3772:
1331:
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
1424:, 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.
737:
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
1910:
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.
615:
568:
724:
665:
1304:
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
458:
396:
322:
2151:
517:
2826:
357:
281:
The description of the key binary relation properties has been formulated with the calculus of relations. The univalence property of functions describes a relation
1300:
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
2909:
2050:
51:
1789:
1250:, "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
836:
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
39:
3223:
3381:
1930:
1899:
1489:
1465:
1401:
2169:
3236:
2559:
1996:
1765:
2821:
3241:
3231:
2968:
2174:
1937:
1181:
28:
2719:
2165:
3377:
1772:
1442:
1718:
1158:
483:
3474:
3218:
2043:
2779:
2472:
2213:
1840:
150:"The basic operations are set-theoretic union, intersection and complementation, the relative multiplication, and conversion."
1388:
Bjarni JĂłnsson (1984). "Maximal
Algebras of Binary Relations". In Kenneth I. Appel; John G. Ratcliffe; Paul E. Schupp (eds.).
3735:
3437:
3200:
3195:
3020:
2441:
2125:
2007:
1628:
1289:
898:
3730:
3513:
3430:
3143:
3074:
2951:
2193:
1745:
1437:
Discrete
Mathematics for Computer Scientists, page 54, EATCS Monographs on Theoretical Computer Science, Springer Verlag,
1421:
1393:
786:
2801:
1810:
Czelakowski, Janusz (2003). "Review: Algebraic
Methods in Philosophical Logic by J. Michael Dunn and Gary M. Hardegree".
3655:
3481:
3167:
2400:
1481:
47:
2806:
3138:
2877:
2135:
2036:
837:
3533:
3528:
573:
526:
135:
of information, so relations are studied with
Boolean arithmetic. Elements of the power set are partially ordered by
1254:
method. Since logical matrices are certain abstract algebras, this led to the use of an algebraic method in logic."
3462:
3052:
2446:
2414:
2105:
1458:
782:
676:
169:
matrix. A relation obtained as the composition of two others is then represented by the logical matrix obtained by
2179:
1264:. The founders of model theory, Ernst Schröder and Leopold Loewenheim, were logicians in the algebraic tradition.
69:(AAL) focus on the process of algebraization itself, like classifying various forms of algebraizability using the
3801:
3752:
3701:
3598:
3096:
3057:
2534:
1597:
1051:
3593:
2208:
1050:
Algebraic logic is, perhaps, the oldest approach to formal logic, arguably beginning with a number of memoranda
34:
What is now usually called classical algebraic logic focuses on the identification and algebraic description of
3796:
3523:
3062:
2914:
2897:
2620:
2100:
1965:
1795:
1162:
1114:
1106:
774:
739:
487:
144:
66:
1569:(1991) "The Origin of Relation Algebras in the Development and Axiomatization of the Calculus of Relations",
620:
3425:
3402:
3363:
3249:
3190:
2836:
2756:
2600:
2544:
2157:
1325:
1002:
982:
977:
796:
1663:
3715:
3442:
3420:
3387:
3280:
3126:
3111:
3084:
3035:
2919:
2854:
2679:
2645:
2640:
2514:
2345:
2322:
1991:& N. Houser (1991) "Nineteenth Century Roots of Algebraic Logic and Universal Algebra", pages 1â36 in
1193:
1086:
1010:
475:
418:
136:
108:
1684:
3645:
3498:
3290:
3008:
2744:
2650:
2509:
2494:
2375:
2350:
1978:
1506:
1239:
956:
778:
424:
362:
288:
170:
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3101:
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3003:
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2688:
2655:
2519:
2307:
2218:
1679:
1230:
1154:
1094:
994:
931:
918:
881:
1921:, IstvĂĄn NĂ©meti and IldikĂł Sain (2001). "Algebraic logic". In Dov M. Gabbay, Franz Guenthner (ed.).
1058:
in 1918. But nearly all of
Leibniz's known work on algebraic logic was published only in 1903 after
1054:
wrote in the 1680s, some of which were published in the 19th century and translated into
English by
38:
appropriate for the study of various logics (in the form of classes of algebras that constitute the
3747:
3638:
3623:
3603:
3560:
3447:
3397:
3323:
3268:
3205:
2998:
2993:
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2189:
2185:
2120:
2115:
1907:
1555:
810:
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3493:
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3255:
3121:
3047:
3030:
2983:
2796:
2705:
2539:
2524:
2484:
2436:
2421:
2409:
2365:
2340:
2110:
2059:
1827:
1658:
1523:
1153:. The "calculus of relations" is arguably the culmination of Leibniz's approach to logic. At the
1090:
1082:
1045:
944:
849:
800:
479:
20:
3273:
2729:
493:
1243:, and Tarski revived interest in relations with his 1941 essay "On the Calculus of Relations".
3711:
3518:
3328:
3318:
3210:
3091:
2926:
2902:
2683:
2667:
2572:
2549:
2426:
2395:
2360:
2255:
2090:
2003:
1926:
1918:
1895:
1819:
1768:
1485:
1461:
1438:
1397:
1368:
1358:
1283:
1027:
1015:
989:
863:
856:
814:
186:
158:
1951:
1702:
3725:
3720:
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3353:
3348:
3333:
3159:
3116:
3013:
2811:
2761:
2335:
2297:
1732:
1646:
The
Mathematical Analysis of Logic, Being an Essay towards a Calculus of Deductive Reasoning
1644:
1602:
1515:
1363:
1277:
1177:
1169:
1031:
998:
973:
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70:
43:
1944:
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3588:
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1531:
1501:
1353:
1211:
1150:
1126:
1110:
1077:
Modern mathematical logic began in 1847, with two pamphlets whose respective authors were
924:
911:
755:
464:
460:
270:
161:
that always exists, contrary to function theory. A given relation may be represented by a
86:
1845:
Handbook of the
History of Logic, Vol. 3: The Rise of Modern Logic from Leibniz to Frege
3691:
3670:
3628:
3608:
3503:
3358:
2956:
2946:
2936:
2931:
2865:
2739:
2615:
2504:
2499:
2477:
2078:
1988:
1876:
Philosophiegeschichte und logische
Analyse / Logical Analysis and History of Philosophy
1757:
1697:
1620:
1297:
1251:
1247:
1234:
1215:
1207:
1189:
1102:
1059:
1055:
833:
824:
The rules of proof are the substitution of equals for equals, and uniform replacement.
414:
269:. The art of putting the right question to elicit a sufficient answer is recognized in
242:
162:
1723:
667:
Schmidt uses this principle as "slipping below negation from the left". For a mapping
3790:
3665:
3343:
2850:
2635:
2625:
2595:
2580:
2250:
1957:
1728:
1592:
1570:
1317:
1305:
1265:
1203:
949:
55:
1760:(1974), "Post Algebras as Semantic Foundations of m-valued Logics", pages 92â142 in
3565:
3412:
3313:
3305:
3185:
3133:
3042:
2978:
2961:
2892:
2751:
2610:
2312:
2095:
1707:
1640:
1566:
1548:
1301:
1261:
1199:
1078:
825:
792:
763:
35:
3675:
3555:
2734:
2724:
2671:
2355:
2275:
2260:
2140:
2085:
1668:
1337:
1219:
960:
845:
818:
1457:, Encyclopedia of Mathematics and its Applications, vol. 132, pages 49 and 57,
2605:
2460:
2431:
2237:
2023:
1947:". Stanford Encyclopedia of Philosophy. Mainly about abstract algebraic logic.
1269:
1223:
1176:
as his version of pure mathematics based on the operations of the calculus as
1022:
936:
867:
2019:
1823:
1724:
Algebra der Logik (Exakte Logik) Dritter Band, Algebra und Logik der Relative
3757:
3660:
2713:
2630:
2590:
2554:
2490:
2302:
2292:
2265:
1309:
1138:
852:
are typically modeled by what are called "Boolean algebras with operators."
182:
166:
90:
1794:. Amsterdam, Netherlands: North-Holland/Elsevier Science BV. Archived from
523:. These equivalences provide alternative formulas for univalent relations (
3742:
3540:
2988:
2693:
2287:
2000:
1551:
1272:
model theory as a major branch of contemporary mathematical logic, also:
1185:
1173:
1063:
193:
1871:
1109:, published "On the Algebra of Logic". Logic turned more algebraic when
3338:
2130:
1912:
1831:
1527:
806:
1260:
discusses the rich historical connections between algebraic logic and
2028:
1960:, 1976, "Algebraic Logic and Predicate Functors" pages 283 to 307 in
1340:
can draw inspiration from, and shed light on, Leibniz's thought, see
1606:
1519:
185:, the theory of questions. In the universe of utterances there are
2882:
2228:
2073:
1791:
From Peirce to Skolem: A Neglected Chapter in the History of Logic
759:
809:, built from terms in the usual way, can be equated if they are
2032:
1192:
in 1918. He treated the logic of relations as derived from the
1818:. Association for Symbolic Logic, Cambridge University Press.
885:
132:
1649:(London, England: Macmillan, Barclay, & Macmillan, 1847).
1237:
on algebraic logic appeared after the 1910â13 publication of
1165:, though De Morgan had anticipated them with his Theorem K.
131:. Whether a given relation holds for two individuals is one
1074:
translated selections from Couturat's volume into English.
1504:(May 1950). "The representation of Relational Algebras".
1392:. Contemporary Mathematics. Vol. 33. Providence/RI:
1180:. The "BooleâSchröder algebra of logic" was developed at
876:, arguably the paradigmatic algebraic logic, can express
832:
In the table below, the left column contains one or more
27:
is the reasoning obtained by manipulating equations with
795:
are built up from variables using primitive and defined
1945:
Propositional Consequence Relations and Algebraic Logic
139:, and lattice of these sets becomes an algebra through
1995:, Colloquia Mathematica Societatis JĂĄnos Bolyai # 54,
332:. The injective property corresponds to univalence of
58:
fall under the umbrella of classical algebraic logic (
1906:
Good introduction for readers with prior exposure to
679:
623:
576:
529:
496:
427:
365:
338:
291:
3684:
3579:
3411:
3304:
3156:
2849:
2772:
2666:
2570:
2459:
2386:
2321:
2236:
2227:
2149:
2066:
165:; then the converse relation is represented by the
718:
659:
609:
562:
511:
452:
390:
351:
316:
610:{\displaystyle {\bar {R}}\subseteq R{\bar {I}}}
563:{\displaystyle R{\bar {I}}\subseteq {\bar {R}}}
1296:In the practice of the calculus of relations,
181:An example of calculus of relations arises in
2044:
719:{\displaystyle f{\bar {A}}={\overline {fA}}.}
16:Reasoning about equations with free variables
8:
1320:of logical vectors; these contribute to the
1089:published the first of several works on the
617:). Therefore, mappings satisfy the formula
1890:J. Michael Dunn; Gary M. Hardegree (2001).
1336:. To see how present-day work in logic and
1157:the calculus of relations was described by
74:
59:
52:representation theorem for Boolean algebras
2870:
2465:
2233:
2051:
2037:
2029:
1854:Leibniz: Philosophical Papers and Letters
1667:, London: Taylor & Walton, link from
1631:, second edition 1932, Dover edition 1960
1067:
758:, as models (interpretations) of certain
698:
684:
683:
678:
643:
642:
625:
624:
622:
596:
595:
578:
577:
575:
549:
548:
534:
533:
528:
498:
497:
495:
441:
426:
373:
364:
343:
337:
328:is the identity relation on the range of
296:
290:
50:and duality. Well known results like the
1892:Algebraic Methods in Philosophical Logic
1616:
1614:
1595:(1941), "On the Calculus of Relations",
1552:The Origins of the Calculus of Relations
1276:Initiated abstract algebraic logic with
1214:all shared Leibniz's dream of combining
1172:developed the calculus of relations and
891:
1954:". Stanford Encyclopedia of Philosophy.
1380:
1071:
660:{\displaystyle {\bar {R}}=R{\bar {I}}.}
1923:Handbook of Philosophical Logic, vol 2
1843:" in Gabbay, D., and Woods, J., eds.,
1333:
828:remains valid, but is seldom employed.
1588:
1586:
1584:
1582:
1433:G. Schmidt & T. Ströhlein (1993)
1341:
1257:
7:
265:α is a homogeneous relation on
1766:Mathematical Association of America
1308:follows from interpretation of the
859:in at least some respects include:
409:But a univalent relation is only a
1685:Principles of the Algebra of Logic
1182:University of California, Berkeley
1137:is represented as a member of the
1099:Principles of the Algebra of Logic
855:Algebraic formalisms going beyond
783:existentially quantified variables
453:{\displaystyle I\subseteq RR^{T}.}
391:{\displaystyle RR^{T}\subseteq I,}
317:{\displaystyle R^{T}R\subseteq I,}
14:
1997:JĂĄnos Bolyai Mathematical Society
1983:The Search for Mathematical Roots
1872:A (Leibnizian) Theory of Concepts
1476:G. Schmidt & M. Winter(2018)
1312:of a staircase. Riguet generated
866:, having the expressive power of
471:for a total, univalent relation.
402:is the identity on the domain of
3770:
1418:The Categories of Charles Peirce
1764:, edited by Aubert Daigneault,
519:for the complement of relation
223:is a direct answer to question
1952:The Algebra of Logic Tradition
1812:The Bulletin of Symbolic Logic
1629:University of California Press
1161:. In particular he formulated
689:
648:
630:
601:
583:
554:
539:
503:
421:. The formula for totality is
46:) and connected problems like
1:
3731:History of mathematical logic
1985:. Princeton University Press.
1746:The Principles of Mathematics
1422:Open Court Publishing Company
1394:American Mathematical Society
1390:Contributions to Group Theory
1149:with properties described by
111:is found in the power set of
3656:Primitive recursive function
1913:Review by Janusz Czelakowski
1484:vol. 2208, Springer Verlag,
1482:Lecture Notes in Mathematics
746:Algebras as models of logics
708:
1894:. Oxford University Press.
1062:discovered it in Leibniz's
762:, making logic a branch of
285:that satisfies the formula
3818:
2720:SchröderâBernstein theorem
2447:Monadic predicate calculus
2106:Foundations of mathematics
1925:(2nd ed.). Springer.
1865:. Oxford University Press.
1762:Studies in Algebraic Logic
1625:A Survey of Symbolic Logic
1459:Cambridge University Press
1322:non-enlargeable rectangles
1196:of two or more variables.
1043:
884:, including the canonical
817:, equate a formula with a
512:{\displaystyle {\bar {R}}}
199:. There are two relations
173:using Boolean arithmetic.
3766:
3753:Philosophy of mathematics
3702:Automated theorem proving
2873:
2827:Von NeumannâBernaysâGödel
2468:
1861:Parkinson, G.H.R (1966).
1839:Lenzen, Wolfgang, 2004, "
1788:Brady, Geraldine (2000).
1598:Journal of Symbolic Logic
1290:LindenbaumâTarski algebra
1105:, a student of Peirce at
899:LindenbaumâTarski algebra
65:Works in the more recent
1966:Harvard University Press
1950:Stanley Burris (2015), "
1852:Loemker, Leroy (1969) ,
1115:composition of relations
1107:Johns Hopkins University
995:Complete Boolean algebra
740:abstract algebraic logic
570:), and total relations (
261:so that the composition
249:. The converse relation
145:composition of relations
67:abstract algebraic logic
3403:Self-verifying theories
3224:Tarski's axiomatization
2175:Tarski's undefinability
2170:incompleteness theorems
1943:Ramon Jansana (2011), "
1863:Leibniz: Logical Papers
1856:(2nd ed.), Reidel.
1703:On the Algebra of Logic
1326:formal concept analysis
1194:propositional functions
1009:First-order logic with
1003:predicate functor logic
983:Monadic Boolean algebra
978:monadic predicate logic
750:Algebraic logic treats
476:complementary relations
141:relative multiplication
3777:Mathematics portal
3388:Proof of impossibility
3036:propositional variable
2346:Propositional calculus
1973:Historical perspective
1847:. North-Holland: 1-84.
1688:, via Internet Archive
1455:Relational Mathematics
1416:Eugene Freeman (1934)
1101:in 1879, and in 1883,
1087:Charles Sanders Peirce
882:axiomatic set theories
775:universally quantified
773:Variables are tacitly
720:
661:
611:
564:
513:
454:
392:
353:
318:
227:. The other relation,
109:heterogeneous relation
3646:Kolmogorov complexity
3599:Computably enumerable
3499:Model complete theory
3291:Principia Mathematica
2351:Propositional formula
2180:BanachâTarski paradox
1979:Ivor Grattan-Guinness
1870:Zalta, E. N., 2000, "
1507:Annals of Mathematics
1314:rectangular relations
1240:Principia Mathematica
779:universe of discourse
721:
662:
612:
565:
514:
455:
393:
354:
352:{\displaystyle R^{T}}
319:
171:matrix multiplication
81:Calculus of relations
3594:ChurchâTuring thesis
3581:Computability theory
2790:continuum hypothesis
2308:Square of opposition
2166:Gödel's completeness
1908:non-classical logics
1680:Alexander Macfarlane
1435:Relations and Graphs
1396:. pp. 299â307.
1155:Hochschule Karlsruhe
1095:Alexander Macfarlane
921:propositional logic
811:logically equivalent
769:In algebraic logic:
752:algebraic structures
677:
621:
574:
527:
494:
425:
413:, while a univalent
363:
336:
289:
3748:Mathematical object
3639:P versus NP problem
3604:Computable function
3398:Reverse mathematics
3324:Logical consequence
3201:primitive recursive
3196:elementary function
2969:Free/bound variable
2822:TarskiâGrothendieck
2341:Logical connectives
2271:Logical equivalence
2121:Logical consequence
1962:The Ways of Paradox
1556:Stanford University
1478:Relational Topology
1113:were combined with
850:nonclassical logics
40:algebraic semantics
3546:Transfer principle
3509:Semantics of logic
3494:Categorical theory
3470:Non-standard model
2984:Logical connective
2111:Information theory
2060:Mathematical logic
1743:B. Russell (1903)
1659:Augustus De Morgan
1453:G. Schmidt (2011)
1091:logic of relatives
1083:Augustus De Morgan
1046:Symbolical algebra
716:
657:
607:
560:
509:
480:Augustus De Morgan
450:
388:
349:
314:
21:mathematical logic
3784:
3783:
3716:Abstract category
3519:Theories of truth
3329:Rule of inference
3319:Natural deduction
3300:
3299:
2845:
2844:
2550:Cartesian product
2455:
2454:
2361:Many-valued logic
2336:Boolean functions
2219:Russell's paradox
2194:diagonal argument
2091:First-order logic
1932:978-0-7923-7126-7
1901:978-0-19-853192-0
1490:978-3-319-74451-3
1466:978-0-521-76268-7
1403:978-0-8218-5035-0
1369:Universal algebra
1284:cylindric algebra
1278:relation algebras
1268:, the founder of
1231:Leopold Löwenheim
1229:Some writings by
1178:primitive notions
1037:
1036:
1028:Combinatory logic
1016:Cylindric algebra
990:First-order logic
932:Ćukasiewicz logic
864:Combinatory logic
857:first-order logic
842:proper extensions
711:
692:
651:
633:
604:
586:
557:
542:
506:
359:, or the formula
159:converse relation
44:deductive systems
3809:
3802:History of logic
3775:
3774:
3726:History of logic
3721:Category of sets
3614:Decision problem
3393:Ordinal analysis
3334:Sequent calculus
3232:Boolean algebras
3172:
3171:
3146:
3117:logical/constant
2871:
2857:
2780:ZermeloâFraenkel
2531:Set operations:
2466:
2403:
2234:
2214:LöwenheimâSkolem
2101:Formal semantics
2053:
2046:
2039:
2030:
1936:
1905:
1879:
1866:
1857:
1848:
1835:
1806:
1804:
1803:
1775:
1755:
1749:
1741:
1735:
1733:Internet Archive
1716:
1710:
1695:
1689:
1677:
1671:
1656:
1650:
1638:
1632:
1618:
1609:
1590:
1577:
1564:
1558:
1546:
1540:
1539:
1498:
1492:
1474:
1468:
1451:
1445:
1431:
1425:
1414:
1408:
1407:
1385:
1364:Computer algebra
1170:Bertrand Russell
1111:binary relations
1068:Parkinson (1966)
1032:relation algebra
999:polyadic algebra
966:Interior algebra
907:sentential logic
892:
878:Peano arithmetic
874:Relation algebra
838:Boolean algebras
756:bounded lattices
735:relation algebra
725:
723:
722:
717:
712:
707:
699:
694:
693:
685:
673:
671:
666:
664:
663:
658:
653:
652:
644:
635:
634:
626:
616:
614:
613:
608:
606:
605:
597:
588:
587:
579:
569:
567:
566:
561:
559:
558:
550:
544:
543:
535:
522:
518:
516:
515:
510:
508:
507:
499:
474:The facility of
459:
457:
456:
451:
446:
445:
411:partial function
405:
401:
398:where this time
397:
395:
394:
389:
378:
377:
358:
356:
355:
350:
348:
347:
331:
327:
323:
321:
320:
315:
301:
300:
284:
264:
252:
233:
203:and α from
202:
130:
120:
102:
89:is found in the
75:Czelakowski 2003
71:Leibniz operator
60:Czelakowski 2003
3817:
3816:
3812:
3811:
3810:
3808:
3807:
3806:
3797:Algebraic logic
3787:
3786:
3785:
3780:
3769:
3762:
3707:Category theory
3697:Algebraic logic
3680:
3651:Lambda calculus
3589:Church encoding
3575:
3551:Truth predicate
3407:
3373:Complete theory
3296:
3165:
3161:
3157:
3152:
3144:
2864: and
2860:
2855:
2841:
2817:New Foundations
2785:axiom of choice
2768:
2730:Gödel numbering
2670: and
2662:
2566:
2451:
2401:
2382:
2331:Boolean algebra
2317:
2281:Equiconsistency
2246:Classical logic
2223:
2204:Halting problem
2192: and
2168: and
2156: and
2155:
2150:Theorems (
2145:
2062:
2057:
2020:Algebraic logic
2016:
1993:Algebraic Logic
1933:
1917:
1902:
1889:
1886:
1884:Further reading
1869:
1860:
1851:
1841:Leibnizâs Logic
1838:
1809:
1801:
1799:
1787:
1784:
1779:
1778:
1756:
1752:
1742:
1738:
1717:
1713:
1696:
1692:
1678:
1674:
1657:
1653:
1639:
1635:
1619:
1612:
1607:10.2307/2268577
1591:
1580:
1565:
1561:
1547:
1543:
1520:10.2307/1969375
1502:Roger C. Lyndon
1500:
1499:
1495:
1475:
1471:
1452:
1448:
1432:
1428:
1415:
1411:
1404:
1387:
1386:
1382:
1377:
1354:Boolean algebra
1350:
1212:A. N. Whitehead
1151:Boolean algebra
1048:
1042:
925:Heyting algebra
912:Boolean algebra
895:Logical system
813:. To express a
799:. There are no
781:. There are no
748:
731:
700:
675:
674:
669:
668:
619:
618:
572:
571:
525:
524:
520:
492:
491:
465:Gunther Schmidt
461:Charles Loewner
437:
423:
422:
403:
399:
369:
361:
360:
339:
334:
333:
329:
325:
292:
287:
286:
282:
279:
271:Socratic method
262:
250:
231:
200:
179:
122:
112:
94:
87:binary relation
83:
25:algebraic logic
17:
12:
11:
5:
3815:
3813:
3805:
3804:
3799:
3789:
3788:
3782:
3781:
3767:
3764:
3763:
3761:
3760:
3755:
3750:
3745:
3740:
3739:
3738:
3728:
3723:
3718:
3709:
3704:
3699:
3694:
3692:Abstract logic
3688:
3686:
3682:
3681:
3679:
3678:
3673:
3671:Turing machine
3668:
3663:
3658:
3653:
3648:
3643:
3642:
3641:
3636:
3631:
3626:
3621:
3611:
3609:Computable set
3606:
3601:
3596:
3591:
3585:
3583:
3577:
3576:
3574:
3573:
3568:
3563:
3558:
3553:
3548:
3543:
3538:
3537:
3536:
3531:
3526:
3516:
3511:
3506:
3504:Satisfiability
3501:
3496:
3491:
3490:
3489:
3479:
3478:
3477:
3467:
3466:
3465:
3460:
3455:
3450:
3445:
3435:
3434:
3433:
3428:
3421:Interpretation
3417:
3415:
3409:
3408:
3406:
3405:
3400:
3395:
3390:
3385:
3375:
3370:
3369:
3368:
3367:
3366:
3356:
3351:
3341:
3336:
3331:
3326:
3321:
3316:
3310:
3308:
3302:
3301:
3298:
3297:
3295:
3294:
3286:
3285:
3284:
3283:
3278:
3277:
3276:
3271:
3266:
3246:
3245:
3244:
3242:minimal axioms
3239:
3228:
3227:
3226:
3215:
3214:
3213:
3208:
3203:
3198:
3193:
3188:
3175:
3173:
3154:
3153:
3151:
3150:
3149:
3148:
3136:
3131:
3130:
3129:
3124:
3119:
3114:
3104:
3099:
3094:
3089:
3088:
3087:
3082:
3072:
3071:
3070:
3065:
3060:
3055:
3045:
3040:
3039:
3038:
3033:
3028:
3018:
3017:
3016:
3011:
3006:
3001:
2996:
2991:
2981:
2976:
2971:
2966:
2965:
2964:
2959:
2954:
2949:
2939:
2934:
2932:Formation rule
2929:
2924:
2923:
2922:
2917:
2907:
2906:
2905:
2895:
2890:
2885:
2880:
2874:
2868:
2851:Formal systems
2847:
2846:
2843:
2842:
2840:
2839:
2834:
2829:
2824:
2819:
2814:
2809:
2804:
2799:
2794:
2793:
2792:
2787:
2776:
2774:
2770:
2769:
2767:
2766:
2765:
2764:
2754:
2749:
2748:
2747:
2740:Large cardinal
2737:
2732:
2727:
2722:
2717:
2703:
2702:
2701:
2696:
2691:
2676:
2674:
2664:
2663:
2661:
2660:
2659:
2658:
2653:
2648:
2638:
2633:
2628:
2623:
2618:
2613:
2608:
2603:
2598:
2593:
2588:
2583:
2577:
2575:
2568:
2567:
2565:
2564:
2563:
2562:
2557:
2552:
2547:
2542:
2537:
2529:
2528:
2527:
2522:
2512:
2507:
2505:Extensionality
2502:
2500:Ordinal number
2497:
2487:
2482:
2481:
2480:
2469:
2463:
2457:
2456:
2453:
2452:
2450:
2449:
2444:
2439:
2434:
2429:
2424:
2419:
2418:
2417:
2407:
2406:
2405:
2392:
2390:
2384:
2383:
2381:
2380:
2379:
2378:
2373:
2368:
2358:
2353:
2348:
2343:
2338:
2333:
2327:
2325:
2319:
2318:
2316:
2315:
2310:
2305:
2300:
2295:
2290:
2285:
2284:
2283:
2273:
2268:
2263:
2258:
2253:
2248:
2242:
2240:
2231:
2225:
2224:
2222:
2221:
2216:
2211:
2206:
2201:
2196:
2184:Cantor's
2182:
2177:
2172:
2162:
2160:
2147:
2146:
2144:
2143:
2138:
2133:
2128:
2123:
2118:
2113:
2108:
2103:
2098:
2093:
2088:
2083:
2082:
2081:
2070:
2068:
2064:
2063:
2058:
2056:
2055:
2048:
2041:
2033:
2027:
2026:
2015:
2014:External links
2012:
2011:
2010:
1989:Irving Anellis
1986:
1970:
1969:
1955:
1948:
1941:
1931:
1919:Hajnal Andréka
1915:
1900:
1885:
1882:
1881:
1880:
1867:
1858:
1849:
1836:
1807:
1783:
1780:
1777:
1776:
1758:Helena Rasiowa
1750:
1736:
1719:Ernst Schröder
1711:
1698:Christine Ladd
1690:
1672:
1651:
1633:
1621:Clarence Lewis
1610:
1578:
1559:
1541:
1514:(3): 707â729.
1493:
1469:
1446:
1426:
1409:
1402:
1379:
1378:
1376:
1373:
1372:
1371:
1366:
1361:
1359:Codd's theorem
1356:
1349:
1346:
1316:by taking the
1298:Jacques Riguet
1294:
1293:
1288:Co-discovered
1286:
1280:
1252:logical matrix
1248:Helena Rasiowa
1235:Thoralf Skolem
1216:symbolic logic
1208:Giuseppe Peano
1190:Clarence Lewis
1163:Schröder rules
1159:Ernst Schröder
1103:Christine Ladd
1097:published his
1072:Loemker (1969)
1060:Louis Couturat
1056:Clarence Lewis
1041:
1038:
1035:
1034:
1025:
1019:
1018:
1013:
1006:
1005:
992:
986:
985:
980:
969:
968:
963:
953:
952:
947:
940:
939:
934:
928:
927:
922:
919:Intuitionistic
915:
914:
909:
902:
901:
896:
890:
889:
871:
830:
829:
822:
804:
790:
747:
744:
730:
727:
715:
710:
706:
703:
697:
691:
688:
682:
656:
650:
647:
641:
638:
632:
629:
603:
600:
594:
591:
585:
582:
556:
553:
547:
541:
538:
532:
505:
502:
486:to introduce
484:Ernst Schröder
449:
444:
440:
436:
433:
430:
415:total relation
387:
384:
381:
376:
372:
368:
346:
342:
313:
310:
307:
304:
299:
295:
278:
275:
243:presupposition
178:
175:
163:logical matrix
157:refers to the
85:A homogeneous
82:
79:
48:representation
29:free variables
15:
13:
10:
9:
6:
4:
3:
2:
3814:
3803:
3800:
3798:
3795:
3794:
3792:
3779:
3778:
3773:
3765:
3759:
3756:
3754:
3751:
3749:
3746:
3744:
3741:
3737:
3734:
3733:
3732:
3729:
3727:
3724:
3722:
3719:
3717:
3713:
3710:
3708:
3705:
3703:
3700:
3698:
3695:
3693:
3690:
3689:
3687:
3683:
3677:
3674:
3672:
3669:
3667:
3666:Recursive set
3664:
3662:
3659:
3657:
3654:
3652:
3649:
3647:
3644:
3640:
3637:
3635:
3632:
3630:
3627:
3625:
3622:
3620:
3617:
3616:
3615:
3612:
3610:
3607:
3605:
3602:
3600:
3597:
3595:
3592:
3590:
3587:
3586:
3584:
3582:
3578:
3572:
3569:
3567:
3564:
3562:
3559:
3557:
3554:
3552:
3549:
3547:
3544:
3542:
3539:
3535:
3532:
3530:
3527:
3525:
3522:
3521:
3520:
3517:
3515:
3512:
3510:
3507:
3505:
3502:
3500:
3497:
3495:
3492:
3488:
3485:
3484:
3483:
3480:
3476:
3475:of arithmetic
3473:
3472:
3471:
3468:
3464:
3461:
3459:
3456:
3454:
3451:
3449:
3446:
3444:
3441:
3440:
3439:
3436:
3432:
3429:
3427:
3424:
3423:
3422:
3419:
3418:
3416:
3414:
3410:
3404:
3401:
3399:
3396:
3394:
3391:
3389:
3386:
3383:
3382:from ZFC
3379:
3376:
3374:
3371:
3365:
3362:
3361:
3360:
3357:
3355:
3352:
3350:
3347:
3346:
3345:
3342:
3340:
3337:
3335:
3332:
3330:
3327:
3325:
3322:
3320:
3317:
3315:
3312:
3311:
3309:
3307:
3303:
3293:
3292:
3288:
3287:
3282:
3281:non-Euclidean
3279:
3275:
3272:
3270:
3267:
3265:
3264:
3260:
3259:
3257:
3254:
3253:
3251:
3247:
3243:
3240:
3238:
3235:
3234:
3233:
3229:
3225:
3222:
3221:
3220:
3216:
3212:
3209:
3207:
3204:
3202:
3199:
3197:
3194:
3192:
3189:
3187:
3184:
3183:
3181:
3177:
3176:
3174:
3169:
3163:
3158:Example
3155:
3147:
3142:
3141:
3140:
3137:
3135:
3132:
3128:
3125:
3123:
3120:
3118:
3115:
3113:
3110:
3109:
3108:
3105:
3103:
3100:
3098:
3095:
3093:
3090:
3086:
3083:
3081:
3078:
3077:
3076:
3073:
3069:
3066:
3064:
3061:
3059:
3056:
3054:
3051:
3050:
3049:
3046:
3044:
3041:
3037:
3034:
3032:
3029:
3027:
3024:
3023:
3022:
3019:
3015:
3012:
3010:
3007:
3005:
3002:
3000:
2997:
2995:
2992:
2990:
2987:
2986:
2985:
2982:
2980:
2977:
2975:
2972:
2970:
2967:
2963:
2960:
2958:
2955:
2953:
2950:
2948:
2945:
2944:
2943:
2940:
2938:
2935:
2933:
2930:
2928:
2925:
2921:
2918:
2916:
2915:by definition
2913:
2912:
2911:
2908:
2904:
2901:
2900:
2899:
2896:
2894:
2891:
2889:
2886:
2884:
2881:
2879:
2876:
2875:
2872:
2869:
2867:
2863:
2858:
2852:
2848:
2838:
2835:
2833:
2830:
2828:
2825:
2823:
2820:
2818:
2815:
2813:
2810:
2808:
2805:
2803:
2802:KripkeâPlatek
2800:
2798:
2795:
2791:
2788:
2786:
2783:
2782:
2781:
2778:
2777:
2775:
2771:
2763:
2760:
2759:
2758:
2755:
2753:
2750:
2746:
2743:
2742:
2741:
2738:
2736:
2733:
2731:
2728:
2726:
2723:
2721:
2718:
2715:
2711:
2707:
2704:
2700:
2697:
2695:
2692:
2690:
2687:
2686:
2685:
2681:
2678:
2677:
2675:
2673:
2669:
2665:
2657:
2654:
2652:
2649:
2647:
2646:constructible
2644:
2643:
2642:
2639:
2637:
2634:
2632:
2629:
2627:
2624:
2622:
2619:
2617:
2614:
2612:
2609:
2607:
2604:
2602:
2599:
2597:
2594:
2592:
2589:
2587:
2584:
2582:
2579:
2578:
2576:
2574:
2569:
2561:
2558:
2556:
2553:
2551:
2548:
2546:
2543:
2541:
2538:
2536:
2533:
2532:
2530:
2526:
2523:
2521:
2518:
2517:
2516:
2513:
2511:
2508:
2506:
2503:
2501:
2498:
2496:
2492:
2488:
2486:
2483:
2479:
2476:
2475:
2474:
2471:
2470:
2467:
2464:
2462:
2458:
2448:
2445:
2443:
2440:
2438:
2435:
2433:
2430:
2428:
2425:
2423:
2420:
2416:
2413:
2412:
2411:
2408:
2404:
2399:
2398:
2397:
2394:
2393:
2391:
2389:
2385:
2377:
2374:
2372:
2369:
2367:
2364:
2363:
2362:
2359:
2357:
2354:
2352:
2349:
2347:
2344:
2342:
2339:
2337:
2334:
2332:
2329:
2328:
2326:
2324:
2323:Propositional
2320:
2314:
2311:
2309:
2306:
2304:
2301:
2299:
2296:
2294:
2291:
2289:
2286:
2282:
2279:
2278:
2277:
2274:
2272:
2269:
2267:
2264:
2262:
2259:
2257:
2254:
2252:
2251:Logical truth
2249:
2247:
2244:
2243:
2241:
2239:
2235:
2232:
2230:
2226:
2220:
2217:
2215:
2212:
2210:
2207:
2205:
2202:
2200:
2197:
2195:
2191:
2187:
2183:
2181:
2178:
2176:
2173:
2171:
2167:
2164:
2163:
2161:
2159:
2153:
2148:
2142:
2139:
2137:
2134:
2132:
2129:
2127:
2124:
2122:
2119:
2117:
2114:
2112:
2109:
2107:
2104:
2102:
2099:
2097:
2094:
2092:
2089:
2087:
2084:
2080:
2077:
2076:
2075:
2072:
2071:
2069:
2065:
2061:
2054:
2049:
2047:
2042:
2040:
2035:
2034:
2031:
2025:
2021:
2018:
2017:
2013:
2009:
2005:
2002:
1998:
1994:
1990:
1987:
1984:
1980:
1977:
1976:
1975:
1974:
1967:
1963:
1959:
1958:Willard Quine
1956:
1953:
1949:
1946:
1942:
1939:
1934:
1928:
1924:
1920:
1916:
1914:
1909:
1903:
1897:
1893:
1888:
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1798:on 2009-04-02
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1729:B. G. Teubner
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1600:
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1594:
1593:Alfred Tarski
1589:
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1572:
1571:Studia Logica
1568:
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1443:3-540-56254-0
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1347:
1345:
1343:
1339:
1335:
1334:Lenzen (2004)
1329:
1327:
1323:
1319:
1318:outer product
1315:
1311:
1307:
1306:N. M. Ferrers
1303:
1299:
1291:
1287:
1285:
1281:
1279:
1275:
1274:
1273:
1271:
1270:set theoretic
1267:
1266:Alfred Tarski
1263:
1259:
1255:
1253:
1249:
1246:According to
1244:
1242:
1241:
1236:
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1227:
1225:
1221:
1217:
1213:
1209:
1205:
1204:Gottlob Frege
1201:
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951:
950:Modal algebra
948:
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788:
787:open formulas
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477:
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467:use the term
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106:
103:for some set
101:
97:
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88:
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72:
68:
63:
61:
57:
56:Stone duality
53:
49:
45:
41:
37:
32:
30:
26:
22:
3768:
3696:
3566:Ultraproduct
3413:Model theory
3378:Independence
3314:Formal proof
3306:Proof theory
3289:
3262:
3219:real numbers
3191:second-order
3102:Substitution
2979:Metalanguage
2920:conservative
2893:Axiom schema
2837:Constructive
2807:MorseâKelley
2773:Set theories
2752:Aleph number
2745:inaccessible
2651:Grothendieck
2535:intersection
2422:Higher-order
2410:Second-order
2356:Truth tables
2313:Venn diagram
2096:Formal proof
1992:
1982:
1972:
1971:
1961:
1922:
1891:
1875:
1862:
1853:
1844:
1815:
1811:
1800:. Retrieved
1796:the original
1790:
1761:
1753:
1744:
1739:
1722:
1714:
1708:Google Books
1701:
1693:
1683:
1675:
1664:Formal Logic
1662:
1654:
1645:
1641:George Boole
1636:
1624:
1596:
1574:
1567:Roger Maddux
1562:
1549:Vaughn Pratt
1544:
1511:
1505:
1496:
1477:
1472:
1454:
1449:
1434:
1429:
1417:
1412:
1389:
1383:
1342:Zalta (2000)
1330:
1321:
1313:
1302:difunctional
1295:
1262:model theory
1258:Brady (2000)
1256:
1245:
1238:
1228:
1200:Hugh MacColl
1198:
1167:
1146:
1142:
1134:
1130:
1122:
1118:
1098:
1079:George Boole
1076:
1049:
943:Modal logic
854:
831:
826:Modus ponens
768:
764:order theory
749:
734:
732:
488:equivalences
473:
468:
408:
280:
266:
258:
254:
246:
245:of question
238:
234:
228:
224:
220:
216:
212:
208:
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196:
189:
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154:
152:
149:
140:
127:
123:
117:
113:
104:
99:
95:
84:
64:
33:
24:
18:
3676:Type theory
3624:undecidable
3556:Truth value
3443:equivalence
3122:non-logical
2735:Enumeration
2725:Isomorphism
2672:cardinality
2656:Von Neumann
2621:Ultrafilter
2586:Uncountable
2520:equivalence
2437:Quantifiers
2427:Fixed-point
2396:First-order
2276:Consistency
2261:Proposition
2238:Traditional
2209:Lindström's
2199:Compactness
2141:Type theory
2086:Cardinality
1878:3: 137-183.
1727:, Leibzig:
1669:Hathi Trust
1420:, page 10,
1338:metaphysics
1220:mathematics
1117:. For sets
819:truth value
801:connectives
729:Abstraction
237:holds when
219:holds when
3791:Categories
3487:elementary
3180:arithmetic
3048:Quantifier
3026:functional
2898:Expression
2616:Transitive
2560:identities
2545:complement
2478:hereditary
2461:Set theory
2024:PhilPapers
2008:0444885439
1802:2009-05-15
1721:, (1895),
1480:, page 8,
1375:References
1224:philosophy
1085:. In 1870
1044:See also:
1023:Set theory
937:MV-algebra
905:Classical
868:set theory
848:and other
797:operations
777:over some
273:dialogue.
253:runs from
187:statements
155:conversion
107:, while a
42:for these
3758:Supertask
3661:Recursion
3619:decidable
3453:saturated
3431:of models
3354:deductive
3349:axiomatic
3269:Hilbert's
3256:Euclidean
3237:canonical
3160:axiomatic
3092:Signature
3021:Predicate
2910:Extension
2832:Ackermann
2757:Operation
2636:Universal
2626:Recursive
2601:Singleton
2596:Inhabited
2581:Countable
2571:Types of
2555:power set
2525:partition
2442:Predicate
2388:Predicate
2303:Syllogism
2293:Soundness
2266:Inference
2256:Tautology
2158:paradoxes
1824:1079-8986
1601:6: 73â89
1310:transpose
1282:Invented
1139:power set
880:and most
844:thereof.
815:tautology
709:¯
690:¯
649:¯
631:¯
602:¯
590:⊆
584:¯
555:¯
546:⊆
540:¯
504:¯
478:inspired
432:⊆
380:⊆
306:⊆
277:Functions
194:questions
183:erotetics
167:transpose
137:inclusion
91:power set
3743:Logicism
3736:timeline
3712:Concrete
3571:Validity
3541:T-schema
3534:Kripke's
3529:Tarski's
3524:semantic
3514:Strength
3463:submodel
3458:spectrum
3426:function
3274:Tarski's
3263:Elements
3250:geometry
3206:Robinson
3127:variable
3112:function
3085:spectrum
3075:Sentence
3031:variable
2974:Language
2927:Relation
2888:Automata
2878:Alphabet
2862:language
2716:-jection
2694:codomain
2680:Function
2641:Universe
2611:Infinite
2515:Relation
2298:Validity
2288:Argument
2186:theorem,
2001:Elsevier
1981:, 2000.
1700:(1883),
1682:(1879),
1661:(1847),
1575:: 421-55
1348:See also
1186:textbook
1174:logicism
1168:In 1903
1127:relation
1064:Nachlass
1011:equality
972:Lewis's
807:Formulas
754:, often
419:function
121:, where
3685:Related
3482:Diagram
3380: (
3359:Hilbert
3344:Systems
3339:Theorem
3217:of the
3162:systems
2942:Formula
2937:Grammar
2853: (
2797:General
2510:Forcing
2495:Element
2415:Monadic
2190:paradox
2131:Theorem
2067:General
1832:3094793
1782:Sources
1623:(1918)
1554:, from
1536:0037278
1528:1969375
1052:Leibniz
1040:History
834:logical
672:,
469:mapping
215:α
177:Example
3448:finite
3211:Skolem
3164:
3139:Theory
3107:Symbol
3097:String
3080:atomic
2957:ground
2952:closed
2947:atomic
2903:ground
2866:syntax
2762:binary
2689:domain
2606:Finite
2371:finite
2229:Logics
2188:
2136:Theory
2006:
1999:&
1929:
1898:
1830:
1822:
1771:
1534:
1526:
1488:
1464:
1441:
1400:
1222:, and
1210:, and
760:logics
490:using
324:where
36:models
3438:Model
3186:Peano
3043:Proof
2883:Arity
2812:Naive
2699:image
2631:Fuzzy
2591:Empty
2540:union
2485:Class
2126:Model
2116:Lemma
2074:Axiom
1938:Draft
1828:JSTOR
1524:JSTOR
1184:in a
1129:over
957:Lewis
846:Modal
793:Terms
417:is a
241:is a
3561:Type
3364:list
3168:list
3145:list
3134:Term
3068:rank
2962:open
2856:list
2668:Maps
2573:sets
2432:Free
2402:list
2152:list
2079:list
2004:ISBN
1927:ISBN
1896:ISBN
1820:ISSN
1769:ISBN
1731:via
1706:via
1486:ISBN
1462:ISBN
1439:ISBN
1398:ISBN
1233:and
1133:and
1125:, a
1121:and
1081:and
1070:and
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3230:of
3178:of
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2473:Set
2022:at
1874:,"
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1188:by
1141:of
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840:or
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