"Let's see what Witters has to say about this"

4.46 GER [→OGD | →P/M] Unter den möglichen Gruppen von Wahrheitsbedingungen gibt es zwei extreme Fälle. In dem einen Fall ist der Satz für sämtliche Wahrheitsmöglichkeiten der Elementarsätze wahr. Wir sagen, die Wahrheitsbedingungen sind t a u t o l o g i s c h. Im zweiten Fall ist der Satz für sämtliche Wahrheitsmöglichkeiten falsch: Die Wahrheitsbedingungen sind k o n t r a d i k t o r i s c h. Im ersten Fall nennen wir den Satz eine Tautologie, im zweiten Fall eine Kontradiktion.


4.461 GER [→OGD | →P/M] Der Satz zeigt was er sagt, die Tautologie und die Kontradiktion, dass sie nichts sagen. Die Tautologie hat keine Wahrheitsbedingungen, denn sie ist bedingungslos wahr; und die Kontradiktion ist unter keiner Bedingung wahr. Tautologie und Kontradiktion sind sinnlos. (Wie der Punkt, von dem zwei Pfeile in entgegengesetzter Richtung auseinandergehen.) (Ich weiß z. B. nichts über das Wetter, wenn ich weiß, dass es regnet oder nicht regnet.) 4.4611 GER [→OGD | →P/M] Tautologie und Kontradiktion sind aber nicht unsinnig; sie gehören zum Symbolismus, und zwar ähnlich wie die „0“ zum Symbolismus der Arithmetik. 4.462 GER [→OGD | →P/M] Tautologie und Kontradiktion sind nicht Bilder der Wirklichkeit. Sie stellen keine mögliche Sachlage dar. Denn jene lässt j e d e mögliche Sachlage zu, diese k e i n e. In der Tautologie heben die Bedingungen der Übereinstimmung mit der Welt—die darstellenden Beziehungen—einander auf, so dass sie in keiner darstellenden Beziehung zur Wirklichkeit steht. 4.463 GER [→OGD | →P/M] Die Wahrheitsbedingungen bestimmen den Spielraum, der den Tatsachen durch den Satz gelassen wird. (Der Satz, das Bild, das Modell, sind im negativen Sinne wie ein fester Körper, der die Bewegungsfreiheit der anderen beschränkt; im positiven Sinne, wie der von fester Substanz begrenzte Raum, worin ein Körper Platz hat.) Die Tautologie lässt der Wirklichkeit den ganzen—unendlichen—logischen Raum; die Kontradiktion erfüllt den ganzen logischen Raum und lässt der Wirklichkeit keinen Punkt. Keine von beiden kann daher die Wirklichkeit irgendwie bestimmen. 4.464 GER [→OGD | →P/M] Die Wahrheit der Tautologie ist gewiss, des Satzes möglich, der Kontradiktion unmöglich. (Gewiss, möglich, unmöglich: Hier haben wir das Anzeichen jener Gradation, die wir in der Wahrscheinlichkeitslehre brauchen.) 4.465 GER [→OGD | →P/M] Das logische Produkt einer Tautologie und eines Satzes sagt dasselbe, wie der Satz. Also ist jenes Produkt identisch mit dem Satz. Denn man kann das Wesentliche des Symbols nicht ändern, ohne seinen Sinn zu ändern. 4.466 GER [→OGD | →P/M] Einer bestimmten logischen Verbindung von Zeichen entspricht eine bestimmte logische Verbindung ihrer Bedeutungen; j e d e b e l i e - b i g e Verbindung entspricht nur den unverbundenen Zeichen. Das heißt, Sätze, die für jede Sachlage wahr sind, können überhaupt keine Zeichenverbindungen sein, denn sonst könnten ihnen nur bestimmte Verbindungen von Gegenständen entsprechen. (Und keiner logischen Verbindung entspricht k e i n e Verbindung der Gegenstände.) Tautologie und Kontradiktion sind die Grenzfälle der Zeichenverbindung, nämlich ihre Auflösung. 4.4661 GER [→OGD | →P/M] Freilich sind auch in der Tautologie und Kontradiktion die Zeichen noch mit einander verbunden, d. h. sie stehen in Beziehungen zu einander, aber diese Beziehungen sind bedeu- tungslos, dem S y m b o l unwesentlich.


4.46 OGD [→GER | →P/M] Among the possible groups of truthconditions 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 OGD [→GER | →P/M] 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. (Like the point from which two arrows go out in opposite directions.) (I know, e.g. nothing about the weather, when I know that it rains or does not rain.) 4.4611 OGD [→GER | →P/M] Tautology and contradiction are, however, not nonsensical; they are part of the symbol- ism, in the same way that “0” is part of the symbolism of Arithmetic. 4.462 OGD [→GER | →P/M] 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 OGD [→GER | →P/M] The truth-conditions determine the range, which is left to the facts by the proposition. (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.) Tautology leaves to reality the whole infinite logical space; contradiction fills the whole logi- cal space and leaves no point to reality. Neither of them, therefore, can in any way determine reality. 4.464 OGD [→GER | →P/M] The truth of tautology is certain, of propositions possible, of contradiction impossible. (Certain, possible, impossible: here we have an indication of that gradation which we need in the theory of probability.) 4.465 OGD [→GER | →P/M] 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 OGD [→GER | →P/M] 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 ev- ery 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.) Tautology and contradiction are the limiting cases of the combination of symbols, namely their dissolution. 4.4661 OGD [→GER | →P/M] Of course the signs are also combined with one another in the tautology and contradiction, i.e. they stand in relations to one another, but these relations are meaningless, unessential to the symbol.

4.46 P/M [→GER | →OGD] Among the possible groups of truthconditions there are two extreme cases. In one of these cases 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 contradictory. In the first case we call the proposition a tautology; in the second, a contradiction. 4.461 P/M [→GER | →OGD] Propositions show what they say: tautolo- gies and contradictions show that they say nothing. A tautology has no truth-conditions, since it is unconditionally true: and a contradiction is true on no condition. Tautologies and contradictions lack sense. (Like a point from which two arrows go out in opposite directions to one another.) (For example, I know nothing about the weather when I know that it is either raining or not raining.) 4.4611 P/M [→GER | →OGD] Tautologies and contradictions are not, however, nonsensical. They are part of the symbolism, much as ‘0’ is part of the symbolism of arithmetic. 4.462 P/M [→GER | →OGD] Tautologies and contradictions are not pictures of reality. They do not represent any possible situations. For the former admit all possible situations, and latter none. In a tautology the conditions of agreement with the world—the representational relations—cancel one another, so that it does not stand in any representational relation to reality. 4.463 P/M [→GER | →OGD] The truth-conditions of a proposition determine the range that it leaves open to the facts. (A proposition, a picture, or a model is, in the negative sense, like a solid body that restricts the freedom of movement of others, and, in the positive sense, like a space bounded by solid substance in which there is room for a body.) A tautology leaves open to reality the whole—the infinite whole—of logical space: a contradiction fills the whole of logical space leaving no point of it for reality. Thus neither of them can determine reality in any way. 4.464 P/M [→GER | →OGD] A tautology’s truth is certain, a proposition’s possible, a contradiction’s impossible. (Certain, possible, impossible: here we have the first indication of the scale that we need in the theory of probability.) 4.465 P/M [→GER | →OGD] The logical product of a tautology and a proposition says the same thing as the proposition. This product, therefore, is identical with the proposition. For it is impossible to alter what is essential to a symbol without altering its sense. 4.466 P/M [→GER | →OGD] What corresponds to a determinate logical combination of signs is a determinate logical combination of their meanings. It is only to the uncombined signs that absolutely any combination corresponds. In other words, propositions that are true for every situation cannot be combinations of signs at all, since, if they were, only determinate combinations of objects could correspond to them. (And what is not a logical combination has no combination of objects corresponding to it.) Tautology and contradiction are the limiting cases—indeed the disintegration—of the combination of signs. 4.4661 P/M [→GER | →OGD] Admittedly the signs are still combined with one another even in tautologies and contradictions—i.e. they stand in certain relations to one another: but these relations have no meaning, they are not essential to the symbol.