Tag: logic

Without criticism no progress

But the most important misunderstandings and muddles arise out of the loose way in which dialecticians speak about contradictions.

They observe, correctly, that contradictions are of the greatest importance in the history of thought—precisely as impor­tant as is criticism. For criticism invariably consists in pointing out some contradiction; either a contradiction within the theory criticized, or a contradiction between the theory and another theory which we have some reason to accept, or a contradiction between the theory and certain facts—or more precisely, between the theory and certain statements of fact. Criticism can never do anything except either point out some such contradiction, or, perhaps, simply contradict the theory (i.e. the criticism may be simply the statement of an antithesis). But criticism is, in a very important sense, the main motive force of any intellectual development. Without contradictions, without criticism, there would be no rational motive for changing our theories: there would be no intellectual progress. [424]

The Munchhausen trilemma of justificationism

Nun entsteht aber, wenn unser Prinzip ernst genommen wird, sogleich folgendes Problem: Wenn man für alles eine Begründung verlangt, muß man auch für die Erkenntnisse, auf die man jeweils die zu begründende Auffassung – bzw. die betreffende Aussagen-Menge – zurückgeführt hat, wieder eine Begründung verlangen. Das führt zu einer Situation mit drei Alternativen, die alle drei unakzeptabel erscheinen, also: zu einem Trilemma, das ich angesichts der Analogie, die zwischen unserer Problematik und dem Problem besteht, das der bekannte Lügenbaron einmal zu lösen hatte, das Münchhausen-Trilemma nennen möchte. Man hat hier offenbar nämlich nur die Wahl zwischen:

1. einem infiniten Regreß, der durch die Notwendigkeit gegeben erscheint, in der Suche nach Gründen immer weiter zurückzugehen, der aber praktisch nicht durchzuführen ist und daher keine sichere Grundlage liefert;
2. einem logischen Zirkel in der Deduktion, der dadurch entsteht, daß man im Begründungsverfahren auf Aussagen zu­rückgreift, die vorher schon als begründungsbedürftig aufgetreten waren, und der ebenfalls zu keiner sicheren Grund­lage führt; und schließlich:
3. einem Abbruch des Verfahrens an einem bestimmten Punkt, der zwar prinzipiell durchführbar erscheint, aber eine willkürliche Suspendierung des Prinzips der zureichenden Begründung involvieren würde.

Da sowohl ein infiniter Regreß als auch ein logischer Zirkel offensichtlich unakzeptabel zu sein scheint, besteht die Neigung, die dritte Möglichkeit, den Abbruch des Verfahrens, schon deshalb zu akzeptieren, weil ein anderer Ausweg aus dieser Situation für unmöglich gehalten wird. Man pflegt in bezug auf Aussagen, bei denen man bereit ist, das Begründungsverfahren abzubrechen, von Selbstevidenz, Selbstbegründung, Fundierung in unmittelbarer Erkenntnis – in Intuition, Erlebnis oder Erfahrung – zu sprechen oder in anderer Weise zu umschreiben, daß man bereit ist, den Begründungsregreß an einem bestimmten Punkt abzubrechen und das Begründungspostulat für diesen Punkt zu suspendieren, indem man ihn als archimedischen Punkt der Erkenntnis deklariert. Das Verfahren ist ganz analog zur Suspendierung des Kausalprinzips durch Einführung einer causa sui. Nennt man aber eine Überzeugung oder Aussage, die selbst nicht zu begründen ist, aber dabei mitwirken soll, alles andere zu begründen, und die als sicher hingestellt wird, obwohl man eigentlich alles – und also auch sie – grundsätzlich bezweifeln kann, eine Behauptung, deren Wahrheit gewiß und die daher nicht der Begründung bedürftig ist: ein Dogma, dann zeigt sich unsere dritte Möglichkeit als das, was man bei einer Lösung des Begründungsproblems am wenigsten erwarten sollte: als Begrün­dung durch Rekurs auf ein Dogma. Die Suche nach dem archimedischen Punkt der Erkenntnis scheint im Dogma­tismus enden zu müssen. An irgendeiner Stelle nämlich muß das Begründungspostulat der klassischen Methodologie auf jeden Fall suspendiert werden. [15-6]

Corroboration and timeless truth

In the logic of science here outlined it is possible to avoid using the concepts ‘true’ and ‘false’. …

Whilst we assume that the properties of physical objects (of ‘genidentical’ objects in Lewin’s sense) change with the passage of time, we decide to use these logical predicates in such a way that the logical properties of statements become timeless: if a statement is a tautology, then it is a tautology once and for all. This same timelessness we also attach to the concepts ‘true’ and ‘false’, in agreement with common usage. It is not common usage to say of a statement that it was perfectly true yesterday but has become false today. If yesterday we appraised a statement as true which to­day we appraise as false, then we implicitly assert today that ; that the statement was false even yesterday—timelessly false—but that we erroneously ‘took it for true’.

Here one can see very clearly the difference between truth and corroboration. The appraisal of a statement as corrobo­rated or as not corroborated is also a logical appraisal and therefore also timeless; for it asserts that a certain logical relation holds between a theoretical system and some system of accepted basic statements. But we can never simply say of a statement that it is as such, or in itself, ‘corroborated’ (in the way in which we may say that it is ‘true’). We can only say that it is corroborated with respect to some system of basic statements—a system accepted up to a particular point in time. ‘The corroboration which a theory has received up to yesterday’ is logically not identical with ‘the corro­boration which a theory has received up to today’. Thus we must attach a subscript, as it were, to every appraisal of cor­roboration—a subscript characterizing the system of basic statements to which the corroboration relates (for example, by the date of its acceptance).

Corroboration is therefore not a ‘truth value’; that is, it cannot be placed on a par with the concepts ‘true’ and ‘false’ (which are free from temporal subscripts); for to one and the same statement there may be any number of different cor­roboration values, of which indeed all can be ‘correct’ or ‘true’ at the same time. For they are values which are logically derivable from the theory and the various sets of basic statements accepted at various times.

The above remarks may also help to elucidate the contrast between my views and those of the pragmatists who pro­pose to define ‘truth’ in terms of the success of a theory—and thus of its usefulness, or of its confirmation or of its corro­boration. If their intention is merely to assert that a logical appraisal of the success of a theory can be no more than an appraisal of its corroboration, I can agree. But I think that it would be far from ‘useful’ to identify the concept of corrobo­ration with that of truth.* [273-5]

* Thus if we were to define ‘true’ as ‘useful’ (as suggested by some pragmatists), or else as ‘successful’ or ‘confirmed’ or ‘corroborated’, we should only have to introduce a new ‘absolute’ and ‘timeless’ concept to play the role of ‘truth’.

Step-by-step approximations to truth

The degree of corroboration of two statements may not be comparable in all cases, any more than the degree of falsi­fiability: we cannot define a numerically calculable degree of corroboration, but can speak only roughly in terms of positive degree of corroboration, negative degrees of corroboration, and so forth. Yet we can lay down various rules; for instance the rule that we shall not continue to accord a positive degree of corroboration to a theory which has been falsified by an inter-subjectively testable experiment based upon a falsifying hypothesis. (We may, however, under cer­tain circumstances accord a positive degree of corroboration to another theory, even though it follows a kindred line of thought. An example is Einstein’s photon theory, with its kinship to Newton’s corpuscular theory of light.) In general we regard an inter-subjectively testable falsification as final (provided it is well tested): this is the way in which the asymme­try between verification and falsification of theories makes itself felt. Each of these methodological points contributes in its own peculiar way to the historical development of science as a process of step by step approximations. [266-7]

Falsifiability and probability statements

How is it possible that probability statements—which are not falsifiable—can be used as falsifiable statements? (The fact that they can be so used is not in doubt: the physicist knows well enough when to regard a probability assumption as falsified.) This question, we find, has two aspects. On the one hand, we must make the possibility of using probability statements understandable in terms of their logical form. On the other hand, we must analyse the rules governing their use as falsifiable statements.

According to section 66, accepted basic statements may agree more or less well with some proposed probability esti­mate; they may represent better, or less well, a typical segment of a probability sequence. This provides the opportunity for the application of some kind of methodological rule; a rule, for instance, which might demand that the agreement between basic statements and the probability estimate should conform to some minimum standard. Thus the rule might draw some arbitrary line and decree that only reasonably representative segments (or reasonably ‘fair samples’) are ‘permitted’, while atypical or non-representative segments are ‘forbidden’. [197]

Why simplicity is so highly desirable

Above all, our theory explains why simplicity is so highly desirable. To understand this there is no need for us to as­sume a ‘principle of economy of thought’ or anything of the kind. Simple statements, if knowledge is our object, are to be prized more highly than less simple ones because they tell us more; because their empirical content is greater; and because they are better testable. [128]

Reproducibility is testability

There is only one way to make sure of the validity of a chain of logical reasoning. This is to put it in the form in which it is most easily testable: we break it up into many small steps, each easy to check by anybody who has learnt the mathe­matical or logical technique of transforming sentences. If after this anybody still raises doubts then we can only beg him to point out an error in the steps of the proof, or to think the matter over again. In the case of the empirical sciences, the situation is much the same. Any empirical scientific statement can be presented (by describing experimental arrange­ments, etc.) in such a way that anyone who has learned the relevant technique can test it. If, as a result, he rejects the statement, then it will not satisfy us if he tells us all about his feelings of doubt or about his feelings of conviction as to his perceptions. What he must do is to formulate an assertion which contradicts our own, and give us his instructions for testing it. If he fails to do this we can only ask him to take another and perhaps a more careful look at our experiment, and think again.

An assertion which owing to its logical form is not testable can at best operate, within science, as stimulus: it can sug­gest a problem. In the field of logic and mathematics, this may be exemplified by Fermat’s problem, and in the field of natural history, say, by reports about sea-serpents. In such cases science does not say that the reports are unfounded; that Fermat was in error or that all the records of observed sea-serpents are lies. Instead, it suspends judgment. [81]

The most basic requirement for any theoretical system

The requirement of consistency plays a special rôle among the various requirements which a theoretical system, or an axiomatic system, must satisfy. It can be regarded as the first of the requirements to be satisfied by every theoretical system, be it empirical or non-empirical.

In order to show the fundamental importance of this requirement it is not enough to mention the obvious fact that a self-contradictory system must be rejected because it is ‘false’. We frequently work with statements which, although actually false, nevertheless yield results which are adequate for certain purposes. (An example is Nernst’s approximation for the equilibrium equation of gases.) But the importance of the requirement of consistency will be appreciated if one realizes that a self-contradictory system is uninformative. It is so because any conclusion we please can be derived from it. Thus no statement is singled out, either as incompatible or as derivable, since all are derivable. A consistent system, on the other hand, divides the set of all possible statements into two: those which it contradicts and those with which it is com­patible. (Among the latter are the conclusions which can be derived from it.) This is why consistency is the most general requirement for a system, whether empirical or non-empirical, if it is to be of any use at all. [72]

The heart of falsification

We must clearly distinguish between falsifiability and falsification. We have introduced falsifiability solely as a criterion for the empirical character of a system of statements. As to falsification, special rules must be introduced which will de­ter­mine under what conditions a system is to be regarded as falsified.

We say that a theory is falsified only if we have accepted basic statements which contradict it. This condition is neces­sary, but not sufficient; for we have seen that non-reproducible single occurrences are of no significance to science. Thus a few stray basic statements contradicting a theory will hardly induce us to reject it as falsified. We shall take it as falsified only if we discover a reproducible effect which refutes the theory. In other words, we only accept the falsification if a low-level empirical hypothesis which describes such an effect is proposed and corroborated. This kind of hypothe­sis may be called a falsifying hypothesis. The requirement that the falsifying hypothesis must be empirical, and so falsi­fiable, only means that it must stand in a certain logical relationship to possible basic statements; thus this requirement only concerns the logical form of the hypothesis. The rider that the hypothesis should be corroborated refers to tests which it ought to have passed—tests which confront it with accepted basic statements.

Thus the basic statements play two different rôles. On the one hand, we have used the system of all logically possible basic statements in order to obtain with its help the logical characterization for which we were looking—that of the form of empirical statements. On the other hand, the accepted basic statements are the basis for the corroboration of hypo­theses. If accepted basic statements contradict a theory, then we take them as providing sufficient grounds for its falsifi­cation only if they corroborate a falsifying hypothesis at the same time. [66-7]

The basic decision in science

I admit that my criterion of falsifiability does not lead to an unambiguous classification. Indeed, it is impossible to decide, by analysing its logical form, whether a system of statements is a conventional system of irrefutable implicit definitions, or whether it is a system which is empirical in my sense; that is, a refutable system. Yet this only shows that my criterion of demarcation cannot be applied immediately to a system of statements […]. The question whether a given system should as such be regarded as a conventionalist or an empirical one is therefore misconceived. Only with reference to the methods applied to a theoretical system is it at all possible to ask whether we are dealing with a conventionalist or an empirical theory. The only way to avoid conventionalism is by taking a decision: the decision not to apply its methods. We decide that if our system is threatened we will never save it by any kind of conventionalist stratagem. [61]