 Section 2 of Tractatus logico filosoficus This is a LibriVox recording. All LibriVox recordings are in the public domain. For more information or to volunteer, please visit LibriVox.org Recording by Jeffrey Edwards Tractatus logico filosoficus by Ludwig Wittgenstein Translated by C. K. Ogden Section 2 4. The thought is the significant proposition. 4.001. The totality of propositions is the language. 4.002. Man possesses the capacity of constructing languages, in which every sense can be expressed, without having an idea how and what each word means, just as one speaks without knowing how the single sounds are produced. Colloquial language is a part of the human organism, and is not less complicated than it. From it, it is humanly possible to gather immediately the logic of language. Language disguises the thought, so that, from the external form of the clothes, one cannot infer the form of the thought they close, because the external form of the clothes is constructed with quite another object, then to let the form of the body be recognized. The silent adjustments to understand colloquial language are enormously complicated. 4.003. Most propositions and questions that have been written about philosophical matters are not false, but senseless. We cannot, therefore, answer questions of this kind at all, but only state their senselessness. Most questions and propositions of the philosophers result from the fact that we do not understand the logic of our language. Bracket. They are of the same kind as the question whether the good is more or less identical than the beautiful. Close bracket. And so, it is not to be wondered at that the deepest problems are really no problems. 4.0031. All philosophy is, quote, critique of language, close quote. Bracket, but not at all in Mothner's sense. Close bracket. Russell's merit is to have shown that the apparent logical form of the proposition need not be its real form. 4.01. The proposition is a picture of reality. The proposition is a model of the reality as we think it is. 4.011. At the first glance the proposition, say, as it stands printed on paper, does not seem to be a picture of the reality of which it treats, but nor does the musical score appear at first sight to be a picture of a musical piece. Nor does our phonetic spelling, bracket, letters, close bracket, seem 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 represent. 4.012. It is obvious that we perceive a proposition of the form A capital R, B, as a picture. Here the sign is obviously a likeness of the signified. 4.013. And if we penetrate to the essence of this pictorial nature, we see that this is not disturbed by apparent irregularities. Bracket, like the use of sharp and flat in the score, close bracket. For these irregularities also picture what they are to express, only in another way. 4.014. The gramophone record, the musical thought, the score, the waves of sound all stand to one another in that pictorial internal relation, which holds between language and the world. To all of them the logical structure is common. Bracket, like the two youths, their two horses and their lilies in the story, they are all in a certain sense one. Close bracket. 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 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. 4.015. The possibility of all similes, of all the images of our language, rests on the logic of representation. 4.016. In order to understand the essence of the proposition, consider hieroglyphic writing, which pictures the facts it describes, and from it came the alphabet without the essence of the representation being lost. 4.02. This we see from the fact that we understand the sense of the propositional sign without having had it explained to us. 4.021. The proposition is a picture of reality, for I know the state of affairs presented by it, if I understand the proposition, and I understand the proposition without its sense having been explained to me. 4.022. The proposition shows its sense. The proposition shows how things stand if it is true, and it says that they do so stand. 4.023. The proposition determines reality to this extent, that one only needs to say quotes yes, or quotes no, to it to make it agree with reality. Reality must therefore be completely described by the proposition. A proposition is the description of a fact. As the description of an object describes it by its external properties, so propositions describe reality by its internal properties. The proposition constructs a world with the help of a logical scaffolding, and therefore one can actually see in the proposition all the logical features possessed by 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. Bracket. One can therefore understand it without knowing whether it is true or not. Close bracket. One understands it if one understands its constituent parts. 4.025. The translation of one language into another is not a process of translating each proposition of the one into a proposition of the other, but only the constituent parts of propositions are translated. Bracket and the dictionary does not only translate substantives, but also adverbs and conjunctions, etc., and it treats them all alike. Close bracket. 4.026. The meaning of the simple signs, bracket, the words, close bracket, must be explained to us if we are to understand them. By means of propositions, we explain ourselves. 4.027. It is essential to propositions that they can communicate a new sense to us. 4.03. A proposition must communicate a new sense with old words. The proposition communicates to us a state of affairs, therefore it must be essentially connected with the state of affairs, and the connection is, in fact, that it is its logical picture. 4.031. In the proposition, a state of affairs is, as it were, put together for the sake of experiment. One can say, instead of, this proposition has such and such a sense, this proposition represents such and such a state of affairs. 4.0311. One name stands for one thing, and another for another thing, and they are connected together, and so the whole, like a living picture, presents the atomic fact. 4.0312. The possibility of propositions is based upon the principle of the representation of objects by signs. My fundamental thought is that the, quote, logical constants, close quote, do not represent, that the logic of the facts cannot be represented. 4.032. The proposition is a picture of its state of affairs, only insofar as it is logically articulated. Bracket. Even the proposition, quotes, ambulo, is composite, for its stem gives a different sense with another termination, or its termination with another stem. Close bracket. 4.04. In the proposition, there must be exactly as many things distinguishable as there are in the state of affairs, which it represents. They must both possess the same logical bracket, mathematical, close bracket, multiplicity, bracket, cf, Hertz mechanics, on dynamic models, close bracket. 4.041. This mathematical multiplicity naturally cannot, in its turn, be represented. One cannot get outside it in the representation. 4.0411. If we tried, for example, to express what is expressed by, quote, bracket, x, close bracket, end symbol, fx, close quote, by putting an index before fx, like, quote, gen, fx, close quote, it would not do. We should not know what was generalized. If we tried to show it by an index, g, like, quote, f, bracket, x, subscript, g, close bracket, close quote, it would not do. We should not know the scope of the generalization. If we were to try it by introducing a mark in the argument places, like, quote, bracket, capital G, comma, capital G, close bracket, end symbol, capital F, bracket, capital G, comma, capital G, close bracket, close quote, it would not do. We could not determine the identity of the variables, etc. All these ways of symbolizing are inadequate, because they have not the necessary mathematical multiplicity. 4.0412. For the same reason, the idealist explanation of the seeing of spatial relations through, quote, spatial spectacles, close quote, does not do, because it cannot explain the multiplicity of these relations. 4.05. Reality is compared with the proposition. 4.06. Propositions can be true or false, only by being pictures of the reality. 4.061. If one does not observe that propositions have a sense independent of the facts, one can easily believe that true and false are two relations between signs and things signified with equal rights. One could, then, for example, say that quotes P signifies in the true way what quotes not symbol P signifies in the false way, etc. 4.062. Can we not make ourselves understood by means of false propositions as hitherto with true ones, so long as we know that they are meant to be false? No. For a proposition is true. If what we assert by means of it is the case, and if by quotes P we mean not symbol P, and what we mean is the case, then quotes P in the new conception is true and not false. 4.0621. That, however, the signs quotes P and quotes not symbol P, can say the same thing is important, for it shows that the sign quotes not symbol corresponds to nothing in reality. That negation occurs in a proposition is no characteristic of its sense. Bracket not symbol not symbol P equals P, close bracket. The proposition quotes P and quotes not symbol P have opposite senses, but to them corresponds one and the same reality. 4.063. An illustration to explain the concept of truth. A black spot on white paper. The form of the spot can be described by saying of each point of the plane 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, bracket not black, close bracket a negative fact. If I indicate a point of the plane, bracket a truth value in Friega's terminology, close bracket. This corresponds to the assumption proposed for judgment, etc., etc. But to be able to say that a point is black or white, I must first know under what conditions a point is called white or black in order to be able to say quotes P is true, bracket or false, close bracket. I must have determined under what conditions I call quotes P true and thereby I determine the sense of the proposition. The point at which the simile breaks down is this. We can indicate a point on the paper without knowing what white and black are. But to a proposition without a sense corresponds nothing at all, for it signifies no thing, bracket, truth value, close bracket, whose properties are called quotes false or quotes true. The verb of the proposition is not quotes is true or quotes is false, as Friega thought, but that which quotes is true must already contain the verb. 4.064 Every proposition must already have a sense. Assertion cannot give it a sense for what it asserts is the sense itself and the same holds of denial, etc. 4.0641 One could say the denial is already related to the logical place determined by the proposition that is denied. The denying proposition determines a logical place other than does the proposition denied. The denying proposition determines a logical place with the help of the logical place of the proposition denied by saying that it lies outside the latter place. That one can deny again the denied proposition shows that what is denied is already a proposition and not merely the preliminary to a proposition. 4.1 A proposition presents the existence and nonexistence of atomic facts. 4.11 The totality of true propositions is the total natural science, bracket, or the totality of the natural sciences. 4.111 Philosophy is not one of the natural sciences. Bracket, the word, quotes, philosophy, must mean something which stands above or below, but not beside the natural sciences. 4.112 The object of philosophy is the logical clarification of thoughts. Philosophy is not a theory, but an activity. A philosophical work consists essentially of elucidations. The result of philosophy is not a number of, quote, philosophical propositions, close quote, but to make propositions clear. Philosophy should make clear and delimit sharply the thoughts which otherwise are, as it were, opaque and blurred. 4.1121 Psychology is no near related to philosophy than is any other natural science. The theory of knowledge is the philosophy of psychology. Does not my study of sign language correspond to the study of thought processes which philosophers held to be so essential to the philosophy of logic? Only they got entangled for the most part in unessential psychological investigations and there is an analogous danger for my method. 4.1122 The Darwinian theory has no more to do with philosophy than has any other hypothesis of natural science. 4.113 Philosophy limits the disputable sphere of natural science. 4.114 It should limit the thinkable and thereby the unthinkable. It should limit the unthinkable from within through the thinkable. 4.115 It will mean the unspeakable by clearly displaying the speakable. 4.116 Everything that can be thought at all can be thought clearly. Everything that can be said can be said clearly. 4.12 Propositions can represent the whole reality, but they cannot represent what they must have in common with reality in order to be able to represent it. The logical form. To be able to represent the logical form, we should have to be able to put ourselves with the propositions outside logic. That is, outside the world. 4.121 Propositions cannot represent the logical form. This mirrors itself in the propositions. That which mirrors itself in language, language cannot represent. That which expresses itself in language, we cannot express by language. The propositions show the logical form of reality. They exhibit it. 4.1211 Thus a proposition, quotes FA, shows that in its sense the object A occurs. Two propositions, quotes FA and quotes GA, that they are both about the same object. If two propositions contradict one another, this is shown by their structure. Similarly, if one follows from another, etc. 4.1212 What can be shown cannot be said. 4.1213 Now we understand our feeling that we are in position of the right logical conception if only all is right in our symbolism. 4.122 We can speak in a certain sense of formal properties of objects and atomic facts, or of properties of the structure of facts, and in the same sense of formal relations and relations of structures. Bracket, instead of property of the structure, I also say, quote, internal property, close quote. Instead of relation of structures, quote, internal relation, close quote. I introduce these expressions in order to show the reason for the confusion, very widespread among philosophers, between internal relations and proper bracket external close bracket relations, close bracket. The holding of such internal properties and relations cannot, however, be asserted by propositions, but it shows itself in the propositions, which present the facts and treat of the objects in question. 4.1221 An internal property of a fact, we also call a feature of this fact. Bracket, in the sense in which we speak of facial features, close bracket. 4.123 A property is internal if it is unthinkable that its object does not possess it. Bracket, this bright blue color and that, stand in the internal relation of bright and darker, aeo ipso. It is unthinkable that these two objects should not stand in this relation. Close bracket. Bracket. Here, to the shifting use of the words quotes property and quotes relation, there corresponds the shifting use of the word quotes object. Close bracket. 4.124 The existence of an internal property of a possible state of affairs is not expressed by a proposition, but it expresses itself in the proposition which presents that state of affairs by an internal property of this proposition. It would be a senseless to ascribe a formal property to a proposition as to deny it the formal property. 4.1241 One cannot distinguish forms from one another by saying that one has this property, the other that. For this assumes that there is a sense in asserting either property of either form. 4.125 The existence of an internal relation between possible states of affairs expresses itself in language by an internal relation between the propositions presenting them. 4.1251 Now this settles the disputed question, quote whether all relations are internal or external. 4.1252 Series which are ordered by internal relations I call formal series. The series of numbers is ordered not by an external but by an internal relation. Similarly, the series of propositions, quote A R B, close quote. Quote bracket there exists X, close bracket, colon A R X and symbol X R B close quote Quote bracket there exists X, Y close bracket, colon A R X and symbol X R Y and symbol Y R B close quote, etc. bracket if B stands in one of these relations to A I call B a successor of A close bracket 4.1256 In the sense in which we speak of formal properties we can now speak also of formal concepts. Bracket I introduce this expression in order to make clear the confusion of formal concepts with proper concepts which runs through the whole of the old logic close bracket that anything falls under a formal concept as an object belonging to it cannot be expressed by a proposition but it is shown in the symbol for the object itself bracket the name shows that it signifies an object the numerical sign that it signifies a number, etc. close bracket formal concepts cannot, like proper concepts be presented by a function for their characteristics the formal properties are not expressed by the functions the expression of a formal property is a feature of certain symbols the sign that signifies the characteristic of a formal concept is therefore a characteristic feature of all symbols whose meanings fall under the concept the expression of the formal concept is therefore a propositional variable in which only this characteristic feature is constant 4.127 the propositional variable signifies the formal concept and its values signify the objects which fall under this concept 4.1271 every variable is the sign of a formal concept for every variable presents a constant form which all its values possess and which can be conceived as a formal property of these values 4.1272 so the variable name quotes X is the proper sign of the pseudo concept object wherever the word quotes object bracket quotes thing quotes entity, etc. close bracket is rightly used it is expressed in logical symbolism by the variable name for example in the proposition there are two objects which close quote by quote bracket there exists X, Y close bracket close quote wherever it is used otherwise in essence as a proper concept word there arise senseless pseudo propositions so one cannot for example say quote there are objects close quote as one says quote there are books close quote nor quote there are 100 objects close quote or quote there are aleph subscript 0 objects close quote and it is senseless to speak of the number of all objects the same holds of the words quotes complex quotes fact quotes function quotes number, etc. they all signify formal concepts and are presented in logical symbolism by variables not by functions or classes bracket as frega and russell thought close bracket expressions like quote one is a number close quote quote there is only one number not close quote and all like them are senseless bracket it is a senseless to say quote there is only one one close quote as it would be to say two plus two is at three o'clock equal to four close bracket four point one two seven two one the formal concept is already given with an object which falls under it one cannot therefore introduce both the objects which fall under a formal concept and the formal concept itself as primitive ideas one cannot therefore e.g. introduce bracket as russell does close bracket the concept of function and also of special functions as primitive ideas or the concept of number and definite numbers four point one two seven three if we want to express in logical symbolism the general proposition quote B is a successor of A close quote we need for this an expression for the general term of the formal series colon A R B bracket there exists X close bracket colon A R X symbol X R B bracket there exists X Y close bracket colon A capital R X and symbol X capital R Y and symbol Y capital R B the general term of a formal series can only be expressed by a variable for the concept symbolized by quote term of this formal series close quote is a formal concept bracket this the way in which they express general propositions like the above is therefore false it contains a vicious circle close bracket we can determine the general term of the formal series by giving its first term and the general form of the operation which generates the following term out of the preceding proposition four point one two seven four the question about the existence of a formal concept is senseless for no proposition can answer such a question bracket for example one cannot ask quote are there unanalyzable subject predicate propositions close quote close bracket four point one two eight the logical forms are a numerical therefore there are in logic no preeminent numbers and therefore there is no philosophical monism or dualism et cetera four point two the sense of a proposition is its agreement and disagreement with the possibilities of the existence and nonexistence of the atomic facts four point two one the simplest proposition the elementary proposition asserts the existence of an atomic fact four point two one one it is a sign of an elementary proposition that no elementary proposition can contradict it four point two two the elementary proposition consists of names it is a connection a concatenation of names four point two two one it is obvious that in the analysis of propositions we must come to elementary propositions which consists of names in immediate combination the question arises here how the propositional connection comes to be four point two two one one even if the world is infinitely complex so that every fact consists of an infinite number of atomic facts and every atomic fact is composed of an infinite number of objects even then there must be objects and atomic facts four point two three the name occurs in the proposition only in the context of the elementary proposition four point two four the names are the simple symbols I indicate them by single letters bracket x comma y comma z close bracket the elementary proposition I write as a function of the names in the form quotes fx comma phi bracket x comma y close bracket close quotes etc or I indicate it by the letters p q r four point two four one if I use two signs with one and the same meaning I express this by putting between them the sign quotes equals symbol quote a equals symbol b close quote means then that the sign quotes a is replaceable by the sign quotes b bracket if I introduce by an equation a new sign quotes b by determining that it shall replace a previously known sign quotes a I write the equation definition bracket like Russell close bracket in the form quote a equals symbol b def close quote a definition is a symbolic rule close bracket four point two four two expressions of the form quote a equals symbol b close quote are therefore only expedience in presentation they assert nothing about the meaning of the signs quotes a and quotes b four point two four three can we understand two names without knowing whether they signify the same thing or different things can we understand a proposition in which two names occur without knowing if they mean the same or different things if I know the meaning of an English and a synonymous German word it is impossible for me not to know that they are synonymous it is impossible for me not to be able to translate them into one another expressions like quote a equals symbol a close quote expressions deduced from these are neither elementary propositions nor otherwise significant signs bracket this will be shown later close bracket four point two five if the elementary proposition is true the atomic fact exists if it is false the atomic fact does not exist four point two six the specification of all true elementary propositions describes the world completely the world is completely described by the specification of all elementary propositions plus the specification which of them are true and which false four point two seven with regard to the existence of n atomic facts there are capital K subscript n equals sign small n over small v equals sign zero bracket n over v close bracket possibilities it is possible for all combinations of atomic facts to exist and the others not to exist four point two eight to these combinations correspond the same number of possibilities of the truth and falsehood and elementary propositions four point three the truth possibilities of the elementary propositions mean the possibilities of the existence and non-existence of the atomic facts four point three one the truth possibilities can be presented by schemata of the following kind bracket quotes t means quotes true quotes f false the rows of t's and f's under the row of the elementary propositions mean their truth possibilities in an easily intelligible symbolism readers note this table consists of three columns p q r t t t f t t t f t t t f f f t f t f t f t f f f f f f readers note this truth table consists of two columns p q t t f t t f t f f f readers note this truth table consists of one column p t f four point four a proposition is the expression of agreement and disagreement with the truth possibilities of the elementary propositions four point four one the truth possibilities of the elementary propositions are the conditions of the truth and falsehood of the propositions four point four one one it seems probable even at first sight that the introduction of the elementary propositions is fundamental for the comprehension of the other kinds of propositions indeed the comprehension of the general propositions depends palpably on that of the elementary propositions four point four two with regard to the agreement and disagreement of a proposition with the truth possibilities of n elementary propositions there are capital K subscript n over sigma over kappa equals zero bracket capital K subscript n over kappa close bracket equals capital L subscript n possibilities four point four three agreement with the truth possibilities can be expressed by coordinating with them in the scheme the mark quotes T bracket true close bracket absence of this mark means disagreement four point four three one the expression of the agreement and disagreement with the truth possibilities of the elementary propositions expresses the truth conditions of the proposition the proposition is the expression of the truth conditions bracket Frigga has therefore quite rightly put them at the beginning as explaining the signs of his logical symbolism only Frigga's explanation of the truth concept is false if quote the true close quote and quote the false close quote were real objects and the arguments in not symbol P etc then the sense of not symbol P could by no means be determined by Frigga's determination four point four four the sign which arises from the coordination of that mark quotes T with the truth possibilities is a propositional sign four point four four one it is clear that to the complex of the signs quotes F and quotes T no object bracket or complex of close bracket corresponds any more than to horizontal and vertical lines or to brackets there are no quote logical objects close quote something analogous holds of course for all signs which express the same as the schemata of quotes T and quotes F four point four four two thus for example readers note the truth table consists of three columns P Q T T T F T T T F blank F F T is a propositional sign bracket Frigga's assertion sign quote inference symbol close quote is logically altogether meaningless in Frigga bracket and Russell close bracket it only shows that these authors hold as true the propositions marked in this way quotes inference symbol belongs therefore to the propositions no more than does the number of the proposition a proposition cannot possibly assert of itself that it is true if the sequence of the truth possibilities in the scheme is once for all determined by a rule of combination then the last column is by itself an expression of the truth conditions if we write this column as a row the propositional sign becomes quote bracket T T hyphen T close bracket P, Q close bracket close quote or more plainly quote bracket T T F T close bracket P, Q close bracket close quote the number of places in the left hand bracket is determined by the number of terms in the right hand bracket close bracket 4.45 for in elementary propositions there are capital L subscript N possible groups of truth conditions the groups of truth conditions which belong to the truth possibilities of a number of elementary propositions can be ordered in a series 4.46 among the possible groups of truth conditions there are two extreme cases in the one case the proposition is true for all the truth possibilities of the elementary propositions we say that the truth conditions are tautological in the second case the proposition is false for all the truth possibilities the truth conditions are self-contradictory in the first case we call the proposition a tautology in the second case a contradiction 4.461 the proposition shows what it says the tautology and the contradiction that they say nothing the tautology has no truth conditions for it is unconditionally true and the contradiction is on no condition true tautology and contradiction are without sense bracket like the point from which two arrows go out in opposite directions close bracket bracket I know for example nothing about the weather when I know that it rains or does not rain close bracket 4.4611 tautology and contradiction are however not nonsensical they are part of the symbolism in the same way that 0 is part of the symbolism of arithmetic 4.462 tautology and contradiction are not pictures of the reality they present no possible state of affairs for the one allows every possible state of affairs the other none in the tautology the conditions of agreement with the world the presenting relations cancel one another so that it stands in no presenting relation to reality 4.463 the truth conditions determine the range which is left to the facts by the proposition bracket the proposition the picture the model are in a negative sense like a solid body which restricts the free movement of another in a positive sense like the space limited by solid substance in which a body may be placed close bracket tautology leaves to reality the whole infinite logical space contradiction fills the whole space and leaves no point to reality neither of them therefore can in any way determine reality 4.464 the truth of tautology is certain of propositions possible of contradiction impossible bracket certain possible impossible here we have an indication of that gradation which we need in the theory of probability close bracket 4.465 the logical product of a tautology and a proposition says the same as the proposition therefore that product is identical with the proposition for the essence of the symbol cannot be altered without altering its sense 4.466 to a definite logical combination of signs corresponds a definite logical combination of their meanings every arbitrary combination only corresponds to the unconnected signs that is propositions which are true for every state of affairs cannot be combinations of signs at all for otherwise there could only correspond to them definite combinations of objects and to no logical combination corresponds no combination of the objects close bracket tautology and contradiction are the limiting cases of the combination of symbols namely their dissolution 4.4661 of course the signs are also combined with one another in the tautology and contradiction in essence they stand in relation to one another but these relations are meaningless unessential to the symbol 4.5 here is to be possible to give the most general form of proposition i.e. to give a description of the propositions of some one sign language so that every possible sense can be expressed by a symbol which falls under the description and so that every symbol which falls under the description can express a sense if the meanings of the names are chosen accordingly it is clear that in the description of the most general form of proposition only what is essential to it may be described otherwise it would not be the most general form that there is a general form is proved by the fact that there cannot be a proposition whose form could not have been foreseen bracket i.e. constructed close bracket the general form of proposition is such and such is the case 4.51 suppose all elementary propositions were given me then we can simply ask what propositions i can build out of them and these are all propositions and so they are limited 4.52 the propositions are everything which follows from the totality of all elementary propositions bracket of course also from the fact that it is the totality of them all close bracket bracket so in some sense one could say that all propositions are generalizations of the elementary propositions close bracket 4.53 the general proposition form is a variable end of section 2 recording by jeffrey edwards section 3 of tractatus logico philosophicus this is a LibriVox recording all LibriVox recordings are in the public domain for more information or to volunteer please visit LibriVox.org recording by jeffrey edwards tractatus logico philosophicus by Ludwig Wittgenstein translated by C. K. Ogden section 3 5 propositions are truth functions of elementary propositions bracket an elementary proposition is a truth function of itself close bracket 5.01 the elementary propositions are the truth arguments of propositions 5.02 it is natural to confuse the arguments of functions with the indices of names for I recognize the meaning of the sign containing it from the argument just as much as from the index in Russell's quote plus subscript C quote C is an index which indicates that the whole sign is the addition sign for cardinal numbers but this way of symbolizing depends on arbitrary agreement and could choose a simple sign instead of quote plus subscript C close quote but in quotes not symbol P quotes P is not an index but an argument the sense of quote not symbol P close quote cannot be understood unless the sense of quotes P has previously been understood bracket in the name Julius Caesar Julius is an index the index is always part of a description of the object to whose name we attach it for example the Caesar of the Julian Jens close bracket the confusion of argument and index is if I am not mistaken at the root of Frigga's theory of the meaning of propositions and functions for Frigga the propositions of logic were names and their arguments the indices of these names 5.1 the 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 in a scheme of the following kind bracket T T T T close bracket bracket P comma Q close bracket tautology if P then P and if Q then Q close bracket square bracket P implies symbol P and symbol Q implies symbol Q close square bracket bracket F T T T close bracket bracket P comma Q close bracket in words not both P and Q square bracket not symbol bracket P and symbol Q close bracket close square bracket bracket T F T T close bracket bracket P comma Q close bracket if Q then P square bracket Q implies symbol P close square bracket bracket T T F T close bracket bracket P comma Q bracket if P then Q square bracket P implies symbol Q close square bracket bracket T T T F close bracket bracket P comma Q close bracket P or Q square bracket P or symbol Q close square bracket bracket F F T T close bracket bracket P comma Q close bracket not Q square bracket not symbol Q close square bracket bracket F T F T close bracket bracket P comma Q close bracket not P square bracket not symbol P close square bracket bracket F T T F close bracket bracket P comma Q close bracket P or Q but not both square bracket P and symbol not symbol Q colon or symbol colon Q and symbol not symbol P close square bracket bracket T F F T close bracket bracket P comma Q close bracket if P then Q and if Q then P square bracket P if and only if symbol Q close square bracket bracket T F T F close bracket bracket P comma Q close bracket P bracket T T F F close bracket bracket P comma Q close bracket Q Bracket, F, F, F, T, close bracket, bracket, P, comma, Q, close bracket, neither P nor Q, square bracket, P, N symbol, not symbol, Q, or P, neither nor symbol, Q, close square bracket, bracket, F, F, T, F, close bracket, bracket, P, comma, Q, close bracket, P and not Q, square bracket, P, N symbol, not symbol, Q, close square bracket, bracket, F, T, F, F, close bracket, bracket, P, comma, Q, close bracket, Q and not P, square bracket, Q, N symbol, not symbol, P, close square bracket, bracket, T, F, F, F, close bracket, bracket, P, comma, Q, close bracket, P and Q, square bracket, P, N symbol, Q, close square bracket, bracket, F, F, F, F, close bracket, bracket, P, comma, Q, close bracket, contradiction, bracket, P and not P, and Q and not Q, close bracket, square bracket, P, N symbol, not symbol, P, N symbol, Q, N symbol, not symbol, Q, close square bracket. Those truth possibilities of its truth arguments, which verify the proposition, I shall call its truth grounds. 5.11. If the truth grounds, which are common to a number of propositions, are all also truth grounds of some one proposition, 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 that of a proposition Q, if all the truth grounds of the second are truth grounds of the first. 5.121. The truth grounds of Q are contained in those of P. P follows from Q. 5.122. If P follows from Q, the sense of quotes P is contained in that of quotes Q. 5.123. If a god creates a world in which certain propositions are true, he creates thereby also a world in which all propositions consequent on them are true. And similarly, he could not create a world in which the proposition quotes P is true without creating all its objects. 5.124. A proposition asserts every proposition which follows from it. 5.1241. Quote P, N symbol, Q, close quote, is one of the propositions which assert quotes P, and at the same time, one of the propositions which assert quotes Q. Two propositions are opposed to one another if there is no significant proposition which asserts them both. Every proposition which contradicts another denies it. 5.13. That the truth of one proposition follows from the truth of other propositions we perceive from the structure of the propositions. 5.131. If the truth of one proposition follows from the truth of others, this expresses itself in relations in which the forms of these propositions stand to one another, and we do not need to put them in these relations first by connecting them with one another in a proposition. For these relations are internal and exist as soon as and by the very fact that the propositions exist. 5.1311. When we conclude from P or symbol Q and not symbol P to Q, the relation between the forms of the propositions quote P or symbol Q, close quote, and quotes not symbol P, is here concealed by the method of symbolizing. But if we write, for example, instead of quote P or symbol Q, close quote, quote P, neither nor symbol Q, and symbol neither nor symbol N symbol, P, neither nor symbol Q, close quote, and instead of quotes not symbol P, quote P, neither nor symbol P, close quote, bracket P, neither nor symbol Q equals neither P, nor Q, close bracket. Then the interconnection becomes obvious. Bracket. The fact that we can infer F A from bracket X, close bracket, and symbol F X shows that generality is present also in the symbol, quote bracket X, close bracket, and symbol F X, close quote, close bracket. 5.132. If P follows from Q, I can conclude from Q to P, infer P from Q. The method of inference is to be understood from the two propositions alone. Only they themselves can justify the inference. Laws of inference, which, as in Friega and Russell, are to justify the conclusions, are senseless, and would be superfluous. 5.133. All inference takes place a priori. 5.134. From an elementary proposition, no other can be inferred. 5.135. In no way can an inference be made from the existence of one state of affairs to the existence of another entirely different from it. 5.136. There is no causal nexus which justifies such an inference. 5.1361. The events of the future cannot be inferred from those of the present. Superstition is the belief in the causal nexus. 5.1362. The freedom of the will consists in the fact that future actions cannot be known now. We could only know them if causality were an inner necessity, like that of logical deduction. The connection of knowledge and what is known is that of logical necessity. Bracket, quote, A knows that P is the case, close quote, is senseless, if P is a tautology, close bracket. 5.1363. If from the fact that a proposition is obvious to us, it does not follow that it is true, then obviousness is no justification for our belief in its truths. 5.14. If a proposition follows from another, then the latter 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. A tautology follows from all propositions. It says nothing. 5.143. Contradiction is something shared by propositions, which no proposition has in common with another. Tautology is that which is shared by all propositions, which have nothing in common with one another. Contradiction vanishes, so to speak, outside. Tautology inside all propositions. Contradiction is the eternal limit of the propositions. Tautology, their substanceless center. 5.15. If T subscript r is the number of the truth grounds of the proposition quotes r, T subscript rs, the number of those truth grounds of the proposition quotes s, which are at the same time truth grounds of quotes r, then we call the ratio T subscript rs colon T subscript r, the measure of the probability which the proposition quotes r gives to the proposition quotes s. 5.151. Suppose in a schema like that above in number 5.101, capital T subscript r is the number of the quotes capital t's in the proposition r, capital T subscript rs, the number of those quotes capital t's in the proposition s, which stands in the same columns as quotes capital t's of the proposition r, then the proposition r gives to the proposition s the probability capital T subscript rs colon capital T subscript r. 5.1511. There's no special object peculiar to probability propositions. 5.152. Propositions which have no truth arguments in common with one another we call independent. Two elementary propositions give to one another the probability one half. If p follows from q, the proposition q gives to the proposition p the probability one. The certainty of logical conclusion is a limiting case of probability. Bracket, application to tautology and contradiction. Close bracket. 5.153. A proposition is in itself neither probable nor improbable. An event occurs or does not occur. There is no middle course. 5.154. In an urn there are equal numbers of white and black balls. Bracket and no others. Close bracket. I draw one ball after another and put them back in the urn. Then I can determine by the experiment that the numbers of the black and white balls which are drawn approximate as the drawing continues, so this is not a mathematical fact. If then I say it is equally probable that I should draw a white and a black ball, this means all the circumstances known to me, bracket, including the natural laws hypothetically assumed, close bracket, give to the occurrence of the one event no more probability than to the occurrence of the other. That is, they give, as can easily be understood from the above explanations, to each the probability one half. What I can verify by the experiment is that the occurrence of the two events is independent of the circumstances with which I have no closer acquaintance. 5.155. The unit of the probability proposition is the circumstances with which I am not further acquainted give to the occurrence of a definite event such and such a degree of probability. 5.156. Probability is a generalization. It involves a general description of a propositional form. Only in default of certainty do we need probability. If we are not completely acquainted with the fact but know something about its form, bracket, a proposition can, indeed, be an incomplete picture of a certain state of affairs, but it is always a complete picture. Close bracket. The probability proposition is, as it were, an extract from other propositions. 5.2. The structures of propositions stand to one another in internal relations. 5.21. We can bring out these internal relations in our manner of expression by presenting a proposition as the result of an operation which produces it from other propositions. Bracket. The basis of the operation. Close bracket. 5.22. The operation is the expression of a relation between the structures of its result and its basis. 5.23. The operation is that which must happen to a proposition in order to make another out of it. 5.231. And that will naturally depend on their formal properties, on the internal similarity of their forms. 5.232. The internal relation, which orders a series, is equivalent to the operation by which one term arises from another. 5.233. The first place in which an operation can occur is where a proposition arises from another in a logically significant way. In essence, where the logical construction of the proposition begins. 5.234. The truth functions of elementary propositions are results of operations which have the elementary propositions as basis. Bracket. I call these operations truth operations. Close bracket. 5.2341. The sense of a truth function of p is a function of the sense of p. Denial, logical addition, logical multiplication, etc., etc., are operations. Bracket. Denial reverses the sense of a proposition. Close bracket. 5.24. An operation shows itself in a variable. It shows how we can proceed from one form of proposition to another. It gives expression to the difference between the forms. Bracket and that which is common to the basis and the result of an operation is the basis themselves. Close bracket. 5.241. The operation does not characterize a form, but only the difference between forms. 5.242. The same operation which makes quotes q from quotes p makes quotes r from quotes q and so on. This can only be expressed by the fact that quotes p, quotes q, quotes r, etc., are variables which give general expression to certain formal relations. 5.25. The occurrence of an operation does not characterize the sense of a proposition. For, an operation does not assert anything. Only its result does, and this depends on the basis of the operation. Bracket. Operation and function must not be confused with one another. Close bracket. 5.251. A function cannot be its own argument, but the result of an operation can be its own basis. 5.252. Only in this way is the progress from term to term in a formal series possible. Bracket. From type to type in the hierarchy of Russell and Whitehead. Close bracket. Bracket. Russell and Whitehead have not admitted the possibility of this progress, but have made use of it all the same. Close bracket. 5.2521. The repeated application of an operation to its own result, I call its successive application. Bracket, quote, capital O prime, capital O prime, capital O prime, A, close quote, is the result of the three-fold successive application of, quote, capital O prime, close quote, to quotes A, close bracket. In a similar sense, I speak of the successive application of several operations to a number of propositions. 5.2522. The general term of the formal series, A comma, capital O prime, A comma, capital O prime, capital O prime, A comma, dot, dot, dot, I write thus, quote, square bracket, A comma, X comma, capital O prime, X, close square bracket, close quote. This expression in brackets is a variable. The first term of the 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 that term of the series which immediately follows X. 5.2523. The concept of the successive application of an operation is equivalent to the concept, quote, and so on, close quote. 5.253. One operation can reverse the effect of another. Operations can cancel one another. 5.254. Operations can vanish. Bracket, for example, denial in quote, not symbol, not symbol, P, close quote. N symbol, not symbol, not symbol, P equals P, close bracket. 5.3. All propositions are results of truth operations on the elementary propositions. The truth operation is the way in which a truth function arises from elementary propositions. According to the nature of truth operations, in the same way as out of elementary propositions arise their truth functions. From truth functions arise a new one. Every truth operation creates from truth functions of elementary propositions another truth function of elementary propositions, in essence a proposition. The result of every truth operation on the results of truth operations on elementary propositions is also the result of one truth operation on elementary propositions. Every proposition is the result of truth operations on elementary propositions. 5.31. The schemata number 4.31 are also significant if quotes P, quotes Q, quotes R, etc., are not elementary propositions, and it is easy to see that the propositional sign in number 4.42 expresses one truth function of elementary propositions, even when quotes P and quotes Q are truth functions 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 clear that there are no such things as quote logical objects, close quote, or quote logical constants, close quote, bracket in the sense of Frigga and Russell, close bracket. 5.41. For all those results of truth operations on truth functions are identical, which are one and the same truth function of elementary propositions. 5.42. That or symbol, implies symbol, etc., are not relations in the sense of right and left, etc., is obvious. The possibility of crosswise definition of the logical quote primitive signs, close quote, of Frigga and Russell shows by itself that these are not primitive signs and that they signify no relations. And it is obvious that the quotes implies symbol, which we define by means of quotes not symbol, and quotes or symbol, is identical with that by which we define quotes or symbol with the help of quotes not symbol, and that this quotes or symbol is the same as the first and so on. 5.43. That from a fact P, an infinite number of others should follow, namely not symbol, not symbol P, not symbol, not symbol, not symbol, not symbol P, etc., is indeed hardly to be believed. And it is no less wonderful that the infinite number of propositions of logic, bracket of mathematics, close bracket, should follow from half a dozen quote primitive propositions, close quote. But the propositions of logic say the same thing. That is nothing. 5.44. Truth functions are not material functions. If, for example, an affirmation can be produced by repeated denial, is the denial in any sense contained in the affirmation? Does quote not symbol not symbol P, close quote, deny quote not symbol P, close quote? Or does it affirm P, or both? The proposition quote not symbol, not symbol P, close quote, does not treat of denial as an object, but the possibility of denial is already prejudged in affirmation. And if there was an object called quotes, not symbol, then quote not symbol, not symbol P, close quote, would have to say something other than quotes P. For the one proposition would then treat of not symbol, the other would not. 5.441. This disappearance of the apparent logical constants also occurs if, quote, not symbol bracket, there exists symbol x, close bracket, n symbol, not symbol fx, close quote, says the same as, quote, bracket, x, close bracket, n symbol, fx, close quote, or quote bracket, there exists symbol x, close bracket, n symbol, fx, and symbol x equals a, close quote, the same as, quote, fa, close quote. 5.442. If a proposition is given to us, then the results of all truth operations which have it as their basis are given with it. 5.45. If there are logical primitive signs, a correct logic must make clear their position relative to one another and justify their existence. The construction of logic out of its primitive signs must become clear. 5.451. If logic has primitive ideas, these must be independent of one another. If a primitive idea is introduced, it must be introduced in all contexts in which it occurs at all. One cannot therefore introduce it for one context and then again for another. For example, if denial is introduced, we must understand it in propositions of the form, quote, not symbol, p, close quote. Just as in propositions like, quote, not symbol, bracket, p, or symbol, q, close bracket, close quote, quote, bracket, there exists symbol, x, close bracket, n symbol, not symbol, fx, close quote, and others. We may not first introduce it for one class of cases and then for another. For it would then remain doubtful whether its meaning in the two cases was the same, and there would be no reason to use the same way of symbolizing in the two cases. Bracket, in short, what frigah, bracket, quote, grund gesetze der arithmetic, close quote, close bracket, has said about the introduction of signs by definitions holds, mutatis, mutandis, for the introduction of primitive signs also, close bracket. 5.452. The introduction of a new expedient in the symbolism of logic must always be an event full of consequences. No new symbol may be introduced in logic in brackets or in the margin, with, so to speak, an entirely innocent face. Bracket, thus in the, quote, principia matematica, close quote, of Russell and Whitehead, there occur definitions and primitive propositions in words. Why suddenly words here? This would need a justification. There was none and can be none for the process is actually not allowed, close bracket. But if the introduction of a new expedient has proved necessary in one place, we must immediately ask, where is this expedient always to be used? Its position in logic must be made clear. 5.453. All numbers in logic must be capable of justification, or rather it must become plain that there are no numbers in logic. There are no preeminent numbers. 5.454. In logic, there is no side-by-side. There can be no classification. In logic, there cannot be a more general and a more special. 5.4541. The solution of logical problems must be neat, for they set the standard of neatness. Man have always thought that there must be a sphere of questions whose answers, a priori, are symmetrical and united into a closed regular structure. A sphere in which the proposition, simplex sigellum verae, is valid. 5.46. When we have rightly introduced the logical signs, the sense of all their combinations has been already introduced with them. Therefore, not only quote P or symbol Q, close quote, but also quote not symbol bracket P or symbol not symbol Q, close bracket, close quote, etc. etc. We should then already have introduced the effect of all possible combinations of brackets, and it would then have become clear that the proper general primitive signs are not quote P or symbol Q, close quote, quote bracket. There exists symbol x, close bracket, n symbol f, x, close quote, etc., but the most general form of their combinations. 5.461. The apparently unimportant fact that the apparent relations like or symbol and imply symbol need brackets, unlike real relations, is of great importance. The use of brackets with these apparent primitive signs shows that these are not the real primitive signs, and nobody of course would believe that the brackets have meaning by themselves. 5.4611. Logical operation signs are punctuations. 5.47. It is clear that everything which can be said beforehand about the form of all propositions at all can be said on one occasion. Four, all logical operations are already contained in the elementary proposition. Four, quote, fa, close quote, says the same as quote bracket. There exists symbol x, close bracket, n symbol f, x, n symbol x equals a, close quote. Where there is a composition, there is argument n function, and where these are, all logical constants already are. One could say, the one logical constant is that which all propositions, according to their nature, have in common with one another. That, however, is the general form of proposition. 5.471. The general form of proposition is the essence of proposition. 5.4711. To give the essence of proposition means to give the essence of all description, therefore 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 in logic. 5.473. Logic must take care of itself. A possible sign must also be able to signify. Everything which is possible in logic is also permitted. Bracket, quote, Socrates is identical, close quote, means nothing because there is no property which is called quotes identical. The proposition is senseless because we have not made some arbitrary determination, not because the symbol is itself unpermissible, close bracket. In a certain sense, we cannot make mistakes in logic. 5.4731. Self-evidence, of which Russell has said so much, can only be discarded in logic by language itself preventing every logical mistake. That logic is a priority consists in the fact that we cannot think illogically. 5.4732. We cannot give a sign the wrong sense. 5.47321. Occam's razor is, of course, not an arbitrary rule nor one justified by its practical success. It simply says that unnecessary elements in a symbolism mean nothing. Signs which serve one purpose are logically equivalent. Signs which serve no purpose are logically meaningless. 5.4733. Frigga says, every legitimately constructed proposition must have a sense. And I say, every possible proposition is legitimately constructed. And if it has no sense, this can only be because we have given no meaning to some of its constituent parts. Bracket. Even if we believe that we have done so, close bracket. Thus, quote, Socrates is identical, close quote, says nothing. Because we have given no meaning to the word, quotes identical as adjective. For when it occurs as the sign of equality, it symbolizes in an entirely different way. The symbolizing relation is another. Therefore, the symbol is in the two cases entirely different. The two symbols have the sign in common with one another only by accident. 5.474. The number of necessary fundamental operations depends only on our notation. 5.475. It is only a question of constructing a system of signs of a definite number of dimensions of a definite mathematical multiplicity. 5.476. It is clear that we are not concerned here with the number of primitive ideas, which must be signified, but with the expression of a rule. 5.5. Every truth function is a result of the successive application of the operation. Bracket dash dash dash dash dash capital T. Close bracket. Bracket xi comma dot dot dot dot. Close bracket to elementary propositions. This operation denies all the propositions in the right hand bracket, 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 the terms in the brackets is indifferent by a sign of the form quotes bracket line over xi. Close bracket close quote. Quotes xi is a variable whose values are the terms of the expression in brackets, and the line over the variable indicates that it stands for all its values in the bracket. Bracket thus if xi has the three values capital P, capital Q, capital R, then bracket line over xi close bracket equals bracket capital P comma capital Q comma capital R close bracket. The values of the variables must be determined. The determination is the description of the propositions which the variable stands for. How the description of the terms of the expression in brackets takes place is unessential. We may distinguish three kinds of description. One, direct enumeration. In this case, we can play simply its constant values instead of the variable. Two, giving a function fx, whose values for all values of x are the propositions to be described. Three, 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. End of section three, according by Jeffrey Edwards.