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)

This paper intends to further the understanding of the formal properties of (higher-order) vagueness by connecting theories of (higher-order) vagueness with more recent work in topology. First, we provide a “translation” of Bobzien's account of columnar higher-order vagueness into the logic of topological spaces. Since columnar vagueness is an essential ingredient of her solution to the Sorites paradox, a central problem of any theory of vagueness comes into contact with the modern mathematical theory of topology. Second, Rumfitt’s recent topological reconstruction (...) of Sainsbury’s theory of prototypically defined concepts is shown to lead to the same class of spaces that characterize Bobzien’s account of columnar vagueness, namely, weakly scattered spaces. Rumfitt calls these spaces polar spaces. They turn out to be closely related to Gärdenfors’ conceptual spaces, which have come to play an ever more important role in cognitive science and related disciplines. Finally, Williamson’s “logic of clarity” is explicated in terms of a generalized topology (“locology”) that can be considered an alternative to standard topology. Arguably, locology has some conceptual advantages over topology with respect to the conceptualization of a boundary and a borderline. Moreover, in Williamson’s logic of clarity, vague concepts with respect to a notion of a locologically inspired notion of a “slim boundary” are (stably) columnar. Thus, Williamson’s logic of clarity also exhibits a certain affinity for columnar vagueness. In sum, a topological perspective is useful for a conceptual elucidation and unification of central aspects of a variety of contemporary accounts of vagueness. (shrink)

nated discussion of the semantics of such verbs. I will call this view.

There is a fairly general consensus that names are Millian (or Russellian) genuine terms, that is, are singular terms whose sole semantic function is to introduce a referent into the propositions expressed by sentences containing the term. This answers the question as to what sort of proposition is expressed by use of sentences containing names. But there is a second serious semantic problem about proper names, that of how the referents of proper names are determined. This is the question that (...) I will discuss in this paper. Various views consistent with Millianism have been proposed as to how the semantic referents of proper names are determined. These views can be classified into (1) description theories and (2) causal theories, but they can also be classified into (3) social practice theories, on which a name’s referent is determined by a social practice involving the referent, and (4) individualistic theories, on which the referent of the use of a name is determined by the speaker’s state of mind. Here I argue against social practice theories of the sorts proposed by Kripke and Evans and in favor of an individualistic approach to name reference. I argue that social practice is irrelevant to determining name reference and that, as a consequence, names have no meanings in natural languages. In the second part of the paper I motivate and propose a new form of individualistic theory which incorporates features of both description theories and Evans’s social practice theory. (shrink)

In “Naming and Necessity” Saul Kripke describes some cases which, he claims, provide counterexamples both to cluster theories and, more generally, to description theories of proper names. My view of these cases is that while they do not provide counterexamples to cluster theories, they can be used to provide evidence against single-description theories. In this paper I shall defend both of the claims involved in my view.

Abstract. On the Direct Reference thesis, proper names are what I call ‘genuine terms’, terms whose sole semantic contributions to the propositions expressed by their use are the terms’ semantic referents. But unless qualified, this thesis implies the false consequence that sentences containing names that fail to refer can never express true or false propositions. (Consider ‘The ancient Greeks worshipped Zeus’, for instance.) I suggest that while names are typically and fundamentally used as genuine terms, there is a small class (...) of names, which I call ‘descriptive names’, whose reference is fixed by commonly associated definite descriptions, and I also suggest that there is an idiom of natural language on which such names can be used as abbreviated definite descriptions in a limited set of sentential contexts, including (1) positive and negative existentials, (2) cognitive ascriptions, and (3) uses of names to talk about myth. Uses of empty descriptive names in such contexts can then be either true or false. Relying on Gregory Currie’s theory of truth in fiction, I also propose an idiom on which fictional names can be used as short for a certain type of description in talk about fiction. Along the way, I provide arguments that names are used as short for descriptions in substantive existential statements as well as in both metamythic and metafictive contexts. I also discuss and criticize alternative views of these matters, including the views of David Braun, Saul Kripke, Peter van Inwagen, and others. (shrink)

Gareth evans has proposed a type of case which shows that kripke's sketch of a causal theory of proper names is in need of modification. Kripke has himself suggested a way in which the modification might proceed, But I argue that this suggestion leads in the wrong direction. I consider a development of kripke's view by michael devitt which may overcome evans' case, But which is shown false by a different sort of case. The latter kind of case also shows (...) that a view of names recently proposed by donnellan is in need of revision. (shrink)

In my 1991 paper, AAnti-Individualism and Privileged Access,@ I argued that externalism in the philosophy of mind is incompatible with the thesis that we have privileged , nonempirical access to the contents of our own thoughts.¹ One of the most interesting responses to my argument has been that of Martin Davies (1998, 2000, and Chapter _ above) and Crispin Wright (2000 and Chapter _ above), who describe several types of cases to show that warrant for a premise does not always (...) transmit to a known deductive consequence of that premise, and who contend that this fact under-mines my argument for incompatibilism. I will try to show here that the Davies/Wright point about transmission of warrant does not adversely affect my argument. (shrink)

According to grice, Semantic concepts like meaning and reference should be explicated in terms of the propositional attitudes. In this paper, I argue that grice's program is mistaken in principle. I first motivate a gricean strategy for defining denotation, Or semantic reference, In terms of rules that govern what speakers may refer to with the terms they use. I then express three paradigm gricean theories of denotation and introduce considerations which show that these theories are false.

1. The primary evidence and motivation for externalism in the philosophy of mind is provided by the semantic facts that support direct reference theories of names, indexi- cal pronouns, and natural kind terms. But many externalists have forgotten their sem- antic roots, or so I shall contend here. I have become convinced of this by a common reaction among externalists to the main argument of my 1991 paper AAnti-Individual- ism and Privileged Access.@ In that argument, I concluded that externalism is (...) incompat- ible with the principle that we can have privileged, non-empirical knowledge of the contents of our own thoughts. The reaction in question amounts to a dismissive denial of one of my argument=s main premises. This premise, which I defended at length in the paper, is that an externalist thesis regarding a cognitive property should hold that possession of the property by a person _logically_, or _conceptually_, implies the existence of objects external to that person. (shrink)

Searle has proposed a "presupposition-Theory" of proper names in which he maintains that names are not short for descriptions and which, He claims, Solves frege's puzzle as to how an identity-Sentence containing co-Referential names can be informative. Two possible interpretations of searle's view are proposed, And it is argued that neither interpretation can be used to solve frege's puzzle and that, On the most plausible interpretation of his view, Searle is committed to the thesis that names are short for descriptions (...) after all. (shrink)

Tyler Burge's (1979) famous thought experiment concerning 'arthritis' is commonly assumed to show that all ascriptions of content to beliefs and other attitudes are dependent for their truth upon facts about the agent's social and linguistic environment. It is also commonly claimed that Burge's argument shows that Putnam's (1975) result regarding natural kind terms applies to all general terms whatever, and hence shows that all such terms have wide meanings.1 But I wish to show here, first, that neither Burge's initial (...) thought experiment nor a second type of example that Burge describes supports either of these conclusions. Second, I will identify the proper conclusion to draw from Burge's discussion and show that this conclusion does not really pose a serious problem for individualism about the mental. And finally, I will argue that Burge's discussion does not in fact provide a conclusive reason for believing its proper conclusion. (shrink)

In this admirable book, Scott Soames provides well defended answers to some of the most difficult and important questions in the philosophy of language, and he does so with characteristic thoroughness, clarity, and rigor. The book's title is appropriate, since it does indeed go ‘beyond rigidity’ in many ways. Among other things, Soames does the following in the course of the book. He persuasively argues that the main thesis of Kripke's Naming and Necessity—that ordinary names are rigid designators—can be extended (...) to the more general thesis that simple proper names are synonymous with neither nonrigid nor rigidified descriptions, and so have no descriptive content whatever. He thoroughly defends and places in the context of a larger semantic theory the Millian thesis that the sole semantic contents of most proper names are the names' referents, so that sentences containing such names semantically express singular Russellian propositions. He provides and defends at length an innovative pragmatic account of why substitution of co-referring proper names in cognitive contexts intuitively fails to be truth preserving, even though it seems to follow from the Millian thesis that such substitution must be semantically valid. And he thoroughly and persuasively defends a semantic theory of natural kind terms that explains how, following Kripke and Putnam, theoretical identities involving such terms can express a posteriori necessities, even though on Soames's theory, and contrary to the Kripke-Putnam view, such terms are neither names of natural kinds nor rigid designators in any interesting sense. (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.

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)

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.

We investigate the algebraic structures of square grid graphs and obtain isomorphism theorems.

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.

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)

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)

En décadas recientes se ha considerado que una formulación adecuada del autoconocimiento (AC) debería ser consistente con la tesis del externalismo (E). Michael McKinsey es uno de los personajes que ha enfatizado con mayor ahínco que la conjunción de ambas posturas es inconsistente. En este trabajo defiendo la idea de que las objeciones presentadas por McKinsey no afectan de manera importante la formulación davidsoniana de la autoridad de la primera persona (AC3) en conjunción con (E); señalo, además, que (...) si no pudiera hacerse una reconstrucción à la Davidson de (AC3) desde la postura de McKinsey sería en virtud de las diferencias metodológicas, semánticas, epistemológicas y ontológicas entre ambos autores, y no debido a que esta reconstrucción en sí misma sea incompatible con el externalismo. (shrink)

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)

In this paper I argue first, that a contrastive account of self-knowledge and the propositional attitudes entails an anti-individualist account of propositional attitude concepts, second, that the final account provides a solution to the McKinsey paradox, and third, that the account has the resources to explain why certain anti-skeptical arguments fail.

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)

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)

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.

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)

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)

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)

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)

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)