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whether a necessary statement is necessarily necessary finds a natural place in probability theory: it is closely connected with the relation between primary and secondary probability statements which plays an important part in probability theory (as shown in appendix *ix, point *13 of the Third Note). Roughly, writing ‘� x’ for ‘x is necessary (or demonstrable)’ and ‘h’ for ‘p(a, ā) = 1’, we may show that and therefore we find that � a ↔ � h, � a → � ‘p(h, h¯) = 1’, which may be taken to mean that � a entails that a is necessarily necessary; and since this means something like � a → � ‘p(‘p(a, ā) = 1’, ‘p(a, ā) = 1’) = 1’, we obtain (secondary) probability statements about (primary) probability statements. But there are of course other possible ways of interpreting the relation between a primary and a secondary probability statement. (Some interpretations would prevent us from treating them as belonging to the same linguistic level, or even to the same language.) Addendum, 1964 I have found since that the following system of three axioms, A, BD, and CD, is equivalent to the six axioms on pp. 337 and 356–7. A (Ea) (Eb)p(a, a) ≠ p(a, b) BD ((d)p(ab, d) = p(c, d)) ↔ (e) (f) (p(a, b) � p(c, b) & p(a, e) � � p(c, e) � p(b, c) & ((p(b, e) � p(f, e) & p(b, f) � p(f, f) � � p(e, f)) → p(a, f)p(b, e) = p(c, e))) CD p(ā, b) = p(b, b) − p(a, b) ↔ (Ec)p(b, b) ≠ p(c, b) appendix *v 367 I also have found since an example not satisfying A2 but satisfying all
368 new appendices the other axioms and Postulate AP (see note 10 on p. 345).—The example on p. 348 can be modified (putting p(2) = 1 2, p(a, b) = 1 whenever p(b) = 0, p(a, b) = p(ab)/p(b) whenever p(b) ≠ 0) so as to obtain a Boolean Algebra proving the independence of C. See also my Conjectures and Refutations, 1963, pp. 388ff.; the third and later German editions of my Logik der Forschung; and Synthese 15, 1963, pp. 107–186, and 21, 1970, p. 107.
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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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10 CORROBORATION, OR HOW A THEORY S
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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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Page 349 and 350: 320 new appendices logical interpre
Page 351 and 352: 322 new appendices It is desirable
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Page 355 and 356: 326 new appendices (3) It will be a
Page 357 and 358: 328 new appendices (7) The derivati
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Page 385 and 386: APPENDIX *v Derivations in the Form
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Page 403 and 404: APPENDIX *vii Zero Probability and
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Page 407 and 408: 378 new appendices (4) p(a) = lim 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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