Modal Frame Correspondences and Fixed-Points

Taking Löb's Axiom in modal provability logic as a running thread, we discuss some general methods for extending modal frame correspondences, mainly by adding fixed-point operators to modal languages as well as their correspondence languages. Our suggestions are backed up by some new results – while we also refer to relevant work by earlier authors. But our main aim is advertizing the perspective, showing how modal languages with fixed-point operators are a natural medium to work with.     

On an Intuitionistic Modal Logic

In this paper we consider an intuitionistic variant of the modal logic S4 (which we call IS4). The novelty of this paper is that we place particular importance on the natural deduction formulation of IS4— our formulation has several important metatheoretic properties. In addition, we study models of IS4— not in the framework of Kirpke semantics, but in the more general framework of category theory. This allows not only a more abstract definition of a whole class of models but also a means of modelling proofs as well as provability.     

On modal μ-calculus with explicit interpolants

This paper deals with the extension of Kozen's μ-calculus with the so-called “existential bisimulation quantifier”. By using this quantifier one can express the uniform interpolant of any formula of the μ-calculus. In this work we provide an explicit form for the uniform interpolant of a disjunctive formula and see that it belongs to the same level of the fixpoint alternation hierarchy of the μ-calculus than the original formula. We show that this result cannot be generalized to the whole logic, because the closure of the third level of the hierarchy under the existential bisimulation quantifier is the whole μ-calculus. However, we prove that the first two levels of the hierarchy are closed. We also provide the μ-logic extended with the bisimulation quantifier with a complete calculus.     

The Epsilon Calculus and Herbrand Complexity

Hilbert's ε-calculus is based on an extension of the language of predicate logic by a term-forming operator e _x . Two fundamental results about the ε-calculus, the first and second epsilon theorem, play a rôle similar to that which the cut-elimination theorem plays in sequent calculus. In particular, Herbrand's Theorem is a consequence of the epsilon theorems. The paper investigates the epsilon theorems and the complexity of the elimination procedure underlying their proof, as well as the length of Herbrand disjunctions of existential theorems obtained by this elimination procedure.     

On Modal <Emphasis Type="Italic">μ</Emphasis>-Calculus and Gödel-Löb Logic

We show that the modal μ-calculus over GL collapses to the modal fragment by showing that the fixpoint formula is reached after two iterations and answer to a question posed by van Benthem in [4]. Further, we introduce the modal μ ^~-calculus by allowing fixpoint constructors for any formula where the fixpoint variable appears guarded but not necessarily positive and show that this calculus over GL collapses to the modal fragment, too. The latter result allows us a new proof of the de Jongh, Sambin Theorem and provides a simple algorithm to construct the fixpoint formula. Presented by Melvin Fitting     

论医疗侵权诉讼中的举证责任倒置规则

医患关系的紧张再次引起了人们对医疗侵权诉讼中举证责任倒置这一规则的关注。在对有关基本知识和人们争议焦点的分析之后,提出了应该在准确理解医疗侵权诉讼中举证责任倒置的适用条件和内容的基础上,结合这一规则设置的目的在当事人之间合理地进行举证责任的分配。     

Amount of General Factor Saturation in the Beck Anxiety Inventory Responses of Outpatients with Anxiety Disorders

To ascertain the amount of general factor saturation underlying the Beck Anxiety Inventory (BAI; Beck and Steer 1993) responses of anxious patients, the BAI was administered to 525 outpatients with DSM-IV-TR anxiety disorders. McDonald’s omega_hierarchical(ω _h) (Zinbarg et al. 2005) was calculated to estimate the amount of general factor saturation; ω _h is the amount of variance explained by a general (second-order) factor underlying a scale composed of multiple items divided by the total amount of variance explained by all of the scale’s items. An iterated principal-factor analysis was first performed and followed by a Schmid-Leiman transformation to determine the amount of variance explained by the second-order factor. ω _h was 0.70 [95% CI 0.66–0.72], and this value was discussed as supporting the current practice of summing the 21 symptom ratings of the BAI to estimate the overall severity of self-reported anxiety.

Correspondence Results for Relational Proof Systems with Application to the Lambek Calculus

We present a general framework for proof systems for relational theories. We discuss principles of the construction of deduction rules and correspondences reflecting relationships between semantics of relational logics and the rules of the respective proof systems. We illustrate the methods developed in the paper with examples relevant for the Lambek calculus and some of its extensions.     

Rhetoric and the Reception Theory of Rationality in the Work of Two Buddhist Philosophers

Although rhetoric is not a category of ancient Indian philosophy, this paper argues that Śāntarakṣita and Kamalaśīla, 2 eighth-century Indian Buddhist philosophers, can nonetheless be seen to embrace a rhetorical conception of rationality. That is, while these thinkers are strong proponents of rational analysis and philosophical argumentation as tools for attaining certainty, they also uphold the contingent nature of all such processes. Drawing on the categories of the New Rhetoric, this paper argues that these Buddhist thinkers understand philosophical argumentation to be directed at a universal audience of rational beings, where this universal audience is not an actual audience but a rhetorical one constructed through the author’s particular and historically contingent conception of what counts as rational. A reception theory of rationality is one that holds that the rationality of an argument depends upon its acceptance by a rational audience. When philosophers recognize the historically contingent nature of what counts as rational, they can embrace a reception theory of rationality that neither reduces the rational to mere opinion nor restricts it to a single, absolute, and timeless standard.

A Deep Inference System for the Modal Logic S5

We present a cut-admissible system for the modal logic S5 in a formalism that makes explicit and intensive use of deep inference. Deep inference is induced by the methods applied so far in conceptually pure systems for this logic. The system enjoys systematicity and modularity, two important properties that should be satisfied by modal systems. Furthermore, it enjoys a simple and direct design: the rules are few and the modal rules are in exact correspondence to the modal axioms. Presented by Heinrich Wansing     

Finitistic and Frequentistic Approximation of Probability Measures with or without <Emphasis Type="Italic">σ</Emphasis>-Additivity

In this paper a theory of finitistic and frequentistic approximations — in short: f-approximations — of probability measures P over a countably infinite outcome space N is developed. The family of subsets of N for which f-approximations converge to a frequency limit forms a pre-Dynkin system . The limiting probability measure over D can always be extended to a probability measure over , but this measure is not always σ-additive. We conclude that probability measures can be regarded as idealizations of limiting frequencies if and only if σ-additivity is not assumed as a necessary axiom for probabilities. We prove that σ-additive probability measures can be characterized in terms of so-called canonical and in terms of so-called full f-approximations. We also show that every non-σ-additive probability measure is f-approximable, though neither canonically nor fully f-approximable. Finally, we transfer our results to probability measures on open or closed formulas of first-order languages.     

The Internal and External Problems of String Theory: A Philosophical View

String theory is at the moment the only advanced approach to a unification of all interactions, including gravity. But, in spite of the more than 30 years of its existence, it does not make any empirically testable predictions, and it is completely unknown which physically interpretable principles could form the basis of string theory. At the moment, “string theory” is no theory at all, but rather a labyrinthic structure of mathematical procedures and intuitions. The only motivations for string theory consist in the mutual incompatibility of the standard model of quantum field theory and of general relativity as well as in the metaphysics of the unification program of physics, aimed at a final unified theory of all interactions, including gravity. The article gives a perspective on the problems leading to and resulting from this situation.

On the Dao in the Commentary of the Book of Change

The existence of the Dao 道 (the Way), according to the Yizhuan 易传 (the Commentary), is something intangible. The connotation of the Dao is the law of change caused by the interaction between yin and yang. The main functions of the Dao are “to change” and “to generate”. The intangible refers to the law of change caused by the interaction between yin and yang, and the law is expressed by the divinatory symbolic system (卦爻符号, the trigrams or hexagrams). It is through the unique permutation of yin and yang lines of a trigram or hexagram that the law of change is explained as a universal model uniting celestial, terrestrial and human laws. The symbolic system is used to express the universal nature of continual generation of life. __________ Translated from Zhongguo Zhexue Shi 中国哲学史 (History of Chinese Philosophy), 2005(4) by Yun Yufen     

Encoding Modal Logics in Logical Frameworks

We present and discuss various formalizations of Modal Logics in Logical Frameworks based on Type Theories. We consider both Hilbert- and Natural Deduction-style proof systems for representing both truth (local) and validity (global) consequence relations for various Modal Logics. We introduce several techniques for encoding the structural peculiarities of necessitation rules, in the typed -calculus metalanguage of the Logical Frameworks. These formalizations yield readily proof-editors for Modal Logics when implemented in Proof Development Environments, such as Coq or LEGO.     

Zhuangzi the Poet: Re-Reading the Peng Bird Image

The image of the Peng bird, which opens the Zhuangzi text, is not the product of metaphysical reasoning. An inspiring example of soaring up and going beyond, the image is used to broaden the outlook of the small mind; its function is thus more therapeutic than instructional. With its rich poetic and experiential content, the image of the Peng refuses to be reduced to an abstract concept, or a mere signifier of certain philosophical position. Misreading of the image results from any attempt to accurately “size up” its philosophical implication by measuring it quantitatively against a spectrum of positions and values. To see only the superficial “inconsistencies” in Zhuangzi’s argument and to read the wind under the Peng’s wings as a handicap that it needs to overcome in order to embark on its “free and easy wandering” is, in the name of logic and “consistency,” to ignore the big picture Zhuangzi presents.     

On the Future of Group Therapy Theory

The author presents a hypothetical view of group therapy theories as they may evolve over the first decades of the New Milennium. Looking at the problem of prediction from the standpoint of postmodernism and social constructivism, the importance of these perspectives as frameworks of the social sciences is emphasized. The influence of general systems theory, Foulkesian group analysis, object relations theory, self psychology, cognitive-behavioral psychology, and neuropsychiatry are briefly examined with respect to the expected evolution of increasingly sophisticated, detailed, and nuanced theories of group psychology and group psychotherapy.     

On the Logic of β-pregroups

In this paper we concentrate mainly on the notion of β-pregroups, which are pregroups (first introduced by Lambek [18] in 1999) enriched with modality operators. β-pregroups were first proposed by Fadda [11] in 2001. The motivation to introduce them was to limit (locally) the associativity in the calculus considered. In this paper we present this new calculus in the form of a rewriting system, prove the very important feature of this system - that in a given derivation the non-expanding rules must always proceed non-contracting ones in order the derivation to be minimal (normalization theorem). We also propose a sequent system for this calculus and prove the cut elimination theorem for it. As an illustration we show how to use β-pregroups for linguistical applications. Special Issue Categorial Grammars and Pregroups Edited by Wojciech Buszkowski and Anne Preller