popper-logic-scientific-discovery

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(An example of an interpretation of ‘pa(x 1)’ in the sense of (3), i.e. of the <strong>logic</strong>al interpretation, is the concept ‘<strong>logic</strong>al probability’ as used by me in a previous publication. 1 ) Now it is possible to proceed with the whole construction from the other end: instead of introducing ‘p(x 1, x 2)’ as primitive concept (primitive functor) of an axiom system s 1 and defining ‘pa(x 1)’ explicitly, we can construct another axiom system s 2 in which ‘pa(x 1)’ appears as (undefined) primitive variable, and we can then proceed to define ‘pa(x 1,x 2)’ explicitly, with the help of ‘pa(x 1)’; as follows. Df 2 p(x 1, x 2) = pa(x 1x 2) pa(x 2) appendix *ii 323 The formulae adopted in s 1 as axioms (and Df 1) now become theorems within s 2, i.e. they can be deduced with the help of the new system of axioms s 2. It can be shown that the two methods described—the choice of s 1 and Df 1, or s 2 and Df 2 respectively—are not equally convenient from the viewpoint of formal axiomatics. The second method is superior to the first in certain respects, the most important of which is that it is possible to formulate in s 2 an axiom of uniqueness which is much stronger than the corresponding axiom of s 1 (if the generality of s 1 is not restricted). This is due to the fact that if pa(x 2) = 0, the value of p(x 1, x 2) becomes indeterminate.* 2 A system of independent axioms, s 2, as described above, is here subjoined. (It is easy to construct a system s 1 with the help of it.) Combined with definition Df 2 it is sufficient for the deduction of the mathematical theory of probability. The axioms can be divided into two groups. Group A is formed by the axioms for the junctional 6 See Popper, l.c., 71, 151. * 2 The absolute system (s 2) has an advantage over the relative system (s 1) only as long as the relative probability p(x, y) is considered as indeterminate if pa(y) = 0. I have since developed a system (see appendix *iv) in which relative probabilitiés are determinate even in case pa(y) = 0. This is why I now consider the relative system superior to the absolute system. (I may also say that I now consider the term ‘axiom of uniqueness’ as badly chosen. What I intended to allude to was, I suppose, something like postulate 2 or axiom A2 of the system of appendix *iv.)
324 new appendices operations—conjunction and negation—of the argument, and is practically an adaptation of the system of postulates for the so-called ‘Algebra of Logic’. 7 Group B gives the axioms peculiar to the measurement of probability. The axioms are: [Here followed—with several misprints—the complicated axiom system which I have since replaced by the simpler one given above.] Christchurch, N.Z., November 20th, 1937. 7 See Huntington, Trans. Amer. Mathem. Soc., 5, 292 (1904), and Whitehead-Russell, Principia Mathematica, I, where the five propositions 22.51, 22.52, 22.68, 24.26, 24.1 correspond to the five axioms of Group A, as given here.
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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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Page 301 and 302: 272 some structural components of a
Page 313 and 314: 284 appendices it is connected with
Page 315 and 316: APPENDIX ii The General Calculus of
Page 317 and 318: 288 appendices —we obtain from (2
Page 319 and 320: APPENDIX iii Derivation of the Firs
Page 321 and 322: 292 appendices (6,1) αF″(σ´ m.
Page 323 and 324: 294 appendices after effect) in the
Page 325 and 326: 296 appendices (b) An analogous met
Page 327 and 328: 298 appendices position), then we a
Page 329 and 330: 300 appendices incoherent photons.
Page 331 and 332: 302 appendices interposition of a f
Page 333 and 334: 304 appendices wave-packets (obtain
Page 335 and 336: 306 appendices slit. This angle φ
Page 337 and 338: 308 appendices latter, if we use hi
Page 339 and 340: 310 new appendices The first two of
Page 341 and 342: APPENDIX *i Two Notes on Induction
Page 343 and 344: 314 new appendices statements’;*
Page 345 and 346: 316 new appendices be the source fr
Page 347 and 348: 318 new appendices with the help of
Page 349 and 350: 320 new appendices logical interpre
Page 351: 322 new appendices It is desirable
Page 355 and 356: 326 new appendices (3) It will be a
Page 357 and 358: 328 new appendices (7) The derivati
Page 359 and 360: 330 new appendices There are three
Page 361 and 362: 332 new appendices that is to say,
Page 363 and 364: 334 new appendices invalid, and sho
Page 365 and 366: 336 new appendices simplified syste
Page 367 and 368: 338 new appendices The following co
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Page 371 and 372: 342 new appendices obtain p(a, c) =
Page 373 and 374: 344 new appendices The second examp
Page 375 and 376: 346 new appendices first consistenc
Page 377 and 378: 348 new appendices S′ ={0, 1, 2,
Page 379 and 380: 350 new appendices B1 is violated b
Page 381 and 382: 352 new appendices always fill the
Page 383 and 384: 354 new appendices It should be men
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
Page 395 and 396: 366 new appendices which is neverth
Page 397 and 398: 368 new appendices the other axioms
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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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