present two novel and comprehensive forms of P-tS for IMLL. Notably, the semantics for IMLL in this paper embodies its resource interpretation through its number-of-uses reading (restricted to atoms). This stands in contrast to the conventional model-theoretic semantics of the logic, underscoring the value that P-tS brings to substructural logics. PubDate: 2024-12-31
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Abstract: We present a logical framework that enables us to define a formal theory of computational trust in which this notion is analysed in terms of epistemic attitudes towards the possible objects of trust and in relation to existing evidence in favour of the trustworthiness of these objects. The framework is based on a quantified epistemic and justification logic featuring a non-standard handling of identities. Thus, the theory is able to account for the hyperintensional nature of computational trust. We present a proof system and a frame semantics for the logic, we prove soundness and completeness results and we introduce the syntactical machinery required to define a theory of trust. PubDate: 2024-12-31
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Abstract: The Book I of Euclid’s Elements begins with three propositiones that ask for the solution of three problems. Unlike other propositiones, these do not assert properties or relationships between geometric objects. In them, some of these objects are assumed as given, and actions are demanded in order to obtain other objects. The solution to this type of problem is a construction, and its foundations can be found in the definitions and postulates of the Geometry of Euclid. Furthermore, each construction is normally accompanied by the resolution of a second problem. This consists in demonstrating that the previously offered solution is correct, i.e., that the presented construction indeed produces what was demanded in the formulation of the initial problem. The first theorem of the Elements is exactly the fourth proposition. We will consider, in a propaedeutic manner, the implications of adopting an approach to geometry where all propositions are interpreted as problems and the solution to any specific construction problem consists in offering a procedure such that, globally, the execution of this procedure on a workspace by a geometer will be simulated by an Euclid machine—which is the central object of analysis of the present article. From this perspective, the Geometry of Euclid would admit an interpretation in which it would be, in addition to a foundational milestone in the history of mathematics, also a foundational milestone in the development of the notion of effective procedure that would only fully come to light in the twentieth century. PubDate: 2024-12-16
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Abstract: New Foundations with Urelements (NFU) is a theory that extends Quine’s original theory (New Foundations) by adding “urelements” (atoms). It was discovered by Jensen in 1969, who proved that NFU is relatively consistent with Peano arithmetic and consequently with Zermelo–Fraenkel set theory (ZF). Jensen’s proof is rather hard to follow, so Boffa introduced a more straightforward method of constructing models for NFU from a model of ZF. However, Boffa’s presentation of his construction is extremely terse with many essential details omitted, since he probably only wanted to emphasize the differences from Jensen’s construction. Our main goal is to elaborate on this construction in great detail, and also to prove that the said construction yields some results that are either not present in the contemporary literature, or are accepted as true at face value. Namely, besides proving that this construction gives a model for NFU, we will also use it to define models for NFU with the Axiom of Infinity, NFU with the Axiom of Choice, and NFU with both of those axioms. PubDate: 2024-12-13
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Abstract: In this paper, we axiomatize modal logic extended with the modal operator $$M\varphi $$ saying that “there are strictly more $$\varphi $$-successors than $$\lnot \varphi $$-successors”, both in the class of image-finite Kripke frames and in the class of all Kripke frames. We follow the proof strategy of van der Hoek (Int J Uncertain Fuzziness Knowl Based Syst 4(1):45–60, 1996.), and prove a characterization result of finite majority structures which are capable of representing finite cardinality measures and a characterization result of finite extended majority structures which are capable of representing cardinality measures. PubDate: 2024-12-03
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Abstract: The logic of the hide and seek game $$\textbf{LHS}$$ was proposed to capture interactions between agents in pursuit-evasion environments. In this paper, we explore a hybrid extension of $$\textbf{LHS}$$ and show that such an extension is beneficial in several aspects. We will show that it improves the technical properties of the resulting logical system, and expands the potential applications of the system. Specifically, we will investigate the expressive power of the hybrid logic of the hide and seek game $${\mathcal {H}}(\textbf{LHS})$$ and its crucial fragment $${\mathcal {H}}(\textbf{LHS}^{-})$$, at both model and frame levels. We will present complete Hilbert-style proof systems for both the logics $${\mathcal {H}}(\textbf{LHS}^{-})$$ and $${\mathcal {H}}(\textbf{LHS})$$, which in fact provides a solution to a known open problem. PubDate: 2024-11-18
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Abstract: The main motivation of this paper is the study of first-order model theoretic properties of structures having their roots in modal logic. We will focus on the connections between ultrafilter extensions and ultrapowers. We show that certain structures (called bounded graphs) are elementary substructures of their ultrafilter extensions, moreover their modal logics coincide. PubDate: 2024-11-15
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Abstract: We study the finite frame property of some extensions of Fitting, Marek, and Truszczyński’s pure logic of necessitation $$\textbf{N}$$. For any natural numbers m, n, we introduce the logic $$\textbf{N}^+\textbf{A}_{m,n}$$ by adding the single axiom scheme $$\Box ^n \varphi \rightarrow \Box ^m \varphi $$ and the rule $$\dfrac{\lnot \Box \varphi }{\lnot \Box \Box \varphi }$$ ($${\text {Ros}}^\Box $$) into $$\textbf{N}$$. We prove the finite frame property of $$\textbf{N}^+\textbf{A}_{m, n}$$ with respect to Fitting, Marek, and Truszczyński’s relational semantics. We also prove that for $$n \ge 2$$, the logic obtained by removing the rule $${\text {Ros}}^\Box $$ from $$\textbf{N}^+\textbf{A}_{0, n}$$ is incomplete with respect to that semantics. PubDate: 2024-11-04
Hybrid journal (It can contain Open Access articles)
