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31:
1217:
1638:
376:
which do not refer to any idea, but rather serve as a form of punctuation in the language (e.g. parentheses). A symbol or string of symbols may comprise a well-formed formula if the formulation is consistent with the formation rules of the language. Symbols of a formal language must be capable of
825:
are semantically complete, but not syntactically complete (for example the propositional logic statement consisting of a single variable "a" is not a theorem, and neither is its negation, but these are not
722:
1796:
681:
647:
606:, or have both. A formal system is used to derive one expression from one or more other expressions. Formal systems, like other syntactic entities may be defined without any
811:
767:
309:
expressed in formal languages are syntactic entities whose properties may be studied without regard to any meaning they may be given, and, in fact, need not be given any.
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372:
of which may be marks or a metalanguage of marks which form a particular pattern. Symbols of a formal language need not be symbols of anything. For instance there are
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1695:
1125:
Wijesekera, Duminda; Ganesh, M.; Srivastava, Jaideep; Nerode, Anil (2001). "Normal forms and syntactic completeness proofs for functional independencies".
245:
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given to them. Syntax is concerned with the rules used for constructing, or transforming the symbols and words of a language, as contrasted with the
1188:
1068:
2868:
3026:
1814:
813:. In another sense, a formal system is syntactically complete iff no unprovable axiom can be added to it as an axiom without introducing an
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2204:
1092:
Hunter, Geoffrey, Metalogic: An
Introduction to the Metatheory of Standard First-Order Logic, University of California Press, 1971, p. 75.
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Syntax is usually associated with the rules (or grammar) governing the composition of texts in a formal language that constitute the
54:. A formal language is identical to the set of its well-formed formulas. The set of well-formed formulas may be broadly divided into
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411:). Which strings of symbols are words is determined by the creator of the language, usually by specifying a set of
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1990:
1967:
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3143:
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2020:
1995:
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334:
792:
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2015:
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212:
127:
51:
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of the formal language which constitute well formed formulas. However, it does not describe their
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2010:
1985:
1755:
1704:
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1314:
921:
137:
107:
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2005:
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1735:
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896:
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182:
132:
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3003:
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2260:
2149:
2144:
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1723:
1647:
1340:
1234:
906:
901:
436:
412:
172:
152:
122:
117:
1138:
789:) iff for each formula A of the language of the system either A or ÂŹA is a theorem of
3435:
3310:
2988:
2495:
2280:
2270:
2240:
2225:
1895:
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1438:
654:
569:
298:
271:
92:
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2687:
2623:
2606:
2537:
2396:
2255:
1957:
1740:
1487:
1309:
887:, which may itself be a formal language, and as such itself is a syntactic entity.
884:
839:
650:
306:
1156:
1036:
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2379:
2369:
2316:
2000:
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1905:
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1152:
945:
865:
814:
527:
511:
477:
361:
27:
Rules used for constructing, or transforming the symbols and words of a language
337:. As in mathematical logic, it is independent of semantics and interpretation.
30:
2250:
2105:
2076:
1882:
1465:
1433:
1398:
17:
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3305:
2358:
2275:
2235:
2199:
2135:
1947:
1937:
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to the sentences of a formal system. The study of interpretations is called
495:
465:
416:
1216:
3387:
3185:
2633:
2338:
1932:
1448:
499:
461:
47:
526:
of words. Propositions are considered to be syntactic entities and also
34:
This diagram shows the syntactic entities which may be constructed from
2983:
1775:
1512:
1443:
507:
365:
302:
55:
1673:
1350:
911:
864:
of a formal system is the assignment of meanings to the symbols, and
377:
being specified without any reference to any interpretation of them.
2527:
1873:
1718:
1542:
1257:
603:
491:
259:
29:
1502:
503:
357:
1677:
1230:
610:
given to it (as being, for instance, a system of arithmetic).
329:
refers to the rules governing the composition of well-formed
798:
754:
668:
665:
634:
631:
1226:
427:
is assigned to it â that is, before it has any meaning.
1662:
456:
of a formal language. It is synonymous with the set of
795:
751:
692:
662:
628:
3329:
3224:
3056:
2949:
2801:
2494:
2417:
2311:
2215:
2104:
2031:
1966:
1881:
1872:
1794:
1711:
1591:
1563:
1556:
1411:
1376:
1328:
1282:
594:). The deductive apparatus may consist of a set of
423:of any of its expressions; it can exist before any
286:of a language which is concerned with its meaning.
805:
761:
716:
675:
641:
586:) consists of a formal language together with a
717:{\displaystyle \Gamma \vdash _{\mathrm {F} S}A}
1689:
1242:
727:Syntactic consequence does not depend on any
239:
8:
614:Syntactic consequence within a formal system
1041:. Cambridge University Press. p. 189.
838:that is sufficiently powerful, such as the
2515:
2110:
1878:
1696:
1682:
1674:
1560:
1325:
1249:
1235:
1227:
1103:"A Note on Interaction and Incompleteness"
395:is a syntactic entity which consists of a
246:
232:
61:
1008:. Cambridge University Press. p. 1.
797:
796:
794:
753:
752:
750:
735:Syntactic completeness of a formal system
701:
700:
691:
664:
663:
661:
630:
629:
627:
415:. Such a language can be defined without
975:. Harvard University Press. p. 82.
407:which are its words (usually called its
938:
842:, can be both consistent and complete.
75:
883:. An interpretation is expressed in a
649:of a set Đ of formulas if there is a
7:
1189:"syntactic completeness from FOLDOC"
1424:Analytic and synthetic propositions
1295:Formal semantics (natural language)
1069:"syntactic consequence from FOLDOC"
444:are a precise description of which
702:
693:
25:
1161:. Elsevier Science. p. 236.
1038:The Cambridge Companion to Carnap
1035:Creath, R.; Friedman, M. (2007).
113:Semantics (programming languages)
3415:
1636:
1215:
1133:(1â2). portal.acm.org: 365â405.
1191:. swif.uniba.it. Archived from
1071:. swif.uniba.it. Archived from
676:{\displaystyle {\mathcal {FS}}}
642:{\displaystyle {\mathcal {FS}}}
1158:Handbook of Mathematical Logic
917:Syntax (programming languages)
832:Gödel's incompleteness theorem
806:{\displaystyle {\mathcal {S}}}
762:{\displaystyle {\mathcal {S}}}
498:. A proposition is identified
266:is anything having to do with
1:
3376:History of mathematical logic
1139:10.1016/S0304-3975(00)00195-X
972:Frege: Philosophy of Language
3301:Primitive recursive function
1127:Theoretical Computer Science
1005:Aristotle and Logical Theory
46:may be broadly divided into
540:Theory (mathematical logic)
198:Programming language theory
193:Natural language processing
3478:
2365:SchröderâBernstein theorem
2092:Monadic predicate calculus
1751:Foundations of mathematics
849:
738:
622:within some formal system
567:
537:
475:
434:
384:
349:
3411:
3398:Philosophy of mathematics
3347:Automated theorem proving
2518:
2472:Von NeumannâBernaysâGödel
2113:
1631:
1508:Necessity and sufficiency
1264:
218:Automated theorem proving
203:Computational linguistics
874:Giving an interpretation
852:Formal semantics (logic)
3048:Self-verifying theories
2869:Tarski's axiomatization
1820:Tarski's undefinability
1815:incompleteness theorems
468:(i.e. what they mean).
3422:Mathematics portal
3033:Proof of impossibility
2681:propositional variable
1991:Propositional calculus
856:Interpretation (logic)
807:
771:syntactically complete
763:
731:of the formal system.
718:
677:
643:
274:without regard to any
178:Propositional calculus
59:
3291:Kolmogorov complexity
3244:Computably enumerable
3144:Model complete theory
2936:Principia Mathematica
1996:Propositional formula
1825:BanachâTarski paradox
1643:Philosophy portal
1224:at Wikimedia Commons
946:Dictionary Definition
808:
764:
719:
683:of A from the set Đ.
678:
644:
620:syntactic consequence
490:expressing something
188:Mathematical notation
33:
3239:ChurchâTuring thesis
3226:Computability theory
2435:continuum hypothesis
1953:Square of opposition
1811:Gödel's completeness
969:Dummett, M. (1981).
912:Syntax (linguistics)
793:
775:deductively complete
749:
741:Completeness (logic)
690:
660:
626:
596:transformation rules
522:, marks, sounds, or
454:well-formed formulas
409:well-formed formulas
335:programming language
316:of a formal system.
314:well-formed formulas
52:well-formed formulas
3462:Philosophy of logic
3393:Mathematical object
3284:P versus NP problem
3249:Computable function
3043:Reverse mathematics
2969:Logical consequence
2846:primitive recursive
2841:elementary function
2614:Free/bound variable
2467:TarskiâGrothendieck
1986:Logical connectives
1916:Logical equivalence
1766:Logical consequence
1305:Philosophy of logic
927:Well-formed formula
876:is synonymous with
819:propositional logic
817:. Truth-functional
588:deductive apparatus
213:Formal verification
128:Well-formed formula
3191:Transfer principle
3154:Semantics of logic
3139:Categorical theory
3115:Non-standard model
2629:Logical connective
1756:Information theory
1705:Mathematical logic
1604:Rules of inference
1573:Mathematical logic
1315:Semantics of logic
922:Mathematical logic
803:
779:maximally complete
759:
714:
673:
639:
341:Syntactic entities
138:Regular expression
60:
44:strings of symbols
3457:Concepts in logic
3429:
3428:
3361:Abstract category
3164:Theories of truth
2974:Rule of inference
2964:Natural deduction
2945:
2944:
2490:
2489:
2195:Cartesian product
2100:
2099:
2006:Many-valued logic
1981:Boolean functions
1864:Russell's paradox
1839:diagonal argument
1736:First-order logic
1671:
1670:
1627:
1626:
1461:Deductive closure
1407:
1406:
1346:Critical thinking
1220:Media related to
1002:Lear, J. (1986).
783:negation complete
618:A formula A is a
374:logical constants
256:
255:
148:Ground expression
108:Semantics (logic)
58:and non-theorems.
16:(Redirected from
3469:
3447:Formal languages
3420:
3419:
3371:History of logic
3366:Category of sets
3259:Decision problem
3038:Ordinal analysis
2979:Sequent calculus
2877:Boolean algebras
2817:
2816:
2791:
2762:logical/constant
2516:
2502:
2425:ZermeloâFraenkel
2176:Set operations:
2111:
2048:
1879:
1859:LöwenheimâSkolem
1746:Formal semantics
1698:
1691:
1684:
1675:
1641:
1640:
1639:
1561:
1326:
1290:Computer science
1251:
1244:
1237:
1228:
1219:
1204:
1203:
1201:
1200:
1185:
1179:
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1176:
1175:
1149:
1143:
1142:
1122:
1116:
1115:
1113:
1112:
1107:
1099:
1093:
1090:
1084:
1083:
1081:
1080:
1065:
1059:
1058:
1056:
1055:
1032:
1026:
1025:
1023:
1022:
999:
993:
992:
990:
989:
966:
960:
954:
948:
943:
870:formal semantics
836:recursive system
821:and first-order
812:
810:
809:
804:
802:
801:
768:
766:
765:
760:
758:
757:
745:A formal system
723:
721:
720:
715:
710:
709:
705:
682:
680:
679:
674:
672:
671:
648:
646:
645:
640:
638:
637:
592:deductive system
580:logical calculus
518:are patterns of
321:computer science
268:formal languages
248:
241:
234:
77:Formal languages
62:
36:formal languages
21:
3477:
3476:
3472:
3471:
3470:
3468:
3467:
3466:
3432:
3431:
3430:
3425:
3414:
3407:
3352:Category theory
3342:Algebraic logic
3325:
3296:Lambda calculus
3234:Church encoding
3220:
3196:Truth predicate
3052:
3018:Complete theory
2941:
2810:
2806:
2802:
2797:
2789:
2509: and
2505:
2500:
2486:
2462:New Foundations
2430:axiom of choice
2413:
2375:Gödel numbering
2315: and
2307:
2211:
2096:
2046:
2027:
1976:Boolean algebra
1962:
1926:Equiconsistency
1891:Classical logic
1868:
1849:Halting problem
1837: and
1813: and
1801: and
1800:
1795:Theorems (
1790:
1707:
1702:
1672:
1667:
1637:
1635:
1623:
1587:
1578:Boolean algebra
1552:
1403:
1394:Metamathematics
1372:
1324:
1278:
1260:
1255:
1213:
1208:
1207:
1198:
1196:
1187:
1186:
1182:
1173:
1171:
1169:
1151:
1150:
1146:
1124:
1123:
1119:
1110:
1108:
1105:
1101:
1100:
1096:
1091:
1087:
1078:
1076:
1067:
1066:
1062:
1053:
1051:
1049:
1034:
1033:
1029:
1020:
1018:
1016:
1001:
1000:
996:
987:
985:
983:
968:
967:
963:
958:Geoffrey Hunter
955:
951:
944:
940:
935:
897:Symbol (formal)
893:
878:constructing a
858:
850:Main articles:
848:
846:Interpretations
823:predicate logic
791:
790:
747:
746:
743:
737:
696:
688:
687:
658:
657:
624:
623:
616:
600:inference rules
590:(also called a
578:(also called a
572:
566:
558:formal language
542:
536:
534:Formal theories
516:token instances
480:
474:
442:Formation rules
439:
433:
431:Formation rules
413:formation rules
393:formal language
389:
387:Formal language
383:
381:Formal language
356:A symbol is an
354:
352:Symbol (formal)
348:
343:
252:
223:
222:
208:Syntax analysis
183:Predicate logic
168:
167:
158:
157:
133:Automata theory
88:
87:
28:
23:
22:
15:
12:
11:
5:
3475:
3473:
3465:
3464:
3459:
3454:
3449:
3444:
3442:Syntax (logic)
3434:
3433:
3427:
3426:
3412:
3409:
3408:
3406:
3405:
3400:
3395:
3390:
3385:
3384:
3383:
3373:
3368:
3363:
3354:
3349:
3344:
3339:
3337:Abstract logic
3333:
3331:
3327:
3326:
3324:
3323:
3318:
3316:Turing machine
3313:
3308:
3303:
3298:
3293:
3288:
3287:
3286:
3281:
3276:
3271:
3266:
3256:
3254:Computable set
3251:
3246:
3241:
3236:
3230:
3228:
3222:
3221:
3219:
3218:
3213:
3208:
3203:
3198:
3193:
3188:
3183:
3182:
3181:
3176:
3171:
3161:
3156:
3151:
3149:Satisfiability
3146:
3141:
3136:
3135:
3134:
3124:
3123:
3122:
3112:
3111:
3110:
3105:
3100:
3095:
3090:
3080:
3079:
3078:
3073:
3066:Interpretation
3062:
3060:
3054:
3053:
3051:
3050:
3045:
3040:
3035:
3030:
3020:
3015:
3014:
3013:
3012:
3011:
3001:
2996:
2986:
2981:
2976:
2971:
2966:
2961:
2955:
2953:
2947:
2946:
2943:
2942:
2940:
2939:
2931:
2930:
2929:
2928:
2923:
2922:
2921:
2916:
2911:
2891:
2890:
2889:
2887:minimal axioms
2884:
2873:
2872:
2871:
2860:
2859:
2858:
2853:
2848:
2843:
2838:
2833:
2820:
2818:
2799:
2798:
2796:
2795:
2794:
2793:
2781:
2776:
2775:
2774:
2769:
2764:
2759:
2749:
2744:
2739:
2734:
2733:
2732:
2727:
2717:
2716:
2715:
2710:
2705:
2700:
2690:
2685:
2684:
2683:
2678:
2673:
2663:
2662:
2661:
2656:
2651:
2646:
2641:
2636:
2626:
2621:
2616:
2611:
2610:
2609:
2604:
2599:
2594:
2584:
2579:
2577:Formation rule
2574:
2569:
2568:
2567:
2562:
2552:
2551:
2550:
2540:
2535:
2530:
2525:
2519:
2513:
2496:Formal systems
2492:
2491:
2488:
2487:
2485:
2484:
2479:
2474:
2469:
2464:
2459:
2454:
2449:
2444:
2439:
2438:
2437:
2432:
2421:
2419:
2415:
2414:
2412:
2411:
2410:
2409:
2399:
2394:
2393:
2392:
2385:Large cardinal
2382:
2377:
2372:
2367:
2362:
2348:
2347:
2346:
2341:
2336:
2321:
2319:
2309:
2308:
2306:
2305:
2304:
2303:
2298:
2293:
2283:
2278:
2273:
2268:
2263:
2258:
2253:
2248:
2243:
2238:
2233:
2228:
2222:
2220:
2213:
2212:
2210:
2209:
2208:
2207:
2202:
2197:
2192:
2187:
2182:
2174:
2173:
2172:
2167:
2157:
2152:
2150:Extensionality
2147:
2145:Ordinal number
2142:
2132:
2127:
2126:
2125:
2114:
2108:
2102:
2101:
2098:
2097:
2095:
2094:
2089:
2084:
2079:
2074:
2069:
2064:
2063:
2062:
2052:
2051:
2050:
2037:
2035:
2029:
2028:
2026:
2025:
2024:
2023:
2018:
2013:
2003:
1998:
1993:
1988:
1983:
1978:
1972:
1970:
1964:
1963:
1961:
1960:
1955:
1950:
1945:
1940:
1935:
1930:
1929:
1928:
1918:
1913:
1908:
1903:
1898:
1893:
1887:
1885:
1876:
1870:
1869:
1867:
1866:
1861:
1856:
1851:
1846:
1841:
1829:Cantor's
1827:
1822:
1817:
1807:
1805:
1792:
1791:
1789:
1788:
1783:
1778:
1773:
1768:
1763:
1758:
1753:
1748:
1743:
1738:
1733:
1728:
1727:
1726:
1715:
1713:
1709:
1708:
1703:
1701:
1700:
1693:
1686:
1678:
1669:
1668:
1666:
1665:
1660:
1650:
1645:
1632:
1629:
1628:
1625:
1624:
1622:
1621:
1616:
1611:
1606:
1601:
1595:
1593:
1589:
1588:
1586:
1585:
1580:
1575:
1569:
1567:
1558:
1554:
1553:
1551:
1550:
1545:
1540:
1535:
1530:
1525:
1520:
1515:
1510:
1505:
1500:
1495:
1490:
1485:
1484:
1483:
1473:
1468:
1463:
1458:
1453:
1452:
1451:
1446:
1436:
1431:
1426:
1421:
1415:
1413:
1409:
1408:
1405:
1404:
1402:
1401:
1396:
1391:
1386:
1380:
1378:
1374:
1373:
1371:
1370:
1365:
1360:
1355:
1354:
1353:
1348:
1338:
1332:
1330:
1323:
1322:
1317:
1312:
1307:
1302:
1297:
1292:
1286:
1284:
1280:
1279:
1277:
1276:
1271:
1265:
1262:
1261:
1256:
1254:
1253:
1246:
1239:
1231:
1222:Syntax (logic)
1212:
1211:External links
1209:
1206:
1205:
1180:
1167:
1144:
1117:
1094:
1085:
1060:
1047:
1027:
1014:
994:
981:
961:
949:
937:
936:
934:
931:
930:
929:
924:
919:
914:
909:
907:Formal grammar
904:
902:Formation rule
899:
892:
889:
862:interpretation
847:
844:
834:shows that no
800:
756:
739:Main article:
736:
733:
729:interpretation
725:
724:
713:
708:
704:
699:
695:
670:
667:
636:
633:
615:
612:
608:interpretation
602:) or a set of
584:logical system
568:Main article:
565:
564:Formal systems
562:
538:Main article:
535:
532:
476:Main article:
473:
470:
437:Formation rule
435:Main article:
432:
429:
425:interpretation
385:Main article:
382:
379:
350:Main article:
347:
344:
342:
339:
276:interpretation
272:formal systems
254:
253:
251:
250:
243:
236:
228:
225:
224:
221:
220:
215:
210:
205:
200:
195:
190:
185:
180:
175:
173:Formal methods
169:
165:
164:
163:
160:
159:
156:
155:
153:Atomic formula
150:
145:
140:
135:
130:
125:
123:Formation rule
120:
118:Formal grammar
115:
110:
105:
100:
95:
89:
85:
84:
83:
80:
79:
73:
72:
26:
24:
18:Logical syntax
14:
13:
10:
9:
6:
4:
3:
2:
3474:
3463:
3460:
3458:
3455:
3453:
3450:
3448:
3445:
3443:
3440:
3439:
3437:
3424:
3423:
3418:
3410:
3404:
3401:
3399:
3396:
3394:
3391:
3389:
3386:
3382:
3379:
3378:
3377:
3374:
3372:
3369:
3367:
3364:
3362:
3358:
3355:
3353:
3350:
3348:
3345:
3343:
3340:
3338:
3335:
3334:
3332:
3328:
3322:
3319:
3317:
3314:
3312:
3311:Recursive set
3309:
3307:
3304:
3302:
3299:
3297:
3294:
3292:
3289:
3285:
3282:
3280:
3277:
3275:
3272:
3270:
3267:
3265:
3262:
3261:
3260:
3257:
3255:
3252:
3250:
3247:
3245:
3242:
3240:
3237:
3235:
3232:
3231:
3229:
3227:
3223:
3217:
3214:
3212:
3209:
3207:
3204:
3202:
3199:
3197:
3194:
3192:
3189:
3187:
3184:
3180:
3177:
3175:
3172:
3170:
3167:
3166:
3165:
3162:
3160:
3157:
3155:
3152:
3150:
3147:
3145:
3142:
3140:
3137:
3133:
3130:
3129:
3128:
3125:
3121:
3120:of arithmetic
3118:
3117:
3116:
3113:
3109:
3106:
3104:
3101:
3099:
3096:
3094:
3091:
3089:
3086:
3085:
3084:
3081:
3077:
3074:
3072:
3069:
3068:
3067:
3064:
3063:
3061:
3059:
3055:
3049:
3046:
3044:
3041:
3039:
3036:
3034:
3031:
3028:
3027:from ZFC
3024:
3021:
3019:
3016:
3010:
3007:
3006:
3005:
3002:
3000:
2997:
2995:
2992:
2991:
2990:
2987:
2985:
2982:
2980:
2977:
2975:
2972:
2970:
2967:
2965:
2962:
2960:
2957:
2956:
2954:
2952:
2948:
2938:
2937:
2933:
2932:
2927:
2926:non-Euclidean
2924:
2920:
2917:
2915:
2912:
2910:
2909:
2905:
2904:
2902:
2899:
2898:
2896:
2892:
2888:
2885:
2883:
2880:
2879:
2878:
2874:
2870:
2867:
2866:
2865:
2861:
2857:
2854:
2852:
2849:
2847:
2844:
2842:
2839:
2837:
2834:
2832:
2829:
2828:
2826:
2822:
2821:
2819:
2814:
2808:
2803:Example
2800:
2792:
2787:
2786:
2785:
2782:
2780:
2777:
2773:
2770:
2768:
2765:
2763:
2760:
2758:
2755:
2754:
2753:
2750:
2748:
2745:
2743:
2740:
2738:
2735:
2731:
2728:
2726:
2723:
2722:
2721:
2718:
2714:
2711:
2709:
2706:
2704:
2701:
2699:
2696:
2695:
2694:
2691:
2689:
2686:
2682:
2679:
2677:
2674:
2672:
2669:
2668:
2667:
2664:
2660:
2657:
2655:
2652:
2650:
2647:
2645:
2642:
2640:
2637:
2635:
2632:
2631:
2630:
2627:
2625:
2622:
2620:
2617:
2615:
2612:
2608:
2605:
2603:
2600:
2598:
2595:
2593:
2590:
2589:
2588:
2585:
2583:
2580:
2578:
2575:
2573:
2570:
2566:
2563:
2561:
2560:by definition
2558:
2557:
2556:
2553:
2549:
2546:
2545:
2544:
2541:
2539:
2536:
2534:
2531:
2529:
2526:
2524:
2521:
2520:
2517:
2514:
2512:
2508:
2503:
2497:
2493:
2483:
2480:
2478:
2475:
2473:
2470:
2468:
2465:
2463:
2460:
2458:
2455:
2453:
2450:
2448:
2447:KripkeâPlatek
2445:
2443:
2440:
2436:
2433:
2431:
2428:
2427:
2426:
2423:
2422:
2420:
2416:
2408:
2405:
2404:
2403:
2400:
2398:
2395:
2391:
2388:
2387:
2386:
2383:
2381:
2378:
2376:
2373:
2371:
2368:
2366:
2363:
2360:
2356:
2352:
2349:
2345:
2342:
2340:
2337:
2335:
2332:
2331:
2330:
2326:
2323:
2322:
2320:
2318:
2314:
2310:
2302:
2299:
2297:
2294:
2292:
2291:constructible
2289:
2288:
2287:
2284:
2282:
2279:
2277:
2274:
2272:
2269:
2267:
2264:
2262:
2259:
2257:
2254:
2252:
2249:
2247:
2244:
2242:
2239:
2237:
2234:
2232:
2229:
2227:
2224:
2223:
2221:
2219:
2214:
2206:
2203:
2201:
2198:
2196:
2193:
2191:
2188:
2186:
2183:
2181:
2178:
2177:
2175:
2171:
2168:
2166:
2163:
2162:
2161:
2158:
2156:
2153:
2151:
2148:
2146:
2143:
2141:
2137:
2133:
2131:
2128:
2124:
2121:
2120:
2119:
2116:
2115:
2112:
2109:
2107:
2103:
2093:
2090:
2088:
2085:
2083:
2080:
2078:
2075:
2073:
2070:
2068:
2065:
2061:
2058:
2057:
2056:
2053:
2049:
2044:
2043:
2042:
2039:
2038:
2036:
2034:
2030:
2022:
2019:
2017:
2014:
2012:
2009:
2008:
2007:
2004:
2002:
1999:
1997:
1994:
1992:
1989:
1987:
1984:
1982:
1979:
1977:
1974:
1973:
1971:
1969:
1968:Propositional
1965:
1959:
1956:
1954:
1951:
1949:
1946:
1944:
1941:
1939:
1936:
1934:
1931:
1927:
1924:
1923:
1922:
1919:
1917:
1914:
1912:
1909:
1907:
1904:
1902:
1899:
1897:
1896:Logical truth
1894:
1892:
1889:
1888:
1886:
1884:
1880:
1877:
1875:
1871:
1865:
1862:
1860:
1857:
1855:
1852:
1850:
1847:
1845:
1842:
1840:
1836:
1832:
1828:
1826:
1823:
1821:
1818:
1816:
1812:
1809:
1808:
1806:
1804:
1798:
1793:
1787:
1784:
1782:
1779:
1777:
1774:
1772:
1769:
1767:
1764:
1762:
1759:
1757:
1754:
1752:
1749:
1747:
1744:
1742:
1739:
1737:
1734:
1732:
1729:
1725:
1722:
1721:
1720:
1717:
1716:
1714:
1710:
1706:
1699:
1694:
1692:
1687:
1685:
1680:
1679:
1676:
1664:
1661:
1658:
1654:
1651:
1649:
1646:
1644:
1634:
1633:
1630:
1620:
1619:Logic symbols
1617:
1615:
1612:
1610:
1607:
1605:
1602:
1600:
1597:
1596:
1594:
1590:
1584:
1581:
1579:
1576:
1574:
1571:
1570:
1568:
1566:
1562:
1559:
1555:
1549:
1546:
1544:
1541:
1539:
1536:
1534:
1531:
1529:
1526:
1524:
1521:
1519:
1516:
1514:
1511:
1509:
1506:
1504:
1501:
1499:
1498:Logical truth
1496:
1494:
1491:
1489:
1486:
1482:
1479:
1478:
1477:
1474:
1472:
1469:
1467:
1464:
1462:
1459:
1457:
1454:
1450:
1447:
1445:
1442:
1441:
1440:
1439:Contradiction
1437:
1435:
1432:
1430:
1427:
1425:
1422:
1420:
1417:
1416:
1414:
1410:
1400:
1397:
1395:
1392:
1390:
1387:
1385:
1384:Argumentation
1382:
1381:
1379:
1375:
1369:
1368:Philosophical
1366:
1364:
1363:Non-classical
1361:
1359:
1356:
1352:
1349:
1347:
1344:
1343:
1342:
1339:
1337:
1334:
1333:
1331:
1327:
1321:
1318:
1316:
1313:
1311:
1308:
1306:
1303:
1301:
1298:
1296:
1293:
1291:
1288:
1287:
1285:
1281:
1275:
1272:
1270:
1267:
1266:
1263:
1259:
1252:
1247:
1245:
1240:
1238:
1233:
1232:
1229:
1225:
1223:
1218:
1210:
1195:on 2001-05-02
1194:
1190:
1184:
1181:
1170:
1168:9780080933641
1164:
1160:
1159:
1154:
1148:
1145:
1140:
1136:
1132:
1128:
1121:
1118:
1104:
1098:
1095:
1089:
1086:
1075:on 2013-04-03
1074:
1070:
1064:
1061:
1050:
1048:9780521840156
1044:
1040:
1039:
1031:
1028:
1017:
1015:9780521311786
1011:
1007:
1006:
998:
995:
984:
982:9780674319318
978:
974:
973:
965:
962:
959:
953:
950:
947:
942:
939:
932:
928:
925:
923:
920:
918:
915:
913:
910:
908:
905:
903:
900:
898:
895:
894:
890:
888:
886:
882:
881:
875:
871:
867:
863:
857:
853:
845:
843:
841:
837:
833:
829:
824:
820:
816:
815:inconsistency
788:
784:
780:
776:
772:
742:
734:
732:
730:
711:
706:
697:
686:
685:
684:
656:
655:formal system
652:
621:
613:
611:
609:
605:
601:
598:(also called
597:
593:
589:
585:
581:
577:
576:formal system
571:
570:Formal system
563:
561:
559:
555:
551:
547:
546:formal theory
541:
533:
531:
529:
525:
521:
517:
513:
509:
505:
501:
500:ontologically
497:
493:
489:
485:
479:
471:
469:
467:
463:
459:
455:
451:
447:
443:
438:
430:
428:
426:
422:
418:
414:
410:
406:
402:
398:
394:
388:
380:
378:
375:
371:
367:
363:
359:
353:
345:
340:
338:
336:
332:
328:
327:
322:
317:
315:
310:
308:
304:
300:
296:
292:
287:
285:
281:
277:
273:
269:
265:
261:
249:
244:
242:
237:
235:
230:
229:
227:
226:
219:
216:
214:
211:
209:
206:
204:
201:
199:
196:
194:
191:
189:
186:
184:
181:
179:
176:
174:
171:
170:
162:
161:
154:
151:
149:
146:
144:
141:
139:
136:
134:
131:
129:
126:
124:
121:
119:
116:
114:
111:
109:
106:
104:
101:
99:
96:
94:
93:Formal system
91:
90:
82:
81:
78:
74:
70:
69:
64:
63:
57:
53:
49:
45:
41:
37:
32:
19:
3413:
3211:Ultraproduct
3058:Model theory
3023:Independence
2959:Formal proof
2951:Proof theory
2934:
2907:
2864:real numbers
2836:second-order
2747:Substitution
2624:Metalanguage
2565:conservative
2538:Axiom schema
2510:
2482:Constructive
2452:MorseâKelley
2418:Set theories
2397:Aleph number
2390:inaccessible
2296:Grothendieck
2180:intersection
2067:Higher-order
2055:Second-order
2001:Truth tables
1958:Venn diagram
1741:Formal proof
1538:Substitution
1358:Mathematical
1319:
1283:Major fields
1214:
1197:. Retrieved
1193:the original
1183:
1172:. Retrieved
1157:
1147:
1130:
1126:
1120:
1109:. Retrieved
1097:
1088:
1077:. Retrieved
1073:the original
1063:
1052:. Retrieved
1037:
1030:
1019:. Retrieved
1004:
997:
986:. Retrieved
971:
964:
952:
941:
885:metalanguage
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324:
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166:Applications
102:
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67:
3321:Type theory
3269:undecidable
3201:Truth value
3088:equivalence
2767:non-logical
2380:Enumeration
2370:Isomorphism
2317:cardinality
2301:Von Neumann
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2082:Quantifiers
2072:Fixed-point
2041:First-order
1921:Consistency
1906:Proposition
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1153:Barwise, J.
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1434:Consequent
1429:Antecedent
1199:2014-10-15
1174:2014-10-15
1111:2014-10-15
1079:2014-10-15
1054:2014-10-15
1021:2014-10-15
988:2014-10-15
933:References
785:or simply
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2241:Inhabited
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2216:Types of
2200:power set
2170:partition
2087:Predicate
2033:Predicate
1948:Syllogism
1938:Soundness
1911:Inference
1901:Tautology
1803:paradoxes
1614:Fallacies
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1599:Logicians
1533:Statement
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1456:Deduction
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1300:Inference
698:⊢
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3388:Logicism
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