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Tarski himself defined truth for atomic sentences in a variant way that does not use any technical terms from semantics, such as the "expressed by" above. This is because he wanted to define these semantic terms in the context of truth. Therefore it would be circular to use one of them in the
180:
These sentences (1 and 2, etc.) have come to be called the "T-sentences". The reason they look trivial is that the object language and the metalanguage are both
English; here is an example where the object language is German and the metalanguage is English:
199:, including the problem that there is no systematic way of deciding whether a given sentence of a natural language is well-formed, and that a natural language is
683:
999:
1674:
129:). In the following, quoted text is use of the object language, while unquoted text is use of the metalanguage; a quoted sentence (such as "
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898:
235:
containing ÂŹ ("not"), ⧠("and"), âš ("or"), â ("for all"), and â ("there exists"), Tarski's inductive definition of truth looks like this:
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79:. Tarski, in "On the Concept of Truth in Formal Languages" (1935), attempted to formulate a new theory of truth in order to resolve the
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203:(that is, it can describe the semantic characteristics of its own elements). But Tarski's approach was extended by
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rendered in the object language. In this way, the metalanguage can be used to talk about the object language;
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for natural languages, which involves treating "truth" as a primitive, rather than a defined, concept. (See
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354:. A contemporary semantic definition of truth would define truth for the atomic sentences as follows:
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430:. It is a rather controversial point whether Tarski's semantic theory should be counted either as a
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definition of truth itself. Tarski's semantic conception of truth plays an important role in
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It is important to note that as Tarski originally formulated it, this theory applies only to
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83:. In the course of this he made several metamathematical discoveries, most notably
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1935) demanded that the object language be contained in the metalanguage.
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160:, holds that any viable theory of truth must entail, for every sentence "
511:
Hale, Bob; Wright, Crispin; Miller, Alexander, eds. (18 February 2017).
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instead of Tarski's logic of totally defined truth predicates) with the
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conception of truth, which is related in different ways to both the
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225:
58:
31:
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2186:
978:
735:
The
Semantic Conception of Truth and the Foundations of Semantics
695:
The
Semantic Conception of Truth and the Foundations of Semantics
590:
Axiomatic
Theories of Truth (Stanford Encyclopedia of Philosophy)
112:
195:. He gave a number of reasons for not extending his theory to
70:
459:
1975) is based on partial logic (a logic of partially defined
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921:
775:
123:) from the language that one is using to do the talking (the
73:
724:
315:)" is true if, and only if, for all objects x, "Fx" is true.
184:(3) 'Schnee ist weiĂ' is true if and only if snow is white.
164:", a sentence of the following form (known as "form (T)"):
137:
for a sentence, such that this name is simply the sentence
177:(2) 'snow is white' is true if and only if snow is white.
95:. Roughly, this states that a truth-predicate satisfying
744:
99:
for the sentences of a given language cannot be defined
580:
Oxford, England: Oxford
University Press, 2012, p. 110.
30:"Convention T" redirects here. Not to be confused with
578:
Quine versus
Davidson: Truth, Reference, and Meaning.
690:, Indianapolis: Hackett 1983, 2nd edition, 152â278.
49:which holds that truth is a property of sentences.
346:) can be reduced to the truth conditions of their
111:To formulate linguistic theories without semantic
2636:
510:
567:Parts of section is adapted from Kirkham, 1992.
326:)" is true if, and only if, there is an object
27:A theory of truth in the philosophy of language
684:"The Concept of Truth in Formalized Languages"
892:
760:
624:Deflationism and Semantic Theories of Truth
441:
106:
1084:
899:
885:
767:
753:
334:These explain how the truth conditions of
515:A Companion to the Philosophy of Language
739:Philosophy and Phenomenological Research
699:Philosophy and Phenomenological Research
672:, 1975. "Outline of a Theory of Truth".
220:Tarski developed the theory to give an
14:
2637:
906:
880:
748:
537:10.1111/b.9780631213260.1999.00015.x
725:Stanford Encyclopedia of Philosophy
716:Internet Encyclopedia of Philosophy
646:Stanford Encyclopedia of Philosophy
24:
598:
25:
2671:
705:
688:Logic, Semantics, Metamathematics
2618:
350:. The simplest constituents are
207:into an approach to theories of
133:") is always the metalanguage's
87:using the same formal technique
607:and Keith Simmons, eds., 1999.
465:strong Kleene evaluation scheme
448:Truth § Kripke's semantics
85:Tarski's undefinability theorem
69:conceptions, is due to work by
583:
570:
561:
504:
193:semantics of first-order logic
13:
1:
2579:History of mathematical logic
497:
2504:Primitive recursive function
288:" is true if, and only if, "
269:" is true if, and only if, "
254:" is true if, and only if, "
7:
611:. Oxford University Press,
470:
382:of values to the variables
243:" is true if, and only if,
239:(1) A primitive statement "
215:truth-conditional semantics
154:material adequacy condition
10:
2676:
1568:SchröderâBernstein theorem
1295:Monadic predicate calculus
954:Foundations of mathematics
721:Tarski's Truth Definitions
642:Tarski's truth definitions
445:
378:) is true (relative to an
224:of truth as follows. (See
29:
2614:
2601:Philosophy of mathematics
2550:Automated theorem proving
2532:
2427:
2259:
2152:
2004:
1721:
1697:
1675:Von NeumannâBernaysâGödel
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1514:
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1316:
1307:
1234:
1169:
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997:
914:
813:
782:
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492:Triune continuum paradigm
426:and also in contemporary
338:sentences (built up from
52:
712:Semantic Theory of Truth
622:Michael K Butler, 2017.
477:Disquotational principle
453:Kripke's theory of truth
442:Kripke's theory of truth
398:)) if the corresponding
143:Tarski's theory of truth
107:Tarski's theory of truth
39:semantic theory of truth
2251:Self-verifying theories
2072:Tarski's axiomatization
1023:Tarski's undefinability
1018:incompleteness theorems
330:for which "Fx" is true.
93:incompleteness theorems
2625:Mathematics portal
2236:Proof of impossibility
1884:propositional variable
1194:Propositional calculus
428:philosophy of language
47:philosophy of language
2660:Theories of deduction
2494:Kolmogorov complexity
2447:Computably enumerable
2347:Model complete theory
2139:Principia Mathematica
1199:Propositional formula
1028:BanachâTarski paradox
693:Alfred Tarski, 1944.
674:Journal of Philosophy
525:John Wiley & Sons
432:correspondence theory
2442:ChurchâTuring thesis
2429:Computability theory
1638:continuum hypothesis
1156:Square of opposition
1014:Gödel's completeness
626:. Pendlebury Press,
222:inductive definition
2596:Mathematical object
2487:P versus NP problem
2452:Computable function
2246:Reverse mathematics
2172:Logical consequence
2049:primitive recursive
2044:elementary function
1817:Free/bound variable
1670:TarskiâGrothendieck
1189:Logical connectives
1119:Logical equivalence
969:Logical consequence
436:deflationary theory
358:An atomic sentence
2645:Mathematical logic
2394:Transfer principle
2357:Semantics of logic
2342:Categorical theory
2318:Non-standard model
1832:Logical connective
959:Information theory
908:Mathematical logic
658:. Bradford Books,
482:Semantics of logic
2655:Theories of truth
2632:
2631:
2564:Abstract category
2367:Theories of truth
2177:Rule of inference
2167:Natural deduction
2148:
2147:
1693:
1692:
1398:Cartesian product
1303:
1302:
1209:Many-valued logic
1184:Boolean functions
1067:Russell's paradox
1042:diagonal argument
939:First-order logic
874:
873:
656:Theories of Truth
410:expressed by the
197:natural languages
16:(Redirected from
2667:
2623:
2622:
2574:History of logic
2569:Category of sets
2462:Decision problem
2241:Ordinal analysis
2182:Sequent calculus
2080:Boolean algebras
2020:
2019:
1994:
1965:logical/constant
1719:
1718:
1705:
1628:ZermeloâFraenkel
1379:Set operations:
1314:
1313:
1251:
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1081:
1062:LöwenheimâSkolem
949:Formal semantics
901:
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461:truth predicates
352:atomic sentences
189:formal languages
167:(1) "P" is true
156:, also known as
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2555:Category theory
2545:Algebraic logic
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2499:Lambda calculus
2437:Church encoding
2423:
2399:Truth predicate
2255:
2221:Complete theory
2144:
2013:
2009:
2005:
2000:
1992:
1712: and
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1665:New Foundations
1633:axiom of choice
1616:
1578:Gödel numbering
1518: and
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1299:
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1179:Boolean algebra
1165:
1129:Equiconsistency
1094:Classical logic
1071:
1052:Halting problem
1040: and
1016: and
1004: and
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998:Theorems (
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652:Richard Kirkham
605:Simon Blackburn
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599:Further reading
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231:For a language
169:if, and only if
121:object language
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103:that language.
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836:Correspondence
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831:Constructivist
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706:External links
704:
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676:, 72: 690â716.
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638:Wilfrid Hodges
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63:correspondence
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2514:Recursive set
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2412:
2410:
2407:
2405:
2402:
2400:
2397:
2395:
2392:
2390:
2387:
2383:
2380:
2378:
2375:
2373:
2370:
2369:
2368:
2365:
2363:
2360:
2358:
2355:
2353:
2350:
2348:
2345:
2343:
2340:
2336:
2333:
2332:
2331:
2328:
2324:
2323:of arithmetic
2321:
2320:
2319:
2316:
2312:
2309:
2307:
2304:
2302:
2299:
2297:
2294:
2292:
2289:
2288:
2287:
2284:
2280:
2277:
2275:
2272:
2271:
2270:
2267:
2266:
2264:
2262:
2258:
2252:
2249:
2247:
2244:
2242:
2239:
2237:
2234:
2231:
2230:from ZFC
2227:
2224:
2222:
2219:
2213:
2210:
2209:
2208:
2205:
2203:
2200:
2198:
2195:
2194:
2193:
2190:
2188:
2185:
2183:
2180:
2178:
2175:
2173:
2170:
2168:
2165:
2163:
2160:
2159:
2157:
2155:
2151:
2141:
2140:
2136:
2135:
2130:
2129:non-Euclidean
2127:
2123:
2120:
2118:
2115:
2113:
2112:
2108:
2107:
2105:
2102:
2101:
2099:
2095:
2091:
2088:
2086:
2083:
2082:
2081:
2077:
2073:
2070:
2069:
2068:
2064:
2060:
2057:
2055:
2052:
2050:
2047:
2045:
2042:
2040:
2037:
2035:
2032:
2031:
2029:
2025:
2024:
2022:
2017:
2011:
2006:Example
2003:
1995:
1990:
1989:
1988:
1985:
1983:
1980:
1976:
1973:
1971:
1968:
1966:
1963:
1961:
1958:
1957:
1956:
1953:
1951:
1948:
1946:
1943:
1941:
1938:
1934:
1931:
1929:
1926:
1925:
1924:
1921:
1917:
1914:
1912:
1909:
1907:
1904:
1902:
1899:
1898:
1897:
1894:
1892:
1889:
1885:
1882:
1880:
1877:
1875:
1872:
1871:
1870:
1867:
1863:
1860:
1858:
1855:
1853:
1850:
1848:
1845:
1843:
1840:
1838:
1835:
1834:
1833:
1830:
1828:
1825:
1823:
1820:
1818:
1815:
1811:
1808:
1806:
1803:
1801:
1798:
1796:
1793:
1792:
1791:
1788:
1786:
1783:
1781:
1778:
1776:
1773:
1769:
1766:
1764:
1763:by definition
1761:
1760:
1759:
1756:
1752:
1749:
1748:
1747:
1744:
1742:
1739:
1737:
1734:
1732:
1729:
1727:
1724:
1723:
1720:
1717:
1715:
1711:
1706:
1700:
1696:
1686:
1683:
1681:
1678:
1676:
1673:
1671:
1668:
1666:
1663:
1661:
1658:
1656:
1653:
1651:
1650:KripkeâPlatek
1648:
1646:
1643:
1639:
1636:
1634:
1631:
1630:
1629:
1626:
1625:
1623:
1619:
1611:
1608:
1607:
1606:
1603:
1601:
1598:
1594:
1591:
1590:
1589:
1586:
1584:
1581:
1579:
1576:
1574:
1571:
1569:
1566:
1563:
1559:
1555:
1552:
1548:
1545:
1543:
1540:
1538:
1535:
1534:
1533:
1529:
1526:
1525:
1523:
1521:
1517:
1513:
1505:
1502:
1500:
1497:
1495:
1494:constructible
1492:
1491:
1490:
1487:
1485:
1482:
1480:
1477:
1475:
1472:
1470:
1467:
1465:
1462:
1460:
1457:
1455:
1452:
1450:
1447:
1445:
1442:
1440:
1437:
1435:
1432:
1430:
1427:
1426:
1424:
1422:
1417:
1409:
1406:
1404:
1401:
1399:
1396:
1394:
1391:
1389:
1386:
1384:
1381:
1380:
1378:
1374:
1371:
1369:
1366:
1365:
1364:
1361:
1359:
1356:
1354:
1351:
1349:
1346:
1344:
1340:
1336:
1334:
1331:
1327:
1324:
1323:
1322:
1319:
1318:
1315:
1312:
1310:
1306:
1296:
1293:
1291:
1288:
1286:
1283:
1281:
1278:
1276:
1273:
1271:
1268:
1264:
1261:
1260:
1259:
1256:
1252:
1247:
1246:
1245:
1242:
1241:
1239:
1237:
1233:
1225:
1222:
1220:
1217:
1215:
1212:
1211:
1210:
1207:
1205:
1202:
1200:
1197:
1195:
1192:
1190:
1187:
1185:
1182:
1180:
1177:
1176:
1174:
1172:
1171:Propositional
1168:
1162:
1159:
1157:
1154:
1152:
1149:
1147:
1144:
1142:
1139:
1137:
1134:
1130:
1127:
1126:
1125:
1122:
1120:
1117:
1115:
1112:
1110:
1107:
1105:
1102:
1100:
1099:Logical truth
1097:
1095:
1092:
1091:
1089:
1087:
1083:
1080:
1078:
1074:
1068:
1065:
1063:
1060:
1058:
1055:
1053:
1050:
1048:
1045:
1043:
1039:
1035:
1031:
1029:
1026:
1024:
1021:
1019:
1015:
1012:
1011:
1009:
1007:
1001:
996:
990:
987:
985:
982:
980:
977:
975:
972:
970:
967:
965:
962:
960:
957:
955:
952:
950:
947:
945:
942:
940:
937:
935:
932:
928:
925:
924:
923:
920:
919:
917:
913:
909:
902:
897:
895:
890:
888:
883:
882:
879:
867:
864:
862:
859:
857:
854:
852:
849:
847:
844:
842:
839:
837:
834:
832:
829:
827:
824:
822:
819:
818:
816:
812:
806:
803:
801:
798:
796:
793:
791:
788:
787:
785:
781:
777:
770:
765:
763:
758:
756:
751:
750:
747:
740:
736:
732:
731:Alfred Tarski
729:
726:
723:(an entry of
722:
719:
717:
713:
710:
709:
700:
696:
692:
689:
685:
681:
680:Alfred Tarski
678:
675:
671:
668:
665:
664:0-262-61108-2
661:
657:
653:
650:
647:
643:
639:
636:
633:
629:
625:
621:
618:
617:0-19-875250-4
614:
610:
606:
603:
602:
591:
586:
579:
573:
564:
548:
546:9780631213260
542:
538:
534:
530:
526:
522:
517:
516:
507:
503:
493:
490:
488:
485:
483:
480:
478:
475:
474:
468:
466:
462:
458:
454:
449:
439:
437:
433:
429:
425:
416:
413:
409:
405:
401:
396:
392:
385:
381:
376:
372:
365:
361:
357:
356:
355:
353:
349:
345:
341:
337:
329:
325:
321:
317:
314:
310:
306:
303:
299:
295:
291:
287:
283:
279:
276:
272:
268:
264:
260:
257:
253:
249:
246:
242:
238:
237:
236:
234:
229:
227:
223:
218:
216:
212:
211:
206:
202:
198:
194:
190:
185:
182:
178:
175:
174:For example,
172:
170:
165:
163:
159:
155:
150:
148:
147:Alfred Tarski
144:
140:
136:
132:
128:
127:
122:
118:
114:
104:
102:
98:
94:
90:
86:
82:
78:
77:Alfred Tarski
75:
72:
68:
64:
60:
50:
48:
44:
40:
33:
19:
2616:
2414:Ultraproduct
2371:
2261:Model theory
2226:Independence
2162:Formal proof
2154:Proof theory
2137:
2110:
2067:real numbers
2039:second-order
1950:Substitution
1827:Metalanguage
1768:conservative
1741:Axiom schema
1685:Constructive
1655:MorseâKelley
1621:Set theories
1600:Aleph number
1593:inaccessible
1499:Grothendieck
1383:intersection
1270:Higher-order
1258:Second-order
1204:Truth tables
1161:Venn diagram
944:Formal proof
865:
841:Deflationary
800:Truth-bearer
795:Propositions
738:
715:
698:
687:
673:
655:
623:
608:
585:
577:
576:Kemp, Gary.
572:
563:
550:. Retrieved
514:
506:
452:
451:
424:modern logic
420:
414:
394:
390:
383:
374:
370:
363:
359:
348:constituents
335:
333:
327:
323:
319:
312:
308:
301:
297:
293:
289:
285:
281:
274:
270:
266:
262:
258:is not true.
255:
251:
244:
240:
232:
230:
219:
208:
200:
186:
183:
179:
176:
173:
166:
161:
158:Convention T
157:
153:
151:
142:
138:
134:
130:
126:metalanguage
124:
120:
117:liar paradox
115:such as the
110:
100:
97:Convention T
91:used in his
81:liar paradox
67:deflationary
56:
38:
36:
18:Convention T
2524:Type theory
2472:undecidable
2404:Truth value
2291:equivalence
1970:non-logical
1583:Enumeration
1573:Isomorphism
1520:cardinality
1504:Von Neumann
1469:Ultrafilter
1434:Uncountable
1368:equivalence
1285:Quantifiers
1275:Fixed-point
1244:First-order
1124:Consistency
1109:Proposition
1086:Traditional
1057:Lindström's
1047:Compactness
989:Type theory
934:Cardinality
805:Truth-maker
670:Saul Kripke
552:28 February
527:. pp.
523:, England:
521:West Sussex
457:Saul Kripke
344:quantifiers
340:connectives
191:, cf. also
2639:Categories
2335:elementary
2028:arithmetic
1896:Quantifier
1874:functional
1746:Expression
1464:Transitive
1408:identities
1393:complement
1326:hereditary
1309:Set theory
861:Redundancy
632:0993594549
498:References
446:See also:
380:assignment
302:B" is true
298:A" is true
294:B" is true
290:A" is true
275:B" is true
271:A" is true
89:Kurt Gödel
2650:Semantics
2606:Supertask
2509:Recursion
2467:decidable
2301:saturated
2279:of models
2202:deductive
2197:axiomatic
2117:Hilbert's
2104:Euclidean
2085:canonical
2008:axiomatic
1940:Signature
1869:Predicate
1758:Extension
1680:Ackermann
1605:Operation
1484:Universal
1474:Recursive
1449:Singleton
1444:Inhabited
1429:Countable
1419:Types of
1403:power set
1373:partition
1290:Predicate
1236:Predicate
1151:Syllogism
1141:Soundness
1114:Inference
1104:Tautology
1006:paradoxes
856:Pragmatic
851:Pluralist
846:Epistemic
826:Consensus
821:Coherence
790:Statement
733:, 1944.
644:. In the
412:predicate
406:bear the
404:variables
152:Tarski's
113:paradoxes
2591:Logicism
2584:timeline
2560:Concrete
2419:Validity
2389:T-schema
2382:Kripke's
2377:Tarski's
2372:semantic
2362:Strength
2311:submodel
2306:spectrum
2274:function
2122:Tarski's
2111:Elements
2098:geometry
2054:Robinson
1975:variable
1960:function
1933:spectrum
1923:Sentence
1879:variable
1822:Language
1775:Relation
1736:Automata
1726:Alphabet
1710:language
1564:-jection
1542:codomain
1528:Function
1489:Universe
1459:Infinite
1363:Relation
1146:Validity
1136:Argument
1034:theorem,
866:Semantic
814:Theories
682:, 1935.
654:, 1992.
640:, 2001.
558:, p. 326
487:T-schema
471:See also
434:or as a
408:relation
226:T-schema
205:Davidson
74:logician
59:semantic
32:T-schema
2533:Related
2330:Diagram
2228: (
2207:Hilbert
2192:Systems
2187:Theorem
2065:of the
2010:systems
1790:Formula
1785:Grammar
1701: (
1645:General
1358:Forcing
1343:Element
1263:Monadic
1038:paradox
979:Theorem
915:General
783:General
389:, ...,
336:complex
210:meaning
45:in the
2296:finite
2059:Skolem
2012:
1987:Theory
1955:Symbol
1945:String
1928:atomic
1805:ground
1800:closed
1795:atomic
1751:ground
1714:syntax
1610:binary
1537:domain
1454:Finite
1219:finite
1077:Logics
1036:
984:Theory
662:
630:
615:
543:
531:â330.
400:values
318:(6) "â
307:(5) "â
250:(2) "ÂŹ
201:closed
101:within
71:Polish
53:Origin
2286:Model
2034:Peano
1891:Proof
1731:Arity
1660:Naive
1547:image
1479:Fuzzy
1439:Empty
1388:union
1333:Class
974:Model
964:Lemma
922:Axiom
776:Truth
609:Truth
369:,...,
300:and "
296:or ("
280:(4) "
273:and "
261:(3) "
171:, P.
41:is a
2409:Type
2212:list
2016:list
1993:list
1982:Term
1916:rank
1810:open
1704:list
1516:Maps
1421:sets
1280:Free
1250:list
1000:list
927:list
660:ISBN
628:ISBN
613:ISBN
554:2024
541:ISBN
342:and
292:or "
135:name
65:and
57:The
2096:of
2078:of
2026:of
1558:Sur
1532:Map
1339:Ur-
1321:Set
737:.
533:doi
529:309
402:of
217:.)
2641::
2482:NP
2106::
2100::
2030::
1707:),
1562:Bi
1554:In
741:4.
714:,
701:4.
697:.
686:.
539:.
519:.
467:.
438:.
324:Fx
313:Fx
304:).
256:A"
228:)
37:A
2562:/
2477:P
2232:)
2018:)
2014:(
1911:â
1906:!
1901:â
1862:=
1857:â
1852:â
1847:â§
1842:âš
1837:ÂŹ
1560:/
1556:/
1530:/
1341:)
1337:(
1224:â
1214:3
1002:)
900:e
893:t
886:v
768:e
761:t
754:v
727:)
666:.
648:.
634:.
619:.
556:.
535::
455:(
417:.
415:F
395:n
391:x
387:1
384:x
375:n
371:x
367:1
364:x
362:(
360:F
328:x
322:(
320:x
311:(
309:x
286:B
284:âš
282:A
277:.
267:B
265:â§
263:A
252:A
247:.
245:A
241:A
233:L
162:P
145:(
139:P
131:P
34:.
20:)
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