130:. For example, a child might truthfully tell their parent "I ate every vegetable on my plate", when there were no vegetables on the child's plate to begin with. In this case, the parent can believe that the child has actually eaten some vegetables, even though that is not true.
939:
Many programming environments have a mechanism for querying if every item in a collection of items satisfies some predicate. It is common for such a query to always evaluate as true for an empty collection. For example:
99:
like "when hell freezes over ..." and "when pigs can fly ...", indicating that not before the given (impossible) condition is met will the speaker accept some respective (typically false or absurd) proposition.
1152:
956:
executes a provided callback function once for each element present in the array, only stopping (if and when) it finds an element where the callback function returns false. Notably, calling the
637:
1072:"All my children are goats" is a vacuous truth when spoken by someone without children. Similarly, "None of my children is a goat" would also be a vacuous truth when spoken by the same person.
280:
50:. It is sometimes said that a statement is vacuously true because it does not really say anything. For example, the statement "all cell phones in the room are turned off" will be
800:
509:
441:
367:
321:
886:
190:
122:
Outside of mathematics, statements in the form of a vacuous truth, while logically valid, can nevertheless be misleading. Such statements make reasonable assertions about
80:
Such statements are considered vacuous truths because the fact that the antecedent is false prevents using the statement to infer anything about the truth value of the
820:
738:
686:
558:
1228:
910:
860:
758:
706:
657:
578:
529:
465:
390:
213:
156:
1271:
1324:
1093:
107:, vacuously true statements are not generally of interest by themselves, but they frequently arise as the base case of proofs by
1087:
1476:
1386:
1509:
1371:
92:("the Eiffel Tower is in Bolivia" in the example) is true or false because the material conditional is defined in that way.
1464:
583:
1491:
1484:
Beer, Ilan; Ben-David, Shoham; Eisner, Cindy; Rodeh, Yoav (1997). "Efficient
Detection of Vacuity in ACTL Formulas".
1170:
1440:
1236:
232:
88:, is true when the antecedent ("Tokyo is in Spain" in the example) is false regardless of whether the conclusion or
96:
54:
when no cell phones are present in the room. In this case, the statement "all cell phones in the room are turned
1487:
Computer Aided
Verification: 9th International Conference, CAV'97 Haifa, Israel, June 22–25, 1997, Proceedings
1188:
964:
767:
1548:
1543:
995:
470:
402:
328:
285:
928:
659:
is the set of all things in the room (including cell phones if they exist in the room), the antecedent
1422:
1057:
are indeed greater than 5), but some of its implications are only vacuously true: for example, when
865:
169:
1203:
1199:
36:
77:. One example of such a statement is "if Tokyo is in Spain, then the Eiffel Tower is in Bolivia".
42:(a universal statement that can be converted to a conditional statement) that is true because the
1263:
1090:– specifically the law that a universal statement is true just in case no counterexample exists:
927:, may attempt to avoid vacuous truths by using alternative conditionals (such as the case of the
1335:
1553:
1174:
108:
1361:
73:
usage refers to a conditional statement (or a universal conditional statement) with a false
1082:
889:
839:
223:
163:
85:
32:
805:
714:
662:
534:
8:
59:
51:
1404:
917:
895:
845:
823:
743:
691:
642:
563:
514:
450:
444:
375:
198:
193:
141:
74:
43:
1196:– another type of true statement that also fails to convey any substantive information
1505:
1367:
1193:
370:
39:
1495:
1485:
1471:
1024:
219:
112:
104:
443:
can be converted to the material conditional form in order to easily identify the
1296:
1183:
924:
831:
116:
70:
1527:
1179:
1165:
835:
47:
16:
Conditional statement which is true because the antecedent cannot be satisfied
1537:
1500:
1160:
1010:
for all items produced by the iterator, or if the iterator produces no items.
913:
467:"all cell phones in the room are turned off", it can be formally written as
159:
127:
1020:
20:
580:
is turned off". This can be written to a material conditional statement
945:
709:
95:
Examples common to everyday speech include conditional phrases used as
89:
81:
1156:
393:
123:
1061:
is the integer 2, the statement implies the vacuous truth that "if
222:) to this basic form (material conditional) include the following
1054:
1147:{\displaystyle \forall x\,P(x)\equiv \neg \exists x\,\neg P(x)}
1050:
949:
24:
960:
method on an empty array will return true for any condition.
84:. In essence, a conditional statement, that is based on the
66:
turned off", which would otherwise be incoherent and false.
979:
or the array is of length zero as shown in these examples:
838:. However, vacuous truths can also appear in, for example,
1002:
function accepts an iterator and a predicate and returns
1483:
1363:
Algorithms and Data
Structures: The Science of Computing
62:
of the two: "all cell phones in the room are turned on
888:
will yield a vacuous truth in any logic that uses the
1096:
898:
868:
848:
808:
770:
746:
717:
694:
665:
645:
586:
566:
537:
517:
473:
453:
405:
378:
331:
288:
235:
201:
172:
144:
916:, then it will also yield a vacuous truth under the
842:, in the same situations as given above. Indeed, if
632:{\displaystyle \forall x\in B:P(x)\Rightarrow Q(x)}
1405:"Built-in Functions – Python 3.10.2 documentation"
1146:
904:
880:
854:
814:
794:
752:
732:
700:
680:
651:
631:
572:
552:
523:
503:
459:
435:
384:
361:
315:
274:
207:
184:
150:
1387:"Array.prototype.every() - JavaScript | MDN"
983:. A less ambiguous way to express this is to say
1535:
1468:. Oxford: Oxford University Press, p. 388.
218:Vacuously true statements that can be reduced (
975:only when all of the elements of an array are
531:is the set of all cell phones in the room and
275:{\displaystyle \forall x:P(x)\Rightarrow Q(x)}
1360:Baldwin, Douglas L.; Scragg, Greg W. (2011),
1359:
58:" would also be vacuously true, as would the
1441:"logic – What precisely is a vacuous truth?"
1027:, illustrate the concept of vacuous truths:
934:
115:, as well as in any other field that uses
1499:
1128:
1103:
1322:
830:Vacuous truths most commonly appear in
133:
1536:
1477:The Cambridge Dictionary of Philosophy
1297:"Definition:Vacuous Truth – ProofWiki"
981:all()==True; all()==False; all()==True
1528:Conditional Assertions: Vacuous truth
1494:. Vol. 1254. pp. 279–290.
1462:Blackburn, Simon (1994). "vacuous",
1318:
1316:
1291:
1289:
1258:
1256:
1254:
1223:
1221:
1219:
923:Other non-classical logics, such as
795:{\displaystyle \forall \xi :Q(\xi )}
1465:The Oxford Dictionary of Philosophy
1323:Edwards, C. H. (January 18, 1998).
504:{\displaystyle \forall x\in A:Q(x)}
436:{\displaystyle \forall x\in A:Q(x)}
362:{\displaystyle \forall x\in A:Q(x)}
316:{\displaystyle \forall x:\neg P(x)}
13:
1129:
1122:
1119:
1097:
771:
587:
474:
406:
332:
298:
289:
236:
14:
1565:
1520:
1492:Lecture Notes in Computer Science
1366:, Cengage Learning, p. 261,
1313:
1286:
1274:from the original on 21 June 2023
1251:
1216:
1171:Paradoxes of material implication
1006:only when the predicate returns
1455:
111:. This notion has relevance in
1433:
1423:"Iterator in std::iter – Rust"
1415:
1397:
1379:
1353:
1264:"Vacuously true - CS2800 wiki"
1141:
1135:
1113:
1107:
989:none of the elements are False
881:{\displaystyle P\Rightarrow Q}
872:
789:
783:
727:
721:
675:
669:
626:
620:
614:
611:
605:
547:
541:
498:
492:
430:
424:
356:
350:
310:
304:
269:
263:
257:
254:
248:
185:{\displaystyle P\Rightarrow Q}
176:
1:
1209:
1189:State of affairs (philosophy)
220:with suitable transformations
826:that has no representatives.
282:, where it is the case that
69:More formally, a relatively
7:
1076:
1014:
10:
1570:
1445:Mathematics Stack Exchange
1053:non-vacuously (since some
929:counterfactual conditional
708:is a cell phone", and the
158:is "vacuously true" if it
1019:These examples, one from
1501:10.1007/3-540-63166-6_28
1480:, 2nd. ed., p. 420.
1204:degeneracy (mathematics)
1200:Triviality (mathematics)
447:. For the above example
1474:(1999). "implication".
1049:." – This statement is
935:In computer programming
97:idioms of improbability
1268:courses.cs.cornell.edu
1233:web.cse.ohio-state.edu
1175:principle of explosion
1148:
906:
882:
856:
816:
796:
754:
734:
702:
682:
653:
633:
574:
554:
525:
505:
461:
437:
386:
363:
317:
276:
224:universally quantified
215:is known to be false.
209:
186:
152:
109:mathematical induction
1391:developer.mozilla.org
1149:
907:
883:
857:
817:
797:
755:
735:
703:
683:
654:
634:
575:
555:
526:
506:
462:
438:
387:
364:
318:
277:
210:
187:
153:
128:do not actually exist
1094:
1083:Definite description
896:
890:material conditional
866:
846:
840:intuitionistic logic
815:{\displaystyle \xi }
806:
768:
744:
733:{\displaystyle Q(x)}
715:
692:
681:{\displaystyle P(x)}
663:
643:
584:
564:
553:{\displaystyle Q(x)}
535:
515:
471:
451:
403:
376:
329:
286:
233:
199:
170:
164:material conditional
142:
134:Scope of the concept
86:material conditional
1239:on 18 November 2023
914:necessary falsehood
822:is restricted to a
802:, where the symbol
1549:Informal fallacies
1544:Mathematical logic
1144:
987:returns True when
918:strict conditional
902:
878:
852:
812:
792:
750:
730:
698:
678:
649:
629:
570:
550:
521:
501:
457:
433:
399:This logical form
382:
359:
313:
272:
205:
182:
148:
1511:978-3-540-63166-8
1427:doc.rust-lang.org
1373:978-1-285-22512-8
1194:Tautology (logic)
1173:, especially the
1031:"For any integer
971:function returns
905:{\displaystyle P}
855:{\displaystyle P}
753:{\displaystyle x}
701:{\displaystyle x}
652:{\displaystyle B}
573:{\displaystyle x}
524:{\displaystyle A}
460:{\displaystyle S}
385:{\displaystyle A}
208:{\displaystyle P}
151:{\displaystyle S}
1561:
1515:
1503:
1472:David H. Sanford
1449:
1448:
1437:
1431:
1430:
1419:
1413:
1412:
1401:
1395:
1394:
1383:
1377:
1376:
1357:
1351:
1350:
1348:
1346:
1341:on 28 April 2021
1340:
1334:. Archived from
1329:
1325:"Vacuously True"
1320:
1311:
1310:
1308:
1307:
1293:
1284:
1283:
1281:
1279:
1260:
1249:
1248:
1246:
1244:
1235:. Archived from
1229:"Vacuously true"
1225:
1153:
1151:
1150:
1145:
1088:De Morgan's laws
1068:
1064:
1048:
1041:
1025:natural language
1009:
1005:
1001:
986:
982:
978:
974:
970:
959:
955:
911:
909:
908:
903:
887:
885:
884:
879:
861:
859:
858:
853:
836:two truth values
821:
819:
818:
813:
801:
799:
798:
793:
759:
757:
756:
751:
739:
737:
736:
731:
707:
705:
704:
699:
687:
685:
684:
679:
658:
656:
655:
650:
638:
636:
635:
630:
579:
577:
576:
571:
559:
557:
556:
551:
530:
528:
527:
522:
510:
508:
507:
502:
466:
464:
463:
458:
442:
440:
439:
434:
391:
389:
388:
383:
368:
366:
365:
360:
322:
320:
319:
314:
281:
279:
278:
273:
214:
212:
211:
206:
191:
189:
188:
183:
157:
155:
154:
149:
113:pure mathematics
105:pure mathematics
1569:
1568:
1564:
1563:
1562:
1560:
1559:
1558:
1534:
1533:
1532:
1523:
1518:
1512:
1458:
1453:
1452:
1439:
1438:
1434:
1421:
1420:
1416:
1409:docs.python.org
1403:
1402:
1398:
1385:
1384:
1380:
1374:
1358:
1354:
1344:
1342:
1338:
1327:
1321:
1314:
1305:
1303:
1295:
1294:
1287:
1277:
1275:
1262:
1261:
1252:
1242:
1240:
1227:
1226:
1217:
1212:
1184:double question
1095:
1092:
1091:
1079:
1066:
1062:
1043:
1036:
1017:
1007:
1003:
999:
984:
980:
976:
972:
968:
967:, the built in
957:
953:
937:
925:relevance logic
897:
894:
893:
867:
864:
863:
862:is false, then
847:
844:
843:
832:classical logic
807:
804:
803:
769:
766:
765:
760:is turned off".
745:
742:
741:
716:
713:
712:
693:
690:
689:
664:
661:
660:
644:
641:
640:
585:
582:
581:
565:
562:
561:
536:
533:
532:
516:
513:
512:
472:
469:
468:
452:
449:
448:
404:
401:
400:
377:
374:
373:
330:
327:
326:
287:
284:
283:
234:
231:
230:
200:
197:
196:
171:
168:
167:
143:
140:
139:
136:
117:classical logic
17:
12:
11:
5:
1567:
1557:
1556:
1551:
1546:
1531:
1530:
1524:
1522:
1521:External links
1519:
1517:
1516:
1510:
1481:
1469:
1459:
1457:
1454:
1451:
1450:
1432:
1414:
1396:
1378:
1372:
1352:
1332:swarthmore.edu
1312:
1285:
1250:
1214:
1213:
1211:
1208:
1207:
1206:
1197:
1191:
1186:
1180:Presupposition
1177:
1168:
1166:Empty function
1163:
1154:
1143:
1140:
1137:
1134:
1131:
1127:
1124:
1121:
1118:
1115:
1112:
1109:
1106:
1102:
1099:
1085:
1078:
1075:
1074:
1073:
1070:
1016:
1013:
1012:
1011:
992:
961:
936:
933:
901:
877:
874:
871:
851:
828:
827:
811:
791:
788:
785:
782:
779:
776:
773:
763:
762:
761:
749:
729:
726:
723:
720:
697:
677:
674:
671:
668:
648:
628:
625:
622:
619:
616:
613:
610:
607:
604:
601:
598:
595:
592:
589:
569:
549:
546:
543:
540:
520:
500:
497:
494:
491:
488:
485:
482:
479:
476:
456:
432:
429:
426:
423:
420:
417:
414:
411:
408:
381:
358:
355:
352:
349:
346:
343:
340:
337:
334:
324:
312:
309:
306:
303:
300:
297:
294:
291:
271:
268:
265:
262:
259:
256:
253:
250:
247:
244:
241:
238:
204:
181:
178:
175:
147:
135:
132:
126:objects which
15:
9:
6:
4:
3:
2:
1566:
1555:
1554:Logical truth
1552:
1550:
1547:
1545:
1542:
1541:
1539:
1529:
1526:
1525:
1513:
1507:
1502:
1497:
1493:
1489:
1488:
1482:
1479:
1478:
1473:
1470:
1467:
1466:
1461:
1460:
1446:
1442:
1436:
1428:
1424:
1418:
1410:
1406:
1400:
1392:
1388:
1382:
1375:
1369:
1365:
1364:
1356:
1337:
1333:
1326:
1319:
1317:
1302:
1301:proofwiki.org
1298:
1292:
1290:
1273:
1269:
1265:
1259:
1257:
1255:
1238:
1234:
1230:
1224:
1222:
1220:
1215:
1205:
1201:
1198:
1195:
1192:
1190:
1187:
1185:
1181:
1178:
1176:
1172:
1169:
1167:
1164:
1162:
1161:empty product
1158:
1155:
1138:
1132:
1125:
1116:
1110:
1104:
1100:
1089:
1086:
1084:
1081:
1080:
1071:
1060:
1056:
1052:
1046:
1039:
1034:
1030:
1029:
1028:
1026:
1023:and one from
1022:
1000:Iterator::all
997:
993:
990:
966:
962:
951:
947:
943:
942:
941:
932:
930:
926:
921:
919:
915:
899:
891:
875:
869:
849:
841:
837:
833:
825:
809:
786:
780:
777:
774:
764:
747:
724:
718:
711:
695:
672:
666:
646:
623:
617:
608:
602:
599:
596:
593:
590:
567:
544:
538:
518:
495:
489:
486:
483:
480:
477:
454:
446:
427:
421:
418:
415:
412:
409:
398:
397:
395:
379:
372:
353:
347:
344:
341:
338:
335:
325:
307:
301:
295:
292:
266:
260:
251:
245:
242:
239:
229:
228:
227:
225:
221:
216:
202:
195:
179:
173:
165:
161:
145:
131:
129:
125:
120:
118:
114:
110:
106:
101:
98:
93:
91:
87:
83:
78:
76:
72:
67:
65:
61:
57:
53:
49:
45:
41:
38:
34:
30:
29:vacuous truth
26:
22:
1486:
1475:
1463:
1456:Bibliography
1444:
1435:
1426:
1417:
1408:
1399:
1390:
1381:
1362:
1355:
1343:. Retrieved
1336:the original
1331:
1304:. Retrieved
1300:
1276:. Retrieved
1267:
1241:. Retrieved
1237:the original
1232:
1058:
1044:
1037:
1032:
1018:
988:
938:
922:
829:
369:, where the
226:statements:
217:
192:, where the
138:A statement
137:
121:
102:
94:
79:
71:well-defined
68:
63:
55:
28:
18:
1345:14 December
1278:15 December
1243:15 December
1021:mathematics
60:conjunction
33:conditional
21:mathematics
1538:Categories
1306:2019-12-15
1210:References
946:JavaScript
710:consequent
445:antecedent
194:antecedent
166:statement
90:consequent
82:consequent
75:antecedent
46:cannot be
44:antecedent
1157:Empty sum
1130:¬
1123:∃
1120:¬
1117:≡
1098:∀
873:⇒
810:ξ
787:ξ
775:ξ
772:∀
615:⇒
594:∈
588:∀
481:∈
475:∀
413:∈
407:∀
339:∈
333:∀
299:¬
290:∀
258:⇒
237:∀
177:⇒
160:resembles
124:qualified
48:satisfied
40:statement
37:universal
1272:Archived
1077:See also
1067:2 > 3
1063:2 > 5
1055:integers
1015:Examples
952:method
1508:
1370:
1047:> 3
1040:> 5
998:, the
965:Python
948:, the
639:where
511:where
1339:(PDF)
1328:(PDF)
1065:then
1042:then
1035:, if
985:all()
969:all()
958:every
954:every
950:array
912:is a
892:; if
834:with
394:empty
31:is a
25:logic
1506:ISBN
1368:ISBN
1347:2019
1280:2019
1245:2019
1202:and
1159:and
1051:true
1008:true
1004:true
996:Rust
977:True
973:True
824:type
740:is "
688:is "
560:is "
52:true
27:, a
23:and
1496:doi
994:In
963:In
944:In
931:).
392:is
371:set
103:In
64:and
35:or
19:In
1540::
1504:.
1490:.
1443:.
1425:.
1407:.
1389:.
1330:.
1315:^
1299:.
1288:^
1270:.
1266:.
1253:^
1231:.
1218:^
1182:,
1069:".
920:.
396:.
162:a
119:.
56:on
1514:.
1498::
1447:.
1429:.
1411:.
1393:.
1349:.
1309:.
1282:.
1247:.
1142:)
1139:x
1136:(
1133:P
1126:x
1114:)
1111:x
1108:(
1105:P
1101:x
1059:x
1045:x
1038:x
1033:x
991:.
900:P
876:Q
870:P
850:P
790:)
784:(
781:Q
778::
748:x
728:)
725:x
722:(
719:Q
696:x
676:)
673:x
670:(
667:P
647:B
627:)
624:x
621:(
618:Q
612:)
609:x
606:(
603:P
600::
597:B
591:x
568:x
548:)
545:x
542:(
539:Q
519:A
499:)
496:x
493:(
490:Q
487::
484:A
478:x
455:S
431:)
428:x
425:(
422:Q
419::
416:A
410:x
380:A
357:)
354:x
351:(
348:Q
345::
342:A
336:x
323:.
311:)
308:x
305:(
302:P
296::
293:x
270:)
267:x
264:(
261:Q
255:)
252:x
249:(
246:P
243::
240:x
203:P
180:Q
174:P
146:S
Text is available under the Creative Commons Attribution-ShareAlike License. Additional terms may apply.