This paper is about a special version of PDL, proposed by Marcus Kracht, for reasoning about sibling ordered trees. It has four basic programs corresponding to the child, parent, left- and right-sibling relations in such trees. The original motivation for this language is rooted in the field of model-theoretic syntax. Motivated by recent developments in the area of semi-structured data, and, especially, in the field of query languages for XML documents, we revisit the language. This renewed interest comes with a (...) special focus on complexity and expressivity aspects of the language, aspects that have so far largely been ignored. We survey and derive complexity results, and spend most of the paper on the most important open question concerning the language: what is its expressive power? We approach this question from two angles: Which first-order properties can be expressed? And which second-order properties? While we are still some way from definitive answers to these questions, we discuss two first-order fragments of the PDL language for ordered trees, and show how the language can be used to express some typical problems, like the boolean circuit and the frontier problem. (shrink)
This paper is the first in a series of three related papers on modal methods in interpretability logics and applications. In this first paper the fundaments are laid for later results. These fundaments consist of a thorough treatment of a construction method to obtain modal models. This construction method is used to reprove some known results in the area of interpretability like the modal completeness of the logic IL. Next, the method is applied to obtain new results: the modal completeness (...) of the logic ILMo, and modal completeness of ILW*. (shrink)
The interpretability logic of a mathematical theory describes the structural behavior of interpretations over that theory. Different theories have different logics. This paper revolves around the question what logic describes the behavior that is present in all theories with a minimum amount of arithmetic; the intersection over all such theories so to say. We denote this target logic by IL.In this paper we present a new principle R in IL. We show that R does not follow from the logic ILP0W* (...) that contains all previously known principles. This is done by providing a modal incompleteness proof of ILP0W*: showing that R follows semantically but not syntactically from ILP0W*. Apart from giving the incompleteness proof by elementary methods, we also sketch how to work with so-called Generalized Veltman Semantics as to establish incompleteness. To this extent, a new version of this Generalized Veltman Semantics is defined and studied. Moreover, for the important principles the frame correspondences are calculated.After the modal results it is shown that the new principle R is indeed valid in any arithmetically theory. The proof employs some elementary results on definable cuts in arithmetical theories. (shrink)
In this note we show that the classical modal technology of Sahlqvist formulas gives quick proofs of the completeness theorems in  : 57-78, 2001) and vastly generalizes them. Moreover, as a corollary, interpolation theorems for the logics considered in  are obtained. We then compare Gregory's modal language enriched with an "actually" operator with the work of Arthur Prior now known under the name of hybrid logic. This analysis relates the "actually" axioms to standard hybrid axioms, yields the decidability (...) results in , and provides a number of complexity results. Finally, we use a bisimulation argument to show that the hybrid language is strictly more expressive than Gregory's language. (shrink)
This paper is the second in a series of three papers. All three papers deal with interpretability logics and related matters. In the first paper a construction method was exposed to obtain models of these logics. Using this method, we obtained some completeness results, some already known, and some new. In this paper, we will set the construction method to work to obtain more results. First, the modal completeness of the logic ILM is proved using the construction method. This is (...) not a new result, but by using our new proof we can obtain new results. Among these new results are some admissible rules for ILM and GL. Moreover, the new proof will be used to classify all the essentially Δ1 and also all the essentially Σ1 formulas of ILM. Closely related to essentially Σ1 sentences are the so-called self provers. A self-prover is a formula φ which implies its own provability, that is φ → □φ. Each formula φ will generate a self prover φ ^ □φ. We will use the construction method to characterize those sentences of GL that generate a self prover that is trivial in the sense that it is Σ1. (shrink)
This paper shows how Aquinas gradually developed his view on angelic speech. His major texts are summarized and compared to those of contemporaries (sections II-III). Next the texts are analyzed, focusing on three issues: the notion of ‘word’ (section IV), the role of the will (section V), and the need of signification (section VI). With regard to each of these topics, Aquinas’ thought evolved, first by juxtaposing and later by integrating Augustinian and Aristotelian viewpoints. Aquinas reaches his mature position in (...) the Summa Theologiae by radically reinterpretating the meaning of ‘word’ and denying that angelic speech has a semiotic structure. (shrink)
The doctrine of being developed by the Franciscan theologian Peter of Oriol 1is highly original. The present contribution will analyse this doctrine from a distinct point of view. It is mainly interested in Aureoli's description of the concept of being as an implicit concept and reads his doctrine of being exclusively in this regard. The interest of the idea that the concept of being is entirely implicit lies in the particularity that the Franciscan also holds the concept of being to (...) be the first concept known by reason. Peter of Oriol hence presents us with a doctrine of implicit knowledge of being as the unannullable condition, the conditio sine qua non of explicit knowledge. (shrink)
Stories about near-death experiences draw much attention from the general public and are extensively discussed by medical doctors and neuroscientists. However, though eschatology belongs to their core business, only few theologians participate in the debate. This article proposes a theological interpretation of NDEs as ‘private revelations’. I first give a critical analysis of the development of the modern, allegedly ‘scientific’, concept of NDE. This concept changes concrete personal testimonies into statistical data that are used as scientific evidence for the existence (...) of an immortal soul. Next, the main criticisms against this concept from neurosciences, study of mysticism and philosophy of mind are discussed. Finally, I argue that ‘private revelation’ is a useful model for a theological understanding of NDEs and that an analogy from Thomas Aquinas’ view on prophetic dreams can help to account for the specific circumstance of imminent death. The interpretation I propose can do justice to the impression NDEs make on people, but can also accept and meet some of the most important criticisms raised against the modern concept of NDEs. (shrink)
The medieval doctrine of God as first known presents a privileged moment in a tradition of classical metaphysics that runs from Plato to Levinas. The presentcontribution analyzes two versions of this doctrine formulated by Bonaventure († 1274) and Henry of Ghent († 1293). In reaction to the preceding discussion inParis, they advance a doctrine of God as first known that distinguishes the relative priority of God within the first known transcendental concepts from the absolutepriority of God over these. Although their (...) two-staged doctrines of God as first known structurally agree, they vary in their strategical embedding. Underlying this variation is a transformation of the concept of reality that abstracts actuality as a standard and criterion to the determination of the first known. As such, thisconcept of reality gives rise to the very idea of neutral existence against which Levinas objects. (shrink)
In deze bijdrage worden twee alternatieve visies op de verhouding van geloof en weten geschetst aan de hand van twee middeleeuwse denkers: Bernardus van Clairvaux en Thomas van Aquino. De auteur betoogt dat zowel de ‘rationalistische’ positie van Thomas van Aquino als de ‘anti-rationalistische’ positie van Bernardus van Clairvaux beide, geloof en rede, als op waarheid betrokken zien, waarmee zicht wordt geboden op het wijsgerige belang van dit vermeende anti-rationalisme.
In this paper we will be concerned with the interpretability logic of PA and in particular with the fact that this logic, which is denoted by ILM, does not have the interpolation property. An example for this fact seems to emerge from the fact that ILM cannot express Σ₁-ness. This suggests a way to extend the expressive power of interpretability logic, namely, by an additional operator for Σ₁-ness, which might give us a logic with the interpolation property. We will formulate (...) this extension, give an axiomatization which is modally complete and arithmetically complete (although for proofs of these theorems we refer to an earlier paper), and investigate interpolation. We show that this logic still does not have the interpolation property. (shrink)
Pendant le symposium à l'occasion des adieux d'Albert Zimmermann, Jan Aertsen, son successeur au Thomas-Institut à Cologne, profitait de cette occasion pour présenter sa conception de la philosophie médiévale. Dans sa contribution à ce symposium, Aertsen décrit un débat dans lequel il prend ensuite position lui-même. Dans le titre de sa contribution, la problématique de ce débat se trouve résumée: «Gibt es eine mittelalterliche Philosophie?». Aertsen répond à cette question en discutant de trois conceptions de la philosophie médiévale. Il prête (...) consécutivement attention à la conception d'Etienne Gilson de la philosophie médiévale comme une «philosophie chrétienne»; à la conception logique-analytique de la philosophie médiévale comme elle a été codifiée dans le Cambridge History of Later Medieval Philosophy; et finalement à la conception d'Alain de Libera, successeur de Gilson à l'École Pratique des Hautes Etudes à Paris et un des auteurs du Cambridge History, qui cherche le propre de la philosophie médiévale dans le développement d'un idéal philosophique, celui de l'«aristotélisme éthique». (shrink)
We establish the bi-modal forgetful projection of the Logic of Proofs and Formal Provability GLA. That is to say, we present a normal bi-modal provability logic with modalities □ and whose theorems are precisely those formulas for which the implicit provability assertions represented by the modality can be realized by explicit proof terms.
"Absolute Beginners" is a multi-approach study of the founding role of the Absolute as the very beginning of knowledge in medieval philosophy (Henry of Ghent, Richard Conington), the subject being addressed from historical, methodological, ...
This study attempts a reconstruction of the philosophy of the unfinished _Opus tripartitum_ of Meister Eckhart. One new feature is the 'hermeneutic approach' to the question of Eckhart's philosophy, another is the recognition that the concept of Unity plays a decisive role in the organisation of his metaphysics, to the extent, indeed, that one can speak of a metaphysics of the One. Eckhart's metaphysics is determined to a contemplation of the divine, which in this thinking of Unity is understood as (...) the principle and end of all things. Yet, at the same time Eckhart's thinking of Unity marks the boundaries of the _ratio naturalis_. In this way his thought combines a supreme trust in the effectivity of reason with a negative theology of the purest sort. By a systematic reading of linked texts of Meister Eckhart the trail of his philosophy of the One is followed down into the German sermons. (shrink)
This book gives a comprehensive overview of central themes of finite model theory â expressive power, descriptive complexity, and zero-one laws â together with selected applications relating to database theory and artificial intelligence, especially constraint databases and constraint satisfaction problems. The final chapter provides a concise modern introduction to modal logic, emphasizing the continuity in spirit and technique with finite model theory. This underlying spirit involves the use of various fragments of and hierarchies within first-order, second-order, fixed-point, and infinitary logics (...) to gain insight into phenomena in complexity theory and combinatorics. The book emphasizes the use of combinatorial games, such as extensions and refinements of the Ehrenfeucht-Fraissé pebble game, as a powerful way to analyze the expressive power of such logics, and illustrates how deep notions from model theory and combinatorics, such as o-minimality and treewidth, arise naturally in the application of finite model theory to database theory and AI. Students of logic and computer science will find here the tools necessary to embark on research into finite model theory, and all readers will experience the excitement of a vibrant area of the application of logic to computer science. (shrink)
L’« effet un peu criant » produit par la juxtaposition qui constitue le concept d’a priori historique ne l’a pas empêché de devenir un concept directeur de l’« épistémologie historique », une orientation philosophique qui introduit la contingence historique au sein des « cadres infrangibles » qui structurent notre expérience. On peut identifier les travaux de cette tradition...
Maarten Boudry and Jerry Coyne have written a piece, forthcoming in Philosophical Psychology, called “Disbelief in Belief,” in which they criticize my recent paper “Religious credence is not factual belief” (2014, Cognition 133). Here I respond to their criticisms, the thrust of which is that we shouldn’t distinguish religious credence from factual belief, contrary to what I say. I respond that their picture of religious psychology undermines our ability to distinguish common religious people from fanatics. My response will appear (...) in the same issue as their paper. (shrink)
Many recent developments in artificial intelligence research are relevant for traditional issues in the philosophy of science. One of the developments in AI research we want to focus on in this article is diagnostic reasoning, which we consider to be of interest for the theory of explanation in general and for an understanding of explanatory arguments in economic science in particular. Usually, explanation is primarily discussed in terms of deductive inferences in classical logic. However, in recent AI research it is (...) observed that a diagnostic explanation is actually quite different from deductive reasoning. In diagnostic reasoning the emphasis is on restoring consistency rather than on deduction. Intuitively speaking, the problem diagnostic reasoning is concerned with is the following. Consider a description of a system in which the normal behavior of the system is characterized and an observation that conflicts with this normal behavior. The diagnostic problem is to determine which of the components of the system can, when assumed to be functioning abnormally, account for the conflicting observation. A diagnosis is a set of allegedly malfunctioning components that can be used to restore the consistency of the system description and the observation. In this article, this kind of reasoning is formalized and we show its importance for the theory of explanation. We will show how the diagnosis nondeductively explains the discrepancy between the observed and the correct system behavior. The article also shows the relevance of the subject for real scientific arguments by showing that examples of diagnostic reasoning can be found in Friedman's Theory of the Consumption Function. Moreover, it places the philosophical implications of diagnostic reasoning in the context of Mill's aprioristic methodology. (shrink)
Basisbegrippen. Een formeel model voor de ontwikkeling van de kunst is een structuur T, <, K, , d, p, q, s, B , waarbij T een verzameling van “tijdstippen” is, < (“is eerder dan”) een relatie op T is, K een verzameling van “mogelijke kunstwerken” is, (“levert commentaar op”) een relatie op K is, d, p, q en s functies van K naar de verzameling van alle deelverzamelingen van K zijn, en B een functie van T naar de verzameling van (...) alle deelverzamelingen van K is. d(x) is de discipline waartoe kunstwerk x behoort, p(x) is het proc´ed´e waarmee x vervaardigd is, q(x) is de kwaliteit van x, s(x) is de stijl van x, en B(t) is de verzameling kunstwerken die op tijdstip t bestaan. (shrink)