3406:
20:
1206:
1627:
365:
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
814:
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
711:
227:
670:
636:
595:, 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
800:
756:
298:
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.
361:
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
1114:
Wijesekera, Duminda; Ganesh, M.; Srivastava, Jaideep; Nerode, Anil (2001). "Normal forms and syntactic completeness proofs for functional independencies".
1785:
824:
271:
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
1177:
1057:
2460:
802:. In another sense, a formal system is syntactically complete iff no unprovable axiom can be added to it as an axiom without introducing an
2543:
1684:
1081:
Hunter, Geoffrey, Metalogic: An
Introduction to the Metatheory of Standard First-Order Logic, University of California Press, 1971, p. 75.
234:
301:
Syntax is usually associated with the rules (or grammar) governing the composition of texts in a formal language that constitute the
43:. A formal language is identical to the set of its well-formed formulas. The set of well-formed formulas may be broadly divided into
2857:
1651:
3015:
1803:
2870:
2193:
820:
1283:
2875:
2865:
2602:
2455:
1808:
1412:
1799:
3011:
1561:
1155:
1035:
1002:
969:
678:
2353:
3108:
2852:
1677:
1645:
400:). Which strings of symbols are words is determined by the creator of the language, usually by specifying a set of
2413:
2106:
1237:
1210:
1847:
3369:
3071:
2834:
2829:
2654:
2075:
1759:
1566:
905:
868:
314:
3364:
3147:
3064:
2777:
2708:
2585:
1827:
542:
131:
3450:
3289:
3115:
2801:
2435:
2034:
101:
3167:
3162:
2772:
2511:
2440:
1769:
1670:
1641:
528:
186:
181:
86:
1181:
3445:
3096:
2686:
2080:
2048:
1739:
1592:
1061:
946:
3435:
3386:
3335:
3232:
2730:
2691:
2168:
1813:
1571:
1496:
1278:
648:
614:
512:
446:
434:
389:
220:
206:
191:
55:
32:
1842:
781:
737:
3227:
3157:
2696:
2548:
2531:
2254:
1734:
1469:
858:
840:
319:
272:
453:
of the formal language which constitute well formed formulas. However, it does not describe their
3430:
3059:
3036:
2997:
2883:
2824:
2470:
2390:
2234:
2178:
1791:
1553:
3349:
3076:
3054:
3021:
2914:
2760:
2745:
2718:
2669:
2553:
2488:
2313:
2279:
2274:
2148:
1979:
1956:
844:
717:
596:
504:
476:
413:
358:
264:
166:
3279:
3132:
2924:
2642:
2378:
2284:
2143:
2128:
2009:
1984:
1607:
409:
268:
176:
1091:
3252:
3214:
3091:
2895:
2735:
2659:
2637:
2465:
2423:
2322:
2289:
2153:
1941:
1852:
1526:
1372:
729:
323:
8:
3381:
3272:
3257:
3237:
3194:
3081:
3031:
2957:
2902:
2839:
2632:
2627:
2575:
2343:
2332:
2004:
1904:
1832:
1823:
1819:
1754:
1749:
1481:
1464:
1444:
1407:
1356:
1351:
1293:
1230:
915:
807:
584:
576:
442:
397:
302:
283:
201:
116:
40:
3440:
3410:
3179:
3142:
3127:
3120:
3103:
2907:
2889:
2755:
2681:
2664:
2617:
2430:
2339:
2173:
2158:
2118:
2070:
2055:
2043:
1999:
1974:
1744:
1693:
1417:
1346:
1303:
910:
126:
96:
2363:
1127:
778:) iff for each formula A of the language of the system either A or ÂŹA is a theorem of
3405:
3345:
3152:
2962:
2952:
2844:
2725:
2560:
2536:
2317:
2301:
2206:
2183:
2060:
2029:
1994:
1889:
1724:
1631:
1602:
1597:
1587:
1521:
1449:
1334:
1151:
1031:
998:
965:
816:
538:
385:
136:
3359:
3354:
3247:
3204:
3026:
2987:
2982:
2967:
2793:
2750:
2647:
2445:
2395:
1969:
1931:
1536:
1262:
1257:
1123:
876:, which may itself be a formal language, and as such itself is a syntactic entity.
362:
309:
65:
3340:
3330:
3284:
3267:
3222:
3184:
3086:
3006:
2813:
2740:
2713:
2701:
2607:
2521:
2495:
2450:
2418:
2219:
2021:
1964:
1914:
1879:
1837:
1382:
1324:
1145:
1025:
992:
959:
885:
811:
546:
508:
438:
393:
375:
340:
279:
256:
196:
171:
121:
28:
24:
934:
16:
Rules used for constructing, or transforming the symbols and words of a language
3325:
3304:
3262:
3242:
3137:
2992:
2590:
2580:
2570:
2565:
2373:
2249:
2138:
2133:
2111:
1712:
1636:
1329:
1223:
895:
890:
425:
401:
326:. As in mathematical logic, it is independent of semantics and interpretation.
161:
141:
111:
106:
3424:
3299:
2977:
2484:
2269:
2259:
2229:
2214:
1884:
1486:
1427:
643:
558:
287:
260:
81:
857:
to the sentences of a formal system. The study of interpretations is called
3199:
3046:
2947:
2939:
2819:
2767:
2676:
2612:
2595:
2526:
2385:
2244:
1946:
1729:
1476:
1298:
873:
828:
639:
295:
3309:
3189:
2368:
2358:
2305:
1989:
1909:
1894:
1774:
1719:
1511:
1506:
1459:
1141:
854:
803:
516:
500:
466:
350:
515:
of words. Propositions are considered to be syntactic entities and also
23:
This diagram shows the syntactic entities which may be constructed from
2239:
2094:
2065:
1871:
1454:
1422:
1387:
3391:
3294:
2347:
2264:
2224:
2188:
2124:
1936:
1926:
1899:
1662:
1516:
1377:
1288:
853:
of a formal system is the assignment of meanings to the symbols, and
484:
454:
405:
366:
being specified without any reference to any interpretation of them.
3376:
3174:
2622:
2327:
1921:
1437:
488:
450:
36:
19:
2972:
1764:
1501:
1432:
496:
354:
291:
44:
1113:
1339:
1205:
900:
602:
657:
623:
599:
given to it (as being, for instance, a system of arithmetic).
2516:
1862:
1707:
1531:
1246:
592:
480:
318:
refers to the rules governing the composition of well-formed
248:
723:
1491:
492:
346:
1215:
416:
is assigned to it â that is, before it has any meaning.
445:
of a formal language. It is synonymous with the set of
1023:
784:
740:
681:
651:
617:
583:). The deductive apparatus may consist of a set of
412:of any of its expressions; it can exist before any
275:of a language which is concerned with its meaning.
794:
750:
705:
664:
630:
3422:
575:) consists of a formal language together with a
706:{\displaystyle \Gamma \vdash _{\mathrm {F} S}A}
1678:
1231:
1084:
716:Syntactic consequence does not depend on any
228:
603:Syntactic consequence within a formal system
1140:
1030:. Cambridge University Press. p. 189.
957:
827:that is sufficiently powerful, such as the
1870:
1685:
1671:
1238:
1224:
1092:"A Note on Interaction and Incompleteness"
384:is a syntactic entity which consists of a
235:
221:
997:. Cambridge University Press. p. 1.
724:Syntactic completeness of a formal system
404:. Such a language can be defined without
990:
964:. Harvard University Press. p. 82.
396:which are its words (usually called its
18:
1134:
831:, can be both consistent and complete.
3423:
1692:
1170:
1017:
984:
872:. An interpretation is expressed in a
1666:
1219:
951:
638:of a set Đ of formulas if there is a
329:
1178:"syntactic completeness from FOLDOC"
1050:
1413:Analytic and synthetic propositions
1284:Formal semantics (natural language)
1058:"syntactic consequence from FOLDOC"
433:are a precise description of which
13:
1107:
834:
787:
743:
691:
682:
654:
620:
522:
419:
369:
14:
3462:
1199:
1150:. Elsevier Science. p. 236.
1027:The Cambridge Companion to Carnap
1024:Creath, R.; Friedman, M. (2007).
552:
102:Semantics (programming languages)
3404:
1625:
1204:
1122:(1â2). portal.acm.org: 365â405.
1180:. swif.uniba.it. Archived from
1060:. swif.uniba.it. Archived from
665:{\displaystyle {\mathcal {FS}}}
631:{\displaystyle {\mathcal {FS}}}
460:
1147:Handbook of Mathematical Logic
1075:
939:
928:
906:Syntax (programming languages)
821:Gödel's incompleteness theorem
795:{\displaystyle {\mathcal {S}}}
751:{\displaystyle {\mathcal {S}}}
487:. A proposition is identified
255:is anything having to do with
1:
3365:History of mathematical logic
1128:10.1016/S0304-3975(00)00195-X
961:Frege: Philosophy of Language
921:
3290:Primitive recursive function
1116:Theoretical Computer Science
994:Aristotle and Logical Theory
35:may be broadly divided into
7:
879:
529:Theory (mathematical logic)
187:Programming language theory
182:Natural language processing
10:
3467:
2354:SchröderâBernstein theorem
2081:Monadic predicate calculus
1740:Foundations of mathematics
838:
727:
611:within some formal system
556:
526:
464:
423:
373:
338:
334:
3400:
3387:Philosophy of mathematics
3336:Automated theorem proving
3318:
3213:
3045:
2938:
2790:
2507:
2483:
2461:Von NeumannâBernaysâGödel
2406:
2300:
2204:
2102:
2093:
2020:
1955:
1861:
1783:
1700:
1620:
1580:
1552:
1545:
1497:Necessity and sufficiency
1400:
1365:
1317:
1271:
1253:
1245:
207:Automated theorem proving
192:Computational linguistics
863:Giving an interpretation
841:Formal semantics (logic)
3037:Self-verifying theories
2858:Tarski's axiomatization
1809:Tarski's undefinability
1804:incompleteness theorems
457:(i.e. what they mean).
3411:Mathematics portal
3022:Proof of impossibility
2670:propositional variable
1980:Propositional calculus
845:Interpretation (logic)
796:
760:syntactically complete
752:
720:of the formal system.
707:
666:
632:
263:without regard to any
167:Propositional calculus
48:
3280:Kolmogorov complexity
3233:Computably enumerable
3133:Model complete theory
2925:Principia Mathematica
1985:Propositional formula
1814:BanachâTarski paradox
1632:Philosophy portal
1213:at Wikimedia Commons
935:Dictionary Definition
797:
753:
708:
672:of A from the set Đ.
667:
633:
609:syntactic consequence
479:expressing something
177:Mathematical notation
22:
3228:ChurchâTuring thesis
3215:Computability theory
2424:continuum hypothesis
1942:Square of opposition
1800:Gödel's completeness
958:Dummett, M. (1981).
901:Syntax (linguistics)
782:
764:deductively complete
738:
730:Completeness (logic)
679:
649:
615:
585:transformation rules
511:, marks, sounds, or
443:well-formed formulas
398:well-formed formulas
324:programming language
305:of a formal system.
303:well-formed formulas
41:well-formed formulas
3451:Philosophy of logic
3382:Mathematical object
3273:P versus NP problem
3238:Computable function
3032:Reverse mathematics
2958:Logical consequence
2835:primitive recursive
2830:elementary function
2603:Free/bound variable
2456:TarskiâGrothendieck
1975:Logical connectives
1905:Logical equivalence
1755:Logical consequence
1294:Philosophy of logic
916:Well-formed formula
865:is synonymous with
808:propositional logic
806:. Truth-functional
577:deductive apparatus
202:Formal verification
117:Well-formed formula
3180:Transfer principle
3143:Semantics of logic
3128:Categorical theory
3104:Non-standard model
2618:Logical connective
1745:Information theory
1694:Mathematical logic
1593:Rules of inference
1562:Mathematical logic
1304:Semantics of logic
911:Mathematical logic
792:
768:maximally complete
748:
703:
662:
628:
330:Syntactic entities
127:Regular expression
49:
33:strings of symbols
3446:Concepts in logic
3418:
3417:
3350:Abstract category
3153:Theories of truth
2963:Rule of inference
2953:Natural deduction
2934:
2933:
2479:
2478:
2184:Cartesian product
2089:
2088:
1995:Many-valued logic
1970:Boolean functions
1853:Russell's paradox
1828:diagonal argument
1725:First-order logic
1660:
1659:
1616:
1615:
1450:Deductive closure
1396:
1395:
1335:Critical thinking
1209:Media related to
991:Lear, J. (1986).
772:negation complete
607:A formula A is a
363:logical constants
245:
244:
137:Ground expression
97:Semantics (logic)
47:and non-theorems.
3458:
3436:Formal languages
3409:
3408:
3360:History of logic
3355:Category of sets
3248:Decision problem
3027:Ordinal analysis
2968:Sequent calculus
2866:Boolean algebras
2806:
2805:
2780:
2751:logical/constant
2505:
2504:
2491:
2414:ZermeloâFraenkel
2165:Set operations:
2100:
2099:
2037:
1868:
1867:
1848:LöwenheimâSkolem
1735:Formal semantics
1687:
1680:
1673:
1664:
1663:
1630:
1629:
1628:
1550:
1549:
1315:
1314:
1279:Computer science
1240:
1233:
1226:
1217:
1216:
1208:
1193:
1192:
1190:
1189:
1174:
1168:
1167:
1165:
1164:
1138:
1132:
1131:
1111:
1105:
1104:
1102:
1101:
1096:
1088:
1082:
1079:
1073:
1072:
1070:
1069:
1054:
1048:
1047:
1045:
1044:
1021:
1015:
1014:
1012:
1011:
988:
982:
981:
979:
978:
955:
949:
943:
937:
932:
859:formal semantics
825:recursive system
810:and first-order
801:
799:
798:
793:
791:
790:
757:
755:
754:
749:
747:
746:
734:A formal system
712:
710:
709:
704:
699:
698:
694:
671:
669:
668:
663:
661:
660:
637:
635:
634:
629:
627:
626:
581:deductive system
569:logical calculus
507:are patterns of
310:computer science
257:formal languages
237:
230:
223:
66:Formal languages
51:
50:
25:formal languages
3466:
3465:
3461:
3460:
3459:
3457:
3456:
3455:
3421:
3420:
3419:
3414:
3403:
3396:
3341:Category theory
3331:Algebraic logic
3314:
3285:Lambda calculus
3223:Church encoding
3209:
3185:Truth predicate
3041:
3007:Complete theory
2930:
2799:
2795:
2791:
2786:
2778:
2498: and
2494:
2489:
2475:
2451:New Foundations
2419:axiom of choice
2402:
2364:Gödel numbering
2304: and
2296:
2200:
2085:
2035:
2016:
1965:Boolean algebra
1951:
1915:Equiconsistency
1880:Classical logic
1857:
1838:Halting problem
1826: and
1802: and
1790: and
1789:
1784:Theorems (
1779:
1696:
1691:
1661:
1656:
1626:
1624:
1612:
1576:
1567:Boolean algebra
1541:
1392:
1383:Metamathematics
1361:
1313:
1267:
1249:
1244:
1202:
1197:
1196:
1187:
1185:
1176:
1175:
1171:
1162:
1160:
1158:
1139:
1135:
1112:
1108:
1099:
1097:
1094:
1090:
1089:
1085:
1080:
1076:
1067:
1065:
1056:
1055:
1051:
1042:
1040:
1038:
1022:
1018:
1009:
1007:
1005:
989:
985:
976:
974:
972:
956:
952:
947:Geoffrey Hunter
944:
940:
933:
929:
924:
886:Symbol (formal)
882:
867:constructing a
847:
839:Main articles:
837:
835:Interpretations
812:predicate logic
786:
785:
783:
780:
779:
742:
741:
739:
736:
735:
732:
726:
690:
689:
685:
680:
677:
676:
653:
652:
650:
647:
646:
619:
618:
616:
613:
612:
605:
589:inference rules
579:(also called a
567:(also called a
561:
555:
547:formal language
531:
525:
523:Formal theories
505:token instances
469:
463:
431:Formation rules
428:
422:
420:Formation rules
402:formation rules
382:formal language
378:
376:Formal language
372:
370:Formal language
345:A symbol is an
343:
341:Symbol (formal)
337:
332:
241:
212:
211:
197:Syntax analysis
172:Predicate logic
157:
156:
147:
146:
122:Automata theory
77:
76:
17:
12:
11:
5:
3464:
3454:
3453:
3448:
3443:
3438:
3433:
3431:Syntax (logic)
3416:
3415:
3401:
3398:
3397:
3395:
3394:
3389:
3384:
3379:
3374:
3373:
3372:
3362:
3357:
3352:
3343:
3338:
3333:
3328:
3326:Abstract logic
3322:
3320:
3316:
3315:
3313:
3312:
3307:
3305:Turing machine
3302:
3297:
3292:
3287:
3282:
3277:
3276:
3275:
3270:
3265:
3260:
3255:
3245:
3243:Computable set
3240:
3235:
3230:
3225:
3219:
3217:
3211:
3210:
3208:
3207:
3202:
3197:
3192:
3187:
3182:
3177:
3172:
3171:
3170:
3165:
3160:
3150:
3145:
3140:
3138:Satisfiability
3135:
3130:
3125:
3124:
3123:
3113:
3112:
3111:
3101:
3100:
3099:
3094:
3089:
3084:
3079:
3069:
3068:
3067:
3062:
3055:Interpretation
3051:
3049:
3043:
3042:
3040:
3039:
3034:
3029:
3024:
3019:
3009:
3004:
3003:
3002:
3001:
3000:
2990:
2985:
2975:
2970:
2965:
2960:
2955:
2950:
2944:
2942:
2936:
2935:
2932:
2931:
2929:
2928:
2920:
2919:
2918:
2917:
2912:
2911:
2910:
2905:
2900:
2880:
2879:
2878:
2876:minimal axioms
2873:
2862:
2861:
2860:
2849:
2848:
2847:
2842:
2837:
2832:
2827:
2822:
2809:
2807:
2788:
2787:
2785:
2784:
2783:
2782:
2770:
2765:
2764:
2763:
2758:
2753:
2748:
2738:
2733:
2728:
2723:
2722:
2721:
2716:
2706:
2705:
2704:
2699:
2694:
2689:
2679:
2674:
2673:
2672:
2667:
2662:
2652:
2651:
2650:
2645:
2640:
2635:
2630:
2625:
2615:
2610:
2605:
2600:
2599:
2598:
2593:
2588:
2583:
2573:
2568:
2566:Formation rule
2563:
2558:
2557:
2556:
2551:
2541:
2540:
2539:
2529:
2524:
2519:
2514:
2508:
2502:
2485:Formal systems
2481:
2480:
2477:
2476:
2474:
2473:
2468:
2463:
2458:
2453:
2448:
2443:
2438:
2433:
2428:
2427:
2426:
2421:
2410:
2408:
2404:
2403:
2401:
2400:
2399:
2398:
2388:
2383:
2382:
2381:
2374:Large cardinal
2371:
2366:
2361:
2356:
2351:
2337:
2336:
2335:
2330:
2325:
2310:
2308:
2298:
2297:
2295:
2294:
2293:
2292:
2287:
2282:
2272:
2267:
2262:
2257:
2252:
2247:
2242:
2237:
2232:
2227:
2222:
2217:
2211:
2209:
2202:
2201:
2199:
2198:
2197:
2196:
2191:
2186:
2181:
2176:
2171:
2163:
2162:
2161:
2156:
2146:
2141:
2139:Extensionality
2136:
2134:Ordinal number
2131:
2121:
2116:
2115:
2114:
2103:
2097:
2091:
2090:
2087:
2086:
2084:
2083:
2078:
2073:
2068:
2063:
2058:
2053:
2052:
2051:
2041:
2040:
2039:
2026:
2024:
2018:
2017:
2015:
2014:
2013:
2012:
2007:
2002:
1992:
1987:
1982:
1977:
1972:
1967:
1961:
1959:
1953:
1952:
1950:
1949:
1944:
1939:
1934:
1929:
1924:
1919:
1918:
1917:
1907:
1902:
1897:
1892:
1887:
1882:
1876:
1874:
1865:
1859:
1858:
1856:
1855:
1850:
1845:
1840:
1835:
1830:
1818:Cantor's
1816:
1811:
1806:
1796:
1794:
1781:
1780:
1778:
1777:
1772:
1767:
1762:
1757:
1752:
1747:
1742:
1737:
1732:
1727:
1722:
1717:
1716:
1715:
1704:
1702:
1698:
1697:
1690:
1689:
1682:
1675:
1667:
1658:
1657:
1655:
1654:
1649:
1639:
1634:
1621:
1618:
1617:
1614:
1613:
1611:
1610:
1605:
1600:
1595:
1590:
1584:
1582:
1578:
1577:
1575:
1574:
1569:
1564:
1558:
1556:
1547:
1543:
1542:
1540:
1539:
1534:
1529:
1524:
1519:
1514:
1509:
1504:
1499:
1494:
1489:
1484:
1479:
1474:
1473:
1472:
1462:
1457:
1452:
1447:
1442:
1441:
1440:
1435:
1425:
1420:
1415:
1410:
1404:
1402:
1398:
1397:
1394:
1393:
1391:
1390:
1385:
1380:
1375:
1369:
1367:
1363:
1362:
1360:
1359:
1354:
1349:
1344:
1343:
1342:
1337:
1327:
1321:
1319:
1312:
1311:
1306:
1301:
1296:
1291:
1286:
1281:
1275:
1273:
1269:
1268:
1266:
1265:
1260:
1254:
1251:
1250:
1243:
1242:
1235:
1228:
1220:
1211:Syntax (logic)
1201:
1200:External links
1198:
1195:
1194:
1169:
1156:
1133:
1106:
1083:
1074:
1049:
1036:
1016:
1003:
983:
970:
950:
938:
926:
925:
923:
920:
919:
918:
913:
908:
903:
898:
896:Formal grammar
893:
891:Formation rule
888:
881:
878:
851:interpretation
836:
833:
823:shows that no
789:
745:
728:Main article:
725:
722:
718:interpretation
714:
713:
702:
697:
693:
688:
684:
659:
656:
625:
622:
604:
601:
597:interpretation
591:) or a set of
573:logical system
557:Main article:
554:
553:Formal systems
551:
527:Main article:
524:
521:
465:Main article:
462:
459:
426:Formation rule
424:Main article:
421:
418:
414:interpretation
374:Main article:
371:
368:
339:Main article:
336:
333:
331:
328:
265:interpretation
261:formal systems
243:
242:
240:
239:
232:
225:
217:
214:
213:
210:
209:
204:
199:
194:
189:
184:
179:
174:
169:
164:
162:Formal methods
158:
154:
153:
152:
149:
148:
145:
144:
142:Atomic formula
139:
134:
129:
124:
119:
114:
112:Formation rule
109:
107:Formal grammar
104:
99:
94:
89:
84:
78:
74:
73:
72:
69:
68:
62:
61:
15:
9:
6:
4:
3:
2:
3463:
3452:
3449:
3447:
3444:
3442:
3439:
3437:
3434:
3432:
3429:
3428:
3426:
3413:
3412:
3407:
3399:
3393:
3390:
3388:
3385:
3383:
3380:
3378:
3375:
3371:
3368:
3367:
3366:
3363:
3361:
3358:
3356:
3353:
3351:
3347:
3344:
3342:
3339:
3337:
3334:
3332:
3329:
3327:
3324:
3323:
3321:
3317:
3311:
3308:
3306:
3303:
3301:
3300:Recursive set
3298:
3296:
3293:
3291:
3288:
3286:
3283:
3281:
3278:
3274:
3271:
3269:
3266:
3264:
3261:
3259:
3256:
3254:
3251:
3250:
3249:
3246:
3244:
3241:
3239:
3236:
3234:
3231:
3229:
3226:
3224:
3221:
3220:
3218:
3216:
3212:
3206:
3203:
3201:
3198:
3196:
3193:
3191:
3188:
3186:
3183:
3181:
3178:
3176:
3173:
3169:
3166:
3164:
3161:
3159:
3156:
3155:
3154:
3151:
3149:
3146:
3144:
3141:
3139:
3136:
3134:
3131:
3129:
3126:
3122:
3119:
3118:
3117:
3114:
3110:
3109:of arithmetic
3107:
3106:
3105:
3102:
3098:
3095:
3093:
3090:
3088:
3085:
3083:
3080:
3078:
3075:
3074:
3073:
3070:
3066:
3063:
3061:
3058:
3057:
3056:
3053:
3052:
3050:
3048:
3044:
3038:
3035:
3033:
3030:
3028:
3025:
3023:
3020:
3017:
3016:from ZFC
3013:
3010:
3008:
3005:
2999:
2996:
2995:
2994:
2991:
2989:
2986:
2984:
2981:
2980:
2979:
2976:
2974:
2971:
2969:
2966:
2964:
2961:
2959:
2956:
2954:
2951:
2949:
2946:
2945:
2943:
2941:
2937:
2927:
2926:
2922:
2921:
2916:
2915:non-Euclidean
2913:
2909:
2906:
2904:
2901:
2899:
2898:
2894:
2893:
2891:
2888:
2887:
2885:
2881:
2877:
2874:
2872:
2869:
2868:
2867:
2863:
2859:
2856:
2855:
2854:
2850:
2846:
2843:
2841:
2838:
2836:
2833:
2831:
2828:
2826:
2823:
2821:
2818:
2817:
2815:
2811:
2810:
2808:
2803:
2797:
2792:Example
2789:
2781:
2776:
2775:
2774:
2771:
2769:
2766:
2762:
2759:
2757:
2754:
2752:
2749:
2747:
2744:
2743:
2742:
2739:
2737:
2734:
2732:
2729:
2727:
2724:
2720:
2717:
2715:
2712:
2711:
2710:
2707:
2703:
2700:
2698:
2695:
2693:
2690:
2688:
2685:
2684:
2683:
2680:
2678:
2675:
2671:
2668:
2666:
2663:
2661:
2658:
2657:
2656:
2653:
2649:
2646:
2644:
2641:
2639:
2636:
2634:
2631:
2629:
2626:
2624:
2621:
2620:
2619:
2616:
2614:
2611:
2609:
2606:
2604:
2601:
2597:
2594:
2592:
2589:
2587:
2584:
2582:
2579:
2578:
2577:
2574:
2572:
2569:
2567:
2564:
2562:
2559:
2555:
2552:
2550:
2549:by definition
2547:
2546:
2545:
2542:
2538:
2535:
2534:
2533:
2530:
2528:
2525:
2523:
2520:
2518:
2515:
2513:
2510:
2509:
2506:
2503:
2501:
2497:
2492:
2486:
2482:
2472:
2469:
2467:
2464:
2462:
2459:
2457:
2454:
2452:
2449:
2447:
2444:
2442:
2439:
2437:
2436:KripkeâPlatek
2434:
2432:
2429:
2425:
2422:
2420:
2417:
2416:
2415:
2412:
2411:
2409:
2405:
2397:
2394:
2393:
2392:
2389:
2387:
2384:
2380:
2377:
2376:
2375:
2372:
2370:
2367:
2365:
2362:
2360:
2357:
2355:
2352:
2349:
2345:
2341:
2338:
2334:
2331:
2329:
2326:
2324:
2321:
2320:
2319:
2315:
2312:
2311:
2309:
2307:
2303:
2299:
2291:
2288:
2286:
2283:
2281:
2280:constructible
2278:
2277:
2276:
2273:
2271:
2268:
2266:
2263:
2261:
2258:
2256:
2253:
2251:
2248:
2246:
2243:
2241:
2238:
2236:
2233:
2231:
2228:
2226:
2223:
2221:
2218:
2216:
2213:
2212:
2210:
2208:
2203:
2195:
2192:
2190:
2187:
2185:
2182:
2180:
2177:
2175:
2172:
2170:
2167:
2166:
2164:
2160:
2157:
2155:
2152:
2151:
2150:
2147:
2145:
2142:
2140:
2137:
2135:
2132:
2130:
2126:
2122:
2120:
2117:
2113:
2110:
2109:
2108:
2105:
2104:
2101:
2098:
2096:
2092:
2082:
2079:
2077:
2074:
2072:
2069:
2067:
2064:
2062:
2059:
2057:
2054:
2050:
2047:
2046:
2045:
2042:
2038:
2033:
2032:
2031:
2028:
2027:
2025:
2023:
2019:
2011:
2008:
2006:
2003:
2001:
1998:
1997:
1996:
1993:
1991:
1988:
1986:
1983:
1981:
1978:
1976:
1973:
1971:
1968:
1966:
1963:
1962:
1960:
1958:
1957:Propositional
1954:
1948:
1945:
1943:
1940:
1938:
1935:
1933:
1930:
1928:
1925:
1923:
1920:
1916:
1913:
1912:
1911:
1908:
1906:
1903:
1901:
1898:
1896:
1893:
1891:
1888:
1886:
1885:Logical truth
1883:
1881:
1878:
1877:
1875:
1873:
1869:
1866:
1864:
1860:
1854:
1851:
1849:
1846:
1844:
1841:
1839:
1836:
1834:
1831:
1829:
1825:
1821:
1817:
1815:
1812:
1810:
1807:
1805:
1801:
1798:
1797:
1795:
1793:
1787:
1782:
1776:
1773:
1771:
1768:
1766:
1763:
1761:
1758:
1756:
1753:
1751:
1748:
1746:
1743:
1741:
1738:
1736:
1733:
1731:
1728:
1726:
1723:
1721:
1718:
1714:
1711:
1710:
1709:
1706:
1705:
1703:
1699:
1695:
1688:
1683:
1681:
1676:
1674:
1669:
1668:
1665:
1653:
1650:
1647:
1643:
1640:
1638:
1635:
1633:
1623:
1622:
1619:
1609:
1608:Logic symbols
1606:
1604:
1601:
1599:
1596:
1594:
1591:
1589:
1586:
1585:
1583:
1579:
1573:
1570:
1568:
1565:
1563:
1560:
1559:
1557:
1555:
1551:
1548:
1544:
1538:
1535:
1533:
1530:
1528:
1525:
1523:
1520:
1518:
1515:
1513:
1510:
1508:
1505:
1503:
1500:
1498:
1495:
1493:
1490:
1488:
1487:Logical truth
1485:
1483:
1480:
1478:
1475:
1471:
1468:
1467:
1466:
1463:
1461:
1458:
1456:
1453:
1451:
1448:
1446:
1443:
1439:
1436:
1434:
1431:
1430:
1429:
1428:Contradiction
1426:
1424:
1421:
1419:
1416:
1414:
1411:
1409:
1406:
1405:
1403:
1399:
1389:
1386:
1384:
1381:
1379:
1376:
1374:
1373:Argumentation
1371:
1370:
1368:
1364:
1358:
1357:Philosophical
1355:
1353:
1352:Non-classical
1350:
1348:
1345:
1341:
1338:
1336:
1333:
1332:
1331:
1328:
1326:
1323:
1322:
1320:
1316:
1310:
1307:
1305:
1302:
1300:
1297:
1295:
1292:
1290:
1287:
1285:
1282:
1280:
1277:
1276:
1274:
1270:
1264:
1261:
1259:
1256:
1255:
1252:
1248:
1241:
1236:
1234:
1229:
1227:
1222:
1221:
1218:
1214:
1212:
1207:
1184:on 2001-05-02
1183:
1179:
1173:
1159:
1157:9780080933641
1153:
1149:
1148:
1143:
1137:
1129:
1125:
1121:
1117:
1110:
1093:
1087:
1078:
1064:on 2013-04-03
1063:
1059:
1053:
1039:
1037:9780521840156
1033:
1029:
1028:
1020:
1006:
1004:9780521311786
1000:
996:
995:
987:
973:
971:9780674319318
967:
963:
962:
954:
948:
942:
936:
931:
927:
917:
914:
912:
909:
907:
904:
902:
899:
897:
894:
892:
889:
887:
884:
883:
877:
875:
871:
870:
864:
860:
856:
852:
846:
842:
832:
830:
826:
822:
818:
813:
809:
805:
804:inconsistency
777:
773:
769:
765:
761:
731:
721:
719:
700:
695:
686:
675:
674:
673:
645:
644:formal system
641:
610:
600:
598:
594:
590:
587:(also called
586:
582:
578:
574:
570:
566:
565:formal system
560:
559:Formal system
550:
548:
544:
540:
536:
535:formal theory
530:
520:
518:
514:
510:
506:
502:
498:
494:
490:
489:ontologically
486:
482:
478:
474:
468:
458:
456:
452:
448:
444:
440:
436:
432:
427:
417:
415:
411:
407:
403:
399:
395:
391:
387:
383:
377:
367:
364:
360:
356:
352:
348:
342:
327:
325:
321:
317:
316:
311:
306:
304:
299:
297:
293:
289:
285:
281:
276:
274:
270:
266:
262:
258:
254:
250:
238:
233:
231:
226:
224:
219:
218:
216:
215:
208:
205:
203:
200:
198:
195:
193:
190:
188:
185:
183:
180:
178:
175:
173:
170:
168:
165:
163:
160:
159:
151:
150:
143:
140:
138:
135:
133:
130:
128:
125:
123:
120:
118:
115:
113:
110:
108:
105:
103:
100:
98:
95:
93:
90:
88:
85:
83:
82:Formal system
80:
79:
71:
70:
67:
64:
63:
59:
58:
53:
52:
46:
42:
38:
34:
30:
26:
21:
3402:
3200:Ultraproduct
3047:Model theory
3012:Independence
2948:Formal proof
2940:Proof theory
2923:
2896:
2853:real numbers
2825:second-order
2736:Substitution
2613:Metalanguage
2554:conservative
2527:Axiom schema
2499:
2471:Constructive
2441:MorseâKelley
2407:Set theories
2386:Aleph number
2379:inaccessible
2285:Grothendieck
2169:intersection
2056:Higher-order
2044:Second-order
1990:Truth tables
1947:Venn diagram
1730:Formal proof
1527:Substitution
1347:Mathematical
1308:
1272:Major fields
1203:
1186:. Retrieved
1182:the original
1172:
1161:. Retrieved
1146:
1136:
1119:
1115:
1109:
1098:. Retrieved
1086:
1077:
1066:. Retrieved
1062:the original
1052:
1041:. Retrieved
1026:
1019:
1008:. Retrieved
993:
986:
975:. Retrieved
960:
953:
941:
930:
874:metalanguage
866:
862:
855:truth values
850:
848:
829:Peano axioms
775:
771:
767:
763:
759:
733:
715:
608:
606:
588:
580:
572:
568:
564:
562:
534:
532:
517:truthbearers
472:
470:
461:Propositions
430:
429:
381:
379:
344:
313:
307:
300:
277:
252:
246:
155:Applications
91:
75:Key concepts
56:
3310:Type theory
3258:undecidable
3190:Truth value
3077:equivalence
2756:non-logical
2369:Enumeration
2359:Isomorphism
2306:cardinality
2290:Von Neumann
2255:Ultrafilter
2220:Uncountable
2154:equivalence
2071:Quantifiers
2061:Fixed-point
2030:First-order
1910:Consistency
1895:Proposition
1872:Traditional
1843:Lindström's
1833:Compactness
1775:Type theory
1720:Cardinality
1642:WikiProject
1512:Proposition
1507:Probability
1460:Description
1401:Foundations
1142:Barwise, J.
945:Metalogic,
817:tautologies
501:abstraction
473:proposition
467:Proposition
351:abstraction
320:expressions
312:, the term
3425:Categories
3121:elementary
2814:arithmetic
2682:Quantifier
2660:functional
2532:Expression
2250:Transitive
2194:identities
2179:complement
2112:hereditary
2095:Set theory
1572:Set theory
1470:Linguistic
1465:Entailment
1455:Definition
1423:Consequent
1418:Antecedent
1188:2014-10-15
1163:2014-10-15
1100:2014-10-15
1068:2014-10-15
1043:2014-10-15
1010:2014-10-15
977:2014-10-15
922:References
774:or simply
640:derivation
388:of finite
132:Production
3441:Metalogic
3392:Supertask
3295:Recursion
3253:decidable
3087:saturated
3065:of models
2988:deductive
2983:axiomatic
2903:Hilbert's
2890:Euclidean
2871:canonical
2794:axiomatic
2726:Signature
2655:Predicate
2544:Extension
2466:Ackermann
2391:Operation
2270:Universal
2260:Recursive
2235:Singleton
2230:Inhabited
2215:Countable
2205:Types of
2189:power set
2159:partition
2076:Predicate
2022:Predicate
1937:Syllogism
1927:Soundness
1900:Inference
1890:Tautology
1792:paradoxes
1603:Fallacies
1598:Paradoxes
1588:Logicians
1522:Statement
1517:Reference
1482:Induction
1445:Deduction
1408:Abduction
1378:Metalogic
1325:Classical
1289:Inference
687:⊢
683:Γ
543:sentences
455:semantics
449:over the
406:reference
273:semantics
3377:Logicism
3370:timeline
3346:Concrete
3205:Validity
3175:T-schema
3168:Kripke's
3163:Tarski's
3158:semantic
3148:Strength
3097:submodel
3092:spectrum
3060:function
2908:Tarski's
2897:Elements
2884:geometry
2840:Robinson
2761:variable
2746:function
2719:spectrum
2709:Sentence
2665:variable
2608:Language
2561:Relation
2522:Automata
2512:Alphabet
2496:language
2350:-jection
2328:codomain
2314:Function
2275:Universe
2245:Infinite
2149:Relation
1932:Validity
1922:Argument
1820:theorem,
1637:Category
1537:Validity
1438:Antinomy
1366:Theories
1330:Informal
1144:(1982).
880:See also
776:complete
477:sentence
451:alphabet
441:are the
410:meanings
292:theorems
284:formulas
87:Alphabet
57:a series
54:Part of
45:theorems
37:nonsense
3319:Related
3116:Diagram
3014: (
2993:Hilbert
2978:Systems
2973:Theorem
2851:of the
2796:systems
2576:Formula
2571:Grammar
2487: (
2431:General
2144:Forcing
2129:Element
2049:Monadic
1824:paradox
1765:Theorem
1701:General
1652:changes
1644: (
1502:Premise
1433:Paradox
1263:History
1258:Outline
571:, or a
513:strings
509:symbols
497:concept
447:strings
439:symbols
435:strings
408:to any
394:symbols
390:strings
355:concept
335:Symbols
288:systems
280:symbols
269:meaning
29:symbols
3082:finite
2845:Skolem
2798:
2773:Theory
2741:Symbol
2731:String
2714:atomic
2591:ground
2586:closed
2581:atomic
2537:ground
2500:syntax
2396:binary
2323:domain
2240:Finite
2005:finite
1863:Logics
1822:
1770:Theory
1554:topics
1340:Reason
1318:Logics
1309:Syntax
1154:
1034:
1001:
968:
762:(also
593:axioms
503:whose
491:as an
359:tokens
315:syntax
296:proofs
253:syntax
92:Syntax
27:. The
3072:Model
2820:Peano
2677:Proof
2517:Arity
2446:Naive
2333:image
2265:Fuzzy
2225:Empty
2174:union
2119:Class
1760:Model
1750:Lemma
1708:Axiom
1581:other
1546:Lists
1532:Truth
1299:Proof
1247:Logic
1095:(PDF)
869:model
545:in a
537:is a
485:false
475:is a
322:in a
249:logic
3195:Type
2998:list
2802:list
2779:list
2768:Term
2702:rank
2596:open
2490:list
2302:Maps
2207:sets
2066:Free
2036:list
1786:list
1713:list
1646:talk
1492:Name
1477:Form
1152:ISBN
1032:ISBN
999:ISBN
966:ISBN
843:and
493:idea
481:true
347:idea
294:and
278:The
39:and
31:and
2882:of
2864:of
2812:of
2344:Sur
2318:Map
2125:Ur-
2107:Set
1388:Set
1124:doi
1120:266
849:An
819:).
758:is
642:in
541:of
539:set
499:or
483:or
437:of
392:of
386:set
353:or
308:In
267:or
259:or
247:In
3427::
3268:NP
2892::
2886::
2816::
2493:),
2348:Bi
2340:In
1118:.
861:.
770:,
766:,
563:A
549:.
533:A
519:.
495:,
471:A
380:A
357:,
349:,
290:,
286:,
282:,
251:,
60:on
3348:/
3263:P
3018:)
2804:)
2800:(
2697:â
2692:!
2687:â
2648:=
2643:â
2638:â
2633:â§
2628:âš
2623:ÂŹ
2346:/
2342:/
2316:/
2127:)
2123:(
2010:â
2000:3
1788:)
1686:e
1679:t
1672:v
1648:)
1239:e
1232:t
1225:v
1191:.
1166:.
1130:.
1126::
1103:.
1071:.
1046:.
1013:.
980:.
788:S
744:S
701:A
696:S
692:F
658:S
655:F
624:S
621:F
236:e
229:t
222:v
Text is available under the Creative Commons Attribution-ShareAlike License. Additional terms may apply.
