A presente investigação questiona a essência teo-lógica dos futuros contingentes. Para o efeito, analisa-se, primeiramente, a argumentação segundo a qual, sob certas con­dições lógicas, teológicas, ontológicas e cosmológicas anti-necessitantes, detetadas por G. W. Leibniz (conciliando a posição de St. Agostinho com a de L. Molina e W. Ockham), a abertura contingente do futuro parece ser compatível com o regime das “verdades contingentes pré-determinadas”, regime enquadrado teologicamente pelo princípio do “futuro melhor” ou do “único futuro verdadeiro”. No entanto, os (...) futuros contingentes incitam, com e contra Aristóteles, ao desenvolvimento de uma lógica temporal e plurivalente, ao modo de J. Łukasiewicz ou A. Prior. Esta lógica garante a abertura do futuro sem o oneroso custo metafísico da adesão a uma teo-lógica omni-determinante. A crítica do determinismo lógico, daí resultante, afigura-se mais coadunável com as condições pós-metafísicas ine­rentes à episteme agnóstica contemporânea, mas, nesse caso, a abertura do futuro impli­caria uma profunda redefinição das próprias ideias e funções de “Deus”, “matéria”, “história” e “verdade”. (shrink)

RESUMO: A presente investigação questiona a essência teo-lógica dos futuros contingentes. Para o efeito, analisa-se, primeiramente, a argumentação segundo a qual, sob certas condições lógicas, teológicas, ontológicas e cosmológicas antinecessitantes, detetadas por G. W. Leibniz, a abertura contingente do futuro parece ser compatível com o regime das "verdades contingentes pré-determinadas", regime enquadrado teologicamente pelo princípio do "futuro melhor" ou do "único futuro verdadeiro". No entanto, os futuros contingentes incitam, com e contra Aristóteles, ao desenvolvimento de uma (...) class='Hi'>lógica temporal e plurivalente, ao modo de J. Łukasiewicz ou A. Prior. Essa lógica garante a abertura do futuro sem o oneroso custo metafísico da adesão a uma teo-lógica omnideterminante. A crítica do determinismo lógico, daí resultante, afigura-se mais coadunável com as condições pós-metafísicas inerentes à episteme agnósticacontemporânea, mas, nesse caso, a abertura do futuro implicaria uma profunda redefinição das próprias ideias e funções de "Deus", "matéria", "história" e "verdade". ABSTRACT: This paper investigates the theological essence of contingent futures. First, it analyses the argument according to which, under certain logical, theological, ontological, and cosmological conditions detected by Leibniz, the contingent openness of the future may be compatible with the regime of "pre- determined contingent truths", theologically grounded in the principle of an "optimal future" or a "single true future". However, the examination of contingent futures has also fostered, both for and against Aristotle's position, the development of a many-valued and temporal logic as proposed by J. Łukasiewicz or A. Prior. This logic safeguards the openness of future without the metaphysically expensive cost of adhering to an all-determining theo-logic. It hence follows that the critique of logical determinism seems to be more appropriate to express the post-metaphysical conditions of our present agnostic mentality. In this case, however, the openness of the future would entail a profound redefinition of the very ideas and functions of "God", "matter", "history", and "truth". (shrink)

Luiz Henrique Lopes dos Santos, em seu texto sobre "Leibniz e a questão dos futuros contingentes”, argumenta em favor de seu diagnóstico segundo o qual, no fundo, a principal diferença entre as doutrinas de Espinosa e Leibniz reside no fato de que o primeiro, diferentemente do segundo, não sabia lógica. Este texto procura objetar à sua posição, respondendo às críticas do autor à posição de Espinosa quanto à liberdade divina. Procurarei mostrar que, sob o aspecto preciso da articulação (...) aí estabelecida entre a distinção vontade/entendimento e a questão dos possíveis jamais realizados, ela não poderia estar disponível para a metafísica de Espinosa, não por ele ter ignorado um princípio básico da lógica modal, como alega o autor, mas porque ele compreende diferentemente o estatuto ontológico da unidade das coisas individuais e, portanto, dos agentes humanos. (shrink)

This article is divided into introduction andd three section. In the first section we examine Leibniz’ termini necesitas-possibilitas. In the second section we propose a minimal modal logic, LLM, arise from the addition of modal priciples. Finally in the last section we examine his complex studie towards the interpretation of modal language in the possible worlds. The resulting interplay between the minimal modallogic and the possible worlds perspective is one of the main charms of semantics.

O filósofo medieval João Buridan (século XIV) desenvolve uma semântica complexa, baseada na distinção entre contexto de enunciação e contexto de avaliação, para determinar as condições de verdade de uma proposição. O objetivo do presente artigo é analisar essa semântica e, mais precisamente, compreender suas implicações relativas às proposições de futuro contingente. Propõe-se aqui que a distinção entre contexto de enunciação e contexto de avaliação permite uma abordagem satisfatória das condições de verdade de proposições de futuro contingente. Defende-se que essa (...) abordagem é satisfatória porque ela evita tanto as consequências desagradáveis advindas da negação do Princípio da Bivalência quanto a adesão ao determinismo.. (shrink)

Some authors have talken about the problem of the future contingents Aristotle exposed in De Interpretatione IX. But most of them do not explain the role of that chapter in his own work. Last analysis always try to find a formal solution. And this is very significative because De Interpretatione is a treatise that belongs to the semantic of the Organon. In this article we show that: 1. The aim of the problem of future contigents is not only formal and (...) the frame of the Aristotle’s solution. 2. Its implication with others important concepts in his philosophy like “substance”, “truth” and “dialectics”. (shrink)

O artigo trata da análise ockhamiana do tema da determinação da verdade nas proposições sobre o futuro contingente, segundo a formulação proposta por Aristóteles em De interpretatione, cap. 9, e de sua relação com o que é proposto sobre este assunto, segundo o próprio Ockham, “de acordo com a verdade e a fé”. A esse respeito, três pontos geralmente são levantados como possíveis decorrências desta leitura de Aristóteles: a assunção de que Ockham discordaria efetivamente da solução aristotélica, porque errônea; a (...) proposta de que Ockham tenha claramente vislumbrado na resposta aristotélica a indicação de uma lógica de três valores e, por fim, a recusa ockhamiana da tese da “necessidade do conseqüente”. Pretendemos mostrar aqui, porém, que apenas o último desses três pontos parece correto. (shrink)

O objetivo deste artigo é associar a lógica de abdução de Peirce à cibernética de sistemas vivos. Sistemas vivos não podem ser entendidos através de uma epistemologia causalista, pois se comportam conforme os efeitos e não conforme as causas. A cibernética analisou o modo pelo qual máquinas não triviais se movem através de retroação, ou circuito de retorno: a cada etapa, o sistema corrige a etapa anterior, dependendo de até onde a etapa anterior pode ir em relação a um (...) possível equilíbrio. A dinâmica implica uma possível inibição da energia excessiva seguida de um possível aumento da energia insuficiente. Cada uma dessas condições é um erro que se corrige automaticamente, atingindo um estado temporário de homeostase. A auto-correção significa que é preciso adicionar à saída um impulso adicional “criativo” que seja singular e imprevisível. A lógica que é capaz de fazer sentido dessa dinâmica é a abdução de Peirce, também chamada retrodução, onde o ato inferencial retroage sobre um fato incompreensível, um “erro” epistemológico, inventando sua causa. O pensar e a auto-poiesis de sistemas vivos percorrem o mesmo caminho, substituindo estratégias ocasionais por causas universais atemporais. A abdução é a evidência de que é possível atingir uma condição de “conhecimento” sem apelar para causalidade. A possibilidade que as “leis da natureza”, possa seguir este caminho conclui o ensaio ao adicionar a noção do físico Lee Smolin do “princípio de precedência”, conforme foi inspirado pela ideia de Peirce da evolução das leis da natureza e seu caráter temporal, ou seja, contingente. (shrink)

In this paper we improve the results of [2] by proving the product f.m.p. for the product of minimal n-modal and minimal n-temporal logic. For this case we modify the finite depth method introduced in [1]. The main result is applied to identify new fragments of classical first-order logic and of the equational theory of relation algebras, that are decidable and have the finite model property.

Una de las más frecuentes críticas dirigidas contra Hegel es que su sistema crea una totalidad finalizante que determina lo que es como absolutamente necesario. La Ciencia de la Lógica, siendo el edificio conceptual de dicho sistema, es el lugar más apropiado para determinar el significado específic..

It has recently been suggested that, for Leibniz, temporal facts globally supervene on causal facts, with the result that worlds differing with respect to their causal facts can be indiscernible with respect to their temporal facts. Such an interpretation is at variance with more traditional readings of Leibniz's causal theory of time, which hold that Leibniz reduces temporal facts to causal facts. In this article, I argue against the global supervenience construal of Leibniz's philosophy of time. On (...) the view of Leibniz defended here, he adopts a non-modal reduction of time to events, a form of reductionism that entails a strong covariation between a world's temporal facts and its causal facts. Consequently, worlds discernible with respect to their temporal facts must be discernible with respect to their causal facts, and worlds discernible with respect to their causal facts must be discernible with respect to their temporal facts. This position strongly favors the standard identificatory reduction of time to causation often imputed to Leibniz. (shrink)

Modal logics, originally conceived in philosophy, have recently found many applications in computer science, artificial intelligence, the foundations of mathematics, linguistics and other disciplines. Celebrated for their good computational behaviour, modal logics are used as effective formalisms for talking about time, space, knowledge, beliefs, actions, obligations, provability, etc. However, the nice computational properties can drastically change if we combine some of these formalisms into a many-dimensional system, say, to reason about knowledge bases developing in time or moving objects. To study (...) the computational behaviour of many-dimensional modal logics is the main aim of this book. On the one hand, it is concerned with providing a solid mathematical foundation for this discipline, while on the other hand, it shows that many seemingly different applied many-dimensional systems (e.g., multi-agent systems, description logics with epistemic, temporal and dynamic operators, spatio-temporal logics, etc.) fit in perfectly with this theoretical framework, and so their computational behaviour can be analyzed using the developed machinery. We start with concrete examples of applied one- and many-dimensional modal logics such as temporal, epistemic, dynamic, description, spatial logics, and various combinations of these. Then we develop a mathematical theory for handling a spectrum of 'abstract' combinations of modal logics - fusions and products of modal logics, fragments of first-order modal and temporal logics - focusing on three major problems: decidability, axiomatizability, and computational complexity. Besides the standard methods of modal logic, the technical toolkit includes the method of quasimodels, mosaics, tilings, reductions to monadic second-order logic, algebraic logic techniques. Finally, we apply the developed machinery and obtained results to three case studies from the field of knowledge representation and reasoning: temporal epistemic logics for reasoning about multi-agent systems, modalized description logics for dynamic ontologies, and spatio-temporal logics. The genre of the book can be defined as a research monograph. It brings the reader to the front line of current research in the field by showing both recent achievements and directions of future investigations (in particular, multiple open problems). On the other hand, well-known results from modal and first-order logic are formulated without proofs and supplied with references to accessible sources. The intended audience of this book is logicians as well as those researchers who use logic in computer science and artificial intelligence. More specific application areas are, e.g., knowledge representation and reasoning, in particular, terminological, temporal and spatial reasoning, or reasoning about agents. And we also believe that researchers from certain other disciplines, say, temporal and spatial databases or geographical information systems, will benefit from this book as well. Key Features: Integrated approach to modern modal and temporal logics and their applications in artificial intelligence and computer science Written by internationally leading researchers in the field of pure and applied logic Combines mathematical theory of modal logic and applications in artificial intelligence and computer science Numerous open problems for further research Well illustrated with pictures and tables. (shrink)

En este artículo, presentamos un sistema de lógica modal que permite representar relaciones entre conjuntos o clases de individuos definidos por una propiedad específica. Introducimos dos operadores modales, [a] y, que se utilizan respectivamente para expresar "para todo A" y "existe un A". Tanto la sintaxis como la semántica del sistema tienen dos niveles que evitan el anidamiento del operador modal. La semántica se basa en una variante de la semántica de Kripke, en donde los operadores modales se indexan (...) sobre fórmulas de lógica proposicional (“pre-fórmulas” en el trabajo). Además, presentamos un conjunto de axiomas y reglas que rigen el sistema, y demostramos que el sistema es correcto y completo con relación a los modelos de Kripke. En la sección final del artículo, discutimos posibles trabajos futuros. Consideramos la posibilidad de combinar nuestro operador modal con otras modalidades, como necesidad o conocimiento. Además, como ejemplo de la utilidad de nuestro operador modal, analizamos brevemente la fórmula de Barcan adaptada de manera conveniente dentro del marco de nuestro sistema. En resumen, proponemos la combinación de nuestro operador modal con otros como una forma más simple y compacta, aunque con un menor poder expresivo, para abordar la lógica modal cuantificada. (shrink)

Deontic logic handles not only deontic modalities, but also alethic and temporal ones. In addition, individuals like authorities and addressees play an important role. R5-D5 is a system handling alethic, deontic and temporal modalities, whose adequacy has been proved in an earlier paper. Similarly for KD*UXY with sets of individuals . The present article is an attempt to construct a general system mixing R5-D5 and KD*UXY.

Scopo del volume è offrire un’introduzione accessibile e rigorosa ai più recenti sviluppi di una fondamentale branca della filosofia del tempo: la filosofia del futuro. Vengono presentate e discusse alcune delle domande chiave del dibattito contemporaneo, ad esempio: il futuro è già scritto o esistono molti cammini alternativi che il tempo è in grado di imboccare? "Esistere" significa semplicemente essere presenti o ci sono veri e propri oggetti futuri? Siamo davvero liberi di scegliere quali azioni compiere e di modificare il (...) corso degli eventi? Il dibattito intorno alle risposte di volta in volta offerte in filosofia esplora un’intrigante zona d’intersezione tra metafisica, logica ed etica, e interessa discipline diverse, come la fisica, la psicologia e l’economia. Non sorprende dunque che la discussione contemporanea sia oramai tanto intricata da risultare ostica ai non addetti ai lavori. Il volume offre gli strumenti necessari per inquadrare le domande sul futuro nella loro reale complessità concettuale, introducendo tutte le indispensabili nozioni tecniche in un linguaggio chiaro e intuitivo. (shrink)

En sus últimos años, Leibniz emplea el concepto de necesidad moral para cualificar la elección divina de lo óptimo. Sin embargo, Leibniz no explica este concepto con precisión. El presente trabajo intenta mostrar que la necesidad moral leibniziana no puede entenderse como una modalidad puramente deóntica, porque ello contraría los fundamentos metafísicos que hacen al carácter teleológico de su ética. El artículo propone también una interpretación del contenido alético de tal noción, basada en la conexión necesaria que Leibniz parece atribuir (...) a la relación entre la voluntad y la tendencia al bien. Dicha necesidad afirmaría que, aun cuando toda elección es contingente, en tanto entraña una infinidad de consideraciones, el Dios leibniziano está metafísicamente necesitado a seguir el criterio de lo mejor. In his last years, Leibniz employs the concept of moral necessity to qualify God's choice of the best. However, he doesn't explain this concept accurately. This paper intends to show that leibnizian moral necessity cannot be conceived as a purely deontic modality, because it would be contrary to the metaphysical grounds which make Leibniz's ethics strictly teleological. This paper also proposes an interpretation of the alethic content of this concept, based on the necessary connection that Leibniz seems to assign to the relationship between free will and the tendency towards good. What this necessity entails is that, for Leibniz, even though the infinity of reasons involved makes every choice contingent, God is metaphysically necessitated to follow the rule of the best. (shrink)

For years philosophers argued for the existence of distinct yet materially coincident things by appealing to modal and temporal properties. For instance, the statue was made on Monday and could not survive being flattened; the lump of clay was made months before and can survive flattening. Such arguments have been thoroughly examined. Kit Fine has proposed a new set of arguments using the same template. I offer a critical evaluation of what I take to be his central lines of (...) reasoning. (shrink)

The Leibniz’s criticism to nominalism provides the metaphysic a more powerful (modal) logic rather than the extensional one of late-medieval period. At the same time he finds a prosecutive way of thinking that allows the systematization of the intelligibility. I maintain that on the formulation of Leibniz’s ontological argument, one can find the cognoscitive subject. Nevertheless, and in a similar way as Hegel, like Leibniz did not elaborate logic of productive action of intelligible possibility, the facticity introduces itself in the (...) subjects of the monada and on the Self. (shrink)

We combine linear temporal logic (with both past and future modalities) with a deontic version of justification logic to provide a framework for reasoning about time and epistemic and normative reasons. In addition to temporal modalities, the resulting logic contains two kinds of justification assertions: epistemic justification assertions and deontic justification assertions. The former presents justification for the agent’s knowledge and the latter gives reasons for why a proposition is obligatory. We present two kinds of semantics for the (...) logic: one based on Fitting models and the other based on neighborhood models. The use of neighborhood semantics enables us to define the dual of deontic justification assertions properly, which corresponds to a notion of permission in deontic logic. We then establish the soundness and completeness of an axiom system of the logic with respect to these semantics. Further, we formalize the Protagoras versus Euathlus paradox in this logic and present a precise analysis of the paradox, and also briefly discuss Leibniz’s solution. (shrink)

The temporal updating of an agent’s beliefs in response to a flow of information is modeled in a simple modal logic that, for every date t, contains a normal belief operator B t and a non-normal information operator I t which is analogous to the ‘only knowing’ operator discussed in the computer science literature. Soundness and completeness of the logic are proved and the relationship between the proposed logic, the AGM theory of belief revision and the notion of plausibility (...) is discussed. (shrink)

Leibnizian senses of temporality are analized, in order to assess that Leibniz’s formal and a priori notion of time is based on his own modal metaphysics. In this way, we can say that time constitutes the notion that allows the modalities articulation (Possibility-Necessity-Contingency) as well as the role of modal concepts allows to control the time.

This paper concerns modal logics appearing from the temporal ordering of domains in two-dimensional Minkowski spacetime. As R. Goldblatt has proved recently, the logic of the whole plane isS4.2. We consider closed or open convex polygons and closed or open domains bounded by simple differentiable curves; this leads to the logics:S4,S4.1,S4.2 orS4.1.2.

The core idea of the ontological proof is to show that the concept of existence is somehow contained in the concept of God, and that therefore God’s existence can be logically derived—without any further assumptions about the external world—from the very idea, or definition, of God. Now, G.W. Leibniz has argued repeatedly that the traditional versions of the ontological proof are not fully conclusive, because they rest on the tacit assumption that the concept of God is possible, i.e. free from (...) contradiction. A complete proof will rather have to consist of two parts. First, a proof of premiseGod is possible. Second, a demonstration of the “remarkable proposition”If God is possible, then God exists. The present contribution investigates an interesting paper in which Leibniz tries to prove proposition. It will be argued that the underlying idea of God as a necessary being has to be interpreted with the help of a distinguished predicate letter ‘E’ as follows:\\). Proposition which Leibniz considered as “the best fruit of the entire logic” can then be formalized as follows:\) \rightarrow \, E)\). At first sight, Leibniz’s proof appears to be formally correct; but a closer examination reveals an ambiguity in his use of the modal notions. According to, the possibility of the necessary being has to be understood in the sense of something which possibly exists. However, in other places of his proof, Leibniz interprets the assumption that the necessary being is impossible in the diverging sense of something which involves a contradiction. Furthermore, Leibniz believes that an »impossible thing«, y, is such that contradictory propositions like \\) and \\) might both be true of y. It will be argued that the latter assumption is incompatible with Leibniz’s general views about logic and that the crucial proof is better reinterpreted as dealing with the necessity, possibility, and impossibility of concepts rather than of objects. In this case, the counterpart of turns out to be a theorem of Leibniz’s second order logic of concepts; but in order to obtain a full demonstration of the existence of God, the counterpart of, i.e. the self-consistency of the concept of a necessary being, remains to be proven. (shrink)

The paper develops a set of quantified temporal alethic boulesic doxastic systems. Every system in this set consists of five parts: a ‘quantified’ part, a temporal part, a modal (alethic) part, a boulesic part and a doxastic part. There are no systems in the literature that combine all of these branches of logic. Hence, all systems in this paper are new. Every system is defined both semantically and proof-theoretically. The semantic apparatus consists of a kind of$$T \times W$$T×Wmodels, (...) and the proof-theoretical apparatus of semantic tableaux. The ‘quantified part’ of the systems includes relational predicates and the identity symbol. The quantifiers are, in effect, a kind of possibilist quantifiers that vary over every object in the domain. The tableaux rules are classical. The alethic part contains two types of modal operators for absolute and historical necessity and possibility. According to ‘boulesic logic’ (the logic of the will), ‘willing’ (‘consenting’, ‘rejecting’, ‘indifference’ and ‘non-indifference’) is a kind of modal operator. Doxastic logic is the logic of beliefs; it treats ‘believing’ (and ‘conceiving’) as a kind of modal operator. I will explore some possible relationships between these different parts, and investigate some principles that include more than one type of logical expression. I will show that every tableau system in the paper is sound and complete with respect to its semantics. Finally, I consider an example of a valid argument and an example of an invalid sentence. I show how one can use semantic tableaux to establish validity and invalidity and read off countermodels. These examples illustrate the philosophical usefulness of the systems that are introduced in this paper. (shrink)

We present a modal logic for the class of subset spaces based on discretely descending chains of sets. Apart from the usual modalities for knowledge and effort the standard temporal connectives are included in the underlying language. Our main objective is to prove completeness of a corresponding axiomatization. Furthermore, we show that the system satisfies a certain finite model property and is decidable thus.

It has recently been suggested that, for Leibniz, temporal facts globally supervene on causal facts, with the result that worlds differing with respect to their causal facts can be indiscernible with respect to their temporal facts. Such an interpretation is at variance with more traditional readings of Leibniz’s causal theory of time, which hold that Leibniz reduces temporal facts to causal facts. In this article, I argue against the global supervenience construal of Leibniz’s philosophy of time. On (...) the view of Leibniz defended here, he adopts a non-modal reduction of time to events, a form of reductionism that entails a strong covariation between a world’s temporal facts and its causal facts. Consequently, worlds discernible with respect to their temporal facts must be discernible with respect to their causal facts, and worlds discernible with respect to their causal facts must be discernible with respect to their temporal facts. This position strongly favors the standard identificatory reduction of time to causation often imputed to Leibniz. (shrink)

In the context of truth-functional propositional many-valued logics, Hájek’s Basic Fuzzy Logic BL [14] plays a major rôle. The completeness theorem proved in [7] shows that BL is the logic of all continuous t -norms and their residua. This result, however, does not directly yield any meaningful interpretation of the truth values in BL per se . In an attempt to address this issue, in this paper we introduce a complete temporal semantics for BL. Specifically, we show that BL (...) formulas can be interpreted as modal formulas over a flow of time, where the logic of each instant is Łukasiewicz, with a finite or infinite number of truth values. As a main result, we obtain validity with respect to all flows of times that are non-branching to the future, and completeness with respect to all finite linear flows of time, or to an appropriate single infinite linear flow of time. It may be argued that this reduces the problem of establishing a meaningful interpretation of the truth values in BL logic to the analogous problem for Łukasiewicz logic. (shrink)

This paper examines the problem of the basis of time’s asymmetry. I hold the view that there is an objective temporal asymmetry in Leibniz’s philosophy of time. I closely examine various asymmetrical phenomena, which can be candidates as an explanation of time’s asymmetry: (1) causation; (2) the flow of time; (3) the modal difference between past and present; (4) counterfactual dependence; and, finally (5) the asymmetry of the world’s progress and its direction and (6) of the progress of rational (...) creatures. I conclude that time’s asymmetry involves an irreducibly pragmatic aspect: the asymmetry of agency itself on which all other asymmetrical phenomena rely. (shrink)

We combine linear temporal logic (with both past and future modalities) with a deontic version of justification logic to provide a framework for reasoning about time and epistemic and normative reasons. In addition to temporal modalities, the resulting logic contains two kinds of justification assertions: epistemic justification assertions and deontic justification assertions. The former presents justification for the agent’s knowledge and the latter gives reasons for why a proposition is obligatory. We present two kinds of semantics for the (...) logic: one based on Fitting models and the other based on neighborhood models. The use of neighborhood semantics enables us to define the dual of deontic justification assertions properly, which corresponds to a notion of permission in deontic logic. We then establish the soundness and completeness of an axiom system of the logic with respect to these semantics. Further, we formalize the Protagoras versus Euathlus paradox in this logic and present a precise analysis of the paradox, and also briefly discuss Leibniz’s solution. (shrink)

I focus on the method designed by the young Leibniz in order to analyze specific legal modalities, the so-called suspensive conditions. Such a method, to analyze the specific conditionals whose if-part is a suspensive condition, gives Leibniz access to the resources of logical analysis of conditionals. I show that the contribution of logic to the law goes hand in hand with an extended complication of the former thanks to which Leibniz achieves at capturing a number of dynamical features of suspensive (...) conditions, but also of resolutory and temporal legal modalities. (shrink)

Prior's three-valued modal logic Q was developed as a philosophically interesting modal logic. Thus, we should be able to modify Q as a temporal logic. Although a temporal version of Q was suggested by Prior, the subject has not been fully explored in the literature. In this paper, we develop a three-valued temporal logic $Q_t $ and give its axiomatization and semantics. We also argue that $Q_t $ provides a smooth solution to the problem of future contingents.

Branching-time temporal logics have proved to be an extraordinarily successful tool in the formal specification and verification of distributed systems. Much of their success stems from the tractability of the model checking problem for the branching time logic CTL, which has made it possible to implement tools that allow designers to automatically verify that systems satisfy requirements expressed in CTL. Recently, CTL was generalised by Alur, Henzinger, and Kupferman in a logic known as "Alternating-time Temporal Logic". The key (...) insight in ATL is that the path quantifiers of CTL could be replaced by "cooperation modalities", of the form $\langle \langle \Gamma \rangle \rangle $, where Γ is a set of agents. The intended interpretation of an ATL formula $\langle \langle \Gamma \rangle \rangle \varphi $ is that the agents Γ can cooperate to ensure that φ holds. In this paper, we extend ATL with knowledge modalities, of the kind made popular in the work of Fagin, Halpern, Moses, Vardi and colleagues. Combining these knowledge modalities with ATL, it becomes possible to express such properties as "group Γ can cooperate to bring about φ iff it is common knowledge in Γ that ψ". The resulting logic -- Alternating-time Temporal Epistemic Logic -- shares the tractability of model checking with its ATL parent, and is a succinct and expressive language for reasoning about game-like multiagent systems. (shrink)

El presente artículo se ocupa de las postura de Aristóteles y Diodoro Crono en torno a la necesidad lógica. Allí se muestra que Aristóteles rechaza la necesidad lógica, mediante la negación de la aplicación universal del principio de bivalencia, para preservar el valor de la deliberación. En segundo lugar se muestra que Diodoro cae en un fatalismo al sostener, mediante sus nociones modales, que hay tipos de eventos lógicamente contingentes, aunque todos los sucesos particulares del mundo son necesarios.

Branching-time temporal logics have proved to be an extraordinarily successful tool in the formal specification and verification of distributed systems. Much of their success stems from the tractability of the model checking problem for the branching time logic CTL, which has made it possible to implement tools that allow designers to automatically verify that systems satisfy requirements expressed in CTL. Recently, CTL was generalised by Alur, Henzinger, and Kupferman in a logic known as Alternating-time Temporal Logic (ATL). The (...) key insight in ATL is that the path quantifiers of CTL could be replaced by cooperation modalities, of the form , where is a set of agents. The intended interpretation of an ATL formula is that the agents can cooperate to ensure that holds (equivalently, that have a winning strategy for ). In this paper, we extend ATL with knowledge modalities, of the kind made popular in the work of Fagin, Halpern, Moses, Vardi and colleagues. Combining these knowledge modalities with ATL, it becomes possible to express such properties as group can cooperate to bring about iff it is common knowledge in that . The resulting logic — Alternating-time Temporal Epistemic Logic (ATEL) — shares the tractability of model checking with its ATL parent, and is a succinct and expressive language for reasoning about game-like multiagent systems. (shrink)

Este artículo es una breve introducción al opúsculo De veritate propositionis de Roberto Grosseteste , acompañada por una traducción inédita de dicho opúsculo, la primera que por primera vez puede ser leído en español. En él Grosseteste comenta el problema de los futuros contingentes tal y como fue expuesto por Aristóteles en Sobre la interpretación, capítulo IX. Las soluciones de Aristóteles y Grosseteste son similares, aunque la idea de necesidad en Grosseteste está más vinculada a la eternidad que a (...) la temporalidad. (shrink)

La logica modale è nata per studiare i ragionamenti su ciò che è possibile e ciò che è necessario. Negli ultimi decenni, a partire dal lavoro di logici e filosofi quali Rudolf Carnap, Saul Kripke e David Lewis, la sua applicazione è stata progressivamente estesa ad altri ambiti, quali il ragionamento sul tempo, sulla conoscenza e sui sistemi di norme. Queste ricerche hanno condotto a un complesso e intrigante dialogo con alcune fondamentali branche della filosofia: la metafisica, l’epistemologia, la filosofia (...) del linguaggio. Lo scopo del volume è offrire una panoramica di questo dialogo al contempo accessibile e rigorosa, pensata sia per gli studenti di un corso universitario, sia per il lettore non specialista. Se da un lato tutte le necessarie nozioni tecniche sono rese accessibili ai non addetti ai lavori, dall'altro si restituisce un'immagine fedele delle sfide concettuali che la ricerca in questi settori è oggi chiamata ad affrontare. (shrink)

Las nociones epistémicas modales se definen como aquellos conceptos epistémicos que, como el de cognoscibilidad o el de indudabilidad, incluyen una nota modal. Segun se defiende en este trabajo, la semántica de mundos posibles y algunas de sus extensiones (especialmente las llevadas a cabo para logica temporal, logica epistemica y logica condicional) son instrumentos adecuados para deshacer el nudo de las intensionalidades superpuestas en estas nociones especialmente esquivas al análisis. Para mostrarlo, se proporcionan una serie de análisis sucesivos de (...) la nocion de cognoscibilidad que a partir de una interpretación naif van salvando una serie de presuposiciones, problemas y paradojas hasta dar con una analisis que se presume satisfactorio.“Modal epistemic notions” are those epistemic concepts which in some way or another has a modal element. These modal epistemic notions, although they could appear intuitively clear, they turn out to be particularly obscure, slippery, when one subjects them to a formal analysis. In this paper we will try to show that possible world semantics (and its extensions for epistemic, temporal and conditional logic) is an appropriate instrument for the explanation ofthese notions. Four successive analysis of the notion of “knowability” are given, ranging from a naive account to an analysis that gets to the bottom ofthe problem. (shrink)

An arithmetical language, whose words are natural numbers written in hexadecimal numeration system, is defined and its applications for the representation, analysis and decision of formulae of some logical and normative systems are described and illustrated.The formulae, operations and relations of the represented system are associated as follows respectively to the numbers and the arithmetical operations and relations of the proposed language:1. Each well-formed-formula of the system is associated to a number of a set of natural numbers between zero (associated (...) to all tautologies and theses) and the binary supremum Φ of the set (Φ is associated to all contradictions and antitheses and his value is 2n-1, n depending on system’s dimensions and structure).2. Negation of a formula and disjonction and conjonction of t wo other more formulae of the system are associated respectively to the binary complement Φ-N(f) of the number associated to the first and to the binary infimun and supremum of the numbers associated to the last.3. The logical relations “f1 implies f2” (f1 -> f2), “f1 is incompat.ible with f2” (f1|f2), “f1 is the contradictory opposite of f2” (f1wf2) and “f1 is alternative to f2” (f1vf2) are true in the represented system if and only if the associated arithmetical relations, respectively “N(f1) absorbs arithmetically N(f2)”, “binary supremum of N(f1) and N(f2) is equal to Φ”, “sum of of N(f1) and N(f2) is equal to Φ” and “binary infimum of N(f1) and N(f2) is equal to 0”, are true.The applications of the proposed arithmetical language as very rapid and simplified tool of analysis and decision method are shown for the following systems: propositional logic, syllogistic, some deontic and alethic modal systems and some normative systems of statutory law:1. In the propositional logic, after fixing the maximum of variables considered, the numbers associated to these variables and to the contradictions are calculated. On this permanent basis, the evaluation of a farmula (especially in the case of many occurrences of many variables) is performed by the calculation of the associated number faster as in the traditional way.2. In the syllogistic, after the calculation of the number associated to each type of premise or conclusion, an arithmetical table of syllogisms shows that a proposed syllogism is valid if and only if the binary supremum of the numbers associated to the premises absorbs arithmetically the number associated to the conclusion. It is well-known that the search of this sort of arithmetization of syllogistic and the study of its possibilitiy has been a recurrent topic in modern logic, from Leibniz to Łukasiewicz and to-day.3. In a deontic equivalent:ial system who includes Von Wright’s deontic system of 1982 far norms of the first order, the calculation of the numbers associated to an types of permission and obligation sentences allows the im mediate arithmetical verification of an the classical relations between those sentences. The same is shown for an alethic modal system including the modlities of the first order of Lewis’s S5.4. For normative systems of statutory law, the arithmetical verification of the logical and normative relations is shown in precedent author’s papers -especially in recent “The ‘Ars Judicandi.’ Programme”-, though not yet in hexadecimal numeration system but only in binary and decimal ones.In all the represented systems, the arithmetical verification of the metalogical properties of the system -consistency and completeness- is performed in a very easy and rapid manner, after the arithmetic representation of the axiomatic basis of the latter by a system of equations. (shrink)

Modernity seemed to be the autumn of eternity. The secularization of European culture provided little sustenance to the concept of eternity with its heavy theological baggage. Yet, our hero would not leave the stage without an outstanding performance of its power and temptation. Indeed, in the first three centuries of the modern period – the subject of the third chapter by Yitzhak Melamed - the concept of eternity will play a crucial role in the great philosophical systems of the period. (...) The first part of this chapter concentrates on the debate about the temporality of God. While most of the great metaphysicians of the seventeenth century – Suarez, Spinoza, Malebranche, and Leibniz – ascribed to God eternal, non-temporal, existence, a growing number of philosophers conceived God as existing in time. For Newton, God’s eternity was simply the fact that “He was, he is, and is to come.” A similar view of God as being essentially in time was endorsed by Pierre Gassendi, Henry More, Samuel Clarke, Isaac Barrow, John Locke, and most probably Descartes as well. In the second part of the chapter we study the concept of eternal truth, and its relation to the emerging notion of Laws of Nature. The third part of the chapter explicates Spinoza’s original understanding of eternity as a modal concept. For Spinoza, eternity is a unique kind of necessary existence: it is existence that is self-necessitated (unlike the existence of other things whose necessity derives from external causes). Eternity is the existence of God or the one substance. Yet, Spinoza claims that if we conceive finite things adequately - “sub specie aeternitatis” – as nothing but modes flowing from the essence of God, even finite things (like our minds) can take part in God’s eternity. The fourth and final part of the chapter is mostly focused on the reception of Spinoza’s original conception of eternity by Leibniz and other eighteenth century philosophers. (shrink)

We present sound, (weakly) complete and cut-free tableau systems for the propositional normal modal logicsS4.3, S4.3.1 andS4.14. When the modality is given a temporal interpretation, these logics respectively model time as a linear dense sequence of points; as a linear discrete sequence of points; and as a branching tree where each branch is a linear discrete sequence of points.Although cut-free, the last two systems do not possess the subformula property. But for any given finite set of formulaeX the superformulae (...) involved are always bounded by a finite set of formulaeX* L depending only onX and the logicL. Thus each system gives a nondeterministic decision procedure for the logic in question. The completeness proofs yield deterministic decision procedures for each logic because each proof is constructive. (shrink)

Durante deacute;cadas, la historia de la filosofiacute;a ha separado a Kierkegaard de Hegel y a Hegel de Kierkegaard, en detrimento tanto de la grandeza especulativa del pensamiento kierkegaardiano como de la vena existencial del sistema de Hegel. En oposicioacute;n a esta desafortunada lectura, el presente artiacute;culo intenta mostrar la profunda convergencia que une internamente el estadio eacute;tico de Kierkegaard con las maacute;s importantes categoriacute;as loacute;gicas de Hegel. Ambos pensadores conciben la idea como el poder real del devenir subjetivo y la (...) existencia como la concrecioacute;n actual de lo ideal. Para ambos, la pura eneacute;rgeia de la libertad, que comienza en la posibilidad abstracta y esteacute;tica de la inmediatez subjetiva, se realiza a siacute; misma como la actual concrecioacute;n de la finitud, capaz de asumir lo temporal y contingente por la fuerza eterna y necesaria del deber. La repeticioacute;n kierkegaardiana no es nada sino el poder de lo ideal, capaz de mediar el flujo de las diferencias finitas en la eterna identidad del sujeto. Sin embargo, tanto para Kierkegaard como para Hegel existe una absoluta contradiccioacute;n, llamada a promover la superacioacute;n de lo eacute;tico. (shrink)

O texto visa apresentar e explicar o papel desempenhado pelas noções de telos e arché na compreensão tanto do movimento dos animais quanto da ação humana realizada no âmbito ético-político. Além disso, pretende evidenciar a importância de estabelecer a distinção modal do tempo em passado, presente e futuro elaborada na Ética a Nicômacos, em oposiçãoà abordagem quantitativa do mesmo empreendida na Física, a fim de poder explicar o silogismo prático.

El primer objetivo de este escrito es evidenciar cómo Aristóteles acepta laaplicación del principio de tercero excluso y de bivalencia en enunciadossobre futuro contingente. La principal consecuencia de esto, contra lasposturas trivalentes, es que enunciados sobre futuro contingente no tienenvalor de verdad indeterminado: son o verdaderos o falsos. Se analizandiversas posturas indeterministas con el objetivo de evidenciar cómoestas dependen de una tesis implausible: i) que la aplicación del principiode tercero excluso y de bivalencia en enunciados sobre futuro deriva endeterminismo. Se (...) propone que el indeterminismo que puede atribuirse a estosenunciados es epistémico, no alético: Un sujeto S no conoce en un tiempot previo a t1 el valor de verdad de enunciados sobre futuro contingente. (shrink)

Importing subsumes several asymmetric ways of combining logics, including modalization and temporalization. A calculus is provided for importing, inheriting the axioms and rules from the given logics and including additional rules for lifting derivations from the imported logic. The calculus is shown to be sound and concretely complete with respect to the semantics of importing as proposed in J. Rasga et al. (100(3):541–581, 2012) Studia Logica.

We present an extension of the mosaic method aimed at capturing many-dimensional modal logics. As a proof-of-concept, we define the method for logics arising from the combination of linear tense operators with an “orthogonal” S5-like modality. We show that the existence of a model for a given set of formulas is equivalent to the existence of a suitable set of partial models, called mosaics, and apply the technique not only in obtaining a proof of decidability and a proof of completeness (...) for the corresponding Hilbert-style axiomatization, but also in the development of a mosaic-based tableau system. We further consider extensions for dealing with the case when interactions between the two dimensions exist, thus covering a wide class of bundled Ockhamist branching-time logics, and present for them some partial results, such as a non-analytic version of the tableau system. (shrink)