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truth values are sets of programs, which can be understood as computational evidence of validity of a formula. For example, the truth value of the statement "for every number there is a prime larger than it" is the set of all programs that take as input a number
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Even though a
Heyting algebra may have many elements, this should not be understood as there being truth values that are neither true nor false, because intuitionistic logic proves
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are treated as false, and strings with content (like "abc"), other numbers, and objects evaluate to true. Sometimes these classes of expressions are called
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exhibits an equivalence of propositions and types, according to which validity is equivalent to inhabitation of a type.
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are truth-valuational in the sense that logical connectives may be interpreted as truth functions. For example,
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of a topological space as intuitionistic truth values, in which case the truth value of a formula expresses
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396:). As opposed to Python, empty containers (Arrays, Maps, Sets) are considered truthy. Languages such as
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But even non-truth-valuational logics can associate values with logical formulae, as is done in
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in the
Boolean domain. Assigning values for propositional variables is referred to as
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Proof that intuitionistic logic has no third truth value, Glivenko 1928
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lacks a complete set of truth values because its semantics, the
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may be assigned a truth value in the subobject classifier.
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For other notions of intuitionistic truth values, see the
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Value indicating the relation of a proposition to truth
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30:"True and false" redirects here. For other uses, see
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864:; this may be expressed as the existence of various
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785:{\displaystyle \neg (p\neq \top \land p\neq \bot )}
108:. Unsourced material may be challenged and removed.
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574:with respect to negation, which is expressed by
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898:conditions, and not directly in terms of the
730:, truth values appear as the elements of the
303:can be evaluated in a context that expects a
546:, whose values are expressed in the form of
534:. This set of two values is also called the
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261:, is a value indicating the relation of a
892:Brouwer–Heyting–Kolmogorov interpretation
828:Brouwer–Heyting–Kolmogorov interpretation
675:the formula holds, not whether it holds.
233:Learn how and when to remove this message
168:Learn how and when to remove this message
982:History of logic § Algebraic period
210:of all important aspects of the article.
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1153:Shramko, Yaroslav; Wansing, Heinrich.
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1164:Stanford Encyclopedia of Philosophy
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299:In some programming languages, any
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1113:"The Elements of JavaScript Style"
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812:is neither true nor false").
667:For example, one may use the
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2747:Computable function
2541:Reverse mathematics
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2112:Free/bound variable
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1327:Cantor's
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1199:
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1169:
1155:"Truth Values"
1148:
1147:External links
1145:
1142:
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1130:
1104:
1073:
1048:
1033:"Truth Values"
1022:
1021:
1019:
1016:
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1012:Verisimilitude
1009:
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989:
984:
979:
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969:
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952:
938:
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876:Main article:
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536:Boolean domain
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376:(of which the
290:
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241:
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202:the key points
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76:
50:
49:
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40:
26:
9:
6:
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3101:
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3087:
3085:
3084:Logical truth
3082:
3080:
3077:
3075:
3072:
3071:
3069:
3054:
3053:inconsistency
3051:
3049:
3048:contradiction
3046:
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3034:
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3024:
3021:
3019:
3016:
3014:
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3004:
2999:
2989:
2986:⊨
2985:
2983:
2980:
2978:
2975:
2974:
2972:
2968:
2963:
2962:Logical truth
2955:
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2809:Recursive set
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2648:
2645:
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2640:
2638:
2635:
2631:
2628:
2627:
2626:
2623:
2619:
2618:of arithmetic
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2559:
2557:
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2542:
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2537:
2534:
2532:
2529:
2526:
2525:from ZFC
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2514:
2508:
2505:
2504:
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2498:
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2458:
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2431:
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2425:
2424:non-Euclidean
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2301:Example
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2167:
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2158:
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2059:
2058:by definition
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2029:
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2024:
2022:
2019:
2018:
2015:
2012:
2010:
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1991:
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1956:
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1945:Kripke–Platek
1943:
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1926:
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1812:
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1797:
1795:
1792:
1790:
1789:constructible
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1622:
1619:
1618:
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1614:
1613:
1610:
1607:
1605:
1601:
1591:
1588:
1586:
1583:
1581:
1578:
1576:
1573:
1571:
1568:
1566:
1563:
1559:
1556:
1555:
1554:
1551:
1547:
1542:
1541:
1540:
1537:
1536:
1534:
1532:
1528:
1520:
1517:
1515:
1512:
1510:
1507:
1506:
1505:
1502:
1500:
1497:
1495:
1492:
1490:
1487:
1485:
1482:
1480:
1477:
1475:
1472:
1471:
1469:
1467:
1466:Propositional
1463:
1457:
1454:
1452:
1449:
1447:
1444:
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1439:
1437:
1434:
1432:
1429:
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1420:
1417:
1415:
1412:
1410:
1407:
1405:
1402:
1400:
1397:
1395:
1394:Logical truth
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1390:
1387:
1386:
1384:
1382:
1378:
1375:
1373:
1369:
1363:
1360:
1358:
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1348:
1345:
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1306:
1304:
1302:
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1285:
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1275:
1272:
1270:
1267:
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1260:
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1235:
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1227:
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1214:
1212:
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1204:
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1192:
1190:
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1177:
1174:
1166:
1165:
1160:
1156:
1151:
1150:
1139:
1134:
1118:
1114:
1108:
1094:on 2015-04-12
1090:
1083:
1077:
1069:
1065:
1063:
1058:
1057:"Truth value"
1052:
1044:
1043:
1038:
1034:
1027:
1023:
1013:
1010:
1008:
1005:
1003:
1000:
998:
995:
993:
990:
988:
985:
983:
980:
978:
977:False dilemma
975:
973:
970:
968:
965:
963:
960:
958:
955:
954:
950:
939:
936:
925:
918:
916:
912:
908:
903:
902:of formulae.
901:
897:
893:
889:
885:
879:
869:
867:
863:
858:[0,1]
856:
855:unit interval
852:
848:
844:
835:
833:
829:
824:
822:
818:
813:
799:
773:
770:
767:
761:
758:
743:
741:
737:
733:
729:
724:
710:
690:
681:
680:realizability
676:
674:
670:
665:
663:
659:
655:
651:
645:
635:
633:
629:
625:
601:
581:
580:
579:
577:
573:
569:
565:
561:
557:
553:
549:
545:
541:
537:
533:
529:
525:
521:
520:
515:
514:
509:
505:
501:
500:
495:
478:
474:
470:
467:
465:
464:
460:
456:
453:
451:
450:
446:
443:
440:
439:
435:
431:
428:
426:
425:
421:
417:
412:
410:
409:
401:
399:
395:
391:
387:
383:
379:
375:
367:
363:
345:
340:
338:
334:
330:
326:
322:
318:
314:
310:
306:
302:
296:
286:
284:
283:
278:
277:
272:
268:
264:
260:
259:logical value
256:
252:
248:
237:
234:
219:
209:
203:
201:
196:
191:
187:
182:
181:
172:
169:
161:
158:February 2012
150:
147:
143:
140:
136:
133:
129:
126:
122:
119: –
118:
117:"Truth value"
114:
113:Find sources:
107:
103:
97:
96:
91:This article
89:
85:
80:
79:
74:
72:
65:
64:
59:
58:
53:
48:
39:
38:
33:
19:
18:Logical value
3079:Propositions
3038:
3018:formal proof
2976:
2911:
2709:Ultraproduct
2698:
2556:Model theory
2521:Independence
2457:Formal proof
2449:Proof theory
2432:
2405:
2362:real numbers
2334:second-order
2245:Substitution
2122:Metalanguage
2063:conservative
2036:Axiom schema
1980:Constructive
1950:Morse–Kelley
1916:Set theories
1895:Aleph number
1888:inaccessible
1794:Grothendieck
1678:intersection
1565:Higher-order
1553:Second-order
1499:Truth tables
1456:Venn diagram
1239:Formal proof
1162:
1133:
1121:. Retrieved
1107:
1096:. Retrieved
1089:the original
1076:
1060:
1051:
1040:
1026:
904:
881:
841:
825:
814:
744:
725:
677:
672:
666:
647:
622:
554:becomes the
548:truth tables
523:
522:(denoted by
517:
511:
503:
502:(denoted by
497:
491:
458:
419:
386:type-checked
381:
373:
341:
324:
320:
313:empty string
298:
280:
274:
258:
254:
244:
229:
213:
197:
195:lead section
164:
155:
145:
138:
131:
124:
112:
100:Please help
95:verification
92:
68:
61:
55:
54:Please help
51:
2977:truth value
2970:Functional:
2819:Type theory
2767:undecidable
2699:Truth value
2586:equivalence
2265:non-logical
1878:Enumeration
1868:Isomorphism
1815:cardinality
1799:Von Neumann
1764:Ultrafilter
1729:Uncountable
1663:equivalence
1580:Quantifiers
1570:Fixed-point
1539:First-order
1419:Consistency
1404:Proposition
1381:Traditional
1352:Lindström's
1342:Compactness
1284:Type theory
1229:Cardinality
957:Agnosticism
896:provability
862:total order
847:fuzzy logic
473:disjunction
434:conjunction
269:, which in
263:proposition
255:truth value
251:mathematics
216:August 2021
3068:Categories
2630:elementary
2323:arithmetic
2191:Quantifier
2169:functional
2041:Expression
1759:Transitive
1703:identities
1688:complement
1621:hereditary
1604:Set theory
1098:2011-03-12
1018:References
562:becomes a
378:complement
344:JavaScript
301:expression
293:See also:
128:newspapers
57:improve it
2988:tautology
2901:Supertask
2804:Recursion
2762:decidable
2596:saturated
2574:of models
2497:deductive
2492:axiomatic
2412:Hilbert's
2399:Euclidean
2380:canonical
2303:axiomatic
2235:Signature
2164:Predicate
2053:Extension
1975:Ackermann
1900:Operation
1779:Universal
1769:Recursive
1744:Singleton
1739:Inhabited
1724:Countable
1714:Types of
1698:power set
1668:partition
1585:Predicate
1531:Predicate
1446:Syllogism
1436:Soundness
1409:Inference
1399:Tautology
1301:paradoxes
845:(such as
777:⊥
774:≠
768:∧
765:⊤
762:≠
753:¬
669:open sets
632:valuation
628:variables
564:bijection
356:undefined
289:Computing
200:summarize
63:talk page
2886:Logicism
2879:timeline
2855:Concrete
2714:Validity
2684:T-schema
2677:Kripke's
2672:Tarski's
2667:semantic
2657:Strength
2606:submodel
2601:spectrum
2569:function
2417:Tarski's
2406:Elements
2393:geometry
2349:Robinson
2270:variable
2255:function
2228:spectrum
2218:Sentence
2174:variable
2117:Language
2070:Relation
2031:Automata
2021:Alphabet
2005:language
1859:-jection
1837:codomain
1823:Function
1784:Universe
1754:Infinite
1658:Relation
1441:Validity
1431:Argument
1329:theorem,
1117:Archived
921:See also
882:Not all
568:permutes
560:negation
556:equality
510:⊤), and
3023:theorem
3006:Formal:
2964: ⊤
2828:Related
2625:Diagram
2523: (
2502:Hilbert
2487:Systems
2482:Theorem
2360:of the
2305:systems
2085:Formula
2080:Grammar
1996: (
1940:General
1653:Forcing
1638:Element
1558:Monadic
1333:paradox
1274:Theorem
1210:General
1161:(ed.).
1123:5 March
1039:(ed.).
987:Paradox
626:become
526:or the
506:or the
390:coerced
142:scholar
3041:
3013:theory
2591:finite
2354:Skolem
2307:
2282:Theory
2250:Symbol
2240:String
2223:atomic
2100:ground
2095:closed
2090:atomic
2046:ground
2009:syntax
1905:binary
1832:domain
1749:Finite
1514:finite
1372:Logics
1331:
1279:Theory
1070:. n.d.
1062:Lexico
819:, the
566:which
528:falsum
513:untrue
468:false
382:truthy
364:, +0,
325:truthy
311:, the
144:
137:
130:
123:
115:
3043:false
2581:Model
2329:Peano
2186:Proof
2026:Arity
1955:Naive
1842:image
1774:Fuzzy
1734:Empty
1683:union
1628:Class
1269:Model
1259:Lemma
1217:Axiom
1157:. In
1092:(PDF)
1085:(PDF)
1035:. In
736:topos
673:where
652:, in
611:) ⇔ ¬
591:) ⇔ ¬
519:false
508:verum
429:true
374:falsy
370:false
321:falsy
282:false
267:truth
247:logic
149:JSTOR
135:books
2704:Type
2507:list
2311:list
2288:list
2277:Term
2211:rank
2105:open
1999:list
1811:Maps
1716:sets
1575:Free
1545:list
1295:list
1222:list
1125:2011
849:and
830:and
572:dual
542:are
499:true
388:and
368:and
352:null
329:Lisp
323:and
317:null
309:zero
276:true
253:, a
249:and
121:news
2391:of
2373:of
2321:of
1853:Sur
1827:Map
1634:Ur-
1616:Set
815:In
726:In
678:In
615:∧ ¬
595:∨ ¬
516:or
492:In
398:PHP
380:is
361:NaN
350:),
342:In
333:nil
285:).
279:or
265:to
245:In
104:by
3070::
2777:NP
2401::
2395::
2325::
2002:),
1857:Bi
1849:In
1066:.
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