The aim of this book is to present the fundamental theoretical results concerning inference rules in deductive formal systems. Primary attention is focused on: admissible or permissible inference rules the derivability of the admissible inference rules the structural completeness of logics the bases for admissible and valid inference rules. There is particular emphasis on propositional non-standard logics (primary, superintuitionistic and modal logics) but general logical consequence relations and classical first-order theories are also considered. The book is basically self-contained and special (...) attention has been made to present the material in a convenient manner for the reader. Proofs of results, many of which are not readily available elsewhere, are also included. The book is written at a level appropriate for first-year graduate students in mathematics or computer science. Although some knowledge of elementary logic and universal algebra are necessary, the first chapter includes all the results from universal algebra and logic that the reader needs. For graduate students in mathematics and computer science the book is an excellent textbook. (shrink)

In this paper1 we study admissible consecutions in multi-modal logics with the universal modality. We consider extensions of multi-modal logic S4n augmented with the universal modality. Admissible consecutions form the largest class of rules, under which a logic is closed. We propose an approach based on the context effective finite model property. Theorem 7, the main result of the paper, gives sufficient conditions for decidability of admissible consecutions in our logics. This theorem also provides an explicit algorithm for recognizing such (...) consecutions. Some applications to particular logics with the universal modality are given. (shrink)

Our investigation is concerned with the finite model property with respect to admissible rules. We establish general sufficient conditions for absence of fmp w. r. t. admissibility which are applicable to modal logics containing K4: Theorem 3.1 says that no logic λ containing K4 with the co-cover property and of width > 2 has fmp w. r. t. admissibility. Surprisingly many, if not to say all, important modal logics of width > 2 are within the scope of this theorem–K4 itself, (...) S4, GL, K4.1, K4.2, S4.1, S4.2, GL.2, etc. Thus the situation is completely opposite to the case of the ordinary fmp–the absolute majority of important logics have fmp, but not with respect to admissibility. As regards logics of width ≤ 2, there exists a zone for fmp w. r. t. admissibility. It is shown that all modal logics A of width ≤ 2 extending S4 which are not sub-logics of three special tabular logics have fmp w.r.t. admissibility. (shrink)

An algorithm recognizing admissibility of inference rules in generalized form (rules of inference with parameters or metavariables) in the intuitionistic calculus H and, in particular, also in the usual form without parameters, is presented. This algorithm is obtained by means of special intuitionistic Kripke models, which are constructed for a given inference rule. Thus, in particular, the direct solution by intuitionistic techniques of Friedman's problem is found. As a corollary an algorithm for the recognition of the solvability of logical equations (...) in H and for constructing some solutions for solvable equations is obtained. A semantic criterion for admissibility in H is constructed. (shrink)

We find an explicit basis for all admissible rules of the modal logic S4. Our basis consists of an infinite sequence of rules which have compact and simple, readable form and depend on increasing set of variables. This gives a basis for all quasi-identities valid in the free modal algebra ℱS4 of countable rank.

The main result of this paper is the following theorem: each modal logic extendingK4 having the branching property belowm and the effective m-drop point property is decidable with respect to admissibility. A similar result is obtained for intermediate intuitionistic logics with the branching property belowm and the strong effective m-drop point property. Thus, general algorithmic criteria which allow to recognize the admissibility of inference rules for modal and intermediate logics of the above kind are found. These criteria are applicable to (...) most modal logics for which decidability with respect to admissibility is known and to many others, for instance, to the modal logicsK4,K4.1,K4.2,K4.3,S4.1,S4.2,GL.2; to all smallest and greatest counterparts of intermediate Gabbay-De-Jong logicsD n; to all intermediate Gabbay-De-Jong logicsD n; to all finitely axiomatizable modal and intermediate logics of finite depth etc. Semantic criteria for recognizing admissibility for these logics are offered as well. (shrink)

This paper offers a brief analysis of the unification problem in modal transitive logics related to the logic S4 : S4 itself, K4, Grz and Gödel-Löb provability logic GL . As a result, new, but not the first, algorithms for the construction of ‘best’ unifiers in these logics are being proposed. The proposed algorithms are based on our earlier approach to solve in an algorithmic way the admissibility problem of inference rules for S4 and Grz . The first algorithms for (...) the construction of ‘best’ unifiers in the above mentioned logics have been given by S. Ghilardi in [ 16 ]. Both the algorithms in [ 16 ] and ours are not much computationally efficient. They have, however, an obvious significant theoretical value a portion of which seems to be the fact that they stem from two different methodological approaches. (shrink)

The aim of this paper is to look from the point of view of admissibility of inference rules at intermediate logics having the finite model property which extend Heyting's intuitionistic propositional logic H. A semantic description for logics with the finite model property preserving all admissible inference rules for H is given. It is shown that there are continuously many logics of this kind. Three special tabular intermediate logics λ, 1 ≥ i ≥ 3, are given which describe all tabular (...) logics preserving admissibility: a tabular logic λ preserves all admissible rules for H iff 7λ has width not more than 2 and is not included in each λ. MSC: 03B55, 03B20. (shrink)

We study unification problem and problem of admissibility for inference rules in minimal Johanssonsʼ logic J and positive intuitionistic logic IPC+. This paper proves that the problem of admissibility for inference rules with coefficients is decidable for the paraconsistent minimal Johanssonsʼ logic J and the positive intuitionistic logic IPC+. Using obtained technique we show also that the unification problem for these logics is also decidable: we offer algorithms which compute complete sets of unifiers for any unifiable formula. Checking just unifiability (...) of formulas with coefficients also works via verification of admissibility. (shrink)

The paper studies Barwise's information frames and answers the John Barwise question: to find axiomatizations for the modal logics generated by information frames. We find axiomatic systems for (i) the modal logic of all complete information frames, (ii) the logic of all sound and complete information frames, (iii) the logic of all hereditary and complete information frames, (iv) the logic of all complete, sound and hereditary information frames, and (v) the logic of all consistent and complete information frames. The notion (...) of weak modal logics is also proposed, and it is shown that the weak modal logics generated by all information frames and by all hereditary information frames are K and K4 respectively. To develop general theory, we prove that (i) any Kripke complete modal logic is the modal logic of a certain class of information frames and that (ii) the modal logic generated by any given class of complete, rarefied and fully classified information frames is Kripke complete. This paper is dedicated to the memory of talented mathematician John Barwise. (shrink)

In terms of formal deductive systems and multi-dimensional Kripke frames we study logical operations know, informed, common knowledge and common information. Based on [6] we introduce formal axiomatic systems for common information logics and prove that these systems are sound and complete. Analyzing the common information operation we show that it can be understood as greatest open fixed points for knowledge formulas. Using obtained results we explore monotonicity, omniscience problem, and inward monotonocity, describe their connections and give dividing examples. Also (...) we find algorithms recognizing these properties for some particular cases. (shrink)

We study the problem of finding a basis for all rules admissible in the intuitionistic propositional logic IPC. The main result is Theorem 3.1 which gives a basis consisting of all rules in semi-reduced form satisfying certain specific additional requirements. Using developed technique we also find a basis for rules admissible in the logic of excluded middle law KC.

The paper deals with a temporal multi-agent logic TMAZ, which imitates taking of decisions based on agents' access to knowledge by their interaction. The interaction is modelled by possible communication channels between agents in special temporal Kripke/hintikka-like models. The logic TMAZ distinguishes local and global decisions-making. TMAZ is based on temporal Kripke/hintikka models with agents' accessibility relations defined on states of all possible time clusters C(i) (where indexes i range over all integer numbers Z). The main result provides a decision (...) algorithm for TMAZ (so, we prove that TMAZ is decidable). This algorithm also solves the satisfiability problem. In the final part of the paper, we consider the admissibility problem for inference rules in TMAZ, and show that this problem is decidable for TMAZ as well. (shrink)

Our paper aims to investigate inference rules for Nelson’s logics and to discuss possible ways to determine admissibility of inference rules in such logics. We will use the technique offered originally for intuitionistic logic and paraconsistent minimal Johannson’s logic. However, the adaptation is not an easy and evident task since Nelson’s logics do not enjoy replacement of equivalences rule. Therefore we consider and compare standard admissibility and weak admissibility. Our paper founds algorithms for recognizing weak admissibility and admissibility itself – (...) for restricted cases, to show the problems arising in the course of study. (shrink)

The paper aims at providing the multi-modal propositional logic LTK with a sound and complete axiomatisation. This logic combines temporal and epistemic operators and focuses on m odeling the behaviour of a set of agents operating in a system on the background of a temporal framework. Time is represented as linear and discrete, whereas knowledge is modeled as an S5-like modality. A further modal operator intended to represent environment knowledge is added to the system in order to achieve the expressive (...) power sufficient to describe the piece of information available to the agents at each moment in the flow of time. (shrink)

We prove that a propositional Linear Temporal Logic with Until and Next has unitary unification. Moreover, for every unifiable in LTL formula A there is a most general projective unifier, corresponding to some projective formula B, such that A is derivable from B in LTL. On the other hand, it can be shown that not every open and unifiable in LTL formula is projective. We also present an algorithm for constructing a most general unifier.

Intensional logic has emerged, since the 1960' s, as a powerful theoretical and practical tool in such diverse disciplines as computer science, artificial intelligence, linguistics, philosophy and even the foundations of mathematics. The present volume is a collection of carefully chosen papers, giving the reader a taste of the frontline state of research in intensional logics today. Most papers are representative of new ideas and/or new research themes. The collection would benefit the researcher as well as the student. This book (...) is a most welcome addition to our series. The Editors CONTENTS PREFACE IX JOHAN VAN BENTHEM AND NATASHA ALECHINA Modal Quantification over Structured Domains PATRICK BLACKBURN AND WILFRIED MEYER-VIOL Modal Logic and Model-Theoretic Syntax 29 RUY J. G. B. DE QUEIROZ AND DOV M. GABBAY The Functional Interpretation of Modal Necessity 61 VLADIMIR V. RYBAKOV Logics of Schemes for First-Order Theories and Poly-Modal Propositional Logic 93 JERRY SELIGMAN The Logic of Correct Description 107 DIMITER VAKARELOV Modal Logics of Arrows 137 HEINRICH WANSING A Full-Circle Theorem for Simple Tense Logic 173 MICHAEL ZAKHARYASCHEV Canonical Formulas for Modal and Superintuitionistic Logics: A Short Outline 195 EDWARD N. ZALTA 249 The Modal Object Calculus and its Interpretation NAME INDEX 281 SUBJECT INDEX 285 PREFACE Intensional logic has many faces. In this preface we identify some prominent ones without aiming at completeness. (shrink)

We know that Prince Volodymyr as Saint of Russia was eventually canonized and proclaimed the saintly saint of God. And in regard to its connection with the Muslim religion, then the domestic authorities from the historical science - such as Karamzin, Tatishchev, Kostomarov, Solovyov, Hrushevsky, Grekov, Tolochko, and even Rybakov and even Gumilev - did not have significant differences. None of them doubted the historical truthfulness of centuries of "trodden" theory of Vladimir's choice of the Great State religion.

A través del método etimológico de Vico es posible ver relaciones entre un enfoque materialista y los temas de la infancia y el mundo primitivo, lo que implica un interés por cuestiones históricas vinculadas al mundo agrícola y religioso de los orígenes y cuestiones cognitivas vinculadas a las relaciones entre tacto, visión e intelecto. La etimología de cogitare como “recoger” y de intelligere como “leer” proporciona las bases para una teoría de la cognición decididamente aplicada a la Dipintura de la (...) Ciencia nueva, que sirve de introducción a la obra y que revela el horizonte gnoseológico de sus pretensiones ya puestas a prueba en sus primeros textos, especialmente en lo que se refiere al tema de la racionalidad y del ingenio humano. (shrink)

"This paper deals with the notion of intelligibility crisis in terms of conflict and harmony. Namely, we will analyze the notion of intelligibility from MacIntyre’s philosophical opus and apply it to the historical case. Intelligibility, according to MacIntyre, is the notion which provides us with contextual meaning and embeds our actions with sense within the specific tradition. Intelligibility crisis is the term that is coined to provide a descriptive account of the phenomenon when we cannot connect ourselves with a new (...) social context in which we find ourselves in. To further elaborate on this and apply it onto an example, we shall use the historical case of the Yugoslavian nation. We will provide analysis between three different contextual narratives – pre-Yugoslav narrative, Yugoslav narrative, and post-Yugoslav narrative. After applying the notion of intelligibility crisis onto this historical case study, we will notice how people of one social narrative lose intelligibility by going into another social narrative. Furthermore, we shall consider the notions of conflicts and harmony as those that are connected to intelligibility. The main argument from the descriptive state of things which was offered would be the following – conflicting sentiments arise when we are not in harmony with the narrative within which we have attained intelligibility. Keywords: intelligibility crisis, Yugoslavia, social narratives, meaning, conflicts, harmony". (shrink)

The first collective monograph on one of Plato’s most intriguing dialogues with interest for readers of ancient philosophy as well as those who study modern theories of language.

Prvi deo ovog rada, objavljen u prethodnom broju casopisa, bio je posvecen razmatranju religijskog porekla dvojnosti 'duha' i 'slova', a zatim i pitanju njenog znacenja u kontekstu filozofije ranog nemackog idealizma. Ovaj, drugi i zakljucni deo rada usredsredjuje se na eticke implikacije tog pitanja, posto je suprotstavljanje 'duha2 Kantovog sistema njegovom 'slovu', uz cesta pozivanja na novozavetno ispunjenje mojsijevskog zakona ljubavlju i verom predstavljalo jednu od osnovnih strategija kriticara Kantove teorije morala. Krajnja konsekvenca ovog pravca kritike Kanta lezi u ukidanju (...) autoriteta transcendentne moralne norme. Ovaj korak, kojim se istovremeno ponistava autoritet teksta kao takvog, a time i stav komentara kao literarnog zanra moze se uociti vec u Hegelovim mladalackim teoloskim spisima, u kojima se zastupa ukidanje transcendencije moralnog zakona njegovim inkorporiranjem u slobodnu subjektivnost koja egzistira u zajednici verujucih hriscana. (shrink)

The purpose of this paper is to shed light on different aspects of the hermeneutical problem in post-Kantian philosophical 'constellation'. In this domain, the problem of the relationship between the text and its commentary is theorized in terms of the antithesis between 'Spirit' and 'Letter', which clearly has religious roots. Therefore, the first part of the paper examines the historical origins of this antithesis, as well as its application in philosophical discussions which developed by the end of the 18th century (...) about the problem of finding the 'true' interpretation to Kant's philosophy. The second part of the text, which is to be published in the next issue of this review, brings the duality of spiritual and literal interpretation into closer connection with the topics of Kant's moral philosophy. Namera ovog rada je da osvetli razlicite aspekte hermeneutickog problema u poslekantovskoj filozofskoj 'konstelaciji'. U ovom podrucju, problem odnosa izmedju teksta i komentara postavlja se u terminima antiteze izmedju 'duha' i 'slova', koja ima prepoznatljive religijske korene. Zato prvi deo rada istrazuje istorijsko poreklo ove antiteze, kao i njenu primenu u filozofskim diskusijama koje su se krajem 18. veka razvile povodom problema pronalazenja 'pravog' tumacenja Kantove filozofije. U drugom delu rada, koji ce biti objavljen u sledecem broju casopisa, dvojnost duhovnog i doslovnog tumacenja dovodi se u uzu vezu sa tematikom Kantove teorije morala. (shrink)

This article analyzes the issues associated with the publication of Martin Heidegger’s Black Notebooks. We attempt to explain the very publication of this document as a kind of experiment the philo...

Name der Zeitschrift: Paragrana Jahrgang: 23 Heft: 2 Seiten: 25-39.