Wittgenstein's Tractatus 9781793632883, 9781793632890, 179363288X

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Wittgenstein’s Tractatus

Wittgenstein’s Tractatus A Student’s Edition Translated with commentary by Duncan Richter

LEXINGTON BOOKS

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Published by Lexington Books An imprint of The Rowman & Littlefield Publishing Group, Inc. 4501 Forbes Boulevard, Suite 200, Lanham, Maryland 20706 www.rowman.com 6 Tinworth Street, London SE11 5AL, United Kingdom Copyright © 2021 The Rowman & Littlefield Publishing Group, Inc. All rights reserved. No part of this book may be reproduced in any form or by any electronic or mechanical means, including information storage and retrieval systems, without written permission from the publisher, except by a reviewer who may quote passages in a review. British Library Cataloguing in Publication Information Available Library of Congress Cataloging-in-Publication Data Library of Congress Control Number: 2020952023 ISBN: 978-1-7936-3288-3 (cloth : alk. paper) ISBN: 978-1-7936-3289-0 (electronic) The paper used in this publication meets the minimum requirements of American National Standard for Information Sciences—Permanence of Paper for Printed Library Materials, ANSI/NISO Z39.48-1992.

Contents

Acknowledgments vii Introduction 1 Tractatus Logico-Philosophicus: Title, Dedication, and Epigraph

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Preface 17 Main Text

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References 143 Index 149 About the Author

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v

Acknowledgments

This book grew out of a blog: tractatusblog.blogspot.com. I am grateful to everyone who has posted comments there. I am grateful also to Cora Diamond and Luciano Bazzocchi, who helped steer me clear of at least one error each, and to an anonymous reviewer for Lexington Books. Troy Siemers helped me with mathematical symbolism, and Stephanie Wilkinson helped with the images. Jana Hodges-Kluck, Sydney Wedbush, and Holly Buchanan of Lexington Books have all been very helpful too. Images used in the book come from Kevin C. Klement’s edition of the Tractatus (available at https://people.umass.edu/klement/tlp/), which he has made available under a Creative Commons Attribution-Share Alike 3.0 United States License. Thanks to him for doing that.

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Introduction

This book aims to do two things: to provide a new and improved translation of Ludwig Wittgenstein’s Tractatus Logico-­Philosophicus, and to provide students of that work with directions to useful secondary sources. These directions, analyses, and comments are provided in brackets, to show that they are not part of Wittgenstein’s text. The book is not intended to stand alone as a complete guide to the Tractatus, but I have made it as user-­friendly as I can. It is meant as a kind of handbook and so can only hope to be a starting point, not the last word on the subject. We should probably begin with some information about Wittgenstein’s life and times. This is especially important as we need to know what he had read (and been told) if we are to know what he is reacting to and talking about in the Tractatus. WITTGENSTEIN Wittgenstein was born into an immensely wealthy family in Vienna, Austria, in 1889. His engineering studies took him to Berlin, Germany, and Manchester, England, but also led to an interest in mathematics which, in turn, led to an interest in the foundations of mathematics. Having sought the advice of the German mathematician and philosopher Gottlob Frege (1848– 1925), Wittgenstein went to Cambridge University to study with Bertrand Russell (1872–1970) in 1911. In 1912 he formally enrolled as a student at the university. Just two years later one of the leading philosophers at Cambridge, G. E. Moore, traveled to Norway to take dictation from Wittgenstein, who had gone there to escape distractions. Later that year (1914) Wittgenstein joined the Austrian army, in which he served throughout the 1

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Introduction

First World War. He finished his book in 1918, during a period of leave, and published it (in German) in 1921 as Logisch-­Philosophische Abhandlung. The English translation, now called Tractatus Logico-­Philosophicus, was published the following year. After this, Wittgenstein quit university life and worked as a school teacher, an architect, and a gardener, before returning to Cambridge and to philosophy in 1929. In his work from then on he criticized many of his earlier ideas. SCHOPENHAUER Perhaps the first source of these ideas was Arthur Schopenhauer (1788–1860). Wittgenstein read Schopenhauer’s The World as Will and Representation as a teenager and accepted much of it as true. Peter Geach writes that: “Wittgenstein himself stated in conversation that when he was young he believed Schopenhauer to have been fundamentally right (though, not surprisingly, he could make nothing of the ‘objectification of the Will’)” (Geach 1957, 558). What this means we will see below. It is not known whether Wittgenstein read Schopenhauer’s The Fourfold Root of the Principle of Sufficient Reason, but there are striking parallels between (some of) what Schopenhauer writes there and (some of) what Wittgenstein writes in the Tractatus. Given Wittgenstein’s interest in Schopenhauer it seems likely that he would have read this early work, which Schopenhauer demands his readers read first in order to understand The World as Will and Representation properly (see Schopenhauer 1969, vol. 1 xiii–xiv). Bryan Magee observes that “No one disputes that Wittgenstein was soaked in Schopenhauer. The point is, though, that he was not soaked in anyone else: there was no other philosopher of the past whose work he knew even passably well” (Magee 1997, 311–312). So what did Schopenhauer believe? He presents his views as a kind of synthesis of the best of Western and Eastern ideas, both ancient and modern, about the ultimate nature of reality and how one ought to live. The world, he says, is representation or idea. When I talk about things around me, for instance, I mean things I am aware of, but all that I am really ever aware of is a) things as they are presented to me by my senses, and b) other ideas or phenomena in my mind. So what I mean by “the world” is really contents, or potential contents, of my mind. The world is a kind of appearance of which I am conscious, an appearance in my consciousness. In this sense the world is little more real than a dream. Its ending when I die will really be no great loss (although, of course, it will seem so to me as the subject whose object is this world). This is the world as representation.



Introduction 3

On the other hand, Schopenhauer argues (following Immanuel Kant), the very idea of representation or appearance (or phenomena) implies the idea of something that is represented, something that exists in itself, not just in the mind. Kant argued that we cannot have knowledge of any such thing, as this would require, impossibly, knowledge without the mind, but Schopenhauer does not quite see it this way. After all, one of the phenomena or appearances in this world is me. Surely I can know what I am. Technically, Schopenhauer admits, I cannot, but he does think that one has something like knowledge here, some kind of insight. Intuitively I know what I am, and what I am is will, Schopenhauer claims. Hands are the will to touch, hold, and grasp made flesh; eyes the will to look and see; tongues the will to taste; and so on. Put together, the human being, like all animals, is the will to live. Our primary drives are reproduction and survival. One problem here is how I can know what anyone else is underneath or behind the appearances. There is a certain plausibility to the idea that I should know what I am even if, strictly speaking, such knowledge is impossible (because the knowing subject can only ever know objects as they appear to it, i.e., appearances, not the subject itself as a subject). But why should I know what you are, except as you appear to me? Schopenhauer’s answer is that the alternative is simply unacceptable. If reality in itself is my will then I am something like God. It is irrationally egocentric to think that I am significantly different from anyone else, or anything else, in the world. So if I am will, the only reasonable thing to believe is that everyone and everything else is too. And not my will, but simply will. The ultimate nature of reality is the will to live, a blind and fundamentally pointless striving to reproduce and survive which manifests itself in the various kinds of things we see around us in the world: human beings, oak trees, clouds, flies, dogs, and so on. This, what appears in these forms, is the world as will. Schopenhauer believed that Kant had proved time, space, and causality to be features, not of reality in itself or the world as will, but of phenomena only. Our minds cannot experience the world except as made up of causally interactive objects in time and space, so these are, inevitably, features of the world as representation, the world as it appears to us. But that is all they are. The will, as thing in itself, is not in space or time, and it neither causes nor is caused by anything else. Nor is the phenomenal world, the world as representation, itself caused, or in time, or in space. Asking where the world is, or at what time it began, or what caused it, makes no sense. True reality, including, so to speak, the real me and you, is eternal. Not being separated by time or space, the real you is the real me. Behind the veil of phenomena, all is one. It follows, as Schopenhauer sees it, that egoism is an enormous mistake. Those who care only about themselves care only about something that is

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Introduction

hardly real at all. They are also bound to be disappointed, because the self they love so much is doomed to die. The altruist, however, in caring about all people equally, evaluates her own importance accurately. And if I love not my life but life itself then death will be nothing much to fear, as life will always go on. Attachment to worldly goods and one’s own life is false and morally bad, while stoical or Buddhist detachment and altruism are correct and morally good. As well as compassion, Schopenhauer values art, because it allegedly has the power to detach us from our worldly concerns and somehow communicate to us things under the aspect of eternity. A still life, for instance, shows us fruit or flowers without reference to questions of ownership, time, or place. And music seems almost to occupy a different realm altogether. The young Wittgenstein believed much, perhaps even most, of this, at least for a while. He denied that he ever believed that the world is will, but he did initially accept the idea of the world as representation. He also seems to have shared much of Schopenhauer’s ethical outlook. For instance, he did not have to fight in the war (he had a hernia) but volunteered to do so anyway, asking to be posted to the front in order to test his fear of death. If he feared death (on Schopenhauer’s view) then he had a wrong view of the world. He gave away his fortune, some to artists and the rest to other members of his family. Throughout his life he took art very seriously (especially music) and lived abstemiously. All this is in line with Schopenhauer’s ethics. FREGE Young Wittgenstein, then, was a Schopenhauerian idealist. Until he encountered the works of Bertrand Russell and Gottlob Frege. Schopenhauer’s work is notable for its commitment to ethics, but there is, arguably, a certain sloppiness to it that might be regarded as unethical. His rejection of solipsism, for instance, is based more on common sense and wishful thinking than rational argument. Frege is rather different. According to Joan Weiner, “What Frege has to offer us is a model of philosophical virtue. Almost everyone who has grappled with Frege’s writings has been moved by Frege’s intellectual honesty” (Weiner 1990, 12). Another virtue of Frege’s is intellectual humility. Schopenhauer’s concerns might be summarized as life, the universe, and everything, but Frege was much less ambitious. His philosophical starting point was the unglamorous but undoubtedly challenging work of trying to get at the foundations of arithmetic. Common sense and wishful thinking might lead many of us to think that there is no need to inquire into the precise nature or basis of mathematics, but Frege was not satisfied with this lack of rigor, and attempted to demon-



Introduction 5

strate that arithmetic could be reduced to logic. This project resulted in some failure, which Frege faced up to, but also to some of the greatest advances in logic since Aristotle. In his book The Foundations of Arithmetic, Frege says that he has followed: three fundamental principles: always to separate sharply the psychological from the logical, the subjective from the objective; never to ask for the meaning of a word in isolation, but only in the context of a proposition; never to lose sight of the distinction between concept and object. (Frege 1980a, x)

These principles are thought to have influenced Wittgenstein considerably. He first met Frege in 1911 and the two corresponded at least until 1920. Wittgenstein studied Frege’s Grundgesetze der Arithmetik (Basic Laws of Arithmetic) during his first year in Manchester (1908–1909) and requested a copy of the same book in 1919 when he was a prisoner of war. He is known to have read other works of Frege’s too, such as his 1918 essay “Thought” (which attacks idealism) and the 1892 “Sense and Reference,” which has become Frege’s best-known work. According to Frege, words with a sense may lack a reference, e.g., “the least rapidly convergent series” (Beaney 1997, 153) or “the only living boy in New York.” There is no such series or boy to refer to, but we know this precisely because we understand the sense of these words. We might say that these words have a meaning (what Frege calls sense) even though there is nothing that they mean (what he calls reference). The internal images evoked by words vary from person to person, and might differ even with the same person and the same word from one time to another. Sense, though, belongs to humanity’s “common store of thoughts,” which it transmits from one generation to another (Beaney 1997, 154). The reference of a word or phrase is more objective still: it is the object or thing itself, which the relevant sign designates. Psychology, at least as Frege sees it, is concerned with the origins of our ideas, their subjective nature and physiological underpinnings. Logic (like mathematics) is concerned with none of this, only with concepts and ideas themselves, and with truth. “A proposition may be thought, and again it may be true; let us never confuse these two things,” Frege writes (Frege 1980a, vi). The distinction between subjective ideas (something like mental pictures, sensible and different for each person) and objective concepts (shared and in principle non-­sensible) “stands or falls with that between psychology and logic” (Frege 1980a, 37). Both sense (because it is shared) and reference belong to the latter, objective category, though reference does so more obviously.

6

Introduction

The distinction between sense and reference cannot strictly be defined, according to Frege. But it is real, and we can grasp it by thinking of the difference between direct and indirect speech, or between non-­fiction and fiction. Reference is ultimately all that matters for logic and science, all that matters to those concerned with truth (sense matters here too, but not quite as ultimately). It is a matter of such things as objects and objective concepts, not subjective ideas. Objects are particulars. They are public. They are determinate or definite. They are such things as Julius Caesar, the number one, and the True. (Controversially, Frege argues that what true sentences refer to is “the True.” In the true sentence “My cat is unusually long,” the words “My cat” refer to my cat, but, Frege believes, the sentence as a whole must also refer to something. According to him, its reference is that which is true: “the True.”) Against empiricists, Frege asserts that the sense of a word is not a mental image. He argues that mental images are irrelevant to everything except, perhaps, the tone of certain words. Even there, though, are there different images associated with the words “but” and “and,” or “dead” and “deceased,” to use one of Michael Dummett’s examples? (See Dummett 1993, 85.) Many words seem to have no mental image associated with them. When an image is associated with a word, it might vary from person to person. And if we know the image, we still need to know what to do with it. “No image can portray the role of the word in the sentence” (Dummett 1993, 158). This is why it is a mistake to consider the meaning of a word in isolation. We should always, Frege argues, take into account its place in the context of a sentence. So, we separate the psychological from the logical because the former is private and therefore irrelevant to shared efforts to discover truth. Frege also rejects John Stuart Mill’s idea that arithmetic is an empirical science based on observation. Mill defends this idea on the grounds that numbers are defined by reference to the fact that groups of things exist which strike the senses a certain way and which can be divided into lesser groups. For instance, three things together in a triangle create a certain impression, and those things can be moved so that they make the impressions associated with the numbers 2 and 1. Frege’s response is memorable: “What a mercy, then, that not everything in the world is nailed down; for if it were, we should not be able to bring off this separation, and 2 + 1 would not be 3!” (Frege 1980a §7). Mill’s view also raises the problem of how we could get the numbers 0, 1, and 777,864 in this way. Mill treats number as a physical property of observed physical objects. But, Frege points out, if I observe a pack of cards, am I to say that what I observe is one (pack) or fifty-two (cards)? It is arbitrary which I say, a matter of convention, not a matter of natural fact. Sense experience alone does not tell us how to conceptualize our experience or make judgments about it. It does not



Introduction 7

tell us whether we have 104 cards or two packs or one pile, nor whether these are cards or color patches or scraps of paper or pieces of trash or what. Judgment involves concepts and, often, their definitions. Without judgment there is no knowledge, no true or false, only raw experience. But then there would be no mathematics, since it involves judgment, knowledge, truth, and falsity. So either everything is hopelessly subjective and there is no knowledge, or else empiricism (not only with regard to arithmetic) is false. RUSSELL Finally, the most immediate influence on Wittgenstein’s thinking before he left to go to war was Bertrand Russell. Russell’s interests were extremely broad and his mind very active. He reconsidered issues and developed new solutions to problems so often that it is difficult to say precisely what Russell thought without specifying the year in which he adhered to that particular view. As far as the Tractatus goes, Russell’s most important works were probably The Principles of Mathematics (1903), Principia Mathematica (co-­ written with Alfred North Whitehead, 1910–1913), and Theory of Knowledge (written in 1913). Some of the ideas contained in these works also show up in the more accessible Introduction to Mathematical Philosophy (1919) and The Problems of Philosophy (1912). One of Russell’s most important ideas for Wittgenstein is the theory of types. This was developed in order to avoid paradoxes that arise when talking about classes, and classes are involved in Russell’s attempt to define numbers. Following Frege, Russell defines the number 2 as the class of all couples (see Russell 1919, 18), the number 3 as the class of all trios, and so on. One problem with this kind of definition, though, is that it brings with it the notion of a class, which is potentially more trouble than it is worth. Russell’s definition of number, he concedes, looks circular. Even if it is not, though, it will still be of very little value if it catches us up in paradox. And this is just what it can appear to do. For anything at all (call it x) and any class at all (call it y) we can ask whether x belongs to y. A spoon, for instance, belongs to the class of spoons, but a barber does not, and a class belongs to the class of classes, but, again, a barber does not. A barber is neither a spoon nor a class. Some classes belong to themselves, such as the class of classes. But what about the class of classes that do not belong to themselves? If it belongs to itself, then it does not belong to itself. If it does not belong to itself, then it does. It looks as though we are stuck in a circle. Paradoxes in logic are no better than circles in definitions. The theory of types aims to rescue us from this problem.

8

Introduction

Russell developed two versions of the theory. In The Principles of Mathematics (1903) he proposed his first, simple version, and in “Mathematical Logic as Based on the Theory of Types” (1908) he proposed the “ramified” version. The basic idea of the theory of types is that classes are not objects. This makes for a simpler ontology (i.e., theory of what there is), and means that it is nonsense to talk about a class being a member of a class in the way that an object (a spoon or a barber, say) can be a member of a class. In the simple theory, there are types of objects (real objects, classes of objects, classes of classes, and so on). In the ramified theory there are also types of properties (properties, properties of properties, etc., as when we say, e.g., shyness is nice or red is a property of objects). A related idea is the axiom of infinity, which postulates the existence of an infinite collection or set. If the number 2 is the class of all couples, then there must be at least one couple. Otherwise we could not talk meaningfully about two. Reference to infinity is certainly useful in mathematics. If it is to be meaningful, on Russell’s view, it seems that we must believe in a class of infinitely many things, even though we cannot know whether such a thing exists. It is quite possible, according to Russell, that a complete analysis of the world would show that all there is can be divided into a finite number of logically indivisible, unanalyzable things. These would be the basic units of reality (“logical atoms”), and knowledge of them would have to come from direct acquaintance. Unlike Frege, Russell was very much an empiricist. He believed that we are acquainted with, i.e., directly aware of, sense data in sensory perception and in introspection. The objects with which we are acquainted include particulars and universals. Particulars include existents and complexes “such as this-­before-­that, this-­above-­that, the-­yellowness-­of-­this” (Russell 1953, 201). Existents include such things as the “this” and “that” referred to here. Universals include “all objects of which no particular is a constituent” (Russell 1953, 201). They are general ideas, such as beardedness, diversity, and so on. We are not acquainted with physical objects or other people’s minds, but we can believe in them on the basis of objects with which we are acquainted. For instance, we may well believe that sense data with which we are acquainted are caused by physical objects that are beyond our immediate experience. If all I smell are smells, all I see are sights, all I hear are sounds, and so on, then I do not directly sense three-­dimensional solid objects. But I can still believe in the existence of such objects, as I might believe that the sounds, smells, and so on (the sense data) that I experience are caused by three-­dimensional objects around me. I can then refer to these objects by name even though I am not directly acquainted with them. According to Russell, any object, physical or otherwise, can be named, a name being “a simple symbol, directly designating an individual which is its



Introduction 9

meaning, and having meaning in its own right, independently of the meanings of all other words” (Russell 1919, 174). However, not every apparent name is really a name. Some are disguised descriptions. For instance, if I want to I can call a particular point in my visual field Ishmael. I could also call my dog Ishmael. But I might also use the apparent name “Ishmael” as shorthand for “the narrator of Moby Dick,” which is a description (if it were a name, this would imply that it referred to a subsistent entity, according to Russell). So it is not always obvious what is a name and what is a description. “The King of France” looks like a name, but, since there is no such person anymore, Russell analyzes these words as a disguised description. Thus “The King of France is bald” turns out to be false, not meaningless, despite its lack of reference to any real person. What the sentence is really doing, according to Russell, is claiming that there is a person who matches the description: a) is King of France, b) is bald. Wittgenstein seems to have been impressed by this theory of descriptions. Exactly what he made of the ideas of Schopenhauer, Frege, and Russell, though, I will leave (to some extent) for later in the text and (to a much greater extent) for others to determine. THE TRANSLATION Three other translations of Wittgenstein’s Tractatus are currently available, but none, I think, is ideal for scholarly purposes. Daniel Kolak’s translation is designed to provide a lyrical and poetic English version of the book, which might make the reading experience more pleasant and in some ways closer to what Wittgenstein wanted, but it does not help those who care about understanding the precise meaning of each sentence. Far more popular with scholars are the translations by C. K. Ogden (available in various editions) and by D. F. Pears and B. F. McGuinness (Routledge & Kegan Paul, 1974). Neither of these is perfect, however. The obvious choice is Ogden’s translation, since Wittgenstein himself helped with the project. However, that translation is neither completely accurate nor completely in accordance with Wittgenstein’s suggestions (although he did approve the whole). Hence the popularity of Pears and McGuinness’s version of the text, which is also more readable than Ogden’s. Unfortunately, this translation is perhaps too readable. At times it is arguably wrong, e.g. at 3.3411 (see Black 1964, 152). At times it goes against Wittgenstein’s explicit directions (e.g., at 6.43). And in general, Pears and McGuinness have been accused of over-­translation (see, e.g., Stokhof 2002, xvii) and leading the reader to interpret the text in one way when the original German is at least equally open to another interpretation. My translation is closer to Ogden’s

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Introduction

but, I hope, with greater fidelity than his to Wittgenstein’s express wishes regarding the translation. I have made a point of being consistent when it comes to the translation of certain key terms. Wittgenstein did not like “state of affairs” for Sachlage. He suggested possibly the Latin status rerum (state of things). I have translated it as “situation,” following Pears and McGuinness, reserving “state of affairs” for “Sachverhalt,” which Ogden has as “atomic fact.” I translate darstellen as “present” and vorstellen as “represent.” Darstellen has to do with representation and performance, as in theater. Vorstellen has to do with presentation or putting forward (as). To do it to yourself is to imagine. Schopenhauer’s Die Welt als Wille und Vorstellung, which surely influenced Wittgenstein’s thinking, is generally translated as The World as Will and Representation. It might be worth using “represent” just in case some reference to Schopenhauer is ever intended by Wittgenstein’s use of vorstellen, Vorstellung, etc. HOW TO READ THE TRACTATUS The Tractatus is usually published with an introduction by Russell, but I have omitted this because Wittgenstein believed that Russell had misunderstood the book and so misrepresented its contents in his essay. It seems better, if possible, to come to the text without possibly misleading ideas about what it contains. Presumably, Wittgenstein intended his readers to begin at the beginning and do their best from there to make sense of what he had written. However, he did also provide some guidance for readers, which might be worth keeping in mind. To Ludwig von Ficker, whom Wittgenstein hoped might publish the book, he wrote that: It is essentially the presentation of a system. And this presentation is extremely compact since I have only recorded in it what – and how it has – really occurred to me. . . . The work is strictly philosophical and simultaneously literary, and yet there is no blathering in it. (Quoted in Nordmann 2005, 48)

He later wrote again to von Ficker, this time saying: [T]he sense of the book is an ethical one. I once wanted to include a sentence in the preface which doesn’t in fact appear there now. But I am writing it to you now because it might serve you as a key: For I wanted to write that my work consists of two parts: the one you have in front of you and all that I have not written. And just that second part is the important one. Because the ethical is delimited by my book as it were from within; and I am convinced that strictly it can only be delimited like that. In short, I believe: Everything that many are



Introduction 11

blathering about today, I settled by being silent about it. And that’s why this book, unless I am very mistaken, says much that you yourself want to say, but perhaps you won’t see that it is said in it. I would now recommend that you read the preface and the conclusion, since these express the sense most immediately. (Quoted in Nordmann 2005, 48)

To Russell, Wittgenstein wrote that, “The main point is the theory of what can be expressed (gesagt) by prop[osition]s – i.e., by language – (and, which comes to the same, what can be thought) and what can not be expressed by prop[osition]s, but only shown (gezeigt); which, I believe, is the cardinal problem of philosophy” (McGuinness 2008, 98). These remarks might help the reader approaching the text for the first time. It is worth noting, however, that, as Ray Monk has said, “More than eighty years after it was published, and despite a vast secondary literature inspired by it, there is still no general agreement about how the book should be read” (Monk 2005, 29–30). FURTHER READING The literature on the Tractatus has grown at a rate that is at times genuinely alarming for those of us who try to keep up with it. Each interpretation is different, but they tend to fall into three groups. The mainstream view is that each part of the book can be understood, but that the parts do not necessarily form a consistent or coherent whole (for instance, one might regard the bulk of the Tractatus as a continuation of Russell’s philosophical project but the ending as a Schopenhauerian graft that does not quite take). There are numerous readings of this general kind, but White 2006 is a good example. Against this kind of view is the “resolute” stance of Cora Diamond and James Conant, according to whom most of the sentences of the Tractatus have no meaning (as Wittgenstein implies at 6.54) and so cannot be understood (although why Wittgenstein wrote them can be). McManus 2006 is probably the closest there is to a book-­length treatment of the whole Tractatus along these lines. The third view is that there is something to be said for, and learned from, each of these other two kinds of view, and that a middle way between them should be sought. McGinn 2006 is a good example of this kind of reading. For more on Wittgenstein himself I recommend Monk 1990, Malcolm 1984, and McGuinness 2005. For additional information about Wittgenstein’s intellectual background see Nordmann 2005, Janik 2006, and Janik and Toulmin 1973.

Tractatus Logico-­Philosophicus

[The title (often abbreviated to TLP) was suggested to Wittgenstein for the English translation of his Logisch-­ Philosophische Abhandlung by G. E. Moore. It is a pretty literal translation of the German into Latin, meaning something like “Logico-­Philosophical Treatise.” It is interesting, though, that Wittgenstein considered a Latin title to be more appropriate than any English one he could think of. Perhaps he liked the fact that this title echoes Spinoza’s Tractatus Theologico-­Politicus, and perhaps he liked the idea of replacing talk of theology and politics with a discussion of logic and philosophy. But we should probably not read too much into the title, since it was not his idea originally and he seems to have accepted it slightly reluctantly, in the absence of any better ideas. See Monk 1990, 206.]

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Dedicated to the memory of my friend David H. Pinsent

[David Hume Pinsent (1891–1918), a descendent of the philosopher David Hume, was a test pilot during World War I. He died in a plane crash.] Motto: . . . und alles, was man weiss, nicht bloss rauschen und brausen, gehört hat, lässt sich in drei Worten sagen. Kürnberger . . . and everything that one knows, that is not mere rumbling and roaring one has heard, can be said in three words. Kürnberger

[Ferdinand Kürnberger (1821–1879) was an Austrian writer. The significance and proper placement of the motto are discussed in Kienzler 2013.]

Preface

This book will perhaps only be understood by one who has himself already at some time thought the thoughts that are expressed herein – or at least similar thoughts. – It is therefore not a textbook. – Its end would be reached if it gave pleasure to one person who read it with understanding.

[Compare Bertrand Russell on Gottlob Frege. Of Frege’s Begriffsschrift Russell writes, “I possessed the book for years before I could make out what it meant. Indeed, I did not understand it until I had myself independently discovered most of what it contained” (Russell 2000, 65).] The book deals with the problems of philosophy and shows – so I believe – that the formulation of questions about these problems is due to misunderstanding the logic of our language. One could put the whole sense of the book perhaps in these words: What can be said at all, can be said clearly; and whereof one cannot speak, thereof one must be silent. The book would thus draw a limit to thinking, or rather – not to thinking, but rather to the expression of thoughts: Because in order to draw a limit to thinking, we would have to be able to think both sides of this limit (we would thus have to be able to think what cannot be thought). So the limit can only be drawn in language, and what lies on the other side of the limit will be simply nonsense.

[The last words of that sentence are einfach Unsinn. Literally they mean “simply nonsense,” not “simple nonsense,” which would be einfacher Unsinn. There has been some disagreement about this, but it is cleared up by Alfred Nordmann (see Nordmann 2005, 82 note 48).

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Tractatus Logico-Philosophicus

Michael Morris and Julian Dodd comment on these last two paragraphs: What we have here is the careful replacement of a form of words that seems to try to do something that cannot be done. But there is a double paradox in this replacement. First, the inappropriate form of words is left on the page: the words were not just omitted after the first draft; they sit there, self-­underminingly, seeming to explain their own impossibility. And, secondly, the supposed improvement seems to be subject to the very fault that has been so carefully explained in connection with the first formulation. We think for a moment that we have avoided paradox, but on reflection we realize that we are no better off. The point of this delay of self-­refutation, we suggest, is to give us time to achieve acquaintance with the world as a limited whole; and the sustaining of this acquaintance is the mystical outlook. (Morris and Dodd 2009, 269)] I do not want to judge how far my efforts coincide with those of other philosophers. Indeed what I have written here makes no particular claim to novelty; and therefore I give no sources, because it is all the same to me if what I have thought has already been thought by another before me. I will mention only that I owe a large part of the stimulus to my thoughts to the great works of Frege and to the work of my friend Mr. Bertrand Russell.

[This has been taken by some commentators to indicate that Wittgenstein’s thinking is closer to Frege’s than it is to Russell’s, since he seems to hold Frege in greater esteem. Ian Proops, though, argues that Russell was more influential than is often thought, both on Wittgenstein’s own ideas and on his understanding of Frege’s. Like G. E. M. Anscombe (who holds that to understand the Tractatus it is very important to study Frege), Proops thinks that Wittgenstein misunderstood Frege on some points, and sees Russell’s influence at work in these cases. Proops quotes several instances of Wittgenstein’s writing in 1912 of “our problems,” “our theory,” and so on, meaning his and Russell’s (Proops 2000, xix). This, Proops thinks, is significant, because it was only in the next year that Wittgenstein wrote his Notes on Logic, much of which was copied directly into the Tractatus. He referred to that work as a summary of what he had done at Cambridge up to that point. However, as Proops notes, even as early as 1913 Wittgenstein wrote about his work in a noticeably less collaborative way, criticizing Russell’s theories. In 1919 Wittgenstein referred to the problems dealt with in the TLP as “our problems” (i.e., his and Russell’s), but to the solutions as his alone (see Proops 2000, xix). In March 1919 (Wittgenstein 1974a, 111), he says that the TLP “upsets all our theory of truth, of classes, of numbers and all the rest.” Proops emphasizes the importance of the Notes on Logic (1913) and Notes Dictated to G. E. Moore (1914) largely because, if Brian McGuinness is



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correct, Wittgenstein referred (in May 1915) to the latter as “essentially . . . definitive” only a few months before he began work on what was later called the Prototractatus (see Proops 2000, xxi, and McGuinness 1989, 35–47). And the Prototractatus is quite close to the Tractatus proper. However, in the same letter in which he said that he regarded the Moore notes “essentially as definitive,” Wittgenstein also wrote that his problems were changing, becoming “more and more lapidary and general” and that his method for dealing with them had “changed dramatically.” As Monk observes (see Monk 1990, 130), Wittgenstein’s work changed more drastically in the next two years, moving from logic to ethics and philosophy in general. Since these are the subjects that he presented the book as being all about (ethics to Ludwig von Ficker, philosophy in general to the readers of the Tractatus’s foreword), we should keep an open mind as to whether remarks copied from the Prototractatus or Notes on Logic have the same point, serve the same function, in the Tractatus as they did in those earlier works.] If this effort has a value then it consists in two things. First in that thoughts are expressed in it, and this value will be the greater the better the thoughts are expressed. The more the nail has been hit on the head. – Here I am aware of falling far short of what is possible. Simply because my ability to accomplish the task is too slight. – May others come and do it better. On the other hand the truth of the thoughts communicated here seems to me unassailable and definitive. I am therefore of the opinion that the problems have in essentials been finally solved. And if I am not wrong in this, then the value of this work now consists secondly in that it shows how little has been achieved by the solving of these problems. L. W. Vienna, 1918.

Main Text

1*

The world is everything that is the case. *The decimal numerals given as numbers of the individual propositions indicate the logical importance of the propositions, the emphasis placed on them in my presentation. The propositions n.1, n.2, n.3, etc., are remarks on proposition no. n; the propositions n.m1, n.m2, etc. remarks on proposition no. n.m; and so on.

[Julian Young says that: In the first book of The World as Will [. . .] Schopenhauer says that the problem of philosophy is to say “what” the world is (WR I: 82). Sometimes he says that it is to solve the “riddle” (Rätsel) of what the world is. Given the rootedness of this word in German folk tales where solving a “riddle” is often a matter of life and death, this suggests that an answer to the question, rather than merely satisfying the curiosity of armchair investigators, will have existential implications, will have an effect on our lives. (Young 2005, 17)

On Wittgenstein’s footnote, Mauro Engelmann writes: According to the numbering system of the Tractatus, all sentences of the book are, directly or indirectly, elucidations of 1, 2, 3, 4, 5 or 6 (of course, 7 is the result of the book). So it must be useful to see how these sentences work as a whole. Let us look at the uebersichtliche presentation of the whole ladder taking into account how its rungs are connected. The connection is italicized and underlined: 1.  The world is everything that is the case. 2. What is the case, the fact, is the existence of atomic facts. 3.  The logical picture of the facts is the thought. 21

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4. The thought is the significant proposition. 5.  Propositions are truth-functions of elementary propositions. 6.  The general form of truth-functions is: [p̄ , ξ̄ , N(ξ̄ )]. 7.  What we cannot speak about we must pass over in silence. A bird’s eye view indicates how each group of aphorisms connects to the next by means of an informal definition or “equivalence.” (Engelmann 2018, 462)

Engelmann goes on to argue that in the Tractatus, obscure ideas, such as world, are gradually replaced by simpler, clearer ones, such as proposition. As we climb Wittgenstein’s ladder we move away from seeming metaphysical profundity toward crystal clear, but empty, logical form: “The general point of the ladder is: if you want to grasp the essence of world and thought, all you really get is an empty form” (Engelmann 2018, 464). Where we might have thought we had metaphysical substance, that is, we end up realizing that all we ever had were definitions, uninformative, and hence dispensable, tautologies. Despite Wittgenstein’s explanatory note, there is still disagreement about the order in which his remarks should be read. Tim Kraft (2016) defends the sequential reading (according to which, as usual, one reads the remarks in the order they appear on the page), while others, such as P. M. S. Hacker (2015), argue that Wittgenstein’s explanation makes it clear that it should be read as a tree with various branches. Reading this way requires paying more attention to the numbers and not simply reading one sentence after another.] 1.1

The world is the totality of facts, not of things.

[Max Black says that this distinction is “the outstanding innovation of Wittgenstein’s ontology” (Black 1964, 27), distinguishing him from all the most famous philosophers from Aristotle to the early Russell. The universe is implicitly not a thing, not something that can be referred to by a name (see Black 1964, 27–28). Black takes Wittgenstein’s references to “the world” to mean the universe, explaining that this use is more common in German than it is in English (see Black 1964, 29). Frege has a notion of fact that is worth bearing in mind here. “A fact is a thought that is true” (Beaney 1997, 342). Thoughts are imperceptible. We may see the sun rise, but we do not in the same sense see that the sun rises. That the sun is rising, is a thought. The sentence “The sun is rising” expresses this thought. Thoughts are not external, perceptible objects, but neither are they private, subjective, individual, psychological ideas. “A third realm must be recognized,” Frege says (Beaney 1997, 337). Frege asked Wittgenstein how “The world is the totality of facts” differs from “The world is everything that is the case.” Wittgenstein’s reply was



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that: “The meaning of these two sentences are one and the same but not the conceptions (Vorstellungen) that I associated with them when I wrote them down” (McGuinness and Haller 1989, 22, quoted in Nordmann 2005, 124 note 70). Alfred Nordmann comments that: “Frege takes this to agree with his own distinction between sense and reference,” but this cannot be right. While Frege does indeed distinguish between what he calls (as Nordmann goes on to quote) “the actual meaning of the sentence” and “the conceptions someone associates with the sentence,” this is not the distinction he makes between sense (Sinn) and reference (Bedeutung). For Frege, Vorstellungen (conceptions or ideas) are purely, indeed necessarily, individual or private, while sense can be common property. See Beaney 1997, 154, for Frege’s explicit distinction between Vorstellung and both Sinn and Bedeuntung. Nordmann suggests that perhaps the meaning that is the same is precisely no meaning at all (see Nordmann 2005, 168). His view is that these two sentences (i.e., “The world is everything that is the case” and “The world is the totality of facts”) have no sense but are able nevertheless to make sense. Russell’s idea of a thing also seems relevant. Russell describes his Logical Atomism, the text of lectures given in early 1918, as “very largely concerned with explaining certain ideas which I learnt from my friend and former pupil Ludwig Wittgenstein” (Russell 1986, 160). At the time Russell had not seen or heard from Wittgenstein since August 1914. Russell explains: The reason that I call my doctrine logical atomism is because the atoms that I wish to arrive at as the sort of last residue in analysis are logical atoms and not physical atoms. Some of them will be what I call “particulars”—such things as little patches of colour or sounds, momentary things—and some of them will be predicates or relations and so on. The point is that the atom I wish to arrive at is the atom of logical analysis, not the atom of physical analysis. (Russell 1986, 161)

Matthew Ostrow notes that 1.1 sounds like a statement of metaphysical realism, and has been taken as such by numerous commentators, while others read Wittgenstein as being a kind of idealist (for instance, a solipsist) (see Ostrow 2002, 21–22). Yet 5.64 suggests that Wittgenstein rejected the realism-idealism dichotomy. Ostrow quotes Wittgenstein later saying to Desmond Lee that the opening of the Tractatus says that: “The world does not consist of a catalogue of things and facts about them (like a catalogue of a show). . . . What the world is is given by description and not by a list of objects” (Wittgenstein 1980c, 119, quoted in Ostrow 2002, 23). At 4.1272 Wittgenstein calls the concept of an object a pseudo-concept. In his Notebooks Wittgenstein wrote that “Properties and relations are objects too” (June 16, 1915). And, as Ostrow notes (Ostrow 2002, 49), in

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1930–1931 Wittgenstein is reported to have said that “ʻObjects’ also include relations: a proposition is not two things connected by a relation. ‘Thing’ and ‘relation’ are on the same level. The objects hang as it were in a chain” (Wittgenstein 1980c, 120). Pasquale Frascolla says that the point about the chain is that there are no such things as “relations” that relate or connect things/ objects (see Frascolla 2007, 79). The work that might be thought to be done by something called “relations” is (already) done by what Wittgenstein calls “objects.” See 2.03 for more on the chain analogy.] 1.11

The world is determined by the facts, and by these being all the facts.

[Peter M. Sullivan points out that this “remark runs straight into the vicious circle principle” if “these being all the facts” is itself meant to be one of the facts that determine the world (Sullivan 2000, 175). As Sullivan notes, Whitehead and Russell define the vicious circle principle thus: “Whatever involves all of a collection must not be one of the collection” (Whitehead and Russell 1910, 37). Possibly Wittgenstein had this in mind when he wrote 1.11.] 1.12

Because the totality of facts determines what is the case and also all that is not the case.

1.13

The facts in logical space are the world.

1.2

The world divides into facts.

1.21

Each can be the case or not be the case and all else stay the same.

[Cf. 5.135, which denies that one can infer the existence of one situation from the existence of another.] 2

What is the case, the fact, is the holding of states of affairs [Sachverhalten].

[I follow Black in translating Bestehen as “holding” rather than “existing” (see Black 1964, 39). He prefers “atomic fact” for Sachverhalt. A Sachverhalt is “the objective counterpart of an unanalysable contingent truth (see, for instance, 4.2211)” (Black 1964, 39–40). However, Black says, Wittgenstein uses Sachverhalt in seemingly inconsistent ways. Most of the time he uses it to mean an actual combination of objects, but he also sometimes uses it to mean a combination that does not exist (e.g., at 2.06 and 4.3). Erik Stenius says that “a Sachverhalt is something that could possibly be the case,” but 2.0124 talks of possible Sachverhalte, which would be odd in

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that case (Stenius 1960, 31). (This objection is Black’s.) Black argues that Sachverhalte should be understood as facts rather than possibilities, at least most of the time (see Black 1964, 41–45). Frascolla suggests that “states of affairs are to be identified with the phenomenal complexes belonging to the various sense realms (visual complexes, auditory complexes, etc.)” (Frascolla 2007, ix). He writes: “states of affairs and facts differ in two ways: first, a state of affairs is merely a possible combination of objects, whereas a minimal fact is an actual combination; second, when a fact is spoken of, it is not necessary that it be thought of as one obtaining state of affairs: several obtaining states of affairs can constitute a fact” (Frascolla 2007, 84). On the next page, though, Frascolla notes that Wittgenstein’s distinction between facts and states of affairs is weakened by his introduction of the term “negative fact” (at 2.06), since now facts are apparently not necessarily actual.] 2.01

The state of affairs is a combination of objects. (Items, things.)

[Hacker comments: “Wittgenstein’s conception of a simple object was, I suspect, an heir to Russell’s notion of a term in The Principles of Mathematics, itself a development of Moore’s notion of a concept” (Hacker 1997, 66). Hacker’s reference is to Moore 1899. In The Principles of Mathematics, Russell writes: Whatever may be an object of thought, or may occur in any true or false proposition, or can be counted as one, I call a term. This, then, is the widest word in the philosophical vocabulary. I shall use as synonymous with it the words unit, individual, and entity. . . . A man, a moment, a number, a class, a relation, a chimaera, or anything else that can be mentioned, is sure to be a term; and to deny that such and such a thing is a term must always be false. (Russell 1903, 43)

Roger White says: It is [. . .] important to bear in mind how little we are told about the objects, and the reader should postpone deciding the extent to which these do represent genuine metaphysical commitments until it is decided how such claims are to be interpreted. It is at any rate safe to say that, rightly or wrongly, Wittgenstein himself did not intend to develop a metaphysics in these opening sections, and in an important sense, these opening paragraphs are meant to be as vacuous as possible: this is as much as can be said about the world without begging any questions as to its detailed nature. (White 2006, 26)

Frascolla: “The conjecture I put forward is that [. . .] objects are to be identified with repeatable phenomenal qualities (qualia, in Nelson Goodman’s sense

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of the word)” (Frascolla 2007, ix). Frascolla finds this interpretation confirmed by Wittgenstein’s remark (Wittgenstein 1980c, 120) that: “ʻAn atomic fact is a combination of objects (entities, things).’ Objects etc. is here used for such things as a colour, a point in visual space etc.” (Frascolla 2007, 78). Robert Fahrnkopf argues that Wittgenstein’s objects include universals. He points out (see Fahrnkopf 1988, 7) that Moore’s notes on Wittgenstein’s lectures (1930–1933) report that Wittgenstein spoke of colors as if they were Russellian individuals (see Fahrnkopf 1988, 7). Fahrnkopf also notes that in the Blue Book Wittgenstein characterizes his Tractatus view as being that redness, roundness, and sweetness are elements or individuals (see Fahrnkopf 1988, 8). Wittgenstein seems to be talking about the universal redness rather than a particular red sense-datum here. Jaakko Hintikka identifies Wittgenstein’s objects as “objects of my experience” (Hintikka 2000, 14). He quotes Frank Ramsey: “Wittgenstein says it is nonsense to believe in anything not given in experience. . . . For to be mine, to be given in experience, is the formal [definitory] property to be a genuine entity. (From item # 004-21-02 of the Ramsey archives of Pittsburgh)” (quoted in Hintikka 2000, 14). Hintikka calls these objects phenomenological entities but denies that they are mere phenomena (see Hintikka 2000, 15). That is, he thinks, they are what is given in immediate experience, but they are not only the contents of our consciousness. We have immediate experience of physical reality (which still remains to be defined), not only of the contents of our own minds. Others, such as David Pears, are less sure: The question, “What kind of thing did [Wittgenstein] take objects to be?” is often made to appear simpler than it really is. Commentators usually ask whether he took them to be material points (point-masses) or sense-data. The Notebooks, which record exploratory work, canvas both possibilities, and in the Tractatus, where he might have been expected to make up his mind and choose between them, he does not do so, and does not even formulate the question to which of the two categories objects belong. (Pears 1987, 89)

Pears concludes: “However, it is really safer to accept his professions of agnosticism about the nature of the objects of the Tractatus, and to take the evidence to show no more than that he allowed for the possibility that they might include relations” (Pears 1987, 142). Colin Johnston brings the very question whether Tractarian objects might include relations into doubt: To argue whether Wittgenstein intended us to take Tractarian objects to be particulars, or to include relations, is a mistake. The Tractarian Wittgenstein does

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not think that any such would-be logical category terms as “particular” or “relation” have a priori application. A suggestion made in advance of the pursuit of truth-functional analysis that Tractarian objects include, or do not include, relations is something Wittgenstein would have seen as “mere playing with words.” (Johnston 2009, 156–157)

See also Marie McGinn, who writes: “What makes it so difficult to understand Wittgenstein’s concept of an object is, I want to claim, precisely that its role is closer to the idea of the meaning of a name than it is to our ordinary notion of what a name refers to, i.e., the bearer of a name” (McGinn 2006, 7 note 8). Later she writes: “Objects are the meanings of the indefinable constituents of a proposition . . .” (McGinn 2006, 113). Similarly, Hidé Ishiguro argues that “the identity of the object referred to is only settled by the use of the name in a set of propositions” (Ishiguro 1969, 21).] 2.011

It is essential to the thing that it can be a constituent part of a state of affairs.

2.012

In logic nothing is accidental: if a thing can occur in a state of affairs then the possibility of the state of affairs must be already prejudged in the thing.

2.0121 It would, as it were, appear as an accident if there were later to be a situation suitable for a thing that could [already] hold for itself, on its own. If things can occur in states of affairs then this [possibility] must already be in them. (Something logical cannot be merely possible. Logic deals with every possibility, and all possibilities are its facts.) As we cannot conceive of spatial objects at all without space, or temporal objects without time, so we can conceive of no thing without the possibility of its uniting with other objects. If I can conceive of an object in the context of a state of affairs then I cannot conceive of it without the possibility of this context. 2.0122 The thing is independent in so far as it can occur in all possible situations, but this form of independence is a form of connection with the state of affairs, a form of dependence. (It is impossible for words to appear in two different ways: alone and in propositions.)

[On objects being independent, see Russell’s Logical Atomism: Particulars have this peculiarity, among the sort of objects that you have to take account of in an inventory of the world, that each of them stands entirely alone

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and is completely self-subsistent. It has that sort of self-subsistence that used to belong to substance, except that it usually only persists through a very short time, so far as our experience goes. That is to say, each particular that there is in the world does not in any way logically depend upon any other particular. (Russell 1986, 179)

Martin Stokhof notes that Wittgenstein’s reference to dependence here “strongly suggests that [objects] cannot be conceived of as material atoms (elementary particles, or wave packets, or whatever), since for such objects the very possibility of an independent existence, however short-lived this may be, cannot be ruled out a priori” (Stokhof 2002, 46–47). Much the same goes for sense data: “For such objects, too, it holds that no logical property prevents their independent occurrence, even if other properties would” (Stokhof 2002, 47).] 2.0123 If I am acquainted with the object then I am also acquainted with all the possibilities of its occurrence in states of affairs. (Each such possibility must be in the nature of the object.) A new possibility cannot be found later.

[In his Letters to Ogden Wittgenstein says: “to know here just means: I know it but I needn’t know anything about it” (Wittgenstein 1973, 59). Presumably he is making use of Russell’s notion of knowledge by acquaintance.] 2.01231 In order to be acquainted with an object, I need not of course [be acquainted with] its external [properties] – but I need to be acquainted with all its internal properties.

[Denis McManus says that objects’ external properties “are their forming particular combinations with other objects, the existence of these combinations being the holding of particular contingent facts” (McManus 2006, 31). He refers to 4.123 in connection with this, although his focus is on what “internal properties” might be. Frascolla says that internal properties are what an object “necessarily can” be, while external properties concern an object’s involvement in states of affairs that happen to obtain (Frascolla 2007, 61 and see also 62). He gives one object’s being taller than another as an example of an external property.] 2.0124 If all objects are given then therewith all possible states of affairs are also given. 2.013

Each thing is, as it were, in a space of possible states of affairs. I can conceive of this space as empty, but I cannot conceive of the thing without the space.

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[Black points out a possible echo of Kant 1969 A 24/B 38 here. (See Black 1964, 50.)] 2.0131 The spatial object must be in infinite space. (A spatial point is an argument-place.) The speck in a visual field of course need not be red, but it must have a color: it has, so to speak, color-space around it. The note must have a pitch, the object of the sense of touch a degree of hardness, etc. 2.014

Objects contain the possibility of all situations.

2.0141 The possibility of its occurrence in states of affairs is the form of an object. 2.02

Objects are simple.

[Fahrnkopf writes: “The [passage] from 2.01 to 2.02, is devoted exclusively to setting out the radical idea that objects are essentially dependent, in that they are not merely a content but also a form” (Fahrnkopf 1988, 42).] 2.0201 Each statement about complexes can be analyzed into a statement about their components and into those propositions that completely describe the complexes.

[Matthew Ostrow says that this remark must be compared with 3.24 (see Ostrow 2002, 27). The “central purpose” of 2.0201, he says, is to make evident the fundamental distinction between complex and object (Ostrow 2002, 28). Complexes cannot be treated as entities or objects. According to 2.0211, a proposition about a nonexistent object is nonsense, but, by 3.24, a proposition about a nonexistent complex is false, not nonsensical. White notes an important difference between the argument from 2.0201– 2.0212 and that running from 3.23–3.24 (see White 2006, 38–40). One starts from the need for the world to have substance, while the latter is based on the demand that sense be determinate. Each concludes that there must be simple objects.] 2.021

Objects make up the substance of the world. Therefore they cannot be composite.

2.0211 If the world had no substance then whether a proposition had sense would depend on whether another proposition was true.

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[Mounce presents Wittgenstein’s reasoning here as follows: . . . whether a proposition has sense cannot be a contingent matter. What is contingent is whether it is true (or false). But in order to be true (or false) a proposition must already possess a sense. The sense of a proposition, in short, must be independent of whether it is in fact true or false. Consequently, there must be a contact between language and the world which is prior to the truth or falsity of what we say. Such a contact is to be found in the relationship between a simple name and a simple object, the relationship being such that the name just stands for the object independently of description. (Mounce 1981, 21)

Severin Schroeder: The meaning of a name is the object it denotes. Hence, if there is no such object, the name will be meaningless, and the sentence in which it occurs will have no sense. But if a name is supposed to stand for a complex object, the decomposition or non-existence of that complex is a real possibility. So to ascertain that the original proposition does have a sense, one would have to check whether the complex in question did in fact exist, i.e., whether another proposition describing that complex was true. (Schroeder 2006, 42)

José Zalabardo: On the standard reading of the substance passage [. . .] the lack of sense that 2.0211 is concerned with is what would occur if the constituents of the proposition failed to refer. But this reading can’t be easily squared with what we know about substance. If the claim that the world has substance concerns the possibilities of combination of objects into states of affairs, it is hard to see how the substance of the world could ensure that the referents of the constituents of propositions don’t go out of existence. If the dependence of sense on truth is avoided by substance, then the source of the dependence must be entirely different. On the reading that I want to recommend, the lack of sense that Wittgenstein is discussing would come about when the referents of the constituents of a proposition cannot be combined into a possible state of affairs. (Zalabardo 2015, 145)] 2.0212 It would then be impossible to draft a picture of the world (true or false). 2.022

It is obvious that even a world quite different from the actual one must have something – a form – in common with it.

2.023

This fixed form consists precisely of the objects.

2.0231 The substance of the world can determine only a form and not any material properties. Because these [material properties] are pre-



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sented only by propositions – are only produced by the configuration of objects. 2.0232 Incidentally: objects are colorless.

[The most literal translation would be something like “By the way: objects are colorless,” but this seems too casual. Nordmann has “By the way: . . .” (Nordmann 2005, 102). Black seems to think that “in a manner of speaking,” “roughly speaking,” “incidentally,” and “in passing” would all be acceptable translations of beiläufig gesprochen (Black 1964, 64). McManus: “My suggestion is that to declare that our talk ultimately rests on an immediate ‘seeing’ of ‘colourless objects’ is one step away from recognizing that the ‘project’ of explaining the ‘possibility’ of ‘meaningful’ talk leaves us nothing to say or think: our ‘experience’ of the pure and simple here is the experience of empty words” (McManus 2006, 124). Given that in his later Cambridge lectures Wittgenstein gave colors as an example of what he meant by “objects,” 2.0232 is an odd statement (see Wittgenstein 1980c, 120). In what sense could colors be colorless? Frascolla offers this account: To put it all in a nutshell, objects do not have any colour, although some of them are colours; they do not occupy any visual place, although some of them are visual places, and they do not have any position in phenomenal time, although some of them are phenomenal times, i.e., in Russell’s jargon, moments of private time. On the other hand, those complexes which have a spatial quale among their constituents do have a spatial location, and likewise all those complexes which have a temporal quale among their constituents (all concrete complexes, as we shall see shortly) do have a position in time. (Frascolla 2007, 80)] 2.0233 Two objects of the same logical form are – apart from their external properties – differentiated from one another only by the fact that they are different.

[G. E. M. Anscombe: “The only ‘external properties’ his [i.e., Wittgenstein’s] simple objects can have, of course, are those of actually occurring in certain facts” (Anscombe 1971, 111). See 5.5302, where, Anscombe says, “he is explicit that it makes sense to say that two objects have all their properties in common.” See 4.123 for more on external properties. Marie McGinn says that Wittgenstein’s point, as she sees it, is that if two objects are really simple then the difference between them cannot be given by definitions. “Red” and “blue” refer to exactly the same kind of thing, namely colors. Of course different things are red than are blue, but these are external

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differences between red and blue. Apart from these, the only difference between them is that they are different colors. (See McGinn 2006, 152.)] 2.02331 Either a thing has properties that no other has, and then one can distinguish it straightaway from the others by means of a description, and refer to it thereby; or else there are several things that have all their properties in common, and then it is altogether impossible to pick out one of them. Because if a thing is not distinguished by anything then I cannot distinguish it, because otherwise it really would have been distinguished all along. 2.024

Substance is what persists [besteht] independently of what is the case.

2.025

It is form and content.

2.0251 Space, time, and color (coloredness) are forms of objects.

[On the translation of Färbigkeit (“coloredness”) see Frascolla 2007, 80–81. Pears and McGuinness have “being coloured” but this is less literal than my translation, does not fit the interpretation of 2.0232 offered above (by Frascolla), and does not fit other uses of the same word made by Wittgenstein. In the Remarks on Colour, Part I §47, for instance, Wittgenstein talks about Färbigkeit as something that can be reduced. Frascolla suggests that he means “something like colour intensity,” not simply having a color (Frascolla 2007, 80).] 2.026

Only if there are objects can there be a fixed form of the world.

[Michael Morris suggests that “the overall point of” 2.02–2.027 is “that there are objects which sustain a fixed form common to all possible worlds” (Morris 2017, 2).] 2.027

The fixed, the persistent, and the object are one.

[Gordon C. F. Bearn notes that 5.621 and 6.421 also talk of things being one. See Bearn 1997, 75.] 2.0271 The object is the fixed, the persistent; the configuration is the changing, the unstable. 2.0272 The configuration of objects forms states of affairs.

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2.03

In a state of affairs, the objects hang one in another like the links in a chain.

[Mounce says: A state of affairs, like a chain, is not just a collection, but a collection that holds together in a determinate way. But what holds together the links of a chain? Nothing, except their fitting into one another. Their fitting into one another is how they hold together. The same point applies to the combination of objects in a state of affairs. That they hold together in a determinate way shows something about their logical form. But logical form is not a further fact about them, that which holds them together. (Mounce 1981, 19)

Anscombe argues that “in the elementary proposition there must be nothing corresponding to bracketing” (Anscombe 1971, 37). That is, the meaning of the proposition must be such that it needs no “collecting” or “punctuating” of terms in the way done by brackets. See 5.461–5.4611. Black says the point is that there is nothing else in a fact (a “bond,” say) that holds its components together. See Black 1964, 66.] 2.031

In a state of affairs the objects relate to each other in a definite way.

2.032

The way that the objects hang together in a state of affairs is the structure of the state of affairs.

[See Black 1964, 66, where he compares this with 2.15 and notes the contrast between “form” and “structure.” Higher up that page he doubts whether the distinction is necessary.] 2.033

The form is the possibility of the structure [or: Form is the possibility of structure].

[Ostrow argues that objects are not just form (i.e., possibility of structure), but also content, as is said of substance at 2.025. “It is constitutive of the object to occur in an atomic fact, but not only in this fact. . . . [T]he object is this thing taken against the background of all the rest of its possibilities of combination with other things” (Ostrow 2002, 25). Otherwise the object would “be understood as dissolving simply into a possibility—as if we could understand the condition of the world apart from any consideration of how things actually stand” (ibid.). A little later he says: While we will no doubt be tempted to bring to bear notions like “particular,” “universal,” or “sense datum” to try to make sense of what he has in mind,

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Wittgenstein will not allow us to rely on any such categories as basic, as clarificatory. Indeed, it would appear to be the reverse: rather than seeking to understand objects in terms of some prior philosophical category, the Tractatus is suggesting that it is only through their possibilities of occurrence that those fundamental categories emerge. The object is, we might say, a primitive notion. (Ostrow 2002, 26–27)

Black notes that we cannot ask “Is it possible that . . . ?” in relation to the “possibility” that Wittgenstein refers to here (Black 1964, 67). This makes interpretation problematic.] 2.034 The structure of a fact consists of the structures of the states of affairs. 2.04

The totality of states of affairs that hold is the world.

2.05

The totality of states of affairs that hold also determines which states of affairs do not hold.

[Black says that 1.12 expresses the same thought (see Black 1964, 70). A state of affairs that holds, the realization of that possibility, is a fact. On this, see 2.] 2.06

The holding and non-holding of states of affairs is reality. (We also call the holding of states of affairs a positive fact, their non-holding a negative one.)

2.061

States of affairs are independent of one another.

[Cf. 1.21.] 2.062

The holding or non-holding of a state of affairs cannot be inferred from the holding or non-holding of another.

2.063

The total reality is the world.

[Ostrow: “It is, it would seem, some version of the ancient problem of the nature of ‘what is not’ that confronts us at the close of the 2.06s” (Ostrow 2002, 34). On this, see Anscombe: “cf. Plato’s Theaetetus 189A: ‘In judging, one judges something; in judging something, one judges something real; so in judging something unreal one judges nothing; but judging nothing, one is not judging at all.’ Wittgenstein returned to the problem presented by this argument again and again throughout his life” (Anscombe 1971, 13 footnote).]

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2.1

We make pictures of facts for ourselves.

[Cf. Heinrich Hertz: “We form for ourselves pictures or symbols of external objects . . .” (Hertz 1956, 1). Hertz is concerned with the question how scientists can represent nature in such a way as to allow for making predictions.] 2.11

A picture represents a situation in logical space, the holding and non-holding of states of affairs.

[Ostrow compares this with 1.13 and says that, “From the start, it would seem, the world is understood always against a larger—logical—backdrop of what is not the case” (Ostrow 2002, 35). He also says that it is a mistake to see Wittgenstein (rightly) as rejecting Frege’s idea that a proposition is a kind of name only to then wonder how a picture represents a state of affairs (see Ostrow 2002, 36–37). Facts should not be reified. They are uses of pictures. Ostrow: “positive and negative fact stand on the same level, a contrast between two uses of a picture” (Ostrow 2002, 38). Ostrow also compares this remark with 2.201, 2.202, and 2.203 (see Ostrow 2002, 80–81). His conclusion is that a picture presents [vorstellt] existent and nonexistent atomic facts, and represents [darstellt] a possibility of such facts.] 2.12

A picture is a model of reality.

[Wittgenstein later commented: “I have inherited this concept of a picture from two sides: first from a drawn picture, second from the model of a mathematician, which already is a general concept. For a mathematician talks of picturing in cases where a painter would no longer use this expression” (Wittgenstein 1979a, 185).] 2.13

In a picture, the elements of the picture correspond to objects.

2.131

In a picture, the elements of the picture stand in for objects.

2.14

A picture consists in its elements relating to one another in a specific way.

2.141

A picture is a fact.

2.15

The elements of a picture’s relating to each other in a specific way represents matters relating to each other just so. Let us call this connection of the elements of a picture its structure, and its possibility the form of representation of the picture [i.e., the form of the picture’s picturing].

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[“Form of representation” is what Ogden has, and is accepted by Black, although he prefers “form of depiction.” (See Black 1964, 81.) Black rejects, though, Pears and McGuinness’s “pictorial form.”] 2.151

The form of representation is the possibility that things are related to each other in the same way as the elements of the picture.

2.1511 This is how a picture is tied to reality: it reaches right up to it.

[Cf. Schopenhauer: “Indeed perceptions through sight ultimately refer to touch, and sight can be regarded as an imperfect touch extending to a distance and making use of the rays of light as long feelers” (Schopenhauer 1974, 81).] 2.1512 It is like a ruler laid against reality.

[Ostrow notes that a ruler does not use itself (see Ostrow 2002, 35). We must apply it. He quotes Wittgenstein (1979a, 185) saying to Waismann later that he might as well have called propositions measuring-rods as pictures.] 2.15121 Only the outermost points of the dividing lines [on the ruler] touch the object to be measured. 2.1513 On this view the picturing relation that makes it a picture also belongs to the picture. 2.1514 The picturing relation consists of the coordination of the elements of the picture with the things [pictured]. 2.1515 These coordinations are as it were the feelers of the picture elements, with which the picture touches reality.

[The feelers here are antennae, “the things which a butterfly has” in Wittgenstein’s words (Wittgenstein 1973, 24).] 2.16

A fact must have something in common with what it depicts in order to be a picture.

2.161

In the picture and the depicted there must be something identical so that one can be a picture of the other at all.

[Ostrow says of standard interpretations that: “Wittgenstein’s answer to the question of how the picture—and hence language—can always be about the world is thus supposedly to be: they share a form.” And yet: “the strategy of



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taking recourse in talk of an isomorphism is empty; it amounts to no more than the claim that depicting the world is possible because the world has the possibility of being depicted” (Ostrow 2002, 39).] 2.17

What a picture must have in common with reality in order to be able to picture it in its way – rightly or falsely – is its form of representation.

[See 2.15 and 2.151 on form of representation.] 2.171

A picture can picture every reality whose form it has. A spatial picture [can represent] everything spatial, a color one everything colored, etc.

[McGinn: “To say that there is agreement in form between a picture and reality is, at bottom, to say no more than that there is a rule of projection whereby the picture, which is essentially structured, is projected onto reality, that is, whereby it represents a possible state of affairs whose internal structure is mirrored in the internal structure of the picture that represents it” (McGinn 2006, 93). She here inserts a footnote quoting Philosophical Grammar, where Wittgenstein writes: “what I said really boils down to this: that every projection must have something in common with what is projected no matter what is the method of projection. But that only means that I am here extending the concept of ‘having in common’ and making it equivalent to the general concept of projection” (Wittgenstein 1974b, 162–163).] 2.172

But a picture cannot picture its form of representation; it exhibits it.

2.173

A picture presents its subject from the outside (its standpoint is its form of presentation), which is why pictures present their subjects rightly or falsely.

2.174

But a picture cannot place itself outside its form of representation.

2.18

What every picture, of whatever form, must have in common with reality in order to be able to picture it at all – rightly or falsely – is the logical form, that is, the form of the reality.

[Ostrow (2002, 47), following Burton Dreben, suggests that the point of the notion of logical form is to ease us into the idea that written propositions are pictures of reality, even though they do not appear to be (as 4.011 acknowledges). Otherwise the concept of pictorial form would seem to have been pointless—Wittgenstein could just have referred to logical form throughout. Ostrow rejects Eli Friedlander’s suggestion that logical form is more general

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and additional to the pictorial form, on the grounds that the nature of the generalization in question would be unclear, and (see Ostrow 2002, 144–145 note 2) that 2.181 implies that it makes sense to speak of cases in which the pictorial form is the logical form, whereas Friedlander treats them as corresponding to very different dimensions of the picture.] 2.181

If the form of representation is the logical form, then the picture is called a logical picture.

2.182 Every picture is also a logical picture. (On the other hand, for instance, not every picture is a spatial one.) 2.19

A logical picture can depict the world.

2.2

A picture has the logical form of representation in common with what it pictures.

2.201

A picture pictures reality by presenting a possibility of the holding and non-holding of states of affairs.

[Chon Tejedor comments: “Propositions, thoughts and other pictures are both bivalent and bipolar: they are determinately either true or false and they are both capable of being true and capable of being false. A picture is true when the possible state it represents obtains and false when the possible state it represents fails to obtain (TLP 2.201, TLP 2.221, TLP 2.222)” (Tejedor 2015, 18).] 2.202

A picture presents a possible situation in logical space.

2.203

A picture contains the possibility of the situation that it presents.

2.21

A picture either agrees with reality or does not; it is correct or incorrect, true or false.

2.22

A picture presents what it presents, independently of its truth or falsehood, by its form of representation.

2.221

What a picture presents is its sense.

2.222 Its truth or falsehood consists in the agreement or disagreement of its sense with reality.

[This contrasts, says Proops, with Russell’s view that truth and falsity are indefinable properties belonging to propositions, the latter being complexes

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which he found it difficult to define (see “The Nature of Truth” in Russell 1994, quoted in Proops 2000, 59–60 note 157).] 2.223

In order to know whether a picture is true or false we must compare it with reality.

2.224

From the picture alone nothing can be known about whether it is true or false.

2.225 An a priori true picture does not exist. 3

A logical picture of facts is a thought.

[A. W. Moore says: There is a tripartite classification that is standardly attributed to the early Wittgenstein. According to this attribution, Wittgenstein acknowledges the following three mutually disjoint categories in the Tractatus: • thoughts; •  tautologies and contradictions; •  nonsensical pseudo-propositions. Thoughts are propositions with a sense, in other words propositions that are bipolar, or in yet other words, propositions that are not only true or false but also such that, if true, they could nevertheless have been false, while, if false, they could nevertheless have been true. (See e.g., TLP 2.2–3, 4, and 4.2.) (Moore 2020, 27)] 3.001

“A state of affairs is thinkable” means: we can imagine it [literally: we can make [for] ourselves a picture of it].

[Wittgenstein says that there is meant to be a kind of pun here, which is why he uses “imagine” for the English translation, since “imagine” comes from “image.” (See Wittgenstein 1973, 24.)] 3.01

The totality of true thoughts is a picture of the world.

3.02

A thought contains the possibility of the situation that it thinks. What is thinkable is also possible.

3.03

We cannot think anything illogical, because we would then have to think illogically.

[Cf. 5.4731 on the impossibility of illogical thought.]

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It used to be said that God could create everything, only nothing that would be contrary to the laws of logic. – We could not say of an “illogical” world how it would look.

[McManus cites the first sentence of this remark as an example (others are in 3.323, 4.002, 4.003, and 5.02) of straightforwardly empirical claims that could not possibly be interpreted as nonsensical (as 6.54 might seem to suggest they should be), even if they are false. See McManus 2006, 59 note 24.] 3.032

One can no more in language present “the logically contradictory” than one can in geometry present via its coordinates a figure that contradicts the laws of space; or give the coordinates of a point that does not exist.

3.0321 We could well present spatially a state of affairs that went against the laws of physics, but not one that went against the laws of geometry. 3.04 An a priori correct thought would be one such that its possibility implied its truth. 3.05

We could only know a priori that a thought were true if its truth could be known from the thought itself (without any object of comparison).

3.1

In a sentence a thought is expressed perceptibly.

[Ogden’s “through the senses” is obscure, it seems to me. Pears and McGuinness have “an expression that can be perceived by the senses.” Black notes that this is “somewhat laboured,” and suggests, “In a sentence the thought expresses itself perceptibly” (see Black 1964, 99). Cf. Frege in “Thought”: “The thought, in itself imperceptible by the senses, gets clothed in the perceptible garb of a sentence, and thereby we are enabled to grasp it. We say a sentence expresses a thought” (Beaney 1997, 328).] 3.11

We use the physically perceptible sign (audible or written, etc.) of the proposition as a projection of a possible situation. The method of projection is the thinking of the proposition’s sense.

[Anscombe says that “Wittgenstein’s use of ‘projection’ is a metaphorical extension of the mathematical use, which may be explained thus: ‘The drawing of straight lines through every point of a given figure, so as to produce a



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new figure each point of which corresponds to a point of the original figure’” (Anscombe 1971, 69 note 1). N. K. Verbin comments: The German can be understood either as explaining what the method of projection is, namely thinking the sense of the proposition, or as explaining what thinking the sense of the proposition is, namely the method of projection, but without specifying what this method is. Given Wittgenstein’s comments on use, given his account of judgement statements and his comments on the metaphysical subject, given his comment that “there is a general rule by which the musician is able to read the symphony out of the score. . . . And the rule is the law of projection . . .” (T 4.0141), I think that we have a strong case for reading proposition 3.11 of the Tractatus according to its latter interpretation. (Verbin 2000, 5)

(For Peter Winch’s influential view on this, with which Verbin is in sympathy, see “Language, Thought and World in Wittgenstein’s Tractatus” in Winch 1987, 3–18.) The “mentalistic” reading of this passage takes it to be explaining what the method of projection is, namely, a mental act or thinking. Apart from the points made by Verbin, other problems with this view are that Wittgenstein explicitly denies that psychology has any special connection to philosophy in 4.1121 (although that passage is ambiguous), and that how thinking would give meaning to words is mysterious. On this see Schroeder 2006 (61 especially), who nonetheless supports a mentalistic reading of the Tractatus. Hacker refers to Winch’s rejection of the mentalistic reading as “ingenious, but mistaken” (Hacker 1999, 128). This, he says, is for three reasons: it is a forced reading of the German, it leaves the method of projection unexplained, and it fails to take into account Prototractatus 3.12 and 3.13, which identify the method of projection with the manner of applying the propositional sign, and applying the propositional sign as thinking its sense. The mentalist thesis is supported also, according to its proponents, by the following passage from a letter written by Wittgenstein to Russell (Wittgenstein 1974a, 125): I don’t know what the constituents of a thought are but I know that it must have such constituents which correspond to the words of Language. Again the kind of relation of the constituents of thought and of the pictured fact is irrelevant. It would be a matter of psychology to find it out. . . . Does a Gedanke consist of words? No! But of psychical constituents that have the same sort of relation to reality as words. What those constituents are I don’t know.

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Against those who interpret 3.11 as Hacker and Norman Malcolm do, Cora Diamond writes: “If the meaningfulness of sentences were mediated by intrinsically meaningful thoughts, as on Malcolm’s view, the elements of those thoughts would not have the same relation to reality as do words. The letter [quoted above] would appear to rule out any view like Malcolm’s” (Diamond 2006, 148). Diamond notes also that Winch reads the very passage from the Prototractatus that Hacker cites as supporting his, opposite, interpretation of Tractatus 3.11. She rejects both Winch’s view that the notion of a method of projection is not being explained in 3.11 and Hacker’s view that thinking a sense is something one does in one’s mind. Instead, she sees Wittgenstein’s idea thus: we make pictures, using methods of depiction in space; these pictures, these representations, in that they are in logical space, are thoughts. In that they are thoughts, they think this or that situation; they think this or that sense. In that they are pictures in a space, the possibility of the representing picture in the space has internal to it the possibility of the represented situation in that space. The logical notion of depiction then explains (in [. . .] 3.12 and 3.13) what Wittgenstein means by the application of the propositional sign: it is used as a picture, and thereby as a projection. The sort of projection involved in our use of propositions is thus tied to the notion of picturing, which itself is a basically projective notion: to use a perceptible sign as a picture is to use it as a projection of a possible situation. (Diamond 2006, 155–156)

The idea of a thought thinking, which sounds odd, and which Diamond treats Hacker as having neglected, comes from 3.02.] 3.12

I call the sign through which we express a thought a propositional sign. And a proposition is a propositional sign in its projective relation to the world.

[Schroeder, giving voice to the mentalistic view, explains Wittgenstein’s distinction here in this way: “A propositional sign is a declarative sentence. A proposition is not an entity distinct from the propositional sign: rather, it is the propositional sign plus something else that makes it meaningful: namely, a mental act of thinking, or meaning” (Schroeder 2006, 60).] 3.13

To a sentence belongs all that belongs to the projection, but not what is projected. Thus the possibility of what is projected [belongs to it], but not it itself. Its sense is therefore not yet contained in a sentence, but the possibility of expressing it is.

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(“The content of a sentence” means the content of a senseful [sinn­ vollen] sentence.) The form of its sense is contained in a sentence, but not its content. 3.14

A propositional sign consists in its elements, the words, relating to each other in a definite way. A propositional sign is a fact.

3.141

A proposition is not a mixture of words. – (In the same way that a musical theme is not a mixture of notes.) A proposition is articulated.

[Wittgenstein says that the main point is that a proposition is a structure, not a mixture (see Wittgenstein 1973, 24). Zalabardo expands on this idea: A proposition, for Wittgenstein, is not a composite item arising from the combination of its constituents. The unity of a proposition is fundamental, not the result of a process of composition. Hence what we think of as the constituents of a proposition are not related to it as components to compound. They are instead features that the proposition has in common with other propositions. (Zalabardo 2015, 115)] 3.142

Only facts can express a sense, a set of names cannot.

3.143

The usual form of expression in writing or printing disguises a propositional sign’s being a fact. Because in a printed sentence, e.g., no essential difference appears between a propositional sign and a word. (This is how it was possible for Frege to call a sentence a complex name.)

3.1431 The essence of a propositional sign becomes very clear if we think of it as made up of spatial objects (such as tables, chairs, books) instead of written signs. The reciprocal spatial position of these things then expresses the sense of the proposition.

[White: The [. . .] two paragraphs—3.143 and 3.1431—are far from helpful. In fact, although one can see what led Wittgenstein to say what he says here, the thought is a bad one. There is no mode of expression that would obviate the

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potential confusion between viewing the propositional sign as a complex object and viewing it as a fact. There is no good reason to suppose that if we used bits of furniture to form propositional signs, people might not take the propositional sign to be the complex object whose parts were those bits of furniture. (White 2006, 53)] 3.1432 Not: “The complex sign ‘aRb’ says that a stands in relation R to b” but rather: That “a” stands in a certain relation to “b” says that aRb.

[Mounce: “In other words, the relation between a proposition and its sense is an internal one. The sense of a proposition is to be found in an arrangement of physical signs; it is not to be found in something that corresponds to that arrangement, some entity over and above it, whether in the empirical or some quasi-empirical world” (Mounce 1981, 25). Fahrnkopf discusses a nominalistic interpretation of this passage and a realistic one. Nominalist readings (e.g., Anscombe’s) take the key point to be that “R” would have no place in an ideal symbolism. Thus relations are not real, and whatever “aRb” tells us might just as well be expressed by, say, “ab” or “ba.” Fahrnkopf writes: according to Wittgenstein’s decimal notation, 3.1432 is a comment on 3.143, and this latter passage is concerned only to make the point that a propositional sign is a fact, not a name; this is also the context of the remark in the “Notes on Logic” which corresponds to 3.1432. On my interpretation, then, the purpose of 3.1432 is only to contrast symbolizing facts with names, and the nominalist tone of this passage—which could have been avoided altogether had Wittgenstein specified that the relation in which “a” stands to “b” consist in their respective relations to “R”—is in any case minimized by the realization that the status of “R” as a name is implied in many other contexts in the Tractatus. (Fahrnkopf 1988, 29)] 3.144

One can describe situations, but not name them. (Names are like points, whereas propositions, having sense, are like arrows.)

[Schroeder: “Both the German word Sinn and the English equivalent ‘sense’ can also mean direction (as in ‘sense of rotation’)” (Schroeder 2006, 55).] 3.2

In propositions thoughts can be so expressed that the objects of the thought match the elements of the propositional sign.

3.201

I call these elements “simple signs” and the proposition “completely analyzed.”

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[Black says that, since we have no way to know when we have reached a complete analysis, this remark does not usefully define “simple sign” (see Black 1964, 108).] 3.202

The simple signs used in a proposition are called names.

3.203

A name means an object. The object is its meaning. (“A” is the same sign as “A.”)

[See Wittgenstein 1953 §39 and §40, where he seems to be criticizing the idea put forward here.] 3.21

The configuration of the simple signs in a propositional sign corresponds to the configuration of objects in a situation.

3.22

In a proposition a name stands for an object.

3.221

I can merely name objects. Signs stand for them. I can only speak of them, I cannot express them. A proposition can only say how a thing is, not what it is.

3.23

The requirement of the possibility of simple signs is the requirement of the definiteness of sense.

[This connects with Frege. See Frege 1903, vol. 2 §56 and §62. Bearn 1997 discusses this on 51–53. Thomas Ricketts: Many commentators have been puzzled by Frege’s doctrine that concepts must have sharp boundaries, that a concept is determinately true or false of each object. This doctrine is, however, an immediate consequence of Frege’s views of first-level variables as unrestricted over objects. As Frege understands it, the logical law, Either Fx or not Fx, refutes the existence of a concept that lacks sharp boundaries. (Ricketts 1985, 6)] 3.24

A proposition that deals with a complex stands in an internal relation to a proposition that deals with a component of the complex. A complex can only be given through its description, and this will match it or not match it. A proposition, in which there is mention of a complex, will, if this complex does not exist, be not nonsensical [unsinnig] but simply false. One can see that a propositional element signifies a complex by a vagueness in the propositions in which it occurs. We know by this

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proposition that something is not yet definite [determinate]. (The notation for generality indeed contains a prototype.) The abbreviation of the symbol of a complex in the form of a simple symbol can be expressed by a definition.

[See 4.123 for the meaning of “internal.” See 4.241 for the meaning of “definition.” Ishiguro criticizes Wittgenstein’s talking about complex objects in the same way as he talks about facts here, because “although the identity of a fact cannot be settled except by settling the identity of the proposition which describes it, the identity of complex objects such as General de Gaulle does not depend on our articulating any one particular description” (Ishiguro 1969, 39). White: “Proposition 3.24 is extraordinarily compressed, with considerable unclarity as to what Wittgenstein meant by ‘determinacy of sense,’ or why he is demanding it” (White 2006, 21). White suggests reading Wittgenstein 1979b, 59–71, to shed light on what is going on here, “without necessarily maintaining that Wittgenstein would subscribe to the detail of what he was saying in these notes.” Determinacy, White says, means specificity. The world cannot be indefinite, even though language can be.] 3.25

There is one and only one complete analysis of a proposition.

3.251

A proposition expresses what it expresses in a definite, clearly specifiable way: a proposition is articulate.

3.26

A name cannot be analyzed further by a definition: it is a primitive sign.

3.261

Every defined sign signifies via the signs through which it can be defined; and the definitions show [weisen] the way. Two signs, one primitive and one defined by primitive signs, cannot signify in the same way. One cannot analyze names through definitions. (Nor any sign that has meaning on its own, independently.)

[The “Nor any” in the last sentence is Wittgenstein’s translation. See Wittgenstein 1973, 59.] 3.262

A sign’s application shows [zeigt] whatever is not expressed in the sign itself. What signs slur over, their application speaks out.

[James Conant argues that the distinction between zeigen and erläutern is important (see Conant 2005, 82 note 49). The former applies only to meaningful

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propositions, while the second can apply to nonsense. I use “show” only for zeigen, i.e., not for erläutern. The application of a sign is here linked with its meaning. Schopenhauer talks of the different applications of the principle of sufficient reason and says that the principle acquires a different meaning in each such application (see Schopenhauer 1974, 2).] 3.263

The meanings of primitive signs can be explained through elucidations. Elucidations [Erläuterungen] are propositions which contain primitive signs. They can thus only be understood if one is already acquainted with the meanings of these signs.

[White suggests that “Erläuterungen” be translated as “illustrative examples” (White 2006, 61). To teach someone the meaning of a name we must use the name in sample sentences, since merely pointing to the object does not define how the name is to be applied. “We must then leave it to chance whether the other catches on to the meanings of those sentences, which is something that can only be done by grasping the meaning of the name” (White 2006, 61). Hacker argues that Tractarian elucidations (Erläuterungen) are confusedly meant to be both ostensive definitions and true assertions about the world (see Hacker 1975). Anscombe suggests that this passage, along with 3.261, provides the best evidence for thinking that the elementary propositions of the Tractatus are simple observation statements, such as “This is a red patch” (see Anscombe 1971, 26). Names and only names are primitive signs. Logical signs, as Wittgenstein indicates elsewhere, are not primitive signs. But (see Anscombe 1971, 27) what elucidates a name need not be an elementary proposition. And from 6.3751 it follows directly that “This is a red patch” cannot be an elementary proposition. Anscombe concludes that elementary propositions are not simple observation statements, and that this explains why Wittgenstein did not refer to observation in connection with them. What they are he cannot say, but they must exist. See, for instance, 5.5562, 3.23, 2.021, 2.0211, and 4.221.] 3.3

Only a proposition has sense; only in the context of a proposition does a name have meaning.

[Wittgenstein here echoes Frege in Foundations §62 (Frege 1980a). Conant writes in connection with 3.3–3.322 that: The methodological import of Frege’s three principles is developed in the Tractatus through the claim that in ordinary language it is often the case that

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the same sign symbolizes in different ways. The distinction between sign [Zeichen] and symbol [Symbol] which this claim presupposes might be summarized as follows: * sign—an orthographic unit, that which the perceptible expressions for propositions have in common (a sign design, inscription, icon, grapheme, etc.) * symbol—a logical unit, that which meaningful propositions have in common (i.e., an item belonging to a given logical category: proper name, first-level function, etc.). (Conant 1998, 235–236)

The three principles that Wittgenstein is thought to have picked up from Frege are to separate the psychological from the logical, to ask for the meaning of a word only in the context of a proposition, and to distinguish concepts from objects. Cf. Schopenhauer: “It is like a word of two meanings; only from the context can we infer what is meant” (Schopenhauer 1974, 95).] 3.31

Every part of a proposition that characterizes its sense I call an expression (a symbol). (The proposition itself is an expression.) The expression is all that is essential for the sense of a proposition that propositions can have in common with each other. An expression marks a form and a content.

3.311

An expression presupposes the forms of all propositions in which it can occur. It is the common characteristic feature of a class of propositions.

3.312

It is thus presented by way of the general form of the propositions that it characterizes. Moreover in this form the expression is constant and everything else is variable.

3.313

An expression is thus presented by way of a variable whose values are the propositions that contain the expression. (In the limiting case the variables become constants, the expression a proposition.) I call such a variable a “propositional variable.”

3.314

An expression has meaning only in a proposition. Every variable can be taken as a propositional variable. (Even a variable name.)

3.315

If we convert a component of a proposition into a variable, then there is a class of propositions which are all the values of the resulting variable proposition. This class still depends in general on what we,

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by arbitrary agreement, mean by the parts of that proposition. But if we convert into variables all those signs whose meaning is arbitrarily determined then a class like this will still always remain. This however is now dependent on no agreement, but only on the nature of the proposition. It corresponds to a logical form – a logical prototype. 3.316

What values a propositional variable may accept is fixed. The fixing of the values is the variable.

3.317

Fixing the values of a propositional variable is specifying the proposi­ tions whose common characteristic the variable is. The fixing is a description of these propositions. The fixing will therefore deal only with symbols, not with their meaning [Bedeuting]. And the only thing essential to the fixing is that it is only a descrip­ tion of symbols and says nothing about the symbolized. How the description of the propositions occurs is unessential.

3.318

Like Frege and Russell, I take a proposition to be a function of the expressions contained in it.

3.32

A sign is what is sensibly perceptible of a symbol.

[See comment on 3.3.] 3.321

Two different symbols can thus have the same sign (written or audible etc.) in common with one another – they then signify in different ways.

3.322

A common characteristic of two objects can never be indicated by our symbolizing them with the same signs, but by two different ways of symbolizing. Because the sign is indeed arbitrary. One could thus also choose two different signs and where would then be what was common in the symbolization?

3.323

In colloquial language it is common for the same word to signify in different ways – and thus belong to different symbols –, or for two words, that signify in different ways, to be applied in a proposition in ways that are the same externally. Thus the word “is” appears as the copula, as the sign of equality, and as the expression for existence; “to exist” as an intransitive verb like “to go”; “identical” as an adjective; we speak about something [an object], but also about something happening [an event]. (In the proposition “Green is green” – where the first word is a person’s name and the last is an adjective – these words do not simply have different meanings but they are different symbols.)

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[Russell: “The is of ‘Socrates is human’ expresses the relation of subject and predicate; the is of ‘Socrates is a man’ expresses identity. It is a disgrace to the human race that it has chosen to employ the same word ‘is’ for these two entirely different ideas—a disgrace which a symbolic logical language of course remedies” (Russell 1919, 172). See also Wittgenstein 1953 §558 and §561, where different uses of the word “is” are discussed. Conant comments: It is perhaps worth elaborating how Wittgenstein’s example in the last paragraph of §3.323 illustrates the point of the first paragraph of §3.323. The propositional sign “Green is green” can be understood to symbolize in three different ways— and hence can be understood as an expression for any one of three different thoughts. One way of noticing how the same sign symbolizes differently in each of these three cases is to focus on the word “is.” In each of the propositions which expresses each of these three different thoughts, the sign “is” symbolizes a different logical relation. In one, the sign “is” symbolizes the copula (a relation between a concept and an object); in another, we have the “is” of identity (a relation between objects); in the third, we have the “is” of co-extensionality (a relation between concepts). The point of the example is to show us that we cannot gather merely from the notation of ordinary language how the sign “is” is symbolizing in a given instance. (Conant 1998, 237)] 3.324

Thus the most fundamental confusions (of which the whole of philosophy is full) easily arise.

3.325 In order to avoid such errors, we must use a symbolism that excludes them by not using the same sign for different symbols and by not using signs that signify in different ways in what appears to be the same way. A symbolism then that obeys logical grammar – logical syntax. (The concept-script of Frege and Russell is one such language, though admittedly it does not yet exclude all errors.)

[I translate Zeichensprache here and elsewhere as symbolism, following Wittgenstein’s request to Ogden that he translate the word that way (see Wittgenstein 1973, 25).] 3.326

In order to recognize the symbol in the sign one must look to the senseful [sinnvollen] use.

[In Letters to Ogden, Wittgenstein writes: “The meaning of this prop[osition] is: that in order to recognize the symbol in a sign we must look at how this sign is used significantly in propositions. I.e., we must observe how the sign

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is used in accordance with the laws of logical syntax. Thus ‘significant’ here means as much as ‘syntactically correct’” (Wittgenstein 1973, 59).] 3.327

A sign determines a logical form only together with its logico-syntactical use.

3.328

If a sign is not used then it is meaningless. That is the point [Sinn] of Occam’s razor. (If everything behaves as if a sign had meaning, then it has meaning.)

[Ogden has “not necessary” and Pears and McGuinness have “useless” for nicht gebraucht. Black points out that the true meaning is “not used” (Black 1964, 134). Michael Kremer comments: On the resolute reading, as I understand it, Wittgenstein’s view of meaning, sense and nonsense in the Tractatus is simply this: meaningful linguistic expressions are those that have a use in the language. The most basic use which we make of language is to say something; expressions that have the same use, or can be used to say the same things, have the same meaning, while expressions that have no use in saying things are meaningless (3.328, 5.47321). (Kremer 2001, 41)] 3.33

In logical syntax the meaning of a sign should never play a role; it must be able to be established without anything thereby being said of the meaning of a sign, only the description of the expressions being presupposed.

[McManus calls this and the following remark about Russell obscure. He writes: “That we should simply strive to make more apparent the difference between different symbols, rather than (per impossibile) stating what the difference between the symbols is (for instance, by discussing the different kinds of things to which they refer), helps explain” this criticism of Russell (McManus 2006, 77).] 3.331

From this remark we get a comprehensive view of Russell’s “Theory of types”: Russell’s error is shown by his having to speak of the meaning of a sign when putting together his rules for signs.

[Black suggests “get a comprehensive view of” for sehen wir in [. . .] hinüber, while Pears and McGuinness have “turn to” and Ogden has “get a further view – into” (see Black 1964, 146). “Hinüber” means “over” and “in” means “in,” so we are seeing into Russell’s theory, getting insight, but also getting

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an overview, as Black suggests, looking over the whole thing (but at it, not to something on the other side). For more on the theory of types, see the introduction. Russell himself was not completely happy with it: “Now the theory of types emphatically does not belong to the finished and certain part of our subject: much of this theory is still inchoate, confused, and obscure. But the need of some doctrine of types is less doubtful than the precise form the doctrine should take” (Russell 1919, 135). Classes are logical fictions, and if they are treated as being real objects, whose names have real signification, then the sentences in which they are treated this way will be devoid of meaning. “The supposition that a class is, or that it is not, a member of itself is meaningless in just this way” (Russell 1919, 137). F. P. Ramsey, following Wittgenstein, objects to this theory. Propositional functions are symbols, while individuals are objects. So talk of functions of functions is not like talk of functions of individuals: For the range of values of a function of individuals is definitely fixed by the range of individuals, an objective totality which there is no [getting?] away from. But the range of arguments to a function of functions is a range of symbols, all symbols which become propositions by inserting in them the name of an individual. And this range of symbols, actual or possible, is not objectively fixed, but depends on our methods of constructing them and requires more precise definition. (Ramsey 1968, 358)

White 2006, 9, says that Wittgenstein explained his objection to Russell more clearly in Philosophical Remarks: “Grammatical conventions cannot be justified by describing what is represented. Any such description already presupposes the grammatical rules. That is to say, if anything is to count as nonsense in the grammar that is to be justified, then it cannot at the same time pass for sense in the grammar of the propositions that justify it (etc.)” (Wittgenstein 1975, 55).] 3.332

No proposition can express something about itself, because a propositional sign cannot be contained in itself (this is the whole “Theory of types”).

3.333

A function can therefore not be its own argument, because a functional sign already contains the prototype of its argument and it cannot contain itself. Let us suppose that the function F(fx) could be its own argument; there would then therefore be a proposition: “F(F(fx))” and in this the outer function F and the inner function F must have different

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meanings, because the inner has the form φ(fx), the outer the form ψ(φ(fx)). Only the letter “F” is common to both functions, but that in itself signifies nothing. This becomes clear immediately if instead of “F(F(u))” we write “(∃φ) : F(φu) . ψu = Fu.” Thus Russell’s Paradox is laid to rest.

[Black says of this reference to Russell’s paradox that Wittgenstein “presumably [means] the variant concerning functions (rather than classes) that are not themselves included in the set of their own values” (Black 1964, 149). Ostrow compares this remark with 3.1432: “Wittgenstein’s aim is once more to bring out how our hold on a notion of logical form is parasitic on how we speak, on what it makes sense to say” (Ostrow 2002, 67). Russell’s belief that a theory of types is needed suggests that he confusedly thinks something like the opposite of this, Ostrow believes. Peter M. Sullivan comments: The appearance of paradox arises when we assign to the inner function “F(fx)” the meaning that fx does not apply to itself, and assume that by that stipulation “F(F(fx)” makes the paradoxical assertion that not applying to itself does not apply to itself. The appearance dissolves when we realize that our stipulation about the inner “F ” is silent about the outer “F,” which for all we’ve said might mean anything or nothing. (Sullivan 2000, 183–184)] 3.334

The rules of logical syntax must be self-evident if one only knows how each one of the signs signifies.

[Winch: “It is important that Wittgenstein writes wie (‘how’) rather than was (‘What’). The what will already have been settled once the how is established” (Winch 1987, 9).] 3.34

A proposition possesses essential and accidental features. The accidental are those features that come from the particular way of producing the propositional sign. The essential are those which alone enable the proposition to express its sense.

3.341

The essential in a proposition is thus that which all propositions that can express the same sense have in common. And likewise in general the essential in a symbol is that which all symbols that can fulfill the same purpose have in common.

3.3411 Thus one could say: The real name is that which all symbols that signify an object have in common. It would then follow that no composition at all is essential for a name.

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[Diamond says that the word eigentliche (“real”) here and in 4.1272 should be translated the same way each time, which is not the case in other translations. See Diamond 2019, 140 note 35.] 3.342

In our notations there is indeed something arbitrary, but this is not arbitrary: if we have determined something arbitrarily then something else must be the case. (This stems from the essence of the notation.)

3.3421 A particular way of symbolizing may be unimportant, but it is always important that this is a possible way of symbolizing. And it is like this in philosophy generally: the particular proves unimportant time and again, but the possibility of each particular gives us an insight into the essence of the world. 3.343

Definitions are rules for translation from one language into another. Every correct symbolism must allow of translation into every other by means of such rules: This is what they must all have in common.

3.344

That which signifies in a symbol is the common feature of all symbols that can take its place following the rules of logical syntax.

3.3441 One can, e.g., express the common feature of all notations for truthfunctions thus: It is common to them that they all – e.g., – can be replaced by the notation “~p” (“not p”) and “p ∨ q” (“p or q”). (This shows the way that a specific possible notation can give us general insight.) 3.3442 The sign for a complex is not arbitrarily resolved by analysis in such a way that its resolution would be different in each sentence structure. 3.4

A proposition determines a place in logical space. The existence of this logical place is guaranteed by the existence of the constituent parts alone, by the existence of the senseful proposition.

3.41

The propositional sign and the logical coordinates: That is the logical place.

3.411

Geometrical and logical place agree in that both are possibilities of existence.

3.42

Although a proposition may determine only one place in logical space, at the same time the whole of logical space must already be given by it.



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(Otherwise negation, logical sum, logical product, etc. would introduce ever new elements – in coordination.) (The logical scaffolding around a picture reaches through the whole logical space. The proposition reaches through the whole logical space.) 3.5

An applied, thought, propositional sign is a thought.

4

A thought is a senseful proposition.

4.001

The totality of sentences is language.

4.002 Humanity possesses the ability to construct languages, whereby every sense can be expressed, without having any inkling how and what each word means. – As one speaks without knowing how the particular sounds are produced. Ordinary language is a part of the human organism and not less complicated than it. It is humanly impossible to gather the logic of language immediately from it. Language disguises thought. Indeed so much so that from the outer form of the clothes one cannot infer the form of the thoughts they clothe; because the outer form of the clothes is made for a wholly different purpose than to let the form of the body be known. The unspoken, silent agreements for understanding ordinary language are enormously complicated.

[Russell on ordinary language: “It is exceedingly difficult to make this point clear as long as one adheres to ordinary language, because ordinary language is rooted in a certain feeling about logic, a certain feeling that our primeval ancestors had, and as long as you keep to ordinary language you find it very difficult to get away from the bias which is imposed upon you by language” (Russell 1986, 205). Frege says his Begriffsschrift is a tool invented for “certain scientific purposes” and that it ought not to be condemned “because it is not suited to others.” He makes no claim, then, that it is better than ordinary language for ordinary purposes. (See Joan Weiner in Reck 2002, 156. Weiner gives the reference to Frege as Begriffsschrift, xi.)] 4.003

Most sentences and questions that have been written about philosophical things are not false but rather nonsensical. So we cannot answer questions of this kind at all, but only ascertain their nonsensicality. Most questions and propositions of philosophers are based on our not understanding the logic of our language.

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(They are of the same kind as the question whether the good is more or less identical than the beautiful.) And it is not surprising that the deepest problems are really no problems.

[Schopenhauer, in the midst of something of a rant about the state of German philosophy, which he regards as dishonest, pretentious, and empty, says that: Moreover, “the Good, the True, and the Beautiful” are much in favour, especially with the sentimental and tender-hearted, as pretended Ideas, although they are simply three very wide and abstract concepts, in that they are drawn from innumerable things and relations, and are consequently very poor in substance, like a thousand other abstracta of a similar kind. (Schopenhauer 1974, 169)] 4.0031 All philosophy is “critique of language.” (Though not in Mauthner’s sense.) Russell’s merit is to have shown that the apparent logical form of a proposition need not be its true form.

[Fritz Mauthner was skeptical about the ability of language to convey truth because, he thought, it only pictures reality, never actually coinciding with nature (see Nordmann 2005, 117–121). For Mauthner, language is conventional and based on metaphor, so it can never really grasp the real world (see Stokhof 2002, 25–27).] 4.01

A proposition is a picture of reality. A proposition is a model of reality as we think it is.

[“Wirklichkeit” might equally be translated as truth here, but standardly isn’t. Its meaning is very close to the English “actuality.”] 4.011

At first glance a sentence – as it exists printed on paper perhaps – seems not to be a picture of the reality with which it deals. But so too do written notes seem at first glance not to be a picture of music, nor our written signs for sounds (letters) to be a picture of our spoken language. And yet these symbolisms prove to be pictures – even in the ordinary sense of the word – of what they present.

[Black says that the bit about pictures even in the ordinary sense “can hardly be defended” (Black 1964, 163). On the same page, above this, he quotes G. E. Moore to the effect that Wittgenstein admitted that when he wrote the TLP he had not noticed that the word “pictures” was vague, but that, nevertheless, “he still . . . thought it ‘useful to say “A proposition is a picture



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or something like one”’ although . . . he was willing to admit that to call a proposition a ‘picture’ was misleading; that propositions are not pictures ‘in any ordinary sense’” (Moore 1959, 263). According to Moore, Wittgenstein said that he merely wished to stress a similarity between the grammar or use of “proposition” and that of “picture.” Frascolla also notes the perplexing nature of what Wittgenstein writes here and in 4.012 (see Frascolla 2007, 17–18).] 4.012

It is obvious that we perceive a proposition of the form “aRb” as a picture. Here the sign is obviously a likeness of the signified.

4.013

And if we delve into the essence of this imagery then we see that it is not disturbed by apparent irregularities (like the use of ♯ and ♭ in musical notation). Because even these irregularities picture what they are meant to express; only in another way.

[“Pictoriality” is suggested by Black for Bildhaftigkeit (see Black 1964, 163). Ogden has “pictorial nature,” Pears and McGuinness have “pictorial character.” I use “imagery” because that is how Wittgenstein translated the same word in 4.015.] 4.014 A gramophone record, a musical thought, musical notation, and sound waves, all stand to one another in that internal picturing relation that holds between language and world. The logical form is common to all of them. (As in the fairy tale with the two youths, their two horses and their lilies. They are all in a certain sense one.)

[“Logical form” is suggested by Black for logische Bau, which literally means something more like Ogden’s “logical structure” (see Black 1964, 163). Since Wittgenstein contrasts structure and form, and refers to logische Bau nowhere but here, Black argues that “form” is more appropriate than “structure” in this case. The fairy tale in question appears to be the story “Golden Children” by the brothers Grimm. (See Nordmann 2005, 114 note 47, and Hacker 1997, 75 note 26.) In this story, Nordmann says, “two youths, two horses, and two lilies mirror each other and yet, in a fairy-tale sense, are ‘literally’ one.” Engelmann links this to the equivalences he identifies in propositions 1, 2, 3, 4, 5, and 6 (see comment on 1 and Engelmann 2018, 463).] 4.0141 In the fact that there is a general rule by which the musician is able to read the symphony out of the score, and that there is a rule by which

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one could reconstruct the symphony from the line on a gramophone record and from this again – by means of the first rule – construct the score, herein lies the internal similarity between these things which at first sight seem to be entirely different. And the rule is the law of projection which projects the symphony into the language of the musical score. It is the rule of translation of this language into the language of the gramophone record.

[This is all Wittgenstein’s translation. (See Wittgenstein 1973, 26.) Cf. 3.11 on projection.] 4.015

The possibility of all similes, of all the imagery of our language, rests on the logic of picturing.

[The clause “of all the imagery of our language” is Wittgenstein’s translation.] 4.016

In order to understand the essence of the proposition, let us consider hieroglyphic writing, which depicts the facts it describes. And from it came the alphabet, without losing the essence of picturing.

[For clarification of the nature of hieroglyphic writing see Jespersen and Reintges 2008.] 4.02

We see this from our understanding the sense of a proposition without its being explained to us.

[Proops argues that Wittgenstein is referring back to 4.01 here, when he says “this.” (See Proops 2000, 103–105.) White confirms this, noting that the numbering system of the Tractatus means that 4.02 should be read as following 4.01, not 4.016. (See White 2006, 17.)] 4.021

A proposition is a picture of reality: Because I am acquainted with the situation it presents if I understand the proposition. And I understand the proposition without its sense having been explained to me.

4.022

A proposition shows [zeigt] its sense. A proposition shows what is the case if it is true. And it says that this is the case.

[Kremer: “when a proposition ‘shows its sense,’ what is shown is how we are to go on using it. To ‘grasp’ what is shown is to possess an ability, an instance of knowledge-how” (Kremer 2001, 62).]

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4.023

Reality must be fixed by a proposition except for a yes or a no. Therefore it must describe reality completely. A proposition is a description of a state of affairs. As the description of an object goes by its external properties, so a proposition describes reality according to its internal properties. A proposition constructs a world with the help of a logical scaffolding and therefore one can actually see in the proposition all the logical features of reality if it is true. One can draw conclusions from a false proposition.

4.024

To understand a proposition means to know what is the case if it is true. (One can therefore understand it without knowing whether it is true.) One understands it if one understands its constituent parts.

4.025

The translation of one language into another does not proceed by one’s translating each sentence of one into a sentence of the other, but only by translating the constituent parts of sentences. (And the dictionary translates not only substantives but also verbs, adjectives, and conjunctions, etc.; and it treats them all the same way.)

[Cf. Wittgenstein 1953 §§1–23, a series of remarks that ends with the observation that: It is interesting to compare the multiplicity of the tools in language and of the ways they are used, the multiplicity of kinds of word and sentence, with what logicians have said about the structure of language. (Including the author of the Tractatus Logico-Philosophicus.)] 4.026

The meanings of simple signs (words) must be explained to us for us to understand them. We make ourselves understood, though, with sentences.

4.027 It is of the essence of a sentence that it can communicate a new sense to us. 4.03

A sentence must communicate a new sense with old terms. A sentence communicates a situation to us, so it must be essen­ tially connected with the situation. And the connection is just that it is its logical picture. A sentence says something only insofar as it is a picture.

4.031

In a sentence a situation is as it were put together by way of a test. Instead of, This sentence has such and such a sense, one can say, This sentence presents such and such a situation.

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[Nordmann discusses various possible translations of this remark, and their various implications. It is ambiguous in the German, the first sentence allowing for the interpretation that a sentence puts together a situation experimentally, or as an experiment, or for the sake of experiment, or in order to be put to the test. (See Nordmann 2005, 108–114.)] 4.0311 A name stands for a thing, another for another thing, and they are connected with each other, so the whole – like a tableau vivant – represents a state of affairs. 4.0312 The possibility of a proposition is based on the principle of the representation of objects by signs. My fundamental thought is that the “logical constants” represent nothing. That the logic of facts does not allow of representation.

[J. Mark Lazenby: “Wittgenstein’s placement of his fundamental idea in a commentary of a commentary of a commentary on a ‘more important’ proposition, is a literary device: an attempt to force the reader to think about the value of this idea in particular, and the value of philosophical truth in general” (Lazenby 2006, 63). Schroeder: It is not clear [. . .] why Wittgenstein should call this claim his ‘fundamental thought.’ After all, its import seems entirely negative. Perhaps it would have been more accurate to call it a ‘ground-clearing’ thought. Rather than laying the foundation of Wittgenstein’s own philosophy of logic, it disposes of alternative views held by Frege and Russell. (Schroeder 2006, 85–86)

Hintikka: “The Tractatus is nothing more and nothing less than Russell’s 1913 theory [as set out in Theory of Knowledge] sans logical forms as objects of acquaintance” (Hintikka 2000, 19). Sections 4.0312 and 5.4 can be seen as evidence of this. This leaves Wittgenstein with a question as to what holds a proposition together if not logical forms conceived as objects of acquaintance? “Wittgenstein’s answer was: A proposition is held together, not by any additional ‘tie’ or ‘glue’ but by the forms of its constituents” (ibid.). See 2.03 for the idea of objects in a state of affairs hanging together like the links in a chain.] 4.032

Only insofar as it is logically articulated is a proposition a picture of a situation. (Even the proposition “ambulo” is composite, because its stem with another ending, or its ending with another stem, gives another sense.)

[Ambulo is Latin for “I walk.”]

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4.04

In a proposition there must be exactly as many things to differentiate as there are in the situation it presents. Both must possess the same logical (mathematical) multiplicity. (Compare Hertz’s Mechanics on dynamic models.)

[J. Alberto Coffa comments: The motivation for the requirement that an appropriate symbolism have the same multiplicity as what it symbolizes is that the other two alternatives have evident drawbacks. If the multiplicity of the symbolic system is smaller than that of what it represents, there will be possible circumstances we will not be able to describe. If the multiplicity is greater, the problem is more familiar—it is called “philosophy.” All of philosophy (up to, and perhaps including, Wittgenstein) had consisted of attempts to say things that cannot be said. Good philosophy attempts to say what can be shown, the sinnlos; bad philosophy attempts to say what cannot even be shown, the unsinnig, the utter nonsense. Most philosophy had been bad philosophy, based on confusions concerning language. These confusions were roughly of the sort displayed by Russell’s paradox: The language we use has a greater multiplicity than what it talks about. We can therefore form expressions whose syntactic appearance is like that of perfectly meaningful claims, but from them we are led to some form of chaos. (Coffa 1993, 156–157)

Allan Janik notes that Hertz treats simplicity as relative rather than absolute. In particular, it is relative to the audience. “Hertz proceeds from the view that even within science it is necessary to construct different representations of the same data depending upon whom you want to talk to. He offers us the analogy with presentations of grammar: pupils learning to master their mother tongue require an altogether different presentation of the rules of grammar than philologists do” (Janik 2006, 51). If simplicity is relative, perhaps multiplicity is too. Nordmann writes: In the Notebooks Wittgenstein struggles with the question of what is required to sharply delineate truth-conditions. He finds that a certain high ideal of precision proves not only unnecessary but actually inappropriate. If one wants to attain a precise measurement of the length of a room, measurements in angstroms are less and not more precise than measurements in meters and centimeters. Indeed, one is far more likely to obtain a definite measurement and fixed value if one doesn’t treat macroscopic objects on subatomic scales [. . .]. Similarly, what is needed for a sharp delineation of truth-conditions is not an analysis in terms of material points or data points, but merely that sentence, thought, and state of affairs have the same multiplicity, i.e., that one distinguishes just as much in the proposition as one means to distinguish in the state of affairs. (Nordmann 2002, 377)]

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Of course one cannot in turn picture this mathematical multiplicity itself. One cannot get outside it to make a picture.

4.0411 Should we want to express, e.g., what we express with “(x) fx” by placing an affix before “fx” – something like “Gen. fx,” it would not suffice – we would not know what was being generalized. Should we want to indicate it by an affix “α” – something like “f(xα)” – it would still not suffice – we would not know the scope of the generality-sign. Should we want to try it by the introduction of a mark in the argument place – something like “(A, A) . F (A, A)” – it would not suffice – we could not fix the identity of the variables. Etc. All these ways of symbolizing do not suffice, because they do not have the necessary mathematical multiplicity. 4.0412 On the same grounds, the idealist explanation of seeing spatial relations by reference to “spatial spectacles” is inadequate because it cannot explain the multiplicity of these relations.

[The idealist sounds rather Kantian here. Black quotes Russell (1913, 491) saying that “The categories of Kant are the coloured spectacles of the mind,” but adds that Wittgenstein might have been thinking of Meinong or Husserl rather than Kant. (See Black 1964, 177.)] 4.05

Reality is compared with a sentence.

[Cf. 2.223 on comparing a picture with reality.] 4.06

A sentence can be true or false only in that it is a picture of reality.

4.061

If one does not notice that a proposition has a sense independent of the facts, then one can easily believe that true and false are relations, with the same rights, between signs and the signified. One could then say, e.g., that “p” signifies in the true way what “~p” signifies in the false way, etc.

[McGinn on 4.061–4.063: [Wittgenstein’s] aim is to show that insofar as Frege holds that true and false propositions designate distinct but equivalent entities, the True and the False, he fails to make the relation between sense and truth and falsity perspicuous. In treating the Bedeutung of true sentences as an equivalent and distinct object from the Bedeutung of false sentences, Wittgenstein believes that Frege fails to make it clear that each proposition with sense essentially has two poles—a true pole and a false pole—each of which excludes the other. (McGinn 2006, 44)]



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4.062 Can’t one make oneself understood with false propositions as one has till now with true ones? Just as long as one knows that they are meant to be false. No! Because a proposition is true if things are as we say they are by means of it; and if by “p” we mean ~p, and things are as we mean, then “p” in the new sense is true and not false. 4.0621 But it is important that the signs “p” and “~p” can say the same thing. Because it shows that the sign “~” corresponds with nothing in reality. That negation occurs in a proposition is still no characteristic [or sign: Merkmal] of its sense (~~p = p). The propositions “p” and “~p” have opposite senses, but one and the same reality corresponds to them. 4.063 A picture to explain the concept of truth: a black spot on white paper; one can describe the form of the spot in that one can answer for each point on the sheet whether it is white or black. To the fact that a point is black corresponds a positive fact, to the fact that a point is white (not black), a negative one. If I indicate a point on the sheet (a Fregean truth-value) then this corresponds to the assumption that is proposed for judgment, etc. etc. In order though to be able to say whether a point is black or white, I must first know when one calls a point black and when one calls it white; in order to be able to say “p” is true (or false), I must have determined under which conditions I call “p” true, and thus I determine the sense of the proposition. The point at which the simile breaks down now is this: we can indicate a point on the paper without knowing what white and black are; to a proposition without sense however nothing whatsoever corresponds, because it signifies no thing (truth-value) whose properties are called false or true; the verb of a proposition is not “is true” or “is false” – as Frege believed – but rather that which “is true” must already contain the verb.

[Anscombe says that Wittgenstein’s reference to “the Fregean Annahme” (assumption), at the end of the first paragraph of 4.063, is really a reference to what Russell says about Frege in Principles of Mathematics Appendix A §477 (see Anscombe 1971, 105–106 note 1). She argues that Russell and Wittgenstein get Frege (in “Function and Concept”) wrong, and mistakenly attribute to him a technical meaning of “assumption.” Assumption in this sense means something like the assertion of a proposition as either true or false, so that the truth-value of the proposition can be thought of as a verb, meaning the checking of an imaginary box next to “is true” or “is false.” Anscombe says that Frege did say that the verb of the proposition is “is true” in the Begriffsschrift, but he never said this of “is false” and he rejected this earlier view of his in “Sense and Reference.”

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Anscombe notes that Wittgenstein’s talk of determining the conditions under which I call a proposition true sounds like verificationism to some people, but says that it is just a reference to truth-conditions. The emphasis is on logic, not epistemology. See Anscombe 1971, 152–153. See comment on 4.442 for Proops on this. He gives reasons for rejecting Anscombe’s account of what “verb” means for Wittgenstein and Frege here (see Proops 2000, 40–42). Proops points to Begriffsschrift §2 as a likely source of Wittgenstein’s belief that Frege’s assertion sign (˫) marks something as an assertion, when in fact, as Frege explains elsewhere, it is the vertical stroke (the judgment stroke) that does this, the horizontal stroke merely marking a potentially assertable proposition, what Wittgenstein appears to be calling an assumption. Proops notes that Wittgenstein links talk of the “assumption” in his Notebooks (January 11, 1915) with a yardstick: “Could we not ask: What has to be added to that yardstick in order for it to assert something about the length of the object? (The yardstick without this addition would be the ‘assumption’ [Annahme])” (Wittgenstein 1979b, 37–38). See Proops 2000, 50–57. A yardstick marks a certain length, as if in readiness for objects one yard long (Proops assumes, for the sake of argument, that it has no finer gradations marked), but does not actually say of its own accord that this or that object is one yard long. Similarly, an unasserted proposition marked only by a horizontal stroke stands, as it were, ready to be asserted as a proposition, but does not assert itself. We might then wonder what needs to be added to it to make it an assertion, an actual proposition rather than mere content. But this content must already have sense. I cannot even consider asserting something unless it is already a proposition. Proops: The Annahme is treated as the notational embodiment of the “showing” aspect of the proposition (picking out a situation while saying nothing about it), while the assertion sign is treated as embodying the proposition’s “truth-claiming” or “saying” aspect (saying of the possible situation thus picked out that it actually obtains). I have wanted to suggest that Wittgenstein’s critique of the assertion sign is best seen as part of an attack on the coherence of such a conception of the proposition. (Proops 2000, 56)

Proops adds: “In the end, then, the thought that a proposition cannot assert its own truth is best seen not as a direct criticism of any view that Frege or Russell actually hold, but as the denial of a crucial presupposition of the coherence of the notion of logical assertion” (Proops 2000, 57). McGinn: The judgement stroke is not itself a function, but it is only by placing the name of a truth-value in the context of a judgement stroke that we move from naming

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an object to expressing something with the bipolarity which Wittgenstein takes to be the defining feature of sense. This is what Wittgenstein means when he says that Frege believes the verb of a proposition is “is true” or “is false”: it is only when we assert, by means of the judgement stroke, that the proposition designates the True that we achieve something with the essential bipolarity of a proposition. (McGinn 2006, 50)] 4.064

Every proposition must already have a sense; assertion cannot give it one, because the sense is the very thing asserted. And the same goes for negation, etc.

[Anscombe says that this is an attack on Frege, but a potentially confusing one, since Frege would agree with it (see Anscombe 1971, 58–59). The problem for him is that he thinks that when one makes a judgment, one “advances from a thought to a truth-value” (see Beaney 1997, 159). Wittgenstein, she says, is attacking this idea. Having a sense means being true or false, so there cannot be propositions that have a sense but are neither true nor false. Frege, Anscombe says (and she argues that Wittgenstein agrees), is wrong. He makes it seem as though it is merely contingent if we construct a proposition with sense and find that it has a truth-value.] 4.0641 One could say: The negation is related already to the logical place that the negated proposition determines. The negating proposition determines another logical place than does the proposition negated. The negating proposition determines a logical place with help from the logical place of the negated proposition, in that it describes it as lying outside this place. That one can again negate the negated proposition shows already that what is negated is already a proposition and not merely the preliminary to a proposition. 4.1

A proposition presents the holding and non-holding of states of affairs.

4.11

The totality of true propositions is the whole of natural science (or the totality of the natural sciences).

[Black criticizes this remark for being incompatible with the more sophisticated 6.341. See Black 1964, 185–186.] 4.111

Philosophy is not one of the natural sciences. (The word “philosophy” must refer to something either over or under, but not standing alongside the natural sciences.)

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The end of philosophy is the logical clarification of thoughts. Philosophy is not a subject but an activity. A philosophical work consists essentially of elucidations. The result of philosophy is not “philosophical propositions” but the clarification of propositions. Philosophy should make clear and distinct thoughts that, without it, are, as it were, unclear and indistinct.

[Schopenhauer writes: “In general the real philosopher will always look for clearness and distinctness; he will invariably try to resemble not a turbid, impetuous torrent, but rather a Swiss lake which by its calm combines great depth with great clearness, the depth revealing itself precisely through the clearness” (Schopenhauer 1974, 4). James Conant and Cora Diamond say: Wittgenstein gives voice to an aspiration that is central to his later philosophy, well before he becomes later Wittgenstein, when he writes in §4.112 of the Tractatus that philosophy is not a matter of putting forward a doctrine or a theory, but consists rather in the practice of an activity—an activity he goes on to characterize as one of elucidation or clarification—an activity which he says does not result in philosophische Sätze, in propositions of philosophy, but rather in das Klarwerden von Sätzem, in our attaining clarity in our relation to the sentences of our language that we call upon to express our thoughts. To say that early Wittgenstein aspired to such a conception of philosophy is not to gainsay that to aspire to practice philosophy in such a manner and to succeed in doing so are not the same thing. (Conant and Diamond 2004, 46)] 4.1121 Psychology is no more closely related to philosophy than is any other natural science. Theory of knowledge is the philosophy of psychology. Does not my study of symbolism correspond to the study of thought-processes, which philosophers held so essential to the philosophy of logic? Only they got entangled mostly in inessential psychological investigations, and there is an analogous danger for my method.

[Mounce: Psychology is irrelevant to philosophy or logic because it is not a psychological process that gives sense to logical form; on the contrary, it is only logical form that can give sense to a psychological process, that can make it, for example, a genuine thought as opposed to a random succession of images. Thus the psychological activity involved in correlating a mark with an object is in itself entirely

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meaningless. What gives it a meaning, what makes it a genuine correlation, is the logical structure into which the mark enters. (Mounce 1981, 32)

Anscombe identifies “Carnap and his school” as people who seem to have fallen into the danger identified here by Wittgenstein (Anscombe 1971, 86).] 4.1122 Darwinian theory has no more to do with philosophy than has any other hypothesis of natural science. 4.113

Philosophy limits the disputable territory of natural science.

4.114

It should delimit the thinkable and therewith the unthinkable. It should limit the unthinkable from inside, by way of the thinkable.

[Echoes here of Kant’s talk of limiting reason in, e.g., §59 of Prolegomena (Kant 2004). At B xxx of the Critique of Pure Reason (Kant 1969) he says that he has had to suspend knowledge in order to make room for faith (Ich musste also das Wissen aufheben, um zu Glauben Platz zu bekommen) and the references in Prolegomena to the limits of reason relate to this idea. We might wonder, therefore, (although perhaps we ought not) whether Wittgenstein is implying that he limits the thinkable in order to make room for some faith in the ineffable.] 4.115

It will refer to the unsayable in that it presents clearly the sayable.

4.116 Everything that can be thought at all can be thought clearly. Everything that can be said can be said clearly. 4.12

The proposition can present the whole of reality, but it cannot present that which it must have in common with reality in order to present it – logical form. In order to present logical form, we would have to be able to put ourselves, along with propositions, outside logic, that is to say outside the world.

[McManus: When we imagine ourselves identifying a logical form that a proposition must possess in order to represent a particular possible fact, we can only latch on to something that might impose a superficially intelligible (pseudo-) requirement by staying “within logic”: that is, by using a proposition that picks out the thusand-so of the imagined logical form—an object’s being in a particular spatial location, say. But such an identification takes for granted that we understand

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how this state of affairs must be represented, that its logical form is such that it must be represented using a proposition that captures an object’s being spatially located. But now the “requirement” exposed is that possible facts that are characterized like this must be . . . characterized like this. Our quest for a genuine requirement must drive us, it seems, “outside logic,” and with it, outside any presupposed frame of reference with which to characterize the world. Our “indication” of the logical form now becomes an inarticulate pointing at a bare that, about which we now find that we cannot ask the sort of question of con-formity which we set out to ask. (McManus 2006, 94–95)

(McManus hyphenates “conformity” in order to bring out the [apparent] importance for intelligibility of shared form between such things as language and world. See ibid., 5.) There is, McManus argues, no similarity as such, nor sharing of the same form as such. Things are only ever like or unlike in some or other particular respect. What is like what then depends on how we look at things, on how we choose to categorize or characterize things.] 4.121

Propositions cannot present logical form, it is reflected in them. What is reflected in language, cannot be presented by it. What expresses itself in language, we cannot express through it. Propositions show the logical form of reality. They display it.

[Ostrow says that this “makes clear that it is the genuine proposition that shows logical form,” a job taken by some commentators to be done by the (pseudo-) propositions of logic (Ostrow 2002, 107).] 4.1211 Thus the proposition “fa” shows that it is about the object a, two propositions “fa” and “ga” show that they are both about the same object. If two propositions contradict each other then their structure shows this; the same applies if one follows from the other. And so on. 4.1212 What can be shown cannot be said.

[Kremer rejects the idea that there are as-it-were sayable things that can only be shown: “this doctrine can only get its content from examples, and in giving such examples we immediately contradict the very doctrine that we are trying to flesh out” (Kremer 2001, 55). That is, if I say that “we cannot say x” then I have just said x, which is precisely what I claimed could not be said. And if no one can ever give an example of some x that supposedly cannot be said, then the claim that such things exist seems empty.]

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4.1213 Now we understand too our feeling that we have a correct logical apprehension only if everything is right in our symbolism.

[For Auffassung here I have “apprehension,” although “comprehension,” “conception,” and “view” would also be all right. Ogden has “conception.” Pears and McGuinness have “point of view.” Wittgenstein 1953 §4 talks of Augustine’s Auffassung of language.] 4.122

We can talk in a certain sense of formal properties of objects and states of affairs or of properties of the structure of facts, and in the same sense of formal relations and relations of structures. (Instead of structural property I say also “internal property”; instead of structural relation, “internal relation.” I introduce these expressions in order to show the basis of the confusion between internal relations and proper (external) relations, which is very widespread among philosophers.) The holding of such internal properties and relations, however, cannot be asserted through propositions, but rather it shows itself in the propositions which present the states of affairs and deal with the objects in question.

4.1221 We can also call an internal property of a fact a feature of that fact. (In the sense in which we speak of facial features.) 4.123

A property is internal if it is unthinkable that its object should not possess it. (This blue color and that stand in the internal relation of lighter and darker eo ipso. It is unthinkable that this pair of objects not stand in this relation.) (Here the shifting use of the word “object” corresponds to the shifting use of the words “property” and “relation.”)

[McGinn: In the later philosophy, it is clear that Wittgenstein thinks that the colour-wheel is itself a part of the symbolism, in the sense that the ordered colour samples of the colour-wheel constitute an instrument of our language, by means of which the logical order of our colour concepts is presented. However, it is not clear that he held this view at the time of writing the Tractatus, where he seems to suggest that the logical order of colour-space will be revealed through the logical analysis of colour terms (see TLP 6.3751). (McGinn 2006, 182)

Schroeder writes that the internal relation between dark, Oxford blue and light, Cambridge blue shows itself “quite literally” (Schroeder 2006, 90).

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This seems to imply that we can literally see a necessary truth or feature of reality. Frege argues against this kind of idea in “Thought” (see Beaney 1997, 325–345). He would say that we literally see objects, but we do not literally see that such-and-such is true, at least not in the same sense of “see.”] 4.124

The existence of an internal property of a possible situation will not be expressed through a proposition, but rather it expresses itself in the proposition that presents the situation, through an internal property of this proposition. It would be equally senseless to ascribe a formal property to a proposition as to deny it.

4.1241 One cannot distinguish forms from each other by saying that the one has this but the other has that property; because this presupposes that it makes sense to assert either property of either form. 4.125 The holding of an internal relation between possible situations expresses itself linguistically through an internal relation between the propositions that present them. 4.1251 Here now the vexed question “whether all relations are internal or external” disappears.

[Black says that this might be a reference to G. E. Moore’s essay “Relations,” which attacks the views of Bradley and others on internal relations (see Black 1964, 198). Hegelian, idealist views on relations certainly were a concern of Russell’s. McGinn says that “This remark and the one following (4.1252) make a clear reference to Russell. Russell had argued against the intelligibility of internal relations and held that all relations are external” (McGinn 2006, 178).] 4.1252 Series which are ordered according to internal relations I call formal series. The series of numbers is ordered not by an external but rather by an internal relation. Equally the series of propositions: “aRb,” “(∃x): aRx . xRb,” “(∃y): aRx . xRy . yRb,” etc. (If b stands in one of these relations to a then I call b a successor of a.)



4.126

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In the sense in which we speak of formal properties, we can now also speak of formal concepts. (I introduce this expression in order to make clear the basis of the confusion of formal concepts with proper concepts, which runs through the whole of the old logic.) That something is an instance of a formal concept cannot be expressed through a proposition. Rather it shows itself in the sign for this object itself. (A name shows that it signifies an object, a numeral that it signifies a number, etc.) Formal concepts cannot, in the way that proper concepts can, be presented by a function. Because of their defining characteristics, formal properties are not expressed through functions. The expression of a formal property is a feature of certain symbols. The sign for the defining characteristics of a formal concept is therefore a characteristic feature of all symbols whose meaning falls under the concept. The expression of a formal concept is therefore a propositional variable, in which only this characteristic feature is constant.

[In the third sentence here I take Black’s suggestion of saying “is an instance of a formal concept” rather than the more literal “falls under a formal concept as an object belonging to it,” as Ogden has it (see Black 1964, 199). It is Black also (ibid.) who suggests “numeral” instead of “numerical sign” as Ogden has it. Black says that Merkmale should be translated as “marks,” since it means criteria or defining properties. But “Because of their marks, formal properties . . .” sounds obscure to me. Wittgenstein says that the term Merkmal here is taken from Frege’s terminology (see Wittgenstein 1973, 28). Richard L. Mendelsohn comments: “Wittgenstein clearly had Frege’s predicament about the concept horse in mind when he spoke about ‘formal concepts.’ . . . Frege’s concept and object are just such formal concepts” (Mendelsohn 2005, 82). Wanting to distinguish objects from concepts, Frege famously denied that the concept horse (an object we can talk about) is a concept. According to Mendelsohn, “there is just no way of coherently expressing this principle (i.e., ‘No concept is an object’) in the symbolism” (Mendelsohn 2005, 81). Frege is thus committed to the view that the principle, which he wants to uphold, is meaningless. Michael Dummett argues that Frege could have avoided the appearance of paradox if he had talked only about kinds of expression and not the things for which expressions stand, but Mendelsohn objects (see Klemke 1968, 269). Predicates and concepts must have analogous properties, Mendelsohn says, as “an immediate consequence of Frege’s general view that the structure of language mirrors the structure of the world” (Mendelsohn 2005, 81).

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Kremer: Wittgenstein responds to this Fregean predicament by distinguishing “proper concepts” from “formal concepts” or “pseudo-concepts,” such as object and function (4.126, 4.1272). “That anything falls under a formal concept,” he tells us, “cannot be expressed by a proposition. But it is shown in the symbol for the object itself.” Words that seem to express formal concepts, like “object” and “function,” when rightly used, will be replaced in Begriffsschrift with variables of appropriate type. (Kremer 2001, 54)

Joan Weiner has a relevant discussion of similarities and differences between Wittgenstein and Frege on what Wittgenstein calls formal concepts in Floyd and Shieh 2001, 47–49.] 4.127 A propositional variable signifies a formal concept and its values [signify] the objects that fall under this concept. 4.1271 Each variable is the sign for a formal concept. Because each variable presents a constant form, which all its values possess, and which can be conceived as a formal property of these values. 4.1272 Thus the variable name “x” is the real sign for the pseudo-concept object. Wherever the word “object” (“thing,” “item,” etc.) is used rightly, it is expressed in the concept-script by a variable name. For example in the proposition “there are two objects, such that . . .” by “(∃x,y). . . .” Wherever it is used otherwise, hence as a real concept word, nonsensical [unsinnige] pseudo-propositions arise. Thus one cannot, e.g., say “There are objects,” as one says “There are books.” Just as little can one say “There are 100 objects” or “There are ‫א‬0 objects.” And it is nonsensical [unsinnig] to speak of the number of all objects. The same goes for the words “complex,” “fact,” “function,” “number,” etc. They all signify formal concepts and are presented in the conceptscript by variables, not by functions or classes. (As Frege and Russell believed.) Expressions like “1 is a number,” “there is only one zero,” and all such are nonsensical [unsinnig]. (It is equally nonsensical [unsinnig] to say “there is only one 1,” as it would be nonsensical [unsinnig] to say: 2 + 2 is at 3 o’clock equal to 4.)



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[Here I have translated eigentliche and eigentliches as “real” to be consistent with the translation of 3.3411. Diamond writes that: The notion of the eigentliche Zeichen, in 4.1272, needs to be seen with the idea of all signs capable of serving a particular purpose and what they therefore have in common, enabling them to do so. A better translation of 4.1272 would use “real sign,” since that makes clearer the idea of what is really doing the sign’s essential work when it is serving that purpose. That translation would bring out the connection with the contrast essential/accidental at 3.34. (Diamond 2019, 140 note 35)

A. W. Moore completes his account of the reading of the early Wittgenstein according to which he distinguishes between thoughts (which have sense), tautologies and contradictions (which lack sense but are not nonsensical), and nonsensical pseudo-propositions (see notes on 3 and 4.4611), by saying that: “Finally, there are pseudo-propositions. These are concatenations of signs that do not belong to either of the first two categories. They are neither true nor false. And, in contrast to tautologies and contradictions, they are nonsensical. (See e.g., TLP 4.1272, 5.4733, and 6.53)” (Moore 2020, 27).] 4.12721 With an object that falls under it, a formal concept is already given. Thus one cannot introduce as primitive ideas the objects of a formal concept and the formal concept itself. Thus one cannot (like Russell) introduce, e.g., the concept of a function and also special functions as primitive ideas; or the concept of number and specific numbers. 4.1273 If we want to express in the concept-script the general proposition: “b is a successor of a,” then we need for this an expression for the general term of the formal series: aRb, (∃x): aRx . xRb, (∃x,y): aRx . xRy . yRb, ... The general term for a formal series can be expressed only by a variable, because the concept “term for this formal series” is a formal concept. (This has been overlooked by Frege and Russell: because of this the way they want to express general propositions like the one above is false; it contains a vicious circle.) We can determine the general term of a formal series by giving its first term and the general form of the operation that produces the next term from the proposition that goes before it. 4.1274 The question of the existence of a formal concept is nonsensical [unsinnig]. Because no proposition can answer such a question.

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(Thus one cannot ask, e.g.: “Are there unanalysable subject-pre­ dicate propositions?”) 4.128

Logical forms are unnumbered [number-less, but not in the sense of too numerous to count]. Therefore there are in logic no pre-eminent numbers, and therefore there is no philosophical monism or dualism, etc.

[Black suggests “anumerical” for zahllos (“unnumbered”) (see Black 1964, 206). He goes on: “It is nonsense to speak of counting logical forms. It is not clear what W. had in mind here: certainly in a universe containing a finite set of objects and a finite set of their combinations, a list could be made of distinct logical forms, which might then be counted. [. . .] Perhaps W. wanted to stress that ‘is a logical form’ is not an authentic predicate such as ‘is a star’” (ibid.)] 4.2

The sense of a proposition is its agreement, and disagreement, with the possibility of the holding and non-holding of states of affairs.

4.21

The simplest proposition, the elementary proposition, asserts the holding of a state of affairs.

4.211

It is a sign of an elementary proposition that no elementary proposition can stand in contradiction to it.

4.22

An elementary proposition consists of names. It is a concatenation, a linking, of names.

[Cf. 2.03 on objects in a state of affairs being like links in a chain.] 4.221

It is obvious that by the analysis of propositions we must come to elementary propositions, which consist of names in immediate combination. Here the question arises of how the combination of propositions comes to be.

[Black: “The questions (i.e., ‘How can names combine to form a sentence?’ and ‘How can objects combine to form a state of affairs?’) are nowhere answered and it is hard to see how any answers, in W.’s view, could be expected. Here perhaps we have instances of irredeemable nonsense” (Black 1964, 208).] 4.2211 Even if the world is infinitely complex, so that each fact consists of infinitely many states of affairs and each state of affairs is composed



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of infinitely many objects, even then there must be objects and states of affairs. 4.23

A name occurs in a proposition only in the context of an elementary proposition.

4.24

Names are simple symbols. I indicate them by single letters (“x,” “y,” “z”). I write an elementary proposition as a function of names, in the form: “fx,” “φ(x,y),” etc. Or else I indicate it by the letters p, q, r.

[Black notes that this seems to conflict with 3.202 and 3.26 (see Black 1964, 209). There, Wittgenstein says that names are signs.] 4.241

If I use two signs with one and the same meaning [Bedeutung], then I express this by putting between them the sign “=.” Thus “a = b” means that the sign “a” is replaceable by the sign “b.” (If I introduce a new sign “b” by an equation, in which I stipulate that it should replace an already known sign “a,” then I write the equation – the definition – (like Russell) in the form: “a = b Def.” The definition is a rule for [using] signs.)

[Ogden has “symbolic rule,” Pears and McGuinness have “rule dealing with signs,” and Black suggests “rule about signs” (see Black 1964, 211).] 4.242

Expressions of the form “a = b” are thus only aids for presentation; they say nothing about the meaning [Bedeutung] of the signs “a” and “b.”

4.243

Can we understand two names without knowing whether they signify the same thing or two different things? – Can we understand a proposition, in which two names occur, without knowing if they mean the same or something different? If I am acquainted with the meaning of an English word and of a synonymous [gleichbedeutenden] German word, then it is impossible that I do not know that they are both synonymous; it is impossible that I cannot translate them by each other. Expressions like “a = a,” or those derived from such expressions, are neither elementary propositions nor otherwise senseful signs [sinnvolle Zeichen]. (This will be shown later.)

[Black presumes that what is to be shown later is so shown at 5.531–5.533 (see Black 1964, 211).]

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4.25

If an elementary proposition is true, then the [relevant] state of affairs holds; if an elementary proposition is false, then the state of affairs does not hold.

4.26

The statement of all true elementary propositions describes the world completely. The world is completely described by the statement of all elementary propositions plus a statement as to which of them are true and which are false.

4.27

As to the holding and non-holding of n states of affairs, there are Kn =

∑nv = 0 (nv) possibilities.

All combinations of states of affairs can hold and the others not hold.

[J. N. Findlay: 4.27 tells us that if we are considering N SACHVERHALTE [states of affairs] each of which is capable of being or not being the case, there will be 2n factual or nonfactual possibilities to which 2n truth/falsehood possibilities will correspond in the case of elementary propositions, that is, one elementary proposition has two truth-possibilities; two propositions have four truth-possibilities; three have eight possibilities; four, sixteen; etc. (Findlay 2008, 88–89)

In other words, as Schroeder notes, Wittgenstein’s formula here amounts to 2n (see Schroeder 2006, 63 note 18).] 4.28

To these combinations there correspond just as many possibilities of truth – and falsehood – for n elementary propositions.

4.3

The truth-possibilities of elementary propositions mean [bedeuten] the possibilities of the holding and non-holding of states of affairs.

4.31

We can present the truth-possibilities by schemata in the following way (“T” means “true,” “F” means “false.” The rows of “T”s and “F”s under the row of elementary propositions indicate [bedeuten] their truth-possibilities in a readily comprehensible symbolism): p q r

p q

p

T F T F T F T F

T F T F

T F

T T F F T T F F

T T T T F F F F

T T F F

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[Schroeder notes that lines 4 and 5 of the first table are wrongly printed in each other’s place in other editions of the Tractatus (see Schroeder 2006, 65 note 20).] 4.4

A proposition is the expression of agreement and disagreement with the truth-possibilities of elementary propositions.

4.41

The truth-possibilities of elementary propositions are the conditions of the truth and falsehood of the propositions.

4.411

It is plausible from the beginning that the introduction of elementary propositions is foundational for the understanding of all other kinds of propositions. Indeed the understanding of all propositions depends palpably on the understanding of the elementary propositions.

[“Plausible” is suggested by Black (see Black 1964, 220). Otherwise we have “It is probable from the beginning . . .” which does not seem to make much sense. Ogden has “It seems probable even at first sight,” while Pears and McGuinness have “It immediately strikes one as probable.” The word I translate as “palpably” is fühlbar—“feel-ably” or sensibly. The Notes on Logic say “obviously” at the corresponding point (Wittgenstein 1979b, 100).] 4.42

As to the agreement and disagreement of a proposition with the truth-possibilities of n elementary propositions, there are K ∑Kk        = 0 ( k) = L possibilities. n

n

n

[Gregory Landini: “Based on n-many propositions there will be p-many (namely, 2n) rows of the truth-table and 2p many truth-functions. Wittgenstein himself computes this at TLP 4.42” (Landini 2007, 142).] 4.43

We can express agreement with truth-possibilities by coordinating the sign “T” (for true) with them in the schema. The absence of this sign means disagreement.

4.431

The expression of agreement and disagreement with the truth-possibilities of elementary propositions expresses the truth-conditions of a proposition. A proposition is the expression of its truth-conditions. (Frege quite rightly therefore put them first [vorausgeschickt] as an explanation of the signs of his concept-script. Only the explanation of the concept of truth that we get from Frege is false: if “the True” and “the False” were really objects and the arguments in ~p

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etc. then Frege’s determination of the sense of “~p” would in no way determine it.)

[Anscombe: “As a criticism of Frege the point can be summarized by saying: ‘If truth-values are the references of propositions, then you do not specify a sense by specifying a truth-value’” (Anscombe 1971, 107). Hacker: if “~” were a name of a genuine function the argument of which is one of the two truth-values, then provided that “p” (e.g., “The sun is cold”) has the same truth-value as “q” (e.g., “The moon is hot”), “~p” would have the same sense as “~q.” The argument turns on the extensionality of functions. In such a case each compound proposition merely expresses the thought that the False falls under the concept of negation. For each such proposition determines the True as the value of the same function for the same argument. But this is absurd by Frege’s own lights. For obviously “~p” is taken to have the same sense as “~q” if and only if “p” has the same sense as “q.” But if so then Frege’s explanation of the negation sign does not determine its sense. (Hacker 1997, 41)] 4.44

The sign that is produced by the coordination of each sign “T” with the truth-possibilities is a propositional sign.

4.441

It is clear that no object (or complex of objects) corresponds to the complex of the signs “F” and “T”; just as little as do horizontal and vertical lines or brackets. – There are no “logical objects.” Of course the same goes for all signs that express the same as the schemata of “T” and “F.”

[Cf. 4.0312, which tells us that the logical constants do not represent.] 4.442

This, e.g., is a propositional sign: “p

q

T

T

T T

F

T

T

F

F

F

T”

(Frege’s “assertion sign” “˫” is logically quite meaningless [ganz bedeutungslos]; for Frege (and Russell) it shows only that these authors hold the propositions thus marked as true. Therefore “˫” belongs as little to the proposition as does the number of a proposition. A proposition cannot possibly assert of itself that it is true.) If the order of the truth-possibilities in a schema is fixed once and for all by a rule of combination, then the last column by itself

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is already an expression of the truth-conditions. If we write this column as a row, then the propositional sign will be: “(TT—T) (p, q)” Or more distinctly “(TTFT) (p, q).” (The number of places inside the brackets on the left is determined by the number of terms in the brackets on the right.)

[Proops treats this, along with 4.064 and 4.063, as containing the core of Wittgenstein’s critique of Frege’s assertion sign (see Proops 2000, 29–57). He argues that Wittgenstein misunderstands Frege’s position, noting that these remarks are virtually identical to ones Wittgenstein wrote in 1913 in his Notes on Logic (see Proops 2000, 31 note 86). Proops says that: what Wittgenstein means by a “proposition” at 4.442 is, in effect, what Frege calls a “Proposition of Begriffsschrift”—i.e., a sentence of Begriffsschrift immediately preceded by the judgement stroke. Wittgenstein understands such signs [. . .] as translatable into English by expressions of the form “S is true” and “S is false,” where “S” is replaceable by a sentence of English. So “is true” and “is false” can also be said to be “verbs” in a derivative sense. (Proops 2000, 38)

Black says that the assertion sign in Russell’s Principia is introduced explicitly to distinguish complete propositions from subordinate propositions contained within them (see Black 1964, 226–227). But, according to Black, this is unnecessary, as the difference is already clear, and so the effect is that the sign indicates simply that the authors are putting the proposition forward as true. Philosophical Investigations §22 says: Frege’s idea that every assertion contains an assumption, which is the thing that is asserted, really rests on the possibility found in our language of writing every statement in the form: “It is asserted that such-and-such is the case.”– But “that such-and-such is the case” is not a sentence in our language – so far it is not a move in the language-game. And if I write, not “It is asserted that . . . ,” but “It is asserted: such-and-such is the case,” the words “It is asserted” simply become superfluous. (Wittgenstein 1953 §22)] 4.45 For n elementary propositions there are Ln possible groups of truthconditions. The groups of truth-conditions that belong to a number of truthpossibilities can be ordered in a series. 4.46

Of all the possible groups of truth-conditions there are two extreme cases. In one case the proposition is true for all truth-possibilities of the elementary propositions. We say that the truth-conditions are tautological.

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In the second case the proposition is false for all truth-possibilities: the truth-conditions are contradictory. In the first case we call the proposition a tautology, in the second case a contradiction. 4.461 A sentence shows what it says, a tautology and a contradiction [show] that they say nothing. A tautology has no truth-conditions because it is unconditionally true; and a contradiction is true under no condition. Tautology and contradiction are senseless [sinnlos]. (Like a point from which two arrows go out in opposite directions to one another.) (I know nothing, e.g., about the weather if I know that it is raining or not raining.) 4.4611 But tautology and contradiction are not meaningless [unsinnig]; they belong to the symbolism, in a way similar to that in which “0” belongs to the symbolism of arithmetic.

[Continuing his account of the standard view that Wittgenstein distinguishes between thoughts, tautologies and contradictions, and nonsensical pseudopropositions (see notes on 3 and 4.1272), Moore writes: Tautologies and contradictions are propositions without a sense. They lack a sense because they lack the bipolarity that thoughts have. Although they are true or false, the true ones, namely the tautologies, are unconditionally true, while the false ones, namely the contradictions, are unconditionally false. Their lacking sense in this very distinctive way is registered by saying that they are senseless, but not nonsensical. (See e.g., TLP 4.46–4.4611.) (Moore 2020, 27)] 4.462

Tautology and contradiction are not pictures of reality. They present no possible situations. Because one lets every possible situation be, and the other none. In tautology the conditions of agreement with the world – the presenting relations – cancel each other out, so that it stands in no presenting relation to reality.

4.463 Truth-conditions define the room to move [Spielraum – literally “play space”] left to the facts by a proposition. (A proposition, a picture, a model, are in a negative sense like a solid body that restricts the free movement of others; in a positive sense [they are] like a space limited by solid substance wherein there is room for a body.) Tautology leaves to reality the whole – endless – logical space; contradiction fills the whole of logical space and leaves reality not a point. Neither of them, therefore, can determine reality in any way.



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[Cf. Notebooks, November 14, 1914 (Wittgenstein 1979b, 30e). Here Wittgenstein expresses a similar idea and illustrates it with diagrams.] 4.464

The truth of tautology is certain, of propositions possible, of contradiction impossible. (Certain, possible, impossible: here we have an indication of the gradation we need in the theory of probability.)

[No article before “tautology” or “contradiction” here, or in 4.463, as per Wittgenstein’s wishes in Letters to Ogden (Wittgenstein 1973, 30). No plural because “there are in fact no contradictions but there is only contradiction, for they all mean the same, i.e., nothing. And the same applies to tautology” (ibid.).] 4.465

The logical product of a tautology and a proposition says the same as the proposition [on its own]. Thus that product is identical with the proposition. Because one cannot change the essence of the symbol without changing its sense.

4.466

To a definite logical combination of symbols corresponds a definite logical combination of their meanings [Bedeutungen]; every arbitrary combination corresponds only to unconnected symbols. That is to say, propositions that are true for every situation cannot after all be combinations of symbols, because otherwise only definite combinations of objects could correspond to them. (And there is no logical combination to which no combination of objects corresponds.) Tautology and contradiction are the limiting cases of the combination of symbols, namely their dissolution.

[Where Ogden has “signs” I have “symbols” here in line with Wittgenstein 1973, 60.] 4.4661 To be sure, even in tautologies and contradictions signs are still combined with one another, i.e., they stand in relations to one another, but these relations are meaningless [bedeutungslos], inessential to the symbol. 4.5

Now it seems possible for the most general propositional form to be given: that is to say, to give a description of the propositions of any symbolism whatsoever, so that every possible sense can be expressed by a symbol that fits the description, and that every symbol that fits the description can express a sense, if the meanings [Bedeutungen] of the names are suitably chosen.

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It is clear that in the description of the most general propositional form only what is essential to it may be described, – otherwise it would not be the most general form. That there is a general propositional form is indicated [bewiesen] by the fact that there may be no proposition whose form one could not have foreseen (i.e., constructed). The general form of the proposition is: Things are thus and so.

[Ostrow: “in the transparent vacuity of this culminating statement we are meant to see the vacuity of the Frege/Russell logic, of any attempt to specify a priori the limits of thought and language” (Ostrow 2002, 114). See also Wittgenstein 1953 §136: “At bottom, giving ‘This is how things are’ as the general form of propositions is the same as giving the definition: a proposition is whatever can be true or false.” Sections 114–137 are also relevant.] 4.51

Suppose that all elementary sentences were given to me: then it can be asked simply: which sentences can I build from them? And those are all sentences and thus are they limited.

4.52

Sentences are everything that follows from the totality of all elementary sentences (and of course from the fact that this is the totality). (Thus one could in a certain sense say that all sentences are generalizations of elementary sentences.)

4.53

The general propositional form is a variable.

5

A proposition is a truth-function of elementary propositions. (An elementary proposition is a truth-function of itself.)

[Janik: If one simply reads the seven propositions that constitute the main ideas of the Tractatus consecutively one quickly comes to the realization that the book’s center is in fact propositions 5 and 6. Proposition 5 tells us that all meaningful sentences are truth functions; whereas 6 tell us that double negation is the general form of all truth functions. The force of this assertion is that all of the propositions of logic are of equal logical significance. (Janik 2006, 61)] 5.01

The elementary propositions are the truth-arguments of the proposition.

5.02

It is natural to confuse the argument of functions with the affixes of names. This is because I recognize just as well from the argument as from the affix the meaning of the sign containing it.

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In Russell’s “+c ,” e.g., “c” is an affix that indicates [hinweist] that the whole sign is the addition sign for cardinal numbers. But this signifying depends on arbitrary agreement and one could choose a simple sign instead of “+c”; in “~p” however, “p” is not an affix, but an argument: the sense of “~p” cannot be understood without the sense of “p” having first been understood. (In the name Julius Caesar “Julius” is an affix. An affix is always a part of a description of an object, to whose name we attach it. E.g., the Caesar of the genus Julius.) The confusion of argument and affix, if I am not mistaken, is at the bottom of Frege’s theory of the meaning [Bedeutung] of propositions and functions. For Frege the propositions of logic were names, and their arguments the affixes of these names.

[Russell uses “+c” in Principia vol. 2 (Whitehead and Russell 1912, 73). Black argues that Wittgenstein gets Frege wrong here (see Black 1964, 239).] 5.1

Truth-functions can be ordered in series. That is the foundation of the theory of probability.

5.101 The truth-functions of every number of elementary propositions can be written out in a schema in the following way: (TTTT) (p, q) Tautology (If p, then p; and if q, then q.) (p ⊃ p . q ⊃ q) (FTTT) (p, q) in words: Not both p and q. (~(p . q)) (TFTT) (p, q) in words: If q, then p. (q ⊃ p) (TTFT) (p, q) in words: If p, then q. (p ⊃ q) (TTTF) (p, q) in words: p or q. (p ∨ q) (FFTT) (p, q) in words: Not q. (~q) (FTFT) (p, q) in words: Not p. (~p) (FTTF) (p, q) in words: p, or q, but not both. (p . ~q: v: q . ~p) (TFFT) (p, q) in words: If p, then q; and if q, then p. (p ≡ q) (TFTF) (p, q) in words: p (TTFF) (p, q) in words: q (FFFT) (p, q) in words: Neither p nor q. (~p . ~q) or (p | q) (FFTF) (p, q) in words: p and not q. (p . ~q) (FTFF) (p, q) in words: q and not p. (q . ~p) (TFFF) (p, q) in words: q and p. (q . p) (FFFF) (p, q) Contradiction (p and not p; and q and not q). (p . ~p . q . ~q) Those truth-possibilities of its truth-arguments that verify a proposition, I will call its truth-grounds.

[Janik: The philosophical significance of the truth table method of representing propositions (as opposed to its significance as a logical decision procedure) is literally

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to show that nothing that is a proposition can be anything other than a tautology, a contradiction or an empirical proposition (T, 5.101). If this is true, not only is the Kantian notion that the propositions of philosophy are synthetic a-priori truths shown to be logically nonsensical, the Fregean notion in the Begriffschrift that logic is based upon privileged propositions designated as axioms turns out to be equally nonsensical. (Janik 2006, 61–62)] 5.11

If the truth-grounds that are common to a number of propositions are the same for a particular proposition, then we say that the truth of this proposition follows from the truth of those propositions.

5.12

In particular, the truth of a proposition “p” follows from the truth of another “q” if all truth-grounds of the second are truth-grounds of the first.

5.121

The truth-grounds of the one are contained in those of the other; p follows from q.

5.122 If p follows from q, then the sense of “p” is contained in the sense of “q.” 5.123

If a god creates a world wherein certain propositions are true then he thereby also creates a world in which all the propositions that follow from them are already true. And similarly he could create no world in which the proposition “p” is true without creating all its objects.

5.124

A proposition affirms every proposition that follows from it.

5.1241 “p. q” is one of the propositions that assert “p” and also one of the propositions that assert “q.” Two propositions are opposed to one another if asserting them both makes no senseful proposition [wenn es keinen sinnvollen Satz gibt]. Every proposition that contradicts another denies it. 5.13

We see from the structure of the propositions that the truth of one proposition follows from the truth of another.

[Zalabardo: What he says at 5.13 is that “[w]‌hen the truth of one proposition follows from the truth of others, we can see this from the structure of the propositions.” He is not saying, in addition, that when the truth of one proposition doesn’t follow from the truth of others, we can also see this from the structure of the proposi-

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tions. In other words, what Wittgenstein is claiming for his structure-inspection method is not that it can determine whether or not the truth of a proposition follows from the truth of others. He is only claiming that the method can determine that the truth of a proposition follows from the truth of others when this is in fact the case. No claim is being made about what the method can do in cases in which the truth of a proposition does not follow from the truth of others. This is a crucial detail. It suggests that, even if Wittgenstein’s structureinspection method is algorithmic, its existence won’t entail that there is a decision procedure for logical consequence. (Zalabardo 2015, 195–196)] 5.131

If the truth of a proposition follows from the truth of another, this is expressed in the relations in which the forms of these propositions stand to one another; and we certainly do not need to put them in these relations first by combining them with one another in a proposition, but rather these relations are internal and hold as soon as, and by the fact that, these propositions hold.

5.1311 When we infer q from p ∨ q and ~p, then the way of symbolizing here veils the relation of the propositional forms of “p ∨ q” and “~p.” But if instead of “p ∨ q,” e.g., we write “p | q .|. p | q” and instead of “~p” “p | p” (p | q = neither p nor q), then the internal connection becomes clear. (The fact that one can infer fa from (x) . fx shows that generality is present in the symbol “(x) . fx” itself.) 5.132 If p follows from q then I can infer from q to p; deduce p from q. The nature of the inference is to be gathered only from the two propositions. Only they themselves can justify the inference. “Laws of inference” that – as in Frege and Russell – are supposed to justify inferences, are senseless [sinnlos], and would be superfluous.

[Ostrow suggests that Wittgenstein’s concern here and elsewhere in the TLP is primarily “to shift our perspective so that we no longer feel any urge to account for why, for example, ‘q’ follows from ‘p’ and ‘p ⊃ q’ in the first place” (Ostrow 2002, 111). Proops says that Frege and Russell seem to use “laws of inference” to mean laws of logic, including axioms, not just the rules for inference in a particular axiom system (see Proops 2000, 80–86). As evidence he cites Frege 1984, 319, and Russell 1994, 515. Proops takes Wittgenstein to think that: “far from expressing truths which lie at the bottom of all valid inferences, the laws of logic are to be viewed as expressing no facts of any kind. Secondly, even if Russell’s conception of logic were correct, even if, that is to say, the laws of logic were not sinnlos, the appeal to logical laws in the explanation of valid

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inference would in any case be superfluous” (Proops 2000, 90). The concept of entailment cannot be explained in other terms, including those of derivability in a sound system, Proops says.] 5.133

All inferences are made a priori.

5.134

No other proposition can be inferred from an elementary proposition.

[Because they are independent. See 4.27 and 4.28.] 5.135

By no means can an inference be made from the existence of one situation to the existence of another quite different situation.

5.136

There is no causal nexus that justifies such an inference.

[Stenius: “By ‘causal nexus’ he obviously means the aprioristic certainty of causal connections” (Stenius 1960, 60). Frascolla connects this remark and the next with 6.37. Wittgenstein is denying “that there is any necessity in the so-called causal nexus between the events of one type, identified as causes, and the events of another type, identified as effects of those causes” (Frascolla 2007, 130). Schopenhauer refers to the causal nexus in The Fourfold Root of the Principle of Sufficient Reason. He writes: it is most important for us clearly to recognize first and foremost that the law of causality relates solely and exclusively to changes of material states, and to nothing else whatever. Consequently, it must not be introduced when these are not mentioned. Thus the law of causality is the regulator of the changes undergone in time by objects of external experience; but all these are material. Every change can take place only through another having preceded it, which is determined according to a rule, but by which it then takes place as having been necessarily brought about. This necessity is the causal nexus. (Schopenhauer 1974, 55–56, italics in the original translation)] 5.1361 We cannot infer the events of the future from those of the present. Superstition is belief in the causal nexus.

[This last sentence would be more literally rendered as “Belief in the causal nexus is superstition,” but despite Wittgenstein’s italics this remains ambiguous, whereas he has told us unambiguously what the sentence means: “I didn’t mean to say that the belief in the causal nexus was one amongst superstitions but rather that superstition is nothing else than the belief in the causal nexus” (Wittgenstein 1973, 31).



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Stokhof: “Wittgenstein’s remarks on causality should not be interpreted as claiming there is no such thing as a causal relationship in the first place. What is denied is that causality is an internal relation between situations” (Stokhof 2002, 99–100). Frascolla interprets Wittgenstein’s remarks here about causality as a rejection of some of Schopenhauer’s views. For instance, Schopenhauer’s idea that “motivation is causality seen from the inside” and that the relation cause-effect can be assimilated to the relation premise-consequence (Frascolla 2007, 133). Janik, however, sees Wittgenstein as making a basically Schopenhauerian point against Russell: for Schopenhauer we can only understand what we perceive in terms of causes, i.e., we understand perception on the basis of causal connection, which physicists designate as laws. Thus Wittgenstein would assert, following Schopenhauer, that the law of causality (i.e., the basis of pre-Kantian metaphysics) is not a law but the form of a law [see 6.32]. For this reason the law of causality cannot be extended to the whole of experience, i.e., the whole of experience as such cannot be represented; its form can only be shown. Thus the law of causality cannot be extended to explain experience as such as he [i.e., Schopenhauer] points out in the chapter explicitly dedicated to the misuse of the law of causality in his dissertation on the fourfold root of the principle of sufficient reason (Werke, III, 116). This is precisely what Russell had done in The Problems of Philosophy with respect to explaining sense data. In an argument that Russell himself admitted was weak, he asserted that the simplest assumption that coheres with our experience is that physical objects cause them. This is precisely what Wittgenstein wants to deny. (Janik 2006, 86)] 5.1362 Freedom of the will consists in the fact that it is impossible now to know future actions. We could only know them if causality were an inner necessity, like that of logical inference. – The connection between knowledge and what is known is that of logical necessity. (“A knows that p is the case” is senseless [sinnlos] if p is a tautology.)

[McGinn says that Black and Anscombe read the last sentence of the first paragraph here as noting the logical connection between “A knows that p” and “p.” However, McGinn reads it instead as follows: “The point of the final sentence of the first paragraph is that our knowledge extends only so far as what is logically entailed by what we know, and no further” (McGinn 2006, 219).] 5.1363 If it does not follow that it is true from a proposition’s being obvious to us, then obviousness is also no justification for our belief in its truth.

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[McGinn: Wittgenstein believes that the problem with any account of logic that treats the propositions of logic as substantial truths, in the way that Frege and Russell do, is that it is forced to rely on a notion of self-evidence to explain our a priori knowledge of their truth. And the problem with any appeal to a notion of selfevidence as a justification for acknowledging a proposition as true is that the truth of a proposition does not follow from its seeming to us to be self-evident. (McGinn 2006, 66–67)] 5.14

If a proposition follows from another, then this latter one says more than the former, the former less than the latter.

5.141 If p follows from q and q from p, then they are one and the same proposition. 5.142

Tautologies follow from all propositions: they say nothing.

5.143 Contradiction is the common property of propositions, what no proposition has in common with another. Tautology is the common property of all propositions that have nothing in common with one another. Contradiction vanishes, so to speak, outside, tautology inside, all propositions. Contradiction is the outer limit of propositions, tautology their substanceless center. 5.15 If Tr is the number of truth-grounds of the proposition “r,” and Trs the number of those truth-grounds of the proposition “s” that are simultaneously truth-grounds of “r,” then we call the ratio Trs: Tr the measure of the probability that the proposition “r” gives to the proposition “s.” 5.151

In a schema like the one above in no. 5.101, let Tr be the number of “T”s in proposition “r”; Trs the number of those “T”s in proposition “s” that stand in the same column as “T”s of the proposition “r.” The proposition “r” then gives the proposition “s” the probability Trs: Tr.

5.1511 There is no special object peculiar to propositions of probability. 5.152

We call propositions that have no truth-arguments in common with one another independent of one another. Propositions independent of one another (e.g., any two elementary propositions) give one another the probability ½.



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If p follows from q, then the proposition “q” gives the proposition “p” the probability 1. The certainty of a logical conclusion is a limiting case of probability. (Application to tautology and contradiction.) 5.153

A proposition is in itself neither probable nor improbable. An event occurs, or it does not occur: there is no middle ground.

5.154 In an urn there are equal numbers of white and black balls (and no others). I draw one ball after another and lay them back again in the urn. Then by this experiment I can determine that the number of black balls drawn and the number of white balls drawn get nearer to one another as the drawing goes on. That is therefore not a mathematical fact. If I now say: It is as probable that I will draw a white ball as a black, then that means: All circumstances known to me (including laws of nature assumed as hypotheses) give to the occurrence of the one event no more probability than to the occurrence of the other. That is, they give – as can be easily gathered from the explanations above – to each the probability ½. What I confirm by the experiment is that the occurrence of both events is independent of the circumstances with which I am no closer acquainted. 5.155

The unit of the proposition of probability is: The circumstances – with which I am otherwise no further acquainted – give to the occurrence of a particular event such and such a degree of probability.

5.156

Hence probability is a generalization. It involves a general description of a propositional form. Only in the absence of certainty do we need probability. – If we are indeed not completely acquainted with a fact but do know something about its form. (A proposition can indeed be an incomplete picture of a certain situation, but it is always a complete picture [i.e., a complete picture of something, or a complete picture in some sense].) A proposition of probability is, as it were, an extract from other propositions.

5.2

The structures of propositions stand in internal relations to one another.

[Cf. 5.13 and 5.131 on internal relations and deductive connections, Black notes (see Black 1964, 259).]

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5.21

We can bring out these internal relations in our means of expression by presenting a proposition as the result of an operation that produces one proposition out of others (the bases of the operation).

5.22

An operation is the expression of a relation between the structures of its result and its bases.

[Black says that “relation” here means internal relation (Black 1964, 259).] 5.23

An operation is that which must happen to a proposition in order to make another out of it.

[Black: “W. wishes to make a distinction between an operation and a function (5.251), yet the difference between the two seems at first nothing more substantial than a difference in point of view (and consequently in terminology). Mathematicians commonly use the terms ‘function’ and ‘operation’ interchangeably” (Black 1964, 259). Black again: “The important point is that W. restricts ‘operation’ to the case where the ‘bases’ and the ‘result’ (5.22) of the operation are internally related” (Black 1964, 260).] 5.231

And that of course will depend on their formal properties, on the internal similarity of their forms.

5.232

An internal relation that orders a series is equivalent to an operation by which one term is generated from another.

[Black points out that 4.1252 tells how a formal series arises (see Black 1964, 260).] 5.233

An operation cannot occur until the point where a proposition is generated from another in a logically meaningful [bedeutungsvolle] way. Thus at the point where the logical construction of the proposition begins.

5.234

The truth-functions of elementary propositions are results of operations that have the elementary propositions as bases. (I call these operations, truth-operations.)

5.2341 The sense of a truth-function of p is a function of the sense of p. Negation, logical addition, logical multiplication, etc., etc. are operations. (Negation reverses the sense of a proposition.)



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5.24

An operation shows itself in a variable: it shows how from one form of proposition one can arrive at another. It gives expression to the difference between the forms. (And what is common to the bases and the result of an operation is just the bases.)

5.241

An operation characterizes no form but only the difference between forms.

5.242

The same operation that produces “q” from “p,” produces “r” from “q” and so on. This can only be expressed by the fact that “p,” “q,” “r,” etc. are variables that give general expression to certain formal relations.

5.25

The occurrence of an operation does not characterize the sense of a proposition. An operation indeed asserts nothing, only its result does, and this depends on the bases of the operation. (Operation and function must not be confused with one another.)

5.251

A function cannot be its own argument, but the result of an operation can be a basis for that operation.

[Cf. 3.333, which also says that a function cannot be its own argument.] 5.252 Only thus is the advance from term to term possible in a formal series (from type to type in the hierarchy of Russell and Whitehead). (Russell and Whitehead did not admit the possibility of this advancing, but they made use of it again and again.)

[See Black 1964, 260, and Anscombe 1971, 130, for more criticism of Russell along these lines.] 5.2521 I call the repeated application of an operation to its own result its successive application (“O’ O’ O’ a” is the result of three successive applications of “O’ ξ” to “a”). In a similar sense I speak of the successive application of multiple operations to a number of propositions. 5.2522 The general term of a formal series a, O’ a, O’ O’ a, . . . I write thus: “[a, x, O’ x].” This bracketed expression is a variable. The first term of the bracketed expression is the beginning of the formal series, the second the form of an arbitrary term x of the series, and the third the form of the term of the series that follows immediately after x.

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5.2523 The concept of successive application of an operation is equivalent to the concept “and so on.” 5.253

One operation can cancel the work of another. Operations can neutralize one another.

5.254

An operation can vanish (e.g., the negation in “~~p”: ~~p = p).

5.3

All propositions are results of truth-operations on elementary propositions. Truth-operations are the way that truth-functions are produced from elementary propositions. In accordance with the essence of truth-operations, a new one arises from truth-functions in the same way that their truth-functions arise from elementary propositions. Every truth-operation begets from truth-functions of elementary propositions another truthfunction of elementary propositions, a proposition. The result of every truth-operation on the results of truth-operations on elementary propositions is again the result of a single truth-operation on elementary propositions. Every proposition is the result of truth-operations on elementary propositions.

5.31

The schemata in no. 4.31 also then have a meaning [Bedeutung], if “p,” “q,” “r,” etc. are not elementary propositions. And it is easy to see that the propositional sign in no. 4.42, even if “p” and “q” are truth-functions of elementary propositions, expresses a single truth-function of elementary propositions.

5.32

All truth-functions are results of the successive application of a finite number of truth-operations to elementary propositions.

5.4

Here it becomes apparent that there are no “logical objects” or “logical constants” (in Frege’s and Russell’s sense).

[Hacker identifies Wittgenstein’s target here: “Frege argued that truth and falsehood were special logical objects named by sentences. Finally, he treated the logical connectives as names of literal functions, viz. ‘not’ as the name of the concept of negation (a unary function), the binary connectives as names of relations, and the quantifiers as names of second-level functions” (Hacker 1997, 35). Russell too “enmeshed himself in confusions” according to Hacker (ibid.). In his The Principles of Mathematics (Russell 1903, vii) Russell gives the name “logical constants” to indefinable logical concepts the dis-



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cussion of which “forms the chief part of philosophical logic” (ibid., v). Russell conceives of these constants as entities with which the mind can be directly acquainted. In Theory of Knowledge, Russell argues that it is a kind of experience or immediate knowledge that allows us to understand such words as “particulars,” “universals,” “relations,” and “predicates” (see Russell 1984, 97). He adds that acquaintance with certain logical objects must be involved in our understanding words such as “or,” “not,” “all,” and “some” (see ibid., 99). By this stage, perhaps thanks to Wittgenstein, Russell denies that these things are entities (see ibid., 97), but he still thinks of them as objects of some kind.] 5.41

Because all results of truth-operations on truth-functions are identical when they are one and the same truth-function of elementary propositions.

5.42

It is obvious that ∨, ⊃, etc. are not relations in the sense that right and left etc. are. The possibility of the crosswise definition of the logical “primitive signs” of Frege and Russell shows already that these are not primitive signs, and much less still signs for relations. And it is obvious that the “⊃” that we define by means of “~” and “∨” is identical with that by which we define “∨” with the help of “~,” and that this “∨” is the same as the first, and so on.

5.43

It is scarcely credible that from a fact p infinitely more others should follow, namely ~~p, ~~~~p, etc. And it is no less remarkable that the infinite number of propositions of logic (of mathematics) follow from half a dozen “primitive propositions.” All propositions of logic say the same thing however. Namely, nothing.

[Wittgenstein suggests leaving out any translation of “von vornherein” (“from the very beginning”) in the first sentence here. See Wittgenstein 1973, 31.] 5.44

Truth-functions are not material functions. If one can produce, e.g., an affirmation by double negation, is then negation – in any sense – contained in the affirmation? Does “~~p” negate ~p, or affirm p; or both? The proposition “~~p” does not deal with negation as with an object; but the possibility of negation is already presupposed in affirmation. And were there an object called “~” then “~~p” would have to say something other than “p.” Since the one proposition would then deal with ~, the other not.

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[Schroeder: “The remark that ‘truth-functions are not material functions’ (TLP 5.44) is merely a paraphrase of the point that logical connectives do not represent things in the world” (Schroeder 2006, 72).] 5.441

This vanishing of the apparent logical constants also occurs if “~(∃x) . fx . x = a” says the same as “(x) . fx,” or “(∃x) . Fx . x = a” the same as “fa.”

5.442

If we are given a proposition then with it we are already given the results of all truth-operations that have it as their basis.

5.45

If there are primitive signs of logic then a correct logic must make clear their position with regard to one another and justify their being. The construction of logic out of its primitive signs must be made clear.

5.451

If logic has primitive concepts then they must be independent of each other. If a primitive concept is introduced then it must be introduced in every combination in which it ever occurs. One cannot therefore introduce it for one combination first and then another time for another. E.g., if negation is introduced then we must now understand it in propositions of the form “~p” in just the same way as in propositions like “~(p ∨ q),” “(∃x) . ~fx” et al. We may not introduce it first for one class of cases and then for another, because it would then remain undecided whether its meaning [Bedeutung] in each case was the same, and there would be no available ground for using the same way of combining signs in both cases. (Briefly, what Frege (Grundgesetze der Arithmetik) has said about the introduction of signs through definitions goes, mutatis mutandis, for the introduction of primitive signs.)

[In Principia Mathematica, Russell and Whitehead first introduce signs for “or” and “it is not the case that,” using “~” only where there are no quantifiers used. When they later introduce quantifiers they have to explain how these work together with the negation sign. Then they define what these combinations of signs mean. White says that Wittgenstein is objecting to this piecemeal approach (see White 2006, 89).] 5.452

The introduction of a new device in the symbolism of logic must always be an important event. No new device may be introduced into logic – with, so to speak, a wholly innocent face – in brackets or in a footnote. (Thus in the Principia Mathematica of Russell and Whitehead there appear definitions and basic laws in words. Why suddenly words



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here? This would need a justification. This is missing and must be missing, since the procedure is actually forbidden.) If, however, the introduction of a new device has proved necessary in one place, then one must ask oneself straightaway: Where must this device now always be used? Its place in logic must now be made clear. 5.453

All numbers in logic must be able to be justified. Or rather: it must be evident that there are no numbers in logic. There are no pre-eminent numbers.

5.454 In logic there is no coexistence, there can be no classification. In logic there cannot be a more general or a more specific. 5.4541 The solutions of logical problems must be simple because they set the standard of simplicity. People have always suspected that there must be a field of questions to which the answers – a priori – are symmetrical and form a closed, regular structure. A field in which the proposition holds: simplex sigillum veri.

[Proops suggests “had an inkling” where I have “suspected,” as Ogden’s “thought” “risks making it sound as though Wittgenstein regarded the idea as some kind of delusion. Pears and McGuinness’s translation: ‘mankind has always had a presentiment,’ is superior to Ogden’s, but a little grandiloquent” (Proops 2000, 27 note 80). But this reminds me of 6.3211, which uses the same root (here it is geahnt, a form of the verb ahnen (to suspect), there it is Ahnung, the noun (suspicion). It is not clear that Wittgenstein thinks there is no delusion here. The Latin means “simplicity is the hallmark of truth.” Black points out that this was a motto of Herman Boerhaave (1668–1738) of Leyden (see Black 1964, 268). Schopenhauer uses the phrase in his dialogue on religion, in which Philalethes says: “Simplex sigillum veri: naked truth must be so simple and intelligible that it can be imparted to everyone in its true shape without adulterating it with myths and fables (a mass of lies)—that is, without disguising it as religion” (Schopenhauer 1970, 106).] 5.46

If one introduced logical signs correctly, then one would also thereby have already introduced the sense of all their combinations; thus not only “p ∨ q” but already also “~(p ∨ ~q)” etc. etc. One would thereby also already have introduced the effect of all possible combinations of brackets. And thereby it would have become clear that the proper general primitive signs are not “p ∨ q,” “(∃x) . fx,” etc., but the most general form of their combinations.

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The apparently unimportant fact that logical pseudo-relations like ∨ and ⊃ need brackets – in contrast to real relations – is of great importance [bedeutungsvoll]. The use of brackets with these seemingly primitive signs indicates [deutet] already indeed that these are not real primitive signs. And surely nobody is going to believe that brackets have an independent meaning.

5.4611 Logical operation signs are punctuation marks.

[Proops: “Wittgenstein’s point is that the logical connectives share with punctuation marks the feature of lacking sense and reference while nonetheless having a meaning in their own right. The point of the comparison with punctuation is to bring out that the logical connectives make a purely structural contribution to the meanings of the sentences in which they figure” (Proops 2000, 15).] 5.47

It is clear that everything that can be said generally in advance about the form of all propositions must be able to be said all at once. Indeed all logical operations are already contained in an elementary proposition. Because “fa” says the same as “(∃x) . fx . x = a.” Where there is complexity [compoundness] there is argument and function, and where these are, are already all logical constants. One could say: the one logical constant is that which all propositions by their nature have in common with one another. That, though, is the general propositional form.

5.471

The general propositional form is the essence of the proposition.

5.4711 Giving the essence of the proposition means giving the essence of all description, thus the essence of the world. 5.472 The description of the most general propositional form is the description of the one and only general primitive sign of logic. 5.473 Logic must take care of itself. A possible sign must be able to signify. Everything that is possible in logic is also allowed. (“Socrates is identical” therefore means [heisst] nothing because there is no property that “identical” denominates. The proposition is nonsensical [unsinnig] because there is some arbitrary definition that we have not made, but not because the symbol in and of itself would be forbidden.) We cannot, in a certain sense, go wrong in logic.



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5.4731 Self-evidence, which Russell spoke so much about, can only become superfluous in logic by language itself preventing each logical mistake. – That logic is a priori consists in the fact that nothing illogical can be thought.

[Black cites Russell 1913, 490 and 492, as good sources for his views on selfevidence (see Black 1964, 274). Cf. 5.1363, which says that self-evidence to us is not justification for our believing something to be true.] 5.4732 We cannot give a sign the wrong sense. 5.47321 Occam’s razor is of course not an arbitrary rule, or one justified by its practical success: it says that unnecessary symbolic units mean nothing [nichts bedeuten]. Signs that fulfill a single purpose are logically equivalent, signs that fulfill no purpose are logically meaningless [bedeutungslos]. 5.4733 Frege says: Every legitimately constructed proposition must have a sense; and I say: Every possible proposition is legitimately constructed, and if it has no sense, then that can only be because we have given some of its parts no meaning. (Even if we believe that we have done so.) So “Socrates is identical” therefore says nothing because we have given no meaning to the word “identical” as an adjective. Since when it occurs as the sign of equality then it signifies [symbolisiert] in a wholly different way – the signifying [bezeichnende] relation is different – thus the symbol too in each case is wholly different; the two symbols have only the sign [das Zeichen] in common with one another, by accident.

[McGinn says that the reference here is to §32 of Frege’s The Basic Laws of Arithmetic (Frege 1893, see McGinn 2006, 242). She adds that “Wittgenstein’s disagreement with Frege amounts to a reassertion of the context principle, and thereby of the priority of the concept of the sense of a proposition over that of what the constituents of a proposition signify” (ibid.). Black also cites §92 of Frege 1903. Diamond: Now, presumably Wittgenstein did not think that you need the Tractatus to tell you that if there is some sign with no meaning in some combination of signs that looks as if it were meant to be a sentence, then the whole combination is not a senseful sentence. In other words, it looks as if, whatever the Tractatus may

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be telling us about what our senseful propositions are, what it is saying about nonsensical ones draws directly on a way of spotting meaninglessness which we had all along. To spot a meaningless sentence by spotting a meaningless word in it is not to apply some general principle discovered for us in the Tractatus for spotting meaninglessness. (Diamond 2006, 162–163)

That is, the Tractatus does not have anything new to tell us about the distinction between sense and nonsense. It does not offer, for instance, a theory of nonsense.] 5.474

The number of necessary fundamental operations depends only on our notation.

5.475

All that matters is to build a symbol system of a definite number of dimensions – of a definite mathematical multiplicity.

[Black points to 4.04 for more on multiplicity (see Black 1964, 275).] 5.476

It is clear that this is not about a number of primitive [or fundamental] concepts, that must be signified, but about the expression of a rule.

5.5

Every truth-function is the result of the successive application of the operation “(-----T) (ξ, ….)” to elementary propositions. This operation negates all the propositions in the right-hand brackets and I call it the negation of these propositions.

5.501

An expression in brackets whose terms are propositions I indicate – if the order of terms in the brackets is indifferent – by a sign of the form “(ξ̄ ).” “ξ” is a variable whose values are the terms of the expression in brackets, and the line over the variables indicates that it represents all its values in the brackets. (Thus if ξ has the three values P, Q, R, then (ξ̄ ) = (P, Q, R).) The values of the variables are to be determined. The determination is the description of the propositions which the variable represents. How the description of the terms of the expression in brackets is done is not essential. We can distinguish three kinds of description: 1. Direct enumeration. In this case we can put in place of the variable simply its constant values. 2. Giving a function fx, whose values for all values of x are the propositions to be described. 3. Giving a formal law, according to which those propositions are constructed. In this case the terms of the expression in brackets are all the terms of a formal series.



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[Friedlander argues that Ogden’s “determination” is preferable to Pears and McGuinness’s “stipulation,” which sounds too arbitrary (see Friedlander 2001, 77). Black suggests “prescribed” and “prescription” (see Black 1964, 276). I follow Ogden.] 5.502 Therefore I write “N(ξ̄ )” instead of “(-----T) (ξ, …..).” N(ξ̄ ) is the negation of all the values of the propositional variable ξ.

[McGinn: “The operation expressed by N(ξ̄ ) is not strictly equivalent to Sheffer’s stroke, which is a two-place operator. N(ξ̄ ) is a multi-grade operator, which jointly negates all the propositions that are the values of the variable ξ, that is, it corresponds to the operation expressed by the sign (-----T) (ξ,…)” (McGinn 2006, 232).] 5.503

It is clear that it is easily expressed how propositions can be constructed with this operation and how propositions are not to be constructed with it, so this must also be capable of exact expression.

5.51 If ξ has only one value then N(ξ̄ ) = ~p (not p), if it has two values then N(ξ̄ ) = ~p . ~q (neither p nor q). 5.511

How can the all-embracing, world-mirroring logic use such special hooks and manipulations? Only by all these being connected into [i.e., so as to form] one infinitely fine network, the great mirror.

[My parenthetical comment follows Wittgenstein 1973, 61. Friedlander 2001, 93 note 4, points to Schopenhauer’s reference to a mirror of the world: “Man . . . is the most complete phenomenon of the will, and, as was shown in the second book, in order to exist, this phenomenon had to be illuminated by so high a degree of knowledge that even a perfectly adequate repetition of the inner nature of the world under the form of representation became possible in it. This is the apprehension of the Ideas, the pure mirror of the world” (Schopenhauer 1969, vol. 1 287–288). Friedlander takes the ideas to be mirrors, but it also seems possible to read this passage as calling our apprehension of the Ideas the mirror of the world. This reading is supported by Schopenhauer 1969, vol. 2 206, where Schopenhauer writes of “the knowing part of consciousness” becoming “the clear mirror of the world.” Cf. Schopenhauer 1969, vol. 2 216 and 380. The intellect is the mirror of the world, according to Schopenhauer. Black calls this image of a mirror the dominant image of the whole of Schopenhauer’s book (see Black 1964, 27). There are forty-two entries under “mirror” in the index to Schopenhauer’s The World as Will and Representation.

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Black says cf. 4.121 (see Black 1964, 277). He notes that Anscombe identifies the great mirror with language in Anscombe 1971, 164.] 5.512 “~p” is true if “p” is false. Thus in the true proposition “~p,” “p” is a false proposition. How can the stroke “~” now bring it to agreement with reality [or: the truth]? What negates in “~p” is however not the “~” but that which is common to all signs of this notation that negate p. Thus the common rule according to which “~p,” “~~~p,” “~p ∨ ~p,” “~p . ~p,” etc. etc. (ad infinitum) are constructed. And this commonality mirrors negation. 5.513 One could say: What is common to all symbols that assert p as well as q, is the proposition “p . q.” What is common to all symbols that assert either p or q, is the proposition “p ∨ q.” And thus one can say: Two propositions are opposed to one another if they have nothing in common with one another, and: Every proposition has only one negative, because there is only one proposition that lies completely outside of it. It is evident also in Russell’s notation that “q: p ∨ ~p” says the same as “q”; that “p ∨ ~p” says nothing. 5.514

If a notation has been set down, then there is in it a rule according to which all propositions negating p are to be constructed, a rule according to which all propositions affirming p are to be constructed, a rule according to which all propositions affirming p or q are to be constructed, and so on. These rules are equivalent to the symbols and in them their sense is mirrored.

5. 515 It must be apparent in our symbols that what is connected with one another by “∨,” “. ,” etc. must be propositions. And this is indeed the case, because the symbol “p” and [the symbol] “q” itself presupposes “∨,” “~,” etc. If the sign “p” in “p ∨ q” does not stand for a complex sign, then it cannot have sense on its own; but then also the signs “p ∨ p,” “p . p,” etc., which have the same sense as “p,” could have no sense. If however “p ∨ p” has no sense, then also “p ∨ q” can have no sense.

[Black writes: “The German text is puzzling and may have been printed incorrectly. I suggest (as an alternative to Pears and McGuinness) that the remark might read: ‘And this is so, for the symbol p in p ∨ q itself presupposes “∨,” “~,” etc.’” (Black 1964, 279).] 5.5151 Must the sign of a negative proposition be constructed with the sign of a positive proposition? Why should one not be able to express a



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negative proposition by means of a negative fact. (For instance: If “a” does not stand in a specific relation to “b,” this could express the fact that aRb is not the case.) But even here the negative proposition is given indirectly by the positive proposition. The positive proposition must presuppose the existence of the negative proposition and vice versa. 5.52

If the values of ξ are all the values of a function fx for all values of x, then N(ξ̄ ) = ~(∃x) . fx.

5.521

I separate the concept all from the truth-function. Frege and Russell introduced generality in connection with the logical product or the logical sum. So it would be hard to understand the propositions “(∃x) . fx” and “(x) . fx,” in which both ideas are contained.

[See Anscombe 1971, 141–143, on this. She says that Frege and Russell did not at all explicitly do what Wittgenstein says here. The relevant Frege paper is “Function and Concept” (in Beaney 1997, 130–148) and Russell offers similar explanations of generality in his work. Frege explains his sign for generality in terms of what it means, and specifically in terms of when it means what he calls “the true.” Wittgenstein believes that the truth of a general proposition is the truth of a logical product. Hence his claim here about what he takes to be implicit in Frege and Russell. Universal propositions (“For all x, . . .”), he thinks, each say that some logical product is true, and particular propositions (“For some x, . . .”) each say that some logical sum is true. White says that Wittgenstein’s target here seems mainly to be what Russell says in Principia Mathematica (see White 2006, 94).] 5.522

What is peculiar to the symbolism of generality is first, that it points to a logical prototype, and secondly, that it emphasizes constants.

[I follow Black on the translation of hinweist as “points to” (see Black 1964, 284). For the idea of a logical prototype, see 3.315.] 5.523

The symbol of generality occurs as an argument.

[Wittgenstein explains: “Here I want to use symbol and not symbolism because I refer to the variable x or y etc. in (∃x, y) . . . and not to the whole complex of symbols as before. I own this is very dark but please leave “symbol” here and don’t make it uniform with 3.24” (Wittgenstein 1973, 49).]

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5.524

If the objects are given, then all objects are thereby also already given to us. If the elementary propositions are given, then all elementary propositions are thereby also given.

5.525

It is incorrect to render the proposition “(∃x) . fx” in words – as Russell does – as “fx is possible.” Certainty, possibility or impossibility of a situation will not be expressed through a proposition, but by the fact that an expression is a tautology, a senseful [sinnvoller] proposition, or a contradiction. That precedent, to which one would always like to appeal, must already lie in the symbol itself.

[Russell implies, e.g., in lecture 5 of Logical Atomism (Russell 1986), that if something is possible then it is sometimes true. The ground for saying it is true would then be just the kind of fact that one wanted to say was possible.] 5.526

One can describe the world completely with completely generalized propositions, which means therefore without initially coordinating any name with a particular object. In order then to get to the usual means of expression one must simply say “And this x is a” after an expression “There is one and only one x, such that. . . .”

5.5261 A completely generalized proposition is composed like every other proposition. (This is shown by the fact that in “(∃x, φ) . φx” we must mention “φ” and “x” separately. Both stand independently in signifying relations to the world, as in an ungeneralized proposition.) A characteristic of a composite symbol: It has something in common with other symbols. 5.5262 The truth or falsehood of every proposition changes something in the general structure of the world. And the range that the totality of elementary propositions would allow its structure is exactly the same as that which completely general propositions delimit. (If an elementary proposition is true, then in any case there is thereby one more true elementary proposition.) 5.53

I express identity of the object by identity of the sign, and not with the aid of an identity sign. [I express] difference of objects by difference of signs.

5.5301 Identity is patently not a relation between objects. This becomes very clear if one considers, e.g., the proposition: “(x) : fx . ⊃ . x = a.” What this proposition says, is simply that only a satisfies the function



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f, and not that only such things satisfy the function f as have a certain relation to a. One could of course say that in fact only a has this relation to a, but in order to express this we would need the identity sign itself. 5.5302 Russell’s definition of “=” is inadequate; because according to it one cannot say that two objects have all their properties in common. (Even if this proposition is never correct, it still has sense.)

[See Principia, vol. 1, definition 13.01 (Whitehead and Russell 1910) for Russell’s definition of identity, which, as Black notes (Black 1964, 292), is based on the principle of the identity of indiscernibles.] 5.5303 Incidentally: To say of two things that they are identical is a nonsense [Unsinn], and to say of one thing that it is identical with itself is to say nothing at all [gar nichts]. 5.531

Therefore I write not “f(a, b) . a = b” but rather “f(a, a)” (or “f(b, b)”). And not “f(a, b) . ~a = b,” but rather “f(a, b).”

5.532

And analogously: Not “(∃x, y) . f(x, y) . x = y,” but rather “(∃x) . f(x, x)”; and not “(∃x, y) . f(x, y) . ~x = y,” but rather “(∃x, y) . f(x, x).” (Therefore instead of the Russellian “(∃x, y) . f(x, y)”: “(∃x, y) . f(x, y) . ∨ . (∃x) . f(x, x).)

5.5321 Instead of “(x): fx ⊃ x = a” we therefore write, e.g., “(∃x) . fx . ⊃ . fa: ~(∃x, y) . fx . fy.” And the proposition “only one x satisfies f( )” reads: “(∃x) . fx: ~(∃x, y) . fx . fy.”

[Anscombe points out that Wittgenstein is here allowing a way of saying that only one thing has f. But in that case, “it is difficult to see how he could avoid a way of admitting formulae which say ‘There are only n things and m functions’ without using either ‘thing’ or ‘function’ as a function” (Anscombe 1971, 149). Yet at 5.535 the number of objects is supposed to be shown by the number of names with different references, not by some statement of how many objects there are. And what can be shown supposedly cannot be said. This looks like a problem, as Anscombe notes.] 5.533

The identity sign is thus not an essential part of the concept script.

5.534

And now we see that pseudo-propositions like “a = a,” “a = b . b = c . ⊃ a = c,” “(x) . x = x,” “(∃x) . x = a,” etc. cannot be written at all in a correct concept script.

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Thereby also all problems that were connected with such pseudopropositions take care of themselves. All the problems that arise from Russell’s “Axiom of Infinity” can be solved here. What the Axiom of Infinity is supposed to say would be expressed in language by there being infinitely many names with different meanings.

[Ogden has “disappear” for erledigen sich, while Pears and McGuinness have “This also disposes of,” but my translation is more literal. It should be read as meaning “are dealt with by themselves” or even “dispatched by themselves.” See also Russell’s Introduction to Mathematical Philosophy, where he says that if this axiom is not the case then “it must be theoretically possible for analysis to reach ultimate subjects, and it is these that give the meaning of ‘particulars’ or ‘individuals’” (Russell 1919, 142). Black notes that it sounds as though Wittgenstein wants to ban certain formulas as pseudo-propositions, but that he says in 6.2 that mathematics consists entirely of pseudo-propositions (see Black 1964, 296). So maybe he doesn’t want to ban them after all.] 5.5351 There are certain cases where one is led into the temptation to use expressions of the form “a = a” or “p ⊃ p” and such. And indeed this happens when one would like to speak of the prototype: Proposition, Thing, etc. Thus Russell in the Principles of Mathematics has rendered the nonsense [Unsinn] “p is a proposition” in symbols with “p ⊃ p” and presented it as a hypothesis in front of certain propositions whose argument places thereby could only be occupied by propositions. (It is therefore already nonsense to put the hypothesis p ⊃ p in front of a proposition in order to ensure that its arguments are of the right form, because the hypothesis for a non-proposition as argument becomes not false but nonsensical [unsinnig], and because the proposition itself becomes nonsensical [unsinnig] with the incorrect kind of argument, therefore it saves itself from incorrect arguments just as well, or as badly, as the senseless [sinnlose] hypothesis hung on it for this purpose.)

[Black suggests that “antecedent” might be a better translation of Hypothese here (see Black 1964, 297).] 5.5352 Equally, people have wanted to express “There are no things” by “~(∃x) . x = x.” But even if this were a proposition, – would it not also be true, if indeed “There were things,” but these were not identical with themselves?

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5.54

In the general propositional form propositions occur in [other] propositions only as bases of truth-operations.

[Conant and Diamond comment: At first it may seem as if Tractatus §5.54 gives us a direct method of criticizing “P entails Q.” “P” and “Q” are intended to be markers of the occurrence of propositions; but “entails” is plainly not a truth-functional connective. So it looks as if §5.54 suggests that “P” and “Q,” as they occur in “P entails Q,” are not being used as propositional signs. Since no other use has been assigned to them, the whole, “P entails Q,” is nonsensical. That argument is inadequate, as the Tractatus indicates. For, immediately after §5.54, Wittgenstein tells us, in §5.541, that a proposition may merely appear to be one in which propositions occur non-truth-functionally. What needs to be done in such a case is that the appearance of non-truth-functional occurrence has to be investigated. We need to attempt to clarify the proposition which appears to be one in which propositions occur non-truth-functionally. (Conant and Diamond 2004, 72–73)] 5.541

At first glance it seems as though there is another way in which a proposition can occur in another. Especially in certain propositional forms of psychology, like “A believes that p is the case,” or “A thinks p,” etc. Here it seems superficially as though the proposition p stands in a kind of relation to an object A. (And in the modern theory of knowledge (Russell, Moore, etc.) those propositions have been understood in just this way.)

[According to Russell, when you have a sentence such as “A believes p,” p cannot stand for a fact, since then you could only ever have true beliefs. Nor can p be a proposition, since propositions do not really exist. What you believe cannot be a logical fiction, but must be something real. So how can we have false beliefs? How can we really believe something that is not the case? For instance, in “A believes Hamlet lives in Finland” “lives in” must be treated as a verb, even though Hamlet does not live in Finland and there is no nonexistent Hamlet who does so, nor a nonexistent Finland in which Hamlet lives. Russell says he does not know how to solve this problem, at least not in Logical Atomism. Earlier, in Theory of Knowledge, he treated propositions as functions of judgments, consisting of objects that the person making the judgment is acquainted with. These objects include relations. But then the relation becomes just another object, so how is it to be related to the other objects, and how are they to be related to it? We cannot simply assume that the objects are related correctly in the judgment, since people judge falsely sometimes. These are the problems that Wittgenstein appears to have pointed out to Russell, and that Russell wrestles with in Logical Atomism.

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Contrast Frege’s view that in the proposition “Copernicus thought that the planetary orbits are circular” “the man and the thought occupy, so to speak, the same stage” (Frege 1980b, 164). Frege takes this sentence to relate two objects, a man (Copernicus) and a thought (that the planetary orbits are circular).] 5.542

It is, though, clear that “A believes that p,” “A thinks p,” “A says p” are of the form “‘p’ says p”: And here it is not a question of a coordination of a fact and an object, but rather of the coordination of facts by way of the coordination of their objects.

[Anscombe says: It is perhaps not quite right to say that “A judges p” is of the form “‘p’ says that p”; what he should have said was that the business part of “A judges that p,” the part that relates to something’s having as its content a potential representation of the fact that p, was of the form ““p” says that p”: “A believes p” or “conceives p” or “says p” must mean “There occurs in A or is produced by A something which is (capable of being) a picture of p.” We should here remember the letter to Russell in which he said he did not know what the constituents of thoughts were, but he was certain that a thought must have constituents corresponding to the words of language. (Anscombe 1971, 88)

Mounce thinks that Anscombe makes a mistake here: failing to distinguish between the contingent fact that a person A happens to have uttered p, and the non-empirical fact that p means p. Of course, the sounds or marks that make up p might have meant something else, or nothing, but given their meaning, it is not contingent that they mean p. According to Mounce: “The point is simply that B can convey to us what A says (or thinks) simply by telling us what sounds he utters. How is this possible? Well, first, because these words possess logical form; and second because, since we ourselves have a grasp of logical form, understand a language, we do not have to be told what these say; we can tell that for ourselves” (Mounce 1981, 86). Mounce also points out that what is believed is not an object in the ordinary sense, because what is believed must make sense.] 5.5421 This shows also that the soul – the subject, etc. – as it is conceived in the contemporary superficial psychology, is a nothing [Unding]. A composite soul would no longer be a soul. 5.5422 The correct explanation of the form of the proposition “A judges that p” must show that it is impossible to judge a nonsense [einen Unsinn]. (Russell’s theory does not satisfy this condition.)

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[Friedlander: “Only an internal connection between the act of thinking or judging and the constitution of the judgment is capable of explaining why a subject cannot judge what is not sense, what is nonsense” (Friedlander 2001, 113). Cf. Wittgenstein 1953 §358. Here Wittgenstein seems to be making the point that one cannot mean a senseless string of words, and so an act of meaning is not what gives sense to (otherwise meaningless) strings of words. On that idea, see 3.11.] 5.5423 To perceive a complex means to perceive that its parts are combined in such and such a way. Perhaps this also explains the fact that one can see the figure as a cube in two ways; and all similar phenomena. Because we quite truly [eben wirklich] see two different facts.

(If I look first at the corners a and only cursorily at b, then a appears in front; and vice versa.)

[Winch sees a: striking structural parallel between the Tractatus discussion of belief in Propositions 5.54 to 5.5423 and that in Philosophical Investigations, Part II, sections x and xi. In both there is a discussion of, and an attempt to resolve, an apparent logical anomaly concerning propositions reporting someone’s belief; and in both, this discussion is followed by a reference to the phenomenon of shifting aspects. (Winch 2001, 209)] 5.55

We must now answer a priori the question about all possible forms of elementary propositions. An elementary proposition consists of names. Since we cannot give the number of names with different meanings [Bedeutung], though, we cannot give the composition of the elementary proposition either.

5.551 Our fundamental principle is that every question that can be decided at all with logic must be decidable without anything else.

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(And if we get into a position where we need to answer such a problem by looking at the world, then this shows that we are on a fundamentally wrong track.) 5.552

The “experience” that we use to understand logic is not that such and such a thing is the case, but rather that something is: but that is no experience at all [eben keine Erfahrung]. Logic is before every experience – that something is thus. It is before the How, not before the What.

[Black points to 6.1222, 3.221, 6.44, 2.0271 (identifying the How with the contingent or changing), 2.024 (identifying the What with Substance), and Wittgenstein 1953 §89 (see Black 1964, 303). Presumably Wittgenstein has Russell in mind here. See comment on 5.4.] 5.5521 And if this were not so, how could we apply logic? One could say: If there would be a logic, even if there were no world, then how could there be a logic, since there is a world? 5.553

Russell said that there were simple relations between different numbers of things (individuals). But between which numbers? And how should this be decided? – By experience? (There is no pre-eminent number.)

[Black (p. 304) gives the reference here as Russell 1956, 206. Here Russell writes: “I see no particular reason to suppose that the simplest relations that occur in the world are (say) of order n, but there is no a priori reason against it.”] 5.554

The giving of any special form would be completely arbitrary.

5.5541 It is supposed to be possible to determine a priori whether I can get in the position, e.g., of having to symbolize with the sign for a 27-termed relation.

[Black says that Wittgenstein “no doubt” has Russell in mind here again, but gives no specific reference and suggests that Wittgenstein is only alleging that Russell supposes this (Black 1964, 304).] 5.5542 But may we then ask such a question at all? Can we erect a symbolic form and not know whether something could correspond to it? Does the question make sense: What must be in order that something can thereby be the case?



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[Wittgenstein says that the correct answer to the initial question “would be, that we may NOT!” (Wittgenstein 1973, 33–34). Black: “The answers to all three questions are clearly intended to be in the negative. But it is ironical to notice that the third question expresses one of the main preoccupations of the Tractatus” (Black 1964, 304).] 5.555

It is clear that we have a concept of the elementary proposition irrespective of its special logical form. However, where one can construct symbols according to a system, there this symbol is what is important logically, and not the individual symbols. And how would it even be possible that I should have to deal with forms in logic that I can invent; rather I must have to deal with what makes it possible for me to invent them.

5.556

There cannot be a hierarchy of forms of elementary propositions. Only what we construct ourselves can we foresee.

5.5561 Empirical reality is limited by the totality of objects. The limit shows up again in the totality of elementary propositions. Hierarchies are, and must be, independent of reality. 5.5562 If we know on purely logical grounds that there must be elementary propositions, then it must be known by everyone who understands the propositions in their unanalyzed form. 5.5563 All propositions of our ordinary language are in fact, just as they are, logically completely in order. – That simplest of things, that we should give here, is not a likeness of the truth, but rather the full truth itself. (Our problems are not abstract, but possibly the most concrete that there are.)

[See Wittgenstein 1953 §97, which refers to this. Propositions are just as OK in ordinary language as they are in any concept-script, “Only it is easier for us to gather their logical form when they are expressed in an appropriate symbolism” (Wittgenstein 1973, 50). On the same page, Wittgenstein says that “That simplest of things” should be an expression parallel to “the highest good” or “the good and the beautiful,” so he might be alluding to a Platonic ideal (or illusion) of pure simplicity. Anscombe points out that this contradicts Russell’s claim in his introduction to the TLP that language only has meaning “in proportion as it approaches to the ideal language which we postulate” (see Anscombe 1971, 91).]

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5.557 The application of logic decides what elementary propositions there are. What lies in the application, logic cannot anticipate. This is clear: Logic must not collide with its application. But logic must be contiguous with its application. Therefore logic and its application may not overlap each other.

[White says that what Wittgenstein writes here and in the following remark show that to discover what objects are would require an empirical investigation (see White 2006, 25). Black suggests “touch” where I have “be contiguous with” and Pears and McGuinness have “be in contact with” (see Black 1964, 306). Logic must, I take it, be in contact with its application at every point, which is what my translation aims to convey.] 5.5571 If I cannot give the elementary propositions a priori, then it must lead to obvious nonsense [offenbarem Unsinn] to try to give them.

[Black points out that wollen could be rendered as “to want” as well as “to try” here (see Black 1964, 306). Maria Balaska notes that: “Section 5.55 starts with ‘We now have to answer a priori the question about all the possible forms of elementary propositions’ and ends with 5.5571, ‘If I cannot say a priori what elementary propositions there are, then the attempt to do so must lead to obvious nonsense’” (Balaska 2019, 128). She says that Friedlander’s is the only reading of the text that both remarks on this and interprets it convincingly. Friedlander (2001, 112–113) connects “the disappearance of the thinking subject” with “the proper understanding of the most general form of the proposition, of what can be given in advance of experience. . . .” He also connects “the reappearance of the subject, that is, the way to speak of the subject in philosophy” with “understanding that the limits of experience [. . .] cannot be anticipated” (all quotations from Balaska 2019, 128). Colin Johnston says of 5.557–5.5571: “There is no saying a priori what forms of elementary propositions there are, and so no saying a priori what types (forms) of names, and so again of objects there are. A demand for an exposition of the logical categories of entity, insofar as that is a request for something to be given a priori, is misguided” (Johnston 2009, 153).] 5.6

The limits of my language mean [bedeuten] the limits of my world.

[Schroeder: note that (both in 5.6 and 5.62c) the limits of language are not said to be the limits of my world, but to mean or indicate them (bedeuten). Logic and language

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reflect the form of the world; they do not produce it (cf. TLP 5.511 and 6.13, where logic is said to mirror the world). Anyway, Wittgenstein nowhere entertains the idea that logic might be subjective in a transcendental sense: imposed upon the world by us. (Schroeder 2006, 96)] 5.61

Logic fills the world; the limits of the world are also its limits. We therefore cannot say in logic: In the world there is this and this, not that. That would seem to presuppose that we exclude certain possibilities and this cannot be the case, since otherwise logic would have to go beyond the limits of the world; that is, if it could consider these limits from the other side too. What we cannot think, we cannot think; we therefore also cannot say what we cannot think.

[McGinn points out that there is a sense of climax here, of the limits of thought having been drawn, which in the preface Wittgenstein said was the aim of the book (see McGinn 2006, 255).] 5.62

This remark provides the key to the resolution of the question, to what extent solipsism is a truth. What solipsism actually [nämlich] means [meint] is completely right, only it cannot be said, but rather shows itself. That the world is my world shows itself by the limits of language (the only language that I understand) meaning the limits of my world.

[Black says that meint should not be translated as “means” (as Pears and McGuinness, Ogden, and I have it) but as “intends” or “wants to say” (see Black 1964, 309). It means something like “thinks,” “believes,” or, perhaps, “has in mind.” On the translation of the parenthetical remark, see Anscombe: “Dr. C. Lewy has found a copy of the first edition of the Tractatus with a correction by Wittgenstein giving ‘the only language that I understand’” (Anscombe 1971, 167 footnote). Wittgenstein might have discussed solipsism because of Russell’s problems with knowledge of other minds and with understanding words’ being able to mean anything other than immediate, private experience (see Glock 1999, 444). Or Wittgenstein could have been thinking of Schopenhauer, or of Otto Weininger (see Haller 1988, 95–96). In The Problems of Philosophy, Russell writes that while solipsism “is not logically impossible, there is no reason whatever to suppose that it is true,” which is similar to Schopenhauer’s dismissal of it (Russell 1912, 10). On understanding language, and the relation between this and the self, see Russell again: “The chief importance of knowledge by description is that it

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enables us to pass beyond the limits of our private experience. In spite of the fact that we can only know truths which are wholly composed of terms which we have experienced in acquaintance, we can yet have knowledge by description of things which we have never experienced” (ibid., 59). Schopenhauer linked the riddle of existence to the connection between inner and outer, the very issue Russell seems concerned with here: I say that the solution to the riddle of the world must come from an understanding of the world itself; and hence that the task of metaphysics is not to pass over experience in which the world exists, but to understand it thoroughly, since inner and outer experience are certainly the principal source of all knowledge. I say, therefore, that the solution to the riddle of the world is possible only through the proper connexion of outer with inner experience, carried out at the right point, and by the combination, thus effected, of these two very heterogeneous sources of knowledge. (Schopenhauer 1969, vol. 1 428)

See TLP 6.5 on “the riddle” and 6.51 on the idea that solipsism cannot be refuted (as Russell and Schopenhauer believed). On solipsism and Wittgenstein generally, see Glock 1999, 446–449, where he argues that Wittgenstein was a solipsist of some kind. On the other hand, compare Rush Rhees, who knew Wittgenstein personally: “Wittgenstein has never held to solipsism, either in the Tractatus or at any other time” (quoted in Magee 1997, 337). If what solipsism means cannot be said, can it be thought? And if not, can there be a meaning here to be correct? See Morris and Dodd: “We suppose for a moment that we have caught what the solipsist means without thinking of something that cannot be said as if it could be said; but a little reflection shows us that the very idea of what is meant is also the idea of what can be said” (Morris and Dodd 2009, 269). In the mid-1930s Wittgenstein wrote, in his “Notes for Lectures on ‘Private Experience’ and ‘Sense Data’”: solipsism teaches us a lesson: It is that thought which is on the way to destroy this error. For if the world is idea it isn’t any person’s idea. (Solipsism stops short of saying this and says that it is my idea.) But then how could I say what the world is if the realm of ideas has no neighbour? What I do comes to defining the word “world.” [. . .] Couldn’t I say: If I had to add the world to my language it would have to be one sign for the whole of language, which sign could therefore be left out. (Wittgenstein 1993, 255)] 5.621

The world and life are one.

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[Schroeder takes this as an expression of idealism that is not argued for but simply taken for granted, as, he says, Schopenhauer largely took his form of idealism for granted (see Schroeder 2006, 95).] 5.63

I am my world. (The microcosm.)

5.631

There is no thinking, representing subject. If I wrote a book The World as I Found It then I would also have to report on my body in it and say which parts are subject to my will and which not, etc., that is to say, this is a means to isolate the subject, or rather to show that in an important sense there is no subject: Of it alone, that is, could there not be talk in this book. –

[Schroeder notes that the “representing subject” referred to here (“das vorstellende Subjekt”) is a Schopenhauerian term (see Schroeder 2006, 101).] 5.632

The subject does not belong to the world, rather it is a limit of the world.

[Schroeder says that Wittgenstein’s view is that there must be a subject if there is experience, but that the subject is not an object of experience (see Schroeder 2006, 97). As such the subject does not belong to the world (of experience), but can be thought of as a kind of boundary of the world or else identified with the world, with its experiences. Strictly speaking nothing can be said about the subject.] 5.633

Where in the world is a metaphysical subject to be found? You say here it is just as with the eye and the field of vision. But you do not really see the eye. And nothing in the field of vision allows the conclusion that it is seen by an eye.

[Cf. Wittgenstein 1953 on the visual room §§398–400. David Weiner says that “The ‘you’ he addresses in 5.633 is Schopenhauer. At issue are Schopenhauer’s two basic metaphors for the metaphysical subject, the eye and the limit. Wittgenstein’s point is that the eye metaphor is misleading, while the metaphor of the limit hits the nail on the head” (Weiner 1992, 60). Weiner explains what he means thus: “According to Wittgenstein, the eye suggests an empirical necessity that does not exist; the limit, on the other hand captures a logical necessity that does exist” (Weiner 1992, 64). Later Weiner adds: By introducing the “You” at 5.633, Wittgenstein suddenly shifts from monologue to dialogue. Up to this point, he has simply been making assertions. His

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voice has been like an oracle that casts out definitive truths to the world at large. Now he suddenly is arguing with an unnamed interlocutor. The reason for this change of voice is that the passage derives from a segment of Wittgenstein’s notebooks in which he is arguing against Schopenhauer. (See Notebooks 1914–1916, 79–80.) (Weiner 1992, 123–124 note 80)] 5.6331 That is, the field of vision does not have a form such as this:

[Luciano Bazzocchi argues that this diagram is wrong, i.e., not what Wittgenstein intended. The problem is that the eye is shown as being inside the visual field when it should be outside. He supports this claim with illustrations from multiple typescripts and a letter written by Wittgenstein. See Bazzocchi 2014. Friedlander: “From the point of view of representation there is no limit whatsoever. This is the point of Wittgenstein’s analogy between the visual field and the field of experience as such” (Friedlander 2001, 116). Compare Schopenhauer, The Fourfold Root: “Thus since our power of vision reaches equally in all directions, we really see everything as a hollow sphere in whose centre is our eye” (Schopenhauer 1974, 96).] 5.634

That no part of our experience is also a priori hangs together with this. Everything that we see could also be otherwise. Everything that we can describe at all could also be otherwise. There is no a priori order of things.

5.64

Here one sees that solipsism, rigorously followed through, coincides with pure Realism. The I of solipsism shrinks to an extensionless point, and the reality coordinated with it remains.

[Schopenhauer on realism: “The aim of realism is just the object without subject; but it is impossible even to conceive such an object clearly” (Schopenhauer 1969, vol. 2 12). Schopenhauer again: The fundamental mistake of all systems is the failure to recognize this truth, namely that the intellect and matter are correlatives, in other words, the one exists only for the other; both stand and fall together; the one is only the other’s reflex. They are in fact really one and the same thing, considered from two

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opposite points of view; and this one thing—here I am anticipating—is the phenomenon of the will or of the thing-in-itself (ibid., 15–16).

Weiner quotes a long passage from Schopenhauer 1969, vol. 2 193, in which Schopenhauer writes that in a sense an identity of the ideal and the real might be affirmed (see Weiner 1992, 78).] 5.641

There is therefore really a sense in which there can be non-psychological talk in philosophy of the I. The I occurs in philosophy through the fact that the “world is my world.” The philosophical I is not the human being, not the human body, or the human soul that psychology deals with, but rather the metaphysical subject, the limit—not a part of the world.

6

The general form of truth-functions is: [pˉ, ξ̄ , N(ξ̄ )]. This is the general form of propositions.

[McManus renders the idea here as that: “Every proposition is an elementary proposition or a (possibly very complex) complex proposition” (McManus 2006, 140). He sees it as problematic because no argument is really given in its favor and because it seems to bring metaphysical commitments with it. If there is a general form of propositions then there is, he suggests (see ibid., 141), a general form of the world (see 2.04). Wittgenstein’s claim is in effect that “all logical incompatibility is a matter of contradiction” (ibid., 153), but this is only a possibility, not something that Wittgenstein has proved must be the case. For instance, if a spot is red it cannot also be blue. But is “This spot is red” a contradiction of “This spot is blue”? Or could it be a kind of empirical knowledge that a spot cannot be both red and blue? Wittgenstein seemingly assumes that the meanings of the logical constants that connect elementary propositions into complex propositions are topic-neutral, the same in all contexts. He later abandoned this assumption (see, e.g., Wittgenstein 1974b, 269, and Wittgenstein 1980b, 38). White refers to 6 as “the central claim to which the book builds up” (see White 2006, 35). Yet Wittgenstein has made a mistake, according to White. He says that Russell silently corrects the text of 6 in his introduction to the Tractatus, giving “a satisfactory informal exposition of what Wittgenstein should have said” (White 2006, 152 note 43). As written, White says, “the formula at proposition 6 is radically incoherent” (ibid., 103). Wittgenstein’s notation gives us a rule for moving from a propositional variable to a proposition, but what it is supposed to do is give us a rule for moving from one proposition to the next proposition in the series. Section 6.001 fixes the problem, or as good as does so, according to White (see ibid., 103–104).

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The rest of the Tractatus up to 7 offers an investigation into, or commentary on, propositions of various kinds: propositions of logic at 6.1, of mathematics at 6.2, of natural science at 6.3, value at 6.4, and the meaning of life at 6.5.] 6.001

This says nothing else than that every proposition is the result of successive applications of the operation N'(ξ̄ ) to the elementary propositions.

6.002

If the general form is given of how a proposition is constructed, then thereby already is given also the general form of how from one proposition, by means of an operation, another one can be produced.

6.01

The general form of the operation Ω'(ηˉ ) is therefore: [ξ̄ , N(ξ̄ )]'(ηˉ ) (= [ηˉ , ξ̄ , N(ξ̄ )]). This is the most general form of transition from one proposition to another.

6.02

And so we come to numbers. I define x = Ω0, x Def. and Ω' Ωv, x = Ωv+1, x Def. According to these rules for signs we therefore write the series x, Ω' x, Ω' Ω' x, Ω' Ω' Ω' x, ….. thus: Ω0, x, Ω0+1, x, Ω0+1+1, x, Ω0+1+1+1, x, ….. Therefore I write “[Ω0, x, Ωv, x, Ωv+1, x]” instead of “[x, ξ, Ω’ ξ].” And I define: 0 + 1 = 1 Def. 0 + 1 + 1 = 2 Def. 0 + 1 + 1 + 1 = 3 Def. (and so on)

[Frascolla: The inductive definition at the outset of 6.02 is given the task of putting the abstract notion of the application of an operation at the bottom of the construction of arithmetic. In sharp opposition to the logicist programme [of Frege and Russell], number is not construed as the number of elements of a class (of the extension of a concept proper), but as the number of applications of a symbolic procedure, whose iteration gives rise to a potentially endless formal series of propositions. (Frascolla 2007, 187)]



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6.021

A number is the exponent of an operation.

6.022

The concept of number is nothing other than what is common to all numbers, the general form of the number. The concept of number is the variable number. And the concept of numerical equality is the general form of all particular numerical equalities.

6.03

The general form of integers is: [0, ξ, ξ + 1].

6.031

The theory of classes is completely superfluous in mathematics. That the generality that we need in mathematics is not the acci­ dental [contingent] kind hangs together with this.

[Russell says, “the class of all couples will be the number 2, according to our definition. At the expense of a little oddity, this definition secures definiteness and indubitableness” (Russell 1919, 18). But then the number 2 depends on the existence of a class of couples, i.e., on the existence of couples. And the number 1,000 depends on the existence of 1,000 things. And so on. But Russell admits that “Logical propositions are such as can be known a priori, without study of the actual world,” and it is not logically necessary that even one thing exists, he says, let alone 1,000, or, even worse, infinity (ibid., 204).] 6.1

The propositions of logic are tautologies.

[This implies, at least as Russell, e.g., uses the word “tautology,” that they are empty or insignificant. And that was the point of using that word, according to Dreben and Floyd 1991.] 6.11

The propositions of logic therefore say nothing. (They are analytical propositions.)

[Black says that this is the only occurrence of the word analytischen in the Tractatus, and suggests that it probably does not mean “true by definition” or “true in virtue of the meanings of its component words” here, nor is it used here in its original, Kantian sense, but rather is simply a synonym for “propositions that say nothing” (Black 1964, 319–320).] 6.111

Theories that allow a proposition of logic to seem to have content are always false. One could e.g., believe that the words “true” and “false” signify two properties among other properties, and then it would seem a remarkable fact that every proposition possesses one of these properties. This now seems to be anything but self-evident,

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just as little self-evident as the proposition “All roses are either yellow or red” would sound, even if it were true. Indeed, that proposition [i.e., a proposition of logic] now takes on completely the character of a natural scientific proposition and this is a sure sign that it has been falsely understood.

[Proops notes that here and in 6.112, 6.1231, and 6.13, Wittgenstein rejects a view called the universalist conception of logic, according to which “logic is a theory of the most general features of reality,” which he found in Frege and Russell (see Proops 2000, 1).] 6.112

The right explanation of a logical proposition must give it a unique position among all propositions.

6.113

It is the peculiar characteristic of logical propositions that one can perceive from the symbol alone that they are true, and this fact contains in itself the whole philosophy of logic. And thus it is also one of the most important facts, that the truth or falsehood of non-logical propositions cannot be perceived from the proposition alone.

[How can a tautology be true, given 4.06 (which says that propositions can only be true or false by being pictures of reality)? Proops argues that Wittgenstein means “true” “only in an honorary sense” (Proops 2000, 4 note 38). After all, at 6.125 Wittgenstein puts “true” in scare quotes, and in his Notes to G. E. Moore he says that “logical propositions are neither true nor false” and refers to “what is called the truth of a logical proposition” (Wittgenstein 1979b, 109 and 108, quoted in Proops).] 6.12

That the propositions of logic are tautologies shows the formal – logical – properties of language, of the world. The fact that its parts connecting together just so gives a tautology characterizes the logic of its parts. For propositions, connected in a specific way, to make a tautology, they must have specific structural properties. That they make a tautology when so connected shows therefore that they have these structural properties.

[I translate bestimmte as “specific,” where Pears and McGuinness have “certain” and Black (1964, 321) suggests “determinate.” “Definite,” which Ogden has, is good too.] 6.1201 That, e.g., the propositions “p” and “~p” in the combination “~(p . ~p)” give a tautology, shows that they contradict one another. That the propositions “p ⊃ q,” “p” and “q” combined with one another in the form “(p ⊃ q) . (p) : ⊃ : (q)” give a tautology, shows that q follows

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from p and p ⊃ q. That “(x) . fx : ⊃ : fa” is a tautology, shows that fa follows from (x) . fx, etc. etc. 6.1202 It is clear that one could use contradictions instead of tautologies to the same end. 6.1203 In order to perceive a tautology as such, one can, in cases in which no sign of generality occurs in the tautology, avail oneself of the following method: I write “TpF,” “TqF,” “TrF,” etc. instead of “p,” “q,” “r,” etc. I express the truth-combinations with brackets, e.g.:

and the coordination of the truth or falsity of the whole proposition and the truth-combinations of the truth-arguments with lines in the following way:

This sign, e.g., would therefore present the proposition p ⊃ q. Now I will investigate on the strength of that whether, e.g., the proposition ~(p . ~p) (the Law of Contradiction) is a tautology. The form “~ξ” gets written in our notation as:

the form “ξ . η” thus:

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So the proposition ~(p . ~q) goes thus:

If we put here “p” instead of “q” and investigate the combination of the outermost T and F with the innermost, then we get the result that the truth of the whole proposition is coordinated with all the truth-combinations of its arguments, its falsity with none of the truth-combinations. 6.121 The propositions of logic demonstrate the logical properties of propositions, by combining them into propositions that say nothing [nichtssagenden Sätzen]. One could also call this method a null method. In a logical proposition, propositions are brought into equilibrium with one another and the state of equilibrium then shows how these propositions must be logically constituted.

[A null method is a method for measuring a current (or other force) by creating a balance so that the measuring apparatus indicates zero. The strength of the current is measured not directly but by looking at what is necessary to bring about a state of equilibrium.] 6.122

From which it follows that we can go on without logical propositions, since we can perceive in an appropriate notation the formal properties of propositions with a simple look at these propositions.

[Black suggests “corresponding” for entsprechend where Pears and McGuinness have “suitable” and I have “appropriate” (see Black 1964, 324). He adds that “In a sense, every notation is suitable” (ibid.). White: “Wittgenstein goes badly astray in his development of this train of thought [i.e., that found in the 6.1s], and subsequent developments in logic [Alonzo Church, 1936] have shown that what he says at 6.122 is demonstrably false” (White 2006, 105). According to White:

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What Wittgenstein overlooks is that once he allows the possibility of there being infinitely many elementary propositions, he has to allow the possibility of quantification over infinite domains. If we then have propositions involving multiple quantifiers ranging over infinite domains, then even the most perspicuous notation may not be able to display the information that a given proposition is a tautology in a form that is surveyable by us. [. . .] This means that in this use of the concept of “showing” at least, “showing” cannot be treated as a straightforward epistemological concept. (White 2006, 107)] 6.1221 If, e.g., two propositions “p” and “q” in the combination “p ⊃ q” give a tautology, then it is clear that q follows from p. That, e.g., “q” follows from “p ⊃ q . p,” we see from these two propositions themselves, but we can also see it by combining them into “p ⊃ q . p : ⊃ : q” and now showing that this is a tautology. 6.1222 This throws light on the question why logical propositions cannot be confirmed by experience, any more than they can be confuted by experience. Not only must a proposition of logic be capable of confutation by no possible experience, but it must also not be confirmable by any such thing. 6.1223 Now it becomes clear why it has often been felt as if “logical truths” were “postulated” by us: Namely, we can postulate them in so far as we can postulate an adequate notation. 6.1224 It becomes clear now also why logic has been called the theory of forms and of inference. 6.123

It is clear: The logical laws must not themselves be subject to further logical laws. (There is not, as Russell thought, a unique law of contradiction for each “type,” rather one is enough, since it is not to be applied to itself.)

[Black quotes Russell’s Principia: “Negation and disjunction and their derivatives must have a different meaning when applied to elementary propositions from that which they have when applied to such propositions as (x). φx or (Ex).φx” (Whitehead and Russell 1910, 127, quoted in Black 1964, 326). See also Russell 1956, 63: “The first difficulty that confronts us [after adopting the “vicious circle principle”] is as to the fundamental principles of logic known under the quaint name of “laws of thought.” “All propositions are either true or false,” for example, has become meaningless. If it were significant, it would be a proposition, and would come under its own scope.”]

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6.1231 The mark of a logical proposition is not its general validity. To be general indeed means only: to happen to be valid for all things. An ungeneralized proposition can indeed be just as tautologous as a generalized one.

[Cf. 6.031 on kinds of generality.] 6.1232 One could call logical general validity essential, in contrast to accidental general validity, for instance of the proposition “All men are mortal.” Propositions like Russell’s “axiom of reducibility” are not logical propositions, and this explains our feeling that, if true, they could be true only by a propitious accident.

[Russell’s Axiom of Reducibility says that any higher-order property or proposition can be reduced to an equivalent first-order one. It means that the same class is determined by two propositional functions that are equivalent. (See Griffin 2003, 299.) Every propositional function is thus logically equivalent to a predicative function. The axiom is introduced because without it the ramified theory of types makes certain mathematical proofs impossible. This kind of thing belongs to logic, in Russell’s view, because it is necessary in order to overcome logical, not just mathematical, paradoxes, e.g., about the class of classes not members of themselves, as well as paradoxes about infinity. Ramsey and Wittgenstein showed that the axiom was not necessary.] 6.1233 A world can be conceived in which the axiom of reducibility does not hold. But it is clear that logic has nothing to do with the question whether our world really is thus or not. 6.124

Logical propositions describe the frame [Gerüst] of the world, or rather they present it. They “deal” with nothing. They presuppose that names have meaning and elementary propositions sense: And this is their connection with the world. It is clear that it must show [anzeigen] something about the world that certain combinations of symbols – which essentially have a specific character – are tautologies. Herein lies the decisive thing. We said that much in the symbols that we use is arbitrary, much not. In logic only this latter expresses anything: That means however that in logic we do not express what we want with the aid of signs, but rather in logic the nature of the essentially necessary signs exhausts itself: If we are acquainted with the logical syntax of some symbolism, then all the propositions of logic are already given.

[Black points out that the image of scaffolding (Gerüst) occurs also at 3.42 and 4.023 (see Black 1964, 331).]

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6.125

It is possible, in fact also according to the old conception of logic, to give in advance a description of all “true” logical propositions.

6.1251 Hence there can also never be surprises in logic. 6.126

One can figure out whether a proposition belongs to logic by figuring out the logical properties of symbols. And this is what we do when we “prove” a logical proposition. Because without concerning ourselves with a sense [Sinn] and a meaning [Bedeutung] we construct the logical proposition from others according to mere rules for signs. The proof of a logical proposition consists in our being able to establish it from other logical propositions with successive applications of certain operations, which produce ever more tautologies from the first one. (In fact from a tautology only tautologies follow.) Of course this way of showing that its propositions are tautologies is thoroughly inessential to logic. Because the propositions, from which the proof starts out, must show, indeed without proof, that they are tautologies.

[Frascolla: If we take the expression “logical proof” in its more general meaning, and if we call “mechanical” any procedure of calculation, understood as a set of effective instructions for manipulating symbols, then in the light of Church’s Theorem of Undecidability of the first-order predicative calculus, Wittgenstein’s thesis is simply false, since no mechanical procedure can exist which enables us to decide, given any arbitrary formula of the first-order predicative calculus (which is included in the calculus of Principia Mathematica), whether it is a tautology or not. (Frascolla 2007, 141)] 6.1261 In logic process and result are equivalent. (Hence no surprises.) 6.1262 Proof in logic is only a mechanical means to make perception of a tautology easier in cases where it is complicated.

[McManus: “ontological distinctions are now [with the right mechanical means, i.e., Begriffsschrift] shown simply in that ‘they’ are shown up for what they are: namely, the confused product of word-play” (McManus 2006, 85).] 6.1263 It would indeed be all too remarkable if one could prove a senseful [sinnvollen] proposition logically from another, and a logical proposition too. It is clear in advance that the logical proof of a senseful proposition and proof in logic must be two completely different things.

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6.1264 A senseful proposition states something, and its proof shows that it is so; in logic every proposition is the form of a proof. Every proposition of logic is a modus ponens presented in signs. (And one cannot express the modus ponens with a proposition.)

[Modus ponens is argument of the form: If p then q, p, therefore q. Why can’t this be expressed in a proposition? Perhaps because modus ponens is an argument form, not an argument.] 6.1265 One can always conceive of logic in such a way that every proposition is its own proof. 6.127

All propositions of logic have equal rights [i.e., (legal) status], there are among them no essential basic laws or derived propositions. Every tautology shows by itself that it is a tautology.

6.1271 It is clear that the number of “logical basic laws” is arbitrary, since one could indeed derive logic from a single basic law, by simply, e.g., forming the logical product of Frege’s basic laws [Grundgesetzen]. (Frege would perhaps say that this basic law is now no longer immediately self-evident. But it is remarkable that so exact a thinker as Frege appealed to the degree of self-evidence as the criterion of a logical proposition.) 6.13

Logic is not a theory [doctrine, science], but rather a mirror-image of the world. Logic is transcendental.

[See also 6.421, where ethics is also said to be transcendental. Wittgenstein’s use of the word “transcendental” gives support to the view that he belongs to the Kantian tradition in philosophy. Sami Pihlström argues that Wittgenstein sees ethics, logic, and religion as playing “the ‘transcendental’ role of providing the fundamental framework through which [. . .] a person sees, or is able to see, the world in general and everything contained in it” (Pihlström 2020, 226). On this view, ethics itself, as well as logic, is “in itself linguistically inexpressible” (Pihlström 2020, 226). Both ethics and logic are transcendental, but neither is transcendent. On the other hand, Pihlström also points out that 6.41 says that the sense of the world must lie outside it, which does at least sound like a reference to transcendence (see Pihlström 2020, 225). Wittgenstein also refers to what is higher (in 6.42) and the mystical (in 6.44), which adds support to the idea that he thinks some things, quite possibly including ethics, are transcendent.

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A different, less Kantian, idea about what Wittgenstein might mean by “transcendental” comes from Diamond. (For her thoughts on similarities and differences between Kant and Wittgenstein in connection with the word “transcendental” see Diamond 2000, 168.) She writes that: The comparison Wittgenstein makes between logic and ethics, in speaking of both as “transcendental” [. . .] has at its heart a contrast: between propositions with a specific subject matter, and logic/ethics, “symbolized,” as it were, by the variable for every particular thing we might say, a variable none of whose values is a proposition with logical or ethical subject matter. There is not, on this view, a “moral vocabulary,” a vocabulary through which we mean moral things. If one wanted to give sense to “moral vocabulary” one might mean: vocabulary we use in saying things that might have applications in moral life, but that excludes no words. (Diamond 1996, 252–253)

Piergiorgio Donatelli, after quoting this passage, says: “The point here, which Diamond develops in many places also as an independent point about moral philosophy, is that ethics is a sphere which we do not sort out by subject matter but by use” (Donatelli 2013, 214). Elsewhere, Diamond says that “just as logic is not, for Wittgenstein, a particular subject, with its own body of truths, but penetrates all thought, so ethics has no particular subject matter; rather, an ethical spirit, an attitude to the world and life, can penetrate any thought or talk” (Diamond 2000, 153). For more on this see the comment on 6.43.] 6.2

Mathematics is a logical method. The propositions of mathematics are equations, hence pseudopropositions.

[Black quotes Ramsey saying that “this is obviously a ridiculously narrow view of mathematics, and confines it to simple arithmetic” (Black 1964, 341).] 6.21

A proposition of mathematics expresses no thoughts.

6.211

In life it is indeed never the mathematical proposition that we need, but rather we use mathematical propositions only in order to infer from propositions that do not belong to mathematics to others that likewise do not belong to mathematics. (In philosophy the question “To what end do we really use this word, that proposition?” leads time and again to valuable insights.)

6.22

The logic of the world, which the propositions of logic show in tautologies, mathematics shows in equations.

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[White: “To understand what Wittgenstein means by ‘equations,’ we need to refer back to 4.241–4.242. There they are described as only ‘representational devices’ and that is what we need to understand if we are to interpret the claim that they are ‘pseudo-propositions’” (White 2006, 109–110).] 6.23

If two expressions are combined with the identity sign, this means that they are substitutable for one another. But whether this is the case must be evident in the two expressions themselves. It is a characteristic of the logical form of two expressions that they are substitutable for one another.

6.231

It is a property of assertion that one can understand it as double negation. It is a property of “1 + 1 + 1 + 1” that one can understand it as “(1 + 1) + (1 + 1).”

6.232

Frege says the two expressions have the same meaning [Bedeutung] but different senses [Sinn]. But the essence of an equation is that it is not necessary in order to show that the two expressions, which the equals sign combines, have the same meaning [Bedeutung], since this can be seen from the two expressions themselves.

6.2321 And, that the propositions of mathematics can be proved, means indeed nothing other than that their correctness can be seen without it being necessary to compare what they express with the objects in order to determine its correctness. 6.2322 The identity of the meaning [Bedeutung] of two expressions cannot be asserted. Because in order to be able to assert something about their meaning, I must be acquainted with their meaning: and in being acquainted with their meaning, I know whether the meaning is the same or different. 6.2323 An equation marks only the standpoint from which I regard the two expressions, namely the standpoint of their equivalence of meaning. 6.233 The question whether one needs intuition to solve mathematical problems must be answered by the fact that language itself here supplies the necessary intuition. 6.2331 The process of calculation brings about precisely this intuition. Calculation is not an experiment. 6.234

Mathematics is a method of logic.



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[Cf. 6.2, which makes the same point.] 6.2341 The essence of the mathematical method is to work with equations. On this method depends the fact that every proposition of mathematics must go without saying. 6.24

The method of mathematics, to get to its equations, is the method of substitution. Since equations express the substitutability of two expressions and we proceed from a number of equations to new equations by substituting other expressions in accordance with the equations.

6.241

So the proof of the proposition 2 x 2 = 4 runs thus: (Ωv)μ'x = Ωv x μ'x Def., Ω2×2'x = (Ω 2)2'x = (Ω 2)1+1'x = Ω 2' Ω 2'x = Ω 1+1' Ω 1+1'x = (Ω'Ω)'(Ω'Ω)'x = Ω'Ω'Ω'Ω'x = Ω 1+1+1+1'x = Ω4'x.

[Black calls Wittgenstein’s supposed proof here eccentric and incomplete (see Black 1964, 343).] 6.3

The exploration of logic means the exploration of all regularity [lawfulness]. And outside logic everything is accidental.

6.31

The so-called law of induction cannot in any event be a logical law, since it is clearly a senseful [sinnvoller] proposition. – And therefore it also cannot be an a priori law.

[Black points out that there is more than one “law of induction” that Wittgenstein might have had in mind here (see Black 1964, 345).] 6.32

The law of causality is not a law, but rather the form of a law.

[Mounce says that the law of causality is the law of sufficient reason, i.e., the idea that everything has a cause (see Mounce 1981, 75–76). This, as he reads Wittgenstein, is not a law because it tells us nothing about the world. So far as two events can be distinguished, they must have some difference, and this difference can always be regarded as causally relevant. Saying “everything has a cause” then is not really reporting on a contingent generality but insisting a priori that every event will be interpreted as caused. Black points out that at 2.033 and 2.151 Wittgenstein links form with possibility (see Black 1964, 345). If he is talking about the possibility of a certain kind of empirical generalization then, Black thinks, this fits with 6.321–6.34, but not with 6.36. He adds that by 6.36 “the ‘law of causality’ has been emptied of any determinate meaning” (ibid., 345).]

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“Law of causality” – that is a generic name. And as we say there are minimum laws in mechanics, – such as that of the least action – so there are in physics laws of causality, laws of the causality form.

[On the law of least action, Black quotes Mach (1942, 460): “Maupertuis really had no principle, properly speaking, but only a vague formula, which was forced to do duty as the expression of different familiar phenomena not really brought under one conception” (quoted in Black 1964, 346). Mach adds that Euler changed the principle (i.e., the law of least action) “into something new and really serviceable” (Mach 1942, 551).] 6.3211 There was indeed also a presentiment that there must be a “law of least action,” before it was known precisely how it went. (Here, as ever, the a priori certain proves to be something purely logical.) 6.33

We do not believe a priori in a law of conservation, but rather we know a priori the possibility of a logical form.

[Black points out that “if form is itself a possibility, the phrase [i.e., “the possibility of a logical form”] shows redundancy” (Black 1964, 346).] 6.34

All these propositions, like the principle of sufficient reason, of continuity in nature, of least expenditure in nature, etc. etc., all these are a priori insights concerning the possible fashioning of propositions of science.

[Black says that in a letter to Russell Wittgenstein treats “principle of sufficient reason” and “law of causation” as synonymous (see Black 1964, 346).] 6.341

Newtonian mechanics, e.g., brings the description of the world to a unified form. Let us think of a white surface with irregular black spots on it. Now we say: Whatever kind of picture these spots produce, I can always describe it as closely as you like by covering the spots with a suitably fine square netting and now say of every square that it is white or black. In this way, I will have brought the description of the spots to a unified form. This form is arbitrary, since I could have used with the same success a net with triangular or hexagonal holes. It is possible that the description would have been simpler with the help of a triangular net; meaning that we could have described the spots more closely with a bigger triangular net than with a finer square one (or vice versa), and so on. Different systems of world description correspond to different nets. Mechanics defines a form of world description by saying: All propositions of the description of



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the world must be obtained from a number of given propositions – the axioms of mechanics – in a given way. In this way it supplies the building stones for the construction of the scientific edifice and says: Whatever edifice you want to build, you must somehow put together with these and only these building stones. (With the system of mechanics, one must be able to write down any arbitrary proposition of physics, as one can [write down] any arbitrary number with the number system.) 6.342

And now we see the relative position of logic and mechanics. (One could also have a net consisting of different kinds of shapes, such as triangles and hexagons.) It says nothing about a picture, such as the one mentioned above, that it can be described by a net of a given form. (Since this goes for every picture of this kind.) However it does characterize the picture that it can be completely described with a specific net of a specific fineness. Thus too it says nothing about the world that it can be described with Newtonian mechanics; but [it does say something] that it can be described in that particular way in which indeed it is described. It also says something about the world that it can be described more simply with one mechanics than with another.

6.343

Mechanics is an attempt to construct according to one plan all true propositions that we need for a description of the world.

6.3431 Through the whole logical apparatus, the physical laws still speak of the objects of the world.

[Wittgenstein says the word “through” at the beginning of this sentence means the same as in “I speak through a tube” (Wittgenstein 1973, 35).] 6.3432 We must not forget that the description of the world with mechanics is always completely general. In it there is never, e.g., talk of particu­ lar material points, but rather always only about any such points.

[Pears and McGuinness have “point-masses” for “material points.”] 6.35

Although the spots in our picture are geometrical figures, geometry can still obviously say absolutely nothing about their actual form and position. But the net is purely geometrical, all its properties can be given a priori. Laws, like the principle of sufficient reason, etc., deal with the net, not with what the net describes.

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6.36

If there were a law of causality, then it could read: “There are laws of nature.” But of course one cannot say that: it shows itself.

6.361

In Hertz’s way of speaking, one could say: Only regular [law-like, lawful] combinations [connections] are thinkable.

[Pears and McGuinness have “subject to law” instead of “regular.” Ogden has “uniform” even though Wittgenstein said this was wrong and told him to look up the English translation of Hertz (see Wittgenstein 1973, 35). Unfortunately, as Black says, the allusion to “Hertz’s terminology” is obscure (see Black 1964, 362).] 6.3611 We cannot compare any process with the “passage of time” – there is no such thing – but rather only with another process (perhaps with the working of the chronometer). Hence the description of a temporal process is only possible if we rely on another process. Exactly the same kind of thing goes for space. Where one, e.g., says that neither of two events (that are mutually exclusive) can occur, because no reason exists why the one rather than the other should occur, there it is really a matter of one’s not being at all able to describe one of the two events without some asymmetry existing. And if there is such an asymmetry, then we can understand this as the reason for the occurrence of the one and the non-occurrence of the other.

[Nordmann criticizes Wittgenstein for being “caught in the present” in the Tractatus (see Nordmann 2005, 128–133). He regards the book as being written in the subjunctive mood. It is, he thinks, a kind of reductio.] 6.36111 The Kantian problem of the right and left hand, that one cannot make cover each other, exists already in a plane, indeed in onedimensional space, where the two congruent figures a and b also cannot be made to cover each other without moving them outside this space: ---o——x--x——o--                                           a          b Right and left hand are actually completely congruent. And the fact that one cannot make one cover the other has nothing to do with it. A right-hand glove could be put on the left hand if one could turn it around in four-dimensional space.

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[This problem is discussed by Kant in the Prolegomena (Kant 2004). Schopenhauer refers to it twice in the Fourfold Root (see Schopenhauer 1974, 40 and 194). His claim is that the difference between a left glove and a right glove can only be seen, not explained from concepts alone. That is, it cannot “be made intelligible except by means of intuition” (Schopenhauer 1974, 194). The idea is that space is thereby shown to be an a priori form of intuition, since such differences in space must be intuited. Wittgenstein writes: The figure [should] be thus -----o——x-----x——o                                                     a               b I.e., the little cross and the circle should be just at the end of each line and not at a distance from it and they should be quite small; for they should just mark that the two ends of each line are different. (Wittgenstein 1973, 61–62)

This is different from the configuration given in 6.36111, but presumably that is not Wittgenstein’s intention. Henk Visser says that what Wittgenstein says about Kant is conspicuously similar to what Mach says on the same issue, and Mach in turn credited Möbius as the source of the idea (see Visser 1982). Black says that according to Kant things such as left and right hands or gloves can be “exactly alike in all spatial respects” and yet do not fit the same space, i.e., they are different. In intuition we get the difference, but the understanding cannot (according to Kant) grasp it. Black: Kant used the argument several times—and to prove opposite conclusions. It was omitted from the second edition of the Critique—because, according to Kemp Smith, Kant had realized it was based “upon a false view of the understanding” (op. cit. p. 165). W. says that the impossibility of making counterparts fill the same space (at least without entry into a higher dimension) leaves their congruence unchallenged. But Kant would readily have agreed: W. does nothing to explain how the congruent counterparts can be numerically distinct, which was Kant’s puzzle. On the face of it, the possibility of non-­identical counterparts does not square with 6.3611 (3)—unless we take W. to be suggesting that the counterparts must have different causal antecedents by which alone they can be distinguished? (And this is now close to Kant’s conclusion.) (Black 1964, 363–364)] 6.362

What can be described can also take place, and what is supposed to be excluded by the law of causality cannot even be described.

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The process of induction consists in our assuming the simplest law that can be brought into unison with our experiences.

[Black says that this resumes the argument of 6.31, but is not consistent with it (see Black 1964, 365).] 6.3631 This process though has no logical, but only a psychological grounding. It is clear that there exists no ground for believing that the simplest case will also be the actual one that comes to be. 6.36311 That the sun will rise tomorrow is an hypothesis; and that means: we do not know whether it will rise. 6.37

There is no force such as to necessitate one thing’s happening on the strength of another thing’s having happened. There is only a logical necessity.

[I.e., the only kind of necessity that there is, is logical necessity. Anscombe says that this view about necessity is a direct consequence of the picture theory, and seems to think that it has nothing else to be said for it (see Anscombe 1971, 80). The picture theory regards sentences as pictures of facts and holds that “nothing but picturable situations can be stated in propositions” (Anscombe 1971, 19). This theory is standardly attributed to Wittgenstein, but not by resolute readers (see, e.g., Kremer 2001, 42).] 6.371

At the root of the whole modern worldview [Weltanschauung] lies the illusion [Täuschung] that the so-­called laws of nature are the explanation of natural phenomena.

6.372

Thus they stop at laws of nature as at something sacrosanct, as the ancients stopped at God or fate. And indeed they are both right, and wrong. The ancients are certainly clearer in so far as they recognize a clear conclusion, whereas in the new system it is supposed to seem as if everything were explained.

6.373

The world is independent of my will.

[According to Schopenhauer, all idea, all object, is phenomenon, but only the will is noumenon or thing-­in-­itself. He writes that: “Now, if this thing-­in-­ itself (we will retain the Kantian expression as a standing formula)—which as such is never object, since all object is its mere appearance or phenomenon,



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and not it itself—is to be thought of objectively, then we must borrow its name and concept from an object . . .” (Schopenhauer 1969, vol. 1 110). Magee says that Schopenhauer was aware of a difficulty here, discusses it at length, and decides only “with some misgiving” to give it a name: will (see Magee 1997, 140–144). Magee thinks the problem would be solved if only Schopenhauer gave the noumenon a less misleading name, such as force or energy. According to Schopenhauer, the will is not a phenomenon, so it can never be a cause. The will does not cause actions. In a sense the will (noumenon) is the body (phenomenon), so acts of the body are acts of the will. The relation between acts of will and bodily acts is one of identity, not cause and effect. “The will as thing-­in-­itself is quite different from its phenomenon, and is entirely free from all the forms of the phenomenon . . .” (Schopenhauer 1969, vol. 1 112). Hence it is independent of time, space, and causality. Plurality cannot apply to it, since this concept belongs to the phenomenal, to the realm of time and space. (Though surely unity cannot apply to it either, for the same reason.) “I wish it had been possible for me by clearness of explanation to dispel the obscurity that clings to the subject-­matter of these thoughts. But I see quite well that the reader’s own observation must help me a great deal, if I am not to remain uncomprehended or misunderstood” (Schopenhauer 1969, vol. 1 145). Anscombe sees the independence of the will and the world here and in 6.374 as undesirable consequences of the picture theory and the theses about modality that it implies. See Anscombe 1971, 80–81.] 6.374

Even if all that we wished happened, then this would still only, so to speak, be a gift of fate, since there is no logical connection between will and world that would guarantee this, and the assumed physical connection itself we could surely not in turn will.

6.375

As there is only a logical necessity, so there is also only a logical impossibility.

[Hacker (1997, 50): “Wittgenstein’s explanation of the nature of logical necessity is, in a sense, the high point of the Tractatus. The nature of necessity had bewildered and bedazzled philosophers for twenty-­five centuries.” Hacker identifies two main types of theory: Platonism, which understands necessary truths to be “descriptions of the properties and relations of special kinds of entities,” and psychologism, which treats them as being about human ideas or ways of thinking. According to Hacker:

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The Tractatus eschewed both psychologism and Platonism. All necessity is logical necessity (TLP, 6.375), and logical necessity is a matter of tautologousness. Tautologies are neither about mental objects nor about abstract Platonic objects. They do not describe the laws of human thinking or the putative super-­ physical laws of abstract entities. Nor do they state what are the most general features of the universe. For tautologies describe nothing, state nothing, are about nothing. (Ibid.)] 6.3751 It is impossible, that is, logically impossible, for, e.g., two colors to be at the same point in the field of vision, since this is excluded by the logical structure of color. Let us consider how this contradiction presents itself in physics. Approximately thus: That a particle cannot have two velocities at the same time; meaning that it cannot be in two places at the same time; meaning that particles in different places at one time cannot be identical. (It is clear that the logical product of two elementary propositions can be neither a tautology nor a contradiction. The expression that a point in the field of vision has two different colors at the same time is a contradiction.) 6.4

All propositions are of equal value.

6.41

The sense of the world must lie outside of it. In the world everything is as it is and happens as it happens; there is no value in it – and if there were, then it [i.e., this value] would be of no value. If there is a value, which is of value, then it must lie outside all happening and being-­so. Since all happening and being-­so is accidental. What makes it non-­accidental cannot lie in the world, since otherwise this would again be accidental. It must lie outside the world.

[Stokhof: “Neither the Tractatus nor the Notebooks contains any argument or reasoning to establish the existence of values or their absolute character. (Analogously, there is no argument for the absolute status of logic either.) In other words, the entire construction is based on a certain kind of experience” (Stokhof 2002, 239).] 6.42

Hence there can also be no propositions of ethics. Propositions can express nothing higher.

6.421

It is clear, that ethics cannot be articulated. Ethics is transcendental. (Ethics and aesthetics are one.)

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6.422

The first thought at the setting up of an ethical law of the form “thou shalt . . .” is: And what then, if I don’t do it? It is clear, however, that ethics has nothing to do with punishment and reward in the ordinary sense. Therefore this question as to the consequences of an act must be irrelevant. – At least these consequences should not be events. Since something must be right in the putting of this question. There must certainly be a kind of ethical reward and ethical punishment, but these must lie in the action itself. (And this too is clear, that the reward must be something agreeable, the punishment something disagreeable.)

[Kremer: In setting up an ethical law we are trying to establish a principle that can serve as both a guide and a justification for our lives. Yet as soon as the principle is set up as a proposition it can be called into question; as a proposition it ceases to have value for us, since it becomes simply another thing that can be intelligibly asserted but also intelligibly denied. If true, it is only contingently true, and therefore it cannot determine for us what we must do—it cannot make what we do “non-­accidental” (6.41). (Kremer 2001, 55)] 6.423

Of the will as the bearer of the ethical, nothing can be said. And the will as a phenomenon is interesting only to psychology.

[This sounds very Schopenhauerian. See Wiggins 2004 on this.] 6.43

If good or evil willing alters the world, then it can only alter the limits of the world, not the facts; not that which can be expressed through language. In short, the world must then thereby become an altogether different one. It must, so to speak, wane or wax as a whole. The world of the happy is a different one from that of the unhappy.

[Pears and McGuinness have “happy man,” though Wittgenstein explicitly asked Ogden to remove the word “man” from the translation (see Wittgenstein 1973, 35). Diamond compares Wittgenstein’s ethical view with those of G. K. Chesterton and William Wordsworth: “Wittgenstein, like some other writers, speaks of two different as it were attitudes to the world as a whole; he refers to them as that of the happy and that of the unhappy. The happy and the unhappy as it were inhabit different worlds (Tr. 6.43; cf. also Notebooks, pp. 73–86)” (Diamond 2000, 153). Kremer compares Wittgenstein’s thinking on ethics with the views of St Paul and St. Augustine. They see it as sinful pride to think that we can justify

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or save ourselves by doing, or choosing to believe, this or that. Humility involves giving up this aspiration. The Tractatus might help us achieve humility if it can show us that our attempts at self-­justification are all futile. See Kremer 2001, 47–52. Schopenhauer contrasts altruism with egoism in a way that comes readily to mind when reading TLP 6.43. Egoism concentrates, while altruism expands, he says. See Schopenhauer 1969, vol. 1 373–374, and Young 2005, 229–231. Young writes: That Wittgenstein’s waxing/waning metaphor so strongly recalls Schopenhauer’s expansion/contraction metaphor makes it look as though Wittgenstein’s person of “good will” is the Schopenhauerian altruist and the person of “bad will” is the Schopenhauerian egoist. In fact, though, I think, only the second half of this equation holds. What Wittgenstein really means by the “good exercise of the will” is a version of asceticism, of Schopenhauer’s “denial of the will.” (Young 2005, 230)

The issue would then be not altruistic willing, but rather giving up willing altogether, as far as that can be done. According to Schopenhauer, we need not only detachment from desire (Stoicism) but the abandonment of desire (Cynicism). Wittgenstein seems to have lived like a Cynic, choosing poverty and asceticism. See Young 2005, 232, where he cites Schopenhauer 1969, vol. 2 155–156. Mounce: “Wittgenstein does not mean that the ethical attitude is itself a matter of temperament. On the contrary, one’s temperament is just another of the facts toward which one has to adopt an ethical attitude” (Mounce 1981, 96). (But, Friedlander asks, “what is an attitude toward the world, and in what sense is it not part of psychology?” [Friedlander 2001, 197–198].) The stuff about the world of the happy is only an analogy, Mounce insists. Anscombe calls the will that alters the limits of the world but effects nothing in it “chimerical” (Anscombe 1971, 172). Will, like intention, she suggests, resides in what we do. See also Wittgenstein 1953 §644, which brings up a similar idea. Anscombe says that Schopenhauer identifies the world with my will, and regards both as bad (see Anscombe 1971, 172 footnote), whereas Wittgenstein sees the world as good and independent of my will. Schopenhauer’s idea of a good will is one that denies itself. Wittgenstein’s good will is not concerned with how things are; it accepts the world as it is, however it is, “and in that sense is like Schopenhauer’s good will” (ibid.). The idea that the will cannot change the facts has a biographical underpinning according to Schroeder (see Schroeder 2006, 102–103). Wittgenstein was manning the searchlight at night on a boat in Russian territory

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and, consequently, a target for the enemy. Clearly he was not in control of his fate at this time. James C. Klagge takes a similar view, pointing out that Wittgenstein only mentions God and death in his notebooks after he was shot at for the first time (on April 29, 1916). See Klagge 2001, 7–12. Hintikka asks of 6.43: Where does this mystical-­sounding view come from? The right answer is: G. E. Moore. A happy person is the one who enjoys the most valuable “unmixed goods.” According to Moore, they include such experiences as the pleasures of personal experience and the contemplation of beautiful objects. But what happens if another person—an unhappy one —does not have the same experience as I do? Moore answers firmly, true to his realist doctrine of the objects of experience, that such a person literally perceives a different object. If we simply add to these Moorean conclusions the Wittgensteinian idea that the objects that constitute the substance of my world are the object of my experience, we must conclude literally that the unhappy philistine has different objects in his world than my objects of experience. (Hintikka 2000, 15)] 6.431

As too at death the world does not change, but rather stops.

6.4311 Death is not an event in life. One does not live through death. If one understands eternity not as an endless period of time but as timelessness, then he who lives in the present lives eternally. Our life is just as endless as our field of vision is limitless.

[Wittgenstein suggested “without limit” instead of “limitless,” but only on the grounds that “limitless” is not normal English. I think it is, so I have left it in. It is a more literal translation of the German grenzenlos.] 6.4312 The temporal immortality of the soul of man, meaning therefore its eternal survival even after death, is not only in no way guaranteed, but in the first place this assumption does not at all do what people have always wanted to achieve with it. Is a riddle thereby solved, because I survive eternally? Is eternal life, on this account, then not just as mysterious as the present one? The solving of the riddle of life in space and time lies outside of space and time. (It is indeed not problems of natural science that are to be solved.)

[“Mystery” might perhaps be better than “riddle” here. Wittgenstein says: “I don’t [wish] that there should be anything ridiculous or profane or frivolous in the word when used in the connection ‘riddle of life’ etc.” (Wittgenstein 1973, 36).]

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How the world is, is completely indifferent for what is higher. God does not disclose [or: manifest, reveal] himself in the world.

6.4321 The facts all belong only to the assignment, not to the correct response to it.

[“Correct response” instead of “solution” because in Letters to Ogden Wittgenstein says the word should be appropriate for, e.g., the digging of a hole when someone tells someone to dig a hole (see Wittgenstein 1973, 36).] 6.44

The mystical is not how the world is, but rather that it is.

6.45

The contemplation of the world sub specie aeterni is its contemplation as a – limited – whole. The feeling of the world as a limited whole is the mystical.

[Wittgenstein says “das mystiche” is an adjective belonging to “Gefühl” here, and considers the translation “the mystical feeling,” although he prefers simply “the mystical” (see Wittgenstein 1973, 36–37).] 6.5

If it requires an answer that one cannot articulate, then one also cannot articulate the question. The riddle does not exist. If a question can be put at all, then it can also be answered.

[The second sentence refers back to 6.4312. In Letters to Ogden, Wittgenstein says (Wittgenstein 1973, 37) that his reference here to “the riddle” “means as much as ‘the riddle “par excellence.”’” Joachim Schulte (in Stern and Szabados 2004, 132) suggests that the reference to a riddle here and in 6.4312 might be allusions to Otto Weininger. Schulte quotes Weininger to the effect that “the deepest problem in the universe” is constituted by the riddle of life together with the riddle of the world (ibid., 128–129). The riddle of the world is said to be the riddle of dualism, while the riddle of life is the riddle of time. Weininger links the fact that life is not reversible with the meaning of life, and claims that “The unidirectionality of time is . . . identical with the fact that the human being is at bottom a being that wills” (Weininger and Burns 2001, 89). As Schulte notes, there is a lot of irony and paradox in Weininger, which makes interpreting such remarks difficult. Wittgenstein discusses the unidirectionality of time in Wittgenstein 1979b, 84e.] 6.51

Skepticism is not irrefutable, but rather manifestly nonsensical [offenbar unsinnig], if it would doubt where nothing can be asked.



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Since doubt can only exist where a question exists; a question only where an answer exists, and this only where something can be said. 6.52

We feel that even if all possible questions of natural science were to be answered, the problems of our life would still not have been touched at all. Of course there would then be no more questions remaining; and just this is the answer.

6.521

The solution of the problem of life is perceived in the vanishing of this problem. (Is not this the reason why people to whom the meaning [Sinn] of life has become clear after long doubt could not then say in what this meaning consists?)

[Anscombe says that this shows that Wittgenstein does not think all thoughts of the meaning of life are nonsense (see Anscombe 1971, 170). After all, how could it become clear unless it could at least be shown? Kremer: “While the discussion of ethics in the 6.4’s suggests a mysticism in which we hold onto the thought that there is a ‘higher realm’ of value that we can grasp in thought though we cannot put it into words, with which we can slake our thirst for justification, in the 6.5’s Wittgenstein unmasks this conception as yet another form of nonsense” (Kremer 2001, 56).] 6.522

There is to be sure the inexpressible. This shows itself, it is the mystical.

[Wittgenstein says that das Mystiche here is the same as in the case of 6.44 but not the same as 6.45 (see Wittgenstein 1973, 37). So perhaps it is not a feeling but rather (the fact) that the world is. Anscombe: “There is indeed much that is inexpressible—which we must not try to state, but must contemplate without words” (Anscombe 1971, 19). Nordmann (2005, 50–51) argues that unaussprechlich should be translated “inexpressible in speech.” It is not the same as “unsayable,” since a proposition can say (as in 5.542’s “‘p’ says p”), but refers rather to the ability (or inability) of a human subject to get something out in language. Section 4.115 is the only place in the Tractatus where Wittgenstein mentions the sayable and the unsayable. Nordmann contrasts the expressible in speech with what is expressible in music, gesture, or conduct. He sees this remark as following from the denial of what needs to be denied in order to avoid the contradiction in 6.41 (see Nordmann 2005, 194). Yet he also sees this remark itself as nonsensical because it fails to establish a subject-­predicate relation, and is therefore ungrammatical. He writes: “That the words ‘there is indeed the inexpressible in speech’ are nonsensical and have no sense makes the point that there is, indeed, the inexpressible in speech” (ibid., 198–199).]

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The right method for philosophy would properly be this: To say nothing other than what can be said, thus propositions of natural science – thus something that has nothing to do with philosophy –, and then always, if another wanted to say something metaphysical, to point out to them that they had given no meaning to certain signs in their propositions. This method would be unsatisfying for the other person – they would not have the feeling that we were teaching them philosophy – but it would be the only strictly correct one.

[Diamond (2000, 155) notes that this is not the method that Wittgenstein uses in the Tractatus (see also Black 1964, 377, where he makes the same point). Anscombe: “The criticism of sentences as expressing no real thought, according to the principles of the Tractatus, could never be of any very simple general form; each criticism would be ad hoc, and fall within the subject-­ matter with which the sentence professed to deal” (Anscombe 1971, 151). Wittgenstein is not, for instance, putting forward a verificationist criterion of meaning. Kremer: there is something curious about 6.53’s description of the “right method of philosophy.” Like the positivist’s verification theory of meaning, it appears to violate its own strictures. The “right method,” we are told, would be to “say nothing except what can be said, therefore propositions of natural science, therefore something that has nothing to do with philosophy.” Yet this proclamation is not a proposition of natural science, nor does it have nothing to do with philosophy. (Kremer 2001, 57–58)] 6.54

My propositions elucidate by whoever understands me perceiving them in the end as nonsensical, when through them – upon them – over them, he has climbed out. (He must, so to speak, throw away the ladder after he has climbed out upon it.) He must overcome these propositions, then he sees the world rightly.

[Peter Sullivan, in “What Is the Tractatus About?” (in Kölbel and Weiss 2004, 32–45), notes two aspects of 6.54: First, Wittgenstein speaks of a reader who has been helped by the book in the way he intends as one who understands him, not one who understands his propositions. In this way he carefully avoids offering any ground for an accusation of double-­think. It is perfectly intelligible that the utterance of nonsense may serve in some context to convey a point which the utterance itself does not express, and in such a case to appreciate the point will be to understand the utterer, not his utterance. Second, Wittgenstein describes here the way an understanding

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reader will be helped, not by his system, but by his propositions – plural. (Kölbel and Weiss 2004, 39–40)

Diamond writes: [T]he Tractatus, in its understanding of itself as addressed to those who are in the grip of philosophical nonsense, and in its understanding of the kind of demands it makes on its readers, supposes a kind of imaginative activity, an exercise of the capacity to enter into the taking of nonsense for sense, of the capacity to share imaginatively the inclination to think that one is thinking something in it. If I could not as it were see your nonsense as sense, imaginatively let myself feel its attractiveness, I could not understand you. And that is a very particular use of imagination. (Diamond 2000, 157–158)

In MS 107 (from 1929), Wittgenstein says that the only way to fight a view is to put oneself completely inside it (“Nur so kann man eine Ansicht bekämpfen, daß man sich ganz in sie hineinversetzt”). Diamond’s suggestion is that this is what Wittgenstein is doing in the Tractatus, but since the view he is fighting is nonsensical, there is no there there. There is no real place to put oneself. Even so, just as confused philosophers think they have a coherent position, so Wittgenstein must use his imagination (and perhaps his memory of his own past confusions) to see things their way as best he can if he is to lead them to a clearer view. The ladder image occurs in Mauthner 1901, 2, and in Schopenhauer 1969, vol. 2 80. See Weiner 1992, 42–43, for more on this. White sets out various ways in which Wittgenstein has “said” things that, he says, cannot be said, e.g., in his remarks on formal concepts and the logical form of propositions that is shared with reality (see White 2006, 115–117). Black quotes Sextus Empiricus comparing a skeptic who proves the non-­ existence of proof to a man who kicks over a ladder after he has used it to climb to a high place (see Black 1964, 377). Morris and Dodd insist that Wittgenstein must be referring to all his propositions as being nonsensical, including these in 6.54: the idea is that the Tractatus’s incoherence—the fact that it is nonsense, if true—reveals, not that we should take up some other philosophical position on the nature of representation, but that the doctrines of the Tractatus and, with these, philosophy itself, are self-­undermining. The text is designed to bring us to adopt another perspective on life altogether; and this other perspective, we suggest, is the perspective of mysticism. (Morris and Dodd 2009, 261)

This perspective, they argue, consists in seeing the world as a limited whole, with its limits visible.

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Cf. Wittgenstein 1980a, 7e: “Anything that I might reach by climbing a ladder does not interest me.”] 7

Whereof one cannot speak, thereof one must be silent.

[Kremer points out that this “strictly speaking, forbids nothing” (Kremer 2001, 57). Ostrow, however, sees an ethical obligation here, since failure to remain silent would indicate a refusal to accept reality or the course of one’s own experience (see Ostrow 2002, 133). (Yet see what Wittgenstein says about talking nonsense in the Lecture on Ethics—it is nonsense, but he respects the tendency to produce it. See Wittgenstein 1993, 44.) Friedlander notes that Wittgenstein talks about speaking (sprechen) not saying (sagen). “What is at stake here is, then, an actual intervention with speech rather than the abstract opposition of the sayable and the unsayable” (Friedlander 2001, 148). He continues, in the next paragraph: Moreover, the opposite of silence is not necessarily speaking with sense but, rather, making noise. Speaking without sense is one way of being noisy. The ending of the Tractatus should therefore be read in conjunction with the epigraph of the book, which places the act of expression against a background of noise: “. . . and whatever a man knows, whatever is not mere rumbling and roaring that he has heard, can be said in three words.” The implication is that the noise of empty talk, whether it be nonsense or mere mindlessness, conceals something. To be silent means primarily not to fall prey to the rumbling and roaring of rumor. Silence is what we need in order to be attentive to what there is, to the showing of truth.

Friedlander goes on to show how Wittgenstein’s views on silence were not simple, at least later in his life. (See ibid., 149–150.) We must not, he seems to have thought, be silent about important matters (e.g., God) just because chatterboxes talk a lot of nonsense about such things. But it still seems important to him not to be one of these chatterboxes. He gave his word to a friend of his (Maurice Drury) that he would not refuse to talk to him about God or religion. It does not follow that he would have no objection to a philosopher publishing works for a general (i.e., wide, impersonal) audience on such subjects. Black quotes Silesius: “Schweig, Allerliebster, schweig: kannst du nur gänzlich schweigen, / So wird dir Gott mehr Gut’s, als du begehrst, erzeigen” (see Black 1964, 378). This is translated thus: “Silence, Beloved, be still; if you be wholly quiet, / God will show you more good than you know how to desire” in Silesius 1986, xi.]

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Index

a priori, 27, 39, 40, 82, 84, 86, 88, 95, 97, 107, 108, 110, 114, 127, 128, 129, 131 accident, 27, 53, 73, 97, 117, 122, 127, 134, 135 aesthetics, 134 affix, 62, 82–83 analysis, 8, 23, 27, 45, 46, 54, 61, 69, 74, 104 Anscombe G. E. M., 18, 31, 33, 34, 40–41, 44, 47, 63–64, 65, 67, 78, 87, 91, 100, 101, 103, 106, 109, 111, 132, 133, 136, 139, 140, 143 application, 42, 46–47, 89, 91, 92, 98, 110, 116, 123 argument, 29, 52, 62, 77–78, 82–83, 88, 91, 96, 101, 104, 119–120 arithmetic, 4–7, 80, 116, 125. See also mathematics atomic fact, 10, 21, 24, 26, 33, 35. See also state of affairs (Sachverhalt). Axiom of Infinity, 8, 104 Axiom of Reducibility, 122 Balaska, Maria, 110, 143 Bazzocchi, Luciano, vii, 114, 143 Beaney, Michael, 143 Bearn, Gordon C. F., 32, 45, 143 Black, Max, passim

brackets, 1, 33, 78, 79, 94, 95, 96, 98, 119 calculation, 123, 126 causality, 3, 86–87, 127–128, 130, 131, 133. See also principle of sufficient reason certainty, 52, 81, 86, 89, 102, 128 clarity, 66, 67. See also elucidation classes, 7–8, 18, 25, 48–49, 52–53, 72, 116–117, 122 Coffa, J. Alberto, 61, 143 color, 26, 29, 31–32, 37, 69, 134 complex, 8, 25, 29–31, 38, 43–46, 54, 72, 74, 78, 96, 100, 101, 107, 115 Conant, James, 11, 46, 47–48, 50, 66, 105, 143 constant, 48, 60, 71, 72, 78, 92–94, 96, 98, 101, 115 contradiction, 39, 73, 74, 80–81, 83–84, 88–89, 102, 115, 119, 121, 134 cube, 107 Darwin, Charles, 67 death, 4, 21, 137 deduction, 85, 87, 88 definition, 6–7, 22, 26, 27, 31, 38–39, 45–47, 52, 54, 65, 71, 75, 80, 82, 92–94, 96, 103, 112, 116–117, 128 149

150

Index

description, 9, 23, 30, 32, 45–46, 49, 51–52, 59, 81–83, 89, 96, 98, 111– 112, 123, 128–130, 133 Diamond, Cora, vii, 11, 42, 54, 66, 73, 97–98, 105, 125, 135, 140, 141, 143 Dodd, Julian, 18, 112, 141, 146 Donatelli, Piergiorgio, 125, 144 Dreben, Burton, 37, 117, 144 Dummett, Michael, 6, 71, 144 elementary proposition, 22, 33, 47, 74–77, 79, 82–83, 86, 88, 90, 92–93, 96, 98, 102, 107, 109–110, 115–116, 121–122, 134 elucidation, 21, 47, 66, 140. See also clarity Engelmann, Mauro Luiz, 21–22, 57, 144 equality, 49, 72, 97, 117, 126 essence, 22, 44, 54, 57, 58, 59, 81, 92, 96, 126–127 eternity, 4, 137 ethics, 4, 19, 124–125, 134–135, 139, 142 existence, 8, 21, 24, 28, 30, 45, 49, 54, 70, 73, 85, 86, 95, 101, 112, 117, 134, 141 expression, 17, 40, 43, 48–51, 61, 69, 71, 72, 73, 75, 77, 79, 90–91, 98, 99, 102, 104, 126–128, 134, 142 external property, 28, 31, 59 facts, 21, 22, 23–28, 31, 34–36, 39, 41, 43–44, 46, 57–58, 60, 62–63, 67–69, 72, 74, 80, 89, 93, 101, 105–107, 132, 135, 136, 138 Fahrnkopf, Robert, 26, 29, 44, 144 fairy tale, 57 fate, 132–133, 137 feature, 53–54, 59, 69, 71 Findlay, J. N., 76, 144 form, 29, 30–33, 35–38, 43, 48, 49, 51, 53, 55–57, 60, 63, 66–70, 72, 74, 81, 89–91, 96, 106, 107–111, 115, 116– 117, 121, 124, 126, 127, 128–129 formal, 69–73, 90–91, 98, 116, 118, 120, 141 formal concepts, 71–73, 141

Frascolla, Pasquale, 24, 25–26, 28, 31, 32, 57, 86, 87, 116, 123, 144 free will, 87 Frege, Gottlob, 1, 4–9, 17, 18, 22, 23, 35, 40, 43, 45, 47, 48, 49, 50, 55, 60, 62–65, 70–73, 77–79, 82–85, 88, 92–94, 97, 101, 106, 116, 118, 124, 126, 143, 144 Friedlander, Eli, 37–38, 99, 107, 110, 114, 136, 142, 144 functions, 48, 49, 52–53, 64, 71–73, 75, 78, 82, 83, 90–94, 96, 98, 101–103, 105, 122 future, 86–87 Geach, Peter, 2, 144 generality, 46, 62, 85, 101, 117, 119, 122, 127 geometry, 40, 129 Glock, Hans-Johann, 111–112, 144 God, 3, 40, 84, 132, 137, 138, 142 Griffin, Nicholas, 122, 144 Hacker, P. M. S., 22, 25, 41–42, 47, 57, 78, 92, 133, 144 Haller, Rudolf, 23, 111, 144 Hertz, Heinrich, 35, 61, 130, 144 hieroglyphic writing, 58 Hintikka, Jaakko, 26, 60, 137, 144 idealism, 4–5, 23, 62, 70, 113 identity, 27, 46, 50, 62, 102–103, 115, 126, 133 immortality, 137 inconceivability, and internality of properties, 69 independence, of propositions, defined, 88 induction, 116, 127, 132 inference, 85–87, 121 internal properties, 28, 59, 69–70 internal relations, 45, 69–70, 87, 89–90 Ishiguro, Hidé, 27, 46, 144 Janik, Allan, 11, 61, 82, 83–84, 87, 145 Jespersen, Bjørn, 58, 145



Index 151

Johnston, Colin, 26–27, 110, 145 Juliet Floyd, 72, 117, 144 Kant, Immanuel, 3, 29, 62, 67, 84, 87, 117, 124–125, 130–131, 132, 145 Kienzler, Wolfgang, 15, 145 Klagge, James C., 137, 145 Klemke, E. D., 71, 145 Kolak, Daniel, 9, 145 Kraft, Tim, 22, 145 Kremer, Michael, 51, 58, 68, 72, 132, 135–136, 139, 140, 142, 145 Landini, Gregory, 77, 145 language, 11, 17, 30, 36, 40–41, 46–47, 49–51, 54–59, 61, 66, 68–69, 71, 79, 82, 97, 100, 104, 106, 109–112, 118, 126, 135, 139 law of least action, 128 laws of nature, 89, 130, 132 Lazenby, J. Mark, 60, 145 life, 4, 21, 112, 116, 125, 137–139, 141 logic, 5–8, 13, 17, 19, 27, 40, 55, 58, 60, 64, 66–68, 71, 74, 82–85, 88, 93–97, 99, 107–111, 116–118, 120– 127, 129, 134 logical constants, 60, 78, 92, 94, 96, 115 logical objects, 78, 92–93 logical space, 24, 35, 38, 42, 54–55, 80 logical syntax, 50–51, 53–54, 122 Mach, Ernst, 128, 131, 145 Magee, Bryan, 2, 112, 133, 145 Malcolm, Norman, 11, 42, 145 mathematics, 1, 4, 5, 7, 8, 35, 40, 61–62, 89, 90, 93, 98, 104, 116, 117, 122, 125–127. See also arithmetic; geometry Mauthner, Fritz, 56, 141, 145 McGinn, Marie, 11, 27, 31–32, 37, 62, 64–65, 69, 70, 87, 88, 97, 99, 111, 145 McGuinness, Brian F., 9, 10, 11, 18–19, 23, 32, 36, 40, 51, 57, 69, 75, 77, 95, 99, 100, 104, 110, 118, 120, 129, 130, 135, 144, 145, 148

McManus, Denis, 11, 28, 31, 40, 51, 67–68, 115, 123, 146 meaning, 5–6, 8–9, 11, 23, 27, 30, 41–42, 45–53, 59, 61, 67, 71, 75, 78, 80–83, 90, 92, 94, 96–98, 104, 106– 107, 109, 111–112, 115, 121–123, 126–127, 140 meaning of life, 116, 138–139 mechanics, 128–129 Mendelsohn, Richard L., 71, 146 microcosm, the, 113 mirror, 37, 57, 71, 99–100, 111, 124 modus ponens, 124 Monk, Ray, 11, 13, 19, 146 Moore, A. W., 39, 73, 80, 146 Moore, G. E., 1, 13, 19, 25, 26, 56–57, 70, 105, 137, 146 Morris, Michael, 18, 32, 112, 141, 146 Mounce, H. O., 30, 33, 44, 66–67, 106, 127, 136, 146 multiplicity, 59, 61, 62, 98 music, 4, 41, 43, 56, 57, 58, 139 mystical, the, 18, 124, 137, 138, 139 names, 8–9, 22, 27, 30, 35, 43–49, 52–53, 60, 64, 71–72, 74–75, 78, 81–83, 92, 102–104, 107, 110, 122, 133 natural science, 6, 61, 65–67, 116, 124, 128, 137, 139–140 necessity, 86–87, 113, 132–134 negation, 55, 63, 65, 78, 82, 90, 92–94, 98–100, 121, 126 Newton, Isaac, 128–129 nonsense, 8, 17, 26, 29, 47, 51–52, 61, 74, 98, 103–104, 106–107, 110, 139–142 Nordmann, Alfred, 10–11, 17, 23, 31, 56, 57, 60, 61, 130, 139, 146 notation, 46, 50, 54, 57, 64, 98, 100, 115, 119–121 Notebooks 1914–1916, 23, 26, 61, 64, 81, 114, 134, 135, 137, 148 null method, 120 numbers, 6–8, 18, 25, 70–74, 83, 95, 108, 116–117

152

Index

objects, 3, 5–6, 8, 23–36, 43–50, 52–53, 59–62, 64–75, 77–78, 81, 83–84, 86, 88, 93, 102–103, 105–106, 109–110, 126, 129, 132–134, 137 Occam’s razor, 51, 97 Ogden, C. K., 9–10, 36, 40, 50, 51, 57, 69, 71, 75, 77, 81, 95, 99, 104, 111, 118, 130, 135, 148 operations, 73, 90–94, 96, 98–99, 105, 116–117, 123 Ostrow, Matthew B., 23, 29, 33–34, 35, 36–38, 53, 68, 82, 85, 142, 146 Pears, David, 9–10, 26, 32, 36, 40, 51, 57, 69, 75, 77, 95, 99, 100, 104, 110, 111, 118, 120, 129, 130, 135, 146, 148 Philosophical Investigations, 79, 107, 148 philosophy, 2, 11, 13, 17, 19, 21, 41, 50, 54, 56, 60, 61, 65–67, 84, 110, 115, 118, 124, 125, 140, 141 physics, 40, 128, 129, 134 pictures, 5, 21, 30, 35–39, 41, 42, 55, 56–60, 62–63, 80, 89, 106, 118, 128–129 picture theory, 36–39, 132–133 Pihlström, Sami, 124, 146 possibility, 27–31, 33–40, 42, 45, 54, 58, 60, 74, 93, 102, 127, 128, 131 primitive ideas, 73 primitive signs, 46–47, 93–96 principle of sufficient reason, 47, 86, 87, 128–129. See also causality probability, 81, 83, 88–89 projection, 37, 40–42, 58 proof, 122, 123–124, 127, 141 Proops, Ian, 18–19, 38–39, 58, 64, 79, 85–86, 95, 96, 118, 146 propositional variable, 48–49, 71–72, 99, 115 prototype, 46, 49, 52, 101, 104 pseudo-concepts, 23, 72 pseudo-propositions, 39, 72–73, 103– 104, 126

psychology, 5, 41, 66, 105–106, 115, 135, 136 Ramsey, F. P., 26, 52, 122, 125, 146 reality, 2–3, 8, 26, 34–39, 41–42, 56, 58–59, 62, 63, 67–68, 70, 80, 100, 109, 114, 118, 141 Reintges, Chris, 58, 145 representation, 2–4, 10, 35–38, 42, 52, 60–61, 67–68, 78, 87, 94, 98–99, 106, 113–114, 126, 141 Ricketts, Thomas, 45, 146 riddle, 21, 112, 137–138 rules, 51–54, 61, 85, 100, 116, 123 Russell, Bertrand, 7–9, passim saying, 11, 21, 45, 51, 58–59, 61, 64, 67–68, 72, 75, 80, 88, 93, 104, 106, 111, 117, 130, 139, 140–142 scaffolding, logical, 55, 59, 122 skepticism, 56, 138, 141 Schopenhauer, Arthur, 2–4, 9, 10, 11, 21, 36, 47–48, 56, 66, 86–87, 95, 99, 111–115, 131–133, 135–136, 141, 147 Schroeder, Severin, 30, 41, 42, 44, 60, 69, 76–77, 94, 110–111, 113, 136, 147 Schulte, Joachim, 138, 147 sense, 5–6, 10–11, 17, 23, 29–31, 38–48, 50–55, 58–60, 62–66, 73–75, 78, 80–81, 84, 90–91, 95–98, 100, 107, 122–124, 126, 127, 134, 142 senseless (sinnlos), 80, 85, 87, 104 series, 5, 70, 73, 79, 83, 90–91, 98, 115, 116 showing, 4, 11, 17, 19, 33, 46–47, 58, 61, 63, 64, 65, 68–69, 71–72, 80, 87, 91, 103, 111, 118, 121, 124–126, 130, 139, 142 sign, 5, 40–57, 59–60, 62–63, 71–73, 75, 77–79, 81–83, 92–103, 105, 108, 112, 116, 119, 122–124, 126, 140 Silesius, Angelus, 142, 147 similes, 58



Index 153

simplicity of objects, 25, 29–31 situation (Sachlage), 10, 24, 27, 29, 35, 38–40, 42, 44–45, 58–61, 64, 70, 80–81, 86–87, 89, 102, 132 Socrates, 50, 96–97 solipsism, 4, 23, 111–112, 114 soul, 106, 115, 137 space, 3, 27, 29, 32, 40, 42, 130–131, 133, 137 state of affairs (Sachverhalt), 10, 24–25, 27–30, 32–35, 37–40, 59–61, 65, 68–69, 74–76. See also atomic fact Stenius, Erik, 24–25, 86, 147 Stokhof, Martin, 9, 28, 56, 87, 134, 147 structure, 33–35, 37, 43, 57, 67–69, 71, 84–85, 89, 90, 95, 102, 134 subject, 2–3, 41, 106–107, 110, 113– 115, 139 substance, 22, 28–30, 32–33, 108, 137 successor, 70, 73 Sullivan, Peter M., 24, 53, 140, 147 superstition, 86 symbol, 8, 35, 46, 48–54, 71–72, 75, 81, 85, 96–98, 100–102, 104, 109, 118, 122–123 tautology, 22, 39, 73, 79–81, 83–84, 87–89, 102, 117–119, 121–125, 134 Tejedor, Chon, 38, 147 theory of types, 7–8, 51–53, 122 thought, 5, 11, 17, 19, 21–22, 38–42, 44, 55, 57, 61, 66–67, 73, 80, 82, 97, 106, 111–112, 125 time, 3–4, 27, 31, 32, 86, 130, 133, 134, 137, 138 totality, 22–24, 34, 39, 52, 55, 65, 82, 102, 109 Toulmin, Stephen, 11, 145 translation, 1, 2, 9–10, 13, 54, 58, 59 truth, 5–7, 18, 19, 24, 30, 38, 40, 56, 60, 62, 63, 64, 70, 76, 77, 81, 84–85, 87–88, 92, 95, 100–102, 109, 112, 114, 118, 119–121, 125, 133, 142 truth-arguments, 82–83, 88, 119 truth-conditions, 61, 64, 77, 79–80

truth-functions, 22, 27, 77, 82–83, 90, 92–94, 98, 101, 105, 115 truth-grounds, 83–84, 88 truth-operations, 90, 92–94, 105 truth-possibilities, 76–80, 83 truth-tables, 77, 83 understanding, 17, 55, 58, 77, 93, 111, 112, 131, 140–141 use, 8, 9, 27, 35, 36, 40–43, 45, 47, 50–51, 57, 59, 69, 72, 75, 96, 104, 105, 119, 122, 125, 128 value, 19, 134–135, 139. See also ethics variable, 45, 48–49, 62, 71–73, 82, 91, 98–99, 101, 115, 117, 125 Verbin, N. K., 41, 147 Visser, Henk, 131, 147 visual field, 29, 114 Weiner, David Avraham, 113–115, 141, 147 Weiner, Joan, 4, 55, 72, 147 Weininger, Otto, 111, 138, 147 White, Roger M., 11, 25, 29, 43, 46, 47, 52, 58, 94, 101, 110, 115, 120–121, 126, 141, 148 Whitehead, Alfred North, 7, 24, 83, 91, 94, 103, 121, 148 Wiggins, David, 135, 148 will, 2–4, 87, 99, 113, 115, 132–133, 135–136, 138 Winch, Peter, 41–42, 53, 107, 148 words, 5–6, 9, 27, 31, 41–42, 43, 49, 59, 72, 94, 106, 107, 111, 117, 125, 139, 142 world, 2–4, 8, 18, 21–25, 27–28, 29–30, 32–40, 42, 44, 46, 47, 54, 56, 57, 59, 67–68, 71, 74, 76, 84, 94, 96, 99, 102, 108, 110–115, 118, 122, 124–125, 128–129, 132–141 Young, Julian, 21, 136, 148 Zalabardo, José L., 30, 43, 84–85, 148

About the Author

Duncan Richter is the author of Historical Dictionary of Wittgenstein’s Philosophy, Anscombe’s Moral Philosophy, Why Be Good?, Wittgenstein at His Word, and Ethics After Anscombe.

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