In this paper, we discuss some questions about compactness in MV-topological spaces. More precisely, we first present a Tychonoff theorem for such a class of fuzzy topological spaces and some consequence of this result, among which, for example, the existence of products in the category of Stone MV-spaces and, consequently, of coproducts in the one of limit cut complete MV-algebras. Then we show that our Tychonoff theorem is equivalent, in ZF, to the Axiom of Choice, classical Tychonoff theorem, and (...) Lowen’s analogous result for lattice-valued fuzzy topology. Last, we show an extension of the Stone–Čech compactification functor to the category of MV-topological spaces, and we discuss its relationship with previous works on compactification for fuzzy topological spaces. (shrink)

The standard presentation of topological spaces relies heavily on (naïve) set theory: a topology consists of a set of subsets of a set (of points). And many of the high-level tools of set theory are required to achieve just the basic results about topological spaces. Concentrating on the mathematical structures, category theory offers the possibility to look synthetically at the structure of continuous transformations between topological spaces addressing specifically how the fundamental notions of point and open come about. As (...) a byproduct of this, one may look at the different approaches to topology from an external perspective and compare them in a unified way. Technically, the category of sober topological spaces can be seen as consisting of (co)algebraic structures in the exact completion of the elementary category of sets and relations. Moreover, the same abstract construction of taking the exact completion, when applied to the category of topological spaces and continuous functions produces an extension of it which is cartesian closed. In other words, there is one general mathematical construction that, when applied to a very elementary category, generates the category of topological spaces and continuous functions, and when applied to that category produces a very suitable category where to deal with all sorts functions spaces. Yet, via such free constructions it is possible to give a new meaning to Marshall Stone’s dictum: “always topologize” as the category of sets and relations is the most natural way to give structure to logic and the category of topological spaces and continuous functions is obtained from it by a good mix of free – i.e. syntactic – constructions. (shrink)

The standard topological representation of a Boolean algebra via the clopen sets of a Stone space requires a nonconstructive choice principle, equivalent to the Boolean Prime Ideal Theorem. In this article, we describe a choice-free topological representation of Boolean algebras. This representation uses a subclass of the spectral spaces that Stone used in his representation of distributive lattices via compact open sets. It also takes advantage of Tarski’s observation that the regular open sets of any topological space form (...) a Boolean algebra. We prove without choice principles that any Boolean algebra arises from a special spectral space X via the compact regular open sets of X; these sets may also be described as those that are both compact open in X and regular open in the upset topology of the specialization order of X, allowing one to apply to an arbitrary Boolean algebra simple reasoning about regular opens of a separative poset. Our representation is therefore a mix of Stone and Tarski, with the two connected by Vietoris: the relevant spectral spaces also arise as the hyperspace of nonempty closed sets of a Stone space endowed with the upper Vietoris topology. This connection makes clear the relation between our point-set topological approach to choice-free Stone duality, which may be called the hyperspace approach, and a point-free approach to choice-free Stone duality using Stone locales. Unlike Stone’s representation of Boolean algebras via Stone spaces, our choice-free topological representation of Boolean algebras does not show that every Boolean algebra can be represented as a field of sets; but like Stone’s representation, it provides the benefit of a topological perspective on Boolean algebras, only now without choice. In addition to representation, we establish a choice-free dual equivalence between the category of Boolean algebras with Boolean homomorphisms and a subcategory of the category of spectral spaces with spectral maps. We show how this duality can be used to prove some basic facts about Boolean algebras. (shrink)

Since antiquity well into the beginnings of the 20th century geometry was a central topic for philosophy. Since then, however, most philosophers of science, if they took notice of topology at all, considered it as an abstruse subdiscipline of mathematics lacking philosophical interest. Here it is argued that this neglect of topology by philosophy may be conceived of as the sign of a conceptual sea-change in philosophy of science that expelled geometry, and, more generally, mathematics, from the central (...) position it used to have in philosophy of science and placed logic at center stage in the 20th century philosophy of science. Only in recent decades logic has begun to loose its monopoly and geometry and topology received a new chance to find a place in philosophy of science. (shrink)

Given an abstract logic L = L(Q i ) i ∈ I generated by a set of quantifiers Q i , one can construct for each type τ a topological space S τ exactly as one constructs the Stone space for τ in first-order logic. Letting T be an arbitrary directed set of types, the set $S_T = \{(S_\tau, \pi^\tau_\sigma)\mid\sigma, \tau \in T, \sigma \subset \tau\}$ is an inverse topological system whose bonding mappings π τ σ are naturally determined (...) by the reduct operation on structures. We relate the compactness of L to the topological properties of S T . For example, if I is countable then L is compact iff for every τ each clopen subset of S τ is of finite type and S τ is homeomorphic to $\underset{lim}S_T$ , where T is the set of finite subtypes of τ. We finally apply our results to concrete logics. (shrink)

As McKinsey and Tarski showed, the Stone representation theorem for Boolean algebras extends to algebras with operators to give topological semantics for propositional modal logic, in which the “necessity” operation is modeled by taking the interior of an arbitrary subset of a topological space. In this article, the topological interpretation is extended in a natural way to arbitrary theories of full first-order logic. The resulting system of S4 first-order modal logic is complete with respect to such topological semantics.

This paper explores relationships between many-valued logic and fuzzy topology from the viewpoint of duality theory. We first show a fuzzy topological duality for the algebras of Łukasiewicz n -valued logic with truth constants, which generalizes Stone duality for Boolean algebras to the n -valued case via fuzzy topology. Then, based on this duality, we show a fuzzy topological duality for the algebras of modal Łukasiewicz n -valued logic with truth constants, which generalizes Jónsson-Tarski duality for modal (...) algebras to the n -valued case via fuzzy topology. We emphasize that fuzzy topological spaces naturally arise as spectrums of algebras of many-valued logics. (shrink)

We show that if we interpret modal diamond as the derived set operator of a topological space, then the modal logic of Stone spaces isK4and the modal logic of weakly scattered Stone spaces isK4G. As a corollary, we obtain thatK4is also the modal logic of compact Hausdorff spaces andK4Gis the modal logic of weakly scattered compact Hausdorff spaces.

We offer a topological treatment of scattered theories intended to help to explain the parallelism between, on the one hand, the theorems provable using Descriptive Set Theory by analysis of the space of countable models and, on the other, those provable by studying a tree of theories in a hierarchy of fragments of infinintary logic. We state some theorems which are, we hope, a step on the road to fully understanding counterexamples to Vaught's Conjecture. This framework is in the early (...) stages of development, and one area for future exploration is the possibility of extending it to a setting in which the spaces of types of a theory are uncountable. (shrink)

In this paper, we study bounded versions of some model-theoretic notions and results. We apply these results to the context of models of bounded arithmetic theories as well as some related complexity questions. As an example, we show that if the theory \(\rm S_2 ^1(PV)\) has bounded model companion then \(\rm NP=coNP\). We also study bounded versions of some other related notions such as Stone topology.

Let T be a simple L-theory and let \ be a reduct of T to a sublanguage \ of L. For variables x, we call an \-invariant set \\) in \ a universal transducer if for every formula \\in L^-\) and every a, $$\begin{aligned} \phi ^-\ L^-\text{-forks } \text{ over }\ \emptyset \ \text{ iff } \Gamma \wedge \phi ^-\ L\text{-forks } \text{ over }\ \emptyset. \end{aligned}$$We show that there is a greatest universal transducer \ and it is type-definable. In (...) particular, the forking topology on \\) refines the forking topology on \\) for all y. Moreover, we describe the set of universal transducers in terms of certain topology on the Stone space and show that \ is the unique universal transducer that is \-type-definable with parameters. If \ is a theory with the wnfcp and T is the theory of its lovely pairs of models we show that \\) and give a more precise description of the set of universal transducers for the special case where \ has the nfcp. (shrink)

For the modal logic |$\textsf {S4}^{C}_{I}$|⁠, we identify the class of completable |$\textsf {S4}^{C}_{I}$|-algebras and prove for them a Stone-type representation theorem. As a consequence, we obtain strong algebraic and topological completeness of the logic |$\textsf {S4}^{C}_{I}$| in the case of local semantic consequence relations. In addition, we consider an extension of the logic |$\textsf {S4}^{C}_{I}$| with certain infinitary derivations and establish the corresponding strong completeness results for the enriched system in the case of global semantic consequence relations.

The non‐forking‐instances topology (NFI topology) is a topology on the Stone space of a theory T that depends on a reduct of T. This topology has been used in [6] to describe the set of universal transducers for (invariants sets that translate forking‐open sets in to forking‐open sets in T). In this paper we show that in contrast to the stable case, the NFI topology need not be invariant over parameters in but a weak (...) version of this holds for any simple T. We also note that for the lovely pair expansions, of theories with the weak non‐finite cover property (wnfcp), the topology is invariant over in. (shrink)

Stone representation theorems are a central ingredient in the metatheory of philosophical logics and are used to establish modal embedding results in a general but indirect and non-constructive way. Their use in logical embeddings will be reviewed and it will be shown how they can be circumvented in favour of direct and constructive arguments through the methods of analytic proof theory, and how the intensional part of the representation results can be recovered from the syntactic proof of those embeddings. (...) Analytic methods will also be used to establish the embedding of subintuitionistic logics into the corresponding modal logics. Finally, proof-theoretic embeddings will be interpreted as a reduction of classes of word problems. (shrink)

The paper analyzes dynamic epistemic logic from a topological perspective. The main contribution consists of a framework in which dynamic epistemic logic satisfies the requirements for being a topological dynamical system thus interfacing discrete dynamic logics with continuous mappings of dynamical systems. The setting is based on a notion of logical convergence, demonstratively equivalent with convergence in Stone topology. Presented is a flexible, parametrized family of metrics inducing the latter, used as an analytical aid. We show maps induced (...) by action model transformations continuous with respect to the Stone topology and present results on the recurrent behavior of said maps. (shrink)

It is a classic result in modal logic, often referred to as Jónsson-Tarski duality, that the category of modal algebras is dually equivalent to the category of descriptive frames. The latter are Kripke frames equipped with a Stone topology such that the binary relation is continuous. This duality generalizes the celebrated Stone duality for boolean algebras. Our goal is to generalize descriptive frames so that the topology is an arbitrary compact Hausdorff topology. For this, instead (...) of working with the boolean algebra of clopen subsets of a Stone space, we work with the ring of continuous real-valued functions on a compact Hausdorff space. The main novelty is to define a modal operator on such a ring utilizing a continuous relation on a compact Hausdorff space.Our starting point is the well-known Gelfand duality between the category ${\sf KHaus}$ of compact Hausdorff spaces and the category $\boldsymbol {\mathit {uba}\ell }$ of uniformly complete bounded archimedean $\ell $ -algebras. We endow a bounded archimedean $\ell $ -algebra with a modal operator, which results in the category $\boldsymbol {\mathit {mba}\ell }$ of modal bounded archimedean $\ell $ -algebras. Our main result establishes a dual adjunction between $\boldsymbol {\mathit {mba}\ell }$ and the category ${\sf KHF}$ of what we call compact Hausdorff frames; that is, Kripke frames equipped with a compact Hausdorff topology such that the binary relation is continuous. This dual adjunction restricts to a dual equivalence between ${\sf KHF}$ and the reflective subcategory $\boldsymbol {\mathit {muba}\ell }$ of $\boldsymbol {\mathit {mba}\ell }$ consisting of uniformly complete objects of $\boldsymbol {\mathit {mba}\ell }$. This generalizes both Gelfand duality and Jónsson-Tarski duality. (shrink)

This thesis is on information dynamics modeled using *dynamic epistemic logic*. It takes the simple perspective of identifying models with maps, which under a suitable topology may be analyzed as *topological dynamical systems*. It is composed of an introduction and six papers. The introduction situates DEL in the field of formal epistemology, exemplifies its use and summarizes the main contributions of the papers.Paper I models the information dynamics of the *bystander effect* from social psychology. It shows how augmenting the (...) standard machinery of DEL with a decision making framework yields mathematically self-contained models of dynamic processes, a prerequisite for rigid model comparison.Paper II extrapolates from Paper I's construction, showing how the augmentation and its natural peers may be construed as maps. It argues that under the restriction of dynamics produced by DEL dynamical systems still falls a collection rich enough to be of interest. Paper III compares the approach of Paper II with *extensional protocols*, the main alternative augmentation to DEL. It concludes that both have benefits, depending on application. In favor of the DEL dynamical systems, it shows that extensional protocols designed to mimic simple, DEL dynamical systems require infinite representations. Paper IV focuses on *topological dynamical systems*. It argues that the *Stone topology* is a natural topology for investigating logical dynamics as, in it, *logical convergence* coinsides with topological convergence. It investigates the recurrent behavior of the maps of Papers II and III, providing novel insigths on their long-term behavior, thus providing a proof of concept for the approach.Paper V lays the background for Paper IV, starting from the construction of metrics generalizing the Hamming distance to infinite strings, inducing the Stone topology. It shows that the Stone topology is unique in making logical and topological convergens coinside, making it the natural topology for logical dynamics. It further includes a metric-based proof that the hitherto analyzed maps are continuous with respect to the Stone topology. Paper VI presents two characterization theorems for the existence of *reduction laws*, a common tool in obtaining complete dynamic logics. In the compact case, continuity in the Stone topology characterizes existence, while a strengthening is required in the non-compact case. The results allow the recasting of many logical dynamics of contemporary interest as topological dynamical systems. (shrink)

We give a characterization of the simple, and of the subdirectly irreducible boolean algebras with operators (including modal algebras), in terms of the dual descriptive frame, or, topological relational structure. These characterizations involve a special binary topo-reachability relation on the dual structure; we call a point u a topo-root of the dual structure if every ultrafilter is topo-reachable from u. We prove that a boolean algebra with operators is simple iff every point in the dual structure is a topo-root; and (...) that it is subdirectly irreducible iff the collection of topo-roots is open and non-empty in the Stone topology on the dual structure iff this collection has non-empty interior in that topology. (shrink)

Let ${\beta(\mathbb{N})}$ denote the Stone–Čech compactification of the set ${\mathbb{N}}$ of natural numbers (with the discrete topology), and let ${\mathbb{N}^\ast}$ denote the remainder ${\beta(\mathbb{N})-\mathbb{N}}$ . We show that, interpreting modal diamond as the closure in a topological space, the modal logic of ${\mathbb{N}^\ast}$ is S4 and that the modal logic of ${\beta(\mathbb{N})}$ is S4.1.2.

For a signature L with at least one constant symbol, an L‐structure is called minimal if it has no proper substructures. Let be the set of isomorphism types of minimal L‐structures. The elements of can be identified with ultrafilters of the Boolean algebra of quantifier‐free L‐sentences, and therefore one can define a Stone topology on. This topology on generalizes the topology of the space of n‐marked groups. We introduce a natural ultrametric on, and show that the (...) Stone topology on coincides with the topology of the ultrametric space iff the ultrametric space is compact iff L is locally finite (that is, L contains finitely many n‐ary symbols for any ). As one of the applications of compactness of the Stone topology on, we prove compactness of certain classes of metric spaces in the Gromov‐Hausdorff topology. This slightly refines the known result based on Gromov's ideas that any uniformly totally bounded class of compact metric spaces is precompact. (shrink)

I present solutions to several questions of Paul Bankston [2] by means of another version of the ultracoproduct construction, and explain the relation of ultracoproduct of compact Hausdorff spaces to other constructions combining topology, algebra and logic.

In this note we introduce and study algebras of type such that is a bounded distributive lattice and ⌝ is an operator that satisfies the condition ⌝ = a ⌝ b and ⌝ 0 = 1. We develop the topological duality between these algebras and Priestley spaces with a relation. In addition, we characterize the congruences and the subalgebras of such an algebra. As an application, we will determine the Priestley spaces of quasi-Stone algebras.

In The Boundary Stones of Thought, Rumfitt defends classical logic against challenges from intuitionistic mathematics and vagueness, using a semantics of pre-topologies on possibilities, and a topological semantics on predicates, respectively. These semantics are suggestive but the characterizations of negation face difficulties that may undermine their usefulness in Rumfitt’s project.

The centrality of the whole/part relation in mathematics is demonstrated through the presentation and analysis of examples from algebra, geometry, functional analysis,logic, topology and category theory.

In traditional semantics for classical logic and its extensions, such as modal logic, propositions are interpreted as subsets of a set, as in discrete duality, or as clopen sets of a Stone space, as in topological duality. A point in such a set can be viewed as a "possible world," with the key property of a world being primeness—a world makes a disjunction true only if it makes one of the disjuncts true—which classically implies totality—for each proposition, a world (...) either makes the proposition true or makes its negation true. This chapter surveys a more general approach to logical semantics, known as possibility semantics, which replaces possible worlds with possibly partial "possibilities." In classical possibility semantics, propositions are interpreted as regular open sets of a poset, as in set-theoretic forcing, or as compact regular open sets of an upper Vietoris space, as in the recent theory of "choice-free Stone duality." The elements of these sets, viewed as possibilities, may be partial in the sense of making a disjunction true without settling which disjunct is true. We explain how possibilities may be used in semantics for classical logic and modal logics and generalized to semantics for intuitionistic logics. The goals are to overcome or deepen incompleteness results for traditional semantics, to avoid the nonconstructivity of traditional semantics, and to provide richer structures for the interpretation of new languages. (shrink)

The interplay between ultrafilters and unbounded subsets of \ with the order \ of strict eventual domination is studied. Among the tools are special kinds of non-principal ultrafilters on \. These include simple P-points; that is, ultrafilters with a base that is well-ordered with respect to the reverse of the order \ of almost inclusion. It is shown that the cofinality of such a base must be either \, the least cardinality of \-unbounded set, or \, the least cardinality of (...) a \-cofinal set. The small uncountable cardinal \ is introduced. Consequences of \ and of \ are explored; in particular, both imply \. Here \ is the reaping number, and is also the least cardinality of a \-base for a free ultrafilter. Both of these inequalities are shown to occur if there exist simple P-points of different cofinalities and there exist simple \-points and \-points), but this is a long-standing open problem. Six axioms on nonprincipal ultrafilters on \ and the relationships between them are discussed along with various models of set theory in which one or more are known to hold. The strongest of these, Axiom 1, is that for every free ultrafilter \ and for every \-unbounded \-chain C of increasing functions in \, C is also unbounded in the ultraproduct \. The other axioms replace one or both quantifiers with “there exists.” The negation of Axiom 3 in a model provides a family of normal sequentially compact spaces whose product is not countably compact. The question of whether such a family exists in ZFC, even with “normal” weakened to “regular”, is a famous unsolved problem of set-theoretic topology, known as the Scarborough–Stone problem. (shrink)

The general framework of this paper is a reformulation of Hilbert’s program using the theory of locales, also known as formal or point-free topology [P.T. Johnstone, Stone Spaces, in: Cambridge Studies in Advanced Mathematics, vol. 3, 1982; Th. Coquand, G. Sambin, J. Smith, S. Valentini, Inductively generated formal topologies, Ann. Pure Appl. Logic 124 71–106; G. Sambin, Intuitionistic formal spaces–a first communication, in: D. Skordev , Mathematical Logic and its Applications, Plenum, New York, 1987, pp. 187–204]. Formal (...) class='Hi'>topology presents a topological space, not as a set of points, but as a logical theory which describes the lattice of open sets. The application to Hilbert’s program is then the following. Hilbert’s ideal objects are represented by points of such a formal space. There are general methods to “eliminate” the use of points, close to the notion of forcing and to the “elimination of choice sequences” in intuitionist mathematics, which correspond to Hilbert’s required elimination of ideal objects. This paper illustrates further this general program on the notion of valuations. They were introduced by Dedekind and Weber [R. Dedekind, H. Weber, Theorie des algebraischen Funktionen einer Veränderlichen, J. de Crelle t. XCII 181–290] to give a rigorous presentation of Riemann surfaces. It can be argued that it is one of the first example in mathematics of point-free representation of spaces [N. Bourbaki, Eléments de Mathématique. Algèbre commutative, Hermann, Paris, 1965, Chapitre 7]. It is thus of historical and conceptual interest to be able to represent this notion in formal topology. (shrink)

Part I of this paper is developed in the tradition of Stone-type dualities, where we present a new topological representation for general lattices (influenced by and abstracting over both Goldblatt's [17] and Urquhart's [46]), identifying them as the lattices of stable compact-opens of their dual Stone spaces (stability refering to a closure operator on subsets). The representation is functorial and is extended to a full duality.In part II, we consider lattice-ordered algebras (lattices with additional operators), extending the Jónsson (...) and Tarski representation results [30] for Boolean algebras with Operators. Our work can be seen as developing, and indeed completing, Dunn's project of gaggle theory [13, 14]. We consider general lattices (rather than Boolean algebras), with a broad class of operators, which we dubb normal, and which includes the Jónsson-Tarski additive operators. Representation of l-algebras is extended to full duality. (shrink)

From a logical point of view, Stone duality for Boolean algebras relates theories in classical propositional logic and their collections of models. The theories can be seen as presentations of Boolean algebras, and the collections of models can be topologized in such a way that the theory can be recovered from its space of models. The situation can be cast as a formal duality relating two categories of syntax and semantics, mediated by homming into a common dualizing object, in (...) this case 2.In the present work, we generalize the entire arrangement from propositional to first-order logic, using a representation result of Butz and Moerdijk. Boolean algebras are replaced by Boolean categories presented by theories in first-order logic, and spaces of models are replaced by topological groupoids of models and their isomorphisms. A duality between the resulting categories of syntax and semantics, expressed primarily in the form of a contravariant adjunction, is established by homming into a common dualizing object, now Sets, regarded once as a boolean category, and once as a groupoid equipped with an intrinsic topology.The overall framework of our investigation is provided by topos theory. Direct proofs of the main results are given, but the specialist will recognize toposophical ideas in the background. Indeed, the duality between syntax and semantics is really a manifestation of that between algebra and geometry in the two directions of the geometric morphisms that lurk behind our formal theory. Along the way, we give an elementary proof of Butz and Moerdijkʼs result in logical terms. (shrink)

Ludics [1] is a novel approach to logic—especially proof-theory. The present introduction emphasises foundational issues.For ages, not a single disturbing idea in the area of “foundations”: the discussion is sort of ossified—as if everything had been said, as if all notions had taken their definite place, in a big cemetery of ideas. One can still refresh the flowers or regild the stone, e.g., prove technicalities, sometimes non-trivial; but the real debate is still: this paper begins with an autopsy, the (...) autopsy of the foundational project.Up to say 1900, the realist/dualist approach to science was dominant; during the last century some domains like physics evolved so as to become completely anti-realist; but this evolution hardly concerned logic.By the turn of the XXth century mathematics was jeopardised by paradoxes, the most famous of them being due to Russell. Hilbert's reaction was to focus on consistency. But the reduction of paradoxes—and therefore of foundations—to solely the antinomies is highly questionable: indeed, the typical paradoxical artifacts are secret sharers, objects satisfying the formal definitions but far astray from the intended meaning, typically the Peano “curve” which contradicts our perception of dimension. Fortunately, topology has been able to show that dimension m is not the same as dimension n … but just for a second, forget this and imagine consistent mathematics in which balls in any dimension are homeomorphic: what a disaster! This exclusive focus on consistency—not to speak of the strategic failure of the Programme—should explain why logic, especially foundations lost contact with other sciences during last century. (shrink)

We investigate mathematical structures that provide natural semantics for families of (quantified) non-classical logics featuring special unary connectives, known as recovery operators, that allow us to ‘recover’ the properties of classical logic in a controlled manner. These structures are known as topological Boolean algebras, which are Boolean algebras extended with additional operations subject to specific conditions of a topological nature. In this study, we focus on the paradigmatic case of negation. We demonstrate how these algebras are well-suited to provide a (...) semantics for some families of paraconsistent Logics of Formal Inconsistency and paracomplete Logics of Formal Undeterminedness. These logics feature recovery operators used to earmark propositions that behave ‘classically’ when interacting with non-classical negations. Unlike traditional semantical investigations, which are carried out in natural language (extended with mathematical shorthand), our formal meta-language is a system of higher-order logic (HOL) for which automated reasoning tools exist. In our approach, topological Boolean algebras are encoded as algebras of sets via their Stone-type representation. We use our higher-order meta-logic to define and interrelate several transformations on unary set operations, which naturally give rise to a topological cube of opposition. Additionally, our approach enables a uniform characterization of propositional, first-order and higher-order quantification, including restrictions to constant and varying domains. With this work, we aim to make a case for the utilization of automated theorem proving technology for conducting computer-supported research in non-classical logics. All the results presented in this paper have been formally verified, and in many cases obtained, using the Isabelle/HOL proof assistant. (shrink)

In this paper, inspired by methods of Bigard, Keimel, and Wolfenstein , we develop an approach to sheaf representations of MV-algebras which combines two techniques for the representation of MV-algebras devised by Filipoiu and Georgescu and by Dubuc and Poveda . Following Davey approach , we use a subdirect representation of MV-algebras that is based on local MV-algebras. This allowed us to obtain: a representation of any MV-algebras as MV-algebra of all global sections of a sheaf of local MV-algebras on (...) the spectruum of its prime ideals; a representation of MV-algebras, having the space of minimal prime ideals compact, as MV-algebra of all global sections of a Hausdorff sheaf of MV-chains on the space of minimal prime ideals, which is a Stone space; an adjunction between the category of all MV-algebras and the category of MV-algebraic spaces, where an MV-algebraic space is a pair , where X is a compact topological space and F is a sheaf of MV-algebras with stalks that are local. (shrink)

Recently, two apparently quite different duality-based approaches to automata minimisation have appeared. One is based on ideas that originated from the controllability-observability duality from systems theory, and the other is based on ideas derived from Stone-type dualities specifically linking coalgebras with algebraic structures derived from modal logics. In the present paper, we develop a more abstract view and unify the two approaches. We show that dualities, or more generally dual adjunctions, between categories can be lifted to dual adjunctions between (...) categories of coalgebras and algebras, and from there to automata with initial as well as final states. As in the Stone-duality approach, algebras are essentially logics for reasoning about the automata. By exploiting the ability to pass between these categories, we show that one can minimize the corresponding automata. We give an abstract minimisation algorithm that has several instances, including the celebrated Brzozowski minimisation algorithm. We further develop three examples that have been treated in previous works: deterministic Kripke frames based on a Stone-type duality, weighted automata based on the self-duality of semimodules, and topological automata based on Gelfand duality. As a new example, we develop alternating automata based on the discrete duality between sets and complete atomic Boolean algebras. (shrink)

Through contact algebras we study theories of mereotopology in a uniform way that clearly separates mereological from topological concepts. We identify and axiomatize an important subclass of closure mereotopologies called unique closure mereotopologies whose models always have orthocomplemented contact algebras , an algebraic counterpart. The notion of MT-representability, a weak form of spatial representability but stronger than topological representability, suffices to prove that spatially representable complete OCAs are pseudocomplemented and satisfy the Stone identity. Within the resulting class of contact (...) algebras the strength of the algebraic complementation delineates two classes of mereotopology according to the key ontological choice between mereological and topological closure operations. All closure operations are defined mereologically if and only if the corresponding contact algebras are uniquely complemented while topological closure operations highly restrict the contact relation but allow not uniquely complemented and nondistributive contact algebras. Each class contains a single ontologically coherent theory that admits discrete models. (shrink)

In lieu of an abstract, here is a brief excerpt of the content:_Russell_ journal (home office): E:CPBRRUSSJOURTYPE2602\REVIEWS.262 : 2007-01-24 01:12 eviews THE DENOTING CENTURY, 1 19 90 05 5– –2 20 00 05 5 Michael Scanlan Philosophy / Oregon State U. Corvallis, or 97331, usa [email protected] Guido Imaguire and Bernard Linsky, eds. On Denoting: 1905–2005. Munich: Philosophia Verlag, 2005. Pp. 451. 98.00. isbn 3-88405-091-5. his anniversary collection of papers connected with Russell’s 1905 publiTcation of “On Denoting” reflects both the almost (...) mythic status that paper has achieved in analytic philosophy and the quite variegated nature of the significance that philosophers have seen in it. The papers in this collection break down roughly into two sorts. There are those which take a historical approach and attempt to explicate various aspects of the content of the OD article. The other sort might be labelled “application” papers in which Russell’s analysis in OD is seen as a tool or a model for subsequent philosophical work, in the spirit of Ramsey’s much quoted “paradigm of philosophy” remark. The volume opens with a reprint of the OD text with pagination indicated for the original Mind printing, for the text in Collected Papers 4 and for the widely referenced Logic and Knowledge Russell collection edited by R. C. Marsh. The Mind and Papers texts use the American convention of single quotation marks within double quotation marks. The Marsh edition (and the text given here) uses the inverse British convention. Readers need to be aware that paper authors here, and elsewhere, vary in which text they cite. The end of the volume has a comprehensive 21-page bibliography of the secondary literature. In between there are thirteen papers. I will be rather brief with the “application” papers, since I believe that Russell readers will tend to favour the more historically oriented papers. Some of the application papers wander quite far afield from Russell’s treatment of definite descriptions. For instance the paper by Thomas Mormann on “Description, Construction and Representation from Russell and Carnap to Stone” takes off from Russell’s assertion that his theory of descriptions was an early version of his subsequent interest in substituting “constructions” for “inferred entities” such as space-time points. The paper is almost exclusively a (worthwhile and mathematically astute) analysis of Russell’s efforts along these lines in Our russell: the Journal of Bertrand Russell Studies n.s. 26 (winter 2006–07): 167–90 The Bertrand Russell Research Centre, McMaster U. issn 0036-01631 _Russell_ journal (home office): E:CPBRRUSSJOURTYPE2602\REVIEWS.262 : 2007-01-24 01:12 168 Reviews Knowledge of the External World and The Analysis of Matter. The author argues that Russell’s constructions did not actually achieve the effects he wanted, but that (curiously) they can be properly developed within the later framework of topological spaces developed by Marshall Stone in 1937. The paper by Guido Imaguire, “Theory of Descriptions and Inferential Semantics”, explicitly eschews an “exegetical” approach and allies itself with a “systematical approach, in which one analyses possible new features and extensions of the proposed theory” (p. 397). The extension considered here is to “inferential semantics”, especially that of Robert Brandom. The spirit of this approach is to take the semantics of a language to be determined by the set of allowed inferences within that language. Imaguire takes as an essential lesson of Russell’s theory of descriptions that definite descriptions and proper names cannot be treated as singular terms of a language in the same manner. He goes on to argue for a criterion for distinguishing proper names from definite descriptions based on inferences allowed between sentences containing them, and he takes this distinction to argue for a modification of Brandom’s treatment of singular terms in his inferential semantics. Elena Tatievskaya’s paper on “Russell’s Theory of Descriptions and Wittgenstein on Internal Properties of Propositions” is another paper that is mainly about something other than Russell and his theory of descriptions. It is a very detailed study of Wittgenstein’s attempt to make good on his claim that the logical symbols of propositional logic notation, such as v and ~, do not correspond to “logical” components of propositions (Tractatus... (shrink)

By analyzing how one obtains the Stone space of the reduced product of an indexed collection of Boolean algebras from the Stone spaces of those algebras, we derive a topological construction, the "reduced coproduct", which makes sense for indexed collections of arbitrary Tichonov spaces. When the filter in question is an ultrafilter, we show how the "ultracoproduct" can be obtained from the usual topological ultraproduct via a compactification process in the style of Wallman and Frink. We prove theorems (...) dealing with the topological structure of reduced coproducts (especially ultracoproducts) and show in addition how one may use this construction to gain information about the category of compact Hausdorff spaces. (shrink)

Many philosophers and psychologists argue that normal adult human beings possess a primitive or 'folk' psychological theory. Recently, however, this theory has come under challenge from the simulation alternative. This alternative view says that human bings are able to predict and explain each others' actions by using the resources of their own minds to simuate the psychological etiology of the actions of others. The thirteen essays in this volume present the foundations of theory of mind debate, and are accompanied by (...) an extensive introduction. (shrink)

Optimism about our ability to enhance societal decision‐making by leaning on Machine Learning (ML) for cheap, accurate predictions has palled in recent years, as these ‘cheap’ predictions have come at significant social cost, contributing to systematic harms suffered by already disadvantaged populations. But what precisely goes wrong when ML goes wrong? We argue that, as well as more obvious concerns about the downstream effects of ML‐based decision‐making, there can be moral grounds for the criticism of these predictions themselves. We introduce (...) and defend a theory of predictive justice, according to which differential model performance for systematically disadvantaged groups can be grounds for moral criticism of the model, independently of its downstream effects. As well as helping resolve some urgent disputes around algorithmic fairness, this theory points the way to a novel dimension of epistemic ethics, related to the recently discussed category of doxastic wrong. (shrink)

A systematic approach to the algebraic and Kripke semantics for logics with restricted structural rules, notably for logics on an underlying non-distributive lattice, is developed. We provide a new topological representation theorem for general lattices, using the filter space X. Our representation involves a galois connection on subsets of X, hence a closure operator $\Gamma$, and the image of the representation map is characterized as the collection of $\Gamma$-stable, compact-open subsets of the filter space . The original lattice ${\cal L}$ (...) is thus imbedded in the complete lattice of stable sets. The representation is shown to be functorial and is extended to a Stone-type duality. ;In the presence of additional operators, the extension problem is addressed in terms of what we call adjoint completions of normal operators. An operator f on a lattice L is normal if f: $L\sp{} \times \cdots \times L\sp{} \to L\sp{}$ is a monotone map, where each $L\sp{}$, $i \in n$ + 1, is either L or $L\sp{op}$ and f sends finite joins of $L\sp{}$ to joins in $L\sp{}$. Every familiar operator, such as $\circ$ , + , $\gets, \to, \neg, \square$ O is normal. By a uniform argument we show that normal operators $f\sb{i}$ on L extend to operators $F\sb{i}$ on $\Gamma X$ that distribute over arbitrary joins or meets and that \sb{i\in I})$ is a universal such extension. ;We also raise and resolve the question of representability of operators on stable sets by relations. We give an extensive number of examples of representation of specific operators and provide explicit constructions of canonical Kripke frames, based on our extension arguments, for a number of logical systems, notably for Intuitionistic and Classical Linear Logics, or Non-commutative Linear Logic and of fragments thereof. (shrink)

Traditionally, pronouns are treated as ambiguous between bound and demonstrative uses. Bound uses are non-referential and function as bound variables, and demonstrative uses are referential and take as a semantic value their referent, an object picked out jointly by linguistic meaning and a further cue—an accompanying demonstration, an appropriate and adequately transparent speaker’s intention, or both. In this paper, we challenge tradition and argue that both demonstrative and bound pronouns are dependent on, and co-vary with, antecedent expressions. Moreover, the semantic (...) value of a pronoun is never determined, even partly, by extra-linguistic cues; it is fixed, invariably and unambiguously, by features of its context of use governed entirely by linguistic rules. We exploit the mechanisms of Centering and Coherence theories to develop a precise and general meta-semantics for pronouns, according to which the semantic value of a pronoun is determined by what is at the center of attention in a coherent discourse. Since the notions of attention and coherence are, we argue, governed by linguistic rules, we can give a uniform analysis of pronoun resolution that covers bound, demonstrative, and even discourse bound readings. Just as the semantic value of the first-person pronoun ‘I’ is conventionally set by a particular feature of its context of use—namely, the speaker—so too, we will argue, the semantic values of other pronouns, including ‘he’, are conventionally set by particular features of the context of use. (shrink)

In our society, some aspects of life are off-limits to commerce. We prohibit the selling of children and the buying of wives, juries, and kidneys. Tainted blood is an inevitable consequence of paying blood donors; even sophisticated laboratory tests cannot supplant the gift-giving relationship as a safeguard of the purity of blood. Like blood, health care is too precious, intimate, and corruptible to entrust to the market.The hospice movement in the United States is approximately 40 years old. During these past (...) four decades, the concept of holistic, multidisciplinary care for patients who are suffering from a terminal illness has evolved from a modest, grassroots constellation of primarily volunteer-run and community-governed endeavors to a multimillion dollar industry where the surviving nonprofits compete with for-profit providers, often publicly traded, managed by M.B.A.-trained executives, and governed by corporate boards. The relatively recent emergence of for-profit hospice reflects an increasing commercialization of health care in the United States, the potentially adverse impact of which has been well documented. (shrink)

This article critically questions the commercialization of hospice care and the ethical concerns associated with the industry's movement toward “market-driven medicine” at the end of life. For example, the article examines issues raised by an influx of for-profit hospice providers whose business model appears at its core to have an ethical conflict of interest between shareholders doing well and terminal patients dying well. Yet, empirical data analyzing the experience of patients across the hospice industry are limited, and general claims that (...) end-of-life patient care is inferior among for-profit providers or even that their business practices are somehow unseemly when compared to nonprofit providers cannot be substantiated. In fact, non-profit providers are not immune to potentially conflicting concerns regarding financial viability (i.e., “no margin, no mission”). Given the limitations of existing empirical data and contrasting ideological commitments of for-profit versus non-profit providers, the questions raised by this article highlight important areas for reflection and further study. Policymakers and regulators are cautioned to keep ethical concerns in the fore as an increasingly commercialized hospice industry continues to emerge as a dominant component of the U.S. health care system. Both practitioners and researchers are encouraged to expand their efforts to better understand how business practices and commercial interests may compromise the death process of the patient and patient's family — a process premised upon a philosophy and ethical tradition that earlier generations of hospice providers and proponents established as a trusted, end-of-life alternative. (shrink)