In this paper an algebraic system of the new type is proposed (namely, a vectorial lattice). This algebraic system is a lattice for the language of Aristotle’s syllogistic and as well as a lattice for the language of Vasiľév’s syllogistic. A lattice for the language of Aristotle’s syllogistic is called a vectorial lattice on cap-semilattice and a lattice for the language of Vasiľév’s syllogistic is called a vectorial lattice on closure cap-semilattice. These constructions are introduced for the first time.

In quest’articolo, articolo prenderò in esame i passi in cui Kant descrive la monadologia leibniziana come un “concetto platonico” del mondo, ossia come una descrizione del mondo intelligibile che non ha nulla a che vedere con la spiegazione del mondo fenomenico. In generale, vorrei mostrare che quest’interpretazione non va contrapposta a quella che lo stesso Kant aveva dato nella prima Critica, dove la monadologia era caratterizzata come un “sistema intellettuale del mondo”. Per fare ciò, risponderò a due domande. La prima (...) riguarda le ragioni che possono aver indotto Kant, in diverse occasioni, a presentare Leibniz come un precursore della teoria della soggettività dello spazio. La seconda riguarda il motivo per cui Kant associa costantemente la monadologia alla teoria platonica delle idee, mostrando come, in entrambi i casi, Kant intenda criticare qualsiasi possibilità di un transito dal mondo sensibile a quello intelligibile. Infine, cercherò di mostrare che, dietro questa lettura deflazionista della monadologia, c’è anche un’implicita autocritica alle tesi che Kant stesso ha sostenuto nella Dissertatio del 1770. (shrink)

The books by Annie Ernaux and Didier Eribon can be interpreted as testimonies of the struggle for belonging. With their rich descriptions of growing up in poverty, of shame and degradation, they make an important contribution to the analysis of orders of belonging. Educational institutions play a significant role in this. In this way, Ernaux and Eribon shed light on the mechanisms through which belonging is created or denied.

Der Beitrag präsentiert wesentliche Bestandteile von Fichtes Wissenschaftslehre mit einigen kritischen Bemerkungen. Als repräsentatives Beispiel seiner philosophischen Position, die zugleich die Grundlage seines wissenschaftlichen Systems bildet, stellt Fichte den Streit zwischen zwei möglichen philosophischen Systeme dar: dem Idealismus und dem Dogmatismus. In Auseinandersetzung mit dem Dogmatismus findet er die Begründung für die idealistische Position durch die Analyse von Begriffen und Phänomenen wie Erfahrung, Bewusstsein, Erkenntnis und schließlich Freiheit. Die Freiheit, verstanden als eine bewusste Entscheidung, nötigt den Philosophen zur Wahl einer (...) konkreten Form von Philosophie, weil sie davon abhängt, was für ein Mensch man ist. (shrink)

The name tells the thing if it's a name. If it doesn’t tell the thing, it isn’t a name. This is the puzzling and enigmatic theory proposed by Cratilo in the homonymous Plato’s dialogue. The thesis in Hermogenes already sounds hermetic, an "oracle" which requires the presence of an interpreter to clarify what remains hidden in the terms of the sentence. According to the disciple of Heraclitus, the names are by nature guaranteed to impart pure truths, that is, they are (...) all right, at least as many as are real names. An imperfectly fitting name isn’t an inappropriate name. It would not be a name at all. In this text we would like to examine the Platonic Cratylus with respect to the meaning that lies behind the words of Cratylus and clarify his position in face of Socrates's refutation. We will therefore concentrate our attention on the last section of the dialogue devoted to the exchange between Cratylus and Socrates, especially pages 427e-440e. (shrink)

Purpose: Tracing the historical roots of Mitterer's non-dualizing philosophy in Austrian philosophers who studied the relationship between object and language around 1900. Method: Discussing the epistemological relevance of the "tertium non datur" principle and disclosing the mutual influence of early language critics Mauthner, Stöhr, and Wahle, who also anticipated many of Wittgenstein's later insights. Findings: Mitterer's philosophy can be considered the endpoint of the Austrian tradition of language criticism. His non-dualizing approach is a methodological constructivism that does not comply (...) with "tertium non datur." Implications: Non-dualizing philosophy can also be applied to media theory. (shrink)

My goal is to illustrate Descartes’ reliance on two quite different and competing interpretations of objective reality by explaining how each is used in defending his causal axioms. The initial criticism comes from Caterus (and is later taken up by Gassendi) who charges that Descartes makes it appear as if the thought in its objective aspect (the intentional entity) is really distinct from the thought qua modification of the mind (i.e., the thought in its formal aspect). This implies that the (...) object-of-the-thought is actually distinct from the thought-of-the-object in which case, (a) Descartes cannot account for the purported relation between the two, and (b) the intentional entity must exemplify properties which belong neither “immaterially” to mental substance nor concretely to physical substance. Descartes rejects Caterus’ assessment of his position: he has not introduced a fourth kind of real entity into the causal order distinct from the mind, its modifications, and the physical object thought of. However, in responding to Caterus, Descartes implicitly appeals to a Suarezian theory of intentionality in which reference to the ostensibly separate reality of the objective entity is reducible to the formal concept: the thought-of-the-object is not really distinct from the object-of-thought. Clearly, Descartes cannot explicitly use the Suarezian theory because it relies on a system of causal explanation (the doctrine of the species “flitting through the air”) which Descartes rejects on scientific grounds. I shall argue that Descartes is committed to a notion of the “form” of the mind, viz., a modification of mind, which should allow for a non-relational modeling of the thought qua modification of the mind and its intentional object, but that he cannot consistently attribute to mental acts enough structure to support his theory. (shrink)

The debate over the civilian use of nuclear power is highly polarised. We argue that a reasonable response to this deep disagreement is to maintain that advocates of both camps should modify their positions. According to the analysis we propose, nuclear power is neither entirely right nor entirely wrong, but rather right and wrong to some degree. We are aware that this non-binary analysis of nuclear power is controversial from a theoretical point of view. Utilitarians, Kantians, and other moral theorists (...) make sharp, binary distinctions between right and wrong acts. However, an important argument speaking in favour of our non-binary analysis is that it better reflects our considered intuitions about the ethical trade-offs we face in discussions of nuclear power. The aim of this article is to make this argument sharp by explaining how it can be rendered compatible with, and supported by, the Capability Approach, which is quickly becoming one of the most influential frameworks for thinking about human development. (shrink)

The present paper is an attempt at the investigation of the nature of polarity contrast in natural languages. Truth conditions for natural language sentences are incomplete unless they include a proper definition of the conditions under which they are false. It is argued that the tertium non datur principle of classical bivalent logical systems is empirically invalid for natural languages: falsity cannot be equated with non-truth. Lacking a direct intuition about the conditions under which a sentence is false, we (...) need an independent foundation of the concept of falsity. The solution I offer is a definition of falsity in terms of the truth of a syntactic negation of the sentence. A definition of syntactic negation is proposed for English. (shrink)

On a bilateralist reading, sequents are interpreted as statements to the effect that, given the assertion of the antecedent it is incoherent to deny the succedent. This interpretation goes against its own ecumenical ambitions, endowing Cut with a meaning very close to that of tertium non datur and thus rendering it intuitionistically unpalatable. This paper explores a top-down route for arguing that, even intuitionistically, a prohibition to deny is as strong as a licence to assert.

Il presente testo intende proporre una riflessione lungo il cammino teoretico proposto da Salomon Maimon ed Emmanuel Lèvinas, prendendo spunto dall’aporia filosofica del dialogo Parmenide di Platone riguardante il tema dell’ exaìphnes. Certamente entro le riflessioni di Lèvinas che in Maimon agisce fortemente una vis giudaica, che ne ispira i temi: per Maimon la risoluzione completa dell’empirico può aver luogo solo nella terra promessa dell’intelletto infinito, per Lèvinas il chiasmo tra l’assolutamente Altro, che si manifesta sottraendosi al tempo stesso nel (...) volto dell’altro, ci chiama continuamente a far esodo da noi stessi. Nel concreto, in ciò che non ha luogo, nella pausa, nell’assenza e nella discontinuità, nel sommamente contingente, si trova il vero trascendentale quale elemento di mediazione che lega la realtà, non nelle categorie dell’intelletto che ci forniscono solo una realtà valida per la Wahrnhemung e la objektive Vorstellung , ovvero una realtà postuma, fittizia. Come tertium non datur tra queste due posizioni - l’intelletto infinito di Maimon o l’esodo perenne da se stessi in Lèvinas – intenderemo porvi lo xynòn, ovvero un cum assente che coniuga in un paradossale intreccio il nominalismo assoluto di Hume con il mondo delle idee di Platone: il luogo assurdo di una patria assente. (shrink)

Here we suggest a formal using of N.A. Vasil’ev’s logical ideas in categorical logic: the idea of “accidental” assertion is formalized with topoi and the idea of the notion of nonclassical negation, that is not based on incompatibility, is formalized in special cases of monoidal categories. For these cases, the variant of the law of “excluded n-th” suggested by Vasil’ev instead of the tertium non datur is obtained in some special cases of these categories. The paraconsistent law suggested by (...) Vasil’ev is also demonstrated with linear and tensor logics but in a form weaker than he supposed. As we have, in fact, many truth-values in linear logic and topos logic, the admissibility of the traditional notion of inference in the categorical interpretation of linear and intuitionistic proof theory is discussed. (shrink)

There are two traditional ways to read Kant's claim that every event necessarily has a cause: the weaker every-event some-cause and the stronger same-cause same-effect causal principles. The focus of the debate about whether and where he subscribes to the SCP has been in the Analogies in the Critique of Pure Reason and in the Metaphysical Foundations of Natural Science. By analysing the arguments and conclusions of both the Analogies and the Postulates as well as the two Latin principles non (...) datur casus and non datur fatum that summarise their results, I will argue for the novel thesis that the SCP is actually demonstrated in the Postulates of the First Critique. (shrink)

Brouwer's criticism of mathematical proofs making essential use of the tertium non datur had a surprisingly late response in logical circles. Among the diverse reactions in the mid 1920s and early 1930s, it is possible to delimit a coherent body of opinions on these questions: (1) whether Brouwer's denial of the tertium non datur meant only the abandonment of this classical law or, beyond that, the affirmation of its negation; (2) whether one or both of these alternatives were (...) logically inconsistent; and (3) whether Brouwer's line of argument was forced to take resort to the very law it was designed to refute. The controversy centred around a series of articles by Marcel Barzin and Alfred Errera who fought against the intuitionistic critique, missed their victory because of conceptual confusions and fallacious reasoning, but emerged unconvinced from the debate in the late 1930s. The controversy is of interest to the historiography of formal logic since it stimulated the clarification not only of the concepts of formal validity, decidability, many-valued systems of logic and non-classical systems generally, but also of the distinction between object and meta-level, and between a formal system and its semantics. Most important, the debate, by putting pressure on the intuitionistic camp to make their ideas more precise, seems to have given the decisive motivation towards Heyting's answer to this demand by his axiomatization of intuitionistic logic in 1930. (shrink)

We outline an intuitionistic view of knowledge which maintains the original Brouwer–Heyting–Kolmogorov semantics for intuitionism and is consistent with the well-known approach that intuitionistic knowledge be regarded as the result of verification. We argue that on this view coreflectionA→KAis valid and the factivity of knowledge holds in the formKA→ ¬¬A‘known propositions cannot be false’.We show that the traditional form of factivityKA→Ais a distinctly classical principle which, liketertium non datur A∨ ¬A, does not hold intuitionistically, but, along with the whole (...) of classical epistemic logic, is intuitionistically valid in its double negation form ¬¬(KA¬A).Within the intuitionistic epistemic framework the knowability paradox is resolved in a constructive manner. We argue that this paradox is the result of an unwarranted classical reading of constructive principles and as such does not have the consequences for constructive foundations traditionally attributed it. (shrink)

ZusammenfassungIn einem typenfreien formalen System, das widerspruchsfrei ist und alle Begriffe der klassischen Mathematik darzustellen vermag, können nicht alle Gesetze der klassischen Aussagenlogik gelten. Es entsteht daher das Problem, die aussagenlogischen Eigenschaften formaler Systeme allgemein zu untersuchen. Unter gewissen Voraussetzungen in aussagenlogischer Hinsicht wird die « aussagenlogische Vollständigkeit » und die « aussagenlogische Widerspruchsfreiheit » eines formalen Systems durch das Tertium non datur beziehungsweise durch die Schlussregel des Ex falso quodlibet charakterisiert. Es werden Entscheidungsverfahren für diejenigen syntaktischen Schlussregeln entwickelt, (...) die unter den vorliegenden aussagenlogischen Voraus‐setzungen in jedem formalen System beziehungsweise in jedem aussagen‐logisch vollständigen formalen System beziehungsweise in jedem aussagen‐logisch widerspruchsfreien formalen System gelten.RésuméDans un système formel sans types sans contradictions et qui pent représenter toutes les notions des mathkmatiques classiques, toutes les lois de la logiqrie propositionelle classique ne peuvent pas êvalables. C'est de là que s'ensuit le problème de discuter généralement les propriétés des systèmes formels en ce qu'elles concernent la logique des propositions. Sous certaines suppositions à I'Cgard des expressions propositionelles le « complet propositionel » et la « non‐contradiction propositionelle » se caractérisent par le tertium non datur ou plutôt par la rkgle d'inférence du ex falso quodlibet. On développe des méthodes de décision pour ces règles d'inférence syntactiques qui sont valables dans chaque système formel sous les suppositions données et pour ces régles d'inférence qui sont valables dans chaque système complet ou consistarit au point de vue de la logiqire des propositions.In a formal system without types which contains no contradictions and is able to represent all notions of classical mathematics, all the laws of classical propositional calculus cannot be valid. From this fact ensues the problem generally to explore the properties of formal systems from the point of view of the propositional calculus. Under certain assumptions conrerninq the propositional expressions the « propositional completeness » and the « propositional consistency » of a formal system is characterized through the tertium Ron datur, respectively through the inference rule of the ex falso quodlibet. There are decision procedures for the syntactical inference rules which in every formal system are valid under the given assumptions and for the syntactical inference rules of every formal system which is complete or consistent with respect to the propositional calculus. (shrink)

Thomas A. Fay Heidegger and the Formalization of Thought 1 Dagfinn F011esdal The Justification of Logic and Mathematics in Husserl's Phenomenology 25 Guillermo E. Rosado Haddock On Husserl's Distinction between State of Affairs and Situation of Affairs.... 35 David Woodruff Smith On Situations and States of Affairs 49 Charles W. Harvey, Jaakko Hintikka Modalization and Modalities................... 59 Gilbert T. Null Remarks on Modalization and Modalities 79 J. N. Mohanty Husserl's Formalism 93 Carl J. Posy Mathematics as a Transcendental Science 107 (...) vi Gian-carlo Rota Mathematics and the Task of Phenomenology 133 John Scalon "Tertium Non Datur: " Husserl's Conception of a Definite Multiplicity..... 139 Thomas M. Seebohm Psychologism Revisited 149 Gerald J. Massey Some Reflections on Psychologism 183 Robert S. Tragesser How Mathematical Foundation all but come about: A Report on Studies Toward a Phenomenological Critique of Godel's Views on Mathematical Intuition.. 195 Kenneth L. Manders On Geometric Intentionality 215 Dallas Willard Sentences which are True in Virtue of their Color... 225 John J. Drummond Willard and Husserl on Logical Form 243 Index of Names 257 Index of Subjects 259 PREFACE The phenomenology of logic and ideal objects is the topic of Husserl's Logical Investigations. This book determined the early development of the so called phenomenological movement. It is still the main source for many phenomenologists, even if they disagree with Husserl's transcendental turn and developed other phenomenological positions or positions beyond phenomenology he early sense. (shrink)

In lieu of an abstract, here is a brief excerpt of the content:Catholic Conscience and Civil Disobedience:The Primacy of TruthAngel Perez-Lopez and Israel Perez-LopezIntroductionSacred Scripture describes different examples of moral conscience dictating civil disobedience. For instance, think of the situation of Daniel (see Dan 6:6–10). In this and many other cases, we always find, above all, a defense of truth and of its primacy over conscience and civil authority.1 In a culture that rapidly abandons Christendom and rejects the Catholic social (...) values once prevalent, this same issue is gaining prominence.Recent civil mandates related to Covid-19 vaccination are raising anew the challenging moral question, among faithful Catholics, concerning the legitimacy of civil disobedience.2 Part of the difficulty consists in the struggle [End Page 773] to harmonize two statements from the Congregation for the Doctrine of the Faith (CDF) concerning the Covid-19 vaccines.On the one hand, it is affirmed that, under certain circumstances, the use of some of these vaccines is morally permissible: "When ethically irreproachable Covid-19 vaccines are not available (e.g. in countries where vaccines without ethical problems are not made available to physicians and patients, or where their distribution is more difficult due to special storage and transport conditions, or when various types of vaccines are distributed in the same country but health authorities do not allow citizens to choose the vaccine with which to be inoculated) it is morally acceptable to receive Covid-19 vaccines that have used cell lines from aborted fetuses in their research and production process."3On the other hand, the CDF also affirms that vaccination should be voluntary; it is not obligatory as a rule: "Practical reason makes evident that vaccination is not, as a rule, a moral obligation and that, therefore, it must be voluntary."4Some Catholics are clinging to the first affirmation to advocate for the primacy of authority in the civil mandate to be vaccinated. Others, instead, are clinging to the second affirmation to uphold the primacy of conscience and the right to civil disobedience. The controversy is reaching a boiling point in the United States. It seems that tertium non datur.In this essay, however, aided by the guidance of our moral magisterium and Thomas Aquinas, we would like to show how these two statements from the CDF are not incompatible. We would like to offer a third option to the ones previously described. Our thesis is that the way to bring about the harmony of both statements consists in underscoring the primacy of truth, in our comprehension of conscience, authority, and civil disobedience.The essay will be divided into two different sections. The first section deals with the general principles at play in the Catholic understanding of conscience and civil disobedience under the primacy of truth. Thus, it focuses on providing some needed information to get rid of the relativistic and political lenses through which many Catholics today are approaching the vaccine mandates. The second section of the essay is instead more particular. It makes a moral evaluation of the recent mandates to be vaccinated. [End Page 774]General PrinciplesToo Political?As Catholics, we should not be so radical as to think that moral theology has nothing to say about these political matters. A legitimate understanding of the separation between church and state does not forbid the Church's magisterial teaching and moral theology to be concerned with this issue.The Church is not only a mother but also a teacher: "The church's motherhood can never in fact be separated from her teaching mission, which she must always carry out as the faithful Bride of Christ, who is the Truth in person."5Moreover, "life in common within the State possesses a relevant moral value and presents specific demands that—according to the law of the Incarnation, though with a completely particular modality—enter in to form a part of the following of Christ. Therefore, moral theology must treat of it."6Thus, we would like to invite our readers to divest from political biases that may be influencing their reasoning. This is not about being Republican or Democrat first and then using one's faith to express our political views. The... (shrink)

THE mind is the locus of various dispositions, of developed sources and motives of action, which are not mere reflex habits but trained abilities and bents, tendencies, liabilities or inhibitions. Human knowing is more an intending of facts or states of affairs than a relation to them. Knowledge is not a predicamental relation. Consciousness is related to its object not as North Pole to South Pole, nor as container to contained, but as matter to form, and to the physical form (...) of the object of consciousness there corresponds the intentional form of the consciousness itself. To know reality is not to acquire it, grasp it, have it, possess it, contemplate it or dwell in it; it is to be informed by it i.e. intentionally to become it. Knowledge is the form rather than the cause of rational behaviour: technique, method, knack, skill, expertise, know-how, flair, strategic sense, prudence, taste, discretion, caution, judiciousness, decorum, conscientiousness, style, discipline, sagacity, discrimination, wisdom, respect for sense. Such principles operate not by description of nor prescription for, but by incarnation in and information of performance. It is, for instance, man’s respect for sense that is the form of a logically clear expression, and ideas are skills in the use of words. This human capacity and achievement is an intentional becoming, an actuation of the subject, bringing him to act. Yet, properly speaking, it is not action but passion. For knowing is an effect of which reality is the cause, and this causality is not a relation: actio est in passo—not in the agent, nor between the agent and the patient, but in the patient as from the agent, an actuation of the subject in dependence on the object. Of course it may be argued that non datur actio in distans, but far from requiring some intellectual ether, this only means that action and passion are not candidates for the application of spatial or temporal predicates. The known need not be close to the knower. True knowledge is the form of our reason when it is truly about matters of fact. It is not, then, a relation between subject and object, but a knowledge of states of affairs or facts constituted by interrelated objects which, to be known, must comprise some states of affairs of whose constituent objects some, at least, are knowing subjects. Inquiry is related to insight as searching to finding, or as trying to succeeding. Knowing is the second act or actualization of the intellect Thomists regard as second potency, first act, or accidental passive potency. But knowing is not a transitive activity. It is an achievement of intention, not an episode. It is a capacity, disposition, proneness or inclination, in so far as it connotes the abiding possibility of its contents’ polymorphic propositional and non-propositional expression in rational behaviour, which is an effect of knowing as an achievement, though it is also rightly said to be informed by know-how rather than caused by it. The act of self-knowledge consciously intends the self, but the self is intended as known and not as conscious. Thus, though true of consciousness, the distinction between consciously judging and consciously being tickled is not in consciousness but in knowledge of consciousness. Knowledge should be studied not in its products, but in its various expressions, especially in linguistic ones. For words and sentences are the most easily inspected class of things that we think in. (shrink)

Being a pragmatic and not a referential approach tosemantics, the dialogical formulation ofparaconsistency allows the following semantic idea tobe expressed within a semi-formal system: In anargumentation it sometimes makes sense to distinguishbetween the contradiction of one of the argumentationpartners with himself (internal contradiction) and thecontradiction between the partners (externalcontradiction). The idea is that externalcontradiction may involve different semantic contextsin which, say A and ¬A have been asserted.The dialogical approach suggests a way of studying thedynamic process of contradictions through which thetwo (...) contexts evolve for the sake of argumentation intoone system containing both contexts.More technically, we show a new, dialogical, way tobuild paraconsistent systems for propositional andfirst-order logic with classical and intuitionisticfeatures (i.e. paraconsistency both with and withouttertium non-datur) and present theircorresponding tableaux. (shrink)

We call a logic regular for a semantics when the satisfaction predicate for at least one of its nontheorems is closed under double negation. Such intuitionistic theories as second-order Heyting arithmetic HAS and the intuitionistic set theory IZF prove completeness for no regular logics, no matter how simple or complicated. Any extensions of those theories proving completeness for regular logics are classical, i.e., they derive the tertium non datur. When an intuitionistic metatheory features anticlassical principles or recognizes that a (...) logic regular for a semantics is nonclassical, it proves explicitly that the logic is incomplete with respect to that semantics. Logics regular relative to Tarski. Beth and Kripke semantics form a large collection that includes propositional and predicate intuitionistic, intermediate and classical logics. These results are corollaries of a single theorem. A variant of its proof yields a generalization of the Gödel-Kreisel Theorem linking weak completeness for intuitionistic predicate logic to Markov's Principle. (shrink)

This paper is the first in a series whose objective is to study notions of large sets in the context of formal theories of constructivity. The two theories considered are Aczel's constructive set theory and Martin-Löf's intuitionistic theory of types. This paper treats Mahlo's π-numbers which give rise classically to the enumerations of inaccessibles of all transfinite orders. We extend the axioms of CZF and show that the resulting theory, when augmented by the tertium non-datur, is equivalent to ZF (...) plus the assertion that there are inaccessibles of all transfinite orders. Finally, the theorems of that extension of CZF are interpreted in an extension of Martin-Löf's intuitionistic theory of types by a universe. (shrink)

In game-theoretical semantics, perfectlyclassical rules yield a strong negation thatviolates tertium non datur when informationalindependence is allowed. Contradictorynegation can be introduced only by a metalogicalstipulation, not by game rules. Accordingly, it mayoccur (without further stipulations) onlysentence-initially. The resulting logic (extendedindependence-friendly logic) explains several regularitiesin natural languages, e.g., why contradictory negation is abarrier to anaphase. In natural language, contradictory negationsometimes occurs nevertheless witin the scope of aquantifier. Such sentences require a secondary interpretationresembling the so-called substitutionalinterpretation of quantifiers.This interpretation is sometimes (...) impossible,and it means a step beyond thenormal first-order semantics, not an alternative to it. (shrink)

A decade ago, Isham and Butterfield proposed a topos-theoretic approach to quantum mechanics, which meanwhile has been extended by Döring and Isham so as to provide a new mathematical foundation for all of physics. Last year, three of the present authors redeveloped and refined these ideas by combining the C*-algebraic approach to quantum theory with the so-called internal language of topos theory (Heunen et al. in arXiv:0709.4364). The goal of the present paper is to illustrate our abstract setup through the (...) concrete example of the C*-algebra M n (ℂ) of complex n×n matrices. This leads to an explicit expression for the pointfree quantum phase space Σ n and the associated logical structure and Gelfand transform of an n-level system. We also determine the pertinent non-probabilisitic state-proposition pairing (or valuation) and give a very natural topos-theoretic reformulation of the Kochen–Specker Theorem.In our approach, the nondistributive lattice ℘(M n (ℂ)) of projections in M n (ℂ) (which forms the basis of the traditional quantum logic of Birkhoff and von Neumann) is replaced by a specific distributive lattice $\mathcal{O}(\Sigma_{n})$ of functions from the poset $\mathcal{C}(M_{n}(\mathbb{C}))$ of all unital commutative C*-subalgebras C of M n (ℂ) to ℘(M n (ℂ)). The lattice $\mathcal{O}(\Sigma_{n})$ is essentially the (pointfree) topology of the quantum phase space Σ n , and as such defines a Heyting algebra. Each element of $\mathcal{O}(\Sigma_{n})$ corresponds to a “Bohrified” proposition, in the sense that to each classical context $C\in\mathcal{C}(M_{n}(\mathbb{C}))$ it associates a yes-no question (i.e. an element of the Boolean lattice ℘(C) of projections in C), rather than being a single projection as in standard quantum logic. Distributivity is recovered at the expense of the law of the excluded middle (Tertium Non Datur), whose demise is in our opinion to be welcomed, not just in intuitionistic logic in the spirit of Brouwer, but also in quantum logic in the spirit of von Neumann. (shrink)

The thesis that the two-valued system of classical logic is insufficient to explanation the various intermediate situations in the entity, has led to the development of many-valued and fuzzy logic systems. These systems suggest that this limitation is incorrect. They oppose the law of excluded middle (tertium non datur) which is one of the basic principles of classical logic, and even principle of non-contradiction and argue that is not an obstacle for things both to exist and to not exist (...) at the same time. However, contrary to these claims, there is no inadequacy in the two-valued system of classical logic in explanation the intermediate situations in existence. The law of exclusion and the intermediate situations in the external world are separate things. The law of excluded middle has been inevitably accepted by other logic systems which are considered to reject this principle. The many-valued and the fuzzy logic systems do not transcend the classical logic. Misconceptions from incomplete information and incomplete research are effective in these criticisms. In addition, it is also effective to move the discussion about the intellectual conception (tasawwur) into the field of judgmental assent (tasdiq) and confusion of the mawhum (imaginable) with the ma‘kûl (intellegible). (shrink)

Compound propositions which can successfully be defended in a quantumdialogue independent of the elementary propositions contained in it, must have this property also independent of the mutual elementary commensur-abilities. On the other hand, formal commensurabilities must be taken into account. Therefore, for propositions which can be proved by P, irrespective of both the elementary propositions and of the elementary commensur-abilities, there exists a formal strategy of success. The totality of propositions with a formal strategy of success in a quantum dialogue (...) form the effective quantum logic. The propositions of the effective quantum logic can be derived from a calculus Q eff which is — on the other hand — equivalent to a lattice L qi.Propositions about measuring results are above all time dependent propositions A(S;t). In a dialogue, different partial propositions will have in general different time values. If one can (accidentally) win a material dialogue, this dialogue can be related to a single time value. For the propositions of the effective quantum logic there exist formal strategies of success, independent of the elementary propositions contained in it. All partial propositions appearing in the dialogue are formally commensurable. Therefore the propositions of effective quantum logic which can be proved by formal dialogues can always be related to a single time. They present a description of the system S considered in which all partial propositions can be related jointly to the state of S.Therefore in the effective quantum logic we have — in the limit of equal time values — a situation which corresponds conceptually to the description of the system (S; ψ) in Hilbert space. Consequently, one would expect that also the lattice L qi — except from the tertium non datur 8 — agrees with the lattice L q of subspaces of Hilbert space. It has been shown that these lattices are in fact isomorphic. (shrink)

In a previous paper [ 21 ] all extensions of Johansson’s minimal logic J with the weak interpolation property WIP were described. It was proved that WIP is decidable over J. It turned out that the weak interpolation problem in extensions of J is reducible to the same problem over a logic Gl, which arises from J by adding tertium non datur. In this paper we consider extensions of the logic Gl. We prove that only finitely many logics over (...) Gl have the Craig interpolation property CIP, the restricted interpolation property IPR or the projective Beth property PBP. The full list of Gl-logics with the mentioned properties is found, and their description is given. We note that IPR and PBP are equivalent over Gl. It is proved that CIP, IPR and PBP are decidable over the logic Gl. (shrink)

In his article on pre-conventions, Celano presents, what the author calls, the Ontological Commitment Thesis and the Normative Bite Thesis. In this short comment, the author argues that the two theses are together both incompatible with the idea that pre-conventions are facts which have causal powers in human behaviour; also, if the ontological thesis is abandoned, normative determination could not be obtained. In other terms, the author argues that either pre-conventions are part of our causal explanation of human behaviour or (...) pre-conventions are abstract entities able to determine human behaviour normatively. In the first case, pre-conventions lack normative meaning, while in the second pre-conventions cannot integrate our causal explanation of human actions. Tertium non datur. (shrink)

By way of their genre, inaugural lectures are condensed and rather broad since they are addressed to a mixed audience. The following theses develop and defend my thoughts on the ‘justified religiou...

Sub titulo supra allato, a. 1987 Cracoviae sat ampla dissertatio lingua polona publici iuris a me facta est, cuius brevis synthesis in hoc articulo datur, poionam linguam nescientium usui imprimis destinata. Thomas Młodzianowski etiam hodie memoria dignus videtur qua multiformis suo tempore activitatis vir pluriumque varii argumenti scriptonim auctor, quae temporum decursu non semel optime sunt notata, pleniorem tamen expositionem nondum acceperunt, quod saltern philosophicos textus attinet, etsi a pluribus historicis hoc expresse postulabatur; alia eins opera convenientibus monographiis iam (...) sunt donata. Cui defectui in philosophico campo obvenire supradicti libri ac huiusce elucubrationis est, quae vitae curriculum, preaecipua eins scripta philosophica eorumque contentum ac momentum exponenda declarandaque intendit. (shrink)

One of the most important pieces of evidence which we possess concerning the judicial rights of Roman provincials, particularly their status in relation to the governor's tribunal, is provided by Cicero's brief outline of those provisions of the lex Rupilia, the Sicilian provincial charter, which dealt with judicial administration. The passage reads as follows: Siculi hoc iure sunt ut, quod civis cum cive agat, domi certet suis legibus, quod Siculus cum Siculo non eiusdem civitatis, ut de eo praetor iudices ex (...) P. Rupili decreto, quod is de decem legatorum sententia statuit, quam legem Rupiliam vocant, sortiatur. Quod privatus a populo petit ant populus a privato, senatus ex aliqua civitate qui iudicet datur, cum alternae civitates reiectae sunt; quod civis Romanus a Siculo petit, Siculus index, quod Siculus a civi Romano, civis Romanus datur; ceterarum rerum selecti iudices ex conventu civium Romanorum nr000ni solent. (shrink)

In lieu of an abstract, here is a brief excerpt of the content:Reviewed by:Volcanus: Recherches comparatistes sur les origines du culte de VulcainJ. LinderskiGérard Capdeville. Volcanus: Recherches comparatistes sur les origines du culte de Vulcain. Rome: Ecole Française de Rome, Palais Farnèse, 1995. viii 1 521 pp. Cloth, no price stated. (Bibliothèque des Ecoles Françaises d’Athènes et de Rome, fasc. 288)In the last twenty years the French School has published every two years one book on Roman religion: monographs dealing with (...) Fortuna, Mercurius, Ops, Penates, Providentia Deorum, Fratres Arvales, l’Association Dionysiaque (cf. JRS 78 [1988] 207–8), the Bacchanalia, religious architecture (Aurea templa), La vie religieuse des matrones, Le délit religieux (cf. CP 79 [1984] 174–77). Now we have a book about Volcanus. First let us remark on what this book is not: it is not a study of the cult of Volcanus, an encyclopedia that would systematically list and discuss all the literary, iconographic, and epigraphical evidence in Rome, Italy, and the provinces. It would be very pleasing to have such a book; perhaps pedestrian, but useful. For information we still have to turn to Georg Wissowa, to his ageless Religion und Kultus der Römer (2d ed. Munich 1912, 229–32); also informative is W. Eisenhut, “Volcanus,” RE Suppl. 14 (1974) 948–62. A rather balanced account of the origines appears in G. Camassa, “Sull’origine [End Page 644] e le funzioni del culto di Volcanus a Roma,” RSI 96 (1984) 811–54. Capdeville’s book we have for speculations, a paradise for those eager to delve into murky origins, and those who are transposed by the dazzling and uncertain art of comparativistic pleading. If bibliography (fifty-one pages, 425–75) stands for learning, it is a learned monograph; it ranges from Rome through Etruria to Crete and Cyprus. Surprisingly we find no Hittites or Turks (the relevance of the latter the reader will see revealed at the end of this piece).The book opens with an evocation of Georges Dumézil. Where do his manes lead the author? They lead to the fish. The monograph “est né... d’une interrogation sur les pisciculi des Volcanalia” (1), a subject which intrigued Dumézil himself (REL 36 [1959] 121–30). At the feast of Volcanalia on 23 August (see the references in A. Degrassi, Fasti anni Numani et Iuliani (5 Inscr. It. XIII.2 [Rome 1963] 500–502), populus pro se in ignem animalia mittit (Varr. LL 6.20). On 7 June (cf. Degrassi 466) the Ludi Piscatorii were celebrated a praetore urbano pro piscatoribus Tiberinis, quorum quaestus non in macellum pervenit, sed fere in aream Volkani, quod id genus pisciculorum vivorum datur ei deo pro animis humanis (Festus 274–76 L); but at the Ludi Piscatorii the offering was not pro animis but rather pro quaestu piscantium (Festus 232 L); cf. J. Le Gall, Recherches sur le culte du Tibre (Paris 1953) 48–50; J. Linderski, Ktema 17 ([1992] 1996) 58. Agni, the Indian god of fire, hated water and fish; this explains the Roman ritual (so Dumézil). Most scholars tacitly assume that pisciculi vivi were offered also on the Volcanalia, but Varro speaks generally of animalia: if he wanted to say pisces he would have said so. Volcanus was a god of destructive fire (so Wissowa, Dumézil, and many other scholars); it makes sense that to propitiate him not only fish, a symbol of water, were burnt, but also other animals (perhaps as a substitute offering; cf., on that sort of sacrifices, G. Capdeville, MEFRA 83 [1971] 283–323). In the context of incendia arcenda an inscription of the Domitianic time (ILS 4914) records an offering on the Volcanalia of vitulus robeus and verres r(obeus). This calls to mind the sacrifice of the rutilae canes to protect the crops from the burning sun (Festus 358 L).Of all of this Capdeville, surprisingly, does not write at all: after their debut on page 1 the pisciculi and the Volcanalia feebly reappear at the very end of the book (420–23), only to be disregarded. Capdeville claims that Volcanus was not at all, and never, a god of “feu destructeur.” But Dumézil too... (shrink)