Tag Archive: logic

Scientific methodology (German edition)

3. Die deduktive Überprüfung der Theorien. Die Methode der kritischen Nachprüfung, der Auslese der Theorien, ist nach unserer Auffassung immer die folgende: Aus der vorläufig unbegründeten Antizipation, dem Einfall, der Hypothese, dem theoretischen System, werden auf logisch-deduktivem Weg Folgerungen abgeleitet; diese werden untereinander und mit anderen Sätzen verglichen, indem man feststellt, welche logischen Beziehungen (z. B. Äquivalenz, Ableitbarkeit, Vereinbarkeit, Widerspruch) zwischen ihnen bestehen.

Dabei lassen sich insbesondere vier Richtungen unterscheiden, nach denen die Prüfung durchgeführt wird: der logische Vergleich der Folgerungen untereinander, durch den das System auf seine innere Widerspruchslosigkeit hin zu unter­suchen ist; eine Untersuchung der logischen Form der Theorie mit dem Ziel, festzustellen, ob es den Charakter einer empirisch-wissenschaftlichen Theorie hat, also z. B. nicht tautologisch ist; der Vergleich mit anderen Theorien, um unter anderem festzustellen, ob die zu prüfende Theorie, falls sie sich in den verschiedenen Prüfungen bewähren sollte, als wissenschaftlicher Fortschritt zu bewerten wäre; schließlich die Prüfung durch „empirische Anwendung“ der abgeleiteten Folgerungen.

Diese letzte Prüfung soll feststellen, ob sich das Neue, das die Theorie behauptet, auch praktisch bewährt, etwa in wis­senschaftlichen Experimenten oder in der technisch-praktischen Anwendung. Auch hier ist das Prüfungsverfahren ein deduktives: Aus dem System werden (unter Verwendung bereits anerkannter Sätze) empirisch moglichst leicht nach­prüf­bare bzw. anwendbare singuläre Folgerungen („Prognosen“) deduziert und aus diesen insbesondere jene ausgewählt, die aus bekannten Systemen nicht ableitbar sind, bzw. mit ihnen in Widerspruch stehen. Über diese – und andere – Folgerungen wird nun im Zusammenhang mit der praktischen Anwendung, den Experimenten usw. entschieden. Fällt die Entscheidung positiv aus, werden die singulären Folgerungen anerkannt, verifiziert, so hat das System die Prüfung vorläufig bestanden; wir haben keinen Anlaß, es zu verwerfen. Fällt eine Entscheidung negativ aus, werden Folgerungen falsifiziert, so trifft ihre Falsifikation auch das System, aus dem sie deduziert wurden.

Die positive Entscheidung kann das System immer nur vorläufig stützen; es kann durch spätere negative Entscheidungen immer wieder umgestoßen werden. Solang ein System eingehenden und strengen deduktiven Nachprüfungen standhält und durch die fortschreitende Entwicklung der Wissenschaft nicht überholt wird, sagen wir, daß es sich bewährt.

Induktionslogische Elemente treten in dem hier skizzierten Verfahren nicht auf; niemals schließen wir von der Geltung der singulären Satze auf die der Theorien. Auch durch ihre verifizierten Folgerungen können Theorien niemals als „wahr“ oder auch nur als „wahrscheinlich“ erwiesen werden.

Infinite learning

Thus every statement (or ‘basic statement’) remains essentially conjectural; but it is a conjecture which can be easily tested. These tests, in their turn, involve new conjectural and testable statements, and so on, ad infinitum; and should we try to establish anything with our tests, we should be involved in an infinite regress. But as I explained in my Logic of Scientific Discovery (especially section 29), we do not establish anything by this procedure: we do not wish to ‘justify’ the ‘acceptance’ of anything, we only test our theories critically, in order to see whether or not we can bring a case against them. [521]

Democracy vs the rule of the majority

[T]here is also a kind of logical argument which can be used to show the inconsistency of any of the particular forms of the theory of sovereignty; more precisely, the logical argument can be given different but analogous forms to combat the theory that the wisest should rule, or else the theories that the best, or the law, or the majority, etc., should rule. One particular form of this logical argument is directed against a too naive version of liberalism, of democracy, and of the principle that the majority should rule; and it is somewhat similar to the well-known ‘paradox of freedom’ which has been used first, and with success, by Plato. In his criticism of democracy, and in his story of the rise of the tyrant, Plato raises implicitly the following question: What if it is the will of the people that they should not rule, but a tyrant instead? The free man, Plato suggests, may exercise his absolute freedom, first by defying the laws and ultimately by defying freedom itself and by clamouring for a tyrant. This is not just a far-fetched possibility; it has happened a number of times; and every time it has happened, it has put in a hopeless intellectual position all those democrats who adopt, as the ultimate basis of their political creed, the principle of the majority rule or a similar form of the principle of sovereignty. On the one hand, the principle they have adopted demands from them that they should oppose any but the majority rule, and therefore the new tyranny; on the other hand, the same principle demands from them that they should accept any decision reached by the majority, and thus the rule of the new tyrant. The inconsistency of their theory must, of course, paralyse their actions. Those of us democrats who demand the institutional control of the rulers by the ruled, and especially the right of dismissing the government by a majority vote, must therefore base these demands upon better grounds than a self-contradictory theory of sovereignty. [ch. 7, 134-5]

Fisher on the logic of null hypotheses

In relation to any experiment we may speak of this hypothesis as the “null hypothesis,” and it should be noted that the null hypothesis is never proved or established, but is possibly disproved, in the course of experimentation. Every ex­periment may be said to exist only in order to give the facts a chance of disproving the null hypothesis.

It might be argued that if an experiment can disprove the hypothesis that the subject possesses no sensory discrimi­nation between two different sorts of object, it must therefore be able to prove the opposite hypothesis, that she can make some such discrimination. But this last hypothesis, however reasonable or true it may be, is ineligible as a null hypothesis to be tested by experiment, because it is inexact. If it were asserted that the subject would never be wrong in her judgements we should again have an exact hypothesis, and it is easy to see that this hypothesis could be dis­proved by a single failure, but could never be proved by any finite amount of experimentation. [16]

Nothing can be proved

The critical attitude, the tradition of free discussion of theories with the aim of discovering their weak spots so that they may be improved upon, is the attitude of reasonableness, of rationality. It makes far-reaching use of both verbal argu­ment and observation—of observation in the interest of argument, however. The Greeks’ discovery of the critical method gave rise at first to the mistaken hope that it would lead to the solution of all the great old problems; that it would establish certainty; that it would help to prove our theories, to justify them. But this hope was a residue of the dogmatic way of thinking; in fact nothing can be justified or proved (outside of mathematics and logic). The demand for rational proofs in science indicates a failure to keep distinct the broad realm of rationality and the narrow realm of rational certainty: it is an untenable, an unreasonable demand. [67]

What argument can do

No argument can force us to accept the truth of any belief. But a valid deductive argument can force us to choose be­tween the truth of its conclusion on the one hand and the falsity of its premises on the other. [10]

Nagel’s obviously objective values

Harris has identified a real problem, rooted in the idea that facts are objective and values are subjective.

Harris rejects this facile opposition in the only way it can be rejected—by pointing to evaluative truths so obvious that they need no defense. For example, a world in which everyone was maximally miserable would be worse than a world in which everyone was happy, and it would be wrong to try to move us toward the first world and away from the second. This is not true by definition, but it is obvious, just as it is obvious that elephants are larger than mice. If someone denied the truth of either of those propositions, we would have no reason to take him seriously.

These cases show that the idea of truth applies to values as much as it does to facts—but as with facts, the idea is not limited to obvious cases. There are many questions of value whose answer is not obvious, and about which people disagree, but that does not mean that they do not have a correct answer. As Harris points out, exactly the same can be said of historical and scientific questions: one should not confuse truth with what is known, or agreed on by everyone.

Decisions cannot be derived from facts

It is important for the understanding of this attitude to realize that these decisions can never be derived from facts (or from statements of facts), although they pertain to facts. The decision, for instance, to oppose slavery does not depend upon the fact that all men are born free and equal, and that no man is born in chains. For even if all were born free, some men might perhaps try to put others in chains, and they may even believe that they ought to put them in chains. And conversely, even if men were born in chains, many of us might demand the removal of these chains. Or to put this matter more precisely, if we consider a fact as alterable—such as the fact that many people are suffering from dis­eases—then we can always adopt a number of different attitudes towards this fact: more especially, we can decide to make an attempt to alter it; or we can decide to resist any such attempt; or we can decide not to take action at all.

All moral decisions pertain in this way to some fact or other, especially to some fact of social life, and all (alterable) facts of social life can give rise to many different decisions. Which shows that the decisions can never be derivable from these facts, or from a description of these facts. [ch. 5, 67]

The relativity of proof

Every proof must proceed from premises; the proof as such, that is to say, the derivation from the premises, can there­fore never finally settle the truth of any conclusion, but only show that the conclusion must be true provided the prem­ises are true. [ch. 11, 260]

Fisher on Bayesianism

[A]dvocates of inverse probability seem forced to regard mathematical probability, not as an objective quantity mea­sured by observable frequencies, but as measuring merely psychological tendencies, theorems respecting which are useless for scientific purposes. [6-7]

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