3783:
1342:
Leibniz had no influence on the rise of algebraic logic because his logical writings were little studied before the
Parkinson and Loemker translations. Our present understanding of Leibniz as a logician stems mainly from the work of Wolfgang Lenzen, summarized in
1435:, quote: By retaining the realistic presuppositions of the plain man concerning the genuineness of external reality, Peirce is able to reinforce the precarious defenses of a conventionalistic theory of nature with the powerful armament of common-sense realism.
748:
structure, based in set theory, was transcended by Tarski with axioms describing it. Then he asked if every algebra satisfying the axioms could be represented by a set relation. The negative answer opened the frontier of
1921:
but without much background in order theory and/or universal algebra; the book covers these prerequisites at length. This book however has been criticized for poor and sometimes incorrect presentation of AAL results.
626:
579:
735:
676:
1315:
relation. Riguet also extended ordering to the heterogeneous context by his note that a staircase logical matrix has a complement that is also a staircase, and that the theorem of
469:
407:
333:
528:
368:
292:
The description of the key binary relation properties has been formulated with the calculus of relations. The univalence property of functions describes a relation
1311:
used the algebraic logic to advance useful concepts: he extended the concept of an equivalence relation (on a set) to the heterogeneous case with the notion of a
1800:
2162:
1261:, "The years 1920-40 saw, in particular in the Polish school of logic, researches on non-classical propositional calculi conducted by what is termed the
847:
or mathematical systems, and the algebraic structure which are its models are shown on the right in the same row. Some of these structures are either
2837:
2920:
2061:
62:
1948:
50:
3234:
1851:
161:"The basic operations are set-theoretic union, intersection and complementation, the relative multiplication, and conversion."
1399:
Bjarni JĂłnsson (1984). "Maximal
Algebras of Binary Relations". In Kenneth I. Appel; John G. Ratcliffe; Paul E. Schupp (eds.).
3392:
1941:
1910:
1500:
1476:
1412:
17:
2180:
3247:
2570:
2007:
1448:
Discrete
Mathematics for Computer Scientists, page 54, EATCS Monographs on Theoretical Computer Science, Springer Verlag,
1821:
Czelakowski, Janusz (2003). "Review: Algebraic
Methods in Philosophical Logic by J. Michael Dunn and Gary M. Hardegree".
1776:
3252:
3242:
2979:
2832:
2185:
1192:
39:
2176:
3388:
1783:
1453:
584:
537:
146:
of information, so relations are studied with
Boolean arithmetic. Elements of the power set are partially ordered by
2730:
1729:
1265:
method. Since logical matrices are certain abstract algebras, this led to the use of an algebraic method in logic."
1169:
860:
494:
3485:
3229:
2054:
687:
180:
matrix. A relation obtained as the composition of two others is then represented by the logical matrix obtained by
1275:. The founders of model theory, Ernst Schröder and Leopold Loewenheim, were logicians in the algebraic tradition.
1173:
498:
80:(AAL) focus on the process of algebraization itself, like classifying various forms of algebraizability using the
2790:
2483:
2224:
1061:
Algebraic logic is, perhaps, the oldest approach to formal logic, arguably beginning with a number of memoranda
45:
What is now usually called classical algebraic logic focuses on the identification and algebraic description of
3746:
3448:
3211:
3206:
3031:
2452:
2136:
2018:
1806:
1639:
3741:
3524:
3441:
3154:
3085:
2962:
2204:
1756:
1580:(1991) "The Origin of Relation Algebras in the Development and Axiomatization of the Calculus of Relations",
1432:
1404:
1300:
909:
797:
631:
1188:
3666:
3492:
3178:
2812:
2411:
1674:
1492:
58:
2002:& N. Houser (1991) "Nineteenth Century Roots of Algebraic Logic and Universal Algebra", pages 1â36 in
3544:
3539:
3149:
2888:
2817:
2146:
2047:
1695:
848:
435:
373:
299:
3473:
3063:
2457:
2425:
2116:
1469:
793:
3812:
3763:
3712:
3609:
3107:
3068:
2545:
2190:
1608:
1062:
2219:
1932:, IstvĂĄn NĂ©meti and IldikĂł Sain (2001). "Algebraic logic". In Dov M. Gabbay, Franz Guenthner (ed.).
1069:
in 1918. But nearly all of
Leibniz's known work on algebraic logic was published only in 1903 after
1065:
wrote in the 1680s, some of which were published in the 19th century and translated into
English by
49:
appropriate for the study of various logics (in the form of classes of algebras that constitute the
3807:
3604:
3534:
3073:
2925:
2908:
2631:
2111:
1976:
1125:
1117:
785:
750:
155:
77:
3436:
3413:
3374:
3260:
3201:
2847:
2767:
2611:
2555:
2168:
1336:
1164:. The "calculus of relations" is arguably the culmination of Leibniz's approach to logic. At the
1013:
993:
988:
807:
253:
504:
3726:
3453:
3431:
3398:
3291:
3137:
3122:
3095:
3046:
2930:
2865:
2690:
2656:
2651:
2525:
2356:
2333:
1254:, and Tarski revived interest in relations with his 1941 essay "On the Calculus of Relations".
1204:
1097:
1021:
486:
429:
147:
119:
3656:
3509:
3301:
3019:
2755:
2661:
2520:
2505:
2386:
2361:
1989:
1962:
1713:
1517:
1250:
967:
789:
181:
1657:
The
Mathematical Analysis of Logic, Being an Essay towards a Calculus of Deductive Reasoning
1655:
3629:
3591:
3468:
3272:
3112:
3036:
3014:
2842:
2800:
2699:
2666:
2530:
2318:
2229:
1955:
1690:
1546:
1165:
1105:
1005:
929:
892:
346:
1241:
1088:
Modern mathematical logic began in 1847, with two pamphlets whose respective authors were
942:
172:
that always exists, contrary to function theory. A given relation may be represented by a
8:
3758:
3649:
3634:
3614:
3571:
3458:
3408:
3334:
3279:
3216:
3009:
3004:
2952:
2720:
2709:
2381:
2281:
2209:
2200:
2196:
2131:
2126:
1918:
1566:
821:
762:
1856:
Handbook of the
History of Logic, Vol. 3: The Rise of Modern Logic from Leibniz to Frege
3787:
3556:
3519:
3504:
3497:
3480:
3284:
3266:
3132:
3058:
3041:
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2807:
2716:
2550:
2535:
2495:
2447:
2432:
2420:
2376:
2351:
2121:
2070:
1887:
Philosophiegeschichte und logische
Analyse / Logical Analysis and History of Philosophy
1838:
1669:
1534:
1101:
1093:
1056:
955:
835:
The rules of proof are the substitution of equals for equals, and uniform replacement.
811:
490:
280:. The art of putting the right question to elicit a sufficient answer is recognized in
31:
2740:
1734:
678:
Schmidt uses this principle as "slipping below negation from the left". For a mapping
3782:
3722:
3529:
3339:
3329:
3221:
3102:
2937:
2913:
2694:
2678:
2583:
2560:
2437:
2406:
2371:
2266:
2101:
2014:
1937:
1906:
1830:
1779:
1496:
1472:
1449:
1408:
1379:
1369:
1294:
1038:
1026:
1000:
874:
867:
825:
197:
169:
1929:
1771:(1974), "Post Algebras as Semantic Foundations of m-valued Logics", pages 92â142 in
3736:
3731:
3624:
3581:
3403:
3364:
3359:
3344:
3170:
3127:
3024:
2822:
2772:
2346:
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1743:
1613:
1526:
1374:
1288:
1180:
1042:
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984:
976:
917:
888:
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852:
421:
81:
54:
3717:
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3644:
3599:
3561:
3463:
3383:
3190:
3117:
3090:
3078:
2984:
2898:
2872:
2827:
2795:
2596:
2398:
2341:
2291:
2256:
2214:
1542:
1512:
1364:
1222:
1161:
1137:
1121:
935:
922:
766:
475:
471:
281:
97:
3702:
3681:
3639:
3619:
3514:
3369:
2967:
2957:
2947:
2942:
2876:
2750:
2626:
2515:
2510:
2488:
2089:
1999:
1768:
1708:
1631:
1468:, Encyclopedia of Mathematics and its Applications, vol. 132, pages 49 and 57,
1308:
1262:
1258:
1245:
1226:
1218:
1200:
1113:
1070:
1066:
844:
425:
173:
1958:". Stanford Encyclopedia of Philosophy. Mainly about abstract algebraic logic.
1187:
as his version of pure mathematics based on the operations of the calculus as
3801:
3676:
3354:
2861:
2646:
2636:
2606:
2591:
2261:
2030:
1968:
1928:
1834:
1739:
1735:
Algebra der Logik (Exakte Logik) Dritter Band, Algebra und Logik der Relative
1603:
1581:
1328:
1316:
1276:
1214:
960:
66:
863:
are typically modeled by what are called "Boolean algebras with operators."
3576:
3423:
3324:
3316:
3196:
3144:
3053:
2989:
2972:
2903:
2762:
2621:
2323:
2106:
1805:. Amsterdam, Netherlands: North-Holland/Elsevier Science BV. Archived from
1718:
1651:
1577:
1559:
1312:
1272:
1210:
1089:
836:
803:
774:
534:. These equivalences provide alternative formulas for univalent relations (
46:
3686:
3566:
2745:
2735:
2682:
2366:
2286:
2271:
2151:
2096:
1679:
1562:
1348:
1283:
model theory as a major branch of contemporary mathematical logic, also:
1230:
971:
856:
829:
1882:
1120:, published "On the Algebra of Logic". Logic turned more algebraic when
2616:
2471:
2442:
2248:
2034:
1923:
1842:
1538:
1280:
1234:
1033:
947:
878:
1271:
discusses the rich historical connections between algebraic logic and
3768:
3671:
2724:
2641:
2601:
2565:
2501:
2313:
2303:
2276:
2039:
1971:, 1976, "Algebraic Logic and Predicate Functors" pages 283 to 307 in
1351:
can draw inspiration from, and shed light on, Leibniz's thought, see
1320:
1149:
193:
177:
101:
1617:
1530:
3753:
3551:
2999:
2704:
2298:
2011:
1196:
1184:
1074:
204:
196:, the theory of questions. In the universe of utterances there are
1802:
From Peirce to Skolem: A Neglected Chapter in the History of Logic
3349:
2141:
817:
1900:
820:, built from terms in the usual way, can be equated if they are
1203:
in 1918. He treated the logic of relations as derived from the
1829:. Association for Symbolic Logic, Cambridge University Press.
2893:
2239:
2084:
1660:(London, England: Macmillan, Barclay, & Macmillan, 1847).
1248:
on algebraic logic appeared after the 1910â13 publication of
770:
1176:, though De Morgan had anticipated them with his Theorem K.
142:. Whether a given relation holds for two individuals is one
1085:
translated selections from Couturat's volume into English.
1515:(May 1950). "The representation of Relational Algebras".
1403:. Contemporary Mathematics. Vol. 33. Providence/RI:
1191:. The "BooleâSchröder algebra of logic" was developed at
896:
887:, arguably the paradigmatic algebraic logic, can express
843:
In the table below, the left column contains one or more
143:
38:
is the reasoning obtained by manipulating equations with
806:
are built up from variables using primitive and defined
1956:
Propositional Consequence Relations and Algebraic Logic
150:, and lattice of these sets becomes an algebra through
2006:, Colloquia Mathematica Societatis JĂĄnos Bolyai # 54,
343:. The injective property corresponds to univalence of
69:
fall under the umbrella of classical algebraic logic (
1917:
Good introduction for readers with prior exposure to
690:
634:
587:
540:
507:
438:
376:
349:
302:
176:; then the converse relation is represented by the
729:
670:
620:
573:
522:
463:
401:
362:
327:
3799:
1511:
756:
621:{\displaystyle {\bar {R}}\subseteq R{\bar {I}}}
574:{\displaystyle R{\bar {I}}\subseteq {\bar {R}}}
1398:
1307:In the practice of the calculus of relations,
192:An example of calculus of relations arises in
2055:
730:{\displaystyle f{\bar {A}}={\overline {fA}}.}
27:Reasoning about equations with free variables
1331:of logical vectors; these contribute to the
1100:published the first of several works on the
628:). Therefore, mappings satisfy the formula
1901:J. Michael Dunn; Gary M. Hardegree (2001).
1820:
1347:. To see how present-day work in logic and
1168:the calculus of relations was described by
85:
70:
63:representation theorem for Boolean algebras
2247:
2062:
2048:
1871:
1865:Leibniz: Philosophical Papers and Letters
1678:, London: Taylor & Walton, link from
1642:, second edition 1932, Dover edition 1960
1078:
769:, as models (interpretations) of certain
339:is the identity relation on the range of
61:and duality. Well known results like the
1903:Algebraic Methods in Philosophical Logic
1627:
1625:
1606:(1941), "On the Calculus of Relations",
1563:The Origins of the Calculus of Relations
1287:Initiated abstract algebraic logic with
1225:all shared Leibniz's dream of combining
1183:developed the calculus of relations and
91:
1965:". Stanford Encyclopedia of Philosophy.
1862:
1082:
671:{\displaystyle {\bar {R}}=R{\bar {I}}.}
14:
3800:
2069:
1934:Handbook of Philosophical Logic, vol 2
1854:" in Gabbay, D., and Woods, J., eds.,
1849:
1344:
839:remains valid, but is seldom employed.
2043:
1880:
1798:
1622:
1599:
1597:
1595:
1593:
1444:G. Schmidt & T. Ströhlein (1993)
1352:
1268:
276:α is a homogeneous relation on
1777:Mathematical Association of America
1319:follows from interpretation of the
870:in at least some respects include:
420:But a univalent relation is only a
24:
1894:
1696:Principles of the Algebra of Logic
1590:
1193:University of California, Berkeley
1148:is represented as a member of the
1110:Principles of the Algebra of Logic
866:Algebraic formalisms going beyond
794:existentially quantified variables
464:{\displaystyle I\subseteq RR^{T}.}
402:{\displaystyle RR^{T}\subseteq I,}
328:{\displaystyle R^{T}R\subseteq I,}
25:
3824:
2024:
2008:JĂĄnos Bolyai Mathematical Society
1994:The Search for Mathematical Roots
1883:A (Leibnizian) Theory of Concepts
1487:G. Schmidt & M. Winter(2018)
1323:of a staircase. Riguet generated
877:, having the expressive power of
482:for a total, univalent relation.
413:is the identity on the domain of
3781:
1429:The Categories of Charles Peirce
1775:, edited by Aubert Daigneault,
1762:
1748:
1723:
1702:
1684:
1663:
1645:
530:for the complement of relation
234:is a direct answer to question
1963:The Algebra of Logic Tradition
1823:The Bulletin of Symbolic Logic
1640:University of California Press
1571:
1553:
1505:
1481:
1458:
1438:
1421:
1392:
1172:. In particular he formulated
739:
700:
659:
641:
612:
594:
565:
550:
514:
432:. The formula for totality is
57:) and connected problems like
13:
1:
3742:History of mathematical logic
1996:. Princeton University Press.
1757:The Principles of Mathematics
1433:Open Court Publishing Company
1405:American Mathematical Society
1401:Contributions to Group Theory
1385:
1160:with properties described by
122:is found in the power set of
3667:Primitive recursive function
1924:Review by Janusz Czelakowski
1495:vol. 2208, Springer Verlag,
1493:Lecture Notes in Mathematics
757:Algebras as models of logics
719:
287:
7:
1905:. Oxford University Press.
1358:
1073:discovered it in Leibniz's
773:, making logic a branch of
296:that satisfies the formula
10:
3829:
2731:SchröderâBernstein theorem
2458:Monadic predicate calculus
2117:Foundations of mathematics
1936:(2nd ed.). Springer.
1876:. Oxford University Press.
1792:
1773:Studies in Algebraic Logic
1636:A Survey of Symbolic Logic
1470:Cambridge University Press
1333:non-enlargeable rectangles
1207:of two or more variables.
1054:
1050:
895:, including the canonical
828:, equate a formula with a
523:{\displaystyle {\bar {R}}}
210:. There are two relations
187:
184:using Boolean arithmetic.
3777:
3764:Philosophy of mathematics
3713:Automated theorem proving
3695:
3590:
3422:
3315:
3167:
2884:
2860:
2838:Von NeumannâBernaysâGödel
2783:
2677:
2581:
2479:
2470:
2397:
2332:
2238:
2160:
2077:
1872:Parkinson, G.H.R (1966).
1850:Lenzen, Wolfgang, 2004, "
1799:Brady, Geraldine (2000).
1609:Journal of Symbolic Logic
1301:LindenbaumâTarski algebra
1116:, a student of Peirce at
910:LindenbaumâTarski algebra
76:Works in the more recent
1977:Harvard University Press
1961:Stanley Burris (2015), "
1863:Loemker, Leroy (1969) ,
1126:composition of relations
1118:Johns Hopkins University
1006:Complete Boolean algebra
751:abstract algebraic logic
581:), and total relations (
272:so that the composition
260:. The converse relation
156:composition of relations
78:abstract algebraic logic
3414:Self-verifying theories
3235:Tarski's axiomatization
2186:Tarski's undefinability
2181:incompleteness theorems
1954:Ramon Jansana (2011), "
1874:Leibniz: Logical Papers
1867:(2nd ed.), Reidel.
1714:On the Algebra of Logic
1337:formal concept analysis
1205:propositional functions
1020:First-order logic with
1014:predicate functor logic
994:Monadic Boolean algebra
989:monadic predicate logic
761:Algebraic logic treats
487:complementary relations
152:relative multiplication
3788:Mathematics portal
3399:Proof of impossibility
3047:propositional variable
2357:Propositional calculus
1984:Historical perspective
1858:. North-Holland: 1-84.
1699:, via Internet Archive
1466:Relational Mathematics
1427:Eugene Freeman (1934)
1112:in 1879, and in 1883,
1098:Charles Sanders Peirce
893:axiomatic set theories
786:universally quantified
784:Variables are tacitly
731:
672:
622:
575:
524:
465:
403:
364:
329:
238:. The other relation,
120:heterogeneous relation
3657:Kolmogorov complexity
3610:Computably enumerable
3510:Model complete theory
3302:Principia Mathematica
2362:Propositional formula
2191:BanachâTarski paradox
1990:Ivor Grattan-Guinness
1881:Zalta, E. N., 2000, "
1518:Annals of Mathematics
1325:rectangular relations
1251:Principia Mathematica
790:universe of discourse
732:
673:
623:
576:
525:
466:
404:
365:
363:{\displaystyle R^{T}}
330:
182:matrix multiplication
92:Calculus of relations
18:Calculus of relations
3605:ChurchâTuring thesis
3592:Computability theory
2801:continuum hypothesis
2319:Square of opposition
2177:Gödel's completeness
1919:non-classical logics
1691:Alexander Macfarlane
1446:Relations and Graphs
1407:. pp. 299â307.
1166:Hochschule Karlsruhe
1106:Alexander Macfarlane
932:propositional logic
822:logically equivalent
780:In algebraic logic:
763:algebraic structures
688:
632:
585:
538:
505:
436:
424:, while a univalent
374:
347:
300:
3759:Mathematical object
3650:P versus NP problem
3615:Computable function
3409:Reverse mathematics
3335:Logical consequence
3212:primitive recursive
3207:elementary function
2980:Free/bound variable
2833:TarskiâGrothendieck
2352:Logical connectives
2282:Logical equivalence
2132:Logical consequence
1973:The Ways of Paradox
1567:Stanford University
1489:Relational Topology
1124:were combined with
861:nonclassical logics
51:algebraic semantics
3557:Transfer principle
3520:Semantics of logic
3505:Categorical theory
3481:Non-standard model
2995:Logical connective
2122:Information theory
2071:Mathematical logic
1754:B. Russell (1903)
1670:Augustus De Morgan
1464:G. Schmidt (2011)
1102:logic of relatives
1094:Augustus De Morgan
1057:Symbolical algebra
727:
668:
618:
571:
520:
491:Augustus De Morgan
461:
399:
360:
325:
32:mathematical logic
3795:
3794:
3727:Abstract category
3530:Theories of truth
3340:Rule of inference
3330:Natural deduction
3311:
3310:
2856:
2855:
2561:Cartesian product
2466:
2465:
2372:Many-valued logic
2347:Boolean functions
2230:Russell's paradox
2205:diagonal argument
2102:First-order logic
1943:978-0-7923-7126-7
1912:978-0-19-853192-0
1501:978-3-319-74451-3
1477:978-0-521-76268-7
1414:978-0-8218-5035-0
1380:Universal algebra
1295:cylindric algebra
1289:relation algebras
1279:, the founder of
1242:Leopold Löwenheim
1240:Some writings by
1189:primitive notions
1048:
1047:
1039:Combinatory logic
1027:Cylindric algebra
1001:First-order logic
943:Ćukasiewicz logic
875:Combinatory logic
868:first-order logic
853:proper extensions
722:
703:
662:
644:
615:
597:
568:
553:
517:
370:, or the formula
170:converse relation
55:deductive systems
16:(Redirected from
3820:
3813:History of logic
3786:
3785:
3737:History of logic
3732:Category of sets
3625:Decision problem
3404:Ordinal analysis
3345:Sequent calculus
3243:Boolean algebras
3183:
3182:
3157:
3128:logical/constant
2882:
2881:
2868:
2791:ZermeloâFraenkel
2542:Set operations:
2477:
2476:
2414:
2245:
2244:
2225:LöwenheimâSkolem
2112:Formal semantics
2064:
2057:
2050:
2041:
2040:
1947:
1916:
1890:
1877:
1868:
1859:
1846:
1817:
1815:
1814:
1786:
1766:
1760:
1752:
1746:
1744:Internet Archive
1727:
1721:
1706:
1700:
1688:
1682:
1667:
1661:
1649:
1643:
1629:
1620:
1601:
1588:
1575:
1569:
1557:
1551:
1550:
1509:
1503:
1485:
1479:
1462:
1456:
1442:
1436:
1425:
1419:
1418:
1396:
1375:Computer algebra
1181:Bertrand Russell
1122:binary relations
1079:Parkinson (1966)
1043:relation algebra
1010:polyadic algebra
977:Interior algebra
918:sentential logic
903:
902:
889:Peano arithmetic
885:Relation algebra
849:Boolean algebras
767:bounded lattices
746:relation algebra
736:
734:
733:
728:
723:
718:
710:
705:
704:
696:
684:
682:
677:
675:
674:
669:
664:
663:
655:
646:
645:
637:
627:
625:
624:
619:
617:
616:
608:
599:
598:
590:
580:
578:
577:
572:
570:
569:
561:
555:
554:
546:
533:
529:
527:
526:
521:
519:
518:
510:
485:The facility of
470:
468:
467:
462:
457:
456:
422:partial function
416:
412:
409:where this time
408:
406:
405:
400:
389:
388:
369:
367:
366:
361:
359:
358:
342:
338:
334:
332:
331:
326:
312:
311:
295:
275:
263:
244:
214:and α from
213:
141:
131:
113:
100:is found in the
86:Czelakowski 2003
82:Leibniz operator
71:Czelakowski 2003
21:
3828:
3827:
3823:
3822:
3821:
3819:
3818:
3817:
3808:Algebraic logic
3798:
3797:
3796:
3791:
3780:
3773:
3718:Category theory
3708:Algebraic logic
3691:
3662:Lambda calculus
3600:Church encoding
3586:
3562:Truth predicate
3418:
3384:Complete theory
3307:
3176:
3172:
3168:
3163:
3155:
2875: and
2871:
2866:
2852:
2828:New Foundations
2796:axiom of choice
2779:
2741:Gödel numbering
2681: and
2673:
2577:
2462:
2412:
2393:
2342:Boolean algebra
2328:
2292:Equiconsistency
2257:Classical logic
2234:
2215:Halting problem
2203: and
2179: and
2167: and
2166:
2161:Theorems (
2156:
2073:
2068:
2031:Algebraic logic
2027:
2004:Algebraic Logic
1944:
1913:
1897:
1895:Further reading
1852:Leibnizâs Logic
1812:
1810:
1795:
1790:
1789:
1767:
1763:
1753:
1749:
1728:
1724:
1707:
1703:
1689:
1685:
1668:
1664:
1650:
1646:
1630:
1623:
1618:10.2307/2268577
1602:
1591:
1576:
1572:
1558:
1554:
1531:10.2307/1969375
1513:Roger C. Lyndon
1510:
1506:
1486:
1482:
1463:
1459:
1443:
1439:
1426:
1422:
1415:
1397:
1393:
1388:
1365:Boolean algebra
1361:
1223:A. N. Whitehead
1162:Boolean algebra
1059:
1053:
936:Heyting algebra
923:Boolean algebra
906:Logical system
824:. To express a
810:. There are no
792:. There are no
759:
742:
711:
709:
695:
694:
689:
686:
685:
680:
679:
654:
653:
636:
635:
633:
630:
629:
607:
606:
589:
588:
586:
583:
582:
560:
559:
545:
544:
539:
536:
535:
531:
509:
508:
506:
503:
502:
476:Gunther Schmidt
472:Charles Loewner
452:
448:
437:
434:
433:
414:
410:
384:
380:
375:
372:
371:
354:
350:
348:
345:
344:
340:
336:
307:
303:
301:
298:
297:
293:
290:
282:Socratic method
273:
261:
242:
211:
190:
133:
123:
105:
98:binary relation
94:
36:algebraic logic
28:
23:
22:
15:
12:
11:
5:
3826:
3816:
3815:
3810:
3793:
3792:
3778:
3775:
3774:
3772:
3771:
3766:
3761:
3756:
3751:
3750:
3749:
3739:
3734:
3729:
3720:
3715:
3710:
3705:
3703:Abstract logic
3699:
3697:
3693:
3692:
3690:
3689:
3684:
3682:Turing machine
3679:
3674:
3669:
3664:
3659:
3654:
3653:
3652:
3647:
3642:
3637:
3632:
3622:
3620:Computable set
3617:
3612:
3607:
3602:
3596:
3594:
3588:
3587:
3585:
3584:
3579:
3574:
3569:
3564:
3559:
3554:
3549:
3548:
3547:
3542:
3537:
3527:
3522:
3517:
3515:Satisfiability
3512:
3507:
3502:
3501:
3500:
3490:
3489:
3488:
3478:
3477:
3476:
3471:
3466:
3461:
3456:
3446:
3445:
3444:
3439:
3432:Interpretation
3428:
3426:
3420:
3419:
3417:
3416:
3411:
3406:
3401:
3396:
3386:
3381:
3380:
3379:
3378:
3377:
3367:
3362:
3352:
3347:
3342:
3337:
3332:
3327:
3321:
3319:
3313:
3312:
3309:
3308:
3306:
3305:
3297:
3296:
3295:
3294:
3289:
3288:
3287:
3282:
3277:
3257:
3256:
3255:
3253:minimal axioms
3250:
3239:
3238:
3237:
3226:
3225:
3224:
3219:
3214:
3209:
3204:
3199:
3186:
3184:
3165:
3164:
3162:
3161:
3160:
3159:
3147:
3142:
3141:
3140:
3135:
3130:
3125:
3115:
3110:
3105:
3100:
3099:
3098:
3093:
3083:
3082:
3081:
3076:
3071:
3066:
3056:
3051:
3050:
3049:
3044:
3039:
3029:
3028:
3027:
3022:
3017:
3012:
3007:
3002:
2992:
2987:
2982:
2977:
2976:
2975:
2970:
2965:
2960:
2950:
2945:
2943:Formation rule
2940:
2935:
2934:
2933:
2928:
2918:
2917:
2916:
2906:
2901:
2896:
2891:
2885:
2879:
2862:Formal systems
2858:
2857:
2854:
2853:
2851:
2850:
2845:
2840:
2835:
2830:
2825:
2820:
2815:
2810:
2805:
2804:
2803:
2798:
2787:
2785:
2781:
2780:
2778:
2777:
2776:
2775:
2765:
2760:
2759:
2758:
2751:Large cardinal
2748:
2743:
2738:
2733:
2728:
2714:
2713:
2712:
2707:
2702:
2687:
2685:
2675:
2674:
2672:
2671:
2670:
2669:
2664:
2659:
2649:
2644:
2639:
2634:
2629:
2624:
2619:
2614:
2609:
2604:
2599:
2594:
2588:
2586:
2579:
2578:
2576:
2575:
2574:
2573:
2568:
2563:
2558:
2553:
2548:
2540:
2539:
2538:
2533:
2523:
2518:
2516:Extensionality
2513:
2511:Ordinal number
2508:
2498:
2493:
2492:
2491:
2480:
2474:
2468:
2467:
2464:
2463:
2461:
2460:
2455:
2450:
2445:
2440:
2435:
2430:
2429:
2428:
2418:
2417:
2416:
2403:
2401:
2395:
2394:
2392:
2391:
2390:
2389:
2384:
2379:
2369:
2364:
2359:
2354:
2349:
2344:
2338:
2336:
2330:
2329:
2327:
2326:
2321:
2316:
2311:
2306:
2301:
2296:
2295:
2294:
2284:
2279:
2274:
2269:
2264:
2259:
2253:
2251:
2242:
2236:
2235:
2233:
2232:
2227:
2222:
2217:
2212:
2207:
2195:Cantor's
2193:
2188:
2183:
2173:
2171:
2158:
2157:
2155:
2154:
2149:
2144:
2139:
2134:
2129:
2124:
2119:
2114:
2109:
2104:
2099:
2094:
2093:
2092:
2081:
2079:
2075:
2074:
2067:
2066:
2059:
2052:
2044:
2038:
2037:
2026:
2025:External links
2023:
2022:
2021:
2000:Irving Anellis
1997:
1981:
1980:
1966:
1959:
1952:
1942:
1930:Hajnal Andréka
1926:
1911:
1896:
1893:
1892:
1891:
1878:
1869:
1860:
1847:
1818:
1794:
1791:
1788:
1787:
1769:Helena Rasiowa
1761:
1747:
1730:Ernst Schröder
1722:
1709:Christine Ladd
1701:
1683:
1662:
1644:
1632:Clarence Lewis
1621:
1589:
1570:
1552:
1525:(3): 707â729.
1504:
1480:
1457:
1437:
1420:
1413:
1390:
1389:
1387:
1384:
1383:
1382:
1377:
1372:
1370:Codd's theorem
1367:
1360:
1357:
1327:by taking the
1309:Jacques Riguet
1305:
1304:
1299:Co-discovered
1297:
1291:
1263:logical matrix
1259:Helena Rasiowa
1246:Thoralf Skolem
1227:symbolic logic
1219:Giuseppe Peano
1201:Clarence Lewis
1174:Schröder rules
1170:Ernst Schröder
1114:Christine Ladd
1108:published his
1083:Loemker (1969)
1071:Louis Couturat
1067:Clarence Lewis
1052:
1049:
1046:
1045:
1036:
1030:
1029:
1024:
1017:
1016:
1003:
997:
996:
991:
980:
979:
974:
964:
963:
958:
951:
950:
945:
939:
938:
933:
930:Intuitionistic
926:
925:
920:
913:
912:
907:
901:
900:
882:
841:
840:
833:
815:
801:
758:
755:
741:
738:
726:
721:
717:
714:
708:
702:
699:
693:
667:
661:
658:
652:
649:
643:
640:
614:
611:
605:
602:
596:
593:
567:
564:
558:
552:
549:
543:
516:
513:
497:to introduce
495:Ernst Schröder
460:
455:
451:
447:
444:
441:
426:total relation
398:
395:
392:
387:
383:
379:
357:
353:
324:
321:
318:
315:
310:
306:
289:
286:
254:presupposition
189:
186:
174:logical matrix
168:refers to the
96:A homogeneous
93:
90:
59:representation
40:free variables
26:
9:
6:
4:
3:
2:
3825:
3814:
3811:
3809:
3806:
3805:
3803:
3790:
3789:
3784:
3776:
3770:
3767:
3765:
3762:
3760:
3757:
3755:
3752:
3748:
3745:
3744:
3743:
3740:
3738:
3735:
3733:
3730:
3728:
3724:
3721:
3719:
3716:
3714:
3711:
3709:
3706:
3704:
3701:
3700:
3698:
3694:
3688:
3685:
3683:
3680:
3678:
3677:Recursive set
3675:
3673:
3670:
3668:
3665:
3663:
3660:
3658:
3655:
3651:
3648:
3646:
3643:
3641:
3638:
3636:
3633:
3631:
3628:
3627:
3626:
3623:
3621:
3618:
3616:
3613:
3611:
3608:
3606:
3603:
3601:
3598:
3597:
3595:
3593:
3589:
3583:
3580:
3578:
3575:
3573:
3570:
3568:
3565:
3563:
3560:
3558:
3555:
3553:
3550:
3546:
3543:
3541:
3538:
3536:
3533:
3532:
3531:
3528:
3526:
3523:
3521:
3518:
3516:
3513:
3511:
3508:
3506:
3503:
3499:
3496:
3495:
3494:
3491:
3487:
3486:of arithmetic
3484:
3483:
3482:
3479:
3475:
3472:
3470:
3467:
3465:
3462:
3460:
3457:
3455:
3452:
3451:
3450:
3447:
3443:
3440:
3438:
3435:
3434:
3433:
3430:
3429:
3427:
3425:
3421:
3415:
3412:
3410:
3407:
3405:
3402:
3400:
3397:
3394:
3393:from ZFC
3390:
3387:
3385:
3382:
3376:
3373:
3372:
3371:
3368:
3366:
3363:
3361:
3358:
3357:
3356:
3353:
3351:
3348:
3346:
3343:
3341:
3338:
3336:
3333:
3331:
3328:
3326:
3323:
3322:
3320:
3318:
3314:
3304:
3303:
3299:
3298:
3293:
3292:non-Euclidean
3290:
3286:
3283:
3281:
3278:
3276:
3275:
3271:
3270:
3268:
3265:
3264:
3262:
3258:
3254:
3251:
3249:
3246:
3245:
3244:
3240:
3236:
3233:
3232:
3231:
3227:
3223:
3220:
3218:
3215:
3213:
3210:
3208:
3205:
3203:
3200:
3198:
3195:
3194:
3192:
3188:
3187:
3185:
3180:
3174:
3169:Example
3166:
3158:
3153:
3152:
3151:
3148:
3146:
3143:
3139:
3136:
3134:
3131:
3129:
3126:
3124:
3121:
3120:
3119:
3116:
3114:
3111:
3109:
3106:
3104:
3101:
3097:
3094:
3092:
3089:
3088:
3087:
3084:
3080:
3077:
3075:
3072:
3070:
3067:
3065:
3062:
3061:
3060:
3057:
3055:
3052:
3048:
3045:
3043:
3040:
3038:
3035:
3034:
3033:
3030:
3026:
3023:
3021:
3018:
3016:
3013:
3011:
3008:
3006:
3003:
3001:
2998:
2997:
2996:
2993:
2991:
2988:
2986:
2983:
2981:
2978:
2974:
2971:
2969:
2966:
2964:
2961:
2959:
2956:
2955:
2954:
2951:
2949:
2946:
2944:
2941:
2939:
2936:
2932:
2929:
2927:
2926:by definition
2924:
2923:
2922:
2919:
2915:
2912:
2911:
2910:
2907:
2905:
2902:
2900:
2897:
2895:
2892:
2890:
2887:
2886:
2883:
2880:
2878:
2874:
2869:
2863:
2859:
2849:
2846:
2844:
2841:
2839:
2836:
2834:
2831:
2829:
2826:
2824:
2821:
2819:
2816:
2814:
2813:KripkeâPlatek
2811:
2809:
2806:
2802:
2799:
2797:
2794:
2793:
2792:
2789:
2788:
2786:
2782:
2774:
2771:
2770:
2769:
2766:
2764:
2761:
2757:
2754:
2753:
2752:
2749:
2747:
2744:
2742:
2739:
2737:
2734:
2732:
2729:
2726:
2722:
2718:
2715:
2711:
2708:
2706:
2703:
2701:
2698:
2697:
2696:
2692:
2689:
2688:
2686:
2684:
2680:
2676:
2668:
2665:
2663:
2660:
2658:
2657:constructible
2655:
2654:
2653:
2650:
2648:
2645:
2643:
2640:
2638:
2635:
2633:
2630:
2628:
2625:
2623:
2620:
2618:
2615:
2613:
2610:
2608:
2605:
2603:
2600:
2598:
2595:
2593:
2590:
2589:
2587:
2585:
2580:
2572:
2569:
2567:
2564:
2562:
2559:
2557:
2554:
2552:
2549:
2547:
2544:
2543:
2541:
2537:
2534:
2532:
2529:
2528:
2527:
2524:
2522:
2519:
2517:
2514:
2512:
2509:
2507:
2503:
2499:
2497:
2494:
2490:
2487:
2486:
2485:
2482:
2481:
2478:
2475:
2473:
2469:
2459:
2456:
2454:
2451:
2449:
2446:
2444:
2441:
2439:
2436:
2434:
2431:
2427:
2424:
2423:
2422:
2419:
2415:
2410:
2409:
2408:
2405:
2404:
2402:
2400:
2396:
2388:
2385:
2383:
2380:
2378:
2375:
2374:
2373:
2370:
2368:
2365:
2363:
2360:
2358:
2355:
2353:
2350:
2348:
2345:
2343:
2340:
2339:
2337:
2335:
2334:Propositional
2331:
2325:
2322:
2320:
2317:
2315:
2312:
2310:
2307:
2305:
2302:
2300:
2297:
2293:
2290:
2289:
2288:
2285:
2283:
2280:
2278:
2275:
2273:
2270:
2268:
2265:
2263:
2262:Logical truth
2260:
2258:
2255:
2254:
2252:
2250:
2246:
2243:
2241:
2237:
2231:
2228:
2226:
2223:
2221:
2218:
2216:
2213:
2211:
2208:
2206:
2202:
2198:
2194:
2192:
2189:
2187:
2184:
2182:
2178:
2175:
2174:
2172:
2170:
2164:
2159:
2153:
2150:
2148:
2145:
2143:
2140:
2138:
2135:
2133:
2130:
2128:
2125:
2123:
2120:
2118:
2115:
2113:
2110:
2108:
2105:
2103:
2100:
2098:
2095:
2091:
2088:
2087:
2086:
2083:
2082:
2080:
2076:
2072:
2065:
2060:
2058:
2053:
2051:
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1809:on 2009-04-02
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1784:0-88385-109-1
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1740:B. G. Teubner
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1605:
1604:Alfred Tarski
1600:
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1582:Studia Logica
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1454:3-540-56254-0
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1350:
1346:
1345:Lenzen (2004)
1340:
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1330:
1329:outer product
1326:
1322:
1318:
1317:N. M. Ferrers
1314:
1310:
1302:
1298:
1296:
1292:
1290:
1286:
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1284:
1282:
1281:set theoretic
1278:
1277:Alfred Tarski
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1270:
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1264:
1260:
1257:According to
1255:
1253:
1252:
1247:
1243:
1238:
1236:
1232:
1228:
1224:
1220:
1216:
1215:Gottlob Frege
1212:
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961:Modal algebra
959:
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799:
798:open formulas
795:
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478:use the term
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121:
117:
114:for some set
112:
108:
103:
99:
89:
87:
83:
79:
74:
72:
68:
67:Stone duality
64:
60:
56:
52:
48:
43:
41:
37:
33:
19:
3779:
3707:
3577:Ultraproduct
3424:Model theory
3389:Independence
3325:Formal proof
3317:Proof theory
3300:
3273:
3230:real numbers
3202:second-order
3113:Substitution
2990:Metalanguage
2931:conservative
2904:Axiom schema
2848:Constructive
2818:MorseâKelley
2784:Set theories
2763:Aleph number
2756:inaccessible
2662:Grothendieck
2546:intersection
2433:Higher-order
2421:Second-order
2367:Truth tables
2324:Venn diagram
2107:Formal proof
2003:
1993:
1983:
1982:
1972:
1933:
1902:
1886:
1873:
1864:
1855:
1826:
1822:
1811:. Retrieved
1807:the original
1801:
1772:
1764:
1755:
1750:
1733:
1725:
1719:Google Books
1712:
1704:
1694:
1686:
1675:Formal Logic
1673:
1665:
1656:
1652:George Boole
1647:
1635:
1607:
1585:
1578:Roger Maddux
1573:
1560:Vaughn Pratt
1555:
1522:
1516:
1507:
1488:
1483:
1465:
1460:
1445:
1440:
1428:
1423:
1400:
1394:
1353:Zalta (2000)
1341:
1332:
1324:
1313:difunctional
1306:
1273:model theory
1269:Brady (2000)
1267:
1256:
1249:
1239:
1211:Hugh MacColl
1209:
1178:
1157:
1153:
1145:
1141:
1133:
1129:
1109:
1090:George Boole
1087:
1060:
954:Modal logic
865:
842:
837:Modus ponens
779:
775:order theory
760:
745:
743:
499:equivalences
484:
479:
419:
291:
277:
269:
265:
257:
256:of question
249:
245:
239:
235:
231:
227:
223:
219:
215:
207:
200:
191:
165:
163:
160:
151:
138:
134:
128:
124:
115:
110:
106:
95:
75:
44:
35:
29:
3687:Type theory
3635:undecidable
3567:Truth value
3454:equivalence
3133:non-logical
2746:Enumeration
2736:Isomorphism
2683:cardinality
2667:Von Neumann
2632:Ultrafilter
2597:Uncountable
2531:equivalence
2448:Quantifiers
2438:Fixed-point
2407:First-order
2287:Consistency
2272:Proposition
2249:Traditional
2220:Lindström's
2210:Compactness
2152:Type theory
2097:Cardinality
1889:3: 137-183.
1738:, Leibzig:
1680:Hathi Trust
1431:, page 10,
1349:metaphysics
1231:mathematics
1128:. For sets
830:truth value
812:connectives
740:Abstraction
248:holds when
230:holds when
3802:Categories
3498:elementary
3191:arithmetic
3059:Quantifier
3037:functional
2909:Expression
2627:Transitive
2571:identities
2556:complement
2489:hereditary
2472:Set theory
2035:PhilPapers
2019:0444885439
1813:2009-05-15
1732:, (1895),
1491:, page 8,
1386:References
1235:philosophy
1096:. In 1870
1055:See also:
1034:Set theory
948:MV-algebra
916:Classical
879:set theory
859:and other
808:operations
788:over some
284:dialogue.
264:runs from
198:statements
166:conversion
118:, while a
53:for these
3769:Supertask
3672:Recursion
3630:decidable
3464:saturated
3442:of models
3365:deductive
3360:axiomatic
3280:Hilbert's
3267:Euclidean
3248:canonical
3171:axiomatic
3103:Signature
3032:Predicate
2921:Extension
2843:Ackermann
2768:Operation
2647:Universal
2637:Recursive
2612:Singleton
2607:Inhabited
2592:Countable
2582:Types of
2566:power set
2536:partition
2453:Predicate
2399:Predicate
2314:Syllogism
2304:Soundness
2277:Inference
2267:Tautology
2169:paradoxes
1835:1079-8986
1612:6: 73â89
1321:transpose
1293:Invented
1150:power set
891:and most
855:thereof.
826:tautology
720:¯
701:¯
660:¯
642:¯
613:¯
601:⊆
595:¯
566:¯
557:⊆
551:¯
515:¯
489:inspired
443:⊆
391:⊆
317:⊆
288:Functions
205:questions
194:erotetics
178:transpose
148:inclusion
102:power set
3754:Logicism
3747:timeline
3723:Concrete
3582:Validity
3552:T-schema
3545:Kripke's
3540:Tarski's
3535:semantic
3525:Strength
3474:submodel
3469:spectrum
3437:function
3285:Tarski's
3274:Elements
3261:geometry
3217:Robinson
3138:variable
3123:function
3096:spectrum
3086:Sentence
3042:variable
2985:Language
2938:Relation
2899:Automata
2889:Alphabet
2873:language
2727:-jection
2705:codomain
2691:Function
2652:Universe
2622:Infinite
2526:Relation
2309:Validity
2299:Argument
2197:theorem,
2012:Elsevier
1992:, 2000.
1711:(1883),
1693:(1879),
1672:(1847),
1586:: 421-55
1359:See also
1197:textbook
1185:logicism
1179:In 1903
1138:relation
1075:Nachlass
1022:equality
983:Lewis's
818:Formulas
765:, often
430:function
132:, where
3696:Related
3493:Diagram
3391: (
3370:Hilbert
3355:Systems
3350:Theorem
3228:of the
3173:systems
2953:Formula
2948:Grammar
2864: (
2808:General
2521:Forcing
2506:Element
2426:Monadic
2201:paradox
2142:Theorem
2078:General
1843:3094793
1793:Sources
1634:(1918)
1565:, from
1547:0037278
1539:1969375
1063:Leibniz
1051:History
845:logical
683:,
480:mapping
226:α
188:Example
3459:finite
3222:Skolem
3175:
3150:Theory
3118:Symbol
3108:String
3091:atomic
2968:ground
2963:closed
2958:atomic
2914:ground
2877:syntax
2773:binary
2700:domain
2617:Finite
2382:finite
2240:Logics
2199:
2147:Theory
2017:
2010:&
1940:
1909:
1841:
1833:
1782:
1545:
1537:
1499:
1475:
1452:
1411:
1233:, and
1221:, and
771:logics
501:using
335:where
47:models
3449:Model
3197:Peano
3054:Proof
2894:Arity
2823:Naive
2710:image
2642:Fuzzy
2602:Empty
2551:union
2496:Class
2137:Model
2127:Lemma
2085:Axiom
1949:Draft
1839:JSTOR
1535:JSTOR
1195:in a
1140:over
968:Lewis
857:Modal
804:Terms
428:is a
252:is a
3572:Type
3375:list
3179:list
3156:list
3145:Term
3079:rank
2973:open
2867:list
2679:Maps
2584:sets
2443:Free
2413:list
2163:list
2090:list
2015:ISBN
1938:ISBN
1907:ISBN
1831:ISSN
1780:ISBN
1742:via
1717:via
1497:ISBN
1473:ISBN
1450:ISBN
1409:ISBN
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1885:,"
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1527:doi
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