A Model of Tolerance

According to the naive theory of vagueness, the vagueness of an expression consists in the existence of both positive and negative cases of application of the expression and in the non-existence of a sharp cut-off point between them. The sorites paradox shows the naive theory to be inconsistent in most logics proposed for a vague language. The paper explores the prospects of saving the naive theory by revising the logic in a novel way, placing principled restrictions on the transitivity of the consequence relation. A lattice-theoretical framework for a whole family of (zeroth-order) “tolerant logics” is proposed and developed. Particular care is devoted to the relation between the salient features of the formal apparatus and the informal logical and semantic notions they are supposed to model. A suitable non-transitive counterpart to classical logic is defined. Some of its properties are studied, and it is eventually shown how an appropriate regimentation of the naive theory of vagueness is consistent in such a logic.     

Harmonious logic: Craig’s interpolation theorem and its descendants

Though deceptively simple and plausible on the face of it, Craig’s interpolation theorem (published 50 years ago) has proved to be a central logical property that has been used to reveal a deep harmony between the syntax and semantics of first order logic. Craig’s theorem was generalized soon after by Lyndon, with application to the characterization of first order properties preserved under homomorphism. After retracing the early history, this article is mainly devoted to a survey of subsequent generalizations and applications, especially of many-sorted interpolation theorems. Attention is also paid tomethodological considerations, since the Craig theorem and its generalizations were initially obtained by proof-theoretic arguments while most of the applications are model-theoretic in nature. The article concludes with the role of the interpolation property in the quest for “reasonable” logics extending first-order logic within the framework of abstract model theory. For Bill Craig, with great appreciation for his fundamental contributions to our subject, and for his perennially open, welcoming attitude and fine personality that enhances every encounter.     

Congruity Effects in Time and Space: Behavioral and ERP Measures

Two experiments investigated whether motion metaphors for time affected the perception of spatial motion. Participants read sentences either about literal motion through space or metaphorical motion through time written from either the ego-moving or object-moving perspective. Each sentence was followed by a cartoon clip. Smiley-moving clips showed an iconic happy face moving toward a polygon, and shape-moving clips showed a polygon moving toward a happy face. In Experiment 1, using an explicit judgment task, participants judged smiley-moving cartoons as related to ego-moving sentences about space and about time, and shape-moving cartoons as related to object-moving sentences. In Experiment 2, participants viewed the same stimuli, but the cartoons were task-irrelevant. Event-related brain potentials revealed an early attentional effect of congruity on cartoons following sentences about space, and a later semantic effect on cartoons following sentences about time. Results are most consistent with accounts that posit differences in the processing of novel and conventional metaphors.     

Algebraic Aspects of Cut Elimination

We will give here a purely algebraic proof of the cut elimination theorem for various sequent systems. Our basic idea is to introduce mathematical structures, called Gentzen structures, for a given sequent system without cut, and then to show the completeness of the sequent system without cut with respect to the class of algebras for the sequent system with cut, by using the quasi-completion of these Gentzen structures. It is shown that the quasi-completion is a generalization of the MacNeille completion. Moreover, the finite model property is obtained for many cases, by modifying our completeness proof. This is an algebraic presentation of the proof of the finite model property discussed by Lafont [12] and Okada-Terui [17].     

A procedural criterion for final derivability in inconsistency-adaptive logics

This paper concerns a (prospective) goal directed proof procedure for the propositional fragment of the inconsistency-adaptive logic ACLuN1. At the propositional level, the procedure forms an algorithm for final derivability. If extended to the predicative level, it provides a criterion for final derivability. This is essential in view of the absence of a positive test. The procedure may be generalized to all flat adaptive logics.     

Nelson's Negation on the Base of Weaker Versions of Intuitionistic Negation

Constructive logic with Nelson negation is an extension of the intuitionistic logic with a special type of negation expressing some features of constructive falsity and refutation by counterexample. In this paper we generalize this logic weakening maximally the underlying intuitionistic negation. The resulting system, called subminimal logic with Nelson negation, is studied by means of a kind of algebras called generalized N-lattices. We show that generalized N-lattices admit representation formalizing the intuitive idea of refutation by means of counterexamples giving in this way a counterexample semantics of the logic in question and some of its natural extensions. Among the extensions which are near to the intuitionistic logic are the minimal logic with Nelson negation which is an extension of the Johansson's minimal logic with Nelson negation and its in a sense dual version — the co-minimal logic with Nelson negation. Among the extensions near to the classical logic are the well known 3-valued logic of Lukasiewicz, two 12-valued logics and one 48-valued logic. Standard questions for all these logics — decidability, Kripke-style semantics, complete axiomatizability, conservativeness are studied. At the end of the paper extensions based on a new connective of self-dual conjunction and an analog of the Lukasiewicz middle value ½ have also been considered.     

The Class of Extensions of Nelson's Paraconsistent Logic

The article is devoted to the systematic study of the lattice εN4_⊥ consisting of logics extending N4_⊥. The logic N4_⊥ is obtained from paraconsistent Nelson logic N4 by adding the new constant ⊥ and axioms ⊥ → p, p → ∼ ⊥. We study interrelations between εN4_⊥ and the lattice of superintuitionistic logics. Distinguish in εN4_⊥ basic subclasses of explosive logics, normal logics, logics of general form and study how they are relate. The author acknowledges support by the Alexander von Humboldt-Stiftung and by Counsil for Grants under RF President, project NSh - 2112.2003.1.     

论道德继承

道德是符合社会之理而维持社会存在的人的行为价值和规范。道德有一般与特殊之分,一般道德的抽象性和特殊道德的具体性决定了二者继承方法的选择,即前者抽象继承,后者批判继承。历史上积累的道德价值和道德规范是道德继承的对象,此对象无论采取何种表述方式,终究可以用语言或者文字的形式表述为一个道德命题。因此,道德继承实质上是对道德命题的继承。道德命题是以善恶价值为内容、对某一道德状况或某一道德要求用逻辑语言所作的判断和陈述。它通常包括形式的或者逻辑的层面和实际的或者经验的层面,前者表征其一般意义,后者表征其特殊意义。理论上,前者采用抽象继承法,后者采用批判继承法,但实际上二者密不可分。因此,现实生活中的道德继承需兼顾两面,即采用一体两面相结合的道德继承模式。     

Affect in the abstract: Abstract mindsets promote sensitivity to affect

Affective information is a key element of abstract, gist-based representations. Given this central role of affect in abstract representations, the authors hypothesized that those in an abstract mindset may show more sensitivity to affective information when attending to, interpreting, or responding to external stimuli. Across 5 studies, the authors present consistent evidence that those who tend to view the world in more abstract versus concrete terms, or who have been experimentally induced into an abstract (versus concrete) mindset, had their attention automatically captured by highly-affective stimuli (Studies 1a, 1b, and 1c), automatically extracted the affective connotation of stimuli presented outside of awareness (Study 2), and behaved more consistently with their affective construal of stimuli (Study 3). These findings suggest that the numerous dispositional and situational factors that influence the level at which one construes the world may also have, heretofore overlooked, consequences for basic affective processing. The present research also emphasizes the interdependence of so-called affective and cognitive processes, bridging major literatures on construal and affect that have developed largely independently.     

An Adaptive Logic Based on Jaśkowskiˈs Approach to Paraconsistency

In this paper, I present the modal adaptive logic AJ ^r (based on S5) as well as the discussive logic D ^r ₂ that is defined from it. D ^r ₂ is a (non-monotonic) alternative for Jaśkowski’s paraconsistent system D ₂ . Like D ₂ , D ^r ₂ validates all single-premise rules of Classical Logic. However, for formulas that behave consistently, D ^r ₂ moreover validates all multiple-premise rules of Classical Logic. Importantly, and unlike in the case of D ₂ , this does not require the introduction of discussive connectives. It is argued that this has clear advantages with respect to one of the main application contexts of discussive logics, namely the interpretation of discussions.*Research for this paper was indirectly supported by the Flemish Minister responsible for Science and Technology (contract BIL1/8). The author is indebted to Leon Horsten, Jo?o Marcos, Jerzy Perzanowski, Liza Verhoeven, and especially to the referee and to Diderik Batens for comments and suggestions.