601:
438:
This is classically provable, as follows: Either an index for a proof of the
Goldbach conjecture exists, or no such index exists. On the one hand, if one does exist, then whatever that index is also functions as a valid
284:
has a proof or it does not. If it does not have a proof, then to assume it has a proof is absurd and anything follows - in particular, it follows that it has a proof. Hence, there is some natural number index
710:
443:
in the above proposition. On the other hand, if no such index exists, then for such an index to also exist is contradictory, and then by explosion anything follows - and in particular it follows that
504:
660:
209:
becomes trivial: "If θ is satisfiable then θ is satisfiable." For illustration purposes, let it be granted that θ is a decidable predicate in arithmetic, meaning for any given number
749:
624:
403:
As noted, the independence of premise principle for fixed φ and any θ follows both from a proof of φ as well as from a rejection of it. Hence, assuming the
362:
An implication is strengthened when the antecedent can be weakened. Of interest here are premises in the form of a negated statements, φ := ¬η.
189:, now in contrast to the two points above, consider the case in which it is not known how to prove or reject φ. A core case is when φ is the formula
466:. Classically, it suffices to draw the same conclusion of interest when starting from two hypotheticals about φ. In the latter framework,
221:
is the index of a formal proof of some mathematical conjecture whose provability is not known. Certainly here, one way to establish
82:
is assumed to be inhabited. That is, part of the theory is at least some term. For the discussion we distinguish one such term as
665:
395:
For existential-quantifier-free φ, theories over intuitionistic logic tend to be well behaved in regard to rules of this nature.
66:, where the principle is not generally valid. Its crucial equivalent special case is discussed below. The principle is valid in
712:. The schema implies the strictly weaker excluded middle for negated propositions (WLEM) through the intuitionistic form of
596:{\displaystyle {\big (}\eta \to (\alpha \lor \beta ){\big )}\to {\big (}(\eta \to \alpha )\lor (\eta \to \beta ){\big )}}
447:
is an index of a proof of the
Goldbach conjecture. In both cases, some index exists that validates the proposition.
300:
for intuitionistic proofs, which should be compared against the classical proof calculus. BHK says that a proof of
262:
719:
633:
776:
Nemoto, Takako; Rathjen, Michael (2019). "The independence of premise rule in intuitionistic set theories".
725:
404:
316:. Here proofs themselves can act as input to functions and, when possible, may be used to construct an
819:
24:
389:
713:
139:
473:
is asserted to exist either which way, and the logic does not demand for it to be explicated.
797:
752:
609:
63:
756:
8:
281:
79:
176:
in particular validates the conclusion of the principle. So in both of these two cases,
777:
374:
297:
217:) can easily be inspected for its truth value. More specifically, θ shall express that
168:
bound in the premise is reused in the conclusion, and it is generally not the a priori
662:, this reduces to the shorter but indeed equivalent so called Dirk Gently’s principle
430:, such that if an index of a proof of the Goldbach conjecture exists, then the number
498:
have analogs in propositional logic. In the intuitionistic calculus, the finite form
339:
has that value. In the proof calculus - like in the weak counterexample - a suitable
335:, together with a function that converts a proof of φ into a proof of θ in which
67:
805:. in S. Buss ed., The Handbook of Proof Theory, North-Holland. pp. 337–405.
16:
Mathematical principle largely used in proof theory and constructive
Mathematics
354:
does not suffice for a generic proof of existence as granted by the principle.
86:. In the theory of the natural numbers, this role may be played by the number
813:
56:
20:
454:
such that one can demonstrate (then aided by φ assumed valid and so also ∃
346:
Indeed, using violating models, it has been established that the premise
722:, obtained by adopting this schema for negated propositions, i.e. with
253:
exactly encodes the proof of a conjecture not yet proven or rejected.
273:
cannot be provable by intuitionistic means: Being able to inspect an
308:
comprises a function that takes a proof of φ and returns a proof of
782:
289:
such that if one assumes the
Goldbach conjecture has a proof, that
280:
For example, consider the following classical argument: Either the
795:
606:
may be understood as expressing that information in the premise
233:
for which it can be shown (then aided by the assumption that
705:{\displaystyle (\alpha \to \beta )\lor (\beta \to \alpha )}
422:
always holds. More concretely, consider the proposition:
261:
The arithmetical example above provides what is called a
62:
The main application of the principle is in the study of
343:
can only be given using more input tied to amenable φ.
388:
It is not known whether this also applies to familiar
249:
is not possible (not yet and possibly never), as such
116:, if φ is established to be true, then if one assumes
728:
668:
636:
612:
507:
172:
that validates it. In the second scenario, the value
434:
is the index of a proof of the
Goldbach conjecture."
630:proposition in a pair of conjuncts it implies. For
407:disjunction axiomatically, the principle is valid.
365:It has been meta-theoretically established that if
245:genuinely satisfies θ. However, explicating a such
743:
704:
654:
618:
595:
59:of φ, while θ can be a predicate depending on it.
811:
799:Gödel's functional ("Dialectica") interpretation
138:, if φ is established to be false, then, by the
277:validating φ → θ would resolve the conjecture.
775:
588:
548:
538:
510:
796:Jeremy Avigad and Solomon Feferman (1999).
296:The issue can also be approached using the
256:
781:
655:{\displaystyle \eta =\alpha \lor \beta }
450:Constructively, one needs to provide an
157:(and indeed any predicate of this form.)
35:θ are sentences in a formal theory and
481:
229:would be to provide a particular index
812:
476:
106:is a predicate that can depend on in.
398:
109:The following is easily established:
98:denote propositions not depending on
13:
744:{\displaystyle \eta =\neg \delta }
735:
14:
831:
755:. The corresponding rule is an
426:"There exists a natural number
201:, in which case the antecedent
769:
699:
693:
687:
681:
675:
669:
583:
577:
571:
565:
559:
553:
543:
533:
521:
518:
142:, any proposition of the form
1:
762:
626:is not required to establish
293:is an index of such a proof.
73:
124:to be provable, there is an
29:independence of premise (IP)
7:
10:
836:
405:law of the excluded middle
183:validates the conclusion.
462:) that θ holds for that
357:
328:must then demonstrate a
25:constructive mathematics
257:In intuitionistic logic
161:In the first scenario,
745:
714:consequentia mirabilis
706:
656:
620:
597:
265:. The existence claim
31:states that if φ and ∃
746:
707:
657:
621:
619:{\displaystyle \eta }
598:
385:has a proof as well.
753:disjunction property
726:
720:Kreisel–Putnam logic
666:
634:
610:
505:
482:Kreisel–Putnam logic
64:intuitionistic logic
486:IP and the shorter
477:Propositional logic
282:Goldbach conjecture
263:weak counterexample
153:formally satisfies
80:domain of discourse
27:, the principle of
741:
702:
652:
616:
593:
410:For example, here
399:In classical logic
298:BHK interpretation
241:satisfies θ) that
213:the proposition θ(
78:As is common, the
51:is provable. Here
43:is provable, then
827:
806:
804:
788:
787:
785:
773:
751:, still has the
750:
748:
747:
742:
711:
709:
708:
703:
661:
659:
658:
653:
625:
623:
622:
617:
602:
600:
599:
594:
592:
591:
552:
551:
542:
541:
514:
513:
497:
421:
384:
372:
353:
327:
315:
307:
272:
228:
208:
200:
156:
145:
131:
123:
50:
42:
835:
834:
830:
829:
828:
826:
825:
824:
820:Predicate logic
810:
809:
802:
792:
791:
774:
770:
765:
757:admissible rule
727:
724:
723:
667:
664:
663:
635:
632:
631:
611:
608:
607:
587:
586:
547:
546:
537:
536:
509:
508:
506:
503:
502:
487:
484:
479:
411:
401:
378:
373:has a proof in
366:
360:
347:
321:
309:
301:
266:
259:
222:
202:
190:
154:
143:
129:
117:
76:
68:classical logic
44:
36:
17:
12:
11:
5:
833:
823:
822:
808:
807:
790:
789:
767:
766:
764:
761:
740:
737:
734:
731:
701:
698:
695:
692:
689:
686:
683:
680:
677:
674:
671:
651:
648:
645:
642:
639:
615:
604:
603:
590:
585:
582:
579:
576:
573:
570:
567:
564:
561:
558:
555:
550:
545:
540:
535:
532:
529:
526:
523:
520:
517:
512:
483:
480:
478:
475:
436:
435:
400:
397:
359:
356:
258:
255:
159:
158:
133:
75:
72:
15:
9:
6:
4:
3:
2:
832:
821:
818:
817:
815:
801:
800:
794:
793:
784:
779:
772:
768:
760:
758:
754:
738:
732:
729:
721:
717:
715:
696:
690:
684:
678:
672:
649:
646:
643:
640:
637:
629:
613:
580:
574:
568:
562:
556:
530:
527:
524:
515:
501:
500:
499:
495:
491:
474:
472:
469:
465:
461:
457:
453:
448:
446:
442:
433:
429:
425:
424:
423:
419:
415:
408:
406:
396:
393:
391:
386:
382:
376:
370:
363:
355:
351:
344:
342:
338:
334:
331:
325:
320:. A proof of
319:
313:
305:
299:
294:
292:
288:
283:
278:
276:
270:
264:
254:
252:
248:
244:
240:
236:
232:
226:
220:
216:
212:
206:
198:
194:
188:
184:
182:
179:
175:
171:
167:
164:
152:
149:
146:holds. Then,
141:
137:
134:
127:
121:
115:
112:
111:
110:
107:
105:
101:
97:
93:
89:
85:
81:
71:
69:
65:
60:
58:
57:free variable
54:
48:
40:
34:
30:
26:
22:
798:
771:
718:
627:
605:
493:
489:
485:
470:
467:
463:
459:
455:
451:
449:
444:
440:
437:
431:
427:
417:
413:
409:
402:
394:
390:set theories
387:
380:
368:
364:
361:
349:
345:
340:
336:
332:
329:
323:
317:
311:
303:
295:
290:
286:
279:
274:
268:
260:
250:
246:
242:
238:
234:
230:
224:
218:
214:
210:
204:
196:
192:
186:
185:
180:
177:
173:
169:
165:
162:
160:
150:
147:
135:
125:
119:
113:
108:
103:
99:
95:
91:
87:
83:
77:
61:
55:cannot be a
52:
46:
38:
32:
28:
21:proof theory
18:
458:θ for some
128:satisfying
783:1911.08027
763:References
375:arithmetic
330:particular
74:Discussion
739:δ
736:¬
730:η
697:α
694:→
691:β
685:∨
679:β
676:→
673:α
650:β
647:∨
644:α
638:η
614:η
581:β
578:→
575:η
569:∨
563:α
560:→
557:η
544:→
531:β
528:∨
525:α
519:→
516:η
140:explosion
90:. Below,
814:Category
383:(¬η → θ)
136:Secondly
102:, while
496:θ) → θ)
420:θ) → θ)
377:, then
326:(φ → θ)
271:(φ → θ)
227:(φ → θ)
187:Thirdly
114:Firstly
49:(φ → θ)
367:¬η → ∃
237:value
803:(PDF)
778:arXiv
628:which
358:Rules
348:φ → ∃
302:φ → ∃
203:φ → ∃
155:φ → θ
144:φ → ψ
130:φ → θ
118:φ → ∃
37:φ → ∃
468:some
235:some
178:some
163:some
94:and
23:and
492:((∃
445:x=7
416:((∃
148:any
19:In
816::
759:.
716:.
392:.
243:it
195:θ(
70:.
786:.
780::
733:=
700:)
688:(
682:)
670:(
641:=
589:)
584:)
572:(
566:)
554:(
549:(
539:)
534:)
522:(
511:(
494:y
490:x
488:∃
471:x
464:x
460:y
456:y
452:x
441:x
432:x
428:x
418:y
414:x
412:∃
381:x
379:∃
371:θ
369:x
352:θ
350:x
341:x
337:x
333:x
324:x
322:∃
318:x
314:θ
312:x
310:∃
306:θ
304:x
291:x
287:x
275:x
269:x
267:∃
251:x
247:x
239:z
231:x
225:x
223:∃
219:x
215:b
211:b
207:θ
205:x
199:)
197:z
193:z
191:∃
181:x
174:a
170:a
166:x
151:x
132:.
126:x
122:θ
120:x
104:θ
100:x
96:ψ
92:φ
88:7
84:a
53:x
47:x
45:∃
41:θ
39:x
33:x
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