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This is an unusual book, both in content and in style. In content it is about logic, though Ludwig Wittgenstein, in discussing this subject, manages to say much about the theory of signs, epistemology, metaphysics, and philosophy in general. Furthermore, Wittgenstein indicates that there are many things that we cannot...

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This is an unusual book, both in content and in style. In content it is about logic, though Ludwig Wittgenstein, in discussing this subject, manages to say much about the theory of signs, epistemology, metaphysics, and philosophy in general. Furthermore, Wittgenstein indicates that there are many things that we cannot say about logic, not because we do not know them or because we cannot find words by which to express them, but because they are literally inexpressible by means of any language. Consequently, we must remain silent about them.

The sentences (or sometimes groups of sentences) in this work are all numbered in accordance with a plan. For example, a certain sentence has the number 3; this follows sentences 1 and 2 in the order of the natural numbers. However, between 2 and 3 are sentences numbered, for example, 2.0122 and 2.151. Sentence 2.0122 is a statement referring to statement 2. Statement 2.151 follows statement 2.1 and is a comment on it; statement 2.1, in turn, is another comment on 2, and follows it. All statements are in this way arranged in a unique, linear order based on the decimal notation, and the reader is able to determine by the number attached to each sentence what general topic is being considered, what special aspect of this general subject is involved, and so on, sometimes to a fourth level of specialization.

Furthermore, because of the unusual meanings associated with many of the terms employed by the author, the editor chose to publish, on facing pages, the original German text and its parallel English translation. This design permits those who are familiar with German to improve their understanding of the text by comparing the English version with its German original, thereby in many cases detecting fine shades of meaning. In a book where such common words as “fact,” “object,” “meaning,” and “truth” occur in great abundance and are employed with somewhat unusual connotations, the parallel translation is of great help. The book also contains a valuable introduction, written by Wittgenstein’s mentor Bertrand Russell, that both summarizes the text and criticizes it on some important points.

Atomic Facts

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For convenience of discussion, one can combine Wittgenstein’s discussion of proposition 1 and the remarks on it with proposition 2 and all of the remarks contained in propositions numbered 2.0. Proposition 1 states that the world is everything that is the case; proposition 2 asserts that what is the case, or the fact, is the existence of atomic facts. The world, then, is made up of atomic facts and is constituted by them. Atomic facts are facts that are incapable of analysis into more elemental facts. This does not mean that atomic facts cannot be analyzed, but only that they cannot be analyzed into other atomic facts. An atomic fact is itself a combination of objects (entities, things), each of whose essence lies in its being a constituent of an atomic fact. However, the objects that are elements of atomic facts cannot themselves be analyzed because they form the substance of the world. If we take advantage of the illustration of an atomic fact that Russell gives in his introduction (Wittgenstein does not give illustrations of atomic facts), we may say that it is what is asserted by the proposition “Socrates is wise.” This contains two objects, Socrates and wise, each of which, in its own unique way, unites with the other to form the atomic fact. Traditional philosophy would call these objects “substances” and “qualities.”

Wittgenstein states that however different from the real world an imaginary world might be, it must have something, its form, in common with the real world. Because the form is given by the objects, we may presume Wittgenstein to be saying that any imaginable world would have to contain substances and qualities, however these might differ from those in the real world. The world is the totality of atomic facts; it also determines the nonexistence of atomic facts, for the nonexistence of an atomic fact is a kind of fact. Reality, therefore, is the totality of atomic facts plus the fact that these are all the atomic facts.


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Beginning with proposition 2.1, continuing more or less explicitly through proposition 4, and extending implicitly through the rest of the book, Wittgenstein examines what is meant by saying that a proposition is a picture of a fact. He describes this picturing relation variously as “modeling,” “standing for,” “representing,” “corresponding with,” “depicting,” and “projecting.” Note, however, that a proposition is itself a fact. By this is not meant the propositional sign that expresses the fact, though Wittgenstein admits that propositions can be expressed perceptibly through the senses, but rather the sense of the proposition. The point is, of course, that the proposition is a logical picture of the fact, not a visual one or an audible one. He says that it represents the fact in “logical space”—a metaphor that he uses repeatedly throughout the book. Its representative character lies in its form or structure, which means a coordination of the elements in the picture with the objects in the fact, and an identity of logical form exhibited by both the picture and the fact.

Thus the proposition “Socrates is wise” pictures the fact of Socrates’ wisdom because “Socrates” represents Socrates, and “wise” represents wisdom, and the form exhibited by “Socrates” and “wise” in the propositional relation is the same as that exhibited by Socrates and wisdom in the fact. That this is a logical form rather than a spatial form is to be seen in the fact that while the sentence “Socrates is wise” has a spatial order of its elements, neither the meaning of the sentence nor the fact asserted by the sentence is spatial; what is common to the meaning and the fact is a logical structure.

More precisely, the proposition does not strictly represent the fact but rather the possibility of the fact, the possibility of the existence and nonexistence of atomic facts. A proposition whose expression mentions a complex is not nonsense if this complex fails to exist but simply false. A proposition represents what it represents independently of its truth or falsity, through its form of representation, through its sense. Furthermore, by virtue of the identity of form that runs through various facts, the picture represents every reality whose form it has; thus “Socrates is wise” also pictures the fact that Plato is human.

“The logical picture of the facts is the thought” (proposition 3). To say that an atomic fact is thinkable means that we can imagine it, and if it is thinkable, it must also be logical, for anything that is “unlogical” could not be expressed at all. Language cannot express anything that “contradicts logic” any more than a spatial figure can represent anything that contradicts the laws of space or can represent the spatial coordinates of a point that does not exist.

The sign through which thoughts are expressed is the propositional sign. Both the proposition and the propositional sign are facts. In the propositional sign, the elements (the words) are combined in a definite way so that the objects of the thoughts correspond to the elements of the propositional sign. The simple signs used in propositional signs are called “names.” Objects can only be named and spoken about; they cannot be asserted. Names cannot be further analyzed; they are primitive signs. They have meanings only in the context of propositions.

A propositional expression presupposes the forms of all propositions that it expresses, and thus it may be said to characterize a form and a content; the form is constant, and everything else is variable. If every constituent part of a proposition is changed into a variable, the logical form (the logical prototype) remains. Thus, to use an example that Russell gives elsewhere, if we change the proposition “Socrates drank hemlock” into the proposition “Coleridge ate opium,” the form of the proposition, “ARB,” remains. This may be called a “propositional variable.”

A Logical Language

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In the language of everyday life, the same word often signifies in two different ways, and different words signify in the same way. For example, the verb “to be” appears sometimes as the sign of equality, sometimes as the expression of existence, sometimes as an intransitive verb, and sometimes as a sign of identity. Words of this kind are the cause of some of the most fundamental confusions in thought, especially in philosophical thought. The only way to avoid these difficulties is to invent a special symbolism—a symbolism that obeys the rules of logical grammar (logical syntax). Such rules follow of themselves if we know how every sign signifies. Russell and the mathematician Gottlob Frege have invented such logical symbolisms, but even these do not exclude the possibility of error.

The great advantage of a logical language is that it calls our attention to formal properties of objects and facts. This is not because propositions express the form of facts, but because they “show it” and do not state it. No proposition is capable of representing its form of representation, for this would require something that is impossible—the picture would have to place itself outside its form of representation. “The proposition shows how things stand, if it is true.” The existence of a structure in a possible state of affairs is not expressed by the proposition that presents the state of affairs; it is shown in the proposition by means of its structure. The identity of form that is exhibited by the proposition and by the fact accounts for the representation of the fact by the proposition, but this does not give the proposition a formal property of representing the fact. It would be as meaningless to ascribe a formal property to a proposition as to deny it of the proposition. It would be equally meaningless to assert that one form has a certain property and another form has a different property, for this assumes that there is sense in asserting that either form has either property. We do not ascribe properties to forms nor do we ascribe forms to propositions or states of affairs.

In this respect, formal concepts differ from proper concepts. The proper concept “man” can be expressed by a propositional function, for example, “x is a man”; but the formal concept “object” cannot be expressed by “x is an object.” In this expression x is a sign of the pseudoconcept object, and to say that a rose is an object (thing, entity) is to utter nonsense. The same holds true of such words as “complex,” “fact,” “function,” and “number,” which should be represented in symbolism by variables, not by proper concepts. Recognizing these as variables shows the absurdity of making such statements as “There are objects” (supposedly patterned after “There are books”) or “There is only one 1” (which according to Wittgenstein is as absurd as it would be to say “2 plus 2 is at 3 o’clock equal to 4”). To summarize, then, the great advantage of a precise logical symbolism is that it prevents us from talking nonsense. The correct use of the symbols, as was said above, follows immediately if we know how every sign signifies.

A further consequence of the notion that “object” is a pseudoconcept is the impossibility of finding some “property” that all “objects” possess. We have already seen that atomic facts are complex and contain objects as elements of a certain structure. These objects are unanalyzable. Consequently, to name a certain atomic fact is to presuppose the truth of a certain atomic proposition, namely, the proposition asserting the relatedness of the constituents of the complex; and this, in turn, presupposes the naming of the constituents (object) themselves. However, according to Wittgenstein, because the concept “object” is a pseudoconcept, there is no way that we can describe the totality of things that can be named. This means that we cannot say anything about the totality of what there is in the world. There is no property, such as self-identity, which all objects possess. To say that if all objects were exactly alike they would be identical, and there could be only one object in the world, is to assert not a logical truth, but an accidental characteristic of the world. Consequently, we cannot use self-identity as a property by which we can “locate” an object. Instead, we signify objects by means of letters, and different objects by different letters.

Types of Propositions

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The simplest proposition, the elementary proposition, asserts the existence of an atomic fact. Such a proposition is a concatenation of names and is incapable of analysis into further propositions. One of Wittgenstein’s important theses is that all propositions are truth-functions of elementary propositions and can be built up from them. (An elementary proposition is a truth-function of itself.) Truth-functions are obtained in the following way: Suppose all elementary propositions were given. Each of these could be either true or false. Therefore a proposition containing three elementary propositions p, q, r could have a truth-function T, T, F or T, F, T, and there would be eight such possible truth-functions. If there were four elementary propositions, there would be sixteen truth-functions. Starting with any group of elementary propositions, the truth-functions formed from them may be arranged in a series.

Of the propositions, two kinds are particularly important. One of them, a “tautology,” is a type of proposition that is true for all the truth-possibilities of the elementary propositions. The other, a “contradiction,” is a proposition that is false for all the truth-possibilities of the elementary propositions. The truth of a tautology is certain; the truth of a contradiction is impossible; the truth of all other propositions is possible. Here we have the serial arrangement of propositions that forms the basis for a theory of probability. An example of a tautology is “It is either raining or it is not raining”; this is always true regardless of whether the p and the not-p that it contains are true or false. A tautology, therefore, “says nothing” about the world, for it is true of all possible states of affairs. An example of a contradiction is “It is both raining and it is not raining”; this is false regardless of whether the p and the not-p that it contains are true or false. A contradiction therefore also “says nothing” about the world, for it is false for all possible states of affairs. Tautologies and contradictions are without sense. “Contradiction is the external limit of the propositions, tautology their substanceless center.”

Logical operations are those that produce propositions from other propositions. For example, “denial” (not), “logical addition” (either/or), “logical multiplication” (and) are all logical operations. Thus, operations do not assert anything; the result of an operation, a proposition, does assert something; and what it asserts depends on the elementary propositions on which it is based. We can thus express the general form of all propositions; this is a propositional variable whose values would be all possible propositions. Wittgenstein states this form in abstract symbols; it means, according to Russell in the introduction, “whatever can be obtained by taking any selection of atomic propositions, negating them all, then taking any selection of the set of propositions now obtained, together with any of the originals—and so on indefinitely.”


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Language cannot express anything that contradicts logic. Wittgenstein now resumes discussion of this topic and points out that we cannot say what we cannot think. We cannot say that there is this in the world but there is not that, for such a statement would imply that logic can exclude certain possibilities from the world; but in such a case, logic would have to “get outside the limits of the world”; that is, it would have to consider these limits from both within and without the world. However, logic cannot go beyond itself: “Logic fills the world: the limits of the world are also its limits.” This principle also has applications for solipsism. Solipsism is correct but cannot be asserted; it can only be “shown.” The subject who knows is the limit of the world; he or she does not belong to it. The best example of this theory is the field of vision: There is nothing in the field of sight that permits us to conclude that it is seen by an eye.

Reality, for Wittgenstein, proves to be very loose-knit. No atomic fact contradicts any atomic fact, and no atomic fact can be inferred from an atomic fact. There is no causal nexus in nature, and belief that there is such a thing is superstition. Induction is a process of assuming the simplest law that can be made to describe the regularities of nature. However, there is no necessity in this process. The only necessity is a logical necessity, and the only impossibility is a logical impossibility; and these presumably do not exist in the world.

The sense of the world lies outside the world. If there were value, it would have to “lie outside all happenings and being-so. For all happenings and being-so is accidental.” As a consequence, ethics and aesthetics cannot be expressed, and are transcendental.

The Role of Philosophy

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What, then, is philosophy? It seems to have two tasks. One is to show that every proposition is a picture of a fact. This cannot be said, for no proposition can say anything about itself. That a proposition has, for example, the subject-predicate form cannot be said in a proposition, and that a proposition has the form “p or q” cannot be said in a proposition. Nor can it be said how a proposition pictures reality. A sentence has no apparent pictorial character. However, neither does a musical score or a phonograph record; nor does a pattern of sound waves obviously picture the sound themselves. Yet all of these stand to that which they represent in a relation that can be seen in the similarity of structure holding between them and the facts. There is a “law of projection” that enables us to translate the picture into the fact, though this law cannot be stated. Because the law cannot be stated, we should not try to do so. Wittgenstein therefore concludes with proposition 7, “Whereof one cannot speak, thereof one must be silent.”

However, there is another task for philosophy. Philosophy is not a theory, like one of the natural sciences, ending in a series of conclusions that can be called “philosophical propositions.” It is an activity, a process of clarification, in which we try to delimit thoughts that are obscure and confused. If philosophy finds that the answers to its questions cannot be expressed, it should realize that its questions have not been properly expressed, for “if a question can be put at all, then it can also be answered.” To doubt where there are no questions is absurd. To insist that the problems of life have not been touched by the sciences and yet to be unable to formulate these problems that remain in a language that is clear enough to permit an answer is really to say that there is no problem left. This is precisely what Wittgenstein says. “The solution of the problem of life is seen in the vanishing of this problem.” The right method of philosophy is to turn all of the things that can be said over to the scientists, who will say them, and then when anyone asks a metaphysical question, to point out that the question is meaningless. Philosophy will then “see the world rightly.”


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Additional Reading

Fann, K. T., ed. Ludwig Wittgenstein: The Man and His Philosophy. New York: Dell, 1967. A collection of articles by friends, students, and scholars of Ludwig Wittgenstein. Included are articles on Wittgenstein as a person, a teacher, and a philosopher, and treatments of various aspects of Wittgenstein’s philosophical work.

Hacker, P. M. S. Wittgenstein. New York: Routledge, 1999. An excellent biographical introduction to the thoughts of the philosopher, clearly presented and requiring no special background. Bibliography.

Hacker, P. M. S. Wittgenstein’s Place in Twentieth-Century Analytic Philosophy. Oxford: Blackwell, 1996. A monumental work by a leading authority of Wittgenstein. This book thoroughly treats philosophical history before, during, and after the time of Wittgenstein.

Hallett, Garth L. Essentialism: A Wittgensteinian Critique. Albany: State University of New York Press, 1991. Strictly speaking, this book is an application of Wittgenstein’s later thought rather than an introduction to it, but Hallett is so faithful to Wittgenstein’s philosophy that the book is in fact a good guide to a correct understanding of it.

Hodges, Michael, and John Lachs. Thinking in the Ruins: Wittgenstein and Santayana on Contingency. Nashville: Vanderbilt University Press, 2000. A comparison of the two quite different philosophers.

Janik, Allan, and Stephen Toulmin. Wittgenstein’s Vienna. New York: Simon and Schuster, 1973. An illustrated survey showing the many connections between Wittgenstein’s philosophical development and twentieth century movements in architecture, literature, music, psychoanalysis, and other fields, in the setting of late nineteenth century Viennese culture.

McGinn, Marie. Wittgenstein and the “Philosophical Investigations.” New York: Routledge, 1997. A very useful and well-written introductory guide to Wittgenstein’s Philosophical Investigations.

Malcolm, Norman. Ludwig Wittgenstein: A Memoir. 2d ed. New York: Oxford University Press, 1984. This book, written by Wittgenstein’s most prominent American philosophical student, is a gem. Malcolm allows the reader to see the force of Wittgenstein’s personality as well as his particular way of practicing philosophy. The second edition includes numerous letters that Wittgenstein wrote to Malcolm.

Monk, Ray. Ludwig Wittgenstein: The Duty of Genius. New York: Free Press, 1990. This is the definitive biography of Wittgenstein. It is thorough and detailed, examining Wittgenstein’s private life as well as his philosophy.

Pitcher, George, ed. Wittgenstein: The “Philosophical Investigations.” Garden City, N.Y.: Doubleday, 1966. Although many of the articles in this collection are rather technical, the book’s first article is a general account of the historical context of Wittgenstein’s philosophy. This is followed by several articles that are book reviews of his Philosophical Investigations.

Sluga, Hans, and David G. Stern, eds. The Cambridge Companion to Wittgenstein. Cambridge, England: Cambridge University Press, 1996. Some of the articles in this collection are rather narrowly focused, but the first two contain general introductions to Wittgenstein’s life, his work, and his critical approach to philosophy.

Stroll, Avrum. Moore and Wittgenstein on Certainty. New York: Oxford University Press, 1994. This volume looks at the relationship between these two philosophers, particularly Wittgenstein’s critical stance on G. E. Moore’s views on certainty based on common sense.

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