Attention is drawn to the fact that what is alternatively known as Dummett logic, Gödel logic, or Gödel-Dummett logic, was actually introduced by Skolem already in 1913. A related work of 1919 introduces implicative lattices, or Heyting algebras in today's terminology.

Recently discovered documents have shown how Gentzen had arrived at the final form of natural deduction, namely by trying out a great number of alternative formulations. What led him to natural deduction in the first place, other than the general idea of studying “mathematical inference as it appears in practice,” is not indicated anywhere in his publications or preserved manuscripts. It is suggested that formal work in axiomatic logic lies behind the birth of Gentzen’s natural deduction, rather than any single (...) decisive influence in the work of others. Various axiomatizations are explored in turn, from the classical axioms of Hilbert and Ackermann to the intuitionistic axiomatization of Heyting. (shrink)

This book gives a comprehensive account of methods in philosophy of education, it also examines their application in the 'real world' of education. It will therefore be of interest to philosophers and educators alike.

In den letzten Jahrzehntel! hat sich das Interesse an der Grund­ legung der Mathematik immer gesteigert. Fanden frtiher die wenigen Forscher, die sich emsthaft mit dieser 'Frage beschaftigten, wenig Be­ achtung, heute ist die Teilnahme sowohl von mathematischer wie von philosophischer Seite fast allgemein. Zu diesem Umschwung hat sieher die CANToRSche Mengenlehre, die gleich nach ihrem Entstehen lebhafte Erorterungen tiber ihre Berechtigung hervorrief, den AnstoB gegeben, und besonders die bei riicksichtsloser Durchfiihrung ihrer Grundgedanken auftretenden Widerspriiche zogen die allgemeine Aufmerksamkeit auf (...) sich. Doch ist die bisweUen noch geauBerte Behauptung, der Zweck -der Grundlagenforschung liege in der Beseitigung der Widersprtiche, verfehlt. In philosophischer und in mathematischer Richtung geht diese weit tiber eine solche Zielsetzung hinaus. Philosophisch untersucht man -das Wesen der mathematischen Erkenntnis, ihre Voraussetzungen und Endziele, ihr Verhaltnis zu anderen Wissensgebieten, insbesondere der Physik, und ihre Abgrenzung gegen diese dem lnhalt und der Methode nacho An diese philosophischen Erorterungen schlieBen sich umfangreiche mathematische Untersuchungen tiber den Aufbau der Mathematik aus den philosophisch gegebenen Voraussetzungen und tiber die Struktur der mathematischen Beweisftihrungen. Einzelne Teilgebiete dieser Unter­ suchungen entwickeln sich schon zu selbstandigen Disziplinen, die sich in ihren Methoden und Problemstellungen von der eigentlichen Grund­ lagenforschung unabhangig machen; ein Beispiel eines solchen neuen Zweiges der Mathematik, der sein Entstehen der Grundlagenforschung verdankt, ist die mathematischp. Logik. Allmahlich haben sich drei J. auptrichtungen gebildet, die je einer eigenen Auffassung tiber das Wesen der Mathematik entsprechen undo je zu verschieden gearteten mathematischen Untersuchungen geftihrt haben. (shrink)

The role of critique in the Anglophone analytical tradition of philosophy of education is outlined and some of its shortcomings are noted, particularly its apparent claim to methodological objectivity in arriving at what are clearly contestable positions about the normative basis of education. Many of these issues can be seen to have a long history within European, and especially German, philosophy of education. In the light of this the discussion moves on to a consideration of similarities and contrasts between the (...) Anglophone and German-inspired deployment of the concept of critical rationality in philosophy of education. The claims to objectivity of the Anglophone tradition are contrasted with a more self-conscious concern for social justice and improvement in other European traditions, which has been followed more recently by a greater scepticism concerning the potential of critique for delivering social justice and improvement in education. This has parallels with the growing Anglophone disillusion with ‘classical’ analytic philosophy of education. This in turn has resulted in a greater awareness of the limitations of critique: its ideological character, its rootedness in specific contexts, its own potential dogmatism and its ambiguities. The various contributions to this volume are briefly described and related to each other. (shrink)

How can philosophy exert its critical function in society and in education if any appeal to independent and even relatively ‘certain’ criteria seems problematic? The epistemological doubts that foundationalist models of justification encounter unavoidably seem to raise this question. In particular, the relativist implications that seem to result from rejecting such models seem to paralyse the critical potential of philosophy of education. In order to explore the possibilities of a conception of educational critique that avoids the pitfalls of foundationalism, I (...) analyse the epistemological dimensions of this much-feared relativism, illustrating this with some characteristic examples. Solving the problems raised will require an interpretation of critique that leaves our daily sense of critique intact, without literally adopting its—foundationalist—basic assumptions. After systematically developing such an alternative interpretation of critical usage, a non-relativist but still non-foundationalist and powerful conception of philosophical critique seems possible. I illustrate results with some examples from philosophy of education. (shrink)

Many philosophers of education emphasise the impossibility to really ‘solve’ philosophical—and with that, educational—problems these days. Philosophers have been trying to give philosophy a new, constructive turn in the face of this insolvability. This paper focuses on irony-based approaches that try to exploit the very uncertainty of philosophical issues to further philosophical understanding. We will first briefly discuss a few highlights of historical uses of irony as a philosophical tool. Then we concentrate on two different interpretations of irony, formulated by (...) Bransen and Rorty, that aim at gaining insight into how we make meaning of the world, while at the same time recognising that such an understanding would be impossible. After discussing some problematic aspects of these interpretations a third interpretation of irony is developed, based on a third view of the nature of meaning-making. Following these three interpretations, we will discuss their philosophical merits and the different kinds of insight they can produce for philosophy of education. (shrink)

Many philosophers of education emphasise the impossibility to really ‘solve’ philosophical—and with that, educational—problems these days. Philosophers have been trying to give philosophy a new, constructive turn in the face of this insolvability. This paper focuses on irony-based approaches that try to exploit the very uncertainty of philosophical issues to further philosophical understanding. We will first briefly discuss a few highlights of historical uses of irony as a philosophical tool. Then we concentrate on two different interpretations of irony, formulated by (...) Bransen and Rorty, that aim at gaining insight into how we make meaning of the world, while at the same time recognising that such an understanding would be impossible. After discussing some problematic aspects of these interpretations a third interpretation of irony is developed, based on a third view of the nature of meaning-making. Following these three interpretations, we will discuss their philosophical merits and the different kinds of insight they can produce for philosophy of education. (shrink)

Differential academic language proficiency is an issue of major educational concern, bearing on problems varying from pupil performance, to social prospects, and citizenship. In this paper we develop a conception of the language‐acquiring subject, and we discuss the consequences for understanding differential language proficiency in schools. Starting from Wittgenstein's meaning‐as‐use theory we show that learning a language requires an activity that relates the subject both to the community of language users, and to the things language is about. In opposition to (...) Luntley, we contend that this does not mean that linguistic development involves linguistic adjustment to the world ‘as it is’. It is argued that, in as far as linguistic development involves a process of adjustment, this concerns conceptions about the world as it is presupposed to be—a ‘world’ that is subjected to doubt and revision time and again. With respect to dealing with differential academic language proficiency, this approach to linguistic development suggests bringing pupils into situations which require active participation in processes of ‘negotiating meaning’, including negotiating the prevailing presuppositions about what the world is like. This also puts novices in a different position—less assimilatory—recognising their co‐constructive potencies at a more fundamental level. (shrink)

In the 1920s Heyting attempted at axiomatizing constructive geometry. Recently, von Plato used different concepts to axiomatize it. He used 14 axioms to formulate constructive apartness geometry, seven of which have occurrences of negation. In this paper we show with the help of ANDP, a theorem prover based on natural deduction, that four new axioms without negation, shorter and more intuitive, can replace seven of von Plato's 14 ones. Thus we obtained a near negation-free new system consisting of 11 (...) axioms. (shrink)

The paper studies Heyting algebras within the framework of computable structure theory. We prove that the class _K_ containing all Heyting algebras with distinguished atoms and coatoms is complete in the sense of the work of Hirschfeldt et al. (Ann Pure Appl Logic 115(1-3):71-113, 2002). This shows that the class _K_ is rich from the computability-theoretic point of view: for example, every possible degree spectrum can be realized by a countable structure from _K_. In addition, there is no (...) simple syntactic characterization of computably categorical members of _K_ (i.e., structures from _K_ possessing a unique computable copy, up to computable isomorphisms). (shrink)

This book presents an English translation of a classic Russian text on duality theory for Heyting algebras. Written by Georgian mathematician Leo Esakia, the text proved popular among Russian-speaking logicians. This translation helps make the ideas accessible to a wider audience and pays tribute to an influential mind in mathematical logic. The book discusses the theory of Heyting algebras and closure algebras, as well as the corresponding intuitionistic and modal logics. The author introduces the key notion of a (...) hybrid that “crossbreeds” topology and order, resulting in the structures now known as Esakia spaces. The main theorems include a duality between the categories of closure algebras and of hybrids, and a duality between the categories of Heyting algebras and of so-called strict hybrids. Esakia’s book was originally published in 1985. It was the first of a planned two-volume monograph on Heyting algebras. But after the collapse of the Soviet Union, the publishing house closed and the project died with it. Fortunately, this important work now lives on in this accessible translation. The Appendix of the book discusses the planned contents of the lost second volume. (shrink)

In this paper, we will give a general description of subdirectly irreducible Heyting algebras with operators under some weak conditions, which includes the finite case, the normal case and the case for Boolean algebras with diamond operator. This can be done by normalizing these operators. This answers the question posed in Wolter [4].

In this paper we introduce a class of inquisitive Heyting algebras as algebraic structures that are isomorphic to algebras of finite antichains of bounded implicative meet semilattices. It is argued that these structures are suitable for algebraic semantics of inquisitive superintuitionistic logics, i.e. logics of questions based on intuitionistic logic and its extensions. We explain how questions are represented in these structures and provide several alternative characterizations of these algebras. For instance, it is shown that a Heyting algebra (...) is inquisitive if and only if its prime filters and filters generated by sets of prime elements coincide and prime elements are closed under relative pseudocomplement. We prove that the weakest inquisitive superintuitionistic logic is sound with respect to a Heyting algebra iff the algebra is what we call a homomorphic p-image of some inquisitive Heyting algebra. It is also shown that a logic is inquisitive iff its Lindenbaum–Tarski algebra is an inquisitive Heyting algebra. (shrink)

For a complete Heyting lattice , we define a category Etale (). We show that the category Etale () is equivalent to the category of the sheaves over , Sh(), hence also with -valued sets, see [2], [1]. The category Etale() is a generalization of the category Etale (X), see [1], where X is a topological space.

An algebra A = ⟨A, ∨, ∧, →, 0, 1⟩ is a semi-Heyting algebra if ⟨A, ∨, ∧, 0, 1⟩ is a bounded lattice, and it satisfies the identities: x ∧ ≈ x ∧ y, x ∧ ≈ x ∧ [ → ], and x → x ≈ 1.

In the framework of algebras with infinitary operations, the equational theory of \(\bigvee _{\kappa }\) -complete Heyting algebras or Heyting \(\kappa \) -frames is studied. A Hilbert style calculus algebraizable in this class is formulated. Based on the infinitary structure of Heyting \(\kappa \) -frames, an equational type completeness theorem related to the \(\langle \bigvee, \wedge, \rightarrow, 0 \rangle \) -structure of frames is also obtained.

After the 1930s, the research into the foundations of mathematics changed.None of its main directions (logicism, formalism and intuitionism) had any longer the pretension to be the only true mathematics.Usually, the determining factor in the change is considered to be Gödel?s work, while Heyting?s role is neglected.In contrast, in this paper I first describe how Heyting directly suggested the abandonment of the big foundational questions and the putting forward of a new kind of foundational research consisting in the (...) isolation of formal, intuitive, logical and platonistic elements within classical mathematics.Furthermore, I describe how Heyting indirectly influenced the abandon?ment of the old directions of foundational research by making out some lists of degrees of evidence that exist within intuitionism. (shrink)

This paper is a contribution toward developing a theory of expansions of semi-Heyting algebras. It grew out of an attempt to settle a conjecture we had made in 1987. Firstly, we unify and extend strikingly similar results of [ 48 ] and [ 50 ] to the (new) equational class DHMSH of dually hemimorphic semi-Heyting algebras, or to its subvariety BDQDSH of blended dual quasi-De Morgan semi-Heyting algebras, thus settling the conjecture. Secondly, we give a criterion for (...) a unary expansion of semi-Heyting algebras to be a discriminator variety and give an algorithm to produce discriminator varieties. We then apply the criterion to exhibit an increasing sequence of discriminator subvarieties of BDQDSH . We also use it to prove that the variety DQSSH of dually quasi-Stone semi- Heyting algebras is a discriminator variety. Thirdly, we investigate a binary expansion of semi-Heyting algebras, namely the variety DblSH of double semi-Heyting algebras by characterizing its simples, and use the characterization to present an increasing sequence of discriminator subvarieties of DblSH . Finally, we apply these results to give bases for “small” subvarieties of BDQDSH , DQSSH , and DblSH. (shrink)

We define and investigate Heyting-valued interpretations for Constructive Zermelo–Frankel set theory . These interpretations provide models for CZF that are analogous to Boolean-valued models for ZF and to Heyting-valued models for IZF. Heyting-valued interpretations are defined here using set-generated frames and formal topologies. As applications of Heyting-valued interpretations, we present a relative consistency result and an independence proof.

A proof of the consistency of Heyting arithmetic formulated in natural deduction is given. The proof is a reduction procedure for derivations of falsity and a vector assignment, such that each reduction reduces the vector. By an interpretation of the expressions of the vectors as ordinals each derivation of falsity is assigned an ordinal less than ε 0, thus proving termination of the procedure.

We present an extension of Heyting arithmetic in finite types called Uniform Heyting Arithmetic that allows for the extraction of optimized programs from constructive and classical proofs. The system has two sorts of first-order quantifiers: ordinary quantifiers governed by the usual rules, and uniform quantifiers subject to stronger variable conditions expressing roughly that the quantified object is not computationally used in the proof. We combine a Kripke-style Friedman/Dragalin translation which is inspired by work of Coquand and Hofmann and (...) a variant of the refined A-translation due to Buchholz, Schwichtenberg and the author to extract programs from a rather large class of classical first-order proofs while keeping explicit control over the levels of recursion and the decision procedures for predicates used in the extracted program. (shrink)

The aim of this paper is to introduce a new approach to the modal operators of necessity and possibility. This approach is based on the existence of two negations in certain lattices that we call bi-Heyting algebras. Modal operators are obtained by iterating certain combinations of these negations and going to the limit. Examples of these operators are given by means of graphs.

In the framework of algebras with infinitary operations, the equational theory of $$\bigvee _{\kappa }$$ ⋁ κ -complete Heyting algebras or Heyting $$\kappa $$ κ -frames is studied. A Hilbert style calculus algebraizable in this class is formulated. Based on the infinitary structure of Heyting $$\kappa $$ κ -frames, an equational type completeness theorem related to the $$\langle \bigvee, \wedge, \rightarrow, 0 \rangle $$ ⟨ ⋁, ∧, →, 0 ⟩ -structure of frames is also obtained.

In this paper, we will study Heyting algebras endowed with tense negative operators, which we call tense H-algebras and we proof that these algebras are the algebraic semantics of the Intuitionistic Propositional Logic with Galois Negations. Finally, we will develop a Priestley-style duality for tense H-algebras.

The variety of Heyting algebras has two well-behaved locally finite reducts, the variety of bounded distributive lattices and the variety of implicative semilattices. The variety of bounded distributive lattices is generated by the →-free reducts of Heyting algebras, while the variety of implicative semilattices is generated by the ∨-free reducts. Each of these reducts gives rise to canonical formulas that generalize Jankov formulas and provide an axiomatization of all superintuitionistic logics. The ∨-free reducts of Heyting algebras give (...) rise to the -canonical formulas that we studied in an earlier work. Here we introduce the -canonical formulas, which are obtained from the study of the →-free reducts of Heyting algebras. We prove that every si-logic is axiomatizable by -canonical formulas. We also discuss the similarities and differences between these two kinds of canonical formulas. One of the main ingredients of these formulas is a designated subset D of pairs of elements of a finite subdirectly irreducible Heyting algebra A. When D=A2, we show that the -canonical formula of A is equivalent to the Jankov formula of A. On the other hand, when D=∅, the -canonical formulas produce a new class of si-logics we term stable si-logics. We prove that there are continuum many stable si-logics and that all stable si-logics have the finite model property. We also compare stable si-logics to splitting and subframe si-logics. (shrink)

In this paper we introduce a logic that we name semi Heyting–Brouwer logic, \, in such a way that the variety of double semi-Heyting algebras is its algebraic counterpart. We prove that, up to equivalences by translations, the Heyting–Brouwer logic \ is an axiomatic extension of \ and that the propositional calculi of intuitionistic logic \ and semi-intuitionistic logic \ turn out to be fragments of \.

La littérature témoigne d’une tendance croissante à soutenir l’intuitionisme par la phénoménologie. Le disciple de Brouwer Arend Heyting est considéré comme un précurseur de cette tendance, parce qu’il usait d’une terminologie phénoménologique en vue de définir la négation intuitioniste, en élaborant la première logique intuitioniste. Dans cet article, l’auteur tente d’explorer — en référence aux matériaux inédits conservés aux Archives Heyting — ce qui, dans la pensée de Heyting, est compatible avec la phénoménologie. Dans la conclusion, l’auteur (...) suggère que Heyting et Husserl, pour autant qu’ils pensent tous deux que la conscience doit être le véritable commencement de la connaissance, partagent une même attitude antipsychologiste coexistant avec une tentative de dépassement du solipsisme. Toutefois, le concept phénoménologique de degré d’évidence ne peut être appliqué à l’échelle de l’évidence de Heyting (incluant les petits nombres naturels, les grands nombres naturels, les suites infinies, le quantificateur universel), d’un côté parce qu’on ne voit pas clairement si la seconde est partagée par tous les intuitionistes, et de l’autre parce que le premier présuppose une évidence révisable qui ne cadre pas avec le point de vue de Heyting. En outre, les conceptions de la nature de la mathématique et de la logique et de leurs relations mutuelles défendues par Husserl et Heyting sont essentiellement différentes. D’un point de vue intuitioniste, la mathématique est le domaine de l’évidence, alors que la logique transcrit ses régularités. D’un point de vue phénoménologique, la mathématique demeure en dehors du domaine de l’évidence. La logique apophantique coïncide avec la mathématique (sans que l’une absorbe l’autre), mais la logique transcendantale se situe à un niveau plus élevé. (shrink)

In this paper we describe the well-founded initial segment of the free Heyting algebra ������α on finitely many, α, generators. We give a complete classification of initial sublattices of ������₂ isomorphic to ������₁ (called 'low ladders'), and prove that for 2 < α < ω, the height of the well-founded initial segment of ������α.

From the point of view of non-classical logics, Heyting's implication is the smallest implication for which the deduction theorem holds. This book studies properties of logical systems having some of the classical connectives and implication in the neighbourhood of Heyt ing's implication. I have not included anything on entailment, al though it belongs to this neighbourhood, mainly because of the appearance of the Anderson-Belnap book on entailment. In the later chapters of this book, I have included material that might (...) be of interest to the intuitionist mathematician. Originally, I intended to include more material in that spirit but I decided against it. There is no coherent body of material to include that builds naturally on the present book. There are some serious results on topological models, second order Beth and Kripke models, theories of types, etc., but it would require further research to be able to present a general theory, possibly using sheaves. That would have postponed pUblication for too long. I would like to dedicate this book to my colleagues, Professors G. Kreisel, M.O. Rabin and D. Scott. I have benefited greatly from Professor Kreisel's criticism and suggestions. Professor Rabin's fun damental results on decidability and undecidability provided the powerful tools used in obtaining the majority of the results reported in this book. Professor Scott's approach to non-classical logics and especially his analysis of the Scott consequence relation makes it possible to present Heyting's logic as a beautiful, integral part of non-classical logics. (shrink)

Two algebraic structures, the contrapositionally complemented Heyting algebra (ccHa) and the contrapositionally |$\vee $| complemented Heyting algebra (c|$\vee $|cHa), are studied. The salient feature of these algebras is that there are two negations, one intuitionistic and another minimal in nature, along with a condition connecting the two operators. Properties of these algebras are discussed, examples are given and comparisons are made with relevant algebras. Intuitionistic Logic with Minimal Negation (ILM) corresponding to ccHas and its extension |${\textrm {ILM}}$|-|${\vee }$| (...) for c|$\vee $|cHas are then investigated. Besides its relations with intuitionistic and minimal logics, ILM is observed to be related to Peirce’s logic and Vakarelov’s logic MIN. With a focus on properties of the two negations, relational semantics for ILM and |${\textrm {ILM}}$|-|${\vee }$| are obtained with respect to four classes of frames, and inter-translations between the classes preserving truth and validity are provided. ILM and |${\textrm {ILM}}$|-|${\vee }$| are shown to have the finite model property with respect to these classes of frames and proved to be decidable. Extracting features of the two negations in the algebras, a further investigation is made, following logical studies of negations that define the operators independently of the binary operator of implication. Using Dunn’s logical framework for the purpose, two logics |$K_{im}$| and |$K_{im-{\vee }}$| are discussed, where the language does not include implication. The |$K_{im}$|-algebras are reducts of ccHas and are different from relevant algebraic structures having two negations. The negations in the |$K_{im}$|-algebras and |$K_{im-{\vee }}$|-algebras are shown to occupy distinct positions in an enhanced form of Dunn’s kite of negations. Relational semantics for |$K_{im}$| and |$K_{im-{\vee }}$| is provided by a class of frames that are based on Dunn’s compatibility frames. It is observed that this class coincides with one of the four classes giving the relational semantics for ILM and |${\textrm {ILM}}$|-|${\vee }$|⁠. (shrink)

In this paper we shall prove that certain subvarieties of the variety of Salgebras (Heyting algebras with successor) has amalgamation. This result together with an appropriate version of Theorem 1 of [L. L. Maksimova, Craig’s theorem in superintuitionistic logics and amalgamable varieties of pseudo-boolean algebras, Algebra i Logika, 16(6):643-681, 1977] allows us to show interpolation in the calculus IPC S (n), associated with these varieties.We use that every algebra in any of the varieties of S-algebras studied in this work (...) has a canonical extension, to show completeness of the calculus IPC S (n) with respect to appropriate Kripke models. (shrink)

We give an example of a variety of Heyting algebras and of a splitting algebra in this variety that is not finitely presentable. Moreover, we show that the corresponding splitting pair cannot be defined by any finitely presentable algebra. Also, using the Gödel-McKinsey-Tarski translation and the Blok-Esakia theorem, we construct a variety of Grzegorczyk algebras with similar properties.

We use the left self‐distributive axiom to introduce and study a special class of weak Heyting algebras, called self‐distributive weak Heyting algebras (SDWH‐algebras). We present some useful properties of SDWH‐algebras and obtain some equivalent conditions of them. A characteristic of SDWH‐algebras of orders 3 and 4 is given. Finally, we study the relation between the variety of SDWH‐algebras and some of the known subvarieties of weak Heyting algebras such as the variety of Heyting algebras, the variety (...) of basic algebras, the variety of subresiduated lattices, the variety of reflexive WH‐algebras (RWH‐algebras), and the variety of transitive WH‐algebras (TWH‐algebras). (shrink)

In this paper we obtain characterizations of subalgebras of Heyting algebras and De Morgan Heyting algebras. In both cases we obtain these characterizations by defining certain equivalence relations on the Priestley-type topological representations of the corresponding algebras. As a particular case we derive the characterization of maximal subalgebras of Heyting algebras given by M. Adams for the finite case.

ZUSAMMENFASSUNGTraditionell wird Demea als das schwächste Glied in Humes berühmten Dialogues concerning Natural Religion angesehen; die Bühne ist ganz dominiert vom optimistischen Theismus, der durch Cleanthes vertreten wird, und den dagegen gerichteten skeptischen Manövern vonseiten Philos. Entgegen diesem traditionellen Bild wird der ›orthodoxe‹ Demea nun verteidigt mit der These: Demea hat – von Hume selbst ungewollt und unbemerkt – das Interessanteste zum religiösen Glauben beizutragen; in ihm deutet sich eine Position jenseits der metaphysischen Phantasien des Theismus einerseits und Philos Destruktionen, (...) die in eine moralische Minimalversion des Glaubens zu münden scheinen, andererseits an. Es wird deutlich, dass diese Verteidigung keine primär exegetischen Ziele verfolgt, sondern auf eine in Humes Figurenkabinett nur treffend personifizierte Konstellation reagiert, die uns sehen lässt, an welchem Punkt theologisch und religionsphilosophisch auch heute zu arbeiten wäre. Dazu wird nach einer begrifflichen Kritik an Cleanthes und Philo anhand von drei konkreten Beispielen herausgearbeitet, wie ein Bild religiösen Glaubens aussehen könnte, das jenseits der metaphysischen Hoffnungen, die Gott zu einer Person erklären, und der reduktionistischen Zugeständnisse, die in ›Gott‹ lediglich den Ausdruck einer moralischen Einstellung zu erkennen meinen, liegt. Demea steht somit für ein postmetaphysisches Bild religiösen Glaubens – und wir sind eingeladen, zu den »friends of Demea« zu gehören.SUMMARYTraditionally, Demea is considered to be the weakest part in Hume's famous Dialogues concerning Natural Religion; the stage is completely dominated by Cleanthes' optimistic theism and by Philo's sceptical moves already critical of the former. Contrary to this traditional approach, however, the ›orthodox‹ Demea will be defended here in maintaining: Demea contributes – neither consciously intended nor recognized by Hume – the most interesting observations concerning religious belief; in him a position is at least alluded lying beyond the metaphysical fantasies of theism on the one hand and Philo's destructions which seem to amount to a moral minimal version on the other hand. It will be clear that this defense is not exegetically orientated; rather, it reacts to a constellation just personalized by Hume's ›casting‹ letting us see at which topics we shall continue to work theologically as well as philosophically today. Accordingly and following a conceptual critique of Cleanthes and Philo it will be elaborated by three concrete examples how a picture of religious belief could look like located beyond metaphysical hopes turning God to a person and reductivist concessions regarding ›God‹ as a mere expression of a moral attitude. Demea, however, represents a postmetaphysical picture of religious belief – and we are invited to belong to the »friends of Demea«. (shrink)

The goal of this paper is to generalize a notion of characteristic (or Jankov) formula by using finite partial Heyting algebras instead of the finite subdirectly irreducible algebras: with every finite partial Heyting algebra we associate a characteristic formula, and we study the properties of these formulas. We prove that any intermediate logic can be axiomatized by such formulas. We further discuss the correlations between characteristic formulas of finite partial algebras and canonical formulas. Then with every well-connected (...) class='Hi'>Heyting algebra we associate a set of characteristic formulas that correspond to each finite relative subalgebra of this algebra. Finally, we demonstrate that in many respects these sets enjoy the same properties as regular characteristic formulas. In the last section we outline an approach how to generalize these obtained results to the broad classes of algebras. (shrink)

The Gödel–Artemov framework offered a formalization of the Brouwer–Heyting–Kolmogorov semantics of intuitionistic logic via classical proofs. In this framework, the intuitionistic propositional logic IPC is embedded in the modal logic S4, S4 is realized in the Logic of Proofs LP, and LP has a provability interpretation in Peano Arithmetic. Self-referential LP-formulas of the type ‘t is a proof of a formula ϕ containing t itself’ are permitted in the realization of S4 in LP, and if such formulas are indeed (...) involved, it is then necessary to use fixed-point arithmetical methods to explain intuitionistic logic via provability. The natural question of whether self-referentiality can be avoided in realization of S4 was answered negatively by Kuznets who provided an S4-theorem that cannot be realized without using directly self-referential LP-formulas. This paper studies the question of whether IPC can be embedded in S4 and then realized in LP without using self-referential formulas. We consider a general class of Gödel-style modal embeddings of IPC in S4 and by extending Kuznetsʼ method, show that there are IPC-theorems such that, under each such embedding, are mapped to S4-theorems that cannot be realized in LP without using directly self-referential formulas. Interestingly, all double-negations of tautologies that are not IPC-theorems, like ¬¬, are shown to require direct self-referentiality. Another example is found in IPC→, the purely implicational fragment of IPC. This suggests that the BHK semantics of intuitionistic logic is intrinsically self-referential. (shrink)

We investigate expansions of Heyting algebras in possession of a unary term describing the filters that correspond to congruences. Hasimoto proved that Heyting algebras equipped with finitely many normal operators have such a term, generalising a standard construction on finite-type boolean algebras with operators. We utilise Hasimoto’s technique, extending the existence condition to a larger class of EHAs and some classes of double-Heyting algebras. Such a term allows us to characterise varieties with equationally definable principal congruences using (...) a single equation. Moreover, in the presence of a dual pseudocomplement operation, discriminator varieties are characterised by a pair of equations. We also prove that a variety of dually pseudocomplemented EHAs with a normal filter term is semisimple if and only if it is a discriminator variety. This generalises two known results, one by Kowalski and Kracht for finite-type varieties of BAOs, and the other by the present author for dually pseudocomplemented Heyting algebras without additional operations. (shrink)

For any $n<\omega $ we construct an infinite $(n+1)$ -generated Heyting algebra whose n-generated subalgebras are of cardinality $\leq m_n$ for some positive integer $m_n$. From this we conclude that for every $n<\omega $ there exists a variety of Heyting algebras which contains an infinite $(n+1)$ -generated algebra, but which contains only finite n-generated algebras. For the case $n=2$ this provides a negative answer to a question posed by G. Bezhanishvili and R. Grigolia in [4].

ZusammenfassungDer Elementarbaustein aller Lebewesen ist die Zelle. Sie besitzt eine reiche innere Struktur mit einer Vielzahl von Organellen fur verschiedene Stoffwechselfunktionen.Zur Analyse der Funktionen dieser Organelle wird die Methode der Zellfraktionierung einge‐setzt. Die Zellen werden dabei aufgebrochen und – meist mit Zentrifugations‐Techniken – in verschiedene Fraktionen aufgetrennt. Mit Hilfe von Markern, d.h. von charakteristischen bio‐chemischen oder morphologischen Merkmalen der einzelnen Organelle werden die Fraktionen identifiziert und dann im Hinblick auf ihre Stoffwechselfunktionen analysiert.Eine prinzipielle Schwierigkeit dieser Analyse besteht in der Tatsache, (...) dass es umso wahr‐scheinlicher ist, dass sich die Funktionen der isolierten Organelle verandern, je sorgfältiger sie gereinigt werden.SummaryThe elementary building block of all living organisms is the cell. It has a rich inner structure with a multitude of organelles for various metabolic functions.The analysis of these functions is made possible by the method of cell fractionation. Cells are homogenized and separated, mainly by centrifugation techniques, into a number of fractions. With the aid of markers, i.e. of characteristic biochemical or morphological properties of individual organelles, the fractions are identified and then analyzed for their metabolic functions.A principal difficulty of this analysis consists in the fact that an alteration of the functions of isolated organelles is the more probable, the more extensively they are purified.RésuméĽélément fondamental de tout être vivant est la cellule. Elle a une riche structure interieure avec une multitude ?on;organelles pour différentes fonctions metaboliques.Ľanalyse de leurs fonctions repose sur la méthode du fractionnement des cellules. Les cellules sont broyées et séparées en différentes fractions, le plus souvent par des techniques de centrifugation. Les fractions sont identifyérs àľaide de marqueurs, c'est‐à‐dire de propriétés biochimiques ou morphologiques charactéristiques pour une organelle individuelle, puis elles sont analysées quant a leurs fonctions métaboliques.Ľune des principales difficultés provient du fait que plus on purifie une organelle, plus on risque ?on;altérer sa fonction. (shrink)

Under some assumptions on an equational theory S , we give a necessary and sufficient condition so that S admits a model completion. These assumptions are often met by the equational theories arising from logic. They say that the dual of the category of finitely presented S-algebras has some categorical stucture. The results of this paper combined with those of [7] show that all the 8 theories of amalgamable varieties of Heyting algebras [12] admit a model completion. Further applications (...) to varieties of modal algebras are given in [8]. (shrink)

The aim of this paper is to give, using the Kripke semantics for intuitionism, a representation of finitely generated free Heyting algebras. By means of the representation we determine in a constructive way some set of "special elements" of such algebras. Furthermore, we show that many algebraic properties which are satisfied by the free algebra on one generator are not satisfied by free algebras on more than one generator.

The main aim of the present paper is to explain a nature of relationships exist between Nelson and Heyting algebras. In the realization, a topological duality theory of Heyting and Nelson algebras based on the topological duality theory of Priestley for bounded distributive lattices are applied. The general method of construction of spaces dual to Nelson algebras from a given dual space to Heyting algebra is described. The algebraic counterpart of this construction being a generalization of the (...) Fidel-Vakarelov construction is also given. These results are applied to compare the equational category N of Nelson algebras and some its subcategories with the equational category H of Heyting algebras. It is proved that the category N is topological over the category H. (shrink)

Since in Heyting Arithmetic all atomic formulas are decidable, a Kripke model for HA may be regarded classically as a collection of classical structures for the language of arithmetic, partially ordered by the submodel relation. The obvious question is then: are these classical structures models of Peano Arithmetic ? And dually: if a collection of models of PA, partially ordered by the submodel relation, is regarded as a Kripke model, is it a model of HA? Some partial answers to (...) these questions were obtained in [6], [3], [1] and [2]. Here we present some results in the same direction, announced in [7]. In particular, it is proved that the classical structures at the nodes of a Kripke model of HA must be models of IΔ1 and that the relation between these classical structures must be that of a Δ1-elementary submodel. MSC: 03F30, 03F55. (shrink)