It is shown that axiomatic extensions of intuitionistic propositional calculus defining univocally new connectives, including those proposed by Gabbay, are strongly complete with respect to valuations in Heyting algebras with additional operations. In all cases, the double negation of such a connective is equivalent to a formula of intuitionistic calculus. Thus, under the excluded third law it collapses to a classical formula, showing that this condition in Gabbay's definition is redundant. Moreover, such connectives can not be interpreted in all Heyting (...) algebras, unless they are already equivalent to a formula of intuitionistic calculus. These facts relativize to connectives over intermediate logics. In particular, the intermediate logic with values in the chain of length n may be "completed" conservatively by adding a single unary connective, so that the expanded system does not allow further axiomatic extensions by new connectives. (shrink)

An extensions by new axioms and rules of an algebraizable logic in the sense of Blok and Pigozzi is not necessarily algebraizable if it involves new connective symbols, or it may be algebraizable in an essentially different way than the original logic. However, extension whose axioms and rules define implicitly the new connectives are algebraizable, via the same equivalence formulas and defining equations of the original logic, by enriched algebras of its equivalente quasivariety semantics. For certain strongly algebraizable logics, all (...) connectives defined implicitly by axiomatic extensions of the logic are explicitly definable. (shrink)

In this paper, we present a prenex form theorem for a version of Independence Friendly logic, a logic with imperfect information. Lifting classical results to such logics turns out not to be straightforward, because independence conditions make the formulas sensitive to signalling phenomena. In particular, nested quantification over the same variable is shown to cause problems. For instance, renaming of bound variables may change the interpretations of a formula, there are only restricted quantifier extraction theorems, and slashed connectives cannot be (...) so easily removed. Thus we correct some claims from Hintikka [8], Caicedo & Krynicki [3] and Hodges [11]. We refine definitions, in particular the notion of equivalence, and sharpen preconditions, allowing us to restore those claims, including the prenex form theorem of Caicedo & Krynicki [3], and, as a side result, we obtain an application to Skolem forms of classical formulas. It is a known fact that a complete calculus for IF-logic is impossible, but with our results we establish several quantifier rules that form a partial calculus of equivalence for a general version of IF-logic reflecting general properties of information flow in games. (shrink)

It is shown that axiomatic extensions of intuitionistic propositional calculus defining univocally new connectives, including those proposed by Gabbay, are strongly complete with respect to valuations in Heyting algebras with additional operations. In all cases, the double negation of such a connective is equivalent to a formula of intuitionistic calculus. Thus, under the excluded third law it collapses to a classical formula, showing that this condition in Gabbay's definition is redundant. Moreover, such connectives can not be interpreted in all Heyting (...) algebras, unless they are already equivalent to a formula of intuitionistic calculus. These facts relativize to connectives over intermediate logics. In particular, the intermediate logic with values in the chain of length n may be "completed" conservatively by adding a single unary connective, so that the expanded system does not allow further axiomatic extensions by new connectives. (shrink)

We prove strong completeness of the □-version and the ◊-version of a Gödel modal logic based on Kripke models where propositions at each world and the accessibility relation are both infinitely valued in the standard Gödel algebra [0,1]. Some asymmetries are revealed: validity in the first logic is reducible to the class of frames having two-valued accessibility relation and this logic does not enjoy the finite model property, while validity in the second logic requires truly fuzzy accessibility relations and this (...) logic has the finite model property. Analogues of the classical modal systems D, T, S4 and S5 are considered also, and the completeness results are extended to languages enriched with a discrete well ordered set of truth constants. (shrink)

We study a class of [0,1][0,1]-valued logics. The main result of the paper is a maximality theorem that characterizes these logics in terms of a model-theoretic property, namely, an extension of the omitting types theorem to uncountable languages.

An extensions by new axioms and rules of an algebraizable logic in the sense of Blok and Pigozzi is not necessarily algebraizable if it involves new connective symbols, or it may be algebraizable in an essentially different way than the original logic. However, extension whose axioms and rules define implicitly the new connectives are algebraizable, via the same equivalence formulas and defining equations of the original logic, by enriched algebras of its equivalente quasivariety semantics. For certain strongly algebraizable logics, all (...) connectives defined implicitly by axiomatic extensions of the logic are explicitly definable. (shrink)

Este trabajo se encuadra dentro de una nueva visión de la lógica de Leibniz, la cual pretende mostrar que sus escritos fueron ricos no solamente en proyectos ambiciosos sino también en desarrollos lógico-matematicos concretos. Se demuestra que su “Caracteristica Numerica” que asigna pares de números a las proposiciones categóricas es una semántiea para la cual la silogística aristotélica es correcta y completa, y que el sistema algebraico presentado en Fundamentos de un Cálculo Lógico es una lógica algebraica similar a la (...) de Boole.This work is a contribution to a new view of Leibniz’s logic, pretending to show that his writings were not only rich in projects, but also in concrete logico-mathematical developments. We prove that his “Numerical Characteristic” assigning pairs of numbers to terms of categorical propositions, is a complete and correct semantics for aristotelian syllogistic, and the algebraic system presented in Fundamentals of Logical Calculus is essentially a complete version of boolean algebraic logic. (shrink)

"Contains a balanced account of recent advances in set theory, model theory, algebraic logic, and proof theory, originally presented at the Tenth Latin American Symposium on Mathematical Logic held in Bogata, Columbia. Traces new interactions among logic, mathematics, and computer science. Features original research from over 30 well-known experts worldwide.".

We generalize a theorem of Mundici relating compactness of a regular logic L to a strong form of normality of the associated spaces of models. Moreover, it is shown that compactness is in fact equivalent to ordinary normality of the model spaces when L has uniform reduction for infinite disjoint sums of structures. Some applications follow. For example, a countably generated logic is countably compact if and only if every clopen class in the model spaces is elementary. The model spaces (...) of L are not normal for vocabularies of uncountable power ωα. It also follows that first-order logic is the only finite-dependence logic having normal model spaces and satisfying at the same time the downward Löwenheim-Skolem theorem and uniform reduction for pairs. (shrink)

It is shown that Friedman's problem, whether there exists a proper extension of first order logic satisfying the compactness and interpolation theorems, has extremely simple positive solutions if one considers extensions by generalized (finitary) propositional connectives. This does not solve, however, the problem of whether such extensions exist which are also closed under relativization of formulas.

In any model theoretic logic, Beth’s definability property together with Feferman-Vaught’s uniform reduction property for pairs imply recursive compactness, and the existence of models with infinitely many automorphisms for sentences having infinite models. The stronger Craig’s interpolation property plus the uniform reduction property for pairs yield a recursive version of Ehrenfeucht-Mostowski’s theorem. Adding compactness, we obtain the full version of this theorem. Various combinations of definability and uniform reduction relative to other logics yield corresponding results on the existence of non-rigid (...) models. (shrink)

We introduce a natural class of quantifiersTh containing all monadic type quantifiers, all quantifiers for linear orders, quantifiers for isomorphism, Ramsey type quantifiers, and plenty more, showing that no sublogic ofL ωω (Th) or countably compact regular sublogic ofL ∞ω (Th), properly extendingL ωω , satisfies the uniform reduction property for quotients. As a consequence, none of these logics satisfies eitherΔ-interpolation or Beth's definability theorem when closed under relativizations. We also show the failure of both properties for any sublogic ofL (...) ∞ω (Th) in which Chang's quantifier or some cardinality quantifierQ α, with α≧1, is definable. (shrink)

Lindström’s theorem obviously fails as a characterization of first-order logic without identity ( $\mathcal {L}_{\omega \omega }^{-} $ ). In this note, we provide a fix: we show that $\mathcal {L}_{\omega \omega }^{-} $ is a maximal abstract logic satisfying a weak form of the isomorphism property (suitable for identity-free languages and studied in [11]), the Löwenheim–Skolem property, and compactness. Furthermore, we show that compactness can be replaced by being recursively enumerable for validity under certain conditions. In the proofs, we (...) use a form of strong upwards Löwenheim–Skolem theorem not available in the framework with identity. (shrink)

En enero de 1974 Zubiri dio un breve curso en la Sociedad de Estudios y Publicaciones de Madrid sobre el tema Tres dimensiones del ser humano: individual, social e histórica. Meses después publicó la última de esas lecciones bajo el título de La dimensión histórica del ser humano. El presente volumen recoge el texto de las tres conferencias, más la versión escrita de la última de ellas. La tesis que Zubiri desarrolla en estas lecciones es que el ser humano es (...) a la vez y necesariamente individual, social e histórico, y que estas dimensiones vienen exigidas por su propia realidad específica, que es pluralizante, continuante y prospectiva. (shrink)

El adiós a las escuelas -- El capitalismo, motor de la modernidad tardía.

En un diálogo continuo con la tradición filosófica, Zubiri va página a página describiendo el acto de la intelección humana y desmontando el cúmulo de hipótesis y teorías que subyacen al llamado «problema del conocimiento». Zubiri consigue descubrir, mediante el recurso a un procedimiento puramente descriptivo, en qué consiste el acto humano por excelencia, la intelección. La intelección humana, dice Zubiri, no es una síntesis trascendental, sino algo más simple a la vez que más radical, la mera actualización de lo (...) real en la inteligencia sentiente. (shrink)

In the absence of Woodin cardinals, fine structural inner models for mild large cardinal hypotheses admit forcing extensions where bounded forcing axioms hold and yet the reals are projectively well-ordered.

The Statesman is a difficult and puzzling Platonic dialogue. In A Stranger's Knowledge Marquez argues that Plato abandons here the classic idea, prominent in the Republic, that the philosopher, qua philosopher, is qualified to rule. Instead, the dialogue presents the statesman as different from the philosopher, the possessor of a specialist expertise that cannot be reduced to philosophy. The expertise is of how to make a city resilient against internal and external conflict in light of the imperfect sociality of human (...) beings and the poverty of their reason. This expertise, however, cannot be produced on demand: one cannot train statesmen like one might train carpenters. Worse, it cannot be made acceptable to the citizens, or operate in ways that are not deeply destructive to the city's stability. Even as the political community requires his knowledge for its preservation, the genuine statesman must remain a stranger to the city. Marquez shows how this impasse is the key to understanding the ambiguous reevaluation of the rule of law that is the most striking feature of the political philosophy of the Statesman. The law appears here as a mere approximation of the expertise of the inevitably absent statesman, dim images and static snapshots of the clear and dynamic expertise required to steer the ship of state across the storms of the political world. Yet such laws, even when they are not created by genuine statesmen, can often provide the city with a limited form of cognitive capital that enables it to preserve itself in the long run, so long as citizens, and especially leaders, retain a “philosophical” attitude towards them. It is only when rulers know that they do not know better than the laws what is just or good (and yet want to know what is just and good) that the city can be preserved. The dialogue is thus, in a sense, the vindication of the philosopher-king in the absence of genuine political knowledge. (shrink)

Insurance is pervasive in many social settings. As a cooperative device based on risk pooling, it serves to attenuate the adverse consequences of various risks by offering policyholders coverage against the losses implied by adverse events in exchange for the payment of premiums. In the insurance industry, the concept of actuarial fairness serves to establish what could be adequate, fair premiums. Accordingly, premiums paid by policyholders should match as closely as possible their risk exposure. Such premiums are the product of (...) the probabilities of losses and the expected losses. This article presents a discussion of the fairness of actuarial fairness through three steps: defining the concept based on its formulation within the insurance industry; determining in which sense it may be about fairness; and raising some objections to the actual fairness of actuarial fairness. The necessity of a normative evaluation of actuarial fairness is justified by the influence of the concept on the current reforms of public insurance systems and the fact that it highlights the question of the repartition of the gains and burdens of social cooperation. (shrink)

On sait que sur la question des pensées imperceptibles, Nicole, bientôt suivi par le bénédictin François Lamy, heurte l’opinion d’Arnauld. L’opposition d’Arnauld à Nicole touchant la question de la grâce est bien connue. C’est plutôt de l’originalité philosophique de Nicole qu’il s’agira dans ce travail. En tirant Nicole de « l’ombre d’Arnauld et de Pascal » et en lui faisant crédit d’une consistance spéculative propre, on retrouvera la finesse de sa psychologie, mais on découvrira également l’esquisse d’une subtile anthropologie.

Cogito, ergo sum. Ce n’est pas tiré de la plume de Descartes que cet énoncé apparaît pour la première fois dans les œuvres du philosophe. C’est en effet Caterus qui, dans la première série d’Objectiones aux Meditationes l’écrit, telle que la postérité scolaire va le considérer, comme un objet textuel autonome et digne par soi-même d’intérêt. Le fait est anecdotique, mais il est révélateur. D’abord parce que ce n’est pas Descartes qui a constitué...

In this article we defend the inferential view of scientific models and idealisation. Models are seen as “inferential prostheses” (instruments for surrogative reasoning) construed by means of an idealisation-concretisation process, which we essentially understand as a kind of counterfactual deformation procedure (also analysed in inferential terms). The value of scientific representation is understood in terms not only of the success of the inferential outcomes arrived at with its help, but also of the heuristic power of representation and their capacity to (...) correct and improve our models. This provides us with an argument against Sugden’s account of credible models: the likelihood or realisticness (their “credibility”) is not always a good measure of their acceptability. As opposed to “credibility” we propose the notion of “enlightening”, which is the capacity of giving us understanding in the sense of an inferential ability. (shrink)

We introduce and consider the inner-model reflection principle, which asserts that whenever a statement \varphi(a) in the first-order language of set theory is true in the set-theoretic universe V, then it is also true in a proper inner model W \subset A. A stronger principle, the ground-model reflection principle, asserts that any such \varphi(a) true in V is also true in some non-trivial ground model of the universe with respect to set forcing. These principles each express a form of width (...) reflection in contrast to the usual height reflection of the Lévy–Montague reflection theorem. They are each equiconsistent with ZFC and indeed \Pi_2-conservative over ZFC, being forceable by class forcing while preserving any desired rank-initial segment of the universe. Furthermore, the inner-model reflection principle is a consequence of the existence of sufficient large cardinals, and lightface formulations of the reflection principles follow from the maximality principle MP and from the inner-model hypothesis IMH. We also consider some questions concerning the expressibility of the principles. (shrink)

Cuerpo físico y cuerpo vivido no deben interpretarse como dos realidades separadas, sino que representan dos aspectos distintos de la misma experiencia corporal descrita en términos fenomenológicos. La experiencia de la corporalidad puede seguir dos direcciones: a) la primera es una dirección ascendente, que implica el desarrollo de las posibilidades de movimiento, acción y expresión del cuerpo en toda su plenitud; b) el otro camino o dirección, en contraste con el anterior, toma una senda descendente, y se halla estrechamente relacionado (...) con la experiencia del dolor, de la limitación, de la resistencia corporal que es experimentada como oposición a la propia voluntad: es el cuerpo vulnerable. Aquí reaparece un nuevo dualismo: el cuerpo poético de la expresión frente al cuerpo vulnerable de la enfermedad y el dolor. Mi contribución trata de cuestionar la idea de que el cuerpo poético de la expresión artística y el cuerpo vulnerable de la vejez, de la discapacidad o de la enfermedad, pertenezcan a esferas completamente separadas. Con este propósito, se intenta poner de relieve los múltiples puntos de intersección entre ambas perspectivas, al tiempo que se sostiene que tales intersecciones pueden llegar a ser interesantes puntos de partida de la investigación interdisciplinar. (shrink)