3533:
1195:
1442:.) However, it is not derivable, because it depends on the structure of the derivation of the premise. Because of this, derivability is stable under additions to the proof system, whereas admissibility is not. To see the difference, suppose the following nonsense rule were added to the proof system:
1075:
1007:. A derivable rule is one whose conclusion can be derived from its premises using the other rules. An admissible rule is one whose conclusion holds whenever the premises hold. All derivable rules are admissible. To appreciate the difference, consider the following set of rules for defining the
421:
This expression states that whenever in the course of some logical derivation the given premises have been obtained, the specified conclusion can be taken for granted as well. The exact formal language that is used to describe both premises and conclusions depends on the actual context of the
1310:
1398:
568:, the premises and conclusion of the inference rules are simply formulae of some language, usually employing metavariables. For graphical compactness of the presentation and to emphasize the distinction between axioms and rules of inference, this section uses the
646:
347:, it preserves a general designation. But a rule of inference's action is purely syntactic, and does not need to preserve any semantic property: any function from sets of formulae to formulae counts as a rule of inference. Usually only rules that are
1505:
1550:. The brittleness of admissibility comes from the way it is proved: since the proof can induct on the structure of the derivations of the premises, extensions to the system add new cases to this proof, which may no longer hold.
710:
1190:{\displaystyle {\begin{matrix}{\begin{array}{c}\\\hline {\mathbf {0} \,\,{\mathsf {nat}}}\end{array}}&{\begin{array}{c}{n\,\,{\mathsf {nat}}}\\\hline {\mathbf {s(} n\mathbf {)} \,\,{\mathsf {nat}}}\end{array}}\end{matrix}}}
965:
This sequence differs from classical logic by the change in axiom 2 and the addition of axiom 4. The classical deduction theorem does not hold for this logic, however a modified form does hold, namely
484:
1224:
1321:
1548:
355:
for determining whether any given formula is the conclusion of a given set of formulae according to the rule. An example of a rule that is not effective in this sense is the infinitary
1315:
Its derivation is the composition of two uses of the successor rule above. The following rule for asserting the existence of a predecessor for any nonzero number is merely admissible:
597:
1440:
1047:
544:. Any derivation has only one final conclusion, which is the statement proved or derived. If premises are left unsatisfied in the derivation, then the derivation is a proof of a
1912:
1510:
In this new system, the double-successor rule is still derivable. However, the rule for finding the predecessor is no longer admissible, because there is no way to derive
719:
can be expressed using just negation (¬), implication (→) and propositional symbols. A well-known axiomatization, comprising three axiom schemata and one inference rule (
2587:
590:
529:. In the rule (schema) above, the metavariables A and B can be instantiated to any element of the universe (or sometimes, by convention, a restricted subset such as
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506:
1448:
1067:
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1811:
653:
218:
1701:
Logic and
Scientific Methods: Volume One of the Tenth International Congress of Logic, Methodology and Philosophy of Science, Florence, August 1995
1218:
is. In this proof system, the following rule, demonstrating that the second successor of a natural number is also a natural number, is derivable:
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3142:
1777:
1744:
1708:
1675:
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1930:
2997:
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It may seem redundant to have two notions of inference in this case, ⊢ and →. In classical propositional logic, they indeed coincide; the
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1935:
855:
851:
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1926:
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1804:
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2233:
1974:
3562:
1305:{\displaystyle {\begin{array}{c}{n\,\,{\mathsf {nat}}}\\\hline {\mathbf {s(s(} n\mathbf {))} \,\,{\mathsf {nat}}}\end{array}}}
3496:
3198:
2961:
2956:
2781:
2202:
1886:
455:
1393:{\displaystyle {\begin{array}{c}{\mathbf {s(} n\mathbf {)} \,\,{\mathsf {nat}}}\\\hline {n\,\,{\mathsf {nat}}}\end{array}}}
328:
takes two premises, one in the form "If p then q" and another in the form "p", and returns the conclusion "q". The rule is
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3274:
3191:
2904:
2835:
2712:
1954:
232:
2562:
1407:. (To prove that this rule is admissible, assume a derivation of the premise and induct on it to produce a derivation of
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2161:
2567:
3557:
2899:
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1896:
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1513:
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2813:
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2175:
1866:
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1969:
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641:{\displaystyle {\begin{array}{c}{\text{Premise }}1\\{\text{Premise }}2\\\hline {\text{Conclusion}}\end{array}}}
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2449:
2416:
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2068:
1979:
1583:
826:. There is however a distinction worth emphasizing even in this case: the first notation describes a
187:
180:
125:
340:), in the sense that if the premises are true (under an interpretation), then so is the conclusion.
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3399:
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402:(and many related areas), rules of inference are usually given in the following standard form:
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2156:
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2016:
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807:
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56:
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329:
1764:
An introduction to many-valued and fuzzy logic: semantics, algebras, and derivation systems
1731:
An introduction to many-valued and fuzzy logic: semantics, algebras, and derivation systems
1500:{\displaystyle {\begin{array}{c}\\\hline {\mathbf {s(-3)} \,\,{\mathsf {nat}}}\end{array}}}
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3457:
3411:
3394:
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3311:
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2148:
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2006:
1964:
1613:
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994:
540:
A proof system is formed from a set of rules chained together to form proofs, also called
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2396:
2386:
2356:
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2011:
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1608:
861:
For some non-classical logics, the deduction theorem does not hold. For example, the
847:
705:{\displaystyle ({\text{Premise }}1),({\text{Premise }}2)\vdash ({\text{Conclusion}})}
374:
144:
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3173:
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62:
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1846:
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In a set of rules, an inference rule could be redundant in the sense that it is
846:
in this case), there is no deduction or inference. This point is illustrated in
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17:
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2192:
1998:
1715:
3518:
3421:
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132:
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Systematic logical process capable of deriving a conclusion from hypotheses
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3301:
2749:
2454:
2048:
422:
derivations. In a simple case, one may use logical formulae, such as in:
343:
Typically, a rule of inference preserves truth, a semantic property. In
3099:
1891:
1554:
1453:
1326:
1229:
1119:
1084:
569:
126:
830:, that is an activity of passing from sentences to sentences, whereas
1789:
106:
2643:
1989:
1834:
592:) instead of a vertical presentation of rules. In this notation,
284:
1793:
311:
consisting of a function which takes premises, analyzes their
1403:
This is a true fact of natural numbers, as can be proven by
842:, implication in this case. Without an inference rule (like
1633:
Boolos, George; Burgess, John; Jeffrey, Richard C. (2007).
195:
153:
119:
74:
1699:; Kees Doets; Daniele Mundici; Johan van Benthem (eds.).
145:
113:
226:
181:
87:
63:
1080:
479:{\displaystyle {\underline {A\quad \quad \quad }}\,\!}
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1451:
1413:
1324:
1227:
1078:
1055:
1020:
656:
600:
578:
560:
Example: Hilbert systems for two propositional logics
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2610:
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1997:
1988:
1910:
1827:
858:to resolve the paradox introduced in the dialogue.
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1499:
1434:
1392:
1304:
1189:
1061:
1041:
704:
640:
584:
500:
478:
443:
1639:. Cambridge: Cambridge University Press. p.
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1334:
1278:
1275:
1267:
1261:
1159:
1151:
497:
475:
139:
1543:{\displaystyle \mathbf {-3} \,\,{\mathsf {nat}}}
1204:is a natural number, and the second states that
409:
351:are important; i.e. rules such that there is an
219:
188:
80:
1805:
521:. Rules of inference are often formulated as
233:
202:
100:
93:
8:
2631:
2226:
1994:
1812:
1798:
1790:
1670:. Cambridge University Press. p. 12.
322:For example, the rule of inference called
29:
1528:
1527:
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1412:
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1366:
1349:
1348:
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1323:
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1026:
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1024:
1019:
694:
677:
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629:
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605:
601:
599:
577:
492:
474:
459:
457:
430:
336:(as well as the semantics of many other
1625:
250:
243:
169:
46:
39:
32:
1768:. Cambridge University Press. p.
1735:. Cambridge University Press. p.
1693:"Logical consequence: a turn in style"
1557:of a proof system. For instance, in a
1553:Admissible rules can be thought of as
1535:
1532:
1529:
1487:
1484:
1481:
1427:
1424:
1421:
1380:
1377:
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1173:
1170:
1167:
1136:
1133:
1130:
1106:
1103:
1100:
1034:
1031:
1028:
386:uses rules of inference to deal with
7:
1716:preprint (with different pagination)
1435:{\displaystyle n\,\,{\mathsf {nat}}}
1042:{\displaystyle n\,\,{\mathsf {nat}}}
852:What the Tortoise Said to Achilles
715:The formal language for classical
415: Premise#n
25:
1667:Theories of Programming Languages
332:with respect to the semantics of
3531:
1521:
1518:
1470:
1467:
1461:
1331:
1264:
1258:
1148:
1092:
856:Bertrand Russell and Peter Winch
854:", as well as later attempts by
838:is simply a formula made with a
467:
466:
465:
315:, and returns a conclusion (or
989:Admissibility and derivability
699:
691:
685:
674:
668:
657:
435:
362:Popular rules of inference in
1:
3492:History of mathematical logic
3417:Primitive recursive function
1200:The first rule states that
3604:
2481:Schröder–Bernstein theorem
2208:Monadic predicate calculus
1867:Foundations of mathematics
1664:John C. Reynolds (2009) .
1604:List of rules of inference
992:
270:Existential generalization
75:Biconditional introduction
3527:
3514:Philosophy of mathematics
3463:Automated theorem proving
2634:
2588:Von Neumann–Bernays–Gödel
2229:
1760:Bergmann, Merrie (2008).
1727:Bergmann, Merrie (2008).
1703:. Springer. p. 290.
1697:Maria Luisa Dalla Chiara
261:Universal generalization
101:Disjunction introduction
88:Conjunction introduction
58:Implication introduction
3164:Self-verifying theories
2985:Tarski's axiomatization
1936:Tarski's undefinability
1931:incompleteness theorems
1636:Computability and logic
1214:is a natural number if
869:can be axiomatized as:
585:{\displaystyle \vdash }
556:the conclusion holds."
418: Conclusion
3563:Propositional calculus
3538:Mathematics portal
3149:Proof of impossibility
2797:propositional variable
2107:Propositional calculus
1544:
1501:
1436:
1394:
1306:
1191:
1069:is a natural number):
1063:
1049:asserts the fact that
1043:
706:
642:
586:
502:
480:
445:
444:{\displaystyle A\to B}
407: Premise#2
405: Premise#1
120:hypothetical syllogism
41:Propositional calculus
3407:Kolmogorov complexity
3360:Computably enumerable
3260:Model complete theory
3052:Principia Mathematica
2112:Propositional formula
1941:Banach–Tarski paradox
1545:
1502:
1437:
1395:
1307:
1192:
1064:
1044:
707:
643:
587:
503:
481:
446:
162:Negation introduction
155:modus ponendo tollens
3355:Church–Turing thesis
3342:Computability theory
2551:continuum hypothesis
2069:Square of opposition
1927:Gödel's completeness
1691:Kosta Dosen (1996).
1584:Argumentation scheme
1569:rule is admissible.
1514:
1449:
1411:
1322:
1225:
1076:
1053:
1018:
850:'s dialogue called "
654:
598:
576:
537:of inference rules.
491:
456:
429:
338:non-classical logics
220:Material implication
171:Rules of replacement
34:Transformation rules
3588:Logical expressions
3509:Mathematical object
3400:P versus NP problem
3365:Computable function
3159:Reverse mathematics
3085:Logical consequence
2962:primitive recursive
2957:elementary function
2730:Free/bound variable
2583:Tarski–Grothendieck
2102:Logical connectives
2032:Logical equivalence
1882:Logical consequence
1594:Inference objection
1589:Immediate inference
717:propositional logic
552:the premises hold,
519:propositional logic
501:{\displaystyle B\!}
388:logical quantifiers
364:propositional logic
353:effective procedure
305:transformation rule
293:deductive reasoning
289:philosophy of logic
133:destructive dilemma
3558:Rules of inference
3307:Transfer principle
3270:Semantics of logic
3255:Categorical theory
3231:Non-standard model
2745:Logical connective
1872:Information theory
1821:Mathematical logic
1540:
1497:
1495:
1432:
1390:
1388:
1302:
1300:
1187:
1185:
1181:
1114:
1059:
1039:
863:three-valued logic
840:logical connective
702:
638:
636:
582:
498:
476:
472:
441:
291:, specifically in
252:Rules of inference
48:Rules of inference
3545:
3544:
3477:Abstract category
3280:Theories of truth
3090:Rule of inference
3080:Natural deduction
3061:
3060:
2606:
2605:
2311:Cartesian product
2216:
2215:
2122:Many-valued logic
2097:Boolean functions
1980:Russell's paradox
1955:diagonal argument
1852:First-order logic
1779:978-0-521-88128-9
1746:978-0-521-88128-9
1710:978-0-7923-4383-7
1677:978-0-521-10697-9
1650:978-0-521-87752-7
1062:{\displaystyle n}
973:if and only if ⊢
818:if and only if ⊢
808:deduction theorem
697:
680:
663:
632:
620:
608:
460:
345:many-valued logic
297:rule of inference
281:
280:
16:(Redirected from
3595:
3536:
3535:
3487:History of logic
3482:Category of sets
3375:Decision problem
3154:Ordinal analysis
3095:Sequent calculus
2993:Boolean algebras
2933:
2932:
2907:
2878:logical/constant
2632:
2618:
2541:Zermelo–Fraenkel
2292:Set operations:
2227:
2164:
1995:
1975:Löwenheim–Skolem
1862:Formal semantics
1814:
1807:
1800:
1791:
1784:
1783:
1767:
1757:
1751:
1750:
1734:
1724:
1718:
1714:
1688:
1682:
1681:
1661:
1655:
1654:
1630:
1559:sequent calculus
1549:
1547:
1546:
1541:
1539:
1538:
1524:
1506:
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1503:
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1496:
1492:
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1038:
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711:
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703:
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678:
664:
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621:
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609:
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591:
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583:
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499:
485:
483:
482:
477:
473:
468:
450:
448:
447:
442:
235:
228:
221:
209:De Morgan's laws
204:
197:
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183:
157:
149:
141:
134:
128:
121:
115:
108:
102:
95:
89:
82:
76:
69:
59:
30:
21:
3603:
3602:
3598:
3597:
3596:
3594:
3593:
3592:
3548:
3547:
3546:
3541:
3530:
3523:
3468:Category theory
3458:Algebraic logic
3441:
3412:Lambda calculus
3350:Church encoding
3336:
3312:Truth predicate
3168:
3134:Complete theory
3057:
2926:
2922:
2918:
2913:
2905:
2625: and
2621:
2616:
2602:
2578:New Foundations
2546:axiom of choice
2529:
2491:Gödel numbering
2431: and
2423:
2327:
2212:
2162:
2143:
2092:Boolean algebra
2078:
2042:Equiconsistency
2007:Classical logic
1984:
1965:Halting problem
1953: and
1929: and
1917: and
1916:
1911:Theorems (
1906:
1823:
1818:
1788:
1787:
1780:
1759:
1758:
1754:
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1689:
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1678:
1663:
1662:
1658:
1651:
1632:
1631:
1627:
1622:
1614:Structural rule
1575:
1563:cut elimination
1512:
1511:
1494:
1493:
1456:
1447:
1446:
1409:
1408:
1387:
1386:
1363:
1362:
1320:
1319:
1299:
1298:
1253:
1252:
1223:
1222:
1184:
1183:
1180:
1179:
1143:
1142:
1116:
1113:
1112:
1087:
1074:
1073:
1051:
1050:
1016:
1015:
1009:natural numbers
997:
995:Admissible rule
991:
963:
946:
929:
911:
885:
804:
787:
769:
739:
652:
651:
635:
634:
626:
625:
614:
613:
596:
595:
574:
573:
562:
489:
488:
461:
454:
453:
427:
426:
417:
413:
408:
406:
396:
384:predicate logic
334:classical logic
245:Predicate logic
239:
203:Double negation
57:
28:
23:
22:
18:Inference rules
15:
12:
11:
5:
3601:
3599:
3591:
3590:
3585:
3580:
3575:
3573:Syntax (logic)
3570:
3568:Formal systems
3565:
3560:
3550:
3549:
3543:
3542:
3528:
3525:
3524:
3522:
3521:
3516:
3511:
3506:
3501:
3500:
3499:
3489:
3484:
3479:
3470:
3465:
3460:
3455:
3453:Abstract logic
3449:
3447:
3443:
3442:
3440:
3439:
3434:
3432:Turing machine
3429:
3424:
3419:
3414:
3409:
3404:
3403:
3402:
3397:
3392:
3387:
3382:
3372:
3370:Computable set
3367:
3362:
3357:
3352:
3346:
3344:
3338:
3337:
3335:
3334:
3329:
3324:
3319:
3314:
3309:
3304:
3299:
3298:
3297:
3292:
3287:
3277:
3272:
3267:
3265:Satisfiability
3262:
3257:
3252:
3251:
3250:
3240:
3239:
3238:
3228:
3227:
3226:
3221:
3216:
3211:
3206:
3196:
3195:
3194:
3189:
3182:Interpretation
3178:
3176:
3170:
3169:
3167:
3166:
3161:
3156:
3151:
3146:
3136:
3131:
3130:
3129:
3128:
3127:
3117:
3112:
3102:
3097:
3092:
3087:
3082:
3077:
3071:
3069:
3063:
3062:
3059:
3058:
3056:
3055:
3047:
3046:
3045:
3044:
3039:
3038:
3037:
3032:
3027:
3007:
3006:
3005:
3003:minimal axioms
3000:
2989:
2988:
2987:
2976:
2975:
2974:
2969:
2964:
2959:
2954:
2949:
2936:
2934:
2915:
2914:
2912:
2911:
2910:
2909:
2897:
2892:
2891:
2890:
2885:
2880:
2875:
2865:
2860:
2855:
2850:
2849:
2848:
2843:
2833:
2832:
2831:
2826:
2821:
2816:
2806:
2801:
2800:
2799:
2794:
2789:
2779:
2778:
2777:
2772:
2767:
2762:
2757:
2752:
2742:
2737:
2732:
2727:
2726:
2725:
2720:
2715:
2710:
2700:
2695:
2693:Formation rule
2690:
2685:
2684:
2683:
2678:
2668:
2667:
2666:
2656:
2651:
2646:
2641:
2635:
2629:
2612:Formal systems
2608:
2607:
2604:
2603:
2601:
2600:
2595:
2590:
2585:
2580:
2575:
2570:
2565:
2560:
2555:
2554:
2553:
2548:
2537:
2535:
2531:
2530:
2528:
2527:
2526:
2525:
2515:
2510:
2509:
2508:
2501:Large cardinal
2498:
2493:
2488:
2483:
2478:
2464:
2463:
2462:
2457:
2452:
2437:
2435:
2425:
2424:
2422:
2421:
2420:
2419:
2414:
2409:
2399:
2394:
2389:
2384:
2379:
2374:
2369:
2364:
2359:
2354:
2349:
2344:
2338:
2336:
2329:
2328:
2326:
2325:
2324:
2323:
2318:
2313:
2308:
2303:
2298:
2290:
2289:
2288:
2283:
2273:
2268:
2266:Extensionality
2263:
2261:Ordinal number
2258:
2248:
2243:
2242:
2241:
2230:
2224:
2218:
2217:
2214:
2213:
2211:
2210:
2205:
2200:
2195:
2190:
2185:
2180:
2179:
2178:
2168:
2167:
2166:
2153:
2151:
2145:
2144:
2142:
2141:
2140:
2139:
2134:
2129:
2119:
2114:
2109:
2104:
2099:
2094:
2088:
2086:
2080:
2079:
2077:
2076:
2071:
2066:
2061:
2056:
2051:
2046:
2045:
2044:
2034:
2029:
2024:
2019:
2014:
2009:
2003:
2001:
1992:
1986:
1985:
1983:
1982:
1977:
1972:
1967:
1962:
1957:
1945:Cantor's
1943:
1938:
1933:
1923:
1921:
1908:
1907:
1905:
1904:
1899:
1894:
1889:
1884:
1879:
1874:
1869:
1864:
1859:
1854:
1849:
1844:
1843:
1842:
1831:
1829:
1825:
1824:
1819:
1817:
1816:
1809:
1802:
1794:
1786:
1785:
1778:
1752:
1745:
1719:
1709:
1683:
1676:
1656:
1649:
1624:
1623:
1621:
1618:
1617:
1616:
1611:
1606:
1601:
1599:Law of thought
1596:
1591:
1586:
1581:
1574:
1571:
1537:
1534:
1531:
1523:
1520:
1508:
1507:
1489:
1486:
1483:
1475:
1472:
1469:
1466:
1463:
1458:
1457:
1454:
1429:
1426:
1423:
1416:
1401:
1400:
1382:
1379:
1376:
1369:
1365:
1364:
1358:
1355:
1352:
1344:
1340:
1336:
1333:
1328:
1327:
1313:
1312:
1294:
1291:
1288:
1280:
1277:
1273:
1269:
1266:
1263:
1260:
1255:
1254:
1248:
1245:
1242:
1235:
1231:
1230:
1198:
1197:
1175:
1172:
1169:
1161:
1157:
1153:
1150:
1145:
1144:
1138:
1135:
1132:
1125:
1121:
1120:
1117:
1108:
1105:
1102:
1094:
1089:
1088:
1085:
1082:
1081:
1058:
1036:
1033:
1030:
1023:
993:Main article:
990:
987:
871:
725:
701:
693:
690:
687:
684:
676:
673:
670:
667:
659:
650:is written as
628:
627:
624:
616:
615:
612:
604:
603:
581:
566:Hilbert system
561:
558:
509:
508:
496:
486:
471:
464:
451:
440:
437:
434:
395:
392:
382:. First-order
380:contraposition
301:inference rule
279:
278:
277:
276:
267:
255:
254:
248:
247:
241:
240:
238:
237:
230:
223:
216:
211:
206:
199:
196:Distributivity
192:
185:
177:
174:
173:
167:
166:
165:
164:
159:
136:
123:
110:
97:
84:
71:
51:
50:
44:
43:
37:
36:
26:
24:
14:
13:
10:
9:
6:
4:
3:
2:
3600:
3589:
3586:
3584:
3581:
3579:
3578:Logical truth
3576:
3574:
3571:
3569:
3566:
3564:
3561:
3559:
3556:
3555:
3553:
3540:
3539:
3534:
3526:
3520:
3517:
3515:
3512:
3510:
3507:
3505:
3502:
3498:
3495:
3494:
3493:
3490:
3488:
3485:
3483:
3480:
3478:
3474:
3471:
3469:
3466:
3464:
3461:
3459:
3456:
3454:
3451:
3450:
3448:
3444:
3438:
3435:
3433:
3430:
3428:
3427:Recursive set
3425:
3423:
3420:
3418:
3415:
3413:
3410:
3408:
3405:
3401:
3398:
3396:
3393:
3391:
3388:
3386:
3383:
3381:
3378:
3377:
3376:
3373:
3371:
3368:
3366:
3363:
3361:
3358:
3356:
3353:
3351:
3348:
3347:
3345:
3343:
3339:
3333:
3330:
3328:
3325:
3323:
3320:
3318:
3315:
3313:
3310:
3308:
3305:
3303:
3300:
3296:
3293:
3291:
3288:
3286:
3283:
3282:
3281:
3278:
3276:
3273:
3271:
3268:
3266:
3263:
3261:
3258:
3256:
3253:
3249:
3246:
3245:
3244:
3241:
3237:
3236:of arithmetic
3234:
3233:
3232:
3229:
3225:
3222:
3220:
3217:
3215:
3212:
3210:
3207:
3205:
3202:
3201:
3200:
3197:
3193:
3190:
3188:
3185:
3184:
3183:
3180:
3179:
3177:
3175:
3171:
3165:
3162:
3160:
3157:
3155:
3152:
3150:
3147:
3144:
3143:from ZFC
3140:
3137:
3135:
3132:
3126:
3123:
3122:
3121:
3118:
3116:
3113:
3111:
3108:
3107:
3106:
3103:
3101:
3098:
3096:
3093:
3091:
3088:
3086:
3083:
3081:
3078:
3076:
3073:
3072:
3070:
3068:
3064:
3054:
3053:
3049:
3048:
3043:
3042:non-Euclidean
3040:
3036:
3033:
3031:
3028:
3026:
3025:
3021:
3020:
3018:
3015:
3014:
3012:
3008:
3004:
3001:
2999:
2996:
2995:
2994:
2990:
2986:
2983:
2982:
2981:
2977:
2973:
2970:
2968:
2965:
2963:
2960:
2958:
2955:
2953:
2950:
2948:
2945:
2944:
2942:
2938:
2937:
2935:
2930:
2924:
2919:Example
2916:
2908:
2903:
2902:
2901:
2898:
2896:
2893:
2889:
2886:
2884:
2881:
2879:
2876:
2874:
2871:
2870:
2869:
2866:
2864:
2861:
2859:
2856:
2854:
2851:
2847:
2844:
2842:
2839:
2838:
2837:
2834:
2830:
2827:
2825:
2822:
2820:
2817:
2815:
2812:
2811:
2810:
2807:
2805:
2802:
2798:
2795:
2793:
2790:
2788:
2785:
2784:
2783:
2780:
2776:
2773:
2771:
2768:
2766:
2763:
2761:
2758:
2756:
2753:
2751:
2748:
2747:
2746:
2743:
2741:
2738:
2736:
2733:
2731:
2728:
2724:
2721:
2719:
2716:
2714:
2711:
2709:
2706:
2705:
2704:
2701:
2699:
2696:
2694:
2691:
2689:
2686:
2682:
2679:
2677:
2676:by definition
2674:
2673:
2672:
2669:
2665:
2662:
2661:
2660:
2657:
2655:
2652:
2650:
2647:
2645:
2642:
2640:
2637:
2636:
2633:
2630:
2628:
2624:
2619:
2613:
2609:
2599:
2596:
2594:
2591:
2589:
2586:
2584:
2581:
2579:
2576:
2574:
2571:
2569:
2566:
2564:
2563:Kripke–Platek
2561:
2559:
2556:
2552:
2549:
2547:
2544:
2543:
2542:
2539:
2538:
2536:
2532:
2524:
2521:
2520:
2519:
2516:
2514:
2511:
2507:
2504:
2503:
2502:
2499:
2497:
2494:
2492:
2489:
2487:
2484:
2482:
2479:
2476:
2472:
2468:
2465:
2461:
2458:
2456:
2453:
2451:
2448:
2447:
2446:
2442:
2439:
2438:
2436:
2434:
2430:
2426:
2418:
2415:
2413:
2410:
2408:
2407:constructible
2405:
2404:
2403:
2400:
2398:
2395:
2393:
2390:
2388:
2385:
2383:
2380:
2378:
2375:
2373:
2370:
2368:
2365:
2363:
2360:
2358:
2355:
2353:
2350:
2348:
2345:
2343:
2340:
2339:
2337:
2335:
2330:
2322:
2319:
2317:
2314:
2312:
2309:
2307:
2304:
2302:
2299:
2297:
2294:
2293:
2291:
2287:
2284:
2282:
2279:
2278:
2277:
2274:
2272:
2269:
2267:
2264:
2262:
2259:
2257:
2253:
2249:
2247:
2244:
2240:
2237:
2236:
2235:
2232:
2231:
2228:
2225:
2223:
2219:
2209:
2206:
2204:
2201:
2199:
2196:
2194:
2191:
2189:
2186:
2184:
2181:
2177:
2174:
2173:
2172:
2169:
2165:
2160:
2159:
2158:
2155:
2154:
2152:
2150:
2146:
2138:
2135:
2133:
2130:
2128:
2125:
2124:
2123:
2120:
2118:
2115:
2113:
2110:
2108:
2105:
2103:
2100:
2098:
2095:
2093:
2090:
2089:
2087:
2085:
2084:Propositional
2081:
2075:
2072:
2070:
2067:
2065:
2062:
2060:
2057:
2055:
2052:
2050:
2047:
2043:
2040:
2039:
2038:
2035:
2033:
2030:
2028:
2025:
2023:
2020:
2018:
2015:
2013:
2012:Logical truth
2010:
2008:
2005:
2004:
2002:
2000:
1996:
1993:
1991:
1987:
1981:
1978:
1976:
1973:
1971:
1968:
1966:
1963:
1961:
1958:
1956:
1952:
1948:
1944:
1942:
1939:
1937:
1934:
1932:
1928:
1925:
1924:
1922:
1920:
1914:
1909:
1903:
1900:
1898:
1895:
1893:
1890:
1888:
1885:
1883:
1880:
1878:
1875:
1873:
1870:
1868:
1865:
1863:
1860:
1858:
1855:
1853:
1850:
1848:
1845:
1841:
1838:
1837:
1836:
1833:
1832:
1830:
1826:
1822:
1815:
1810:
1808:
1803:
1801:
1796:
1795:
1792:
1781:
1775:
1771:
1766:
1765:
1756:
1753:
1748:
1742:
1738:
1733:
1732:
1723:
1720:
1717:
1712:
1706:
1702:
1698:
1694:
1687:
1684:
1679:
1673:
1669:
1668:
1660:
1657:
1652:
1646:
1642:
1638:
1637:
1629:
1626:
1619:
1615:
1612:
1610:
1609:Logical truth
1607:
1605:
1602:
1600:
1597:
1595:
1592:
1590:
1587:
1585:
1582:
1580:
1577:
1576:
1572:
1570:
1568:
1564:
1560:
1556:
1551:
1445:
1444:
1443:
1414:
1406:
1367:
1338:
1318:
1317:
1316:
1271:
1233:
1221:
1220:
1219:
1217:
1213:
1210:
1207:
1203:
1155:
1123:
1072:
1071:
1070:
1056:
1021:
1014:
1010:
1006:
1002:
996:
988:
986:
984:
980:
976:
972:
968:
962:
958:
954:
950:
945:
941:
937:
933:
927:
923:
919:
915:
909:
905:
901:
897:
893:
889:
883:
879:
875:
870:
868:
864:
859:
857:
853:
849:
848:Lewis Carroll
845:
841:
837:
833:
829:
825:
821:
817:
813:
809:
803:
799:
795:
791:
785:
781:
777:
773:
767:
763:
759:
755:
751:
747:
743:
737:
733:
729:
724:
722:
718:
713:
688:
682:
679:Premise
671:
665:
662:Premise
648:
622:
619:Premise
610:
607:Premise
593:
579:
571:
567:
559:
557:
555:
551:
547:
543:
538:
536:
533:) to form an
532:
528:
527:metavariables
524:
520:
516:
515:
494:
487:
469:
462:
452:
438:
432:
425:
424:
423:
419:
416:
412:
403:
401:
394:Standard form
393:
391:
389:
385:
381:
377:
376:
375:modus tollens
371:
370:
365:
360:
358:
354:
350:
346:
341:
339:
335:
331:
327:
326:
320:
318:
314:
310:
306:
302:
298:
294:
290:
286:
275:
274:instantiation
271:
268:
266:
265:instantiation
262:
259:
258:
257:
256:
253:
249:
246:
242:
236:
231:
229:
224:
222:
217:
215:
214:Transposition
212:
210:
207:
205:
200:
198:
193:
191:
189:Commutativity
186:
184:
182:Associativity
179:
178:
176:
175:
172:
168:
163:
160:
158:
156:
150:
148:
147:modus tollens
142:
137:
135:
129:
124:
122:
116:
111:
109:
103:
98:
96:
90:
85:
83:
77:
72:
70:
67:
64:elimination (
60:
55:
54:
53:
52:
49:
45:
42:
38:
35:
31:
19:
3529:
3327:Ultraproduct
3174:Model theory
3139:Independence
3089:
3075:Formal proof
3067:Proof theory
3050:
3023:
2980:real numbers
2952:second-order
2863:Substitution
2740:Metalanguage
2681:conservative
2654:Axiom schema
2598:Constructive
2568:Morse–Kelley
2534:Set theories
2513:Aleph number
2506:inaccessible
2412:Grothendieck
2296:intersection
2183:Higher-order
2171:Second-order
2117:Truth tables
2074:Venn diagram
1857:Formal proof
1763:
1755:
1730:
1722:
1700:
1686:
1666:
1659:
1635:
1628:
1566:
1552:
1509:
1402:
1314:
1215:
1211:
1208:
1205:
1201:
1199:
1004:
1000:
998:
982:
978:
974:
970:
966:
964:
960:
956:
952:
948:
943:
939:
935:
931:
925:
921:
917:
913:
907:
903:
899:
895:
891:
887:
881:
877:
873:
860:
844:modus ponens
843:
835:
831:
823:
819:
815:
811:
810:states that
805:
801:
797:
793:
789:
783:
779:
775:
771:
765:
761:
757:
753:
749:
745:
741:
735:
731:
727:
721:modus ponens
720:
714:
649:
594:
563:
553:
549:
548:statement: "
546:hypothetical
545:
541:
539:
535:infinite set
531:propositions
514:modus ponens
512:
511:This is the
510:
420:
414:
410:
404:
400:formal logic
397:
373:
369:modus ponens
367:
361:
342:
325:modus ponens
323:
321:
309:logical form
304:
300:
296:
282:
272: /
263: /
251:
154:
151: /
146:
143: /
130: /
127:Constructive
117: /
104: /
91: /
78: /
66:modus ponens
65:
61: /
47:
33:
3437:Type theory
3385:undecidable
3317:Truth value
3204:equivalence
2883:non-logical
2496:Enumeration
2486:Isomorphism
2433:cardinality
2417:Von Neumann
2382:Ultrafilter
2347:Uncountable
2281:equivalence
2198:Quantifiers
2188:Fixed-point
2157:First-order
2037:Consistency
2022:Proposition
1999:Traditional
1970:Lindström's
1960:Compactness
1902:Type theory
1847:Cardinality
1565:holds, the
867:Łukasiewicz
542:derivations
317:conclusions
227:Exportation
114:Disjunctive
107:elimination
94:elimination
81:elimination
3552:Categories
3248:elementary
2941:arithmetic
2809:Quantifier
2787:functional
2659:Expression
2377:Transitive
2321:identities
2306:complement
2239:hereditary
2222:Set theory
1620:References
1001:admissible
930:(LA4) ⊢ ((
912:(CA3) ⊢ (¬
770:(CA3) ⊢ (¬
696:Conclusion
631:Conclusion
572:notation (
525:employing
140:Absorption
3583:Inference
3519:Supertask
3422:Recursion
3380:decidable
3214:saturated
3192:of models
3115:deductive
3110:axiomatic
3030:Hilbert's
3017:Euclidean
2998:canonical
2921:axiomatic
2853:Signature
2782:Predicate
2671:Extension
2593:Ackermann
2518:Operation
2397:Universal
2387:Recursive
2362:Singleton
2357:Inhabited
2342:Countable
2332:Types of
2316:power set
2286:partition
2203:Predicate
2149:Predicate
2064:Syllogism
2054:Soundness
2027:Inference
2017:Tautology
1919:paradoxes
1579:Inference
1519:−
1468:−
1405:induction
1005:derivable
886:(LA2) ⊢ (
828:deduction
740:(CA2) ⊢ (
689:⊢
580:⊢
470:_
436:→
349:recursive
234:Tautology
3504:Logicism
3497:timeline
3473:Concrete
3332:Validity
3302:T-schema
3295:Kripke's
3290:Tarski's
3285:semantic
3275:Strength
3224:submodel
3219:spectrum
3187:function
3035:Tarski's
3024:Elements
3011:geometry
2967:Robinson
2888:variable
2873:function
2846:spectrum
2836:Sentence
2792:variable
2735:Language
2688:Relation
2649:Automata
2639:Alphabet
2623:language
2477:-jection
2455:codomain
2441:Function
2402:Universe
2372:Infinite
2276:Relation
2059:Validity
2049:Argument
1947:theorem,
1573:See also
1555:theorems
1013:judgment
872:(CA1) ⊢
726:(CA1) ⊢
523:schemata
517:rule of
366:include
287:and the
3446:Related
3243:Diagram
3141: (
3120:Hilbert
3105:Systems
3100:Theorem
2978:of the
2923:systems
2703:Formula
2698:Grammar
2614: (
2558:General
2271:Forcing
2256:Element
2176:Monadic
1951:paradox
1892:Theorem
1828:General
752:)) → ((
723:), is:
570:sequent
3209:finite
2972:Skolem
2925:
2900:Theory
2868:Symbol
2858:String
2841:atomic
2718:ground
2713:closed
2708:atomic
2664:ground
2627:syntax
2523:binary
2450:domain
2367:Finite
2132:finite
1990:Logics
1949:
1897:Theory
1776:
1743:
1707:
1674:
1647:
1561:where
947:(MP)
894:) → ((
788:(MP)
378:, and
357:ω-rule
313:syntax
3199:Model
2947:Peano
2804:Proof
2644:Arity
2573:Naive
2460:image
2392:Fuzzy
2352:Empty
2301:union
2246:Class
1887:Model
1877:Lemma
1835:Axiom
1695:. In
1011:(the
920:) → (
902:) → (
778:) → (
760:) → (
564:In a
330:valid
307:is a
285:logic
3322:Type
3125:list
2929:list
2906:list
2895:Term
2829:rank
2723:open
2617:list
2429:Maps
2334:sets
2193:Free
2163:list
1913:list
1840:list
1774:ISBN
1741:ISBN
1705:ISBN
1672:ISBN
1645:ISBN
942:) →
938:) →
554:then
319:).
295:, a
3009:of
2991:of
2939:of
2471:Sur
2445:Map
2252:Ur-
2234:Set
1770:114
1737:100
1641:364
1567:cut
1003:or
985:).
977:→ (
934:→ ¬
916:→ ¬
876:→ (
865:of
774:→ ¬
744:→ (
730:→ (
411:...
398:In
303:or
283:In
3554::
3395:NP
3019::
3013::
2943::
2620:),
2475:Bi
2467:In
1772:.
1739:.
1643:.
1206:s(
981:→
969:⊢
959:⊢
955:→
951:,
924:→
910:))
906:→
898:→
890:→
880:→
834:→
822:→
814:⊢
800:⊢
796:→
792:,
782:→
768:))
764:→
756:→
748:→
734:→
712:.
550:if
390:.
372:,
359:.
299:,
3475:/
3390:P
3145:)
2931:)
2927:(
2824:∀
2819:!
2814:∃
2775:=
2770:↔
2765:→
2760:∧
2755:∨
2750:¬
2473:/
2469:/
2443:/
2254:)
2250:(
2137:∞
2127:3
1915:)
1813:e
1806:t
1799:v
1782:.
1749:.
1713:.
1680:.
1653:.
1536:t
1533:a
1530:n
1522:3
1488:t
1485:a
1482:n
1474:)
1471:3
1465:(
1462:s
1428:t
1425:a
1422:n
1415:n
1381:t
1378:a
1375:n
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1357:t
1354:a
1351:n
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1339:n
1335:(
1332:s
1293:t
1290:a
1287:n
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1276:)
1272:n
1268:(
1265:s
1262:(
1259:s
1247:t
1244:a
1241:n
1234:n
1216:n
1212:)
1209:n
1202:0
1174:t
1171:a
1168:n
1160:)
1156:n
1152:(
1149:s
1137:t
1134:a
1131:n
1124:n
1107:t
1104:a
1101:n
1093:0
1057:n
1035:t
1032:a
1029:n
1022:n
983:B
979:A
975:A
971:B
967:A
961:B
957:B
953:A
949:A
944:A
940:A
936:A
932:A
928:)
926:A
922:B
918:B
914:A
908:C
904:A
900:C
896:B
892:B
888:A
884:)
882:A
878:B
874:A
836:B
832:A
824:B
820:A
816:B
812:A
802:B
798:B
794:A
790:A
786:)
784:A
780:B
776:B
772:A
766:C
762:A
758:B
754:A
750:C
746:B
742:A
738:)
736:A
732:B
728:A
700:)
692:(
686:)
683:2
675:(
672:,
669:)
666:1
658:(
623:2
611:1
495:B
463:A
439:B
433:A
68:)
20:)
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