The original motivation of D. Gabbay’s concept of Fibring concerned the combination of logics, and initially it involved the syntactic introduction of modals into formulations of intuitionistic logic in which modals are syntactically absent. We show, using the notion of structural modals that there are many modals of intuitionism, and logics for subjunctive and epistemic conditionals which are not syntactically evident in our best formulations of them. We discuss some cases when the attempt to make them syntactically (...) evident can have undesirable consequences. (shrink)

We prove completeness and decidability results for a family of combinations of propositional dynamic logic and unimodal doxastic logics in which the modalities may interact. The kind of interactions we consider include three forms of commuting axioms, namely, axioms similar to the axiom of perfect recall and the axiom of no learning from temporal logic, and a Church–Rosser axiom. We investigate the influence of the substitution rule on the properties of these logics and propose a new semantics for (...) the test operator to avoid unwanted side effects caused by the interaction of the classic test operator with the extra interaction axioms. (shrink)

In this paper we propose a very general de nition of combination of logics by means of the concept of sheaves of logics. We first discuss some properties of this general definition and list some problems, as well as connections to related work. As applications of our abstract setting, we show that the notion of possible-translations semantics, introduced in previous papers by the first author, can be described in categorial terms. Possible-translations semantics constitute illustrative cases, since they (...) provide a new semantical account for abstract logical systems, particularly for many-valued and paraconsistent logics. (shrink)

In recent years combinations of tense and modality have moved intothe focus of logical research. From a philosophical point of view, logical systems combining tense and modality are of interest because these logics have a wide field of application in original philosophical issues, for example in the theory of causation, of action, etc. But until now only methods yielding completeness results for propositional languages have been developed. In view of philosophical applications, analogous results with respect to languages of predicate (...) logic are desirable, and in this paper I present two such results. The main developments in this area can be split into two directions, differing in the question whether the ordering of time is world-independent or not. Semantically, this difference appears in the discussion whether T x W-frames or Kamp-frames (resp. Ockham-frames) provide a suitable semantics for combinations of tense and modality. Here, two calculi are presented, the first adequate with respect to Kamp-semantics, the second to T x Wsemantics. (Both calculi contain an appropriate version of Gabbay's irreflexivity rule.) Furthermore, the proposed constructions of canonical frames simplify some of those which have hitherto been discussed. (shrink)

This paper generalises and complements the work on combining temporal logics started by Finger and Gabbay [11, 10]. We present proofs of transference of soundness, completeness and decidability for the temporalisation of logics T for any flow of time, eliminating the original restriction that required linear time for the transference of those properties through logic combination. We also generalise such results to the external application of a multi-modal system containing any number of connectives with arbitrary arity, that (...) respect normality.This generalisation over generic flows of time propagates to other combinations of logics that can be interpreted in terms of temporalisations. In this way, the independent combination of temporal logics is studied over generic flows of time. We show the transfer of soundness, completeness and decidability for independent combination of temporal logics. Finally, we also discuss the independent combination of any finite number of normal multi-modal logics. (shrink)

We extend meet-combination of logics for capturing the consequences that are common to both logics. With this purpose in mind we define meet-combination of consequence systems. This notion has the advantage of accommodating different ways of presenting the semantics and the deductive calculi. We consider consequence systems generated by a matrix semantics and consequence systems generated by Hilbert calculi. The meet-combination of consequence systems generated by matrix semantics is the consequence system generated by their product. (...) On the other hand, the meet-combination of consequence systems generated by Hilbert calculi is the consequence system generated by their interconnection. We investigate preservation of several properties. Capitalizing on these results we show that interconnection provides an axiomatization for the product. Illustrations are given for intuitionistic and modal logics, Łukasiewicz logic and some paraconsistent logics. (shrink)

The general methodology of "algebraizing" logics is used here for combining different logics. The combination of logics is represented as taking the colimit of the constituent logics in the category of algebraizable logics. The cocompleteness of this category as well as its isomorphism to the corresponding category of certain first-order theories are proved.

This paper investigates modular combinations of temporal logic systems. Four combination methods are described and studied with respect to the transfer of logical properties from the component one-dimensional temporal logics to the resulting combined two-dimensional temporal logic. Three basic logical properties are analyzed, namely soundness, completeness, and decidability. Each combination method comprises three submethods that combine the languages, the inference systems, and the semantics of two one-dimensional temporal logic systems, generating families of two-dimensional temporal languages with varying (...) expressivity and varying degrees of transfer of logical properties. The temporalization method and the independent combination method are shown to transfer all three basic logical properties. The method of full join of logic systems generates a considerably more expressive language but fails to transfer completeness and decidability in several cases. So a weaker method of restricted join is proposed and shown to transfer all three basic logical properties. (shrink)

본고는 라이프니츠의 논리학과 직접적으로 연관되어 있는 증명을 위한 사고의 조합에 관해 연구한 논문이다. 그의 논리학 개념에는 다양한 측면들이 있으며, 그러한 측면들은 그의 여러 가 지 논리학적 구상들과 기획들에 반영되어 있다. 이 논문은 이러한 구상들과 기획들이 중요한 철 학적 함의들을 갖고 있음을, 그리고 그것들이 보편학으로서의 그의 논리학을 구성함을, 그리고 그의 논리학이 기호들의 조작과 사고의 조합에 근거한 증명법임을 보이고자 한다. 라이프니츠는 우선적으로 자신의 논리 체계를 정립하기 위해 다양한 수학적인 방법들을 도입했다. 그리고 그 는 인간 사고의 알파벳과 새로운 백과사전과 같은 예비적인 논리적 기획들을 (...) 감행했다. 여기서 전자는 복합 개념들을 분석하여 도달한 단순 개념들의 기호들의 목록이며, 후자는 모든 학문들 에 포함된 단순 개념들과 정의들을 산출한다. 이러한 기획들을 토대로, 그는 그의 논리 체계를 건설하고자 하였다. 이러한 논리 체계는 모든 학문적인 명제들을 증명하며 보편 기호법과 추리 계산법의 사용을 가능하게 한다. 비록 라이프니츠가 명확한 논리 체계를 건설하는 데 실패했다 할지라도, 그는 최초로 논리적 기호법을 고안하고자 노력했으며, 그의 논리학 개념은 부울과 프 레게에게 현대 논리학을 형성하도록 영감을 주었다. (shrink)

The general methodology of "algebraizing" logics is used here for combining different logics. The combination of logics is represented as taking the colimit of the constituent logics in the category of algebraizable logics. The cocompleteness of this category as well as its isomorphism to the corresponding category of certain first-order theories are proved.

A graph-theoretic account of fibring of logics is developed, capitalizing on the interleaving characteristics of fibring at the linguistic, semantic and proof levels. Fibring of two signatures is seen as a multi-graph (m-graph) where the nodes and the m-edges include the sorts and the constructors of the signatures at hand. Fibring of two models is a multi-graph (m-graph) where the nodes and the m-edges are the values and the operations in the models, respectively. Fibring of two deductive systems is (...) an m-graph whose nodes are language expressions and the m-edges represent the inference rules of the two original systems. The sobriety of the approach is confirmed by proving that all the fibring notions are universal constructions. This graph-theoretic view is general enough to accommodate very different fibrings of propositional based logics encompassing logics with non-deterministic semantics, logics with an algebraic semantics, logics with partial semantics and substructural logics, among others. Soundness and weak completeness are proved to be preserved under very general conditions. Strong completeness is also shown to be preserved under tighter conditions. In this setting, the collapsing problem appearing in several combinations of logic systems can be avoided. (shrink)

This paper shows Hilbert system (C+J)-, given by del Cerro and Herzig (1996) is semantically incomplete. This system is proposed as a proof theory for Kripke semantics for a combination of intuitionistic and classical propositional logic, which is obtained by adding the natural semantic clause of classical implication into intuitionistic Kripke semantics. Although Hilbert system (C+J)- contains intuitionistic modus ponens as a rule, it does not contain classical modus ponens. This paper gives an argument ensuring that the system (C+J)- (...) is semantically incomplete because of the absence of classical modus ponens. Our method is based on the logic of paradox, which is a paraconsistent logic proposed by Priest (1979). (shrink)

This paper presents a short survey of recent developments in stit theories, with an emphasis on combinations of stit and deontic logic, and those of stit and epistemic logic.

Although a very recent topic in contemporary logic, the subject of combinations of logics has already shown its deep possibilities. Besides the pure philosophical interest offered by the possibility of defining mixed logic systems in which distinct operators obey logics of different nature, there are also several pragmatical and methodological reasons for considering combined logics. We survey methods for combining logics (integration of several logic systems into a homogeneous environment) as well as methods for decomposing (...) class='Hi'>logics, showing their interesting properties and applications. (shrink)

Motivated by applications in software engineering, we propose two forms of combination of logics: synchronization on formulae and synchronization on models. We start by reviewing satisfaction systems, consequence systems, one-step derivation systems and theory spaces, as well as their functorial relationships. We define the synchronization on formulae of two consequence systems and provide a categorial characterization of the construction. For illustration we consider the synchronization of linear temporal logic and equational logic. We define the synchronization on models of (...) two satisfaction systems and provide a categorial characterization of the construction. We illustrate the technique in two cases: linear temporal logic versus equational logic; and linear temporal logic versus branching temporal logic. Finally, we lift the synchronization on formulae to the category of logics over consequences systems. (shrink)

We investigate transfer of interpolation in such combinations of modal logic which lead to interaction of the modalities. Combining logics by taking products often blocks transfer of interpolation. The same holds for combinations by taking unions, a generalization of Humberstone's inaccessibility logic. Viewing first-order logic as a product of modal logics, we derive a strong counterexample for failure of interpolation in the finite variable fragments of first-order logic. We provide a simple condition stated only in terms of frames (...) and bisimulations which implies failure of interpolation. Its use is exemplified in a wide range of cases. (shrink)

This Handbook documents the main trends in current research between logic and language, including its broader influence in computer science, linguistic theory and cognitive science. The history of the combined study of Logic and Linguistics goes back a long way, at least to the work of the scholastic philosophers in the Middle Ages. At the beginning of this century, the subject was revitalized through the pioneering efforts of Gottlob Frege, Bertrand Russell, and Polish philosophical logicians such as Kazimierz Ajdukiewicz. Around (...) 1970, the landmark achievements of Richard Montague established a junction between state-of-the-art mathematical logic and generative linguistic theory. Over the subsequent decades, this enterprise of Montague Grammar has flourished and diversified into a number of research programs with empirical and theoretical substance. This appears to be the first Handbook to bring logic-language interface to the fore. Both aspects of the interaction between logic and language are demonstrated in the book i.e. firstly, how logical systems are designed and modified in response to linguistic needs and secondly, how mathematical theory arises in this process and how it affects subsequent linguistic theory. The Handbook presents concise, impartial accounts of the topics covered. Where possible, an author and a commentator have cooperated to ensure the proper breadth and technical content of the papers. The Handbook is self-contained, and individual articles are of the highest quality. (shrink)

This chapter shows how necessary it is for any science, including a science of evidence, to be able to classify phenomena of interest. It presents an evidence classification scheme that is ‘substance blind’, meaning that the classes of individual items of evidence identified are recurrent and apply regardless of the substance or content of the evidence. There are also substance-blind combinations of evidence that are also recurrent. The chapter shows how substance-blindness occurs as a matter of course involving concepts encountered (...) throughout science, logic, and mathematics. (shrink)

In this paper a principle of substitutivity of logical equivalents salve veritate and a version of Leibniz’s law are formulated and each is shown to cause problems when combined with naive truth theories.

The paper investigates the propriety of applying the form versus matter distinction to arguments and to logic in general. Its main point is that many of the currently pervasive views on form and matter with respect to logic rest on several substantive and even contentious assumptions which are nevertheless uncritically accepted. Indeed, many of the issues raised by the application of this distinction to arguments seem to be related to a questionable combination of different presuppositions and expectations; this holds (...) in particular of the vexed issue of demarcating the class of logical constants. I begin with a characterization of currently widespread views on form and matter in logic, which I refer to as 'logical hylomorphism as we know it'—LHAWKI, for short—and argue that the hylomorphism underlying LHAWKI is mereological. Next, I sketch an overview of the historical developments leading from Aristotelian, non-mereological metaphysical hylomorphism to mereological logical hylomorphism (LHAWKI). I conclude with a reassessment of the prospects for the combination of hylomorphism and logic, arguing in particular that LHAWKI is not the only and certainly not the most suitable version of logical hylomorphism. In particular, this implies that the project of demarcating the class of logical constants as a means to define the scope and nature of logic rests on highly problematic assumptions. (shrink)

This Handbook is a combination of authoritative exposition, comprehensive survey, and fundamental research exploring the underlying unifying themes in the various areas. The intended audience is graduate students and researchers in the areas of computing and logic, as well as other people interested in the subject. We assume as background some mathematical sophistication. Much of the material will also be of interest to logicians and mathematicians.

The paper is devoted to an approach to analytic tableaux for propositional logic, but can be successfully extended to other logics. The distinguishing features of the presented approach are:(i) a precise set-theoretical description of tableau method; (ii) a notion of tableau consequence relation is defined without help of a notion of tableau, in our universe of discourse the basic notion is a branch;(iii) we define a tableau as a finite set of some chosen branches which is enough to check; (...) hence, in our approach a tableau is only a way of choosing a minimal set of closed branches;(iv) a choice of tableau can be arbitrary, it means that if one tableau starting with some set of premisses is closed in the defined sense, then every branch in the power set of the set of formulas, that starts with the same set, is closed. (shrink)

Seeking to better understand how nonprofit organizations (NPOs) manage hybridity, we investigated what distinguishes NPOs that combine multiple logics in productive and unproductive ways. We collected and analyzed data from six case studies of NPOs delivering social services in Australia. Our findings reveal that organizational members of NPOs take a _perspective_ on their hybrid nature which comprises four elements: motivational framing, actor engagement, resourcing attitude, and governance orientation. NPOs that combine multiple logics in productive and unproductive ways, respectively, (...) are distinguished by (1) a compelling or confused motivational framing for combining logics; (2) actors having active and shared, or passive and isolated, engagement with multiple logics; (3) attitudes toward resourcing multiple logics that are either coherent or competitive; and (4) a governance orientation toward multiple logics as opportunities to leverage or problems to resist. Our findings contribute to the literature by deepening understanding of the interplay between complex constellations of multiple logics in NPOs, including religious and professional logics. We also develop a model of organizational perspectives on hybridity and their implications for distinguishing NPOs that productively harness tensions between logics. (shrink)

We present a simple theory of actions against the background of branching time, based on which we propose two versions of an extended stit theory, one equipped with particular actions and the other with sets of such actions. After reporting some basic results of a formal development of such a theory, we briefly explore its connection to a version of branching ETL.

Satisfaction systems and reductions between them are presented as an appropriate context for analyzing the satisfiability and the validity problems. The notion of reduction is generalized in order to cope with the meet-combination of logics. Reductions between satisfaction systems induce reductions between the respective satisfiability problems and (under mild conditions) also between their validity problems. Sufficient conditions are provided for relating satisfiability problems to validity problems. Reflection results for decidability in the presence of reductions are established. The validity (...) problem in the meet-combination is proved to be decidable whenever the validity problem for the components are decidable. Several examples are discussed, namely, involving modal and intuitionistic logics, as well as the meet-combination of modal logic and intuitionistic logic. (shrink)

The probabilization of a logic system consists of enriching the language (the formulas) and the semantics (the models) with probabilistic features. Such an operation is said to be exogenous if the enrichment is done on top, without internal changes to the structure, and is called endogenous otherwise. These two different enrichments can be applied simultaneously to the language and semantics of a same logic. We address the problem of studying the transference of metaproperties, such as completeness and decidability, to the (...) exogenous probabilization of an abstract logic system. First, we setup the necessary framework to handle the probabilization of a satisfaction system by proving transference results within a more general context. In this setup, we define a combination mechanism of logics through morphisms and prove sufficient condition to guarantee completeness and decidability. Then, we demonstrate that probabilization is a special case of this exogenous combination method, and that it fulfills the general conditions to obtain transference of completeness and decidability. Finally, we motivate the applicability of our technique by analyzing the probabilization of the linear temporal logic over Markov chains, which constitutes an endogenous probabilization. The results are obtained first by studying the exogenous semantics, and then by establishing an equivalence with the original probabilization given by Markov chains. (shrink)

Logical pluralism is the view that there is more than one correct logic. A key objection to logical pluralism is that it collapses into monism. The core of the Collapse Objection is that only the pluralist’s strongest logic does any genuine normative work; since a logic must do genuine normative work, this means that the pluralist is really a monist, who is committed to her strongest logic being the one true logic. This paper considers a neglected question in the collapse (...) debate: what is it for a logic to do genuine normative work? As well as having wider upshot for the connection between logic and normativity, grappling with this question provides a new response to the Collapse Objection on behalf of the pluralist. I suggest that we should allow logics to generate pro tanto reasons in a way that bears not just on combinations of attitudes but on how an agent’s attitudes are based on one another. This motivates adopting normative principles that allow the pluralist’s weaker logics to earn their normative keep. Rather than being ad hoc, these principles capture a sense in which good reasoning goes beyond the consistency of an agent’s attitudes. Good reasoning is also concerned with how an agent’s attitudes are based on one another. (shrink)

The generation of the pitch control signal in a wind turbine (WT) is not straightforward due to the nonlinear dynamics of the system and the coupling of its internal variables; in addition, they are subjected to the uncertainty that comes from the random nature of the wind. Fuzzy logic has proved useful in applications with changing system parameters or where uncertainty is relevant as in this one, but the tuning of the fuzzy logic controller (FLC) parameters is neither straightforward nor (...) an easy task. On the other hand, reinforcement learning (RL) allows systems to automatically learn, and this capability can be exploited to tune the FLC. In this work, a WT pitch control architecture that uses RL to tune the membership functions and scale the output of a fuzzy controller is proposed. The RL strategy calculates the fuzzy controller gains in order to reduce the output power error of the WT according to the wind speed. Different reward mechanisms based on the output power error have been considered. Simulation results with different wind profiles show that this architecture performs better (123.7 W) in terms of power errors than an FLC without RL (133.2 W) or a simpler PID (208.8 W). Even more, it provides a smooth response and outperforms other hybrid controllers such as RL-PID and radial basis function neural network control. (shrink)

This is Part 1 of a paper on fibred semantics and combination of logics. It aims to present a methodology for combining arbitrary logical systems L i , i ∈ I, to form a new system L I . The methodology `fibres' the semantics K i of L i into a semantics for L I , and `weaves' the proof theory (axiomatics) of L i into a proof system of L I . There are various ways of doing (...) this, we distinguish by different names such as `fibring', `dovetailing' etc, yielding different systems, denoted by L F I , L D I etc. Once the logics are `weaved', further `interaction' axioms can be geometrically motivated and added, and then systematically studied. The methodology is general and is applied to modal and intuitionistic logics as well as to general algebraic logics. We obtain general results on bulk, in the sense that we develop standard combining techniques and refinements which can be applied to any family of initial logics to obtain further combined logics. The main results of this paper is a construction for combining arbitrary, (possibly not normal) modal or intermediate logics, each complete for a class of (not necessarily frame) Kripke models. We show transfer of recursive axiomatisability, decidability and finite model property. Some results on combining logics (normal modal extensions of K) have recently been introduced by Kracht and Wolter, Goranko and Passy and by Fine and Schurz as well as a multitude of special combined systems existing in the literature of the past 20-30 years. We hope our methodology will help organise the field systematically. (shrink)

Recent work on analyticity distinguishes two kinds, metaphysical and epistemic. This paper argues that the distinction allows for a new view in the philosophy of logic according to which the claims of logic are metaphysically analytic and have distinctive modal profiles, even though their epistemology is holist and in many ways rather Quinean. It is argued that such a view combines some of the more attractive aspects of the Carnapian and Quinean approaches to logic, whilst avoiding some famous problems.

ABSTRACT Theories of sets such as Zermelo Fraenkel set theory are usually presented as the combination of two distinct kinds of principles: logical and set-theoretic principles. The set-theoretic principles are imposed ‘on top’ of first-order logic. This is in agreement with a traditional view of logic as universally applicable and topic neutral. Such a view of logic has been rejected by the intuitionists, on the ground that quantification over infinite domains requires the use of intuitionistic rather than classical logic. (...) In the following, I consider constructive set theories, which use intuitionistic rather than classical logic, and argue that they manifest a distinctive interdependence or an entanglement between sets and logic. In fact, Martin-Löf type theory identifies fundamental logical and set-theoretic notions. Remarkably, one of the motivations for this identification is the thought that classical quantification over infinite domains is problematic, while intuitionistic quantification is not. The approach to quantification adopted in Martin-Löf’s type theory is subtly interconnected with its predicativity. I conclude by recalling key aspects of an approach to predicativity inspired by Poincaré, which focuses on the issue of correct quantification over infinite domains and relate it back to Martin-Löf type theory. (shrink)

This paper presents two general results of decidability concerning logics based on an indeterministic metric tense logic, which can be applied to, among others, logics combining knowledge, time and agency. We provide a general Kripke semantics based on a variation of the notion of synchronized Ockhamist frames. Our proof of the decidability is by way of the finite frame property, applying subframe transformations and a variant of the filtration technique.

A graph-theoretic account of logics is explored based on the general notion of m-graph (that is, a graph where each edge can have a finite sequence of nodes as source). Signatures, interpretation structures and deduction systems are seen as m-graphs. After defining a category freely generated by a m-graph, formulas and expressions in general can be seen as morphisms. Moreover, derivations involving rule instantiation are also morphisms. Soundness and completeness theorems are proved. As a consequence of the generality of (...) the approach our results apply to very different logics encompassing, among others, substructural logics as well as logics with nondeterministic semantics, and subsume all logics endowed with an algebraic semantics. (shrink)

Although it is frequently said that logic is a purely formal discipline lacking any content for special philosophical subdisciplines, I argue in this essay that the concepts of predication, and of the properties of objects presupposed by standard first-order logic are sufficient to address many of the traditional problems of ontology. The concept of an object's having a property is extended to provide an intensional definition of the existence of an object as the object's possessing a maximally consistent property (...) class='Hi'>combination, consisting, for any property or its complement. Nonexistent objects by the proposed definition are those that either lack both some property and its complement, or have both a property and its complement included in its corresponding property combination. The definition in turn makes it possible to offer solutions based on purely logical concepts of such longstanding metaphysical problems as why there is something rather than nothing, and why there exists exactly one logically contingent actual world, itself a maximally consistent combination of true predication instances or facts. Additionally, the ontology upholds an argument in support of modal actualism and against modal realism, treating all logically possible worlds other than the actual world as predicationally incomplete nonexistent objects or semantic fictions. (shrink)

The construction of a systematic philosophical foundation for logic is a notoriously difficult problem. In Part One I suggest that the problem is in large part methodological, having to do with the common philosophical conception of “providing a foundation”. I offer an alternative to the common methodology which combines a strong foundational requirement with the use of non-traditional, holistic tools to achieve this result. In Part Two I delineate an outline of a foundation for logic, employing the new methodology. The (...) outline is based on an investigation of why logic requires a veridical justification, i.e., a justification which involves the world and not just the mind, and what features or aspect of the world logic is grounded in. Logic, the investigation suggests, is grounded in the formal aspect of reality, and the outline proposes an account of this aspect, the way it both constrains and enables logic, logic's role in our overall system of knowledge, the relation between logic and mathematics, the normativity of logic, the characteristic traits of logic, and error and revision in logic. (shrink)

The combination of logic and game theory provides a fine-grained perspective on information and interaction dynamics, a Theory of Play. In this paper we lay down the main components of such a theory, drawing on recent advances in the logical dynamics of actions, preferences, and information. We then show how this fine-grained perspective has already shed new light on the long-term dynamics of information exchange, as well as on the much-discussed question of extensive game rationality.

We combine intuitionistic logic and classical logic into a new, first-order logic called polarized intuitionistic logic. This logic is based on a distinction between two dual polarities which we call red and green to distinguish them from other forms of polarization. The meaning of these polarities is defined model-theoretically by a Kripke-style semantics for the logic. Two proof systems are also formulated. The first system extends Gentzenʼs intuitionistic sequent calculus LJ. In addition, this system also bears essential similarities to Girardʼs (...) LC proof system for classical logic. The second proof system is based on a semantic tableau and extends Dragalinʼs multiple-conclusion version of intuitionistic sequent calculus. We show that soundness and completeness hold for these notions of semantics and proofs, from which it follows that cut is admissible in the proof systems and that the propositional fragment of the logic is decidable. (shrink)

This paper combines personal reminiscences of the philosopher John Corcoran with a discussion of certain conflicts between historians of logic and philosophers of logic. Some mistaken claims about the history of the Bolzano-Weierstrass Theorem are analyzed in detail and corrected.

In this paper we deal with two different approaches to logic, the Boolean and the Fregean. In particular, we give some reasons to explain certain ignorance of the Boolean approach at some Philosophy Departments, particularly in the teaching of logic. After giving some reasons for the convenience of combining both approaches, we recommend material on the Boolean approach to be included in the mentioned contexts.

This stimulating book covers a wide range of topics concerning Kant and the history of logic, with the overall goal of specifying how Kant's various conceptions of logic developed out of that history. A first chapter on methodology argues that Kant is properly understood as negotiating a middle way between eclecticism and systematic philosophy. This is combined with the author's views on the proper method for our reconstruction of Kant's philosophy of logic, where her idea is that Kant's own critical (...) eclecticism will provide the key to the latter. One issue that is never addressed is why this is not a sort of category error: even if Kant's own views on methodology had featured a... (shrink)

What is the source of logical and mathematical truth? This book revitalizes conventionalism as an answer to this question. Conventionalism takes logical and mathematical truth to have their source in linguistic conventions. This was an extremely popular view in the early 20th century, but it was never worked out in detail and is now almost universally rejected in mainstream philosophical circles. Shadows of Syntax is the first book-length treatment and defense of a combined conventionalist theory of logic and mathematics. It (...) argues that our conventions, in the form of syntactic rules of language use, are perfectly suited to explain the truth, necessity, and a priority of logical and mathematical claims, as well as our logical and mathematical knowledge. (shrink)

Heinrich Behmann (1891-1970) obtained his Habilitation under David Hilbert in Göttingen in 1921 with a thesis on the decision problem. In his thesis, he solved - independently of Löwenheim and Skolem's earlier work - the decision problem for monadic second-order logic in a framework that combined elements of the algebra of logic and the newer axiomatic approach to logic then being developed in Göttingen. In a talk given in 1921, he outlined this solution, but also presented important programmatic remarks on (...) the significance of the decision problem and of decision procedures more generally. The text of this talk as well as a partial English translation are included. (shrink)