A hundred years ago, Ramsey (1927) devised a new system of notation to more perspicuously represent the idea that ‘p’ and ‘¬¬p’ express the same fact. Recently, Monroy Pérez (2023) has argued that we can apply Ramsey’s insight to solve a recalcitrant problem for logical elitism — the thesis, most famously defended by Sider (2011), that the world’s fundamental metaphysical structure includes logical structure. Here, I discuss the general role of language in the framing of logical elitism and present (...) four arguments that show that the Ramseyan strategy for defending logical elitism fails. First, it does not eliminate all arbitrariness even for zeroth-order classical logic (§3). Second, it is a case of a more general strategy that exploits the fact that classical negation is an involution; then, choosing the Ramseyan proposal among the other cases is arbitrary (§4). Third, there are proposals to solve the riddle that are simpler than every case of the involutory strategy (§5). The discussion has consequences for logical elitism at large: in my fourth argument I contend, contrary to a fairly favoured opinion, that logical elitism should not be thought of as a thesis about languages, but as a thesis about joints in metaphysical structure (§6). (shrink)

Notations are central for understanding mathematical discourse. Readers would like to read notations that transport the meaning well and prefer notations that are familiar to them. Therefore, authors optimize the choice of notations with respect to these two criteria, while at the same time trying to remain consistent over the document and their own prior publications. In print media where notations are fixed at publication time, this is an over-constrained problem. In living documents notations can be adapted at reading (...) time, taking reader preferences into account. We present a representational infrastructure for notations in living mathematical documents. Mathematical notations can be defined declaratively. Author and reader can extensionally define the set of available notation definitions at arbitrary document levels, and they can guide the notation selection function via intensional annotations. We give an abstract specification of notation definitions and the flexible rendering algorithms and show their coverage on paradigmatic examples. We show how to use this framework to render OpenMath and Content-MathML to Presentation-MathML, but the approach extends to arbitrary content and presentation formats. We discuss prototypical implementations of all aspects of the rendering pipeline. (shrink)

This paper investigates provability and non-provability of well-foundedness of ordinal notations in weak theories of bounded arithmetic. We define a notion of well-foundedness on bounded domains. We show that T21 and S22 can prove the well-foundedness on bounded domains of the ordinal notations below 0 and Γ0. As a corollary, the class of polynomial local search problems, PLS, can be augmented with cost functions that take ordinal values below 0 and Γ0 without increasing the class PLS.

The terms of a mathematical problem become precise and concise if they are expressed in an appropriate notation, therefore notations are useful to mathematics. But are notations only useful, or also essential? According to prevailing view, they are not essential. Contrary to this view, this paper argues that notations are essential to mathematics, because they may play a crucial role in mathematical discovery. Specifically, since notations may consist of symbolic notations, diagrammatic notations, or a mix of symbolic and diagrammatic (...) notations, notations may play a crucial role in mathematical discovery in different ways. Some examples are given to illustrate these ways. (shrink)

Notations are central for understanding mathematical discourse. Readers would like to read notations that transport the meaning well and prefer notations that are familiar to them. Therefore, authors optimize the choice of notations with respect to these two criteria, while at the same time trying to remain consistent over the document and their own prior publications. In print media where notations are fixed at publication time, this is an over-constrained problem. In living documents notations can be adapted at reading (...) time, taking reader preferences into account. We present a representational infrastructure for notations in living mathematical documents. Mathematical notations can be defined declaratively. Author and reader can extensionally define the set of available notation definitions at arbitrary document levels, and they can guide the notation selection function via intensional annotations. We give an abstract specification of notation definitions and the flexible rendering algorithms and show their coverage on paradigmatic examples. We show how to use this framework to render OpenMath and Content-MathML to Presentation-MathML, but the approach extends to arbitrary content and presentation formats. We discuss prototypical implementations of all aspects of the rendering pipeline. (shrink)

This paper first defines the concept of an iconic notation for a property P by a notation providing decision criteria for P. This definition distinguishes an iconic notation from a symbolic notation. The notion of an iconic proof is then defined by an algorithmic translation of a symbolic notation into an iconic notation. The defined concepts are illustrated by examples from mathematics and monadic logic. The definitions and examples then serve as a background for a discussion of the decision (...) class='Hi'>problem that asks for the possibility of an algorithmic translation of first-order formulas into a proper iconic notation for the whole realm of first-order logic. © 2021 Elsevier B.V., All rights reserved. (shrink)

Stereotypically, computation involves intrinsic changes to the medium of representation: writing new symbols, erasing old symbols, turning gears, flipping switches, sliding abacus beads. Perspectival computation leaves the original inscriptions untouched. The problem solver obtains the output by merely alters his orientation toward the input. There is no rewriting or copying of the input inscriptions; the output inscriptions are numerically identical to the input inscriptions. This suggests a loophole through some of the computational limits apparently imposed by physics. There can (...) be symbol manipulation without inscription manipulation because symbols are complex objects that have manipulatable elements besides their inscriptions. Since a written symbol is an ordered pair of consisting of a shape and the reader's orientation to that inscription, the symbol can be changed by changing the orientation rather than inscription. Although there are the usual physical limits associated with reading the answer, the computation is itself instantaneous. This is true even when the sub-calculations are algorithmically complex, exponentially increasing or even infinite. (shrink)

Wittgenstein composed five original musical fragments during his transitional middle period, in which he employs musical notation as a means by which to convey his philosophical thoughts. This is an overlooked aspect of the importance of aesthetics, and musical thinking in particular, in the development of Wittgenstein’s philosophy. We explain and evaluate the way the music interlinks with Wittgenstein’s philosophical thoughts. We show the direct relation of these musical examples as precursors to some of Wittgenstein’s most celebrated ideas (the push (...) beyond the inner/outer picture, the problem of rule-following, and aspect perception). We also show how important broad aesthetic notions such as style and composition concerning the very possibility of philosophizing were to Wittgenstein. (shrink)

Which function is computed by a Turing machine will depend on how the symbols it manipulates are interpreted. Further, by invoking bizarre systems of notation it is easy to define Turing machines that compute textbook examples of uncomputable functions, such as the solution to the decision problem for first-order logic. Thus, the distinction between computable and uncomputable functions depends on the system of notation used. This raises the question: which systems of notation are the relevant ones for determining whether (...) a function is computable? These are the acceptable notations. I argue for a use criterion of acceptability: the acceptable notations for a domain of objects D\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathcal {D}}$$\end{document} are the notations that we can use for the usual D\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathcal {D}}$$\end{document}-activities. Acceptable notations for natural numbers are ones that we can count with. Acceptable notations for logical formulas are the ones that we can use in inference and logical analysis. And so on. This criterion makes computability a relative notion—whether a function is computable depends on which notations are acceptable, which is relative to our interests and abilities. I argue that this is not a problem, however, since there is independent reason for taking the extension of computable function to be relative to contingent facts. Similarly, the fact that the use criterion makes a mathematical distinction depend on practical concerns is also not a problem because it dovetails with similar phenomena in other areas of computability theory, namely the roles of notation in computation over real numbers and of Extended Church’s Thesis in complexity theory. (shrink)

This paper investigates provability and non-provability of well-foundedness of ordinal notations in weak theories of bounded arithmetic. We define a notion of well-foundedness on bounded domains. We show that T21 and S22 can prove the well-foundedness on bounded domains of the ordinal notations below 0 and Γ0. As a corollary, the class of polynomial local search problems, PLS, can be augmented with cost functions that take ordinal values below 0 and Γ0 without increasing the class PLS.

We propose a reconstruction of the constellation of problems and philosophical positions on the nature and number of the primitives of logic in four authors of the nineteenth century logical scene: Peano, Padoa, Frege and Peirce. We argue that the proposed reconstruction forces us to recognize that it is in at least four different senses that a notation can be said to be simpler than another, and we trace the origins of these four senses in the writings of these authors. (...) We conclude that Frege, and even more so Peirce, developed new notations not to make drawing logical conclusions easier but in order to answer the needs of logical analysis. (shrink)

Le point de départ de l'article est la théorie de la signification et de la référence des termes scientifiques présentée par Hilary Putnam dans son article « The Meaning of 'Meaning7 » et quelques autres essais de Mind, Language and Reality. Dans la partie critique du texte la thèse et les arguments de Putnam, ainsi que sa prétention d'éviter le problème de rincommensurabilité des théories rivales, sont évalués à la lumière de la distinction que nous introduisons entre référence et dénotation. (...) Dans sa partie constructive le texte propose les grandes lignes d'un traitement de la référence des termes scientifiques qui s'appuie sur notre conception de la structure de la théorie physique, le pragmatisme constructif.The paper's starting point is Hilary Putnam's theory of meaning and reference for scientific terms, as presented in " The Meaning of 'Meaning' " and several other essays from his Mind, Language and Reality. A critique of Putnam's thesis and arguments, as well as of his claim to avoid the problem of incommensurability of competing theories, is offered in the light of our own distinction between reference and denotation. In the process we develop an outline of a theory of reference for scientific terms that builds on our view of the structure of physical theory: constructive pragmatism. (shrink)

The following are some preliminary remarks that will set the stage for my individual replies to Timothy Pawl, Thomas McCall, A. J. Cotnoir, and Sara L. Uckelman’s responses to my paper ‘Christ – A Contradiction’. In that paper I advance and defend a contradictory Christology which solves the fundamental ‘problem’ of Christology by holding that Christ is a contradictory being: it is true that Christ is mutable and it is false that Christ is mutable; it is true that Christ (...) is immutable and it is false that Christ is immutable; and so on for other fundamental-problem properties. Familiarity with ‘Christ – A Contradiction’ is assumed. (shrink)

The article is an overview of the main philosophical ideas of Gottlob Frege. The author stresses that Frege has built important tools for the analysis of language: the logic of sentences and the logic of predicates. He stresses also the importance of Frege’s work on the foundations of mathematics. He argues that his greatest achievement is the fact that thanks to Frege language has become main object of investigations in contemporary philosophy, and that theory of meaning has replaced the traditional (...) theory of knowledge. Frege’s ideas, taken together, constitute a well-defined method, a particular style of approach to philosophical problems and of their resolution. The effectiveness and enduring pertinence of Frege’s method is demonstrated by means of an analysis of several typical category mistakes, frequent in philosophical argumentation. Key words FREGE, LANGUAGE. (shrink)

This paper consists principally of selections from a much longer work on the semantics of English. It discusses some problems concerning how to represent grammatical modifiers (e.g. slowly in x drives slowly) in a logically perspicuous notation. A proposal of Reichenbach's is given and criticized; then a new theory (apparently discovered independently by myself, Romain Clark, and Richard Montague and Hans Kamp) is given, in which grammatical modifiers are represented by operators added to a first-order predicate calculus. Finally some problems (...) concerning applications of adjectives to that-clauses and gerundive-clauses are discussed. (shrink)

In the course of daily life we solve problems often enough that there is a special term to characterize the activity and the right to expect a scientific theory to explain its dynamics. The classical view in psychology is that to solve a problem a subject must frame it by creating an internal representation of the problem‘s structure, usually called a problem space. This space is an internally generable representation that is mathematically identical to a graph structure (...) with nodes and links. The nodes can be annotated with useful information, and the whole representation can be distributed over internal and external structures such as symbolic notations on paper or diagrams. If the representation is distributed across internal and external structures the subject must be able to keep track of activity in the distributed structure. Problem solving proceeds as the subject works from an initial state in this mentally supported space, actively construction possible solution paths, evaluating them and heuristically choosing the best. Control of this exploratory process is not well understood, as it is not always systematic, but various heuristic search algorithms have been proposed and some experimental support has been provided for them. (shrink)

When people make sense of situations, illustrations, instructions and problems they do more than just think with their heads. They gesture, talk, point, annotate, make notes and so on. What extra do they get from interacting with their environment in this way? To study this fundamental problem, I looked at how people project structure onto geometric drawings, visual proofs, and games like tic tac toe. Two experiments were run to learn more about projection. Projection is a special capacity, similar (...) to perception, but less tied to what is in the environment. Projection, unlike pure imagery, requires external structure to anchor it, but it adds ‘mental’ structure to the external scene much like an augmented reality system adds structure to an outside scene. A person projects when they look at a chessboard and can see where a knight may be moved. Because of the cognitive costs of sustaining and extending projection, humans make some of their projections real. They create structure externally. They move the piece, they talk, point, notate, represent. Much of our interactivity during sense making and problem solving involves a cycle of projecting then creating structure. (shrink)

Several philosophical problems concerning the object of criticism in dance are identified and analyzed as preliminary to an eventual theory of evaluation of dance. Basic to philosophical adequacy is understanding the artform as it is actually practiced and appreciated, recognizing its complexity as a performing artform using unique human bodies as instruments. ;Definitions of "dance" proposed by philosophers, dance historians, and others are inadequate to specify necessary and sufficient conditions of dance, to distinguish dance from other human non-art phenomena, and (...) to distinguish dance from other artforms. Definitions can be adequate for a specific purpose, using descriptive characteristics and standards for evaluation and appreciation by an audience. ;The ontological status of dance is best understood as consisting of the primary media of movement by human bodies and music and secondary media of the visual dimensions of costumes, scenery, and lighting. Improved understanding of the artform is possible with this more comprehenive recognition of its complex multi-media status, as shown in an analysis of "appropriateness" of the various media, a typical criterion for evaluation. ;The identity of a work of art in dance can be established using a notational system in conjunction with a standard for acceptable compliance with that notation. "Substantial similarity," as determined by lay observers, the test for copyright infringement, provides a useful standard for dance. Other theories have required too much of an identity standard; its purpose is to identify a particular performance as one of a certain work, not to teach dancers to perform the work, nor to enable directors to produce it, nor to provide standards for a good performance of the work. ;A necessary condition of the proper object of criticism is perceivability. The considerable interest in the dance world in production and other factors not perceivable on stage during performance is best understood as a interest in the on-going skills of artists, as opposed to the evaluation of an aesthetic object. (shrink)

Engraved portable objects from Upper Palaeolithic and earlier sites are argued to be cognitive tools designed to store information for the purposes of calculation, record-keeping, or communication. This paper reviews the surprisingly long intellectual history of comparisons between these ancient objects and message sticks: marked graphic devices traditionally used for long-distance communication in Indigenous Australia. I argue that, while such comparisons have often been misguided, more cautious applications of ethnographic analogy may yield useful insights. A systematic analysis of historical observations (...) together with more recent fieldwork, indicate that Australian message sticks are primarily tools of social cognition, as opposed to cognition tout court, and rely on orality and other context to become meaningful. Further, the practice of message stick communication may help clarify ongoing problems in the interpretation of Upper Palaeolithic objects including their possible role in aggregation activities, the distinction between decoration and notation, and the interplay between graphic sequences and speech. (shrink)

Some 55 years after the publication of Russell's seminal ‘On Denoting’, Quine adopted the theory of definite descriptions presented therein to formulate his perspicuous lingua franca for the sciences. This paper illustrates how, in so doing, Quine's lingua franca inherited an old antinomy at the core of Russell's theory of definite descriptions, specifically, the lack of congruency between notation and metaphysical commitments when the variable assumes the role of fundamental singular reference. In this context, the paper contends that the phrase (...) ‘to be is purely and simply to be the value of the variable’ is far from the straightforward solution Quine initially envisioned for his lingua franca. Additionally, it argues that it is unlikely that an attempt to account for a determinate object theory, such as those propounded by Russell and Quine, can prove successful when grounded in a referential theory of quantificational logic where the variable, seen as a simple range of values, holds fundamental singular referent‐position. The paper concludes by suggesting that Meinong's object theory could resolve the problem of the variable, albeit at the cost of potentially undesirable ontological and logical commitments. Alternatively, Kit Fine's recent work on other antinomies in the variable may provide a promising approach to exploring how to salvage the variable as a formal foundation for ontology. (shrink)

This chapter argues that the account of logic that Wittgenstein offers in the _Notes_ is, as he plainly knew himself, incomplete. One of the issues that he knew still required further thought was how to represent the relation of identity. Russell's definition of identity, eliminating identity, and the notational problem are discussed.

This work has been selected by scholars as being culturally important, and is part of the knowledge base of civilization as we know it. This work was reproduced from the original artifact, and remains as true to the original work as possible. Therefore, you will see the original copyright references, library stamps (as most of these works have been housed in our most important libraries around the world), and other notations in the work. This work is in the public domain (...) in the United States of America, and possibly other nations. Within the United States, you may freely copy and distribute this work, as no entity (individual or corporate) has a copyright on the body of the work. As a reproduction of a historical artifact, this work may contain missing or blurred pages, poor pictures, errant marks, etc. Scholars believe, and we concur, that this work is important enough to be preserved, reproduced, and made generally available to the public. We appreciate your support of the preservation process, and thank you for being an important part of keeping this knowledge alive and relevant. (shrink)

The notation and terminology of this paper follow [2], and are dual to those of [6] and [7]. If L is a language in the narrow sense, Cn may be any consequence operation on sets of sentences of L that includes classical sentential logic. Henceforth when we talk of the language L we intend to include reference to some fixed, though unspecified, operation Cn. X is a deductive system if X = Cn(X). Sentences x, z that are logically equivalent with (...) respect to Cn – that is x ∈ Cn({z}) and z ∈ Cn({x}) – are identified. If X and Z are systems we often write X x instead of x ∈ X and Z X instead of X ⊆ Z. If X = Cn({x}) for some sentence x, X is (finitely) axiomatizable. The set theoretical intersection of X and Z has the logical force of disjunction, and is written X ∨ Z; Cn(X ∪ Z), the smallest system to include both X and Z, is written XZ. If K is a family of systems, [K] and [K] may be defined in an analogous way. The logically strongest system S is the set of all sentences of L; the weakest system T is defined as Cn(∅). The autocomplement Z of Z is defined to be the strongest system to complement T, namely the system [{Y : Y ∨ Z = T}]. More generally we may define X − Z as [{Y : X Y ∨ Z}]. In terms of this operation to remainder, Z is identical with T − Z. The class of all deductive systems forms a distributive lattice under the operations of concatenation and ∨; and indeed a Brouverian algebra (a relatively authocomplemented lattice with unit) under concatenation, ∨ – and T. (shrink)

This work has been selected by scholars as being culturally important, and is part of the knowledge base of civilization as we know it. This work was reproduced from the original artifact, and remains as true to the original work as possible. Therefore, you will see the original copyright references, library stamps (as most of these works have been housed in our most important libraries around the world), and other notations in the work. This work is in the public domain (...) in the United States of America, and possibly other nations. Within the United States, you may freely copy and distribute this work, as no entity (individual or corporate) has a copyright on the body of the work. As a reproduction of a historical artifact, this work may contain missing or blurred pages, poor pictures, errant marks, etc. Scholars believe, and we concur, that this work is important enough to be preserved, reproduced, and made generally available to the public. We appreciate your support of the preservation process, and thank you for being an important part of keeping this knowledge alive and relevant. (shrink)

This work has been selected by scholars as being culturally important, and is part of the knowledge base of civilization as we know it. This work was reproduced from the original artifact, and remains as true to the original work as possible. Therefore, you will see the original copyright references, library stamps (as most of these works have been housed in our most important libraries around the world), and other notations in the work. This work is in the public domain (...) in the United States of America, and possibly other nations. Within the United States, you may freely copy and distribute this work, as no entity (individual or corporate) has a copyright on the body of the work. As a reproduction of a historical artifact, this work may contain missing or blurred pages, poor pictures, errant marks, etc. Scholars believe, and we concur, that this work is important enough to be preserved, reproduced, and made generally available to the public. We appreciate your support of the preservation process, and thank you for being an important part of keeping this knowledge alive and relevant. (shrink)

The Problem of Truth’s Value arises when we try to reconcile an account of why we should care about truth with an account of what truth is. Some theories, called “normativist” here, build value into the very nature of truth. To be true is, in part, to be fit for belief. They risk severing the connection between a claim’s being true and its saying things are as they really are. Other theories, called “Aristotelian” here, leave all normativity out of (...) the account of what truth is. They leave it mysterious why we should prefer true beliefs to false ones or even ignorance. This chapter outlines how the Problem of Truth’s Value arises for normativist and Aristotelian theories. It also sketches the solution defended over the course of the rest of the book. The main proposal is that truth’s value derives from the moral value of caring about truth, not from special value inherent in states of true belief. That proposal, the Strong Virtue Theory of truth’s value, has two parts. The first claims we morally ought to be Truthful people. The second claims that _truth_ itself does not confer special value on beliefs. The chapter also describes the methodology to be followed throughout the book and describes notational conventions to be used. (shrink)

This work has been selected by scholars as being culturally important, and is part of the knowledge base of civilization as we know it. This work was reproduced from the original artifact, and remains as true to the original work as possible. Therefore, you will see the original copyright references, library stamps (as most of these works have been housed in our most important libraries around the world), and other notations in the work. This work is in the public domain (...) in the United States of America, and possibly other nations. Within the United States, you may freely copy and distribute this work, as no entity (individual or corporate) has a copyright on the body of the work. As a reproduction of a historical artifact, this work may contain missing or blurred pages, poor pictures, errant marks, etc. Scholars believe, and we concur, that this work is important enough to be preserved, reproduced, and made generally available to the public. We appreciate your support of the preservation process, and thank you for being an important part of keeping this knowledge alive and relevant. (shrink)

Since its original publication in 1929, Martin Heidegger’s provocative book on Kant’s Critique of Pure Reason has attracted much attention both as an important contribution to twentieth-century Kant scholarship and as a pivotal work in Heidegger’s own development after Being and Time. This fifth, enlarged edition includes marginal notations made by Heidegger in his personal copy of the book and four new appendices—Heidegger's postpublication notes on the book, his review of Ernst Cassirer's Philosophy of Symbolic Forms, Heidegger's response to reviews (...) by rudolf Odebrecht and Cassirer, and an essay "On the History of the Philosophical Chair since 1866." The work is significant not only for its illuminating assessment of Kant’s thought but also for its elaboration of themes first broached in Being and Time, especially the problem of how Heidegger proposed to enact his destruction of the metaphysical tradition and the role that his reading of Kant would play therein. (shrink)

This work has been selected by scholars as being culturally important, and is part of the knowledge base of civilization as we know it. This work was reproduced from the original artifact, and remains as true to the original work as possible. Therefore, you will see the original copyright references, library stamps (as most of these works have been housed in our most important libraries around the world), and other notations in the work. This work is in the public domain (...) in the United States of America, and possibly other nations. Within the United States, you may freely copy and distribute this work, as no entity (individual or corporate) has a copyright on the body of the work. As a reproduction of a historical artifact, this work may contain missing or blurred pages, poor pictures, errant marks, etc. Scholars believe, and we concur, that this work is important enough to be preserved, reproduced, and made generally available to the public. We appreciate your support of the preservation process, and thank you for being an important part of keeping this knowledge alive and relevant. (shrink)

In a recent paper, Nils Kürbis argues that bilateral natural deduction systems in which assertions and denials figure as hypothetical assumptions are unintelligible. In this paper, I respond to this claim on two counts. First, I argue that, if we think of bilateralism as a tool for articulating discursive norms, then supposition of assertions and denials in the context of bilateral natural deduction systems is perfectly intelligible. Second, I show that, by transposing such systems into sequent notation, one can make (...) perfect sense of them without talking about supposition at all, just talking in terms of relations of committive consequence. I conclude by providing some motivation for adopting this normative interpretation of bilateralism on which this response to Kürbis’s argument is based. (shrink)

This work has been selected by scholars as being culturally important, and is part of the knowledge base of civilization as we know it. This work was reproduced from the original artifact, and remains as true to the original work as possible. Therefore, you will see the original copyright references, library stamps (as most of these works have been housed in our most important libraries around the world), and other notations in the work. This work is in the public domain (...) in the United States of America, and possibly other nations. Within the United States, you may freely copy and distribute this work, as no entity (individual or corporate) has a copyright on the body of the work. As a reproduction of a historical artifact, this work may contain missing or blurred pages, poor pictures, errant marks, etc. Scholars believe, and we concur, that this work is important enough to be preserved, reproduced, and made generally available to the public. We appreciate your support of the preservation process, and thank you for being an important part of keeping this knowledge alive and relevant. (shrink)

Intuitively, it seems that S 1 is “more random” or “less regular” than S 2. In other words, it seems more plausible (in some sense) that S 1 (as opposed to S 2) was generated by a random process ( e.g. , by tossing a fair coin eight times, and recording an H for a heads outcome and a T for a tails outcome). We will use the notation x σ 1 ą σ 2y to express the claim that xstring (...) σ 1 is more random than string σ 2y. And, we take it to be intuitively clear that — on any plausible definition of such a relation — we should have S 1 ą S 2. (shrink)

The constantly accelerating progress of contemporary natural science is indissolubly associated with the development and use of mathematics and with the processes of mathematical modeling of the phenomena of nature. The essence of this diverse and highly fertile interaction of mathematics and natural science and the dialectics of this interaction can only be disclosed through analysis of the nature of theoretical notions in general. Today, above all in the ranks of materialistically minded researchers, it is generally accepted that theory possesses (...) a value of its own. Contrary to positivist concepts of the nature of our knowledge, the meaning and significance of theory do not consist merely of the recording of experimental data, their classification and notation in abbreviated form. The meaning of theory is considerably more significant and is revealed, above all, in its explanatory and predictive functions. Contemporary theoretical knowledge is quite advanced and comprises a highly complex system, relatively self-contained and capable of internal development. Mathematics is the most interesting and distinctive phenomenon in the system of theoretical knowledge, in the system of science in general. It is in its dialectical development that the internal force and dynamics of the development of theory, in the very broadest sense of that word, find expression. Paul Lafargue tells us that Karl Marx held that "science only attains perfection when it succeeds in making use of mathematics". In many fields of research the formulation of new ideas and concepts rests upon mathematics, its concepts and notions, and "is suggested" by the latter. "Mathematics," writes F. Dyson, "is the main source of the notions and principles out of which new theories are created". The basic significance of mathematics to theory has been noted by outstanding thinkers over the course of the entire history of development of science. Today the expressive phrase, "the mathematization of knowledge," has come into general currency as a description of the principal direction of growth of theoretical notions in natural science. However, with respect to the growth of modern theoretical physics in general, the statement that it develops primarily by the method of the mathematical hypothesis has general validity. For example, the principal findings and discoveries of quantum theory and particle physics, starting from the corpuscularwave dualism and terminating with the omega minus hyperon and hypothetical quarks, were obtained or made "at the tip of a mathematician's pen.". (shrink)

The book is not only dedicated to linguists, but also to readers who are not familiar with notations developed in linguistics. The first part of the study presents philosophical justifications for linguistic settlements. These are based on the phenomenological reduction of Edmund Husserl, Karl R. Popper's falsificationism, the moderate rationalism of science of Izydora Dąmbska and Andrzej Bogusławski's lack of the nomological explanation in linguistics. The second part presents a re-examination of the solutions proposed in the field of linguistics, some (...) new philosophical explanations and a discussion of the truth of linguistic propositions. (shrink)

This work has been selected by scholars as being culturally important, and is part of the knowledge base of civilization as we know it. This work was reproduced from the original artifact, and remains as true to the original work as possible. Therefore, you will see the original copyright references, library stamps (as most of these works have been housed in our most important libraries around the world), and other notations in the work. This work is in the public domain (...) in the United States of America, and possibly other nations. Within the United States, you may freely copy and distribute this work, as no entity (individual or corporate) has a copyright on the body of the work. As a reproduction of a historical artifact, this work may contain missing or blurred pages, poor pictures, errant marks, etc. Scholars believe, and we concur, that this work is important enough to be preserved, reproduced, and made generally available to the public. We appreciate your support of the preservation process, and thank you for being an important part of keeping this knowledge alive and relevant. (shrink)

This work has been selected by scholars as being culturally important, and is part of the knowledge base of civilization as we know it. This work was reproduced from the original artifact, and remains as true to the original work as possible. Therefore, you will see the original copyright references, library stamps (as most of these works have been housed in our most important libraries around the world), and other notations in the work. This work is in the public domain (...) in the United States of America, and possibly other nations. Within the United States, you may freely copy and distribute this work, as no entity (individual or corporate) has a copyright on the body of the work. As a reproduction of a historical artifact, this work may contain missing or blurred pages, poor pictures, errant marks, etc. Scholars believe, and we concur, that this work is important enough to be preserved, reproduced, and made generally available to the public. We appreciate your support of the preservation process, and thank you for being an important part of keeping this knowledge alive and relevant. (shrink)

This work has been selected by scholars as being culturally important, and is part of the knowledge base of civilization as we know it. This work was reproduced from the original artifact, and remains as true to the original work as possible. Therefore, you will see the original copyright references, library stamps (as most of these works have been housed in our most important libraries around the world), and other notations in the work. This work is in the public domain (...) in the United States of America, and possibly other nations. Within the United States, you may freely copy and distribute this work, as no entity (individual or corporate) has a copyright on the body of the work. As a reproduction of a historical artifact, this work may contain missing or blurred pages, poor pictures, errant marks, etc. Scholars believe, and we concur, that this work is important enough to be preserved, reproduced, and made generally available to the public. We appreciate your support of the preservation process, and thank you for being an important part of keeping this knowledge alive and relevant. (shrink)

This work has been selected by scholars as being culturally important, and is part of the knowledge base of civilization as we know it. This work was reproduced from the original artifact, and remains as true to the original work as possible. Therefore, you will see the original copyright references, library stamps (as most of these works have been housed in our most important libraries around the world), and other notations in the work.This work is in the public domain in (...) the United States of America, and possibly other nations. Within the United States, you may freely copy and distribute this work, as no entity (individual or corporate) has a copyright on the body of the work.As a reproduction of a historical artifact, this work may contain missing or blurred pages, poor pictures, errant marks, etc. Scholars believe, and we concur, that this work is important enough to be preserved, reproduced, and made generally available to the public. We appreciate your support of the preservation process, and thank you for being an important part of keeping this knowledge alive and relevant. (shrink)

This work has been selected by scholars as being culturally important, and is part of the knowledge base of civilization as we know it. This work was reproduced from the original artifact, and remains as true to the original work as possible. Therefore, you will see the original copyright references, library stamps (as most of these works have been housed in our most important libraries around the world), and other notations in the work. This work is in the public domain (...) in the United States of America, and possibly other nations. Within the United States, you may freely copy and distribute this work, as no entity (individual or corporate) has a copyright on the body of the work. As a reproduction of a historical artifact, this work may contain missing or blurred pages, poor pictures, errant marks, etc. Scholars believe, and we concur, that this work is important enough to be preserved, reproduced, and made generally available to the public. We appreciate your support of the preservation process, and thank you for being an important part of keeping this knowledge alive and relevant. (shrink)

Magdalena Krasińska A Work of Music in Roman Ingarden’s AestheticsThe theme taken in this article is an ontological concept of the musical work of Roman Ingarden developed on the basis of phenomenological aesthetics. According to the eidetic reduction Ingarden rejects any colloquial opinions about how the existence of a musical composition, so that he holds that musical work is not the same nor the mental experience, or the performance, or to the music notations. Making a critique of the idealistic, psychologistic (...) and materialistic positions to explain how the existence of the musical work, Ingarden classified it as a object of purely intentional, where the musical notation is the ontological basis. Currently notation don’t allow for consolidation of all the moments of the work, especially not sound moments, so musical composition as every being intentional, is the scheme required to complete and define more precisely where are the ontological gaps; each performance of a musical work is the complete of potential spaces, specification of works. A few comments on so constructed theory publishes musicologist Zofia Lissa. The main objection against Ingarden is the fact that his concept aspires to be universal, while he created it solely on the basis constituted in the scores ‘classical’ works of modern European music, while ignoring other musical phenomena, such as avant-garde music or folklore. Finally raised is a matter of the ontology of newest pieces of music, belonging to the so-called popular music, that do not have ontological basis in the form of musical notation, what in view of the Ingarden’s theory does problematic the issue of their identity. Keywords: ontological concept of the musical work, phenomenological aesthetics, Roman Ingarden, identity of the musical work. (shrink)

This work has been selected by scholars as being culturally important, and is part of the knowledge base of civilization as we know it. This work was reproduced from the original artifact, and remains as true to the original work as possible. Therefore, you will see the original copyright references, library stamps (as most of these works have been housed in our most important libraries around the world), and other notations in the work. This work is in the public domain (...) in the United States of America, and possibly other nations. Within the United States, you may freely copy and distribute this work, as no entity (individual or corporate) has a copyright on the body of the work. As a reproduction of a historical artifact, this work may contain missing or blurred pages, poor pictures, errant marks, etc. Scholars believe, and we concur, that this work is important enough to be preserved, reproduced, and made generally available to the public. We appreciate your support of the preservation process, and thank you for being an important part of keeping this knowledge alive and relevant. (shrink)

This work has been selected by scholars as being culturally important, and is part of the knowledge base of civilization as we know it. This work was reproduced from the original artifact, and remains as true to the original work as possible. Therefore, you will see the original copyright references, library stamps (as most of these works have been housed in our most important libraries around the world), and other notations in the work. This work is in the public domain (...) in the United States of America, and possibly other nations. Within the United States, you may freely copy and distribute this work, as no entity (individual or corporate) has a copyright on the body of the work. As a reproduction of a historical artifact, this work may contain missing or blurred pages, poor pictures, errant marks, etc. Scholars believe, and we concur, that this work is important enough to be preserved, reproduced, and made generally available to the public. We appreciate your support of the preservation process, and thank you for being an important part of keeping this knowledge alive and relevant. (shrink)

Here is an informal treatment of many-valued logic, especially propositional logic, using Polish notation. The relations of n-valued logics with 2-valued logics are considered, and numerous systems are drawn on for illustrations. The book is written with an eye to scientific applications of many-valued logics, and considers empirical application of formal calculi. A short section on the relation of formal and "dialectical" logics appears. The work is introductory in nature.—P. S.

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In this paper I shall give a solution of the decision problem for the Lewis systems S2 and S4; i.e., I shall establish a constructive method for deciding whether an arbitrary given sentence of one of these systems is provable. The method is laborious to apply, since, in order to decide by means of it whether a given sentence is provable, it is necessary to construct a (usually very large) finite matrix. The argument will perhaps be of general interest, (...) however, because it does not seem to depend too closely on the special features of these particular systems, so that it may be possible to apply it in order to solve the decision problem for other such systems.Section II presents the decision method for S2, and Section III for S4. In Section IV, I shall establish a certain correspondence between S4 and topology, which will provide a solution for a decision problem in topology; this correspondence also enables us to settle a previously unsolved problem with regard to the Lewis systems.In treating of this system, I shall use the notation of Lewis, with the single exception that I shall use the symbol “≣”, instead of Lewis's symbol “=”, for strict equivalence. I shall use the symbol “=” to denote identity. Thus “p≣q” is a formula of S2, while “x=y” is the statement asserting that x and y are identical. I shall refer to the rules, primitive sentences and theorems of S2 by the names and numbers used by Lewis. Whenever a theorem stated by Lewis involves the symbol “=”, I shall of course suppose that this symbol has been replaced throughout by “≣”: thus, for example, I shall take 19.82 to be “(◇p∨◇q) ≣ ◇(p∨q)”, instead of “(◇p∨◇q) ≣ ◇(p∨q)”. (shrink)

In Bermúdez 2013 I argued against David Lewis’s well-known and widely accepted claim that Newcomb’s problem and the prisoner’s dilemma are really notational variants of a single problem. Mark Walker’s paper in this journal takes issue with my argument. In this note I show how Walker’s criticisms are misplaced. The problems with Walker’s argument point to more general and independently interesting conclusions about, first, the relation between deliberation and decision and, second, the differences between the prisoner’s dilemma, (...) which is a strategic choice problem, and Newcomb’s problem, which is a parametric choice problem. (shrink)

The new notation for mental-models theory proposed by Johnson-Laird and Byrne (1991; 1993) makes predictions that are inconsistent with those made previously by those authors. Notably, two frequently observed reasoning fallacies are not predicted by the theory. Moreover, the assumption made in the domain of syllogistic reasoning, that arbitrary numbers of individuals are used to represent premises, is also shown to cause problems for the prediction of problem difficulty.

This work has been selected by scholars as being culturally important, and is part of the knowledge base of civilization as we know it. This work was reproduced from the original artifact, and remains as true to the original work as possible. Therefore, you will see the original copyright references, library stamps (as most of these works have been housed in our most important libraries around the world), and other notations in the work. This work is in the public domain (...) in the United States of America, and possibly other nations. Within the United States, you may freely copy and distribute this work, as no entity (individual or corporate) has a copyright on the body of the work. As a reproduction of a historical artifact, this work may contain missing or blurred pages, poor pictures, errant marks, etc. Scholars believe, and we concur, that this work is important enough to be preserved, reproduced, and made generally available to the public. We appreciate your support of the preservation process, and thank you for being an important part of keeping this knowledge alive and relevant. (shrink)