The first-order logical theory Th $({\mathbb{N}},x + 1,F(x))$ is proved to be complete for the class ATIME-ALT $(2^{O(n)},O(n))$ when $F(x) = 2^{x}$ , and the same result holds for $F(x) = c^{x}, x^{c} (c \in {\mathbb{N}}, c \ge 2)$ , and F(x) = tower of x powers of two. The difficult part is the upper bound, which is obtained by using a bounded Ehrenfeucht–Fraïssé game.

While logical theories of information attitudes, such as knowledge, certainty and belief, have flourished in the past two decades, formalization of other facets of rational behavior have lagged behind significantly. One intriguing line of research concerns the concept of intention. I will discuss one approach to tackling the notion within a logical framework, based on a database perspective.

While the epistemic significance of disagreement has been a popular topic in epistemology for at least a decade, little attention has been paid to logical disagreement. This monograph is meant as a remedy. The text starts with an extensive literature review of the epistemology of (peer) disagreement and sets the stage for an epistemological study of logical disagreement. The guiding thread for the rest of the work is then three distinct readings of the ambiguous term ‘logical disagreement’. (...) Chapters 1 and 2 focus on the Ad Hoc Reading according to which logical disagreements occur when two subjects take incompatible doxastic attitudes toward a specific proposition in or about logic. Chapter 2 presents a new counterexample to the widely discussed Uniqueness Thesis. Chapters 3 and 4 focus on the Theory Choice Reading of ‘logical disagreement’. According to this interpretation, logical disagreements occur at the level of entire logical theories rather than individual entailment-claims. Chapter 4 concerns a key question from the philosophy of logic, viz., how we have epistemic justification for claims about logical consequence. In Chapters 5 and 6 we turn to the Akrasia Reading. On this reading, logical disagreements occur when there is a mismatch between the deductive strength of one’s background logic and the logical theory one prefers (officially). Chapter 6 introduces logical akrasia by analogy to epistemic akrasia and presents a novel dilemma. Chapter 7 revisits the epistemology of peer disagreement and argues that the epistemic significance of central principles from the literature are at best deflated in the context of logical disagreement. The chapter also develops a simple formal model of deep disagreement in Default Logic, relating this to our general discussion of logical disagreement. The monograph ends in an epilogue with some reflections on the potential epistemic significance of convergence in logical theorizing. (shrink)

Includes the complete text of Dewey's Studies in Logical Theory and The Child and the Curriculum.

"The first book that provides a systematic and rigorous logical theory of causality"--.

First published in 1952, professor’s Strawson’s highly influential _Introduction_ _to Logical Theory_ provides a detailed examination of the relationship between the behaviour of words in common language and the behaviour of symbols in a logical system. He seeks to explain both the exact nature of the discipline known as Formal Logic, and also to reveal something of the intricate logical structure of ordinary unformalised discourse.

Weak ontic necessity is the ontic necessity expressed by “should/ought to” in English. An example of it is “I should be dead by now”. A feature of this necessity is that whether it holds at the present world is irrelevant to whether its prejacent holds at the present world. In this paper, by combining premise semantics and update semantics for conditionals, we present a logical theory for conditional weak ontic necessity based on context update. A context is a (...) set of ordered defaults, determining expected possible states of the present world. Sentences are evaluated with respect to contexts. When evaluating the conditional weak ontic necessity with respect to a context, we first update the context with the antecedent, then check whether the consequent holds with respect to the updated context. We compare this theory to some related works. The logic’s expressivity is studied. (shrink)

The content of this book provides a unified and coherent treatment of a number of important issues in the philosophy of language and logical theory.

Lewis, D. Semantic analyses for dyadic deontic logic.--Salomaa, A. Some remarks concerning many-valued propositional logics.--Chellas, B. F. Conditional obligation.--Jeffrey, R.C. Remarks on interpersonal utility theory.--Hintikka, J. On the proper treatment of quantifiers in Montague semantics.--Mayoh, B.H. Extracting information from logical proofs.--Åqvist, L. A new approach to the logical theory of actions and causality.--Pörn, I. Some basic concepts of action.--Bouvère, K. de. Some remarks concerning logical and ontological theories.--Hacking, I. Combined evidence.--Äberg, C. Solution to a problem (...) raised by Stig Kanger and a set theoretical statement equivalent to the axiom of choice.--Lindström, P. On characterizing elementary logic.-- Scott, D. Rules and derived rules.--Hansson, B. A program for pragmatics.--Hermerén, G. Models.--Fenstad, J.E. Remarks on logic and probability.--Stenlund, S. Analytic and synthetic arithmetical statements. (shrink)

This book reclaims logic as a branch of philosophy, offering a self-contained and complete introduction to the three traditional systems of classical logic and the philosophical issues that surround those systems. The exposition is lucid, clear, and engaging. Practical methods are favored over the traditional, and creative approaches over the merely mechanical. The author’s guiding principle is to introduce classical logic in an intellectually honest way, and not to shy away from difficulties and controversies where they arise. Relevant philosophical issues, (...) such as the relation between the meaning and the referent of a proper name, logical versus metaphysical possibility, and the conceptual content of an expression, are discussed throughout. In this way, the book is not only an introduction to the three main systems of classical logic, but also an introduction to the philosophy of classical logic. (shrink)

part 1. Fuzzy logic theory 1 -- part 2. Applications and advanced topics of fuzzy logic.

The anti-exceptionalist debate brought into play the problem of what are the relevant data for logical theories and how such data affects the validities accepted by a logical theory. In the present paper, I depart from Laudan's reticulated model of science to analyze one aspect of this problem, namely of the role of logical data within the process of revision of logical theories. For this, I argue that the ubiquitous nature of logical data is (...) responsible for the proliferation of several distinct methodologies for logical theories. The resulting picture is coherent with the Laudanean view that agreement and disagreement between scientific theories take place at different levels. From this perspective, one is able to articulate other kinds of divergence that considers not only the inferential aspects of a given logical theory, but also the epistemic aims and the methodological choices that drive its development. (shrink)

In 1995, in his seminal paper introducing the abstract argumentation framework, Dung has also established the first relationship between this framework and a logical framework (in this case: logic programming). Since that time, a lot of work have pursued this path, proposing different definitions, uses and exhibiting distinct relationships between argumentation and logic. In this paper, we present a survey of existing works about this topic and more especially those that address the following question: “How logic has been used (...) for capturing various aspects or parts of Dung’s argumentation”. This survey covers many different approaches but is not intended to be totally exhaustive due to the huge quantity of papers in this scope. Moreover, due to the fact that each approach has its own specificities, sometimes antagonistic with the other approaches, and is also justified by its own context of definition or use, the aim of this survey is not to identify one approach as being better than another. (shrink)

This study concerns logical systems considered as theories. By searching for the problems which the traditionally given systems may reasonably be intended to solve, we clarify the rationales for the adequacy criteria commonly applied to logical systems. From this point of view there appear to be three basic types of logical systems: those concerned with logical truth; those concerned with logical truth and with logical consequence; and those concerned with deduction per se as well (...) as with logical truth and logical consequence. Adequacy criteria for systems of the first two types include: effectiveness, soundness, completeness, Post completeness, "strong soundness" and strong completeness. Consideration of a logical system as a theory of deduction leads us to attempt to formulate two adequacy criteria for systems of proofs. The first deals with the concept of rigor or "gaplessness" in proofs. The second is a completeness condition for a system of proofs. An historical note at the end of the paper suggests a remarkable parallel between the above hierarchy of systems and the actual historical development of this area of logic. (shrink)

A central problem in applying logical knowledge representation formalisms to traditional robotics is that the treatment of belief change is categorical in the former, while probabilistic in the latter. A typical example is the fundamental capability of localization where a robot uses its noisy sensors to situate itself in a dynamic world. Domain designers are then left with the rather unfortunate task of abstracting probabilistic sensors in terms of categorical ones, or more drastically, completely abandoning the inner workings of (...) sensors to black-box probabilistic tools and then interpreting their outputs in an abstract way. Building on a first-principles approach by Bacchus, Halpern and Levesque, and a recent continuous extension to it by Belle and Levesque, we provide an axiomatization that shows how localization can be realized wrt a basic action theory, thereby demonstrating how such capabilities can be enabled in a single logical framework. We then show how the framework can also enable localization for multiple agents, where an agent can appeal to the sensing already performed by another agent and the knowledge of their relative positions to localize itself. (shrink)

In 1995, in his seminal paper introducing the abstract argumentation framework, Dung has also established the first relationship between this framework and a logical framework (in this case: logic programming). Since that time, a lot of work have pursued this path, proposing different definitions, uses and exhibiting distinct relationships between argumentation and logic. In this paper, we present a survey of existing works about this topic and more especially those that address the following question: “How logic has been used (...) for capturing various aspects or parts of Dung’s argumentation”. This survey covers many different approaches but is not intended to be totally exhaustive due to the huge quantity of papers in this scope. Moreover, due to the fact that each approach has its own specificities, sometimes antagonistic with the other approaches, and is also justified by its own context of definition or use, the aim of this survey is not to identify one approach as being better than another. (shrink)

Judgment aggregation theory, or rather, as we conceive of it here, logical aggregation theory generalizes social choice theory by having the aggregation rule bear on judgments of all kinds instead of merely preference judgments. It derives from Kornhauser and Sager’s doctrinal paradox and List and Pettit’s discursive dilemma, two problems that we distinguish emphatically here. The current theory has developed from the discursive dilemma, rather than the doctrinal paradox, and the final objective of the paper (...) is to give the latter its own theoretical development along the line of recent work by Dietrich and Mongin. However, the paper also aims at reviewing logical aggregation theory as such, and it covers impossibility theorems by Dietrich, Dietrich and List, Dokow and Holzman, List and Pettit, Mongin, Nehring and Puppe, Pauly and van Hees, providing a uniform logical framework in which they can be compared with each other. The review goes through three historical stages: the initial paradox and dilemma, the scattered early results on the independence axiom, and the so-called canonical theorem, a collective achievement that provided the theory with its specific method of analysis. The paper goes some way towards philosophical logic, first by briefly connecting the aggregative framework of judgment with the modern philosophy of judgment, and second by thoroughly discussing and axiomatizing the ‘general logic’ built in this framework. (shrink)

This is the first book that integrates nonmonotonic reasoning and belief change into a single framework from an artificial intelligence logic point-of-view.

Logical pluralism has been in vogue since JC Beall and Greg Restall 2006 articulated and defended a new pluralist thesis. Recent criticisms such as Priest 2006a and Field 2009 have suggested that there is a relationship between their type of logical pluralism and the meaning-variance thesis for logic. This is the claim, often associated with Quine 1970, that a change of logic entails a change of meaning. Here we explore the connection between logical pluralism and meaning-variance, both (...) in general and for Beall and Restall's theory specifically. We argue that contrary to what Beall and Restall claim, their type of pluralism is wedded to meaning-variance. We then develop an alternative form of logical pluralism that circumvents at least some forms of meaning-variance. (shrink)

The Mīmāṃsā school of Indian philosophy has for its main purpose the interpretation of injunctions that are found in a set of sacred texts, the Vedas. In their works, Mīmāṃsā authors provide some of the most detailed and systematic examinations available anywhere of statements with a deontic force; however, their considerations have generally not been registered outside of Indological scholarship. In the present article we analyze the Mīmāṃsā theory of Vedic injunctions from a logical and philosophical point of (...) view. The theory at issue can be regarded as a system of reasoning based on certain fundamental principles, such as the distinction between strong and weak duties, and on a taxonomy of ritual actions. We start by reconstructing the conceptual framework of the theory and then move to a formalization of its core aspects. Our contribution represents a new perspective to study Mīmāṃsā and outlines its relevance, in general, for deontic reasoning. (shrink)

Modal logics, originally conceived in philosophy, have recently found many applications in computer science, artificial intelligence, the foundations of mathematics, linguistics and other disciplines. Celebrated for their good computational behaviour, modal logics are used as effective formalisms for talking about time, space, knowledge, beliefs, actions, obligations, provability, etc. However, the nice computational properties can drastically change if we combine some of these formalisms into a many-dimensional system, say, to reason about knowledge bases developing in time or moving objects. To study (...) the computational behaviour of many-dimensional modal logics is the main aim of this book. On the one hand, it is concerned with providing a solid mathematical foundation for this discipline, while on the other hand, it shows that many seemingly different applied many-dimensional systems (e.g., multi-agent systems, description logics with epistemic, temporal and dynamic operators, spatio-temporal logics, etc.) fit in perfectly with this theoretical framework, and so their computational behaviour can be analyzed using the developed machinery. We start with concrete examples of applied one- and many-dimensional modal logics such as temporal, epistemic, dynamic, description, spatial logics, and various combinations of these. Then we develop a mathematical theory for handling a spectrum of 'abstract' combinations of modal logics - fusions and products of modal logics, fragments of first-order modal and temporal logics - focusing on three major problems: decidability, axiomatizability, and computational complexity. Besides the standard methods of modal logic, the technical toolkit includes the method of quasimodels, mosaics, tilings, reductions to monadic second-order logic, algebraic logic techniques. Finally, we apply the developed machinery and obtained results to three case studies from the field of knowledge representation and reasoning: temporal epistemic logics for reasoning about multi-agent systems, modalized description logics for dynamic ontologies, and spatio-temporal logics. The genre of the book can be defined as a research monograph. It brings the reader to the front line of current research in the field by showing both recent achievements and directions of future investigations (in particular, multiple open problems). On the other hand, well-known results from modal and first-order logic are formulated without proofs and supplied with references to accessible sources. The intended audience of this book is logicians as well as those researchers who use logic in computer science and artificial intelligence. More specific application areas are, e.g., knowledge representation and reasoning, in particular, terminological, temporal and spatial reasoning, or reasoning about agents. And we also believe that researchers from certain other disciplines, say, temporal and spatial databases or geographical information systems, will benefit from this book as well. Key Features: Integrated approach to modern modal and temporal logics and their applications in artificial intelligence and computer science Written by internationally leading researchers in the field of pure and applied logic Combines mathematical theory of modal logic and applications in artificial intelligence and computer science Numerous open problems for further research Well illustrated with pictures and tables. (shrink)

Many philosophers have attempted to offer a logical theory of objects, employing different techniques. Thus R. Carnap tried to “reconstruct” logically the world by using the modern symbolic logic, while N. Goodman “constructed” the world with the help of the calculus of individuals or the logic of part-whole relations. W.V. Quine helped to steer the attention toward the question of ontological commitment and toward a theory of objects produced by a logical analysis of natural languages. Recently, (...) there have been controversial discussions concerning the metaphysical and logical aspects of objects in actual and possible worlds as well as new contributions to the theory of nonexistent objects, rekindling the old Russell-Meinong controversy. It seems that even the proposed theories of events, such as that of R. M. Martin, maintain the great significance of the notion of an object. Then there is a plethora of the so-called intensional objects, such as the alleged objects of propositional attitudes, the story of which is tied with the names of Frege, Russell, Carnap and many others. Is it possible to give an all-embracing theory of objects which would satisfy all the parties involved? Although such possibility may be very questionable, the instrumental role of logic for the development of the metaphysical theories of objects is indisputable. Two factors are, however, involved in the constitution of a logical theory of objects: the basis or the given data from which the theory starts; the logical apparatus employed, i.e., the logical axioms and rules of inference, just as the relevant semantical rules which are formulated in the metalanguage. While the logical apparatus is usually very clear, at least in the case of the formally well-construed theories, the given basis of a metaphysical theory T—as we may call it—might contain many tacit assumptions which would need a lot of clarification. It seems that philosophers fight more about the presuppositions and assertions of their systems than about the logic involved. I would like to discuss the interplay between the above two factors, starting with the realm of the given objects. As to the second factor—the logical apparatus involved—the contributions of first-order predicate logic and of the calculus of individuals will be considered. (shrink)

This book presents novel formalizations of three of the most important medieval logical theories: supposition, consequence and obligations. In an additional fourth part, an in-depth analysis of the concept of formalization is presented - a crucial concept in the current logical panorama, which as such receives surprisingly little attention.Although formalizations of medieval logical theories have been proposed earlier in the literature, the formalizations presented here are all based on innovative vantage points: supposition theories as algorithmic hermeneutics, theories (...) of consequence analyzed with tools borrowed from model-theory and two-dimensional semantics, and obligations as logical games. For this reason, this is perhaps the first time that these medieval logical theories are made fully accessible to the modern philosopher and logician who wishes to obtain a better grasp of them, but who has always been held back by the lack of appropriate ‘translations' into modern terms.Moreover, the book offers a reflection on the very nature of logic, a reflection that is prompted by the comparisons between medieval and modern logic, their similarities and dissimilarities. It is thus a contribution not only to the history of logic, but also to the philosophy of logic, the philosophy of language and semantics.The analysis of medieval logic is also relevant for the modern philosopher and logician in that, being the unifying methodology used across all disciplines at that time, logic really provided unity to science. It thus presents a unified model of scientific investigation, where logic plays the aggregating role. (shrink)

There is at present a certain dispute about counterfactuals taking place. What is at issue is whether counterfactuals with necessarily false antecedents are all true. Some hold that such counterfactuals are vacuously true, appearances notwithstanding. Let us call such people vacuists. Others hold that some counterfactuals with necessarily false antecedents are true; some are false: it just depends on their contents. Let us call such people non-vacuists. As a notable representative of the vacuists, I will take Tim Williamson. On the (...) other side, I will take the position defended by Berto, French, Priest, and Ripley. I will argue (unsurprisingly) that the better choice is Non-Vacuism. That, however, is a subsidiary aim of this paper. The main point is to illustrate the method of theory-choice. (shrink)

Russell's involuted path in the development of his theory of logical types from 1903 to 1910-13 is examined and explained in terms of the development in his early philosophy of the notion of a logical subject vis-a-vis the problem of the one and many; i.e., the problem for russell, first, of a class-as-one as a logical subject as opposed to a class as many, and, secondly, of a propositional function as a single and separate logical (...) subject as opposed to existing only in the many propositions that are its values. (shrink)

Speculative logical theory, as provided in Hegel’s Science of Logic, consists of three main parts: the logic of being, the logic of essence, and the logic of the concept. The peculiar character of each logic’s starting point determines the most general character of each logic’s development. The present paper aims at making explicit the character of the starting-point of the third logic, the logic of the concept. This starting-point is exemplified by the category of universality. It is shown (...) (a) that the fundamental determination of this category is the harmonious unit y of self-identity and full determinacy and (b) that this unity has necessarily the logical structure of “double shining.” The latter is described in detail and justification is provided as to why it is preferred from “single shining.” I conclude the paper by defending the structure of “double shining” against certain objections raised against it by Friedrike Schick and Christian Iber. (shrink)

The search for the backbone of the types of rationality inherent in different cultures keeps on to be an open problem, which remains relevant to the need of closer intercultural interaction in the global world. At the same time, the analysis of the logic of language as the basis for the study of rationality types continues to occupy an important place. Meanwhile, the studies of grammatical structures and language models from the point of view of their connection to a certain (...) type of thinking and influence on the method of philosophizing are still quantitatively inferior to the researches of lexical aspects. Expansion of this type of research with the involvement of a diverse cultural material over time will allow us to reach the level of establishing regularities of a more general nature. This article contributes to the development of this issue. The author deals with the approaches to the defnition of the paradigm of Japanese rationality proposed by researchers of grammatical features of the Japanese language, on the one hand, and the “logic of place” concept and the “absolutely contradictory identity” principle of prominent philosopher Nishida Kitaro, on the other hand. A special attention is drawn to the structural similarity of the grammatical form highlighted by linguists, in which the decisive role belongs to the predicate, and to Nishida’s logical model reflecting specifcs of subject-predicate relationship as well as perception of opposition in terms of “contradictory self-identity.” In the conclusion of the article, the author demonstrates the relationship of this model with the certain idea of subject, the type of epistemology overcoming dualism, the processual and cosmocentric comprehension of the world that can be traced in Nishida’s statements. (shrink)

This paper deals with Popper's little-known work on deductive logic, published between 1947 and 1949. According to his theory of deductive inference, the meaning of logical signs is determined by certain rules derived from ?inferential definitions? of those signs. Although strong arguments have been presented against Popper's claims (e.g. by Curry, Kleene, Lejewski and McKinsey), his theory can be reconstructed when it is viewed primarily as an attempt to demarcate logical from non-logical constants rather than (...) as a semantic foundation for logic. A criterion of logicality is obtained which is based on conjunction, implication and universal quantification as fundamental logical operations. (shrink)

This paper accomplishes three goals. First, the essay demonstrates that Edmund Husserl’s theory of meaning consciousness from his 1901 Logical Investigations is internally inconsistent and falls apart upon closer inspection. I show that Husserl, in 1901, describes non-intuitive meaning consciousness as a direct parallel or as a ‘mirror’ of intuitive consciousness. He claims that non-intuitive meaning acts, like intuitions, have substance and represent their objects. I reveal that, by defining meaning acts in this way, Husserl cannot account for (...) our experiences of countersensical, absurd, or impossible meanings. Second, I examine how Husserl came to recognize this 1901 mistake in his 1913/14 Revisions to the Sixth Logical Investigation. I discuss how he accordingly reformulates his understanding of non-intuitive meaning acts from the ground up in those Revisions, where this also allows for him to properly account for the experience of impossible meanings. Instead of describing them as mirrors of intuitions, Husserl takes non-intuitive meaning acts to be modifications of intuitions, where they have no substance and do not represent their objects. Finally, in the conclusion to this essay, I demonstrate how this fundamental change to his understanding of meaning consciousness forced Husserl to revise other central tenets of his philosophy, such that the trajectory of his thought can only be properly understood in light of these revisions to his theory of non-intuitive meaning consciousness. (shrink)

But how is is that thought (viz., sense, imagination, and thought proper) is sometimes followed by action, sometimes not; sometimes by movement, sometimes not? What happens seems parallel to the case of thinking and inferring about the immovable objects of science. There the end is the truth seen (for, when one conceives the two premises, one at once conceives and comprehends the conclusion), but here the two premises result in a conclusion which is an action…Now that the action is the (...) conclusion is clear. But the premises of action are of two kinds, of the good and of the possible. Aristotle, De Motu Animalium, 701a, 6–10, 23–25. (shrink)

This chapter offers a logical, linguistic, and philosophical account of modern quantification theory. Contrasting the standard approach to quantifiers (according to which logical quantifiers are defined by enumeration) with the generalized approach (according to which quantifiers are defined systematically), the chapter begins with a brief history of standard quantifier theory and identifies some of its logical, linguistic, and philosophical strengths and weaknesses. It then proceeds to a brief history of generalized quantifier theory and explains (...) how it overcomes the weaknesses of the standard theory. One of the main philosophical advantages of the generalized theory is its philosophically informative criterion of logicality. The chapter describes the work done so far in this theory, highlights some of its central logical results, offers an overview of its main linguistic contributions, and discusses its philosophical significance. (shrink)