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This paper enlarges classical syllogistic logic with assertions having to do with comparisons between the sizes of sets. So in addition to assertions like All x are y and Some x are y, we also have There are at least as many x as y, and There are more x than y. Our work also allows all nouns to be complemented. We thus obtain sentences equivalent to No x are y and At least half of the universe are x. We work on finite models exclusively. We formulate a syllogistic logic for our language. The main result is a soundness/completeness theorem. The logic has a rule of ex falso quodlibet, and reductio ad absurdum is admissible. There are efficient algorithms for proof search and model construction, and the logic has been implemented.

This paper proposes a formalization of the class of sentences quantified by most, which is also interpreted as proportion of or majority of depending on the domain of discourse. We consider sentences of the form “Most A are B”, where A and B are plural nouns and the interpretations of A and B are infinite subsets of  N . There are two widely used semantics for Most A are B: (i)  C(A∩B)>C(A∖B)  and (ii)  C(A∩B)>C(A)2 , where C(X) denotes the cardinality of a given finite set X. Although (i) is more descriptive than (ii), it also produces a considerable amount of insensitivity for certain sets. Since the quantifier most has a solid cardinal behaviour under the interpretation majority and has a slightly more statistical behaviour under the interpretation proportional of, we consider an alternative approach in deciding quantity-related statements regarding infinite sets. For this we introduce a new semantics using natural density for sentences in which interpretations of their nouns are infinite subsets of  N , along with a list of the axiomatization of the concept of natural density. In other words, we take the standard definition of the semantics of most but define it as applying to finite approximations of infinite sets computed to the limit.

Bipolar neutrosophic matrices (BNM) are obtained by bipolar neutrosophic sets. Each bipolar neutrosophic number represents an element of the matrix. The matrices are representable multi-dimensional arrays (3D arrays). The arrays have nested list data type. Some operations, especially the composition is a challenging algorithm in terms of coding because there are so many nested lists to manipulate. This paper presents a Python tool for bipolar neutrosophic matrices. The advantage of this work, is that the proposed Python tool can be used also for fuzzy matrices, bipolar fuzzy matrices, intuitionistic fuzzy matrices, bipolar intuitionistic fuzzy matrices and single valued neutrosophic matrices.

In recent years, "mathematical orientations on real-life problems", which continue to increase, began to make a significant impact. Information systems for many decision-making problems consist of uncertain, incomplete, indeterminate and indiscernible structures and components. Classical set theory and interpretation methods fail to represent, express and solve the problems of these types or cause to make wrong decisions. For this reason, in this study, we provide definitions and methods to present information and problem representations in more detail and precision. This paper introduces three new topologies called covering-based rough fuzzy, covering-based rough intuitionistic fuzzy and covering-based rough neutrosophic nano topology. Some fundamental definitions such as open set, closed set, interior, closure and basis are given. Neutrosophic definitions and properties are mainly investigated. We give some real life applications of covering-based rough neutrosophic nano topology in the final part of the paper and an explanatory example of decision making application by defining core point. INDEX TERMS Approximation space, core point, covering-based topology, fuzzy nano topology, fuzzy sets, intuitionistic nano topology, intuitionistic sets, neutrosophic nano topology, neutrosophic sets, rough decision making, location selection problem.

This paper presents a novel social choice theory based multi-criteria decision method under neutrosophic environment. Our hybrid method consists of classical methods and an aggregation operator used in social choice theory. In addition to this, we also use distributed in-determinacy form function in our method to provide a more sensitive indeterminacy approach towards accuracy functions. We also consider reciprocal property for all individuals. This provides, as in intuitionistic fuzzy decision making theory, a consistent decision making for each individual. The solution approach presented in this paper in group decision making is treated under neutrosophic individual preference relations. These new approaches seem to be more consistent with natural human behaviour, hence should be more acceptable and implementable. Moreover , the use of a similar approach to develop some deeper soft degrees of consensus is outlined. Finally, we give a Python implementation of our work in the Appendix section. Mathematics Subject Classification (2010). 91B06, 91B14, 91B10, 03B52.

This paper enlarges classical syllogistic logic with assertions having to do with comparisons between the sizes of sets. So it concerns a logical system whose sentences are of the following forms: All x are y and Some x are y, There are at least as many x as y, and There are more x than y. Here x and y range over subsets (not elements) of a given infinite set. Moreover, x and y may appear complemented (i.e., as x and y), with the natural meaning. We formulate a logic for our language that is based on the classical syllogistic. The main result is a soundness/completeness theorem. There are efficient algorithms for proof search and model construction.

This paper extends the logic CARD of comparisons between the sizes of sets to the logic CARD with IA which contains intersecting adjectives. We prove completeness of the logics CARD and CARD with IA. We also give algorithmic analysis and some algebraic properties of the logics.

The main goal of this paper is to construct Bé-zier surface modeling for neutrosophic data problems. We show how to build the surface model over a data sample from agriculture science after the theoretical structure of the modeling is introduced. As a sampler application for agriculture systems, we give a visualization of Bézier surface model of an estimation of a given yield of bean seeds grown in a field over a period.

Neutrosophic set concept is defined with membership, non-membership and indeterminacy degrees. This concept is the solution and representation of the problems with various fields. In this paper, a geometric model is introduced for Neutrosophic data problem for the first time. This model is based on neutrosophic sets and neutrosophic relations. Neutrosophic control points are defined according to these points, resulting in neutrosophic B` ezier curves.

In this paper, we study the logic of language of L(More, IA). The logic contains the quantifier called '' more '' which makes cardinality comparisons can not be expressed in the language of the first order logic. The sentence forms are basically the form of ''There are more y than x .'' with x and y being common plural nouns. The sentence forms of common plural nouns combined with intersecting adjectives are ''There are more b y than a x. " with the intersecting adjectives a and b. We focus on derivation algorithms of the sentences having this type of quantifier and algorithms of construction of counter-models when the derivations are not provided.

This paper presents labeled digraph representations of logics of All, Some, No,
More and At Least by using algebraic properties of derivations of the logics. The logics of
All, Some and No come from the Aristotalian syllogisms. The syllogisms contain sentence
forms of All p are q, Some p are q, No p are q where p and q are plural nouns. The logics
of At least and More have respectively the form of “There are at least as many p as q”
and “There are more p than q” where p and q are plural nouns. Especially, labeled digraph
representations of logics of At least and More are a crucial position since languages of the
logics are not expressible in first order language.

The purpose of this paper is to contribute to the natural logic program which invents logics in natural language. This study presents two logics: a logical system called) , (  R containing transitive verbs and a more expressive logical system) , , (IA   R containing both transitive verbs and intersective adjectives. The paper offers three different set-theoretic semantics which are equivalent for the logics.

The aim of this paper is to provide a contribution to the natural logic program which explores logics in natural language. The paper offers two logics called  R(∀,∃)  and  G(∀,∃)  for dealing with inference involving simple sentences with transitive verbs and ditransitive verbs and quantified noun phrases in subject and object position. With this purpose, the relational logics (without Boolean connectives) are introduced and a model-theoretic proof of decidability for they are presented. In the present paper we develop algebraic semantics (bounded meet semi-lattice) of the logics using congruence theory.

In this paper, we define the neutrosophic valued (and generalized or G) metric spaces for the first time. Besides, we newly determine a mathematical model for clustering the neutrosophic big data sets using G-metric. Furthermore, relative weighted neutrosophic-valued distance and weighted cohesion measure, is defined for neutrosophic big data set. We offer a very practical method for data analysis of neutrosophic big data although neutrosophic data type (neutrosophic big data) are in massive and detailed form when compared with other data types.

We introduce refined concepts for neutrosophic quantum computing such as neutrosophic quantum states and transformation gates, neutrosophic Hadamard matrix, coherent and decoherent superposition states, entanglement and measurement notions based on neutrosophic quantum states. We also give some observations using these principles. We present a number of quantum computational matrix transformations based on neutrosophic logic and clarify quantum mechanical notions relying on neutrosophic states. The paper is intended to extend the work of Smarandache by introducing a mathematical framework for neutrosophic quantum computing and presenting some results.

We introduce oracle Turing machines with neutrosophic values allowed in the oracle information and then give some results when one is permitted to use neutrosophic sets and logic in relative computation. We also introduce a method to enumerate the elements of a neutrosophic subset of natural numbers.

In this paper we introduce some new graphs obtained from bipartite posets. We show that lower-minimal
graph of a bipartite poset is isomorphic to upper-maximal graph of dual of the poset by using set representations of the posets by using set representations of the posets.

This article presents a technical construction of reasoning and counter-models for some sentences called
fragments as in [9] in English. Speaking English and logical inferences are brought together in computer based
approach to natural language. Not only the inferences in the language [7] are given but also counter-model
constructions in case of no inference from input sentences. Approach of this construction considers usage of
minimal number of set elements.

This paper proposes a formalization of the class of sentences quantified by most, which is also interpreted as proportion of or majority of depending on the domain of discourse. We consider sentences of the form "Most A are B ", where A and B are plural nouns and the interpretations of A and B are infinite subsets of N. There are two widely used semantics for Most A are B : (i) C(A ∩ B) > C(A \ B) and (ii) C(A ∩ B) > C(A) 2 , where C(X) denotes the cardinality of a given finite set X. Although (i) is more descriptive than (ii), it also produces a considerable amount of insensitivity for certain sets. Since the quantifier most has a solid cardinal behaviour under the interpretation majority and has a slightly more statistical behaviour under the interpretation proportional of, we consider an alternative approach in deciding quantity-related statements regarding infinite sets. For this we introduce a new semantics using natural density for sentences in which interpretations of their nouns are infinite subsets of N, along with a list of the axiomatization of the concept of natural density. In other words, we take the standard definition of the semantics of most but define it as applying to finite approximations of infinite sets computed to the limit.

The draft paper is still under review. There might be something to correct.

All comments are welcome.

Abstract:
This paper considers the quantified simple sentences by \textit{Most}, sometimes referred to as proportional, sometimes the majority. The sentence form: \textit{Most A are B} where \textit{A} and \textit{B} are plural nouns. A and B range over elements of P(N). Moreover, A and B may appear complemented (i.e., as Non−A and Non−B). Two different but equivalent semantics are for \textit{Most A are B} as (i) C(A∩B)>C(A∖B) and (ii) C(A∩B)>C(A)2 where C(X) is the cardinality of the set X. Both semantics work well on finite sets but exhibit problematic behaviors on infinite sets since division is undefined on cardinal arithmetic. Although semantics (i) is more descriptive than semantics (ii), it also produces insensitivity for certain sets. \textquotedblleft Most" has a solid cardinal structure under the interpretation of the majority, and has the more statistical structure with proportional interpretation, and this statistical interpretation provides more flexible range of motion. For all these reasons, we introduce a new semantics with natural density for the sentences ranging over N. We also give an axiomatization of this logic.