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probability 205 express all we know about it by means of a formally singular probability statement which looks like an indefinite prediction about the particular event in question.* 2 Thus I do not object to the subjective interpretation of probability statements about single events, i.e. to their interpretation as indefinite predictions—as confessions, so to speak, of our deficient knowledge about the particular event in question (concerning which, indeed, nothing follows from a frequency statement). I do not object, that is to say, so long as we clearly recognize that the objective frequency statements are fundamental, since they alone are empirically testable. I reject, however, any interpretation of these formally singular probability statements—these indefinite predictions—as statements about an objective state of affairs, other than the objective statistical state of affairs. What I have in mind is the view that a statement about the probability 1/6 in dicing is not a mere confession that we know nothing definite (subjective theory), but rather an assertion about the next throw—an assertion that its result is objectively both indeterminate and undetermined— something which as yet hangs in the balance.* 3 I regard all attempts at this kind of objective interpretation (discussed at length by Jeans, among others) as mistaken. Whatever indeterministic airs these interpretations may give themselves, they all involve the metaphysical idea that not only can we deduce and test predictions, but that, in addition, nature is more or less ‘determined’ (or ‘undetermined’); so that the success (or failure) of predictions is to be explained not by the laws from which they are deduced, but over and above this by the fact that * 2 At present I think that the question of the relation between the various interpretations of probability theory can be tackled in a much simpler way—by giving a formal system of axioms or postulates and proving that it is satisfied by the various interpretations. Thus I regard most of the considerations advanced in the rest of this chapter (sections 71 and 72) as being superseded. See appendix *iv, and chapters *ii, *iii, and *v of my Postscript. But I still agree with most of what I have written, provided my ‘reference classes’ are determined by the conditions defining an experiment, so that the ‘frequencies’ may be considered as the result of propensities. * 3 I do not now object to the view that an event may hang in the balance, and I even believe that probability theory can best be interpreted as a theory of the propensities of events to turn out one way or another. (See my Postscript.) But I should still object to the view that probability theory must be so interpreted. That is to say, I regard the propensity interpretation as a conjecture about the structure of the world.
206 some structural components of a theory of experience nature is actually constituted (or not constituted) according to these laws.* 4 72 THE THEORY OF RANGE In section 34 I said that a statement which is falsifiable to a higher degree than another statement can be described as the one which is logically more improbable; and the less falsifiable statement as the one which is logically more probable. The logically less probable statement entails 1 the logically more probable one. Between this concept of logical probability and that of objective or formally singular numerical probability there are affinities. Some of the philosophers of probability (Bolzano, von Kries, Waismann) have tried to base the calculus of probability upon the concept of logical range, and thus upon a concept which (cf. section 37) coincides with that of logical probability; and in doing so, they also tried to work out the affinities between logical and numerical probability. Waismann 2 has proposed to measure the degree of interrelatedness between the logical ranges of various statements (their ratios, as it were) by means of the relative frequencies corresponding to them, and thus to treat the frequencies as determining a system of measurement for ranges. I think it is feasible to erect a theory of probability on this foundation. Indeed we may say that this plan amounts to the same thing as correlating relative frequencies with certain ‘indefinite predictions’ —as we did in the foregoing section, when defining formally singular probability statements. It must be said, however, that this method of defining probability is only practicable when a frequency theory has already been constructed. Otherwise one would have to ask how the frequencies used in defining the system of measurement were defined in their turn. If, however, a frequency theory is at our disposal already, then the introduction of the theory of range becomes really superfluous. But in spite of this objec- * 4 This somewhat disparaging characterization fits perfectly my own views which I now submit to discussion in the ‘Metaphysical Epilogue’ of my Postscript, under the name of ‘the propensity interpretation of probability’. 1 Usually (cf. section 35). 2 Waismann, Logische Analyse des Wahrscheinlichkeitsbegriffes, Erkenntnis 1, 1930, p. 128 f.
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The Logic of Scientific Discovery
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Logik der Forschung first published
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CONTENTS Translators’ Note xii Pr
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contents ix 7 37 Logical Ranges. No
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iii Derivation of the First Form of
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translators’ note xiii In these n
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PREFACE TO THE FIRST EDITION, 1934
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There is nothing more necessary to
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preface, 1959 xix Language analysts
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xxii preface, 1959 perception or kn
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preface, 1959 xxiii essence of phil
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preface, 1959 xxv elaborate and fam
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ACKNOWLEDGMENTS, 1960 and 1968 I wi
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1 A SURVEY OF SOME FUNDAMENTAL PROB
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a survey of some fundamental proble
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trivial; for metaphysics has usuall
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non-contradictory, a possible world
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2 ON THE PROBLEM OF A THEORY OF SCI
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on the problem of a theory of scien
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Part II Some Structural Components
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38 some structural components of a
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5 THE PROBLEM OF THE EMPIRICAL BASI
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284 appendices it is connected with
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APPENDIX ii The General Calculus of
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288 appendices —we obtain from (2
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APPENDIX iii Derivation of the Firs
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292 appendices (6,1) αF″(σ´ m.
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294 appendices after effect) in the
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296 appendices (b) An analogous met
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298 appendices position), then we a
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300 appendices incoherent photons.
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302 appendices interposition of a f
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304 appendices wave-packets (obtain
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306 appendices slit. This angle φ
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308 appendices latter, if we use hi
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310 new appendices The first two of
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APPENDIX *i Two Notes on Induction
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314 new appendices statements’;*
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316 new appendices be the source fr
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318 new appendices with the help of
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320 new appendices logical interpre
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322 new appendices It is desirable
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324 new appendices operations—con
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326 new appendices (3) It will be a
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328 new appendices (7) The derivati
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330 new appendices There are three
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332 new appendices that is to say,
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334 new appendices invalid, and sho
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336 new appendices simplified syste
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338 new appendices The following co
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340 new appendices able to show tha
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342 new appendices obtain p(a, c) =
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344 new appendices The second examp
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346 new appendices first consistenc
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348 new appendices S′ ={0, 1, 2,
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350 new appendices B1 is violated b
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352 new appendices always fill the
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354 new appendices It should be men
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APPENDIX *v Derivations in the Form
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358 new appendices (26) (Eb) (Ea) p
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360 new appendices (Applying B2 twi
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362 new appendices form: it is unco
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364 new appendices In order to prov
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366 new appendices which is neverth
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368 new appendices the other axioms
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370 new appendices between rain and
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372 new appendices in the widest po
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APPENDIX *vii Zero Probability and
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376 new appendices favourable possi
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378 new appendices (4) p(a) = lim p
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380 new appendices The same point m
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382 new appendices inferences from
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384 new appendices the required a a
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386 new appendices ‘universe’
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388 new appendices All this does no
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390 new appendices The fine-structu
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APPENDIX *viii Content, Simplicity,
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394 new appendices each entry repre
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396 new appendices But formula (*)
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398 new appendices (−) d(a 1)p(a
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400 new appendices thus becomes som
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APPENDIX *ix Corroboration, the Wei
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404 new appendices I will now brief
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406 new appendices of the brevity o
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408 new appendices who identify, ex
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410 new appendices Kemeny.) Therefo
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412 new appendices DEGREE OF CONFIR
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414 new appendices (4.5) C(x 1x 2 o
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416 new appendices strongly upon P(
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418 new appendices (vii) If C(x) =
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420 new appendices to be slightly a
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422 new appendices length, to take
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424 new appendices A THIRD NOTE ON
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426 new appendices (3) P(a) = P(a,
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428 new appendices behaviour of E a
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430 new appendices either if δ is
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432 new appendices statistical inte
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434 new appendices them, there is a
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436 new appendices statements e, f,
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438 new appendices simply deceive o
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APPENDIX *x Universals, Disposition
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442 new appendices One can easily s
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444 new appendices statement such a
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446 new appendices more abstract. T
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448 new appendices natural laws see
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450 new appendices indirect proof.
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452 new appendices physical theory
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454 new appendices is deducible fro
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456 new appendices A little reflect
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458 new appendices differ (if at al
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460 new appendices the world. And a
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462 new appendices the phenomenalis
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APPENDIX *xi On the Use and Misuse
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466 new appendices a gravitational
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468 new appendices with, its co-ord
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470 new appendices based upon his f
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472 new appendices moving freely be
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474 new appendices of insuperable b
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476 new appendices Einstein.) The m
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478 new appendices photographic pla
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480 new appendices My impression is
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482 new appendices letter were to b
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484 new appendices statistical desc
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486 new appendices
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488 new appendices
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490 name index Carnap, R. 7n, 13n,
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492 name index Mises, R. 133, 135n,
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SUBJECT INDEX Italicized page numbe
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496 subject index Binomial formula
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498 subject index Cosmology, its pr
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500 subject index Empirical basis s
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502 subject index and ad hoc hypoth
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504 subject index 150&nt, see also
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506 subject index Modus Ponens 71n,
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508 subject index 319-20; indutive
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510 subject index 142&n, 154n, 174&
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512 subject index experiment 474-5,
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