In this paper we provide a detailed proof-theoretical analysis of a natural deduction system for classical propositional logic that (i) represents classical proofs in a more natural way than standard Gentzen-style natural deduction, (ii) admits of a simple normalization procedure such that normal proofs enjoy the Weak Subformula Property, (iii) provides the means to prove a Non-contamination Property of normal proofs that is not satisfied by normal proofs in the Gentzen tradition and is useful for applications, especially in formal argumentation, (...) (iv) naturally leads to defining a notion of depth of a proof, to the effect that, for every fixed natural k, normal k-depth deducibility is a tractable problem and converges to classical deducibility as k tends to infinity. (shrink)

The problem of rationality is nowadays studied in an explicit fashion mostly by philosophers of science, the prevailing assumption being that science is rationality par excellence, so that an analysis of science will yield an understanding of rationality. It is therefore with great interest that one opens this book whose suggestive title gives the impression of approaching the problem in a more original way, namely from the point of view of logic. However, one finds the logic in question to be (...) conventional formal logic, and the problem touched upon only in those parts of the book dealing with the question of the role of formalization and formal methods in rationality, which parts constitute only a small portion of the work. The author advocates a view he calls metaformalism, which he describes obscurely as "the grammar of the connectivity and integration of viewpoints, the technical embodiment of magnanimity and love" ; sociologically as stemming from "the writings of Polanyi... Popper... Butterfield, Conant, Kuhn, Agassi, Feyerabend, Hanson, Sellars, Wartofsky and... Patrick Heelan" ; and insightfully as such that it "would bring a qualitative change to modern logic—a science which has been widening itself more than it has been deepening in roots. It would be the kind of technical development which would underwrite the view that logic cannot be fully understood without being seen in its relationship with rational competence and human subjectivity. A ’system logic’ would perhaps point to this relationship more strongly than propositional logic". (shrink)

In this article, I will discuss possible differences between ecosystems and organisms on the basis of their intrinsic complexity. As the concept of complexity still remains highly debated, I propose here a practical and original way to measure the complexity of an ecosystem or an organism. For this purpose, I suggest using the concept of logical depth (LD) in a specific manner, in order to take into account the difficulty as well as the time needed to generate the (...) studied object. I illustrate this method with fully controlled Daisyworld simulations (i.e., simulations based on the Gaia hypothesis) that have often been proposed to mimic living systems. The method consists of the following sequential stages: (1) identification of the shortest program able to numerically model the studied system (also called the Kolmogorov–Solomonoff complexity); (2) running the program, once if there are no stochastic components in the system, several times if stochastic components are there; and (3) computing the time needed to generate the system with LD complexity. This measure is supposed to estimate the system complexity. It appears in this study that LD estimations fit well with the intuition we have of complex systems, with higher complexities being found for more realistic Daisyworlds. With such a method of capturing the time needed to build a system, we expect to detect in future studies any quantified differences between complex ecosystems and organisms. (shrink)

The paper essentially shows that the paraconsistent logicDR satisfies the depth relevance condition. The systemDR is an extension of the systemDK of [7] and the non-triviality of a dialectical set theory based onDR has been shown in [3]. The depth relevance condition is a strengthened relevance condition, taking the form: If DR- AB thenA andB share a variable at the same depth, where the depth of an occurrence of a subformulaB in a formulaA is roughly the (...) number of nested ''s required to reach the occurrence ofB inA. The method of proof is to show that a model structureM consisting of {M 0 , M1, ..., M}, where theM i s are all characterized by Meyer''s 6-valued matrices (c. f, [2]), satisfies the depth relevance condition. Then, it is shown thatM is a model structure for the systemDR. (shrink)

Relevant logics infamously have the property that they only validate a conditional when some propositional variable is shared between its antecedent and consequent. This property has been strengthened in a variety of ways over the last half-century. Two of the more famous of these strengthenings are the strong variable sharing property and the depth relevance property. In this paper I demonstrate that an appropriate class of relevant logics has a property that might naturally be characterized as the supremum of (...) these two properties. I also show how to use this fact to demonstrate that these logics seem to be constructive in previously unknown ways. (shrink)

This paper proves the finite axiomatizability of transitive modal logics of finite depth and finite width w.r.t. proper-successor-equivalence. The frame condition of the latter requires, in a rooted transitive frame, a finite upper bound of cardinality for antichains of points with different sets of proper successors. The result generalizes Rybakov’s result of the finite axiomatizability of extensions of$\mathbf {S4}$of finite depth and finite width.

Logic for Philosophy is an introduction to logic for students of contemporary philosophy. It is suitable both for advanced undergraduates and for beginning graduate students in philosophy. It covers (i) basic approaches to logic, including proof theory and especially model theory, (ii) extensions of standard logic that are important in philosophy, and (iii) some elementary philosophy of logic. It emphasizes breadth rather than depth. For example, it discusses modal logic and counterfactuals, but does not prove the central metalogical results (...) for predicate logic (completeness, undecidability, etc.) Its goal is to introduce students to the logic they need to know in order to read contemporary philosophical work. It is very user-friendly for students without an extensive background in mathematics. In short, this book gives you the understanding of logic that you need to do philosophy. (shrink)

Formal symptoms of relevance usually concern the propositional variables shared between the antecedent and the consequent of provable conditionals. Among the most famous results about such symptoms are Belnap’s early results showing that for sublogics of the strong relevant logic R, provable conditionals share a signed variable between antecedent and consequent. For logics weaker than R stronger variable sharing results are available. In 1984, Ross Brady gave one well-known example of such a result. As a corollary to the main result (...) of the paper, we give a very simple proof of a related but strictly stronger result. (shrink)

This paper argues that self-deception cannot be explained without employing a depth-psychological ("psychodynamic") notion of the unconscious, and therefore that mainstream academic psychology must make space for such approaches. The paper begins by explicating the notion of a dynamic unconscious. Then a brief account is given of the "paradoxes" of self-deception. It is shown that a depth-psychological self of parts and subceptive agency removes any such paradoxes. Next, several competing accounts of self-deception are considered: an attentional account, a (...) constructivist account, and a neo-Sartrean account. Such accounts are shown to face a general dilemma: either they are able only to explain unmotivated errors of self-perception--in which case they are inadequate for their intended purpose--or they are able to explain motivated self-deception, but do so only by being instantiation mechanisms for depth-psychological processes. The major challenge to this argument comes from the claim that self-deception has a "logic" different to other-deception--the position of Alfred Mele. In an extended discussion it is shown that any such account is explanatorily adequate only for some cases of self-deception--not by any means all. Concluding remarks leave open to further empirical work the scope and importance of depth-psychological approaches. (shrink)

This paper explores different interpretations of the word ‘deep’ as it is used by mathematicians, with a large number of examples illustrating various criteria for depth. Most of the examples are theorems with ‘historical depth’, in the sense that many generations of mathematicians contributed to their proof. Some also have ‘foundational depth’, in the sense that they support large mathematical theories. Finally, concepts from mathematical logic suggest that it may be possible to order certain theorems or problems (...) according to ‘logical depth’. (shrink)

Philosophical Logic is a clear and concise critical survey of nonclassical logics of philosophical interest written by one of the world's leading authorities on the subject. After giving an overview of classical logic, John Burgess introduces five central branches of nonclassical logic, focusing on the sometimes problematic relationship between formal apparatus and intuitive motivation. Requiring minimal background and arranged to make the more technical material optional, the book offers a choice between an overview and in-depth study, and it balances (...) the philosophical and technical aspects of the subject.The book emphasizes the relationship between models and the traditional goal of logic, the evaluation of arguments, and critically examines apparatus and assumptions that often are taken for granted. Philosophical Logic provides an unusually thorough treatment of conditional logic, unifying probabilistic and model-theoretic approaches. It underscores the variety of approaches that have been taken to relevantistic and related logics, and it stresses the problem of connecting formal systems to the motivating ideas behind intuitionistic mathematics. Each chapter ends with a brief guide to further reading.Philosophical Logic addresses students new to logic, philosophers working in other areas, and specialists in logic, providing both a sophisticated introduction and a new synthesis. (shrink)

Weak enough relevant logics are often closed under depth substitutions. To determine the breadth of logics with this feature, we show there is a largest sublogic of R closed under depth substitutions and that this logic can be recursively axiomatized.

The interpretation given by Craik and Lockhart of the findings by Hyde and Jenkins involving supposed depth of incidental-task processing on subsequent word recall is brought into question by the tenets of logical learning theory. It is shown that Craik and Lockhart overlooked the possible role of oppositionality in this research. An alternative explanation relying on an oppositional context and predication is offered. Two experiments present evidence supporting the hypothesis that oppositionality in an incidental task facilitates subsequent word (...) recall . In both experiments, the importance of taking a subject's meaningful understanding of the task instruction into consideration is highlighted. The discussion contrasts Boolean "binary" disjunction with the logic of oppositionality. It is shown how oppositionality allows us to conceptualize a testable theory of human agency. (shrink)

The paper attempts to reconstruct some notions of Naess's semantics, and at the same time to relate them to more recent developments. On Naess's view, there is no such thing as a language in the sense of a shared structure which determines clear-cut literal meanings like Fregean Gedanken or propositions. We use words, and try to interpret each other; but there is no a priori or intuitive basis for secure and precise knowledge about language. Interpretation or understanding, as well as (...) thinking and perception, involve discrimination at higher or lower levels. There is neither a ceiling of crystal-clear propositions nor a floor of 'given' sensations or 'objective' stimuli independent of categorizations and discriminations, which might serve as a bridge for translation or interpretation between the conceptual-perceptual schemes of individuals. The notions of 'concept', 'conceptual framework', 'situation', 'situation type' and 'proposition' are defined, with some reliance on Barwise and Perry's situation semantics. Two notions of accessibility of situation types are defined; one as constructibility of the type at a given level of discrimination, i.e. in terms of objects discerned and concepts discriminated; the other as intelligibility of the type. Intelligibility entails constructibility, but not conversely. A type of situation may be constructible at some person's level of discrimination, but not coherently intelligible to him. On this basis, a logic of senses or truth-conditions for uses of expressions is developed, and a logic of interpretations of uses. The central notions of Naessian semantics are defined: ambiguity and direction of interpretation, level of discrimination and depth of intention. (shrink)

We examine the relationship between the logics of nonsense of Bochvar and Halldén and the containment logics in the neighborhood of William Parry’s A I. We detail two strategies for manufacturing containment logics from nonsense logics—taking either connexive and paraconsistent fragments of such systems—and show how systems determined by these techniques have appeared as Frederick Johnson’s R C and Carlos Oller’s A L. In particular, we prove that Johnson’s system is precisely the intersection of Bochvar’s B 3 and Graham Priest’s (...) non-symmetrized connexive logic and that Oller’s system lies just beneath the intersection of B 3 and Priest’s paraconsistent L P. We conclude by examining Oller’s system in more depth, giving it a characterization in terms of L P and showing that it plays the same role to Harry Deutsch’s paraconsistent containment logic S that Aleksandr Zinov'ev’s S 1 plays with respect to A I. (shrink)

When we say “I know why he was late”, we know not only the fact that he was late, but also an explanation of this fact. We propose a logical framework of “knowing why” inspired by the existing formal studies on why-questions, scientific explanation, and justification logic. We introduce the Kyi\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${{\mathcal {K}}{}\textit{y}}_i$$\end{document} operator into the language of epistemic logic to express “agent i knows why φ\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} (...) \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\varphi $$\end{document}” and propose a Kripke-style semantics of such expressions in terms of knowing an explanation of φ\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\varphi $$\end{document}. We obtain two sound and complete axiomatizations w.r.t. two different model classes depending on different assumptions about introspection. Finally we connect our logic with justification logic technically by providing an alternative semantics and an in-depth comparison on various design choices. (shrink)

We consider the class of pointed varieties of algebras having a lattice term reduct and we show that each such variety gives rise in a natural way, and according to a regular pattern, to at least three interesting logics. Although the mentioned class includes several logically and algebraically significant examples (e.g. Boolean algebras, MV algebras, Boolean algebras with operators, residuated lattices and their subvarieties, algebras from quantum logic or from depth relevant logic), we consider here in greater detail Abelian (...) ℓ-groups, where such logics respectively correspond to: i) Meyer and Slaney’s Abelian logic [31]; ii) Galli et al.’s logic of equilibrium [21]; iii) a new logic of “preservation of truth degrees”. (shrink)

This article interrogates the persistence of critical frameworks informed by depth-models of hermeneutics, and the repercussions the equation of “depth” with meaningfulness has for the appreciation of the “shallow” aesthetics of post-classical action cinema. Oppositions such as depth/surface, body/mind, and proximity/distance associated with a hermeneutics of depth are not neutral, but rather exist in a “violent hierarchy”. This ensures that works or styles that foreground surface are automatically deemed to be meaningless. One influential example of this (...) logic is Fredric Jameson's dismissal of postmodern superficiality in favour of modernist depth. In contrast, this article will explore alternative models of postmodern superficiality. Taking up Catherine Constable's reading of Linda Hutcheon, I will endeavour to demonstrate the benefits of adopting a postmodern critical perspective when engaging with texts that resist or problematise depth-models. Through a close reading of a key se... (shrink)

This paper partly answers the question “what a frame may be exactly like when it characterizes a pretabular logic in NExtK4”. We prove the pretabularity crieria for the logics of finite depth in NExtK4. In order to find out the criteria, we create two useful concepts—“pointwise reduction” and “invariance under pointwise reductions”, which will remain important in dealing with the case of infinite depth.

The Handbook of Modal Logic contains 20 articles, which collectively introduce contemporary modal logic, survey current research, and indicate the way in which the field is developing. The articles survey the field from a wide variety of perspectives: the underling theory is explored in depth, modern computational approaches are treated, and six major applications areas of modal logic (in Mathematics, Computer Science, Artificial Intelligence, Linguistics, Game Theory, and Philosophy) are surveyed. The book contains both well-written expository articles, suitable for (...) beginners approaching the subject for the first time, and advanced articles, which will help those already familiar with the field to deepen their expertise. Please visit: http://people.uleth.ca/~woods/RedSeriesPromo_WP/PubSLPR.html - Compact modal logic reference - Computational approaches fully discussed - Contemporary applications of modal logic covered in depth. (shrink)

Kaneko-Suzuki developed epistemic logics of shallow depths with multiple players for investigations of game theoretical problems. By shallow depth, we mean that nested occurrences of belief operators of players in formulae are restricted, typically to be of finite depths, by a given epistemic structure. In this paper, we develop various methods of surgical operations (cut and paste) of epistemic world models. An example is a bouquet-making, i.e., tying several models into a bouquet. Another example is to engraft a model (...) to some branches of another model. By these methods, we obtain various meta-theorems on semantics and syntax on epistemic logics. To illustrate possible uses of our meta-theorems, we present one game theoretical theorem, which is also a meta-theorem in the sense of logic. (shrink)

This print edition of Read's account of logical thought includes the original publication's diagrams and tables. In this excellent book, Read commences by offering an overview of past attitudes and definitions of logic. Individual chapters consider the various means by which logical processes are conceived and developed in the mind. Philosophical arguments, spatial reasoning and mathematical forms of logic are discussed in great depth, with illustrations appended where deemed necessary. Read, an academic and philosopher, employs his decades (...) long experience of teaching students about the logical disciplines to write a detailed and comprehensive accounts of logic's aspects. Throughout this work, he attempts to retain a clarity of expression which makes the concepts discussed understandable for readers previously unversed in the types of logical expression. Read also sets out the principles behind setting up a hypothesis, and highlights the common logical fallacies which crop up in human reasoning. Extensive attention is given to the process of syllogism, by which sound conclusions are drawn from the premises offered. Throughout the book, Read is attentive in defining and explaining the esoteric terminology native to logic. As well as allowing the reader swift comprehension of the various subtopics, this explanatory style confers improved flow to the book. At the conclusion we are offered many questions corresponding to each chapter: these are intended to test the reader's understanding, and enforce the mental habits of utilising logical and critical thought. First published in 1898 and subject to several editions with revisions and refinements thereafter, Logic Deductive and Inductive retains a good reputation among academics and logicians. Its successful balance between striking a concise introductory tone with an embrace of the core elements of the subject has served to preserve its popularity into the modern day. (shrink)

In the Western philosophical tradition logical investigation and philosophical advance have been inextricably linked, each having stimulated and shaped the other. In Logic and Philosophy William H. Brenner examines a broad range of logical concepts and methods as they relate to the larger context of philosophical investigation and thus bring to light the philosophical depth of logic and its relevance to philosophy in general.

A concise yet rigorous introduction to logic and discrete mathematics. This book features a unique combination of comprehensive coverage of logic with a solid exposition of the most important fields of discrete mathematics, presenting material that has been tested and refined by the authors in university courses taught over more than a decade. The chapters on logic - propositional and first-order - provide a robust toolkit for logical reasoning, emphasizing the conceptual understanding of the language and the semantics of (...) classical logic as well as practical applications through the easy to understand and use deductive systems of Semantic Tableaux and Resolution. The chapters on set theory, number theory, combinatorics and graph theory combine the necessary minimum of theory with numerous examples and selected applications. Written in a clear and reader-friendly style, each section ends with an extensive set of exercises, most of them provided with complete solutions which are available in the accompanying solutions manual. Key Features: Suitable for a variety of courses for students in both Mathematics and Computer Science. Extensive, in-depth coverage of classical logic, combined with a solid exposition of a selection of the most important fields of discrete mathematics Concise, clear and uncluttered presentation with numerous examples. Covers some applications including cryptographic systems, discrete probability and network algorithms. Logic and Discrete Mathematics: A Concise Introduction is aimed mainly at undergraduate courses for students in mathematics and computer science, but the book will also be a valuable resource for graduate modules and for self-study. (shrink)

To understand logic is, first and foremost, to understand logical consequence. This Element provides an in-depth, accessible, up-to-date account of and philosophical insight into the semantic, model-theoretic conception of logical consequence, its Tarskian roots, and its ideas, grounding, and challenges. The topics discussed include: the passage from Tarski's definition of truth to his definition of logical consequence, the need for a non-proof-theoretic definition, the idea of a semantic definition, the adequacy conditions of preservation of truth, formality, (...) and necessity, the nature, structure, and totality of models, the logicality problem that threatens the definition of logical consequence, a general solution to the logicality, formality, and necessity problems/challenges, based on the isomorphism-invariance criterion of logicality, philosophical background and justification of the isomorphism-invariance criterion, and major criticisms of the semantic definition and the isomorphism-invariance criterion. (shrink)

In this paper we prove the pretabularity criteria for the logics of infinite depth in NExtK4. Then we use the criteria to resolve the problems of pretabular logics in NExtQ4 and prove that there is a continuum of pretabular logics in NExtQ4 just like NExtK4.

Giving justice to Maximus any philosophy wich does not include mysticism will be false as philosophy. Our metaphysics must be mystical in order to be rational. In Maximus’ doctrine, then, Christ comes not to destroy but to fulfill the metaphysics of mystery elaborated by the philosophers. For him there can be no separation between philosophy and theology, or between natural and revealed theology. Thereby, Christology and liturgical mysticism are not additional to a neoplatonic, aristotelian, and other methaphysics. Maximus concern was (...) to continue, not the philosophical tradition of the Aristotelian commentators, but the theological one of the Fathers. He was not an Aristotelian commentator himself. The union and distinction are basic logical concepts in Maximus’ thinking, but the Chalcedonian logic is the application of these concepts. Only in this way one can talk about Christianization of aristotelian logic. St. Maximus the Confessor synthesized Aristotelianism influences with those of Platonism in order to exceed the daring speculations of cosmology origeniene. He had an extraordinary ability to combine metaphysical requirements with the effort of defining the faith dogma, and the monastic experiences with the depth thinking, succeeding to propose a new conception in which converge all cultural and religious influences. This study is analyzing the relationship between logoi and energeia (the intentional or “logical” energeia and the ontology of divine energy as ontological “logic”) within the maximian cosmology, by referring to the palamite theology. The concept of logoi for St. Maximus play a role similar in many respects to that of energy (energeiai) in Cappadocian Fathers, but the functional similarity it should not lead to the identification rationales with the energies. (shrink)

Major figures of twentieth-century philosophy were enthralled by the revolution in formal logic, and many of their arguments are based on novel mathematical discoveries. Hilary Putnam claimed that the Löwenheim-Skølem theorem refutes the existence of an objective, observer-independent world; Bas van Fraassen claimed that arguments against empiricism in philosophy of science are ineffective against a semantic approach to scientific theories; W. V. O. Quine claimed that the distinction between analytic and synthetic truths is trivialized by the fact that any theory (...) can be reduced to one in which all truths are analytic. This book dissects these and other arguments through in-depth investigation of the mathematical facts undergirding them. It presents a systematic, mathematically rigorous account of the key notions arising from such debates, including theory, equivalence, translation, reduction, and model. The result is a far-reaching reconceptualization of the role of formal methods in answering philosophical questions. (shrink)

"This book constitutes the proceedings of the 1994 European Workshop on Logics in Artificial Intelligence, held at York, UK in September 1994. The 24 papers presented were selected from a total of 79 submissions; in addition there are two abstracts of invited talks and one full paper of the invited presentation by Georg Gottlob. The papers point out that, with the depth and maturity of formalisms and methodologies available in AI today, logics provide a formal basis for the study (...) of the whole field of AI. The volume offers sections on nonmonotonic reasoning, automated reasoning, logic programming, knowledge representation, and belief revision."--PUBLISHER'S WEBSITE. (shrink)

Introduction 1.1 GOALS This book is devoted to an in-depth investigation of some of the properties of Logical Form (LF). In particular, the primary aim of ...

'This sentence is false' - is that true? The 'Liar paradox' embodied in those words exerted a particular fascination on the logicians of the Western later Middle Ages, and, along with similar 'insoluble' problems, forms the subject of the first group of articles in this volume. In the following parts Professor Spade turns to medieval semantic theory, views on the relationship between language and thought, and to a study of one particular genre of disputation, that known as 'obligationes'. The focus (...) is on the Oxford scholastics of the first half of the 14th century, and it is the name of William of Ockham which dominates these pages - a thinker with whom Professor Spade finds himself in considerable philosophical sympathy, and whose work on logic and semantic theory has a depth and richness that have not always been sufficiently appreciated. (shrink)

This paper presents a generalization of Fine’s completeness theorem for transitive logics of finite width, and proves the Kripke completeness of transitive logics of finite “suc-eq-width”. The frame condition for each finite suc-eq-width axiom requires, in rooted transitive frames, a finite upper bound of cardinality for antichains of points with different proper successors. The paper also presents a generalization of Rybakov’s completeness theorem for transitive logics of prefinite width, and proves the Kripke completeness of transitive logics of prefinite “suc-eq-width”. The (...) frame condition for each prefinite suc-eq-width axiom requires, in rooted transitive frames, a finite upper bound of cardinality for antichains of points that have a finite lower bound of depth and have different proper successors. We will construct continuums of transitive logics of finite suc-eq-width but not of finite width, and continuums of those of prefinite suc-eq-width but not of prefinite width. This shows that our new completeness results cover uncountably many more logics than Fine’s theorem and Rybakov’s theorem respectively. (shrink)

Wittgenstein’s later remarks on music, those written after his return to Cambridge in 1929 in increasing intensity, frequency, and elaboration, occupy a unique place in the annals of the philosophy of music, which is rarely acknowledged or discussed in the scholarly literature. These remarks reflect and emulate the spirit and subject matter of Romantic thinking about music, but also respond to it critically, while at the same time they interweave into Wittgenstein’s forward thinking about the philosophic entanglements of language and (...) the mind, and also his pervasive pessimism as a philosopher of culture. In this essay I explore and explicate some of the major tenets of this unique position. I argue that Wittgenstein appropriates the Romantic focus on the specificity of musical expression by means of the idea that gesture consists in complex vertical interrelations between language games. Understanding what a musical passage is about logically presupposes a myriad of correlate moves in the entire range of our language-games. Wittgenstein explicates the notion of musical aboutness in terms of intransitive understanding, which expresses an internal relation conjoining musical gesture and our culture, our entire life in practice, whereupon the related concepts cannot be identified independently of the relation which holds them together. Wittgenstein responds to the Romantic focus on the unique knowledge of human life which is afforded by musical experience with his idiosyncratic later notion of Menschenkenntnis. I conclude that, in the context of Wittgenstein’s late work, ineffability pertaining to musical meaning is not a shortcoming, but rather constitutional of the type of games, which admit what Wittgenstein calls ‘imponderable evidence’, or indefiniteness. (shrink)