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a planet rather than to a point of the plane, and that the possible positions do not overlap. 3 Somewhat differently from the example of the preceding appendix, we now replace the various curves which are the usual geometrical representations of our theories by ‘quasi curves’ (of a width approximately equal to ε); that is to say, by sets, or chains, of squares. As a result of all this, the number of the possible theories becomes finite. We now consider the representation of a theory with d parameters which in the continuous case was represented by a d-dimensional continuum whose points (d-tuples) each represented a curve. We find that we can still use a similar representation, except that our d-dimensional continuum will be replaced by a d-dimensional arrangement of ddimensional ‘cubes’ (with the side ε). Each chain of these cubes will now represent one ‘quasi curve’, and thus one of the possibilities favourable to the theory; and the d-dimensional arrangement will represent the set of all ‘quasi-curves’ compatible with, or favourable to, the theory. But we can now say that the theory with fewer parameters—that is to say, the set of quasi curves which is represented by an arrangement of fewer dimensions—will not only have fewer dimensions, but will also contain a smaller number of ‘cubes’; that is, of favourable possibilities. Thus we are justified in applying the results of the preceding section: if a 1 has fewer parameters than a 2, we can assert that, in a sufficiently large but finite universe, we shall have and therefore (*) p(a 1)
396 new appendices But formula (*) remains valid when we assume that ε tends to zero, which in the limit amounts to replacing the finite universe by an infinite one. We arrive, therefore, at the following theorem. (1) If the number of parameters of a 1 is smaller than the number of parameters of a 2, then the assumption p(a 1)>p(a 2) contradicts the laws of the calculus of probability. Writing ‘d F(a)’, or more simply ‘d(a)’, for the dimension of the theory a (with respect to the field of application F) we can formulate our theorem as follows. (1) If d(a 1)p(a 2)’ is incompatible with ‘d(a 1)
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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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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) =
Page 373 and 374: 344 new appendices The second examp
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Page 379 and 380: 350 new appendices B1 is violated b
Page 381 and 382: 352 new appendices always fill the
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Page 385 and 386: APPENDIX *v Derivations in the Form
Page 387 and 388: 358 new appendices (26) (Eb) (Ea) p
Page 389 and 390: 360 new appendices (Applying B2 twi
Page 391 and 392: 362 new appendices form: it is unco
Page 393 and 394: 364 new appendices In order to prov
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Page 397 and 398: 368 new appendices the other axioms
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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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Page 417 and 418: 388 new appendices All this does no
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Page 421 and 422: APPENDIX *viii Content, Simplicity,
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Page 435 and 436: 406 new appendices of the brevity o
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Page 439 and 440: 410 new appendices Kemeny.) Therefo
Page 441 and 442: 412 new appendices DEGREE OF CONFIR
Page 443 and 444: 414 new appendices (4.5) C(x 1x 2 o
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Page 447 and 448: 418 new appendices (vii) If C(x) =
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Page 453 and 454: 424 new appendices A THIRD NOTE ON
Page 455 and 456: 426 new appendices (3) P(a) = P(a,
Page 457 and 458: 428 new appendices behaviour of E a
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Page 461 and 462: 432 new appendices statistical inte
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Page 465 and 466: 436 new appendices statements e, f,
Page 467 and 468: 438 new appendices simply deceive o
Page 469 and 470: APPENDIX *x Universals, Disposition
Page 471 and 472: 442 new appendices One can easily s
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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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