In the recent years, problem-solving become a central topic that discussed by educators or researchers in mathematics education. it’s not only as the ability or as a method of teaching. but also, it is a little in reviewing about the components of the support to succeed in problem-solving, such as student's belief and attitude towards mathematics, algebraic thinking skills, resources and teaching materials. In this paper, examines the algebraic thinking skills as a foundation for problem-solving, and (...) learning cycle as a breath of continuous learning. In this paper, learning cycle to be used is a modified type of 5E based on beliefs. (shrink)

This paper aims to clarify Merleau-Ponty’s contribution to an embodied-enactive account of mathematical cognition. I first identify the main points of interest in the current discussions of embodied higher cognition and explain how they relate to Merleau-Ponty and his sources, in particular Husserl’s late works. Subsequently, I explain these convergences in greater detail by more specifically discussing the domains of geometry and algebra and by clarifying the role of gestalt psychology in Merleau-Ponty’s account. Beyond that, I explain how, for Merleau-Ponty, (...) mathematical cognition requires not only the presence and actual manipulation of some concrete perceptible symbols but, more strongly, how it is fundamentally linked to the structural transformation of the concrete configurations of symbolic systems to which these symbols appertain. Furthemore, I fill a gap in the literature by explaining Merleau-Ponty’s claim that these structural transformations are operated through motor intentionality. This makes it possible, in turn, to contrast Merleau-Ponty’s approach to ontologically idealistic and realistic views on mathematical objects. On Merleau-Ponty’s account, mathematical objects are relational entities, that is, gestalts that necessarily imply situated cognizers to whom they afford a specific type of engagement in the world and on whom they depend in their eventual structural transformations. I argue that, by attributing a strongly constitutive role to phenomenal configurations and their motor transformation in mathematical thinking, Merleau-Ponty contributes to clarifying the worldly, historical, and socio-cultural aspects of mathematical truths without compromising what we perceive as their universality, certainty, and necessity. (shrink)

This article introduces the mathematical models of the thinking laws in the internal structure of consciousness, the spatial and temporal features of the thinking laws, and the phenomenon of resonance as a general feature of the cognitive process. The article will focus on the logical order and space-time existence of the thinking laws, by interrelating such mathematical concepts as Boolessche Algebra, Set theory, Crowd round of Abel, and ordinal number. Finally, the article discusses how thinking laws (...) can a naturalized theory of consciousness, and how they, together with established principles of consciousness, can play important roles in the process of cognition. (shrink)

How does Aristotle think about sentences like ‘Every x is y’ in the Prior Analytics? A recently popular answer conceives of these sentences as expressing a mereological relationship between x and y: the sentence is true just in case x is, in some sense, a part of y. I argue that the motivations for this interpretation have so far not been compelling. I provide a new justification for the mereological interpretation. First, I prove a very general algebraic soundness and (...) completeness result that unifies the most important soundness and completeness results to date. Then I argue that this result vindicates the mereological interpretation. In contrast to previous interpretations, this argument shows how Aristotle’s conception of predication in mereological terms can do important logical work. (shrink)

In the philosophy of the analytical tradition, set theory and formal logic are familiar formal tools. I think there is no deep reason why the philosopher’s tool kit should be restricted to just these theories. It might well be the case—to generalize a dictum of Suppes concerning philosophy of science—that the appropriate formal device for doing philosophy is mathematics in general; it may be set theory, algebra, topology, or any other realm of mathematics. In this paper I want to employ (...) elementary topological considerations to shed new light on the intricate problem of the relation of qualities and similarity. Thereby I want to make plausible the general thesis that topology might be a useful device for matters epistemological. (shrink)

The existence of singularities alerts that one of the highest priorities of a centennial perspective on general relativity should be a careful re-thinking of the validity domain of Einstein’s field equations. We address the problem of constructing distinguishable extensions of the smooth spacetime manifold model, which can incorporate singularities, while retaining the form of the field equations. The sheaf-theoretic formulation of this problem is tantamount to extending the algebra sheaf of smooth functions to a distribution-like algebra sheaf in which (...) the former may be embedded, satisfying the pertinent cohomological conditions required for the coordinatization of all of the tensorial physical quantities, such that the form of the field equations is preserved. We present in detail the construction of these distribution-like algebra sheaves in terms of residue classes of sequences of smooth functions modulo the information of singular loci encoded in suitable ideals. Finally, we consider the application of these distribution-like solution sheaves in geometrodynamics by modeling topologically-circular boundaries of singular loci in three-dimensional space in terms of topological links. It turns out that the Borromean link represents higher order wormhole solutions. (shrink)

We develop a simple framework called ‘natural topology’, which can serve as a theoretical and applicable basis for dealing with real-world phenomena.Natural topology is tailored to make pointwise and pointfree notions go together naturally. As a constructive theory in BISH, it gives a classical mathematician a faithful idea of important concepts and results in intuitionism. -/- Natural topology is well-suited for practical and computational purposes. We give several examples relevant for applied mathematics, such as the decision-support system Hawk-Eye, and various (...) real-number representations. -/- We compare classical mathematics (CLASS), intuitionistic mathematics (INT), recursive mathematics (RUSS), Bishop-style mathematics (BISH) and formal topology, aiming to reduce the mutual differences to their essence. To do so, our mathematical foundation must be precise and simple. There are links with physics, regarding the topological character of our physical universe. -/- Any natural space is isomorphic to a quotient space of Baire space, which therefore is universal. We develop an elegant and concise ‘genetic induction’ scheme, and prove its equivalence on natural spaces to a formal-topological induction style. The inductive Heine-Borel property holds for ‘compact’ or ‘fanlike’ natural subspaces, including the real interval [g, h]. Inductive morphisms respect this Heine-Borel property, inversely. This partly solves the continuous-function problem for BISH, yet pointwise problems persist in the absence of Brouwer’s Thesis. -/- By inductivizing the definitions, a direct correspondence with INT is obtained which allows for a translation of many intuitionistic results into BISH. We thus prove a constructive star-finitary metrization theorem which parallels the classical metrization theorem for strongly paracompact spaces. We also obtain non-metrizable Silva spaces, in infinite-dimensional topology. Natural topology gives a solid basis, we think, for further constructive study of topological lattice theory, algebraic topology and infinite-dimensional topology. The final section reconsiders the question of which mathematics to choose for physics. Compactness issues also play a role here, since the question ‘can Nature produce a non-recursive sequence?’ finds a negative answer in CTphys . CTphys , if true, would seem at first glance to point to RUSS as the mathematics of choice for physics. To discuss this issue, we wax more philosophical. We also present a simple model of INT in RUSS, in the two-players game LIfE. (shrink)

Consider the set of inferences that are acceptable to use in all our theory building endeavors. Call this set of inferences the universal theory building toolkit, or just ’the toolkit’ for short. It is clear that the toolkit is tightly connected to logic in a variety of ways. Beall, for example, has argued that logic just is the toolkit. This paper avoids making a stand on that issue and instead investigates reasons for thinking that, logic or not, the toolkit (...) is substructural. It is presented as a dialogue for the simple reason that it summarizes a range of dialogues on this subject that the author has had with various folks over the past few years. The method I use to investigate the toolkit is inspired in both philosophical and technical details by Alasdair Urquhart’s work on semantics for relevance logics from the early 1970s. (shrink)

Consider the set of inferences that are acceptable to use in all our theory building endeavors. Call this set of inferences the universal theory building toolkit, or just ’the toolkit’ for short. It is clear that the toolkit is tightly connected to logic in a variety of ways. Beall, for example, has argued that logic just is the toolkit. This paper avoids making a stand on that issue and instead investigates reasons for thinking that, logic or not, the toolkit (...) is substructural. It is presented as a dialogue for the simple reason that it summarizes a range of dialogues on this subject that the author has had with various folks over the past few years. The method I use to investigate the toolkit is inspired in both philosophical and technical details by Alasdair Urquhart’s work on semantics for relevance logics from the early 1970s. (shrink)

The Belnap–Dunn logic is a well-known and well-studied four-valued logic, but until recently little has been known about its extensions, i.e. stronger logics in the same language, called super-Belnap logics here. We give an overview of several results on these logics which have been proved in recent works by Přenosil and Rivieccio. We present Hilbert-style axiomatizations, describe reduced matrix models, and give a description of the lattice of super-Belnap logics and its connections with graph theory. We adopt the point of (...) view ofAlgebraic Logic, exploring applications of the general theory of algebraization of logics to the super-Belnap family. In this respect we establish a number of new results, including a description of the algebraic counterparts, Leibniz filters, and strong versions of super-Belnap logics, as well as the classification of these logics within the Leibniz and Frege hierarchies. (shrink)

This paper investigates a generalization of Boolean algebras which I call agglomerative algebras. It also outlines two conceptions of propositions according to which they form an agglomerative algebra but not a Boolean algebra with respect to conjunction and negation.

Katie Steele and H. Orri Stefánsson argue that, to reflect an agent’s limited awareness, the algebra of propositions on which that agent’s credences are defined should be relativised to their awareness state. I argue that this produces insurmountable difficulties. But the project of relativising the agent’s algebra to reflect their partial perspective need not be abandoned: the algebra can be relativised, not to the agent’s awareness state, but to what we might call their subjective modality.

We present an algebraic account of the Tongan kinship terminology (TKT) that provides an insightful journey into the fabric of Tongan culture. We begin with the ethnographic account of a social event. The account provides us with the activities of that day and the centrality of kin relations in the event, but it does not inform us of the conceptual system that the participants bring with them. Rather, it is a slice in time of an ongoing dynamic process that (...) links behavior with a conceptual system of kin relations and vice versa. To understand this interplay, we need an account of the underlying conceptual system that is being activated during the event. Thus, we introduce a formal, algebraically based account of TKT. This account brings to the fore the underlying logic of TKT and allows us to distinguish between features of the kinship system that arise from the logic of TKT as a generative structure and features that must have arisen through cultural intervention. (shrink)

Hyperboolean algebras are Boolean algebras with operators, constructed as algebras of complexes (or, power structures) of Boolean algebras. They provide an algebraic semantics for a modal logic (called here a {\em hyperboolean modal logic}) with a Kripke semantics accordingly based on frames in which the worlds are elements of Boolean algebras and the relations correspond to the Boolean operations. We introduce the hyperboolean modal logic, give a complete axiomatization of it, and show that it lacks the finite model property. (...) The method of axiomatization hinges upon the fact that a "difference" operator is definable in hyperboolean algebras, and makes use of additional non-Hilbert-style rules. Finally, we discuss a number of open questions and directions for further research. (shrink)

We define emergence algebraically in the context of discrete dynamical systems modeled as transformation semigroups. Emergence happens when a quotient structure (coarse-grained dynamics) is not a substructure of the original system. We survey small groups to show that algebraic emergence is neither ubiquitous nor rare. Then, we describe connections with hierarchical decompositions and explore some of the philosophical implications of the algebraic constraints.

In this review, we present a rigorous construction of an algebraic method for quantum unstable states, also called Gamow states. A traditional picture associates these states to vectors states called Gamow vectors. However, this has some difficulties. In particular, there is no consistent definition of mean values of observables on Gamow vectors. In this work, we present Gamow states as functionals on algebras in a consistent way. We show that Gamow states are not pure states, in spite of their (...) representation as Gamow vectors. We propose a possible way out to the construction of averages of observables on Gamow states. The formalism is intended to be presented with sufficient mathematical rigor. (shrink)

The logico-algebraic study of Lewis's hierarchy of variably strict conditional logics has been essentially unexplored, hindering our understanding of their mathematical foundations, and the connections with other logical systems. This work aims to fill this gap by providing a comprehensive logico-algebraic analysis of Lewis's logics. We begin by introducing novel finite axiomatizations for varying strengths of Lewis's logics, distinguishing between global and local consequence relations on Lewisian sphere models. We then demonstrate that the global consequence relation is strongly (...) algebraizable in terms of a specific class of Boolean algebras with a binary operator representing the counterfactual implication; in contrast, we show that the local consequence relation is generally not algebraizable, although it can be characterized as the degree-preserving logic over the same algebraic models. Further, we delve into the algebraic semantics of Lewis's logics, developing two dual equivalences with respect to particular topological spaces. In more details, we show a duality with respect to the topological version of Lewis's sphere models, and also with respect to Stone spaces with a selection function; using the latter, we demonstrate the strong completeness of Lewis's logics with respect to sphere models. Finally, we draw some considerations concerning the limit assumption over sphere models. (shrink)

An elementary algebra identifies conceptual and corresponding applicational limitations in John Kemeny and Paul Oppenheim’s (K-O) 1956 model of theoretical reduction in the sciences. The K-O model was once widely accepted, at least in spirit, but seems afterward to have been discredited, or in any event superceeded. Today, the K-O reduction model is seldom mentioned, except to clarify when a reduction in the Kemeny-Oppenheim sense is not intended. The present essay takes a fresh look at the basic mathematics of K-O (...) comparative vocabulary theoretical term reductions, from historical and philosophical standpoints, as a contribution to the history of the philosophy of science. The K-O theoretical reduction model qualifies a theory replacement as a successful reduction when preconditions of explanatory adequacy and comparable systematicization are met, and there occur fewer numbers of theoretical terms identified as replicable syntax types in the most economical statement of a theory’s putative propositional truths, as compared with the theoretical term count for the theory it replaces. The challenge to the historical model developed here, to help explain its scope and limitations, involves the potential for equivocal theoretical meanings of multiple theoretical term tokens of the same syntactical type. (shrink)

The purpose of this study is to compile a selection of the various formalisms found in conversation theory to introduce readers to Pask's discursive algebra. In this way, the text demonstrates how concept sharing and concept formation by means of the interaction of two participants may be formalized. The approach taken in this study is to examine the formal notation system used by Pask and demonstrate how such formalisms may be used to represent concept sharing and concept formation through conversation. (...) The compilation of the discursive algebra using the framework provided by conversation theory could potentially be used as an auxiliary framework to study conversational interactions in multi-agent systems theory. (shrink)

Multialgebras have been much studied in mathematics and in computer science. In 2016 Carnielli and Coniglio introduced a class of multialgebras called swap structures, as a semantic framework for dealing with several Logics of Formal Inconsistency that cannot be semantically characterized by a single finite matrix. In particular, these LFIs are not algebraizable by the standard tools of abstract algebraic logic. In this paper, the first steps towards a theory of non-deterministic algebraization of logics by swap structures are given. (...) Specifically, a formal study of swap structures for LFIs is developed, by adapting concepts of universal algebra to multialgebras in a suitable way. A decomposition theorem similar to Birkhoff’s representation theorem is obtained for each class of swap structures. Moreover, when applied to the 3-valued algebraizable logics J3 and Ciore, their classes of algebraic models are retrieved, and the swap structures semantics become twist structures semantics. This fact, together with the existence of a functor from the category of Boolean algebras to the category of swap structures for each LFI, suggests that swap structures can be seen as non-deterministic twist structures. This opens new avenues for dealing with non-algebraizable logics by the more general methodology of multialgebraic semantics. (shrink)

We introduce Pura Vida Neutrosophic Algebra, an algebraic structure consisting of neutrosophic numbers equipped with two binary operations namely addition and multiplication. The addition can be calculated sometimes with the function min and other times with the max function. The multiplication operation is the usual sum between numbers. Pura Vida Neutrosophic Algebra is an extension of both Tropical Algebra (also known as Min-Plus, or Min-Algebra) and Max-Plus Algebra (also known as Max-algebra). Tropical and Max-Plus algebras are algebraic structures (...) included in semirings and their operations can be used in matrices and vectors. Pura Vida Neutrosophic Algebra is included in Neutrosophic semirings and can be used in Neutrosophic matrices and vectors. (shrink)

The aim of this article is to investigate the roles of commutative diagrams (CDs) in a specific mathematical domain, and to unveil the reasons underlying their effectiveness as a mathematical notation; this will be done through a case study. It will be shown that CDs do not depict spatial relations, but represent mathematical structures. CDs will be interpreted as a hybrid notation that goes beyond the traditional bipartition of mathematical representations into diagrammatic and linguistic. It will be argued that one (...) of the reasons why CDs form a good notation is that they are highly mathematically tractable: experts can obtain valid results by ‘calculating’ with CDs. These calculations, take the form of ‘diagram chases’. In order to draw inferences, experts move algebraic elements around the diagrams. It will be argued that these diagrams are dynamic. It is thanks to their dynamicity that CDs can externalize the relevant reasoning and allow experts to draw conclusions directly by manipulating them. Lastly, it will be shown that CDs play essential roles in the context of proof as well as in other phases of the mathematical enterprise, such as discovery and conjecture formation. (shrink)

We give a complete axiomatization of the identities of the basic game algebra valid with respect to the abstract game board semantics. We also show that the additional conditions of termination and determinacy of game boards do not introduce new valid identities.En route we introduce a simple translation of game terms into plain modal logic and thus translate, while preserving validity both ways, game identities into modal formulae.The completeness proof is based on reduction of game terms to a certain ‘minimal (...) canonical form’, by using only the axiomatic identities, and on showing that the equivalence of two minimal canonical terms can be established from these identities. (shrink)

El objetivo de este artículo es presentar una demostración de un teorema clásico sobre álgebras booleanas y ordenes parciales de relevancia actual en teoría de conjuntos, como por ejemplo, para aplicaciones del método de construcción de modelos llamado “forcing” (con álgebras booleanas completas o con órdenes parciales). El teorema que se prueba es el siguiente: “Todo orden parcial se puede extender a una única álgebra booleana completa (salvo isomorfismo)”. Donde extender significa “sumergir densamente”. Tal demostración se realiza utilizando cortaduras de (...) Dedekind siguiendo el texto “Set Theory” de Jech, y otras ideas propias del autor de este artículo. Adicionalmente, se formulan algunas versiones débiles del axioma de elección relacionadas con las álgebras booleanas, las cuales son también de gran importancia para la investigación en teoría de conjuntos y teoría de modelos, pues estas son poderosas técnicas de construcción de modelos, como por ejemplo, el teorema de compacidad (permite construir modelos no estándar, etc) y el teorema del ultrafiltro, que permite construir ultraproductos (pueden ser usados para investigar problemas de cardinales grandes, etc). Se presentan algunas referencias de problemas abiertos sobre el tema. -/- The objective of this paper is to present a demonstration of a classical theorem on boolean algebras and partial orders of current relevance in set theory, as for example, for applications of model construction method called forcing" (with boolean algebras complete or with partial orders). The theorem to be proved is as follows: Any partial order can be extended to a single complete boolean algebra (up to isomorphism)". Where to extend means embed densely". Such a demonstration is done using Dedekind's cuts following the text Set Theory" of Jech, and other ideas of the author of this article. In addition, some weak versions of the axiom of choice related to boolean algebras are formulated, which are also of great importance for the research in set theory and model theory, since this are powerful model construction techniques, such as the compactness theorem (allows the construction of non-standard models, etc.) and the ultralter theorem, which allows the construction of ultraproducts (can be used to investigate problems of large cardinals, etc). Some references of open problems on the subject are presented. (shrink)

Robert Batterman’s ontological insights (2002, 2004, 2005) are apt: Nature abhors singularities. “So should we,” responds the physicist. However, the epistemic assessments of Batterman concerning the matter prove to be less clear, for in the same vein he write that singularities play an essential role in certain classes of physical theories referring to certain types of critical phenomena. I devise a procedure (“methodological fundamentalism”) which exhibits how singularities, at least in principle, may be avoided within the same classes of formalisms (...) discussed by Batterman. I show that we need not accept some divergence between explanation and reduction (Batterman 2002), or between epistemological and ontological fundamentalism (Batterman 2004, 2005). Though I remain sympathetic to the ‘principle of charity’ (Frisch (2005)), which appears to favor a pluralist outlook, I nevertheless call into question some of the forms such pluralist implications take in Robert Batterman’s conclusions. It is difficult to reconcile some of the pluralist assessments that he and some of his contemporaries advocate with what appears to be a countervailing trend in a burgeoning research tradition known as Clifford (or geometric) algebra. In my critical chapters (2 and 3) I use some of the demonstrated formal unity of Clifford algebra to argue that Batterman (2002) equivocates a physical theory’s ontology with its purely mathematical content. Carefully distinguishing the two, and employing Clifford algebraic methods reveals a symmetry between reduction and explanation that Batterman overlooks. I refine this point by indicating that geometric algebraic methods are an active area of research in computational fluid dynamics, and applied in modeling the behavior of droplet-formation appear to instantiate a “methodologically fundamental” approach. I argue in my introductory and concluding chapters that the model of inter-theoretic reduction and explanation offered by Fritz Rohrlich (1988, 1994) provides the best framework for accommodating the burgeoning pluralism in philosophical studies of physics, with the presumed claims of formal unification demonstrated by physicists choices of mathematical formalisms such as Clifford algebra. I show how Batterman’s insights can be reconstructed in Rohrlich’s framework, preserving Batterman’s important philosophical work, minus what I consider are his incorrect conclusions. (shrink)

This paper introduces the novel concept of Neutro-BCK-algebra. In Neutro-BCK-algebra, the outcome of any given two elements under an underlying operation (neutro-sophication procedure) has three cases, such as: appurtenance, non-appurtenance, or indeterminate.

Recent research on algebraic models of _quasi-Nelson logic_ has brought new attention to a number of classes of algebras which result from enriching (subreducts of) Heyting algebras with a special modal operator, known in the literature as a _nucleus_. Among these various algebraic structures, for which we employ the umbrella term _intuitionistic modal algebras_, some have been studied since at least the 1970s, usually within the framework of topology and sheaf theory. Others may seem more exotic, for their (...) primitive operations arise from algebraic terms of the intuitionistic modal language which have not been previously considered. We shall for instance investigate the variety of _weak implicative semilattices_, whose members are (non-necessarily distributive) meet semilattices endowed with a nucleus and an implication operation which is not a relative pseudo-complement but satisfies the postulates of Celani and Jansana’s strict implication. For each of these new classes of algebras we establish a representation and a topological duality which generalize the known ones for Heyting algebras enriched with a nucleus. (shrink)

Abstract. Let REL(O*E) be the relation algebra of binary relations defined on the Boolean algebra O*E of regular open regions of the Euclidean plane E. The aim of this paper is to prove that the canonical contact relation C of O*E generates a subalgebra REL(O*E, C) of REL(O*E) that has infinitely many elements. More precisely, REL(O*,C) contains an infinite family {SPPn, n ≥ 1} of relations generated by the relation SPP (Separable Proper Part). This relation can be used to define (...) point-free concept of connectedness that for the regular open regions of E coincides with the standard topological notion of connectedness, i.e., a region of the plane E is connected in the sense of topology if and only if it has no separable proper part. Moreover, it is shown that the contact relation algebra REL(O*E, C) and the relation algebra REL(O*E, NTPP) generated by the non-tangential proper parthood relation NTPP, coincide. This entails that the allegedly purely topological notion of connectedness can be defined in mereological terms. (shrink)

Neutrosophic theory and its applications have been expanding in all directions at an astonishing rate especially after of the introduction the journal entitled “Neutrosophic Sets and Systems”. New theories, techniques, algorithms have been rapidly developed. One of the most striking trends in the neutrosophic theory is the hybridization of neutrosophic set with other potential sets such as rough set, bipolar set, soft set, hesitant fuzzy set, etc. The different hybrid structures such as rough neutrosophic set, single valued neutrosophic rough set, (...) bipolar neutrosophic set, single valued neutrosophic hesitant fuzzy set, etc. are proposed in the literature in a short period of time. Neutrosophic set has been an important tool in the application of various areas such as data mining, decision making, e-learning, engineering, law, medicine, social science, and some more. -/- This book explores the emerging field of Neutrosophic Algebraic Structures, focusing on both their theoretical foundations and practical applications. We apply innovative algorithmic methods to investigate the complex interactions of neutrosophic elements, such as neutrosophic numbers, sets, and functions, within algebraic systems. Our goal is to show how neutrosophic structures challenge and expand traditional algebraic approaches, offering solutions to problems across diverse fields like computer science, engineering, artificial intelligence, and decision-making. (shrink)

The topics approached in the 52 papers included in this book are: neutrosophic sets; neutrosophic logic; generalized neutrosophic set; neutrosophic rough set; multigranulation neutrosophic rough set (MNRS); neutrosophic cubic sets; triangular fuzzy neutrosophic sets (TFNSs); probabilistic single-valued (interval) neutrosophic hesitant fuzzy set; neutro-homomorphism; neutrosophic computation; quantum computation; neutrosophic association rule; data mining; big data; oracle Turing machines; recursive enumerability; oracle computation; interval number; dependent degree; possibility degree; power aggregation operators; multi-criteria group decision-making (MCGDM); expert set; soft sets; LA-semihypergroups; single valued (...) trapezoidal neutrosophic number; inclusion relation; Q-linguistic neutrosophic variable set; vector similarity measure; fundamental neutro-homomorphism theorem; neutro-isomorphism theorem; quasi neutrosophic triplet loop; quasi neutrosophic triplet group; BE-algebra; cloud model; Maclaurin symmetric mean; pseudo-BCI algebra; hesitant fuzzy set; photovoltaic plan; decision-making trial and evaluation laboratory (DEMATEL); Choquet integral; fuzzy measure; clustering algorithm; and many more. (shrink)

The topics approached in this collection of papers are: neutrosophic sets; neutrosophic logic; generalized neutrosophic set; neutrosophic rough set; multigranulation neutrosophic rough set (MNRS); neutrosophic cubic sets; triangular fuzzy neutrosophic sets (TFNSs); probabilistic single-valued (interval) neutrosophic hesitant fuzzy set; neutro-homomorphism; neutrosophic computation; quantum computation; neutrosophic association rule; data mining; big data; oracle Turing machines; recursive enumerability; oracle computation; interval number; dependent degree; possibility degree; power aggregation operators; multi-criteria group decision-making (MCGDM); expert set; soft sets; LA-semihypergroups; single valued trapezoidal neutrosophic number; (...) inclusion relation; Q-linguistic neutrosophic variable set; vector similarity measure; fundamental neutro-homomorphism theorem; neutro-isomorphism theorem; quasi neutrosophic triplet loop; quasi neutrosophic triplet group; BE-algebra; cloud model; fuzzy measure; clustering algorithm; and many more. (shrink)

How is what we believe related to how we act? That depends on what we mean by ‘believe’. On the one hand, there is what we're sure of: what our names are, where we were born, whether we are sitting in front of a screen. Surety, in this sense, is not uncommon — it does not imply Cartesian absolute certainty, from which no possible course of experience could dislodge us. But there are many things that we think that we are (...) not sure of. For example, you might think that it will rain sometime this month, but not be sure that it will. Both what we're sure of and what we think have important normative connections to action. But the connections are quite different. This paper explores these issues with respect to assertion, inquiry, and decision making. We conclude by arguing that there is no theoretically significant notion of ‘full belief’ intermediate in strength between thinking and being sure. (shrink)

In this book the authors introduced the notions of soft neutrosophic algebraic structures. These soft neutrosophic algebraic structures are basically defined over the neutrosophic algebraic structures which means a parameterized collection of subsets of the neutrosophic algebraic structure. For instance, the existence of a soft neutrosophic group over a neutrosophic group or a soft neutrosophic semigroup over a neutrosophic semigroup, or a soft neutrosophic field over a neutrosophic field, or a soft neutrosophic LA-semigroup over a neutrosophic (...) LAsemigroup, or a soft neutosophic loop over a neutrosophic loop. It is interesting to note that these notions are defined over finite and infinite neutrosophic algebraic structures. These structures are even bigger than the classical algebraic structures. This book contains five chapters. Chapter one is about the introductory concepts. In chapter two the notions of soft neutrosophic group, soft neutrosophic bigroup and soft neutrosophic N-group are introduced and many fantastic properties are given with illustrative examples. (shrink)

In this paper we recall, improve, and extend several definitions, properties and applications of our previous 2019 research referred to NeutroAlgebras and AntiAlgebras (also called NeutroAlgebraic Structures and respectively AntiAlgebraic Structures). Let <A> be an item (concept, attribute, idea, proposition, theory, etc.). Through the process of neutrosphication, we split the nonempty space we work on into three regions {two opposite ones corresponding to <A> and <antiA>, and one corresponding to neutral (indeterminate) <neutA> (also denoted <neutroA>) between the opposites}, which may (...) or may not be disjoint – depending on the application, but they are exhaustive (their union equals the whole space). A NeutroAlgebra is an algebra which has at least one NeutroOperation or one NeutroAxiom (axiom that is true for some elements, indeterminate for other elements, and false for the other elements). A Partial Algebra is an algebra that has at least one Partial Operation, and all its Axioms are classical (i.e. axioms true for all elements). Through a theorem we prove that NeutroAlgebra is a generalization of Partial Algebra, and we give examples of NeutroAlgebras that are not Partial Algebras. We also introduce the NeutroFunction (and NeutroOperation). (shrink)

This is the full version of the Delphi Report on critical thinking and critical thinking instruction at the post-secondary level.

The aim of this paper is to show that every topological space gives rise to a wealth of topological models of the modal logic S4.1. The construction of these models is based on the fact that every space defines a Boolean closure algebra (to be called a McKinsey algebra) that neatly reflects the structure of the modal system S4.1. It is shown that the class of topological models based on McKinsey algebras contains a canonical model that can be used to (...) prove a completeness theorem for S4.1. Further, it is shown that the McKinsey algebra MKX of a space X endoewed with an alpha-topologiy satisfies Esakia's GRZ axiom. (shrink)

In this book, the authors introduce the notion of Super linear algebra and super vector spaces using the definition of super matrices defined by Horst (1963). Many theorems on super linear algebra and its properties are proved. Some theorems are left as exercises for the reader. These new class of super linear algebras which can be thought of as a set of linear algebras, following a stipulated condition, will find applications in several fields using computers. The authors feel that such (...) a paradigm shift is essential in this computerized world. (shrink)

We propose an axiomatic approach to constructing the dynamics of systems, in which one the main elements 9e8 is the consciousness of a subject. The main axiom is the statements that the state of consciousness is completely determined by the results of measurements performed on it. In case of economic systems we propose to consider an offer of transaction as a fundamental measurement. Transactions with delayed choice, discussed in this paper, represent a logical generalization of incomplete transactions and allow for (...) a more rigorous approach to studying the properties of the algebra of economic measurements. Considering the behavior of an object as a sequence of actions and choices allows extending the obtained results to random systems, which include the subject's consciousness as one of its elements. The specifics, on which the description of fundamental measurements is based, allows easily transferring the formalism of Schwinger's theory of selective measurements (and the consequent quantum-mechanical formalism) to the economic and social systems. (shrink)

By the middle of the seventeenth century we that find that algebra is able to offer proofs in its own right. That is, by that time algebraic argument had achieved the status of proof. How did this transformation come about?

There are empirical grounds to doubt the effectiveness of a common and intuitive approach to teaching debiasing strategies in critical thinking courses. We summarize some of the grounds before suggesting a broader taxonomy of debiasing strategies. This four-level taxonomy enables a useful diagnosis of biasing factors and situations, and illuminates more strategies for more effective bias mitigation located in the shaping of situational factors and reasoning infrastructure—sometimes called “nudges” in the literature. The question, we contend, then becomes how best (...) to teach the construction and use of such infrastructures. (shrink)

We deepen the study of conjoined and disjoined conditional events in the setting of coherence. These objects, differently from other approaches, are defined in the framework of conditional random quantities. We show that some well known properties, valid in the case of unconditional events, still hold in our approach to logical operations among conditional events. In particular we prove a decomposition formula and a related additive property. Then, we introduce the set of conditional constituents generated by $n$ conditional events and (...) we show that they satisfy the basic properties valid in the case of unconditional events. We obtain a generalized inclusion-exclusion formula, which can be interpreted by introducing a suitable distributive property. Moreover, under logical independence of basic unconditional events, we give two necessary and sufficient coherence conditions. The first condition gives a geometrical characterization for the coherence of prevision assessments on a family F constituted by n conditional events and all possible conjunctions among them. The second condition characterizes the coherence of prevision assessments defined on $F\cup K$, where $K$ is the set of conditional constituents associated with the conditional events in $F$. Then, we give some further theoretical results and we examine some examples and counterexamples. Finally, we make a comparison with other approaches and we illustrate some theoretical aspects and applications. (shrink)

Why do people create extra representations to help them make sense of situations, diagrams, illustrations, instructions and problems? The obvious explanation— external representations save internal memory and com- putation—is only part of the story. I discuss seven ways external representations enhance cognitive power: they change the cost structure of the inferential landscape; they provide a structure that can serve as a shareable object of thought; they create persistent referents; they facilitate re- representation; they are often a more natural representation of (...) structure than mental representations; they facilitate the computation of more explicit encoding of information; they enable the construction of arbitrarily complex structure; and they lower the cost of controlling thought—they help coordinate thought. Jointly, these functions allow people to think more powerfully with external representations than without. They allow us to think the previously unthinkable. (shrink)

Here algebraic Routley-Meyer semantics is addressed for two fuzzy versions of the logic of relevant implication R. To this end, two versions R t and R T of R and their fuzzy extensions FRt and FRT , respectively, are first discussed together with their algebraic semantics. Next algebraic Routley-Meyer semantics for these two fuzzy extensions is introduced. Finally, it is verified that these logics are sound and complete over the semantics.

The objective of this paper is to study algebraic properties of neutrosophic matrices, where a necessary and sufficient condition for the invertibility of a square neutrosophic matrix is presented by defining the neutrosophic determinant. On the other hand, this work introduces the concept of neutrosophic Eigen values and vectors with an easy algorithm to compute them. Also, this article finds a necessary and sufficient condition for the diagonalization of a neutrosophic matrix.

In this book we define some new notions of soft neutrosophic algebraic structures over neutrosophic algebraic structures. We define some different soft neutrosophic algebraic structures but the main motivation is two-fold. Firstly the classes of soft neutrosophic group ring and soft neutrosophic semigroup ring defined in this book is basically the generalization of two classes of rings: neutrosophic group rings and neutrosophic semigroup rings. These soft neutrosophic group rings and soft neutrosophic semigroup rings are defined over neutrosophic (...) group rings and neutrosophic semigroup rings respectively. This is basically the collection of parameterized subneutrosophic group ring and subneutrosophic semigroup ring of a neutrosophic group ring and neutrosophic semigroup ring respectively. These soft neutrosophic algebraic structures are even bigger than the corresponding classical algebraic structures. It is interesting to see that it wider the area of research for the researcher of algebraic structures. (shrink)

Modifying the descriptive and theoretical generalizations of Relativized Minimality, we argue that a significant subset of weak island violations arise when an extracted phrase should scope over some intervener but is unable to. Harmless interveners seem harmless because they can support an alternative reading. This paper focuses on why certain wh-phrases are poor wide scope takers, and offers an algebraic perspective on scope interaction. Each scopal element SE is associated with certain operations (e.g., not with complements). When a wh-phrase (...) scopes over some SE, the operations associated with that SE are performed in its denotation domain. The requisite operations may or may not be available in a domain, however. We present an empirical analysis of a variety of wh-phrases. It is argued that the wh-phrases that escape all weak islands (i.e., can scope over any intervener) are those that range over individuals, the reason being that all Boolean operations are defined for their domain. Collectives, manners, amounts, numbers, etc. all denote in domains with fewer operations and are thus selectively sensitive to scopal interveners—a “semantic relativized minimality effect”. (shrink)

I argue that on Aristotle’s account practical thinking is thinking whose origin (archē) is a desire that has as its object the very thing that one reasons about how to promote. This feature distinguishes practical from productive reasoning since in the latter the desire that initiates it is not (unless incidentally) a desire for the object that one productively reasons about. The feature has several interesting consequences: (a) there is only a contingent relationship between the desire that one (...) practically reasons about how to satisfy and the action one decides on; (b) practical thinking and action cannot be separated from the agent, whereas productive thinking and production can be outsourced to someone else. The view has consequences also for the distinction between action and production. Finally, I illustrate the usefulness and correctness of my account of practical thinking by using it to shed new light on Aristotle’s claim that the virtuous agent must decide on her virtuous actions ‘for themselves’. (shrink)

Eugenic thinking involves distinguishing between sorts or kinds of people in terms of the perceived desirable or undesirable traits that those people are likely to transmit to future generations. While eugenics itself is often thought of as an ideology that generated a social movement of global influence from roughly 1900 to 1945, eugenic thinking both pre-dates this period and continues to inform a range of contemporary debates and social policies, including those concerning prenatal screening, transhumanism, population control, and (...) disability. Various sciences, including the then-nascent sciences of psychology and psychiatry, played important roles in early twentieth-century eugenics, and their legacy is reflected in contemporary eugenic thinking and policies. Understanding the history of eugenics and the role of these early sciences of cognition in that history is useful in reflecting further on ongoing forms of eugenic thinking to which the contemporary cognitive sciences are relevant. (shrink)

This paper defends the view, put roughly, that to think that p is to guess that p is the answer to the question at hand, and that to think that p rationally is for one’s guess to that question to be in a certain sense non-arbitrary. Some theses that will be argued for along the way include: that thinking is question-sensitive and, correspondingly, that ‘thinks’ is context-sensitive; that it can be rational to think that p while having arbitrarily low (...) credence that p; that, nonetheless, rational thinking is closed under entailment; that thinking does not supervene on credence; and that in many cases what one thinks on certain matters is, in a very literal sense, a choice. Finally, since there are strong reasons to believe that thinking just is believing, there are strong reasons to think that all this goes for belief as well. (shrink)