Translation Of Logic Research Articles

Different negations and fuzziness in Web resources and resource description and their mathematical foundation

In Web resources and resource descriptions, many resources and descriptions of resources have different negations and fuzziness. In this paper, we investigate and analyze the cognition and characteristics of “negation” in philosophy, logic, linguistics and other knowledge fields, and based on the distinction between the concept of “contradiction” and the concept of “opposition”, and “clear concept” and “fuzzy concept”, we propose that there are three different kinds of negation in knowledge from the conceptual level, namely “contradictory negation”, “opposite negation” and “intermediary negation”. In order to establish a mathematical basis that can fully reflect the internal relations, properties and laws of these different negations and fuzziness, we propose a “Scoi: sets with contradictory negation, opposite negation and intermediary negation” and “Lcoi&PLcoi: logic with contradictory negation, opposite negation and intermediary negation”, and the operational properties of Scoi and the formal deduction properties of Lcoi have been proven. The soundness theorem, completeness theorem and compactness theorem for Lcoi are proved under the three-valued semantic interpretation of Lcoi, and the soundness and completeness for Lcoi&PLcoi are discussed under the given an infinite-valued semantics interpretation of Lcoi&PLcoi. Furthermore, we also demonstrated that descriptions of objective things such as “x is neither a positive nor a negative integer” and “he is neither rich nor poor” typle of statements are “intermediary negation” in Scoi and Lcoi&PLcoi. It shows that “intermediary negation” in Scoi and Lcoi&PLcoi has objectively.In order to enable RDF(Resource Description Framework) to describe Web resources with different negations and fuzziness, we extend RDF based on Scoi and Lcoi&PLcoi, and propose a resource description framework RDFcoi which can fully handle fuzzy attribute values and their different negations in attributes of resources. How to reflect the uncertainty of the relationship between Web resources and its attribute values, we believe that the relationship between a resource and fuzzy attribute value in one of its attributes should be a composite mapping from the resource domain to the fuzzy attribute value set and then to [0, 1], and a mathematical representation is given in this paper. Thus, RDFcoi differ from other RDF extensions. RDFcoi maintains the basic syntax and normative of RDF, and extends the basic syntax and semantics of RDF. Where, the concepts of RDFcoi triple with three kinds of negation, RDFcoi formula and RDFcoi graph are defined. RDFcoi vocabulary is determined and an interpretation is given. RDFcoi-entailment is defined and its properties are discussed. In addition, we also study the reasoning regimes of RDFcoi. By using the given logical translation method, a well-typed and interoperable RDFcoi graph can be transformed into a set of formulas in Lcoi&PLcoi, thus establishing “interoperability” between RDFcoi and Lcoi&PLcoi. From the perspective of reasoning mechanism of RDFcoi-entailment, RDFcoi reasoning can be carried out in Lcoi&PLcoi, and RDFcoi has the same reasoning mechanism as formal inference in Lcoi&PLcoi.

AFFINE LOGIC FOR CONSTRUCTIVE MATHEMATICS

AbstractWe show that numerous distinctive concepts of constructive mathematics arise automatically from an “antithesis” translation of affine logic into intuitionistic logic via a Chu/Dialectica construction. This includes apartness relations, complemented subsets, anti-subgroups and anti-ideals, strict and non-strict order pairs, cut-valued metrics, and apartness spaces. We also explain the constructive bifurcation of some classical concepts using the choice between multiplicative and additive affine connectives. Affine logic and the antithesis construction thus systematically “constructivize” classical definitions, handling the resulting bookkeeping automatically.

Inquisitive logic as an epistemic logic of knowing how

In this paper, we present an alternative interpretation of propositional inquisitive logic as an epistemic logic of knowing how. In our setting, an inquisitive logic formula α being supported by a state is formalized as knowing how to resolve α (more colloquially, knowing how α is true) holds on the S5 epistemic model corresponding to the state. Based on this epistemic interpretation, we use a dynamic epistemic logic with both know-how and know-that operators to capture the epistemic information behind the innocent-looking connectives in inquisitive logic. We show that the set of valid know-how formulas corresponds precisely to the inquisitive logic. The main result is a complete axiomatization with intuitive axioms using the full dynamic epistemic language. Moreover, we show that the know-how operator and the dynamic operator can both be eliminated without changing the expressivity over models, which is consistent with the modal translation of inquisitive logic existing in the literature. We hope our framework can give an intuitive alternative interpretation to various concepts and technical results in inquisitive logic, and also provide a powerful and flexible tool to handle both the inquisitive reasoning and declarative reasoning in an epistemic context.

Chinese Research on Mathematical Logic and the Foundations of Mathematics

This paper outlines the Chinese research on mathematical logic and the foundations of mathematics. Firstly, it presents the introduction and spread of mathematical logic in China, especially the teaching and translation of mathematical logic initiated by Bertrand Russell’s lectures in the country. Secondly, it outlines the Chinese research on mathematical logic after the founding of the People’s Republic of China. The research in this period experienced a short revival under the criticism of the Soviet Union, explorations under the heavy influence of the Cultural Revolution, and the vigorous development of mathematical logic teaching and research after the period of “Reform and Opening Up” that started in the late 1970s, and the full integration of Chinese mathematical logic research into the international academic circle in the new century after 2000. In the third part, it focuses on the unique and original results of the Chinese mathematical logic research teams from the following three aspects: medium logic, lattice implication algebras and their lattice-valued systems of logic, and Chinese notation of logical constants, which can be used as a substantive supplement to the relevant literature on the history of mathematical logic in China. The last part is a reflection on the shortcomings of contemporary Chinese research on mathematical logic and the foundations of mathematics.

Meaning-Preserving Translations of Non-classical Logics into Classical Logic: Between Pluralism and Monism

In order to prove the validity of logical rules, one has to assume these rules in the metalogic. However, rule-circular ‘justifications’ are demonstrably without epistemic value (sec. 1). Is a non-circular justification of a logical system possible? This question attains particular importance in view of lasting controversies about classical versus non-classical logics. In this paper the question is answered positively, based on meaning-preserving translations between logical systems. It is demonstrated that major systems of non-classical logic, including multi-valued, paraconsistent, intuitionistic and quantum logics, can be translated into classical logic by introducing additional intensional operators into the language (sec. 2–5). Based on this result it is argued that classical logic is representationally optimal. In sec. 6 it is investigated whether non-classical logics can be likewise representationally optimal. The answer is predominantly negative but partially positive. Nevertheless the situation is not symmetric, because classical logic has important ceteris paribus advantages as a unifying metalogic.

A FIRST-ORDER FRAMEWORK FOR INQUISITIVE MODAL LOGIC

AbstractWe present a natural standard translation of inquisitive modal logic $\mathrm{InqML}$ into first-order logic over the natural two-sorted relational representations of the intended models, which captures the built-in higher-order features of $\mathrm{InqML}$ . This translation is based on a graded notion of flatness that ties the inherent second-order, team-semantic features of $\mathrm{InqML}$ over information states to subsets or tuples of bounded size. A natural notion of pseudo-models, which relaxes the non-elementary constraints on the intended models, gives rise to an elementary, purely model-theoretic proof of the compactness property for $\mathrm{InqML}$ . Moreover, we prove a Hennessy-Milner theorem for $\mathrm{InqML}$ , which crucially uses $\omega $ -saturated pseudo-models and the new standard translation. As corollaries we also obtain van Benthem style characterisation theorems.

A critical study into the extent that co-operative learning promotes a greater confidence and a more accurate rendering of syntax in unseen translations for Year 12 IB Higher Latinists

The ability to translate unseen passages is a skill tested in both Latin A-level and the International Baccalaureate (IB) Diploma. Higher IB candidates are expected to translate a passage of 105–125 words of Latin poetry (in this case, Ovid's Metamorphoses) as the first of their externally marked papers. The International Baccalaureate Organization (IBO) uses unseen passages to ‘measure the [students’] ability to understand and translate texts in the original language’ (2014, p. 25). The passages are marked according to two criteria: (a) meaning; (b) vocabulary and grammar. In order to access the highest grades, the students must provide a ‘logical translation [in which] errors do not impair the meaning’ and ‘render vocabulary appropriately and grammar accurately and effectively’ (IBO, 2014, pp. 28–9).

FIBRED ALGEBRAIC SEMANTICS FOR A VARIETY OF NON-CLASSICAL FIRST-ORDER LOGICS AND TOPOLOGICAL LOGICAL TRANSLATION

AbstractLawvere hyperdoctrines give categorical algebraic semantics for intuitionistic predicate logic. Here we extend the hyperdoctrinal semantics to a broad variety of substructural predicate logics over the Typed Full Lambek Calculus, verifying their completeness with respect to the extended hyperdoctrinal semantics. This yields uniform hyperdoctrinal completeness results for numerous logics such as different types of relevant predicate logics and beyond, which are new results on their own; i.e., we give uniform categorical semantics for a broad variety of non-classical predicate logics. And we introduce an analogue of Lawvere–Tierney topology and cotopology in the hyperdoctrinal setting, which gives a unifying perspective on different logical translations, in particular allowing for a uniform treatment of Girard’s exponential translation between linear and intuitionistic logics and of Kolmogorov’s double negation translation between intuitionistic and classical logics. In the hyerdoctrinal conception, type theories are categories, logics over type theories are functors, and logical translations between them, then, are natural transformations, in particular Lawvere–Tierney topologies and cotopologies on hyperdoctrines. The view of logical translations as hyperdoctrinal Lawvere–Tierney topologies and cotopologies has not been elucidated before, and may be seen as a novel contribution of the present work. From a broader perspective, this work may be regarded as taking first steps towards interplay between algebraic and categorical logics; it is, technically, a combination of substructural (or Lambekian) algebraic logic and hyperdoctrinal (or Lawverian) categorical logic, as the hyperdoctrinal completeness theorem is shown via the integration of the Lindenbaum–Tarski algebra construction with the syntactic category construction. As such this work lays a foundation for further interactions between algebraic and categorical logics.

General Bi-Tri Logic SPWM for Current Source Converter With Optimized Zero-State Replacement

This article proposes an optimized zero-state replacement (ZSR) method for general bi-tri logic SPWM (BTSPWM) to deal with the common-mode voltage (CMV) issue. The BTSPWM can transform the well-developed modulation strategies for voltage source converters (VSCs) into current source converter (CSC) system through the logic translation, which can utilize the analysis concepts developed for VSC. With BTSPWM, all- <sc xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">off</small> gating signals can be generated through the logic translation, which should be replaced with the redundant zero-states to avoid open-circuit, which provides the flexibility for ZSR. Existing ZSR strategies are mainly focused on switching time minimization while the CMV excited by the replaced zero-states is not considering. To address the CMV issue, optimized ZSR methods for general BTSPWM are proposed to address the CMV while not influence the regular PWM features. With the proposed methods, both the CMV peak value and the third-order component can be effectively reduced. Moreover, the proposed method can be easily applied to a parallel CSC system due to the inherent modularity. The performances of the proposed methods have been experimentally verified on both single and parallel CSC systems.

Towards a Notion of Basis for Knowledge-Based Systems—Applications

In the paradigm of Knowledge-Based Systems (KBS), the design of methods to simplify the reasoning leads to more efficient processes. A point of view that provides valuable insights is the algebraic one. In this work, a notion of basis (and dimension) for Knowledge Bases in Propositional Logic associated with knowledge forgetting is introduced. It is based on ideas that come from the translation of such logic in (Computer) Algebra, particularly from the interpretation of variable forgetting. In this paper, the concept of weak base is defined as a set of variables sufficient to decide the consistency using variable forgetting. Several applications of weak bases are presented in order to show their usefulness in KBS reasoning and to justify their study and use in solving problems within this topic.

Artificial Intelligence Applications in Investments

This paper investigates the increasing use of artificial intelligence in the field of investments today. It seeks to explore how artificial intelligence strategies can help investors make better investment decisions as well as highlight the risks posed by the use of artificial intelligence. In the digital era, transformational technology is powering new forms of automation that are more universal and smarter than ever before. The technology has not only changed significant operational aspects such as logistics, manufacturing, and warehousing but also the investment industry at large. Technologies associated with artificial intelligence help in performing tasks commonly associated with human beings such as logical operations, visual operations, speech recognition, visual perceptions, language translation and decision-making. Today, Computer technology has the capabilities of trouncing human intelligence in solving complex computations in shorter periods. The same technique can be replicated in investment in problem solving, decision-making and risk management. For instance, artificial intelligence tools helps in making faster precise assessment of potential borrowers at minimal cost which accounts for better informed and data backed decision that reduces human bias. Keywords: Artificial intelligence; Investments

A Unified Translation of Linear Temporal Logic to ω-Automata

We present a unified translation of linear temporal logic (LTL) formulas into deterministic Rabin automata (DRA), limit-deterministic Büchi automata (LDBA), and nondeterministic Büchi automata (NBA). The translations yield automata of asymptotically optimal size (double or single exponential, respectively). All three translations are derived from one single Master Theorem of purely logical nature. The Master Theorem decomposes the language of a formula into a positive Boolean combination of languages that can be translated into ω-automata by elementary means. In particular, Safra’s, ranking, and breakpoint constructions used in other translations are not needed. We further give evidence that this theoretical clean and compositional approach does not lead to large automata per se and in fact in the case of DRAs yields significantly smaller automata compared to the previously known approach using determinisation of NBAs.

Multiple conclusion linear logic: cut elimination and more

Abstract Full intuitionistic linear logic (FILL) was first introduced by Hyland and de Paiva, and went against current beliefs that it was not possible to incorporate all of the linear connectives, e.g. tensor, par and implication, into an intuitionistic linear logic. Bierman showed that their formalization of FILL did not enjoy cut elimination as such, but Bellin proposed a small change to the definition of FILL regaining cut elimination and using proof nets. In this note we adopt Bellin’s proposed change and give a direct proof of cut elimination for the sequent calculus. Then we show that a categorical model of FILL in the basic dialectica category is also a linear/non-linear model of Benton and a full tensor model of Melliès’ and Tabareau’s tensorial logic. We give a double-negation translation of linear logic into FILL that explicitly uses par in addition to tensor. Lastly, we introduce a new library to be used in the proof assistant Agda for proving properties of dialectica categories.

Modal translation of substructural logics

ABSTRACTIn an article dating back in 1992, Kosta Došen initiated a project of modal translations in substructural logics, aiming at generalising the well-known Gödel–McKinsey–Tarski translation of intuitionistic logic into S4. Došen's translation worked well for (variants of) BCI and stronger systems (BCW, BCK), but not for systems below BCI. Dropping structural rules results in logic systems without distribution. In this article, we show, via translation, that every substructural (indeed, every non-distributive) logic is a fragment of a corresponding sorted, residuated (multi) modal logic. At the conceptual and philosophical level, the translation provides a classical interpretation of the meaning of the logical operators of various non-distributive propositional calculi. Technically, it allows for an effortless transfer of results, such as compactness and the Löwenheim-Skolem property and it opens up new directions for deducing properties of substructural logic systems by establishing appropriate transfer theorems.

Conservative translations of four-valued logics in modal logic

Following a proposal by Kooi and Tamminga, we introduce a conservative translation manual for every four-valued truth-functional propositional logic into a modal logic. However, the application of this translation does not preserve the intuitive reading of the truth-values for every four-valued logic. In order to solve this problem, we modify the translation manual and prove its conservativity by exploiting the method of generalized truth-values.

Translating JSON Schema logics into OWL axioms for unified data validation on a digital manufacturing platform

Abstract JSON (JavaScript Object Notation) is a prevalent data format used in cloud-based platforms that support composable digital manufacturing workflows. The current work presents a method to translate the logics found in JSON Schema into OWL axioms, in order to facilitate ontology-based unified data validation with JSON data. The specific contributions of this paper include the demonstration of using a formal ontology for the logic translation and data validation, a technique for disambiguating implicit relations found in JSON Schema as explicit OWL properties, and mapping JSON Schema validation keywords to equivalent OWL expressions.

A Kuroda-style j-translation

A nucleus is an operation on the collection of truth values which, like double negation in intuitionistic logic, is monotone, inflationary, idempotent and commutes with conjunction. Any nucleus determines a proof-theoretic translation of intuitionistic logic into itself by applying it to atomic formulas, disjunctions and existentially quantified subformulas, as in the Gödel–Gentzen negative translation. Here we show that there exists a similar translation of intuitionistic logic into itself which is more in the spirit of Kuroda’s negative translation. The key is to apply the nucleus not only to the entire formula and universally quantified subformulas, but to conclusions of implications as well.

Lattice logic as a fragment of (2-sorted) residuated modal logic

ABSTRACTCorrespondence and Shalqvist theories for Modal Logics rely on the simple observation that a relational structure is at the same time the basis for a model of modal logic and for a model of first-order logic with a binary predicate for the accessibility relation. If the underlying set of the frame is split into two components, , and , then frames are at the same time the basis for models of non-distributive lattice logic and of two-sorted, residuated modal logic. This suggests that a reduction of the first to the latter may be possible, encoding Positive Lattice Logic (PLL) as a fragment of Two-Sorted, Residuated Modal Logic. The reduction is analogous to the well-known Gödel-McKinsey-Tarski translation of Intuitionistic Logic into the S4 system of normal modal logic. In this article, we carry out this reduction in detail and we derive some properties of PLL from corresponding properties of First-Order Logic. The reduction we present is extendible to the case of lattices with operators, making use of recent results by this author on the relational representation of normal lattice expansions.

A logic-algebraic tool for reasoning with Knowledge-Based Systems

A detailed exposition of foundations of a logic-algebraic model for reasoning with knowledge bases specified by propositional (Boolean) logic is presented. The model is conceived from the logical translation of usual derivatives on polynomials (on residue rings) which is used to design a new inference rule of algebro-geometric inspiration. Soundness and (refutational) completeness of the rule are proved. Some applications of the tools introduced in the paper are shown.

English Proverbs into Arabic through Machine Translation

Proverbs are sentences used by people to convey wisdom, truth or morals that have been handed down for generations. Translating them from one language into another through machine translation has been a matter of interest because of their importance in keeping the religious, cultural and historical aspects. The accuracy of machine translation systems has become one of the most important issues nowadays as the demand for online translation is increasing. Therefore, the current study aimed at investigating the linguistic obstacles that machine translation might face in carrying out the same translation from English proverbs into Arabic. It also attempted to confirm the importance of human intervention in solving the issue of accuracy. To achieve these objectives, a group of English proverbs were randomly selected, translated into Arabic by ‘Google translate’ and analysed qualitatively by the researchers. On the one hand, the findings revealed that ‘Google translate’ faced some linguistic obstacles in carrying out the same meaning of English proverbs into Arabic. Lexically, words that have multiple meanings present countless challenges and difficulties to online translation. On the other hand, the different linguistic structures between English and Arabic make the process of translation sometimes a bit challenging as both languages belong to unrelated families. Simply put, machine translation is expected to encounter various problems in providing meaningful logical translation. A detailed discussion is provided and recommendations for further research are given based on the analysis.