Basic Connectives Research Articles

Some implication operators on bounded lattices

In this paper, first we give a construction method for implications by an arbitrary element on bounded lattice. We present some implications obtained from binary operations on the bounded lattices, that is, from basic logic connectives, viz., t-norms, t-conorms and negations. Finally, we examine the main properties of the constructed implications.

한국어 분석 도구를 활용한 국어교과서 텍스트의 장르 비교 분석

This study aimed to investigate the characteristics of three different genres, including argumentative, expository, and narrative texts that were selected from high school Korean textbooks revised by the 2015 National Curriculum with Auto-Kohesion, a Korean text analysis tool. The indices used for the current study included basic measures (the number of words, the number of sentences, the average sentence length, the average word length), word frequency (word frequency for contents words and all words), lexical diversity (type-token ratio for content words), personal pronouns (the first person pronouns, the second person pronouns, the third person pronouns, all pronouns), connectives (causal, additive, and all connectives), syntactic complexity (the ratio of modifiers, the ratio of constituents), and reference cohesion measures. The analysis demonstrated that the argumentative and expository texts were not significantly different in all measures. However, the differences between the two genres versus narratives were substantial; a greater use of personal pronouns and lower frequency words as well as more words and sentences was identified in the narrative texts, whereas the argumentative and expository texts included the higher text cohesion and the sentence constituent ratio along with the longer average words and sentences. The argumentative texts have been shown to include more personal pronouns than the other two genres. The current study provides some implications for designing Korean textbooks and analyzing the characteristics of different genres.

An Analysis of Text Difficulty across Grades in Korean Middle School English Textbooks Using Coh-Metrix

This study aims to investigate the text difficulty of the reading materials of Korean middle school English textbooks with Coh-Metrix, a software developed by the Institute for Intelligent Systems at the University of Memphis to analyze the linguistic and psycholinguistic features of English text and textbooks with a wide range of indices on cohesion and language. In this study, the textbook corpus consisted of the text files extracted from 13 English textbooks. These files were used for analyzing the text difficulty among grades with Coh-Metrix. The Coh-Metrix indices selected for this study contained basic counts, word frequency, word features, lexical diversity, pronouns, connectives, readability, syntax complexity, syntax similarity, reference cohesion, semantic cohesion, and situation model measures. The results showed that there were significant differences among grades for basic counts, word features, first pronouns, causal and temporal connectives, readability, reference and semantic cohesion, the number of words before main verbs, syntactic similarity, and situation model measures. The differences among grades, however, were not significant for word frequency, lexical diversity, second and third person pronouns, additive connectives, and NP density measures. The findings have educational implications for textbook design and language learning for English learners.

Importation lattices

In recent years, many works have appeared that propose order from basic fuzzy logic connectives. However, all of them assume the connectives to possess some kind of monotonicity, which succinctly implies that the underlying set is already endowed with an order. In this work, given a set P≠∅, we define an algebra based on an implicative-type function I without assuming any order-theoretic properties, either on P or I. Terming it the importation algebra, since the law of importation becomes one of the main axioms in this algebra, we show that such algebras can impose an order on the underlying set P to obtain importation posets. In the case P=[0,1], it is well-known that only R-implications obtained from left-continuous t-norms lead us to richer order-theoretic structures. However, we show here that we can obtain new order-theoretic structures on it even from other families of fuzzy implications, in fact, even when I is not a fuzzy implication, and that one can recover the usual order on [0,1] even from fuzzy implications that do not have the classical ordering property. We also give some sufficient conditions under which such importation posets become lattices. Finally, we present some further possible explorations.

Equality propositional logic and its extensions

We introduce a new formal logic, called equality propositional logic. It has two basic connectives, $boldsymbol{wedge}$ (conjunction) and $equiv$ (equivalence). Moreover, the $Rightarrow$ (implication) connective can be derived as $ARightarrow B:=(Aboldsymbol{wedge}B)equiv A$. We formulate the equality propositional logic and demonstrate that the resulting logic has reasonable properties such as Modus Ponens(MP) rule, Hypothetical Syllogism(HS) rule and completeness, etc. Especially, we provide two ways to prove the completeness of this logic system. We also introduce two extensions of equality propositional logic. The first one is involutive equality propositional logic, which is equality propositional logic with double negation. The second one adds prelinearity which is rich enough to enjoy the strong completeness property. Finally, we introduce additional connective $Delta$(delta) in equality propositional logic and demonstrate that the resulting logic holds soundness and completeness.

A new similarity measure for Pythagorean fuzzy sets

One of the methods of studying on two sets is to calculate the similarity of two sets. Triangular norms and conorms generalize the basic connectives between fuzzy sets, intuitionistic fuzzy sets, Pythagorean fuzzy sets. In this paper we used triangular conorms (S-norm). The advantage of using S-norm is that the similarity order does not change using different norms. In fact, we are looking for a new definition for calculating the similarity of two Pythagorean fuzzy sets. To achieve this goal, using an S-norm, we first present a formula for calculating the similarity of two Pythagorean fuzzy values, so that they are truthful in similarity properties. Following that, we generalize a formula for calculating the similarity of the two Pythagorean fuzzy sets which prove truthful in similarity conditions. Finally, we give some examples of this method.

Generalized Correspondence Analysis for Three-Valued Logics

Correspondence analysis is Kooi and Tamminga’s universal approach which generates in one go sound and complete natural deduction systems with independent inference rules for tabular extensions of many-valued functionally incomplete logics. Originally, this method was applied to Asenjo–Priest’s paraconsistent logic of paradox LP. As a result, one has natural deduction systems for all the logics obtainable from the basic three-valued connectives of LP (which is built in the $$ \{\vee ,\wedge ,\lnot \} $$ -language) by the addition of unary and binary connectives. Tamminga has also applied this technique to the paracomplete analogue of LP, strong Kleene logic $$ \mathbf K_3 $$ . In this paper, we generalize these results for the negative fragments of LP and $$ \mathbf K_3 $$ , respectively. Thus, the method of correspondence analysis works for the logics which have the same negations as LP or $$ \mathbf K_3 $$ , but either have different conjunctions or disjunctions or even don’t have them as well at all. Besides, we show that correspondence analyses for the negative fragments of $$ \mathbf K_3 $$ and LP, respectively, are also suitable without any changes for the negative fragments of Heyting’s logic $$ \mathbf G_3 $$ and its dual $$ \mathbf DG_3 $$ (which have different interpretations of negation than $$ \mathbf K_3 $$ and LP).

The Psychology of Uncertainty and Three-Valued Truth Tables.

Psychological research on people's understanding of natural language connectives has traditionally used truth table tasks, in which participants evaluate the truth or falsity of a compound sentence given the truth or falsity of its components in the framework of propositional logic. One perplexing result concerned the indicative conditional if A then C which was often evaluated as true when A and C are true, false when A is true and C is false but irrelevant“ (devoid of value) when A is false (whatever the value of C). This was called the “psychological defective table of the conditional.” Here we show that far from being anomalous the “defective” table pattern reveals a coherent semantics for the basic connectives of natural language in a trivalent framework. This was done by establishing participants' truth tables for negation, conjunction, disjunction, conditional, and biconditional, when they were presented with statements that could be certainly true, certainly false, or neither. We review systems of three-valued tables from logic, linguistics, foundations of quantum mechanics, philosophical logic, and artificial intelligence, to see whether one of these systems adequately describes people's interpretations of natural language connectives. We find that de Finetti's (1936/1995) three-valued system is the best approximation to participants' truth tables.

The First Philosophical Word

Abstract All questions of thought lead back to philosophy. However, there remains a lack of clarity with regard to the preconditions of philosophy, especially the genesis of philosophy, that is, what is the first philosophical topic is not much clearer. It is often thought that philosophizing stems from being, or a state of existence, a legend from Greek philosophy. This paper attempts to reanalyze the precondition or the critical point of “how thinking is possible” by an archaeology of thought, so as to pinpoint the first word that initiated human thought. The first thinking word must be a philosophical word. In examining historical conditions, oracle bone scripts and basic connectives in mathematical logic, this paper argues that human thought stemmed from negative words that are the first philosophical words. The trajectory is that negatives, upon entering the human lexicon, inspired the concept of possibility; only with various possibilities, consciousness begins to think freely, put forward issues worth being reflected and deliberated and provide different alternative answers. In short, the advent of the negative “no” marked the existence of the critical condition of thinking, i. e., thinking and philosophy started when the human being could say no, which is therefore the first term and the primary topic of philosophy.

Guiding reinvention of conventional tools of mathematical logic: students’ reasoning about mathematical disjunctions

Motivated by the observation that formal logic answers questions students have not yet asked, we conducted exploratory teaching experiments with undergraduate students intended to guide their reinvention of truth-functional definitions for basic logical connectives. We intend to reframe the relationship between reasoning and logic by showing how logic emerges within students’ mathematical activity. This activity entails reflecting on and systematizing their own language use across diverse semantic content. We present categories of students’ untrained strategies for assessing the truth-values for mathematical disjunctions. Students’ initial reasoning heavily reflected content-specific and pragmatic factors in ways inconsistent with the norms and conventions of mathematical logic. Despite this, all student groups reinvented the standard truth-functional definition for simple disjunctions. We demonstrate how this learning depended upon particular forms of reasoning about logic. We also contrast various strategies for assessing quantified disjunctions and their different affordances in students’ mathematical activity.

A map of dependencies among three-valued logics

Three-valued logics arise in several fields of computer science, both inspired by concrete problems (such as in the management of the null value in databases) and theoretical considerations. Several three-valued logics have been defined. They differ by their choice of basic connectives, hence also from a syntactic and proof-theoretic point of view. Different interpretations of the third truth value have also been suggested. They often carry an epistemic flavor. In this work, relationships between logical connectives on three-valued functions are explored. Existing theorems of functional completeness have laid bare some of these links, based on specific connectives. However we try to draw a map of such relationships between conjunctions, negations and implications that extend Boolean ones. It turns out that all reasonable connectives can be defined from a few of them and so all known three-valued logics appear as a fragment of only one logic. These results can be instrumental when choosing, for each application context, the appropriate fragment where the basic connectives make full sense, based on the appropriate meaning of the third truth-value.

Graded properties of unary and binary fuzzy connectives

The paper studies basic graded properties of unary and binary fuzzy connectives, i.e., unary and binary operations on the set of truth degrees of a background fuzzy logic extending the logic MTL of left-continuous t-norms. The properties studied in this paper are graded generalizations of monotony, Lipschitz continuity, null and unit elements, idempotence, commutativity, and associativity. The paper elaborates the initial study presented in previous papers and focuses mainly on parameterization of graded properties by conjunction-multiplicities of subformulae in the defining formulae, preservation of graded properties under compositions and slight variations of fuzzy connectives, the values of graded properties for basic connectives of the ground logic, and the dependence of the values on the ground logic. The results are proved in the formal framework of higher-order fuzzy logic MTL, also known as Fuzzy Class Theory (FCT). General theorems provable in FCT are illustrated on several semantic examples.

On Constructive Connectives and Systems

Canonical inference rules and canonical systems are defined in the framework of non-strict single-conclusion sequent systems, in which the succeedents of sequents can be empty. Important properties of this framework are investigated, and a general non-deterministic Kripke-style semantics is provided. This general semantics is then used to provide a constructive (and very natural), sufficient and necessary coherence criterion for the validity of the strong cut-elimination theorem in such a system. These results suggest new syntactic and semantic characterizations of basic constructive connectives.

Genetic Code of Electrical Operational Procedures: Lock-Out Systems, Simulators, and Training

In electrical installations, each working zone and each source node present a kit of operating instructions as a logic gene. Previous papers have introduced a new approach and a language of a ternary code for operational procedures (Parise program). This paper deals with some comments on definitions and the syntax of the operational language. The same status of switching defines working zones or nodes by double operating value, free value, and packed value, thereby discriminating between decision and execution, and executed act from verification. The recommended command is a double click. The instructions have to be assembled like in a genetic cryptogram in the context of an electrical power system and of its architecture and constitution. They describe the complete and reversible evolution of the operational procedures for safe transition from a one set of switching configuration to another ( <i xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">waves of safe sets</i> ). Transitions maps are introduced to assist in the elaboration of operational procedures and to help the training of operators, their competence, and performance. These maps are specialized for the operational procedures more than the Karnaugh maps of digital electronics. On the basis of the logic gates, the basic equivalent mechanical connectives are defined to guide the operational procedures. A switching simulator based on electronic gates can be very useful for verifying the effects of any set of switching operations, adopting for commands the double click.

Randomization of classical inference patterns and its application

By means of randomization, the concept of D-randomized truth degree of formulas in two-valued propositional logic is introduced, and it is proved that the set of values of D-randomized truth degree of formulas has no isolated point in [0,1]. The concepts of D-logic pseudo-metric and D-logic metric space are also introduced and it is proved that there is no isolated point in the space. The new built D-randomized concepts are extensions of the corresponding concepts in quantified logic. Moreover, it is proved that the basic logic connectives are continuous operators in D-logic metric space. Lastly, three different types of approximate reasoning patterns are proposed.

Uncertainty Modeling by Bilattice-Based Squares and Triangles

In this paper, Ginsberg's/Fitting's theory of bilattices, and in particular the associated constructs of bilattice-based squares and triangles, is introduced as an attractive framework for the representation of uncertain and potentially conflicting information, paralleling Goguen's L-fuzzy set theory. We recall some of the advantages of bilattice-based frameworks for handling fuzzy sets and systems, provide the related structures with adequately defined graded versions of the basic logical connectives, and study their properties and relationships

A BILATTICE-BASED FRAMEWORK FOR HANDLING GRADED TRUTH AND IMPRECISION

We present a family of algebraic structures, called rectangular bilattices, which serve as a natural accommodation and powerful generalization to both intuitionistic fuzzy sets (IFSs) and interval-valued fuzzy sets (IVFSs). These structures are useful on one hand to clarify the exact nature of the relationship between the above two common extensions of fuzzy sets, and on the other hand provide an intuitively attractive framework for the representation of uncertain and potentially conflicting information. We also provide these structures with adequately defined graded versions of the basic logical connectives, and study their properties and relationships. Application potential and intuitive appeal of the proposed framework are illustrated in the context of preference modeling.

Correct models of fuzzy IF–THEN rules are continuous

We show that a system of fuzzy IF–THEN rules being modeled correctly works as a partially given (fuzzy) function. Its behavior is determined by a chosen structure for IF–THEN rules which assigns meaning to fuzzy sets in the IF and THEN parts as well as to basic connectives. This partial function can be extended to a whole domain of fuzzy sets with the help of the Compositional Rule of Inference which plays a role of a mechanism for computing the dependent functional values. We investigate the question whether this function is continuous. Two new notions are introduced: a correct model of fuzzy IF–THEN rules in a structure and a continuous model of fuzzy IF–THEN rules with respect to given data. We prove that correctness of models of fuzzy IF–THEN rules is connected with the problem of solvability of the respective system of fuzzy relation equations. In this and only this case a model of fuzzy IF–THEN rules is continuous. As a side result we have established a new criterion of a solvability of a system of fuzzy relation equations with sup - * -composition.

Possibility Theory, Probability Theory and Multiple-Valued Logics: A Clarification

There has been a long-lasting misunderstanding in the literature of artificial intelligence and uncertainty modeling, regarding the role of fuzzy set theory and many-valued logics. The recurring question is that of the mathematical and pragmatic meaningfulness of a compositional calculus and the validity of the excluded middle law. This confusion pervades the early developments of probabilistic logic, despite early warnings of some philosophers of probability. This paper tries to clarify this situation. It emphasizes three main points. First, it suggests that the root of the controversies lies in the unfortunate confusion between degrees of belief and what logicians call “degrees of truth”. The latter are usually compositional, while the former cannot be so. This claim is first illustrated by laying bare the non-compositional belief representation embedded in the standard propositional calculus. It turns out to be an all-or-nothing version of possibility theory. This framework is then extended to discuss the case of fuzzy logic versus graded possibility theory. Next, it is demonstrated that any belief representation where compositionality is taken for granted is bound to at worst collapse to a Boolean truth assignment and at best to a poorly expressive tool. Lastly, some claims pertaining to an alleged compositionality of possibility theory are refuted, thus clarifying a pervasive confusion between possibility theory axioms and fuzzy set basic connectives.

Temporal Causal Abduction

The paper is devoted to presentation of the idea of Temporal Causal Networks (TCN). Temporal Causal Networks constitute a tool for representing and dealing with causal dependencies propagation over time. A temporal causal network is a causal network incorporating explicit representation of time during which its symptoms/nodes are valid, not valid, or unknown. The atemporal causal structure is basically an AND/OR/NOT causal graph, i.e. a causal graph incorporating basic logical connectives for the representation of different types of causal dependencies. The presented approach uses a specific time constraints propagation algorithm to determine possible system behavior in time. The main application includes simulation, monitoring and elements of diagnostic reasoning for dynamic systems with explicit time representation.