Tag Archive: logic

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]

Fisher on significance tests

In considering the appropriateness of any proposed experimental design, it is always needful to forecast all possible results of the experiment, and to have decided without ambiguity what interpretation shall be placed upon each one of them. Further, we must know by what argument this interpretation is to be sustained. …

It is open to the experimenter to be more or less exacting in respect of the smallness of the probability he would require before he would be willing to admit that his observations have demonstrated a positive result. It is obvious that an experiment would be useless of which no possible result would satisfy him. Thus, if he wishes to ignore results having probabilities as high as 1 in 20—the probabilities being of course reckoned from the hypothesis that the phenomenon to be demonstrated is in fact absent … . It is usual and convenient for the experimenters to take 5 per cent. as a standard level of significance, in the sense that they are prepared to ignore all results which fail to reach this standard, and, by this means to eliminate from further discussion the greater part of the fluctuations which chance causes have intro­duced into their experimental results. No such selection can eliminate the whole of the possible effects of chance co­incidence, and if we accept this convenient convention, and agree that an event which would occur by chance only once in 70 trials is decidedly “significant”, in the statistical sense, we thereby admit that no isolated experiment, how­ever significant in itself, can suffice for the experimental demonstration of any natural phenomenon; for the “one chance in a million” will undoubtedly occur, with no less and no more than its appropriate frequency, however surprised we may be that it should occur to us. In order to assert that a natural phenomenon is experimentally demonstrable we need, not an isolated record, but a reliable method of procedure. In relation to the test of significance we may say that a pheno­menon is experimentally demonstrable when we know how to conduct an experiment which will rarely fail to give us a statistically significant result. [12-4]

Weak statistical tests

The distinction between the strong and the weak use of significance tests is logical or epistemological; it is not a statistical issue. The weak use of significance tests asks merely whether the observations are attributable to “chance” (i.e., no relation exists) when a weak theory can only predict some sort of relation, but not what or how much. The strong use of significance tests asks whether observations differ significantly from the numerical values that a strong theory predicts, and it leads to the fourth figure of the syllogism—p ⊃ q, ~q , infer ~p—which is formally valid, the logician’s modus tollens (“destroying mode”). Psychologists should work hard to formulate theories that, even if somewhat weak, permit derivation of numerical point values or narrow ranges, yielding the possibility of modus tollens refutations. [422]

Induction, philosophy’s toughest zombie

Science is an exercise in inductive reasoning: we are making observations and trying to infer general rules from them. Induction can never be certain. In contrast, deductive reasoning is easier: you deduce what you would expect to ob­serve if some general rule were true and then compare it with what you actually see. The problem is that, for a scientist, deductive arguments don’t directly answer the question that you want to ask.

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