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]
Tag: falsifiability
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-
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]
On auxiliary hypotheses
As regards auxiliary hypotheses we propose to lay down the rule that only those are acceptable whose introduction does not diminish the degree of falsifiability or testability of the system in question, but, on the contrary, increases it. […] If the degree of falsifiability is increased, then introducing the hypothesis has actually strengthened the theory: the system now rules out more than it did previously: it prohibits more. We can also put it like this. The introduction of an auxiliary hypothesis should always be regarded as an attempt to construct a new system; and this new system should then always be judged on the issue of whether it would, if adopted, constitute a real advance in our knowledge of the world. [62]
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]
The testability of facts
Higher level empirical statements have always the character of hypotheses relative to the lower level statements deducible from them: they can be falsified by the falsification of these less universal statements. But in any hypothetical deductive system, these less universal statements are themselves still strictly universal statements, in the sense here understood. Thus they too must have the character of hypotheses—a fact which has often been overlooked in the case of lower-level universal statements.
I shall say even of some singular statements that they are hypothetical, seeing that conclusions may be derived from them (with the help of a theoretical system) such that the falsification of these conclusions may falsify the singular statements in question. [55]
Assumptional contraband
Scientific theories are perpetually changing. This is not due to mere chance but might well be expected, according to our characterization of empirical science.
Perhaps this is why, as a rule, only branches of science—and these only temporarily—ever acquire the form of an elaborate and logically well-constructed system of theories. In spite of this, a tentative system can usually be quite well surveyed as a whole, with all its important consequences. This is very necessary; for a severe test of a system presupposes that it is at the time sufficiently definite and final in form to make it impossible for new assumptions to be smuggled in. In other words, the system must be formulated sufficiently clearly and definitely to make every new assumption easily recognizable for what it is: a modification and therefore a revision of the system. [50]
Falsifying universal statements
Statements in which only universal names and no individual names occur will here be called ‘strict’ or ‘pure’. Most important among them are the strictly universal statements which I have already discussed. In addition to these, I am especially interested in statements of the form ‘there are black ravens’, which may be taken to mean the same as ‘there exists at least one black raven’. Such statements will be called strictly or purely existential statements (or ‘there-is’ statements).
The negation of a strictly universal statement is always equivalent to a strictly existential statement and vice versa. For example, ‘not all ravens are black’ says the same thing as ‘there exists a raven which is not black’, or ‘there are non-
The theories of natural science, and especially what we call natural laws, have the logical form of strictly universal statements; thus they can be expressed in the form of negations of strictly existential statements or, as we may say, in the form of non-existence statements (or ‘there-is-not’ statements). For example, the law of the conservation of energy can be expressed in the form: ‘There is no perpetual motion machine’, or the hypothesis of the electrical elementary charge in the form: ‘There is no electrical charge other than a multiple of the electrical elementary charge’.
In this formulation we see that natural laws might be compared to ‘proscriptions’ or ‘prohibitions’. They do not assert that something exists or is the case; they deny it. They insist on the non-existence of certain things or states of affairs, proscribing or prohibiting, as it were, these things or states of affairs: they rule them out. And it is precisely because they do this that they are falsifiable. If we accept as true one singular statement which, as it were, infringes the prohibition by asserting the existence of a thing (or the occurrence of an event) ruled out by the law, then the law is refuted. [47-8]
Science’s methodological rules
Two simple examples of methodological rules may be given. They will suffice to show that it would be hardly suitable to place an inquiry into method on the same level as a purely logical inquiry.
(1) The game of science is, in principle, without end. He who decides one day that scientific statements do not call for any further test, and that they can be regarded as finally verified, retires from the game.
(2) Once a hypothesis has been proposed and tested, and has proved its mettle, it may not be allowed to drop out without ‘good reason’. A ‘good reason’ may be, for instance: replacement of the hypothesis by another which is better testable; or the falsification of one of the consequences of the hypothesis. (The concept ‘better testable’ will later be analysed more fully.) …
In establishing these rules we may proceed systematically. First a supreme rule is laid down which serves as a kind of norm for deciding upon the remaining rules, and which is thus a rule of a higher type. It is the rule which says that the other rules of scientific procedure must be designed in such a way that they do not protect any statement in science against falsification. [32-3]
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