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)

This paper describes an attempt to develop a program for teaching history and philosophy of mathematics to inservice mathematics teachers. I argue briefly for the view that philosophical positions and epistemological accounts related to mathematics have a significant influence and a powerful impact on the way mathematics is taught. But since philosophy of mathematics without history of mathematics does not exist, both philosophy and history of mathematics are necessary components of programs for the training of preservice as well as inservice (...) mathematics teachers. (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)

Both healthcare professionals and the healthcare system must defend each patient's health individually while simultaneously seeking to protect the population's health in general. Nowadays, there is an important increase in the cost of healthcare supply, mainly due to the developments of medical science, the public's expectations and the demographic ageing. Since healthcare resources are not unlimited, it is obvious that immoderate consumption of them by certain patients limits the use of the same funds by others. Therefore, we have to seek (...) an optimal distribution of the existing resources in order to manage a constriction of expenses, especially under the circumstances of the modern economic crisis. The criteria of effectiveness and efficiency should be used. Health policies focus on both the public's behaviour and the rules of medical practice. Under the modern challenges the physician's role is particularly important for the protection of the patient's health and the promotion of public health. (shrink)

We present an algebraic theory of structured objects based on and generalizing Aczel's theory of form systems. Notions of identity of structured objects and of transformations of systems of such objects are discussed. A generalization of Aczel's representation theorem is proven.

The contribution of this paper lies with providing a systematically specified and intuitive interpretation pattern and delineating a class of relational structures and models providing a natural interpretation of logical operators on an underlying propositional calculus of Positive Lattice Logic and subsequently proving a generic completeness theorem for the related class of logics, sometimes collectively referred to as Generalized Galois Logics.

This article addresses and resolves some issues of relational, Kripke-style, semantics for the logics of bounded lattice expansions with operators of well-defined distribution types, focusing on the case where the underlying lattice is not assumed to be distributive. It therefore falls within the scope of the theory of Generalized Galois Logics, introduced by Dunn, and it contributes to its extension. We introduce order-dual relational semantics and present a semantic analysis and completeness theorems for non-distributive lattice logic with n -ary additive (...) or multiplicative operators, with negation operators modally interpreted as impossibility and unnecessity, as well as with implication connectives. Order-dual relational semantics shares with the generalized Kripke frames, or the bi-approximation semantics approach, the use of both a satisfaction and a co-satisfaction relation, but it also responds to the recently voiced concerns of Craig, Haviar and Conradie about the relative non-intuitiveness of the 2-sorted semantics of the aforementioned approaches. In this article, we provide a standard interpretation of modalities and natural interpretations of both negation and implication, despite the absence of distribution. Thereby, our results contribute in creating the necessary background for research in non-distributive logics with modalities variously interpreted as dynamic, temporal etc, by analogy to the classical case. (shrink)

This article presents an approach to the semantics of non-distributive propositional logics that is based on a lattice representation theorem that delivers a canonical extension of the lattice. Our approach supports both a plain Kripke-style semantics and, by restriction, a general frame semantics. Unlike the framework of generalized Kripke frames, the semantic approach presented in this article is suitable for modeling applied logics, as it respects the intended interpretation of the logical operators. This is made possible by restricting admissible interpretations.

One of the main problems of the orthoframe approach to quantum logic was that orthomodularity could not be captured by any first-order condition. This paper studies an elementary and natural class of orthomodular frames that can work around this limitation. Set-theoretically, the frames we propose form a natural subclass of the orthoframes, where is an irreflexive and symmetric relation on X. More specifically, they are partially-ordered orthoframes with a designated subset. Our frame class contains the canonical orthomodular frame of the (...) logic and therefore characterises orthomodular quantum logic. To prove soundness, a further restriction is placed on admissible valuations which requires that they be point-generated in addition to the requirement that they return stable sets. (shrink)

It is argued that the philosophical and epistemological beliefs about the nature of mathematics have a significant influence on the way mathematics is taught at school. In this paper, the philosophy of mathematics of the NCTM's Standards is investigated by examining is explicit assumptions regarding the teaching and learning of school mathematics. The main conceptual tool used for this purpose is the model of two dichotomous philosophies of mathematics-absolutist versus- fallibilist and their relation to mathematics pedagogy. The main conclusion is (...) that a fallibilist view of mathematics is assumed in the Standards and that most of its pedagogical assumptions and approaches are based on this philosophy. (shrink)

Name der Zeitschrift: Klio Jahrgang: 99 Heft: 1 Seiten: 380-385.

One of the main characteristics of the post-modern society is the democratized diffusion of knowledge even on issues of which the average citizen could hardly be aware: social, cultural, ecological and economical risks. However, many philosophers and thinkers (Chomsky, Barber, Zizek, Sartori) argue that we live rather in information societies than in knowledge ones, because the content of information on political actions of major importance are ideologically-oriented and therefore misleading. In the present paper we study two examples of political and (...) scientific rhetoric, trying to interpret not only their political and moral determinants but also their possible impact on international welfare. Aristotle and modern semiotic theories will provide the methodological tools of thisanalysis. (shrink)

ABSTRACTCorrespondence and Shalqvist theories for Modal Logics rely on the simple observation that a relational structure is at the same time the basis for a model of modal logic and for a model of first-order logic with a binary predicate for the accessibility relation. If the underlying set of the frame is split into two components,, and, then frames are at the same time the basis for models of non-distributive lattice logic and of two-sorted, residuated modal logic. This suggests that (...) a reduction of the first to the latter may be possible, encoding Positive Lattice Logic as a fragment of Two-Sorted, Residuated Modal Logic. The reduction is analogous to the well-known Gödel-McKinsey-Tarski translation of Intuitionistic Logic into the S4 system of normal modal logic. In this article, we carry out this reduction in detail and we derive some properties of PLL from corresponding properties of First-Order Logic. The reduction we present is extendible to the case of lattices with operators, making use of recent results by this author on the relational representation of normal lattice expansions. (shrink)

In an article dating back in 1992, Kosta Došen initiated a project of modal translations in substructural logics, aiming at generalising the well-known Gödel–McKinsey–Tarski translation of intuitionistic logic into S4. Došen's translation worked well for (variants of) BCI and stronger systems (BCW, BCK), but not for systems below BCI. Dropping structural rules results in logic systems without distribution. In this article, we show, via translation, that every substructural (indeed, every non-distributive) logic is a fragment of a corresponding sorted, residuated (multi) (...) modal logic. At the conceptual and philosophical level, the translation provides a classical interpretation of the meaning of the logical operators of various non-distributive propositional calculi. Technically, it allows for an effortless transfer of results, such as compactness and the Löwenheim-Skolem property and it opens up new directions for deducing properties of substructural logic systems by establishing appropriate transfer theorems. (shrink)

We present dualities for implicative and residuated lattices. In combination with our recent article on a discrete duality for lattices with unary modal operators, the present article contributes in filling in a gap in the development of Orłowska and Rewitzky’s research program of discrete dualities, which seemed to have stumbled on the case of non-distributive lattices with operators. We discuss dualities via truth, which are essential in relating the non-distributive logic of two-sorted frames with their sorted, residuated modal logic, as (...) well as full Stone duality for residuated lattices. Our results have immediate applications to the semantics of related substructural logical calculi. (shrink)

We introduce game-theoretic semantics for systems without the conveniences of either a De Morgan negation, or of distribution of conjunction over disjunction and conversely. Much of game playing rests on challenges issued by one player to the other to satisfy, or refute, a sentence, while forcing him/her to move to some other place in the game’s chessboard-like configuration. Correctness of the game-theoretic semantics is proven for both a training game, corresponding to Positive Lattice Logic and for more advanced games for (...) the logics of lattices with weak negation and modal operators. (shrink)

We present a Hilbert style axiomatization and an equational theory for reasoning about actions and capabilities. We introduce two novel features in the language of propositional dynamic logic, converse as backwards modality and abstract processes specified by preconditions and effects, written as \({\varphi \Rightarrow \psi}\) and first explored in our recent paper (Hartonas, Log J IGPL Oxf Univ Press, 2012), where a Gentzen-style sequent calculus was introduced. The system has two very natural interpretations, one based on the familiar relational semantics (...) and the other based on type semantics, where action terms are interpreted as types of actions (sets of binary relations). We show that the proof systems do not distinguish between the two kinds of semantics, by completeness arguments. Converse as backwards modality together with action types allow us to produce a new purely equational axiomatization of Dynamic Algebras, where iteration is axiomatized independently of box and where the fixpoint and Segerberg induction axioms are derivable. The system also includes capabilities operators and our results provide then a finitary Hilbert-style axiomatization and a decidable system for reasoning about agent capabilities, missing in the KARO framework. (shrink)

In this paper, we address the issue of privacy protection in context aware services, through the use of entropy as a means of measuring the capability of locating a user’s whereabouts and identifying personal selections. We present a framework for calculating levels of abstraction in location and personal preferences reporting in queries to a context aware services server. Finally, we propose a methodology for determining the levels of abstraction in location and preferences that should be applied in user data reporting (...) during service provision, according to her personal privacy settings. (shrink)

This paper is an investigation into the logic of information flow. The basic perspective is that logic flows in virtue of constraints and that constraints classify channels connecting particulars In this paper we explore some logics intended to model reasoning in the case of idealized information flow, that is, where the constraints involved are exceptionless. We look at this as a step toward the far more challenging task of understanding the logic of imperfect information flow, that is where the constraints (...) admit of exceptional connections. This paper continues and amplifies work presented by the same authors in [10]. (shrink)

In Plato’s Theaetetus , Socrates is portrayed as a midwife to the intellect, a metaphor for his task as a dialectician as he seeks to help give birth to wisdom. Thus it is that the author refers to Plato as the midwife’s apprentice. This volume represents an attempt to provide a more manageable account of the author’s two volume magnum opus, An Examination of Plato’s Doctrines . An accessible and lucid introduction to Plato’s ideas is provided which nonetheless challenges traditional (...) interpretations. In particular the author is concerned to offer an interpretation of the significance of what Plato said. The chapters are arranged by topic, for ease of comprehension. (shrink)

Individual actions designed to address issues of public concern is a common theme in the discourse on how to mobilize resources and target efforts toward sustainable practices. We contribute to this area by developing and empirically validating a multidimensional scale for civic engagement; synthesizing and testing the adequacy of the theory of planned behavior and the value–belief–norm theory in explaining civic engagement; and considering how an individual’s orientation, identity, and beliefs motivate moral thinking and action. The focus is on the (...) important social issues of global warming and climate change, income inequality, and world poverty, and hunger. We follow both correlational and configurational approaches to examine symmetric and asymmetric causal relationships, respectively. The findings from a sample of 819 US citizens reveal that the TPB and VBN theory can adequately explain civic engagement, after we control for the influence of past experience. In addition, while belief in a just world inhibits the occurrence of adverse consequences and the formation of positive attitudes, social value orientation, and moral identity facilitate them. Notably, at least two causal conditions need to be present for adverse confsequences to emerge, while moral identity is almost a necessary condition for the development of positive attitudes. We conclude with a discussion of important implications for researchers and practitioners. (shrink)

During the COVID-19 pandemic, the Greek authorities enforced a vaccination mandate for healthcare workers (HCWs). At the same time, multiple concerns were raised about the epidemiological profile of Greece in addition to the ethical status of the harsh measures and their impact on employees, organizations, society, and public health. According to the World Health Organization (WHO), considerations regarding the evidence of vaccine safety and effectiveness, necessity, and proportionality should be clearly evaluated by before imposing mandatory vaccination policies. We discuss the (...) issues regarding the mechanics of the transmission and contraction of SARS-CoV-2, the toxicity of COVID-19 vaccines, and the impact of the suspension of HCWs who did not vaccinate versus the potential expected benefits in addition to whether the vaccine mandates were justified considering the overall epidemiological context. (shrink)