Every Finitely Reducible Logic has the Finite Model Property with Respect to the Class of ♦-Formulae

In this paper a unified framework for dealing with a broad family of propositional multimodal logics is developed. The key tools for presentation of the logics are the notions of closure relation operation and monotonous relation operation. The two classes of logics: FiRe-logics (finitely reducible logics) and LaFiRe-logics (FiRe-logics with local agreement of accessibility relations) are introduced within the proposed framework. Further classes of logics can be handled indirectly by means of suitable translations. It is shown that the logics from these classes have the finite model property with respect to the class of -formulae, i.e. each -formula has a -model iff it has a finite -model. Roughly speaking, a -formula is logically equivalent to a formula in negative normal form without occurrences of modal operators with necessity force. In the proof we introduce a substantial modification of Claudio Cerrato's filtration technique that has been originally designed for graded modal logics. The main core of the proof consists in building adequate restrictions of models while preserving the semantics of the operators used to build terms indexing the modal operators.     

An Operational Logic of Proofs with Positive and Negative Information

The logic of proofs was introduced by Artemov in order to analize the formalization of the concept of proof rather than the concept of provability. In this context, some operations on proofs play a very important role. In this paper, we investigate some very natural operations, paying attention not only to positive information, but also to negative information (i.e. information saying that something cannot be a proof). We give a formalization for a fragment of such a logic of proofs, and we prove that our fragment is complete and decidable.     

Predicate Logics on Display

The paper provides a uniform Gentzen-style proof-theoretic framework for various subsystems of classical predicate logic. In particular, predicate logics obtained by adopting van Behthem's modal perspective on first-order logic are considered. The Gentzen systems for these logics augment Belnap's display logic by introduction rules for the existential and the universal quantifier. These rules for x and x are analogous to the display introduction rules for the modal operators and and do not themselves allow the Barcan formula or its converse to be derived. En route from the minimal modal predicate logic to full first-order logic, axiomatic extensions are captured by purely structural sequent rules.     

Logic and Philosophy of Logic in Wittgenstein

This essay discusses Wittgenstein's conception of logic, early and late, and some of the types of logical system that he constructed. The essay shows that the common view according to which Wittgenstein had stopped engaging in logic as a philosophical discipline by the time of writing Philosophical Investigations is mistaken. It is argued that, on the contrary, logic continued to figure at the very heart of later Wittgenstein's philosophy; and that Wittgenstein's mature philosophy of logic contains many interesting thoughts that have gone widely unnoticed.     

Translating the ICAP Theory of Cognitive Engagement Into Practice

ICAP is a theory of active learning that differentiates students’ engagement based on their behaviors. ICAP postulates that I nteractive engagement, demonstrated by co‐generative collaborative behaviors, is superior for learning to C onstructive engagement, indicated by generative behaviors. Both kinds of engagement exceed the benefits of A ctive or P assive engagement, marked by manipulative and attentive behaviors, respectively. This paper discusses a 5‐year project that attempted to translate ICAP into a theory of instruction using five successive measures: (a) teachers’ understanding of ICAP after completing an online module, (b) their success at designing lesson plans using different ICAP modes, (c) fidelity of teachers’ classroom implementation, (d) modes of students’ enacted behaviors, and (e) students’ learning outcomes. Although teachers had minimal success in designing Constructive and Interactive activities, students nevertheless learned significantly more in the context of Constructive than Active activities. We discuss reasons for teachers’ overall difficulty in designing and eliciting Interactive engagement.     

论八卦是八个逻辑范式

基于太极代数,本文证明八卦是八个逻辑范式,八卦中包含四对矛盾关系,其中"六子"构成辩证逻辑组。八卦是生命生产和思想生产都必须共同遵循的变化法则。学界似有这样的倾向,以为《周易》中只有类推逻辑而没有演绎逻辑,本文证明这种观点是不能成立的。八卦本质上就是演绎逻辑的,卦象的本质是逻辑法则。因此,基于卦象的联想或推理不能脱离八卦的逻辑内涵;否则,想象的灵活性必将导致卦象上的混淆,甚至使八卦沦为象数游戏的工具。     

Philosophy,Logic, Science,History

Analytic philosophy is sometimes said to have particularly close connections to logic and to science, and no particularly interesting or close relation to its own history. It is argued here that although the connections to logic and science have been important in the development of analytic philosophy, these connections do not come close to characterizing the nature of analytic philosophy, either as a body of doctrines or as a philosophical method. We will do better to understand analytic philosophy—and its relationship to continental philosophy—if we see it as a historically constructed collection of texts, which define its key problems and concerns. It is true, however, that analytic philosophy has paid little attention to the history of the subject. This is both its strength—since it allows for a distinctive kind of creativity—and its weakness—since ignoring history can encourage a philosophical variety of “normal science.”     

Cognitive technology to capture deep computational concepts with combinators

Computational activity is now recognized as a natural science, and computational and information processes have been discovered in the deep structures of many areas. Computations in the natural world were present long before the invention of computers, but a remarkable shift in understanding its fundamental nature occurs, in fact, before our eyes. The present moment, in fact, is a transition from the concept of computer science as an artificial science to the understanding that information processes are abundant in nature. Computing is recognized as a natural science that studies natural and artificial information processes.In everyday computing, operations are performed on the individual generators, with little attention paid to their internal structure. However, many common operations consist of more primitive constructions connected by a combination mode. The interaction of information processes and corresponding structures is carried out in an environment of “applicative interaction”, their applications to each other, and the study of the properties of this environment allows us to understand the nature of the computations.In the present work, the main attention is paid to elucidating the technological features of computations with individual generators, or objects. Their interaction is considered in an applicative environment, which allows us to elucidate the internal structure of ordinary operations, the knowledge of which allows us to understand their properties. The choice of initial constant generators, considered as generic ones and expressed by combinators, is discussed. These initial generators are used as the main “building blocks” that occur within the larger blocks of the applicative environment in interaction with each other. As a result of the interaction, constructions arise that give representative sets of ordinary operators and embedded computing systems.