Classical Logic Research Articles

Symbolic Automata: Omega-Regularity Modulo Theories

Symbolic automata are finite state automata that support potentially infinite alphabets, such as the set of rational numbers, generally applied to regular expressions and languages over finite words. In symbolic automata (or automata modulo A ), an alphabet is represented by an effective Boolean algebra A , supported by a decision procedure for satisfiability. Regular languages over infinite words (so called ω-regular languages) have a rich history paralleling that of regular languages over finite words, with well-known applications to model checking via Büchi automata and temporal logics. We generalize symbolic automata to support ω-regular languages via transition terms and symbolic derivatives , bringing together a variety of classic automata and logics in a unified framework that provides all the necessary ingredients to support symbolic model checking modulo A . In particular, we define: (1) alternating Büchi automata modulo A ( ABW A ) as well (non-alternating) nondeterministic Büchi automata modulo A ( NBW A ); (2) an alternation elimination algorithm Æ that incrementally constructs an NBW A from an ABW A , and can also be used for constructing the product of two NBW A ; (3) a definition of linear temporal logic modulo A , LTL ( A ), that generalizes Vardi’s construction of alternating Büchi automata from LTL, using (2) to go from LTL modulo A to NBW A via ABW A . Finally, we present RLTL ( A ), a combination of LTL ( A ) with extended regular expressions modulo A that generalizes the Property Specification Language (PSL). Our combination allows regex complement , that is not supported in PSL but can be supported naturally by using transition terms. We formalize the semantics of RLTL ( A ) using the Lean proof assistant and formally establish correctness of the main derivation theorem.

The Duality of λ-Abstraction

In this paper, we develop and study the following perspective -- just as higher-order functions give exponentials, higher-order continuations give coexponentials. From this, we design a language that combines exponentials and coexponentials, producing a duality of lambda abstraction. We formalise this language by giving an extension of a call-by-value simply-typed lambda-calculus with covalues, coabstraction, and coapplication. We develop the semantics of this language using the axiomatic structure of continuations, which we use to produce an equational theory, that gives a complete axiomatisation of control effects. We give a computational interpretation to this language using speculative execution and backtracking, and use this to derive the classical control operators and computational interpretation of classical logic, and encode common patterns of control flow using continuations. By dualising functional completeness, we further develop duals of first-order arrow languages using coexponentials. Finally, we discuss the implementation of this duality as control operators in programming, and develop some applications.

A note on formalizing discussive logic

Discussive logic was introduced by Jaskowski as a logic of discussion. In this note we show that some natural translation-based formalizations of discussive logic in modal logic do not yield a paraconsistent logic but rather classical logic. Some alternative modal formalizations of discussive logic that avoid the collapse into classical logic are put forward.

Role of Rough Neutrosophic Attribute Reduction with Deep Learning-Based Enhanced Kidney Disease Diagnosis

The kidneys have an important role in keeping blood pressure, electrolyte sense, and acid-base sense of body balance to remove toxins from our body. Malfunction is responsible for irrelevant to life-threatening diseases, along with malfunction in the other functional organs. As a result, scholars worldwide have committed to finding methods for effectively diagnosing and accurately treating chronic kidney disease. As machine learning (ML) classifier is widely deployed in the healthcare field for diagnoses, also CKD is now involved in the collection of disorders that could be predicted through the ML classifier. Neutrosophic logic (NL) can be employed as a form of logic that expands classical, fuzzy, and intuitionistic fuzzy logic (IF) by integrating a third constituent: indeterminacy. It enables data handling and representation with three dissimilar membership functions: truth (T), indeterminacy (I), and false (F). The complete set is independent and may differ in the interval [0, 1], providing a convoluted strategy to handle, data incompleteness, vagueness and uncertainty. This makes NL especially relevant in complicated systems where data might be partially unknown, ambiguous, or inconsistent. This article employs a Rough Neutrosophic Attribute Reduction with Deep Learning based Enhanced Kidney Disease Diagnosis (RNSAR-DLKDD) technique. Initially, the RNSAR-DLKDD technique reduces the attributes via the RNSAR technique. Followed by, the detection and classification of kidney disease take place using long short-term memory (LSTM) model. Finally, the hyperparameter selection process is carried out via crow search algorithm (CSA). To highlight the performance of the RNSAR-DLKDD technique, a series of experiments were involved. The extensive results inferred the betterment of the RNSAR-DLKDD technique over other models

Was Aristotle a non-classical logician?

This paper discusses the possible classification of Aristotle’s syllogistic as a non-classical logical system, positing Aristotle himself as a non-classical logician. Initially, we find compelling arguments for this thesis, particularly regarding the expressive power and the rules governing logical inference inherent in Aristotle’s approach. My analysis nevertheless addresses two significant counterarguments. The first, the special case objection, posits that Aristotle’s syllogistic can be framed as a classical logic which deals with canonical syllogistic forms. I argue that this objection is insufficient, as it is possible to point cases in which his system seems to differ from classical logic. The second counterargument, the formalisation gap objection, highlights that Aristotle’s syllogistic resists straightforward modern logical interpretations. This latter objection is evaluated as more compelling and substantial. In particular, a distinction between two concepts is proposed which could help us understand what Aristotle was aiming at in his theory of inference: the notions of ‘to follow from’ and ‘to be a conclusion of’. While the former aligns with the usual sense formal validity, the second requires an inferential structure connecting the premises to the conclusion, explaining why Aristotle excluded inferences like from A ⊢ A syllogisms despite acknowledging that A follows from A.

Complexity of Nonassociative Lambek Calculus with classical logic

Complexity of Nonassociative Lambek Calculus with classical logic

On the non-substantiality of logic: a case study

AbstractOne of the goals of the natural sciences– for example biology– is to provide new information about certain phenomena with previously unknown nature. Their contribution to our knowledge is substantial. From this perspective, logic is seemingly not substantial. Sometimes, logic’s insubstantiality is taken for granted while explaining the alleged insubstantiality of other notions. For example, according to truth deflationism, truth is a non-substantial notion in the sense of being a logical property. However, it is not fully clear to what such an insubstantiality amounts. It is also debatable whether logic really is insubstantial. In this paper, we aim to clarify this issue by proposing a formal way of looking at it. In particular, we used the notion of conservativity, which has already been used by truth deflationism, for a similar aim. We show that if insubstantiality is read in terms of conservativity, then classical logic is substantial. We then argue that such a verdict of substantiality can be resisted if precise stances on certain prima facie unrelated issues of philosophy of logic are taken, or an anti-exceptionalist view is adopted.

Polytime embedding of intuitionistic modal logics into their one-variable fragments

Abstract We prove that propositional intuitionistic modal logics $\textbf{FS}$ (also known as $\textbf{IK}$) and $\textbf{MIPC}$ (also known as $\textbf{IS5}$) are polynomial-time embeddable into, and hence polynomial-time equivalent to, their own one-variable fragments. It follows that the one-variable fragment of $\textbf{MIPC}$ is coNEXPTIME-complete. The method of proof applies to a wide range of intuitionistic modal logics characterizable by two-dimensional frames, among them intuitionistic analogues of such classical modal logics as $\textbf{K4}$ and $\textbf{S4}$.

The Logic of Potential Infinity

Abstract Michael Dummett argues that acceptance of potentially infinite collections requires that we abandon classical logic and restrict ourselves to intuitionistic logic. In this paper we examine whether Dummett is correct. After developing two detailed accounts of what, exactly, it means for a concept to be potentially infinite (based on ideas due to Charles McCarty and Øystein Linnebo, respectively), we construct a Kripke structure that contains a natural number structure that satisfies both accounts. This model supports a logic much stronger than intuitionistic logic, demonstrating that Dummett was wrong. We conclude by briefly examining ways to extend the account(s) in question to indefinitely extensible concepts such as Cardinal, Ordinal, and Set.

Enhancing Culinary Operations Through Fuzzy Logic: A Case Study in the Catering Industry

This study aimed to analyze the business impact of the catering sector using the fuzzy logic method. The research was conducted at a catering company in Istanbul, utilizing document review and participant observation methods to evaluate the business impact. The nominal prioritization method was used to identify critical business processes, and a model along with a mathematical formula was developed for calculating the business impact. The Fuzzy Logic Designer Toolbox in MATLAB was utilized for this calculation. The study identified eight critical business processes: (1) material supply, (2) material storage, (3) pre-preparation process, (4) cooking process, (5) portioning, (6) shipping, (7) hygiene and food safety, and (8) customer relationship management. The business impact was assessed using classical and fuzzy logic methods, and the results were compared. The fuzzy logic method provided a more flexible and comprehensive assessment, managing uncertainty and variability more effectively than classical logic. Overall, it proved to be more effective in optimizing business processes, offering a more dynamic and holistic approach to improving and prioritizing these processes.

The Structural Role of the Subject in Categorical Propositions and Its Existential Import

Logic, as a system designed to distinguish between right and wrong, necessitates that all judgments pertain to a “thing.” In categorical propositions, this judgment is anchored in the “thing,” which is represented by the term “subject.” The subject serves as the foundational element upon which the entire logical system is built. Within the classical tradition of logic, it is widely accepted that only affirmative propositions presuppose the existence of their subjects. However, for a judgment to be formed within a proposition, the subject must refer to an existent. This study examines the debates within the classical tradition regarding the existence of the subject in propositions and questions its existential significance based on these discussions. The existence of the subject holds pivotal importance in classical logic for several reasons. For example, to establish a contradiction between two propositions, the “unity of subject”—one of the eight conditions for contradiction—requires that the same subject be used in both propositions. Furthermore, the study explores the discussions between Farabi and Ibn Sina regarding the actuality and possibility of the individuals to which the subject refers, evaluating their approaches to the possible individuals denoted by the subject. At its core, analyzing the ontological implications of the subject is essential for assessing the coherence and consistency of the logical system.

The Structure of Paradoxes in a Logic of Sentential Operators

AbstractAny language $$\mathcal {L}$$ L of classical logic, of first- or higher-order, is expanded with sentential quantifiers and operators. The resulting language $$\mathcal {L}^+\!$$ L + , capable of self-reference without arithmetic or syntax encoding, can serve as its own metalanguage. The syntax of $$\mathcal {L}^+$$ L + is represented by directed graphs, and its semantics, which coincides with the classical one on $$\mathcal {L}$$ L , uses the graph-theoretic concepts of kernels and semikernels. Kernels provide an explosive semantics, while semikernels generalize this to situations where paradoxes do not lead to explosion, thus distinguishing them from contradictions. Paradoxes arise only at the metalevel due to specific interpretations of the operators, but they can be avoided: $$\mathcal {L}^+$$ L + can express paradoxes but remains free from them. For an expansion $$\mathcal {L}^+$$ L + of any FOL language $$\mathcal {L}$$ L , with the non-explosive semantics, a complete reasoning system is obtained by extending Gentzen’s classical sequent calculus with two rules for the sentential quantifiers. Adding (cut) yields a complete system for the explosive semantics. The novel semantics and self-referential capabilities seem promising for a further extension of classical logic towards one capable also of consistently expressing its own syntax and truth theory.

Insight into soft binary piecewise lambda operation: a new operation for soft sets

AbstractThe notion of soft set operations is one of the key concept for soft set theory as the theory has been progressing, both theoretically and practically, based on this notion. As proposing new soft set operations, deriving their algebraic properties, and studying the algebraic structure of soft sets from the perspective of soft set operations offer a comprehensive understanding of their applications as well as the appreciation of how soft set algebra can be applied to classical and nonclassical logic, in this study, a new soft set operation, called the “soft binary piecewise lambda operation" is proposed. Since one of the the main objective of abstract algebra is to analyze the properties of the operations defined on a set to classify the algebraic structures, the operation’s full properties and its distributions over other soft set operations are investigated to reveal which algebraic structures the operation forms individually, and together with other soft set operations in the collection of soft sets over the universe. It is showed that the operation forms a noncommutative semigroup and a right-left system, besides semi-rings and near-semi-rings together with certain types of soft set operations under certain conditions in the collection of soft sets over the universe. Since such in-depth analyses advance our knowledge of the applications of soft sets over a range of field, this novel operation may serve as an inspiration to create new perspectives for addressing issues related to parametric data, soft set-based cryptography, or decision-making techniques in practical settings, business, and technology.

Truthmaker Semantics and Natural Language Semantics

ABSTRACTTruthmaker semantics is a non‐classical logical framework that has recently garnered significant interest in philosophy, logic, and natural language semantics. It redefines the propositional connectives and gives rise to more fine‐grained entailment relations than classical logic. In its model theory, truth is not determined with respect to possible worlds, but with respect to truthmakers, such as states or events. Unlike possible worlds, these truthmakers may be partial; they may be either coherent or incoherent; and they are understood to be exactly or wholly relevant to the truth of the sentences they verify. Truthmaker semantics generalises collective, fusion‐based theories of conjunction; alternative‐based theories of disjunction; and nonstandard negation semantics. This article provides a gentle introduction to truthmaker semantics aimed at linguists; describes applications to various natural language phenomena such as imperatives, ignorance implicatures, and negative events; and discusses its similarities and differences to related frameworks such as event semantics, situation semantics, alternative semantics, and inquisitive semantics.

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Realizing Bidirectional Photocurrent in Monolithic Dual‐Mode Device for Neuromorphic Vision and Logically‐Encrypted Transmission

AbstractDue to significant response contradictions, unidirectional carrier transmission of typical semiconductor p‐n junctions limits integrated sensors and artificial synapses in a monolithic device. In this work, a bipolar p‐n junction designed by employing the hydrogel/p‐GaN local contact interface with p‐GaN/(In,Ga)N heterojunction, demonstrating the bidirectional photocurrent and sensor/artificial synapse dual‐mode device successfully. After modifying Au nanoparticles, the negative (positive) photocurrent is increased by 900% (300%) under 365 nm (520 nm) light to achieve a more balanced bipolar carrier dynamic. Such a regulation of photocurrent is found to be attributed to the promotion of hydrogen evolution reaction (HER) at the hydrogel/Au/p‐GaN interface. Thus, the device achieves the ultrahigh paired‐pulse facilitation (PPF) index of 243% under self‐powered condition. Moreover, the implementation of plasticity from short‐term to long‐term demonstrates its adaptability in neural morphology visual recognition. Beyond independent working mode, the function of five classic logic gates is achieved under three‐terminal operation. Finally, an encrypted wireless communication system using dual‐channel light signals demonstrates the extensive application potential of the monolithic device. Therefore, this work presents a viable construction blueprint to meet the demands of next‐generation all‐in‐one optoelectronics for intricate application scenarios.

Sequential merging and construction of rankings as cognitive logic

We introduce and evaluate a cognitively inspired formal reasoning approach that sequentially applies a combination of a belief merging operator and a ranking construction operator. The approach is inspired by human propositional reasoning, which is understood here as a sequential process in which the agent constructs a new epistemic state in each task step according to newly acquired information. Formally, we model epistemic states by Spohn's ranking functions. The posterior representation of the epistemic state is obtained by merging the prior ranking function and a ranking function constructed from the new piece of information. We denote this setup as the sequential merging approach. The approach abstracts from the concrete merging operation and abstracts from the concrete way of constructing a ranking function according to new information. We formally show that sequential merging is capable of predicting with theoretical maximum achievable accuracy. Various instantiations of our approach are benchmarked on data from a psychological experiment, demonstrating that sequential merging provides formal reasoning methods that are cognitively more adequate than classical logic.

Disjunction and access to knowledge: Educational implications

In classical logic, it is possible to derive ‘either p or q’ from ‘p’ (where p and q are sentences with any content). This is a cognitive problem, since people often tend not to make inferences of that kind. This paper analyzes the solution the theory of mental models gives for this problem. Based on that solution, the paper proposes to use tasks with inferences such as that mentioned in the educational context. The idea is that those tasks can allow assessing certain learnings. In particular, they can reveal in some cases whether students understand the relations between the semantic contents assigned to p and q. This is because, following the theory of mental models, whenever q implies p, the inference must be accepted.

Fundamental Laws and Quantum Dynamics

Background: Recently, we obtained a unitary theory of quantum mechanics and general relativity, where a quantum particle is a continuous distribution of matter in the two conjugate spaces of the coordinates and momentum, quantized by the equality of the mass parameter describing the relativistic dynamics of the matter, with the mass as an integral of the matter density. However, in this framework, we have not explicitly revealed the connection between our new theory and the fundamental laws of quantum mechanics. Methods: We analyzed in detail the three fundamental laws of quantum mechanics, explicitly describing experimental data: 1) Planck’s law of the blackbody electromagnetic radiation of a system of electrically charged harmonic oscillators, 2) Einstein’s law of the photon energy proportionality with the photon frequency, and 3) de Broglie’s law of the quantum particle as an oscillator in space. Results: We reobtained the two dynamical equations, in the conjugate spaces of the coordinates and momentum, as functions of the Lagrangian system, unlike the Schrödinger equation, depending on the Hamiltonian. Conclusion: According to the fundamental laws of quantum mechanics, a quantum particle is a continuous distribution of matter with an intrinsic mass, unlike the conventional quantum mechanics for the state occupation probabilities of punctual entities moving with the light velocity and getting an apparent mass only by collisions with some bosons pervading the whole universe. According to these laws, we obtained a quantum theory in agreement with common sense, classical logic, and general relativity.

Absurdity as the impossible command in natural deduction

AbstractIn this paper, we propose a new approach to absurdity in the context of natural deduction for intuitionistic and classical logic. It combines the aspects of both the logical approach, which treats absurdity as a propositional constant, and the structural approach, which treats absurdity as a structural punctuation mark signalling the dead end of derivations. In particular, we will treat absurdity as an impossible command, that is, a speech act composed of an imperative force indicator, and the false propositional constant, that is, a proposition that cannot be true by definition. In return, we obtain a framework that constitutes a middle ground between the logical and structural approaches. For example, it allows us to consider the ex falso quodlibet rule as a kind of structural rule, specifically, a semi‐structural rule, and at the same time also maintain that negation can be reduced to implication of absurdity, specifically, to its propositional content.