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)
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.
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)
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 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)
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.
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 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 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 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)
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)
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)
In terms of formal deductive systems and multi-dimensional Kripke frames we study logical operations know, informed, common knowledge and common information. Based on  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)
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)
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.
"This personification of wisdom with golden hair and a radiant aura echoes both the eternal feminine and the world soul. Rooted in Christian and Jewish mysticism, Eastern Orthodox iconography, Greek philosophy, and European romanticism, the Sophiology that suffuses Solovyov's philosophical and artistic works is both intellectually sophisticated and profoundly inspiring. Judith Deutsch Kornblatt brings together key texts from Solovyov's writings about Sophia: poetry, fiction, drama, and philosophy, all extensively annotated and some available in English for the first time (with assistance (...) from the translators Boris Jakim and Laury Magnus)."--Amazon website. (shrink)
We investigate logical consequence in temporal logics in terms of logical consecutions. i.e., inference rules. First, we discuss the question: what does it mean for a logical consecution to be 'correct' in a propositional logic. We consider both valid and admissible consecutions in linear temporal logics and discuss the distinction between these two notions. The linear temporal logic LDTL, consisting of all formulas valid in the frame 〈L, ≤, ≥〉 of all integer numbers, is the prime object of our investigation. (...) We describe consecutions admissible LDTL in a semantic way—via consecutions valid in special temporal Kripke/Hintikka models. Then we state that any temporal inference rule has a reduced normal form which is given in terms of uniform formulas of temporal degree 1. Using these facts and enhanced semantic techniques we construct an algorithm, which recognizes consecutions admissible in LDTL. Also, we note that using the same technique it follows that the linear temporal logic L (N) of all natural numbers is also decidable w.r.t. inference rules. So, we prove that both logics LDTL and L (N) are decidable w.r.t. admissible consecutions. In particular, as a consequence, they both are decidable (Known fact), and the given deciding algorithms are explicit. (shrink)
Recently, psychologists have explored moral concepts including obligation, blame, and ability. While little empirical work has studied the relationships among these concepts, philosophers have widely assumed such a relationship in the principle that “ought” implies “can,” which states that if someone ought to do something, then they must be able to do it. The cognitive underpinnings of these concepts are tested in the three experiments reported here. In Experiment 1, most participants judge that an agent ought to keep a promise (...) that he is unable to keep, but only when he is to blame for the inability. Experiment 2 shows that such “ought” judgments correlate with judgments of blame, rather than with judgments of the agent’s ability. Experiment 3 replicates these findings for moral “ought” judgments and finds that they do not hold for nonmoral “ought” judgments, such as what someone ought to do to fulfill their desires. These results together show that folk moral judgments do not conform to a widely assumed philosophical principle that “ought” implies “can.” Instead, judgments of blame play a modulatory role in some judgments of obligation. (shrink)
One of the trademarks of Nicolai Hartmann’s ontology is his theory of levels of reality. Hartmann drew from many sources to develop his version of the theory. His essay “Die Anfänge des Schichtungsgedankens in der alten Philosophie” testifies of the fact that he drew from Plato, Aristotle, and Plotinus. But this text was written relatively late in Hartmann’s career, which suggests that his interest in the theories of levels of the ancients may have been retrospective. In “Nicolai Hartmann und seine (...) Zeitgenossen,” Martin Morgenstern puts the emphasis on contemporaries of Hartmann: Émile Boutroux, Max Scheler, Heinrich Rickert, Karl Jaspers, and Arnold Gehlen. But there is another plausible source for Hartmann’s conception of levels that has so far remained overlooked in the literature. Hartmann studied with and was influenced by Nikolai Lossky. Lossky has a theory of levels that he adopted from Vladimir Solovyov. Solovyov presents his theory of levels, among other places, in Oпpaвдaнie дoбpa, where he says that the five principal stages of the cosmogonic process of ascension toward universal perfection, which are given in experience, are the mineral or inorganic realm, the vegetal realm, the animal realm, the realm of natural humanity, and the realm of spiritual or divine humanity. This theory appears to bear significant similarities with the theory of levels of reality that Hartmann will develop a few decades later. Solovyov was widely read in Russia and it would be unlikely that Hartmann was not at least minimally acquainted with his work. Chances are that Hartmann came into contact with it in some details. An intellectual lineage could thus likely be traced from Hartmann back to Solovyov. In this paper, I document and discuss this possible lineage. (shrink)
Vladimir Soloviev et Jacques Maritain sont des philosophes pour aujourd'hui et pour demain. Ils ont laissé des œuvres considérables par leur originalité, leur profondeur et la qualité de leur style. Célèbres, puis un peu oubliés, ces deux talentueux penseurs ont cherché la vérité. Ni Soloviev ni Maritain n'ont eu d'ailleurs des carrières académiques classiques, et la liberté que donne une vie dispensée d'obligations administratives a probablement favorisé leur remarquable créativité. L'un et l'autre ont aimé non seulement le Christ, mais (...) aussi son Eglise tout en ayant des appartenances confessionnelles différentes. "L'ouvrage que l'on présente ici est le fruit de deux colloques tenus successivement à Moscou et à Kiev en 2006. Il s'agissait de faire mieux connaître Soloviev en Occident et Maritain en Russie ou en Ukraine. C'est une manière de dialogue entre deux inspirations que l'on pourrait rattacher à Platon dans le cas de Soloviev et à Aristote avec Maritain, mais aussi et surtout à un personnalisme chrétien qui les caractérise.". (shrink)
Vladimir Solov'ev (1853-1900- is regarded as the most original and systematic of the Russian philosophers in the 19th century. He has once again become the subject of international scholarly attention both in Slavic countries and the West. This volume contains selected papers presented at the international conference on Vladimir Solov'ev held at Nijmegen University, the Netherlands, in September 1998. The scope of this conference was wide-ranging, dealing with theological, metaphysical, philosophical and historical themes. Though Solov'ev's broad intellectual activity (...) defies any strict attempt at categorisation, the editors have classified its major themes under the dual characteristic of reconciliation and polemics. Solov'ev was passionately committed to the reconciliation of all beings under the idea of all-unity, which he attemted to achieve by engaging in uncompromising polemics with his contemporaries, The thirty contributors to this volume are specialists from Russia, Ukraine, Bulgaria, Western Europe and the United States. The volume makes a significant contribution to the intellectual reassesment of Vladimir Solov'ev since the rediscovery of his philosophical heritage in his own homeland in the 1980s. (shrink)
The main article is devoted to the historical and philosophical reconstruction of controversy between Vladimir Solovyov and the authors of the “Faith and Reason” - a magazine of the Kharkov Theological Seminary. This controversy took its place in the “theological and journalistic” or the “theocratic” period of Solovyov’s works. Particular attention is paid to the disputes of Solovyov and T. Stoyanov, A.P. Shost'in and the French Orthodox priest Fr. Vladimir Gette on the theory of dogmatic development in the (...) church. In the context of this controversy, the arguments for the “defense” of Solovyov's position, cited in the magazine “Orthodox Review” by a theologian and Konstantin Leontyev's follower Ivan Kristi are also analyzed. The reception of Solovyov's theocratic ideas and reaction to his ecclesiastical views in both the Catholic and Orthodox circles of Russian and Western society is shown. Especially it concerns the criticism of Solovyov’s ideas in the pages of the French magazines “L’Univers”, “L'Union Chrétienne”, “Revue d’Eglise greque-unie”, etc. The evolution of Solovyov's views on the problem of the union of Eastern and Western churches, the renewal of church communication between Orthodoxy and Catholicism, the main result of which was his fundamental but unfinished work “The History and Future of the Theocracy” was demonstrated. A conclusion about the “superficiality” of the judgments of the majority of Vladimir Solovyov's ideological opponents, as well as later interpreters of his legacy, following the French Jesuit Michel d'Erbigny, who tried to present him as a “Russian Newman” who converted from Orthodoxy into the Catholic faith is drawn. It is shown that Solovyov’s projects of the “religion of the Holy Spirit” and the “Universal Church”, created on its basis, should be considered primarily in the context of his own philosophical quest, and not in connection with the confessional and ideological divergences of his time. (shrink)
Arguably, the existence of bald-faced (i.e. knowingly undisguised) lies entails that not all lies are intended to deceive. Two kinds of bald-faced lies exist in the literature: those based on some common knowledge that implies that you are lying and those that involve tell-tale signs (e.g. blushing) that show that you are lying. I designed the tell-tale sign bald-faced lies to avoid objections raised against the common knowledge bald-faced lies but I now see that they are more problematic than what (...) I initially thought. Therefore, I will discuss these lies in more detail, refine the existing cases, and resolve some anticipated objections. I conclude that tell-tale sign bald-faced lies are genuine lies not intended to deceive. (shrink)
According to one influential argument put forward by, e.g. Chisholm and Feehan, Pfister, Meibauer, Dynel, Keiser, and Harris, asserting requires intending to give your hearer a reason to believe what you say (first premise) and, because liars must assert what they believe is false (second premise), liars necessarily intend to cause their hearer to believe as true what the liars believe is false (conclusion). According to this argument, that is, all genuine lies are intended to deceive. ‘Lies’ not intended to (...) deceive are not genuine lies because they do not involve assertions and you need to assert in order to lie. In this paper, I reject this argument by arguing that the first premise is false: intending to give your hearer a reason to believe what you say is not necessary for asserting. (shrink)