{"collection":"philosophy-ethics","full_text_available":true,"graph_url":"https://unchartedknowledge.5gfusion.net/v2/api/page/wikisource_Tractatus_Logico_Philosophicus_5/graph","html_url":"https://unchartedknowledge.5gfusion.net/v2/page/wikisource_Tractatus_Logico_Philosophicus_5","id":"wikisource_Tractatus_Logico_Philosophicus_5","license":"Public Domain / CC-BY-SA (see source)","rights_basis":"wikisource_site_inclusion_policy","source":"wikisource","source_url":"https://en.wikisource.org/wiki/Tractatus_Logico-Philosophicus/5","summary":null,"tags":["Pages_using_automatic_headers","Subpages"],"text":"5 Propositions are truth-functions of elementary propositions.\n\n(An elementary proposition is a truth-function of itself.)\n\n5.01 The elementary propositions are the truth-arguments of propositions.\n\n5.02 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.\n\nIn Russell's \"+c\", for example, \"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 one could choose a simple sign instead of \"+c\": but in \"~p\" \"p\" is not an index but an argument; the sense of \"~p\" cannot be understood, unless the sense of \"p\" has previously been understood. (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, e.g. The Caesar of the Julian gens.)\n\nThe confusion of argument and index is, if I am not mistaken, at the root of Frege's theory of the meaning of propositions and functions. For\nFrege the propositions of logic were names and\ntheir arguments the indices of these names.\n\n5.1 The truth-functions can be ordered in\nseries.\nThat is the foundation of the theory of probability.\n\n5.101 The truth-functions of every number of elementary propositions can be written in a schema of\nthe following kind:\n\n(T T T T) (p, q) Tautology (if p then p, and if q then q) [p\n\n\u2283\n\n{\\displaystyle \\supset }\n\np q\n\n\u2283\n\n{\\displaystyle \\supset }\n\nq]\n\n(F T T T) (p, q) in words: Not both p and q. [\n\n\u223c\n\n{\\displaystyle \\sim }\n\n(p.q)]\n\n(T F T T) (p, q) \u201e \u201e If q then p. [q\n\n\u2283\n\n{\\displaystyle \\supset }\n\np]\n\n(T T F T) (p, q) \u201e \u201e If p then q. [p\n\n\u2283\n\n{\\displaystyle \\supset }\n\nq]\n\n(T T T F) (p, q) \u201e \u201e p or q. [p\n\n\u2228\n\n{\\displaystyle \\lor }\n\nq]\n\n(F F T T) (p, q) \u201e \u201e Not q. [p\n\n\u2228\n\n{\\displaystyle \\lor }\n\nq]\n\n(F T F T) (p, q) \u201e \u201e Not p [p\n\n\u2228\n\n{\\displaystyle \\lor }\n\np]\n\n(F T T F) (p, q) \u201e \u201e p or q, but not both. [p.\n\n\u2228\n\n{\\displaystyle \\lor }\n\nq\u00a0: v\u00a0: q.\n\n\u2228\n\n{\\displaystyle \\lor }\n\np]\n\n(T F F T) (p, q) \u201e \u201e If p then q\u00a0; and if q, then p [p\n\n\u2261\n\n{\\displaystyle \\equiv }\n\nq]\n\n{T F T F) (p, q) \u201e \u201e p\n\n(T T F F) (p, q) \u201e \u201e q\n\n(F F F T) (p, q) \u201e \u201e Neither p nor q. [\n\n\u2228\n\n{\\displaystyle \\lor }\n\np.\n\n\u2228\n\n{\\displaystyle \\lor }\n\nq or p|q]\n\n(F F T F) (p, q) \u201e \u201e p and not q. [p.\n\n\u2228\n\n{\\displaystyle \\lor }\n\nq]\n\n(F T F F) (p, q) \u201e \u201e q and not p [q.\n\n\u2228\n\n{\\displaystyle \\lor }\n\np]\n\n(T F F F) (p, q) \u201e \u201e p and q. [p.q]\n\n(F F F F) (p, q) Contradiction (p and notp and q and not q.)[p.\n\n\u2228\n\n{\\displaystyle \\lor }\n\np.q.\n\n\u2228\n\n{\\displaystyle \\lor }\n\nq]\n\nThose truth-possibilities of its truth-arguments,\nwhich verify the proposition, I shall call its truth-grounds.\n\n5.11 If the truth-grounds which are common to a\nnumber of propositions are all also truth-grounds\nof some one proposition, we say that the truth of\nthis proposition follows from the truth of those\npropositions.\n\n5.12 In particular the truth of a proposition p follows\nfrom that of a proposition q, if all the truth-grounds of the second are truth-grounds of the\nfirst.\n\n5.121 The truth-grounds of q are contained in those\nof p; p follows from q.\n\n5.122 If p follows from q, the sense of \"p\" is contained\nin that of \"q\".\n\n5.123 If a god creates a world in which certain propositions are true, he creates thereby also a world\nin which all propositions consequent on them are\ntrue. And similarly he could not create a world\nin which the proposition \"p\" is true without\ncreating all its objects.\n\n5.124 A proposition asserts every proposition which\nfollows from it.\n\n5.1241 \"p.q\" is one of the propositions which assert\n\"p\" and at the same time one of the propositions\nwhich assert \"q\".\n\nTwo propositions are opposed to one another\nif there is no significant proposition which asserts\nthem both.\n\nEvery proposition which contradicts another,\ndenies it.\n\n5.13 That the truth of one proposition follows from\nthe truth of other propositions, we perceive from\nthe structure of the propositions.\n\n5.131 If the truth of one proposition follows from the\ntruth of others, this expresses itself in relations in\nwhich the forms of these propositions stand to one\nanother, and we do not need to put them in these\nrelations first by connecting them with one another\nin a proposition; for these relations are internal,\nand exist as soon as, and by the very fact that,\nthe propositions exist.\n\n5.1311 When we conclude from pvq and ~p to q the\nrelation between the forms of the propositions\n\"pvq\" and \"~p\" is here concealed by the method\nof symbolizing. But if we write, e.g. instead of\n\"pvq\" \"p|q .|. p|q\" and instead of \"~p\"\n\"p|p\" (p|q = neither p nor q), then the inner connexion becomes obvious.\n\n(The fact that we can infer fa from (x)fx shows\nthat generality is present also in the symbol \"(x).fx\".\n\n5.132 If p follows from q, I can conclude from qp to p;\ninfer p from q.\n\nThe method of inference is to be understood\nfrom the two propositions alone.\n\nOnly they themselves can justify the inference.\n\nLaws of inference, which\u2014as in Frege and\nRussell\u2014are to justify the conclusions, are senseless and would be superfluous.\n\n5.133 All inference takes place a priori.\n\n5.134 From an elementary proposition no other can\nbe inferred.\n\n5.135 In no way can an inference be made from the\nexistence of one state of affairs to the existence of\nanother entirely different from it.\n\n5.136 There is no causal nexus which justifies such\nan inference.\n\n5.1361 The events of the future cannot be inferred from\nthose of the present.\n\nSuperstition is the belief in the causal nexus.\n\n5.1362 The freedom of the will consists in the fact that\nfuture actions cannot be known now. We could\nonly know them if causality were an inner necessity,\nlike that of logical deduction.\u2014The connexion\nof knowledge and what is known is that of logical\nnecessity.\n\n(\"A knows that p is the case*' is senseless if p\nis a tautology.)\n\n5.1363 If from the fact that a proposition is obvious\nto us it does not follow that it is true, then obviousness is no justification for our belief in its truth.\n\n5.14 If a proposition follows from another, then the\nlatter says more than the former, the former less\nthan the latter.\n\n5.141 If p follows from q and q from p then they are\none and the same proposition.\n\n5.142 A tautology follows from all propositions: it\nsays nothing.\n\n5.143 Contradiction is something shared by propositions, which no proposition has in common with\nanother. Tautology is that which is shared by\nall propositions, which have nothing in common\nwith one another.\n\nContradiction vanishes so to speak outside,\ntautology inside all propositions.\n\nContradiction is the external limit of the propositions, tautology their substanceless centre.\n\n5.15 If Tr is the number of the truth-grounds of the\nproposition \"r\", Trs the number of those truth-grounds of the proposition \"s\" which are at the\nsame time truth-grounds of \"r\", then we call the\nratio Trs:Tr the measure of the probability which\nthe proposition \"r\" gives to the proposition \"s\".\n\n5.151 Suppose in a schema like that above in No.\n5.101 Tr is the number of the \"T\"'s in the proposition r, Trs the number of those \"T\"'s in\nthe proposition s, which stand in the same columns\nas \"T\"'s of the proposition r; then the proposition r gives to the proposition s the probability\nTrs: Tr.\n\n5.1511 There is no special object peculiar to probability\npropositions.\n\n5.152 Propositions which have no truth-arguments\nin common with one another we call independent.\n\nTwo elementary propositions give to one another\nthe probability \u00bd.\n\nIf p follows from q, the proposition q gives\nto the proposition p the probability 1. The\ncertainty of logical conclusion is a limiting case\nof probability.\n\n(Application to tautology and contradiction.)\n\n5.153 A proposition is in itself neither probable nor improbable. An event occurs or does not occur,\nthere is no middle course.\n\n5.154 In an urn there are equal numbers of white\nand black balls (and no others). I draw one\nball after another and put them back in the\nurn. Then I can determine by the experiment\nthat the numbers of the black and white balls\nwhich are drawn approximate as the drawing\ncontinues.\n\nSo this is not a mathematical fact.\n\nIf then, I say, It is equally probable that\nI should draw a white and a black ball, this\nmeans, All the circumstances known to me (including the natural laws hypothetically assumed)\ngive to the occurrence of the one event no more\nprobability than to the occurrence of the other.\nThat is they give\u2014as can easily be understood from the above explanations\u2014to each the\nprobability \u00bd.\n\nWhat I can verify by the experiment is that\nthe occurrence of the two events is independent\nof the circumstances with which I have no closer\nacquaintance.\n\n5.155 The unit of the probability proposition is: The\ncircumstances\u2014with which I am not further acquainted\u2014give to the occurrence of a definite event\nsuch and such a degree of probability.\n\n5.156 Probability is a generalization.\nIt involves a general description of a propositional form. Only in default of certainty do we\nneed probability.\n\nIf we are not completely acquainted with a fact,\nbut know something about its form.\n\n(A proposition can, indeed, be an incomplete\npicture of a certain state of affairs, but it is always a complete picture.)\n\nThe probability proposition is, as it were, an\nextract from other propositions.\n\n5.2 The structures of propositions stand to one\nanother in internal relations.\n\n5.21 We can bring out these internal relations in\nour manner of expression, by presenting a proposition as the result\nof an operation which produces it from other propositions (the bases of the\noperation).\n\n5.22 The operation is the expression of a relation\nbetween the structures of its result and its\nbases.\n\n5.23 The operation is that which must happen to a\nproposition in order to make another out of it.\n\n5.231 And that will naturally depend on their formal\nproperties, on the internal similarity of their\nforms.\n\n5.232 The internal relation which orders a series is\nequivalent to the operation by which one term\narises from another.\n\n5.233 The first place in which an operation can occur\nis where a proposition arises from another in a\nlogically significant way; i.e. where the logical\nconstruction of the proposition begins.\n\n5.234 The truth-functions of elementary proposition,\nare results of operations which have the elementary propositions as bases. (I call these\noperations, truth-operations.)\n\n5.2341 The sense of a truth-function of p is a function\nof the sense of p.\n\nDenial, logical addition, logical multiplication,\netc. etc., are operations.\n\n(Denial reverses the sense of a proposition.)\n\n5.24 An operation shows itself in a variable; it shows\nhow we can proceed from one form of proposition\nto another.\n\nIt gives expression to the difference between\nthe forms.\n\n(And that which is common to the bases, and\nthe result of an operation, is the bases themselves.)\n\n5.241 The operation does not characterize a form but\nonly the difference between forms.\n\n5.242 The same operation which makes \"q\" from\n\"p\", makes \"r\" from \"q\", and so on. This\ncan only be expressed by the fact that \"p\", \"q\",\n\"r\", etc., are variables which give general expression to certain formal relations.\n\n5.25 The occurrence of an operation does not characterize the sense of a proposition.\n\nFor an operation does not assert anything; only\nits result does, and this depends on the bases of\nthe operation.\n\n(Operation and function must not be confused\nwith one another.)\n\n5.251 A function cannot be its own argument, but\nthe result of an operation can be its own\nbasis.\n\n5.252 Only in this way is the progress from term\nto term in a formal series possible (from type\nto type in the hierarchy of Russell and Whitehead). (Russell and Whitehead have not admitted\nthe possibility of this progress but have made use\nof it all the same.)\n\n5.2521 The repeated application of an operation to\nits own result I call its successive application\n(\"O' O' O' a\" is the result of the threefold successive application of \"O' \u03be\" to \"a\").\n\nIn a similar sense I speak of the successive\napplication of several operations to a number of\npropositions.\n\n5.2522 The general term of the formal series a, O', a, O' O' a, ... I write thus: \"[a, x, O' x]\". This\nexpression in brackets is a variable. The first\nterm of the expression is the beginning of the formal series, the second the form of an arbitrary\nterm x of the series, and the third the form\nof that term of the series which immediately\nfollows x.\n\n5.2523 The concept of the successive application of\nan operation is equivalent to the concept \"and\nso on\".\n\n5.253 One operation can reverse the effect of another.\nOperations can cancel one another.\n\n5.254 Operations can vanish (e.g. denial in \"~ ~ p\".\n~ ~ p = p).\n\n5.3 All propositions are results of truth-operations\non the elementary propositions.\n\nThe truth - operation is the way in which a\ntruth - function arises from elementary propositions.\n\nAccording to the nature of truth-operations,\nin the same way as out of elementary propositions arise their truth-functions, from truth-functions arises a new one. Every truth-operation\ncreates from truth-functions of elementary propositions another truth-function of elementary\npropositions, i.e., a proposition. The result of\nevery truth-operation on the results of truth-operations on elementary propositions is also\nthe result of one truth-operation on elementary\npropositions.\n\nEvery proposition is the result of truth-operations on elementary propositions.\n\n5.31 The Schemata No. 4.31 are also significant, if\n\"p\", \"q\", \"r\", etc. are not elementary propositions.\n\nAnd it is easy to see that the propositional\nsign in No. 4.42 expresses one truth-function of\nelementary propositions even when \"p\" and\n\"q\" are truth-functions of elementary propositions.\n\n5.32 All truth-functions are results of the successive application of a finite number of truth-operations\nto elementary propositions.\n\n5.4 Here it becomes clear that there are no such\nthings as \"logical objects\" or \"logical constants\"\n(in the sense of Frege and Russell).\n\n5.41 For all those results of truth-operations on truth-functions are identical, which are one and the same\ntruth-function of elementary propositions.\n\n5.42 That \u2228, \u2283 etc., are not relations in the sense of\nright and left, etc., is obvious.\n\nThe possibility of crosswise definition of the\nlogical \"primitive signs\" of Frege and Russell\nshows by itself that these are not primitive signs\nand that they signify no relations.\n\nAnd it is obvious that the \"\u2283\" which we define\nby means of \"~\" and \"\u2228\" is identical with that\nby which we define \"\u2228\" with the help of \"~\", and\nthat this \"\u2228\" is the same as the first, and\nso on.\n\n5.43 That from a fact p an infinite number of others\nshould follow, namely ~ ~p, ~ ~ ~ ~p etc., is\nindeed hardly to be believed, and it is no less\nwonderful that the infinite number of propositions\nof logic (of mathematics) should follow from half\na dozen \"primitive propositions\".\n\nBut all propositions of logic say the same thing.\nThat is, nothing.\n\n5.44 Truth-functions are not material functions.\n\nIf e.g. an affirmation can be produced by\nrepeated denial, is the denial\u2014in any sense\u2014contained in the affirmation?\n\nDoes \"~ ~p\" deny ~p, or does it affirm p;\nor both?\n\nThe proposition \"~ ~p\" does not treat of\ndenial as an object, but the possibility of denial is\nalready prejudged in affirmation.\n\nAnd if there was an object called \"~\", then \"~ ~p\" would have to say something other than\n\"p\". For the one proposition would then treat\nof ~ , the other would not.\n\n5.441 This disappearance of the apparent logical\nconstants also occurs if \"\n\n\u00a0\n(\n\u2203\nx\n)\n\u00a0\nf\nx\n\n{\\displaystyle ~(\\exists x)~fx}\n\n\" says the\nsame as \"\n\n\u00a0\n(\nx\n)\n\u00a0\nf\nx\n\n{\\displaystyle ~(x)~fx}\n\n\", or \"\n\n\u00a0\n(\n\u2203\nx\n)\n.\nf\nx\n.\nx\n=\na\n\n{\\displaystyle ~(\\exists x).fx.x=a}\n\n\" the same\nas fa.\n\n5.442 If a proposition is given to us then the results\nof all truth-operations which have it as their basis\nare given with it.\n\n5.45 If there are logical primitive signs a correct logic\nmust make clear their position relative to one\nanother and justify their existence. The construction of logic out of its primitive signs must become\nclear.\n\n5.451 If logic has primitive ideas these must be\nindependent of one another. If a primitive idea\nis introduced it must be introduced in all contexts\nin which it occurs at all. One cannot therefore\nintroduce it for one context and then again for\nanother. For example, if denial is introduced,\nwe must understand it in propositions of the form\n\"~p\" just as in propositions like \"~(pvq)\",\n\"\n\n\u00a0\n(\n\u2203\nx\n)\n.\n\u00a0\nf\nx\n\n{\\displaystyle ~(\\exists x).~fx}\n\n\" and others. We may not first\nintroduce it for one class of cases and then for\nanother, for it would then remain doubtful whether\nits meaning in the two cases was the same, and\nthere would be no reason to use the same way of\nsymbolizing in the two cases.\n\n(In short, what Frege (\"Grundgesetze der\nArithmetik\") has said about the introduction of\nsigns by definitions holds, mutatis mutandis, for\nthe introduction of primitive signs also.)\n\n5.452 The introduction of a new expedient in the\nsymbolism of logic must always be an event full\nof consequences. No new symbol may be introduced in logic in brackets or in the margin\u2014with,\nso to speak, an entirely innocent face. (Thus in the \"Principia Mathematica\" of\nRussell and Whitehead there occur definitions\nand primitive propositions in words. Why suddenly words here? This would need a justification. There was none, and can be none for the\nprocess is actually not allowed.)\n\nBut if the introduction of a new expedient has\nproved necessary in one place, we must immediately ask: Where is this expedient always to be\nused? Its position in logic must be made\nclear.\n\n5.453 All numbers in logic must be capable of justification.\n\nOr rather it must become plain that there are\nno numbers in logic.\n\nThere are no pre-eminent numbers.\n\n5.454 In logic there is no side by side, there can be no classification.\n\nIn logic there cannot be a more general and a\nmore special.\n\n5.4541 The solution of logical problems must be neat\nfor they set the standard of neatness.\n\nMen have always thought that there must be a sphere of questions whose answers\u2014a priori\u2014are symmetrical and united into a closed regular structure.\n\nA sphere in which the proposition, simplex\nsigillum veri, is valid.\n\n5.46 When we have rightly introduced the logical\nsigns, the sense of all their combinations has been\nalready introduced with them: therefore not only\n\"pvq\" butalso \"~(pv~q)\", etc. etc. We should\nthen already have introduced the effect of all\npossible combinations of brackets\u00a0; and it would\nthen have become clear that the proper general\nprimitive signs are not \"pvq\", \"\n\n\u00a0\n(\n\u2203\nx\n)\n.\nf\nx\n\n{\\displaystyle ~(\\exists x).fx}\n\n\", etc., but the most general form of their combinations.\n\n5.461 The apparently unimportant fact that the apparent relations like\n\n\u2228\n\n{\\displaystyle \\lor }\n\nand\n\n\u2283\n\n{\\displaystyle \\supset }\n\nneed brackets\u2014unlike real relations\u2014is of great importance.\n\nThe use of brackets with these apparent primitive signs shows that these are not the real\nprimitive signs; and nobody of course would\nbelieve that the brackets have meaning by themselves.\n\n5.4611 Logical operation signs are punctuations.\n\n5.47 It is clear that everything which can be said\nbeforehand about the form of all propositions at\nall can be said on one occasion.\n\nFor all logical operations are already contained\nin the elementary proposition. For \"\n\nf\na\n\n{\\displaystyle fa}\n\n\" says\nthe same as \"\n\n\u00a0\n(\n\u2203\nx\n)\n.\nf\nx\n.\nx\n=\na\n\n{\\displaystyle ~(\\exists x).fx.x=a}\n\n\".\n\nWhere there is composition, there is argument\nand function, and where these are, all logical\nconstants already are.\n\nOne could say: the one logical constant is that\nwhich all propositions, according to their nature,\nhave in common with one another.\n\nThat however is the general form of proposition.\n\n5.471 The general form of proposition is the essence\nof proposition.\n\n5.4711 To give the essence of proposition means to\ngive the essence of all description, therefore the\nessence of the world.\n\n5.472 The description of the most general propositional form is the description of the one and only\ngeneral primitive sign in logic.\n\n5.473 Logic must take care of itself.\n\nA possible sign must also be able to signify.\nEverything which is possible in logic is also\npermitted. (\"Socrates is identical \" means nothing because there is no property which is called\n\"identical\". The proposition is senseless because\nwe have not made some arbitrary determination,\nnot because the symbol is in itself unpermissible.)\n\nIn a certain sense we cannot make mistakes in\nlogic.\n\n5.4731 Self-evidence, of which Russell has said so\nmuch, can only be discarded in logic by language\nitself preventing every logical mistake. That\nlogic is a priori consists in the fact that we cannot\nthink illogically.\n\n5.4732 We cannot give a sign the wrong sense.\n\n5.47321 Occam's razor is, of course, not an arbitrary rule\nnor one justified by its practical success. It simply\nsays that unnecessary elements in a symbolism\nmean nothing.\n\nSigns which serve one purpose are logically\nequivalent, signs which serve no purpose are\nlogically meaningless.\n\n5.4733 Frege says: Every legitimately constructed\nproposition must have a sense; and I say: Every\npossible proposition is legitimately constructed,\nand if it has no sense this can only be because\nwe have given no meaning to some of its constituent\nparts.\n\n(Even if we believe that we have done so.)\n\nThus \"Socrates is identical\" says nothing,\nbecause we have given no meaning to the word\n\"identical\" as adjective. For when it occurs as\nthe sign of equality it symbolizes in an entirely\ndifferent way\u2014the symbolizing relation is another\u2014therefore the symbol is in the two cases entirely\ndifferent; the two symbols have the sign in common with one another only by accident.\n\n5.474 The number of necessary fundamental operations depends only on our notation.\n\n5.475 It is only a question of constructing a system\nof signs of a definite number of dimensions\u2014of\na definite mathematical multiplicity.\n\n5.476 It is clear that we are not concerned here with\na number of primitive ideas which must be signified\nbut with the expression of a rule.\n\n5.5 Every truth-function is a result of the successive\napplication of the operation (- - - - T)(\n\n\u03be\n\n{\\displaystyle \\xi }\n\n, . . . .) to\nelementary propositions.\n\nThis operation denies all the propositions in\nthe right-hand bracket and I call it the negation\nof these propositions.\n\n5.501 An expression in brackets whose terms are\npropositions I indicate\u2014if the order of the terms\nin the bracket is indifferent\u2014by a sign of the form\n\"\n\n(\n\n\u03be\n\u00af\n\n)\n\n{\\displaystyle ({\\bar {\\xi }})}\n\n\". \"\n\n\u03be\n\n{\\displaystyle \\xi }\n\n\" is a variable whose values are the\nterms of the expression in brackets, and the line\nover the variable indicates that it stands for all\nits values in the bracket.\n\n(Thus if\n\n\u03be\n\n{\\displaystyle \\xi }\n\nhas the 3 values P, Q, R, then\n\n(\n\n\u03be\n\u00af\n\n)\n\n{\\displaystyle ({\\bar {\\xi }})}\n\n=(P,Q, R).)\n\nThe values of the variables must be determined.\n\nThe determination is the description of the propositions which the variable stands for.\n\nHow the description of the terms of the expression in brackets takes place is unessential.\n\nWe may distinguish 3 kinds of description:\n\nDirect enumeration. In this case we can place simply its constant values instead of the variable.\n\nGiving a function \"\n\nf\nx\n\n{\\displaystyle fx}\n\n\" whose values for all values of \"\n\nx\n\n{\\displaystyle x}\n\n\" are the propositions to be described.\n\nGiving a formal law, according to which those propositions are constructed. In this case the terms of the expression in brackets are all the\n\nterms of a formal series.\n\n5.502 Therefore I write instead of \"(- - - - T)(\n\n\u03be\n\n{\\displaystyle \\xi }\n\n, . . . .)\", \"\n\nN\n(\n\n\u03be\n\u00af\n\n)\n\n{\\displaystyle N({\\bar {\\xi }})}\n\n\".\n\nN\n(\n\n\u03be\n\u00af\n\n)\n\n{\\displaystyle N({\\bar {\\xi }})}\n\nis the negation of all the values of the\npropositional variable\n\n\u03be\n\n{\\displaystyle \\xi }\n\n.\n\n5.503 As it is obviously easy to express how propositions can be constructed by means of this operation\nand how propositions are not to be constructed by\nmeans of it, this must be capable of exact expression.\n\n5.51 If\n\n\u03be\n\n{\\displaystyle \\xi }\n\nhas only one value, then\n\nN\n(\n\n\u03be\n\u00af\n\n)\n\n{\\displaystyle N({\\bar {\\xi }})}\n\n=\n\n\u223c\np\n\n{\\displaystyle \\sim p}\n\n(not\n\np\n\n{\\displaystyle p}\n\n),\nif it has two values then\n\nN\n(\n\n\u03be\n\u00af\n\n)\n\n{\\displaystyle N({\\bar {\\xi }})}\n\n=\n\n\u223c\np\n.\n\u223c\nq\n\n{\\displaystyle \\sim p.\\sim q}\n\n(neither\n\np\n\n{\\displaystyle p}\n\nnor\n\nq\n\n{\\displaystyle q}\n\n).\n\n5.511 How can the all-embracing logic which mirrors\nthe world use such special catches and manipulations? Only because all these are connected into\nan infinitely fine network, to the great mirror.\n\n5.512 \"\n\n\u223c\np\n\n{\\displaystyle \\sim p}\n\n\" is true if \"\n\np\n\n{\\displaystyle p}\n\n\" is false. Therefore in the\ntrue proposition \"\n\n\u223c\np\n\n{\\displaystyle \\sim p}\n\n\" \"\n\np\n\n{\\displaystyle p}\n\n\" is a false proposition.\nHow then can the stroke \"\n\n\u223c\n\n{\\displaystyle \\sim }\n\n\" bring it into\nagreement with reality?\n\nThat which denies in \"\n\n\u223c\np\n\n{\\displaystyle \\sim p}\n\n\" is however not\n\"\n\n\u223c\n\n{\\displaystyle \\sim }\n\n\" but that which all signs of this notation,\nwhich deny\n\np\n\n{\\displaystyle p}\n\n, have in common.\n\nHence the common rule according to which\n\"\n\n\u223c\np\n\n{\\displaystyle \\sim p}\n\n\", \"\n\n\u223c\u223c\u223c\np\n\n{\\displaystyle \\sim \\sim \\sim p}\n\n\", \"\n\n\u223c\np\n\u2228\n\u223c\np\n\n{\\displaystyle \\sim p\\lor \\sim p}\n\n\", \"\n\n\u223c\np\n.\n\u223c\np\n\n{\\displaystyle \\sim p.\\sim p}\n\n\",\netc. etc. (to infinity) are constructed. And this\nwhich is common to them all mirrors denial.\n\n5.513 We could say: What is common to all symbols,\nwhich assert both\n\np\n\n{\\displaystyle p}\n\nand\n\nq\n\n{\\displaystyle q}\n\n, is the proposition\n\"\n\np\n.\nq\n\n{\\displaystyle p.q}\n\n\". What is common to all symbols, which\nassert either\n\np\n\n{\\displaystyle p}\n\nor\n\nq\n\n{\\displaystyle q}\n\n, is the proposition \"\n\np\n\u2228\nq\n\n{\\displaystyle p\\lor q}\n\n\".\n\nAnd similarly we can say: Two propositions\nare opposed to one another when they have\nnothing in common with one another; and every\nproposition has only one negative, because there\nis only one proposition which lies altogether\noutside it.\n\nThus in Russell's notation also it appears evident\nthat \"\n\nq\n:\np\n\u2228\n\u223c\np\n\n{\\displaystyle q:p\\lor \\sim p}\n\n\" says the same as \"\n\nq\n\n{\\displaystyle q}\n\n\"; that\n\"\n\np\n\u2228\n\u223c\np\n\n{\\displaystyle p\\lor \\sim p}\n\n\" says nothing.\n\n5.514 If a notation is fixed, there is in it a rule according to which all the propositions denying\n\np\n\n{\\displaystyle p}\n\nare\nconstructed, a rule according to which all the\npropositions asserting\n\np\n\n{\\displaystyle p}\n\nare constructed, a rule\naccording to which all the propositions asserting\n\np\n\n{\\displaystyle p}\n\nor\n\nq\n\n{\\displaystyle q}\n\nare constructed, and so on. These rules\nare equivalent to the symbols and in them their\nsense is mirrored.\n\n5.515 It must be recognized in our symbols that what\nis connected by \"\n\n\u2228\n\n{\\displaystyle \\lor }\n\n\", \"\n\n.\n\n{\\displaystyle .}\n\n\", etc., must be propositions.\n\nAnd this is the case, for the symbols \"\n\np\n\n{\\displaystyle p}\n\n\" and\n\"\n\nq\n\n{\\displaystyle q}\n\n\" presuppose \"\n\n\u2228\n\n{\\displaystyle \\lor }\n\n\", \"\n\n\u00a0\n\n{\\displaystyle ~}\n\n\", etc. If the sign\n\"\n\np\n\n{\\displaystyle p}\n\n\" in \"\n\np\n\u2228\nq\n\n{\\displaystyle p\\lor q}\n\n\" does not stand for a complex sign,\nthen by itself it cannot have sense; but then also\nthe signs \"\n\np\n\u2228\np\n\n{\\displaystyle p\\lor p}\n\n\", \"\n\np\n.\np\n\n{\\displaystyle p.p}\n\n\" etc. which have the\nsame sense as \"\n\np\n\n{\\displaystyle p}\n\n\" have no sense. If, however,\n\"\n\np\n\u2228\np\n\n{\\displaystyle p\\lor p}\n\n\" has no sense, then also \"\n\np\n\u2228\nq\n\n{\\displaystyle p\\lor q}\n\n\" can have no sense.\n\n5.5151 Must the sign of the negative proposition be\nconstructed by means of the sign of the positive?\nWhy should one not be able to express the\nnegative proposition by means of a negative fact?\n(Like: if \"\n\na\n\n{\\displaystyle a}\n\n\" does not stand in a certain relation\nto \"\n\nb\n\n{\\displaystyle b}\n\n\", it could express that \"\n\na\nR\nb\n\n{\\displaystyle aRb}\n\n\" is not the case.)\n\nBut here also the negative proposition is indirectly constructed with the positive.\n\nThe positive proposition must presuppose the\nexistence of the negative proposition and conversely.\n\n5.52 If the values of\n\n\u03be\n\n{\\displaystyle \\xi }\n\nare the total values of a function\n\nf\nx\n\n{\\displaystyle fx}\n\nfor all values of\n\nx\n\n{\\displaystyle x}\n\n, then\n\nN\n\n\u03be\n\u00af\n\n=\u223c\n(\n\u2203\nx\n)\n.\nf\nx\n\n{\\displaystyle N{\\bar {\\xi }}=\\sim (\\exists x).fx}\n\n.\n\n5.521 I separate the concept all from the truth-function.\n\nFrege and Russell have introduced generality\nin connexion with the logical product or the logical sum. Then it would be difficult to understand\nthe propositions \"\n\n\u00a0\n(\n\u2203\nx\n)\n.\nf\nx\n\n{\\displaystyle ~(\\exists x).fx}\n\n\" and \"\n\n\u00a0\n(\nx\n)\n.\nf\nx\n\n{\\displaystyle ~(x).fx}\n\n\" in which\nboth ideas lie concealed.\n\n5.522 That which is peculiar to the \"symbolism of\ngenerality\" is firstly, that it refers to a logical\nprototype, and secondly, that it makes constants\nprominent.\n\n5.523 The generality symbol occurs as an argument.\n\n5.524 If the objects are given, therewith are all objects also given.\n\nIf the elementary propositions are given, then\ntherewith all elementary propositions are also\ngiven.\n\n5.525 It is not correct to render the proposition\n\"\n\n\u00a0\n(\n\u2203\nx\n)\n.\nf\nx\n\n{\\displaystyle ~(\\exists x).fx}\n\n\"\u2014as Russell does\u2014in words \"fx is possible\".\n\nCertainty, possibility or impossibility of a state\nof affairs are not expressed by a proposition but\nby the fact that an expression is a tautology, a\nsignificant proposition or a contradiction.\n\nThat precedent to which one would always\nappeal, must be present in the symbol itself.\n\n5.526 One can describe the world completely by\ncompletely generalized propositions, i.e., without\nfrom the outset co-ordinating any name with a\ndefinite object.\n\nIn order then to arrive at the customary way\nof expression we need simply say after an expression \"there is one and only one x, which. . . .\":\nand this x is a,\n\n5.5261 A completely generalized proposition is like\nevery other proposition composite. (This is shown\nby the fact that in \"\n\n\u00a0\n(\n\u2203\nx\n,\n\u03d5\n)\n.\n\u03d5\nx\n\n{\\displaystyle ~(\\exists x,\\phi ).\\phi x}\n\n\" we must mention\n\"\n\n\u03d5\n\n{\\displaystyle \\phi }\n\n\" and \"\n\nx\n\n{\\displaystyle x}\n\n\" separately. Both stand independently in signifying relations to the world\nas in the ungeneralized proposition.)\n\nA characteristic of a composite symbol: it has\nsomething: in common with other symbols.\n\n5.5262 The truth or falsehood of every proposition alters\nsomething in the general structure of the world.\nAnd the range which is allowed to its structure by\nthe totality of elementary propositions is exactly\nthat which the completely general propositions\ndelimit.\n\n(If an elementary proposition is true, then, at\nany rate, there is one more elementary proposition\ntrue.)\n\n5.53 Identity of the object I express by identity of\nthe sign and not by means of a sign of identity.\nDifference of the objects by difference of the\nsigns.\n\n5.5301 That identity is not a relation between objects is\nobvious. This becomes very clear if, for example,\none considers the proposition \"\n\n(\nx\n)\n:\nf\nx\n\u2283\n.\nx\n=\na\n\n{\\displaystyle (x):fx\\supset .x=a}\n\n\"\nWhat this proposition says is simply that only\n\na\n\n{\\displaystyle a}\n\nsatisfies the function\n\nf\n\n{\\displaystyle f}\n\n, and not that only such\nthings satisfy the function\n\nf\n\n{\\displaystyle f}\n\nwhich have a certain\nrelation to\n\na\n\n{\\displaystyle a}\n\n.\n\nOne could of course say that in fact only\n\na\n\n{\\displaystyle a}\n\nhas this relation to\n\na\n\n{\\displaystyle a}\n\nbut in order to express\nthis we should need the sign of identity itself.\n\n.5302 Russell's definition of \"\n\n=\n\n{\\displaystyle =}\n\n\" won't do; because\naccording to it one cannot say that two objects\nhave all their properties in common. (Even if\nthis proposition is never true, it is nevertheless\nsignificant.)\n\n5.5303 Roughly speaking: to say of two things that\nthey are identical is nonsense, and to say of one\nthing that it is identical with itself is to say\nnothing.\n\n5.531 I write therefore not \"\n\nf\n(\na\n,\nb\n)\n.\na\n=\nb\n\n{\\displaystyle f(a,b).a=b}\n\n\" but \"\n\nf\n(\na\n,\na\n)\n\n{\\displaystyle f(a,a)}\n\n\"\n(or \"\n\nf\n(\nb\n,\nb\n)\n\n{\\displaystyle f(b,b)}\n\n\"). And not \"\n\nf\n(\na\n,\nb\n)\n.\n\u223c\na\n=\nb\n\n{\\displaystyle f(a,b).\\sim a=b}\n\n\" but \"\n\nf\n(\na\n,\nb\n)\n\n{\\displaystyle f(a,b)}\n\n\"\n\n5.532 And analogously: not \"\n\n(\n\u2203\nx\n,\ny\n)\n.\nf\n(\nx\n,\ny\n)\n.\nx\n=\ny\n\n{\\displaystyle (\\exists x,y).f(x,y).x=y}\n\n\", but \"\n\n(\n\u2203\nx\n)\n.\nf\n(\nx\n,\nx\n)\n\n{\\displaystyle (\\exists x).f(x,x)}\n\n\"; and not \"\n\n(\n\u2203\nx\n,\ny\n)\n.\nf\n(\nx\n,\ny\n)\n.\n\u223c\nx\n=\ny\n\n{\\displaystyle (\\exists x,y).f(x,y).\\sim x=y}\n\n\" but \"\n\n(\n\u2203\nx\n,\ny\n)\n.\nf\n(\nx\n,\ny\n)\n\n{\\displaystyle (\\exists x,y).f(x,y)}\n\n\".\n\n(Therefore instead of Russell's \"\n\n(\n\u2203\nx\n,\ny\n)\n.\nf\n(\nx\n,\ny\n)\n\n{\\displaystyle (\\exists x,y).f(x,y)}\n\n\": \"\n\n(\n\u2203\nx\n,\ny\n)\n.\nf\n(\nx\n,\ny\n)\n\u2228\n(\n\u2203\nx\n)\n.\nf\n(\nx\n,\nx\n)\n\n{\\displaystyle (\\exists x,y).f(x,y)\\lor (\\exists x).f(x,x)}\n\n\".)\n\n5.5321 Instead of \"\n\n(\nx\n)\n.\nf\nx\n\u2283\nx\n=\na\n\n{\\displaystyle (x).fx\\supset x=a}\n\n\" we therefore write e.g. \"\n\n(\n\u2203\nx\n)\n.\nf\nx\n.\n\u2283\n.\nf\na\n:\u223c\n(\n\u2203\nx\n,\ny\n)\n.\nf\nx\n.\nf\ny\n\n{\\displaystyle (\\exists x).fx.\\supset .fa:\\sim (\\exists x,y).fx.fy}\n\n\"\n\nAnd the proposition \"only one\n\nx\n\n{\\displaystyle x}\n\nsatisfies\n\nf\n(\n)\n\n{\\displaystyle f()}\n\n\"\nreads: \"\n\n(\n\u2203\nx\n)\n.\nf\nx\n:\u223c\n(\n\u2203\nx\n,\ny\n)\n.\nf\nx\n.\nf\ny\n\n{\\displaystyle (\\exists x).fx:\\sim (\\exists x,y).fx.fy}\n\n\".\n\n5.533 The identity sign is therefore not an essential\nconstituent of logical notation.\n\n5.534 And we see that apparent propositions like:\n\"\n\na\n=\na\n\n{\\displaystyle a=a}\n\n\", \"\n\na\n=\nb\n.\nb\n=\nc\n.\n\u2283\na\n=\nc\n\n{\\displaystyle a=b.b=c.\\supset a=c}\n\n\", \"\n\n(\nx\n)\n.\nx\n=\nx\n\n{\\displaystyle (x).x=x}\n\n\", \"\n\n(\n\u2203\nx\n)\n.\nx\n=\na\n\n{\\displaystyle (\\exists x).x=a}\n\n\", etc. cannot be written in a correct logical notation at all.\n\n5.535 So all problems disappear which are connected with such pseudo-propositions.\n\nThis is the place to solve all the problems which\narise through Russell's \"Axiom of Infinity\".\n\nWhat the axiom of infinity is meant to say\nwould be expressed in language by the fact that\nthere is an infinite number of names with different\nmeanings.\n\n5.5351 There are certain cases in which one is tempted\nto use expressions of the form *\"\n\na\n=\na\n\n{\\displaystyle a=a}\n\n\" or \"\n\np\n\u2283\np\n\n{\\displaystyle p\\supset p}\n\n\".\nAs, for instance, when one would speak of the\narchetype Proposition, Thing, etc. So Russell in the\nPrinciples of Mathematics has rendered the nonsense \"p\nis a proposition\" in symbols by \"\n\np\n\u2283\np\n\n{\\displaystyle p\\supset p}\n\n\" and has\nput it as hypothesis before certain propositions to\nshow that their places for arguments could only\nbe occupied by propositions.\n\n(It is nonsense to place the hypothesis\n\np\n\u2283\np\n\n{\\displaystyle p\\supset p}\n\nbefore a proposition in order to ensure that its\narguments have the right form, because the\nhypothesis for a non-proposition as argument\nbecomes not false but meaningless, and because\nthe proposition itself becomes senseless for arguments of the wrong kind, and therefore it survives\nthe wrong arguments no better and no worse\nthan the senseless hypothesis attached for this\npurpose.)\n\n5.5352 Similarly it was proposed to express \"There are\nno things\" by \"\n\n\u223c\n(\n\u2203\nx\n)\n.\nx\n=\nx\n\n{\\displaystyle \\sim (\\exists x).x=x}\n\n\" But even if this\nwere a proposition\u2014would it not be true if indeed\n\"There were things\", but these were not identical\nwith themselves?\n\n5.54 In the general propositional form, propositions\noccur in a proposition only as bases of the truth-operations.\n\n5.541 At first sight it appears as if there were also a\ndifferent way in which one proposition could occur\nin another.\n\nEspecially in certain propositional forms of\npsychology, like \"A thinks, that\n\np\n\n{\\displaystyle p}\n\nis the case\",\nor \"A thinks\n\np\n\n{\\displaystyle p}\n\n\", etc.\n\nHere it appears superficially as if the proposition\n\np\n\n{\\displaystyle p}\n\nstood to the object A in a kind of relation.\n\n(And in modern epistemology (Russell, Moore,\netc.) those propositions have been conceived in\nthis way.)\n\n5.542 But it is clear that \"A believes that\n\np\n\n{\\displaystyle p}\n\n\", \"A\nthinks\n\np\n\n{\\displaystyle p}\n\n\", \"A says\n\np\n\n{\\displaystyle p}\n\n\", are of the form \"'\n\np\n\n{\\displaystyle p}\n\n' says\n\np\n\n{\\displaystyle p}\n\n\": and here we have no co-ordination of a fact\nand an object, but a co-ordination of facts by\nmeans of a co-ordination of their objects.\n\n5.5421 This shows that there is no such thing as the\nsoul\u2014the subject, etc.\u2014as it is conceived in contemporary superficial psychology.\n\nA composite soul would not be a soul any longer.\n\n5.5422 The correct explanation of the form of the\nproposition \"\n\np\n\n{\\displaystyle p}\n\nA judges\n\np\n\n{\\displaystyle p}\n\n\" must show that it is\nimpossible to judge a nonsense. (Russell's theory\ndoes not satisfy this condition.)\n\n5.5423 To perceive a complex means to perceive that its constituents are combined in such and such a\nway.\n\nThis perhaps explains that the figure\n\ncan be seen in two ways as a cube; and all similar\nphenomena. For we really see two different facts.\n\n(If I fix my eyes first on the corners a and only\nglance at b, a appears in front and b behind, and\nvice versa.)\n\n5.55 We must now answer a priori the question\nas to all possible forms of the elementary propositions.\n\nThe elementary proposition consists of names.\nSince we cannot give the number of names with\ndifferent meanings, we cannot give the composition\nof the elementary proposition.\n\n5.551 Our fundamental principle is that every question\nwhich can be decided at all by logic can be decided\noff-hand.\n\n(And if we get into a situation where we need\nto answer such a problem by looking at the world,\nthis shows that we are on a fundamentally wrong\ntrack.)\n\n5.552 The \"experience\" which we need to understand logic is not that such and such is the\ncase, but that something is; but that is no experience.\n\nLogic precedes every experience\u2014that something is so.\n\nIt is before the How, not before the What.\n\n5.5521 And if this were not the case, how could\nwe apply logic? We could say: if there\nwere a logic, even if there were no world,\nhow then could there be a logic, since there is a\nworld?\n\n5.553 Russell said that there were simple relations\nbetween different numbers of things (individuals).\nBut between what numbers? And how should\nthis be decided\u2014by experience?\n\n(There is no pre-eminent number.)\n\n5.554 The enumeration of any special forms would\nbe entirely arbitrary.\n\n5.5541 How could we decide a priori whether, for\nexample, I can get into a situation in which I need\nto symbolize with a sign of a 27-termed relation?\n\n5.5542 May we then ask this at all? Can we set out\na sign form and not know whether anything can\ncorrespond to it?\n\nHas the question sense: what must there be\nin order that anything can be the case?\n\n5.555 It is clear that we have a concept of the\nelementary proposition apart from its special\nlogical form.\n\nWhere, however, we can build symbols\naccording to a system, there this system is the\nlogically important thing and not the single\nsymbols.\n\nAnd how would it be possible that I should\nhave to deal with forms in logic which I can\ninvent: but I must have to deal with that which\nmakes it possible for me to invent them.\n\n5.556 There cannot be a hierarchy of the forms of the\nelementary propositions. Only that which we\nourselves construct can we foresee.\n\n5.5561 Empirical reality is limited by the totality of\nobjects. The boundary appears again in the\ntotality of elementary propositions. The hierarchies are and must be independent\nof reality.\n\n5.5562 If we know on purely logical grounds, that\nthere must be elementary propositions, then this\nmust be known by everyone who understands\npropositions in their unanalysed form.\n\n5.5563 All propositions of our colloquial language are\nactually, just as they are, logically completely\nin order. That simple thing which we ought to\ngive here is not a model of the truth but the complete\ntruth itself.\n\n(Our problems are not abstract but perhaps the\nmost concrete that there are.)\n\n5.557 The application of logic decides what elementary\npropositions there are.\n\nWhat lies in its application logic cannot anticipate.\n\nIt is clear that logic may not conflict with its\napplication.\n\nBut logic must have contact with its application.\n\nTherefore logic and its application may not\noverlap one another.\n\n5.5571 If I cannot give elementary propositions a\npriori then it must lead to obvious nonsense to\ntry to give them.\n\n5.6 The limits of my language mean the limits of my world.\n\n5.61 Logic fills the world: the limits of the world\nare also its limits.\n\nWe cannot therefore say in logic: This and\nthis there is in the world, that there is not.\n\nFor that would apparently presuppose that we\nexclude certain possibilities, and this cannot be\nthe case since otherwise logic must get outside\nthe limits of the world: that is, if it could\nconsider these limits from the other side\nalso. What we cannot think, that we cannot think:\nwe cannot therefore say what we cannot\nthink.\n\n5.62 This remark provides a key to the question, to\nwhat extent solipsism is a truth.\n\nIn fact what solipsism means, is quite correct,\nonly it cannot be said, but it shows itself.\n\nThat the world is my world, shows itself in\nthe fact that the limits of the language (the\nlanguage which I understand) mean the limits of\nmy world.\n\n5.621 The world and life are one.\n\n5.63 I am my world. (The microcosm.)\n\n5.631 The thinking, presenting subject; there is no\nsuch thing.\n\nIf I wrote a book \"The world as I found it\",\nI should also have therein to report on my body\nand say which members obey my will and which\ndo not, etc. This then would be a method of\nisolating the subject or rather of showing that in\nan important sense there is no subject: that is to\nsay, of it alone in this book mention could not be\nmade.\n\n5.632 The subject does not belong to the world but\nit is a limit of the world.\n\n5.633 Where in the world is a metaphysical subject to\nbe noted?\n\nYou say that this case is altogether like that of\nthe eye and the field of sight. But you do not\nreally see the eye.\n\nAnd from nothing in the field of sight can it be\nconcluded that it is seen from an eye.\n\n5.6331 For the field of sight has not a form like this:\n\n5.634 This is connected with the fact that no part of our experience is also a priori.\n\nEverything we see could also be otherwise.\n\nEverything we can describe at all could also be otherwise.\n\nThere is no order of things a priori.\n\n5.64 Here we see that solipsism strictly carried out coincides with pure realism. The I in solipsism shrinks to an extensionless point and there remains the reality co-ordinated with it.\n\n5.641 There is therefore really a sense in which in philosophy we can talk of a non-psychological I.\n\nThe I occurs in philosophy through the fact that the \"world is my world\".\n\nThe philosophical I is not the man, not the human body or the human soul of which psychology treats, but the metaphysical subject, the limit\u2014not a part of the world.","title":"Tractatus Logico-Philosophicus/5","tradition":"philosophy-ethics","translated_text":null,"word_count":7572}