4844:
468:
3600:, it is not true that every formula is logically equivalent to a prenex formula. The negation connective is one obstacle, but not the only one. The implication operator is also treated differently in intuitionistic logic than classical logic; in intuitionistic logic, it is not definable using disjunction and negation.
449:
277:
3520:
3456:
3210:
3578:
3393:
3331:
3268:
3146:
2905:
3799:
3662:
3088:
3027:
2966:
2847:
2790:
1842:
and applying the rules for disjunction and negation above. As with the rules for disjunction, these rules require that the variable quantified in one subformula does not appear free in the other subformula.
1568:
345:
528:(in classical logic) to some formula in prenex normal form. There are several conversion rules that can be recursively applied to convert a formula to prenex normal form. The rules depend on which
1635:
4241:
4203:
4162:
4124:
2214:
2176:
2110:
2072:
2027:
1989:
1923:
1885:
1489:
1356:
1223:
1413:
1280:
841:
803:
625:
587:
4083:
4045:
4004:
3966:
919:
881:
758:
720:
2526:
2407:
1811:
1144:
1840:
4302:
4273:
2606:
2327:
1768:
1739:
1704:
1675:
680:
2254:
1085:
3274:
This is not the only prenex form equivalent to the original formula. For example, by dealing with the consequent before the antecedent in the example above, the prenex form
3832:
2136:
1949:
945:
651:
4524:
4501:
4458:
4435:
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3698:
185:
156:
127:
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4356:
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2737:
2717:
2697:
1033:
993:
353:
1053:
1013:
969:
193:
2796:
By recursively applying the rules starting at the innermost subformulas, the following sequence of logically equivalent formulas can be obtained:
3463:
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4695:
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3527:
3342:
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3217:
3095:
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2973:
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485:
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1783:: two that remove quantifiers from the antecedent and two that remove quantifiers from the consequent. These rules can be derived by
507:
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285:
41:
4898:
2440:
is less than any natural number, it must be less than the smallest natural number (zero). The latter statement is true because
4758:
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4661:
521:
489:
4370:
will only deal with a theory whose formulae are written in prenex normal form. The concept is essential for developing the
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4765:
31:
4208:
4170:
4129:
4091:
2181:
2143:
2077:
2039:
1994:
1956:
1890:
1852:
1494:
3607:
illustrates why some formulas have no intuitionistically-equivalent prenex form. In this interpretation, a proof of
1425:
1292:
1159:
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1573:
478:
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1228:
808:
770:
592:
554:
86:
4050:
4012:
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3933:
886:
848:
725:
687:
4407:
2461:
2342:
3588:
of the two universal quantifier with the same scope doesn't change the meaning/truth value of the statement.
4808:
4803:
2221:
1790:
1422:
is equal to 0, while the formula on the right has no free variables and is false in any nontrivial ring. So
78:
74:
1816:
4790:
4583:
4371:
4278:
4249:
2739:
are quantifier-free formulas and no two of these formulas share any free variable. Consider the formula
2541:
2262:
1744:
1715:
1680:
1651:
656:
70:
4828:
4813:
82:
2231:
1090:
1058:
4823:
4817:
4798:
4571:
4375:
3597:
2532:
2333:
1780:
525:
97:
3811:
2115:
1928:
924:
630:
545:
541:
4893:
4860:
3604:
3585:
2455:, which is very important for the meaning of the formula. Consider the following two statements:
1150:
529:
54:
35:
4395:
4506:
4483:
4440:
4417:
4852:
4738:
4715:
4705:
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4667:
4389:
3837:
3707:
3674:
2445:
161:
132:
103:
4463:
4340:
4315:
1846:
The rules for removing quantifiers from the antecedent are (note the change of quantifiers):
4888:
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4681:
4623:
444:{\displaystyle \forall x((\exists y\phi (y))\lor ((\exists z\psi (z))\rightarrow \rho (x)))}
4549:
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3894:
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2722:
2702:
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1018:
978:
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4843:
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93:
3596:
The rules for converting a formula to prenex form make heavy use of classical logic. In
272:{\displaystyle \forall x\exists y\forall z(\phi (y)\lor (\psi (z)\rightarrow \rho (x)))}
4367:
2225:
1038:
998:
954:
58:
2627:
is less than zero. The latter statement says that if there exists some natural number
4913:
4593:
4567:
2452:
2436:
is less than zero. Both statements are true. The former statement is true because if
972:
2643:
is less than zero. Both statements are false. The former statement doesn't hold for
2420:
is less than zero. The latter statement says that there exists some natural number
4833:
467:
4734:
17:
4687:
An
Introduction to Mathematical Logic and Type Theory: To Truth Through Proof
1784:
50:
1418:
because the formula on the left is true in any ring when the free variable
4381:
4750:
3515:{\displaystyle \forall z(\forall x((\phi \lor \psi )\rightarrow \rho ))}
3451:{\displaystyle \forall z((\exists x(\phi \lor \psi ))\rightarrow \rho )}
3205:{\displaystyle \forall x(\forall z((\phi \lor \psi )\rightarrow \rho ))}
492: in this section. Unsourced material may be challenged and removed.
4566:
is a sentence that does not contain any quantifier. This fact allowed
4392:
presupposes that all formulae have been recast in prenex normal form.
3573:{\displaystyle \forall z\forall x((\phi \lor \psi )\rightarrow \rho )}
3388:{\displaystyle \forall z((\phi \lor \exists x\psi )\rightarrow \rho )}
3326:{\displaystyle \forall z\forall x((\phi \lor \psi )\rightarrow \rho )}
3263:{\displaystyle \forall x\forall z((\phi \lor \psi )\rightarrow \rho )}
3141:{\displaystyle \forall x((\phi \lor \psi )\rightarrow \forall z\rho )}
2900:{\displaystyle (\exists x(\phi \lor \psi ))\rightarrow \forall z\rho }
3794:{\displaystyle \exists y(\exists x\phi \rightarrow \psi ),\qquad (2)}
3657:{\displaystyle (\exists x\phi )\rightarrow \exists y\psi \qquad (1)}
3083:{\displaystyle \forall x(\neg (\phi \lor \psi )\lor \forall z\rho )}
3022:{\displaystyle (\forall x\neg (\phi \lor \psi ))\lor \forall z\rho }
2961:{\displaystyle \neg (\exists x(\phi \lor \psi ))\lor \forall z\rho }
2842:{\displaystyle (\phi \lor \exists x\psi )\rightarrow \forall z\rho }
2785:{\displaystyle (\phi \lor \exists x\psi )\rightarrow \forall z\rho }
4613:
2655:, but not less than zero. The latter statement doesn't hold for
187:
are quantifier-free formulas with the free variables shown then
4754:
4663:
Classical
Mathematical Logic: The Semantic Foundations of Logic
461:
2033:
The rules for removing quantifiers from the consequent are:
340:{\displaystyle \phi (y)\lor (\psi (z)\rightarrow \rho (x))}
4406:, a special case of the prenex normal form that has every
3922:
The rules for converting a formula to prenex form that do
4690:. Springer Science & Business Media. pp. 111–.
3919:
then formula (1) will not be equivalent to formula (2).
3804:
on the other hand, produces a single concrete value of
454:
is logically equivalent but not in prenex normal form.
2611:
The former statement says that for any natural number
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4414:, so that all sentences can be rewritten in the form
4398:
for geometry is a logical system whose sentences can
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to a formula in prenex normal form. For example, if
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4789:
4711:Introduction to Mathematical Logic, Fourth Edition
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121:
4236:{\displaystyle \exists x(\phi \rightarrow \psi )}
4198:{\displaystyle \phi \rightarrow (\exists x\psi )}
4157:{\displaystyle \exists x(\phi \rightarrow \psi )}
4119:{\displaystyle (\forall x\phi )\rightarrow \psi }
3733:. In this case it is allowable for the value of
2209:{\displaystyle \forall x(\phi \rightarrow \psi )}
2171:{\displaystyle \phi \rightarrow (\forall x\psi )}
2105:{\displaystyle \exists x(\phi \rightarrow \psi )}
2067:{\displaystyle \phi \rightarrow (\exists x\psi )}
2022:{\displaystyle \forall x(\phi \rightarrow \psi )}
1984:{\displaystyle (\exists x\phi )\rightarrow \psi }
1918:{\displaystyle \exists x(\phi \rightarrow \psi )}
1880:{\displaystyle (\forall x\phi )\rightarrow \psi }
1563:{\displaystyle (\exists x'(x'^{2}=1))\land (0=x)}
65:, followed by a quantifier-free part, called the
4619:"tied or bound up in front", past participle of
3915:can be constructed without knowledge of such an
2451:Note that the placement of brackets implies the
1484:{\displaystyle (\exists x(x^{2}=1))\land (0=x)}
1351:{\displaystyle (\exists x(x^{2}=1))\land (0=x)}
1218:{\displaystyle (\exists x(x^{2}=1))\land (0=y)}
682:(meaning that at least one individual exists),
4766:
4666:. Princeton University Press. pp. 108–.
1630:{\displaystyle \exists x'(x'^{2}=1\land 0=x)}
8:
1408:{\displaystyle \exists x(x^{2}=1\land 0=x)}
1275:{\displaystyle \exists x(x^{2}=1\land 0=y)}
836:{\displaystyle \exists x(\phi \land \psi )}
798:{\displaystyle (\exists x\phi )\land \psi }
620:{\displaystyle \forall x(\phi \land \psi )}
582:{\displaystyle (\forall x\phi )\land \psi }
4773:
4759:
4751:
4078:{\displaystyle \phi \lor (\forall x\psi )}
4040:{\displaystyle \forall x(\phi \lor \psi )}
3999:{\displaystyle (\forall x\phi )\lor \psi }
3961:{\displaystyle \forall x(\phi \lor \psi )}
3808:and a function that converts any proof of
914:{\displaystyle \exists x(\phi \lor \psi )}
876:{\displaystyle (\exists x\phi )\lor \psi }
753:{\displaystyle \forall x(\phi \lor \psi )}
715:{\displaystyle (\forall x\phi )\lor \psi }
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2521:{\displaystyle \forall n\in \mathbb {N} }
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2402:{\displaystyle \exists n\in \mathbb {N} }
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508:Learn how and when to remove this message
355:
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163:
134:
105:
4660:Richard L. Epstein (18 December 2011).
4605:
3737:to be computed from the given value of
1596:
1520:
4337:does not appear as a free variable of
4312:does not appear as a free variable of
3667:is a function which, given a concrete
2416:is less than any natural number, then
1806:{\displaystyle \phi \rightarrow \psi }
1835:{\displaystyle \lnot \phi \lor \psi }
282:is in prenex normal form with matrix
7:
4570:to prove that Euclidean geometry is
4297:{\displaystyle \exists x\lnot \phi }
4268:{\displaystyle \lnot \forall x\phi }
2601:{\displaystyle \rightarrow (x<0)}
2322:{\displaystyle \rightarrow (x<0)}
1763:{\displaystyle \exists x\lnot \phi }
1734:{\displaystyle \lnot \forall x\phi }
1699:{\displaystyle \forall x\lnot \phi }
1670:{\displaystyle \lnot \exists x\phi }
675:{\displaystyle \lnot \forall x\bot }
490:adding citations to reliable sources
69:. Together with the normal forms in
1570:and then put in prenex normal form
4731:Fundamentals of Mathematical Logic
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2412:The former statement says that if
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627:under (mild) additional condition
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197:
25:
4612:The term 'prenex' comes from the
2249:{\displaystyle n\in \mathbb {N} }
1645:The rules for negation say that
1139:{\displaystyle (\exists x'\phi )}
4842:
4625:(archived as of May 27, 2011 at
1149:For example, in the language of
1080:{\displaystyle (\exists x\phi )}
951:The equivalences are valid when
466:
4899:Normal form (natural deduction)
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3644:
477:needs additional citations for
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3827:{\displaystyle \exists x\phi }
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2131:{\displaystyle \exists x\top }
2099:
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2016:
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1944:{\displaystyle \exists x\top }
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940:{\displaystyle \exists x\top }
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646:{\displaystyle \exists x\top }
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27:Formalism of first-order logic
1:
3926:in intuitionistic logic are:
2659:, because the natural number
4714:. CRC Press. pp. 109–.
3887:can be used to construct a
2453:scope of the quantification
2112:(under the assumption that
1925:(under the assumption that
1491:will be first rewritten as
1035:, one can rename the bound
921:under additional condition
536:Conjunction and disjunction
4936:
4404:universal–existential form
1087:and obtain the equivalent
4840:
4646:Hinman, P. (2005), p. 111
4637:Hinman, P. (2005), p. 110
4519:{\displaystyle \exists b}
4496:{\displaystyle \exists a}
4453:{\displaystyle \forall v}
4430:{\displaystyle \forall u}
1779:There are four rules for
458:Conversion to prenex form
87:automated theorem proving
3856:{\displaystyle \psi (y)}
3726:{\displaystyle \psi (y)}
3693:{\displaystyle \phi (x)}
2444:makes the implication a
532:appear in the formula.
180:{\displaystyle \rho (x)}
151:{\displaystyle \psi (z)}
122:{\displaystyle \phi (y)}
4809:Disjunctive normal form
4804:Conjunctive normal form
4473:{\displaystyle \ldots }
4351:{\displaystyle \,\phi }
4326:{\displaystyle \,\psi }
2671:is not less than zero.
2222:range of quantification
79:conjunctive normal form
75:disjunctive normal form
4729:Hinman, Peter (2005),
4584:Arithmetical hierarchy
4560:
4540:
4520:
4497:
4474:
4454:
4431:
4412:existential quantifier
4372:arithmetical hierarchy
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4120:
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3700:, produces a concrete
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2220:For example, when the
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1009:
989:
965:
941:
915:
877:
837:
799:
754:
716:
676:
647:
621:
583:
445:
341:
273:
181:
152:
123:
4829:Canonical normal form
4814:Algebraic normal form
4561:
4559:{\displaystyle \phi }
4541:
4539:{\displaystyle \phi }
4521:
4498:
4475:
4455:
4432:
4353:
4328:
4299:
4270:
4238:
4200:
4159:
4121:
4080:
4042:
4001:
3963:
3906:
3904:{\displaystyle \psi }
3882:
3880:{\displaystyle \phi }
3858:
3829:
3796:
3728:
3695:
3659:
3575:
3517:
3453:
3390:
3328:
3265:
3207:
3143:
3085:
3024:
2963:
2902:
2844:
2787:
2734:
2732:{\displaystyle \rho }
2714:
2712:{\displaystyle \psi }
2694:
2692:{\displaystyle \phi }
2603:
2523:
2404:
2324:
2251:
2211:
2173:
2133:
2107:
2069:
2024:
1986:
1946:
1920:
1882:
1837:
1808:
1765:
1736:
1701:
1672:
1632:
1565:
1486:
1410:
1358:is not equivalent to
1353:
1277:
1220:
1141:
1082:
1050:
1030:
1028:{\displaystyle \psi }
1010:
990:
988:{\displaystyle \psi }
971:does not appear as a
966:
942:
916:
878:
838:
800:
755:
717:
677:
648:
622:
584:
446:
342:
274:
182:
153:
124:
83:canonical normal form
4920:Normal forms (logic)
4824:Blake canonical form
4818:Zhegalkin polynomial
4799:Negation normal form
4550:
4530:
4507:
4484:
4464:
4441:
4418:
4408:universal quantifier
4386:completeness theorem
4376:analytical hierarchy
4341:
4316:
4279:
4250:
4209:
4171:
4130:
4092:
4051:
4013:
3972:
3934:
3895:
3871:
3838:
3812:
3748:
3708:
3675:
3614:
3598:intuitionistic logic
3592:Intuitionistic logic
3528:
3464:
3400:
3343:
3281:
3218:
3154:
3096:
3035:
2974:
2913:
2855:
2803:
2746:
2723:
2703:
2683:
2542:
2533:logically equivalent
2462:
2343:
2334:logically equivalent
2263:
2232:
2224:is the non-negative
2182:
2144:
2116:
2078:
2040:
1995:
1957:
1929:
1891:
1853:
1817:
1791:
1745:
1716:
1681:
1652:
1574:
1495:
1426:
1362:
1293:
1229:
1160:
1091:
1059:
1039:
1019:
1015:does appear free in
999:
979:
955:
925:
887:
849:
809:
771:
726:
688:
657:
653:, or, equivalently,
631:
593:
555:
526:logically equivalent
486:improve this article
354:
286:
194:
162:
133:
104:
98:logically equivalent
4791:Propositional logic
530:logical connectives
71:propositional logic
4894:Modal clausal form
4871:Prenex normal form
4861:Skolem normal form
4556:
4536:
4516:
4493:
4470:
4450:
4427:
4362:Use of prenex form
4348:
4323:
4294:
4265:
4233:
4195:
4154:
4116:
4075:
4037:
3996:
3958:
3901:
3877:
3853:
3824:
3791:
3723:
3690:
3654:
3605:BHK interpretation
3570:
3512:
3448:
3385:
3336:can be obtained:
3323:
3260:
3202:
3138:
3080:
3019:
2958:
2897:
2839:
2782:
2729:
2709:
2689:
2598:
2518:
2399:
2319:
2246:
2206:
2168:
2128:
2102:
2064:
2019:
1981:
1941:
1915:
1877:
1832:
1803:
1760:
1731:
1696:
1667:
1627:
1560:
1481:
1405:
1348:
1272:
1225:is equivalent to
1215:
1136:
1077:
1045:
1025:
1005:
985:
961:
937:
911:
873:
833:
795:
750:
712:
672:
643:
617:
579:
441:
337:
269:
177:
148:
119:
36:predicate calculus
4907:
4906:
4744:978-1-56881-262-5
4721:978-0-412-80830-2
4706:Elliott Mendelson
4697:978-94-015-9934-4
4684:(17 April 2013).
4673:978-1-4008-4155-4
4390:first-order logic
4358:in (2) and (4)).
2336:to the statement
2256:), the statement
2178:is equivalent to
2074:is equivalent to
1991:is equivalent to
1887:is equivalent to
1741:is equivalent to
1677:is equivalent to
1048:{\displaystyle x}
1008:{\displaystyle x}
964:{\displaystyle x}
883:is equivalent to
805:is equivalent to
722:is equivalent to
589:is equivalent to
518:
517:
510:
92:Every formula in
81:), it provides a
16:(Redirected from
4927:
4889:Beta normal form
4846:
4775:
4768:
4761:
4752:
4747:
4725:
4701:
4682:Peter B. Andrews
4677:
4647:
4644:
4638:
4635:
4629:
4610:
4565:
4563:
4562:
4557:
4545:
4543:
4542:
4537:
4525:
4523:
4522:
4517:
4502:
4500:
4499:
4494:
4479:
4477:
4476:
4471:
4459:
4457:
4456:
4451:
4436:
4434:
4433:
4428:
4384:'s proof of his
4357:
4355:
4354:
4349:
4333:in (1) and (3);
4332:
4330:
4329:
4324:
4303:
4301:
4300:
4295:
4274:
4272:
4271:
4266:
4242:
4240:
4239:
4234:
4204:
4202:
4201:
4196:
4163:
4161:
4160:
4155:
4125:
4123:
4122:
4117:
4084:
4082:
4081:
4076:
4046:
4044:
4043:
4038:
4005:
4003:
4002:
3997:
3967:
3965:
3964:
3959:
3910:
3908:
3907:
3902:
3886:
3884:
3883:
3878:
3862:
3860:
3859:
3854:
3834:into a proof of
3833:
3831:
3830:
3825:
3800:
3798:
3797:
3792:
3732:
3730:
3729:
3724:
3699:
3697:
3696:
3691:
3663:
3661:
3660:
3655:
3579:
3577:
3576:
3571:
3521:
3519:
3518:
3513:
3457:
3455:
3454:
3449:
3394:
3392:
3391:
3386:
3332:
3330:
3329:
3324:
3269:
3267:
3266:
3261:
3211:
3209:
3208:
3203:
3147:
3145:
3144:
3139:
3089:
3087:
3086:
3081:
3028:
3026:
3025:
3020:
2967:
2965:
2964:
2959:
2906:
2904:
2903:
2898:
2848:
2846:
2845:
2840:
2791:
2789:
2788:
2783:
2738:
2736:
2735:
2730:
2718:
2716:
2715:
2710:
2698:
2696:
2695:
2690:
2607:
2605:
2604:
2599:
2561:
2527:
2525:
2524:
2519:
2478:
2408:
2406:
2405:
2400:
2359:
2328:
2326:
2325:
2320:
2282:
2255:
2253:
2252:
2247:
2245:
2215:
2213:
2212:
2207:
2177:
2175:
2174:
2169:
2137:
2135:
2134:
2129:
2111:
2109:
2108:
2103:
2073:
2071:
2070:
2065:
2028:
2026:
2025:
2020:
1990:
1988:
1987:
1982:
1950:
1948:
1947:
1942:
1924:
1922:
1921:
1916:
1886:
1884:
1883:
1878:
1841:
1839:
1838:
1833:
1812:
1810:
1809:
1804:
1787:the implication
1769:
1767:
1766:
1761:
1740:
1738:
1737:
1732:
1705:
1703:
1702:
1697:
1676:
1674:
1673:
1668:
1636:
1634:
1633:
1628:
1605:
1604:
1603:
1587:
1569:
1567:
1566:
1561:
1529:
1528:
1527:
1511:
1490:
1488:
1487:
1482:
1450:
1449:
1414:
1412:
1411:
1406:
1383:
1382:
1357:
1355:
1354:
1349:
1317:
1316:
1281:
1279:
1278:
1273:
1250:
1249:
1224:
1222:
1221:
1216:
1184:
1183:
1145:
1143:
1142:
1137:
1129:
1121:
1107:
1086:
1084:
1083:
1078:
1054:
1052:
1051:
1046:
1034:
1032:
1031:
1026:
1014:
1012:
1011:
1006:
994:
992:
991:
986:
970:
968:
967:
962:
946:
944:
943:
938:
920:
918:
917:
912:
882:
880:
879:
874:
842:
840:
839:
834:
804:
802:
801:
796:
759:
757:
756:
751:
721:
719:
718:
713:
681:
679:
678:
673:
652:
650:
649:
644:
626:
624:
623:
618:
588:
586:
585:
580:
513:
506:
502:
499:
493:
470:
462:
450:
448:
447:
442:
346:
344:
343:
338:
278:
276:
275:
270:
186:
184:
183:
178:
157:
155:
154:
149:
128:
126:
125:
120:
21:
4935:
4934:
4930:
4929:
4928:
4926:
4925:
4924:
4910:
4909:
4908:
4903:
4875:
4866:Herbrandization
4853:Predicate logic
4847:
4838:
4785:
4779:
4745:
4728:
4722:
4708:(1 June 1997).
4704:
4698:
4680:
4674:
4659:
4656:
4651:
4650:
4645:
4641:
4636:
4632:
4611:
4607:
4602:
4589:Herbrandization
4580:
4548:
4547:
4528:
4527:
4505:
4504:
4482:
4481:
4462:
4461:
4439:
4438:
4416:
4415:
4396:Tarski's axioms
4364:
4339:
4338:
4314:
4313:
4277:
4276:
4248:
4247:
4207:
4206:
4169:
4168:
4128:
4127:
4090:
4089:
4049:
4048:
4011:
4010:
3970:
3969:
3932:
3931:
3893:
3892:
3869:
3868:
3836:
3835:
3810:
3809:
3746:
3745:
3741:. A proof of
3706:
3705:
3704:and a proof of
3673:
3672:
3671:and a proof of
3612:
3611:
3594:
3526:
3525:
3462:
3461:
3398:
3397:
3341:
3340:
3279:
3278:
3216:
3215:
3152:
3151:
3094:
3093:
3033:
3032:
2972:
2971:
2911:
2910:
2853:
2852:
2801:
2800:
2744:
2743:
2721:
2720:
2701:
2700:
2681:
2680:
2677:
2540:
2539:
2460:
2459:
2341:
2340:
2261:
2260:
2230:
2229:
2180:
2179:
2142:
2141:
2114:
2113:
2076:
2075:
2038:
2037:
1993:
1992:
1955:
1954:
1927:
1926:
1889:
1888:
1851:
1850:
1815:
1814:
1789:
1788:
1777:
1743:
1742:
1714:
1713:
1679:
1678:
1650:
1649:
1643:
1595:
1591:
1580:
1572:
1571:
1519:
1515:
1504:
1493:
1492:
1441:
1424:
1423:
1374:
1360:
1359:
1308:
1291:
1290:
1241:
1227:
1226:
1175:
1158:
1157:
1122:
1100:
1089:
1088:
1057:
1056:
1037:
1036:
1017:
1016:
997:
996:
977:
976:
953:
952:
923:
922:
885:
884:
847:
846:
807:
806:
769:
768:
724:
723:
686:
685:
655:
654:
629:
628:
591:
590:
553:
552:
538:
514:
503:
497:
494:
483:
471:
460:
352:
351:
284:
283:
192:
191:
160:
159:
131:
130:
102:
101:
94:classical logic
59:bound variables
53:as a string of
28:
23:
22:
15:
12:
11:
5:
4933:
4931:
4923:
4922:
4912:
4911:
4905:
4904:
4902:
4901:
4896:
4891:
4885:
4883:
4877:
4876:
4874:
4873:
4868:
4863:
4857:
4855:
4849:
4848:
4841:
4839:
4837:
4836:
4831:
4826:
4821:
4811:
4806:
4801:
4795:
4793:
4787:
4786:
4780:
4778:
4777:
4770:
4763:
4755:
4749:
4748:
4743:
4726:
4720:
4702:
4696:
4678:
4672:
4655:
4652:
4649:
4648:
4639:
4630:
4604:
4603:
4601:
4598:
4597:
4596:
4591:
4586:
4579:
4576:
4555:
4535:
4515:
4512:
4492:
4489:
4469:
4449:
4446:
4426:
4423:
4410:preceding any
4402:be written in
4363:
4360:
4347:
4322:
4306:
4305:
4293:
4290:
4287:
4284:
4264:
4261:
4258:
4255:
4244:
4232:
4229:
4226:
4223:
4220:
4217:
4214:
4194:
4191:
4188:
4185:
4182:
4179:
4176:
4165:
4153:
4150:
4147:
4144:
4141:
4138:
4135:
4115:
4112:
4109:
4106:
4103:
4100:
4097:
4086:
4074:
4071:
4068:
4065:
4062:
4059:
4056:
4036:
4033:
4030:
4027:
4024:
4021:
4018:
4007:
3995:
3992:
3989:
3986:
3983:
3980:
3977:
3957:
3954:
3951:
3948:
3945:
3942:
3939:
3900:
3876:
3852:
3849:
3846:
3843:
3823:
3820:
3817:
3802:
3801:
3790:
3787:
3784:
3780:
3777:
3774:
3771:
3768:
3765:
3762:
3759:
3756:
3753:
3722:
3719:
3716:
3713:
3689:
3686:
3683:
3680:
3665:
3664:
3653:
3650:
3647:
3643:
3640:
3637:
3634:
3631:
3628:
3625:
3622:
3619:
3593:
3590:
3582:
3581:
3569:
3566:
3563:
3560:
3557:
3554:
3551:
3548:
3545:
3542:
3539:
3536:
3533:
3523:
3511:
3508:
3505:
3502:
3499:
3496:
3493:
3490:
3487:
3484:
3481:
3478:
3475:
3472:
3469:
3459:
3447:
3444:
3441:
3438:
3435:
3432:
3429:
3426:
3423:
3420:
3417:
3414:
3411:
3408:
3405:
3395:
3384:
3381:
3378:
3375:
3372:
3369:
3366:
3363:
3360:
3357:
3354:
3351:
3348:
3334:
3333:
3322:
3319:
3316:
3313:
3310:
3307:
3304:
3301:
3298:
3295:
3292:
3289:
3286:
3272:
3271:
3259:
3256:
3253:
3250:
3247:
3244:
3241:
3238:
3235:
3232:
3229:
3226:
3223:
3213:
3201:
3198:
3195:
3192:
3189:
3186:
3183:
3180:
3177:
3174:
3171:
3168:
3165:
3162:
3159:
3149:
3137:
3134:
3131:
3128:
3125:
3122:
3119:
3116:
3113:
3110:
3107:
3104:
3101:
3091:
3079:
3076:
3073:
3070:
3067:
3064:
3061:
3058:
3055:
3052:
3049:
3046:
3043:
3040:
3030:
3018:
3015:
3012:
3009:
3006:
3003:
3000:
2997:
2994:
2991:
2988:
2985:
2982:
2979:
2969:
2957:
2954:
2951:
2948:
2945:
2942:
2939:
2936:
2933:
2930:
2927:
2924:
2921:
2918:
2908:
2896:
2893:
2890:
2887:
2884:
2881:
2878:
2875:
2872:
2869:
2866:
2863:
2860:
2850:
2838:
2835:
2832:
2829:
2826:
2823:
2820:
2817:
2814:
2811:
2808:
2794:
2793:
2781:
2778:
2775:
2772:
2769:
2766:
2763:
2760:
2757:
2754:
2751:
2728:
2708:
2688:
2676:
2673:
2609:
2608:
2597:
2594:
2591:
2588:
2585:
2582:
2579:
2576:
2573:
2570:
2567:
2564:
2560:
2556:
2553:
2550:
2547:
2529:
2528:
2517:
2514:
2511:
2508:
2505:
2502:
2499:
2496:
2493:
2490:
2487:
2484:
2481:
2477:
2473:
2470:
2467:
2410:
2409:
2398:
2395:
2392:
2389:
2386:
2383:
2380:
2377:
2374:
2371:
2368:
2365:
2362:
2358:
2354:
2351:
2348:
2330:
2329:
2318:
2315:
2312:
2309:
2306:
2303:
2300:
2297:
2294:
2291:
2288:
2285:
2281:
2277:
2274:
2271:
2268:
2244:
2240:
2237:
2226:natural number
2218:
2217:
2205:
2202:
2199:
2196:
2193:
2190:
2187:
2167:
2164:
2161:
2158:
2155:
2152:
2149:
2139:
2127:
2124:
2121:
2101:
2098:
2095:
2092:
2089:
2086:
2083:
2063:
2060:
2057:
2054:
2051:
2048:
2045:
2031:
2030:
2018:
2015:
2012:
2009:
2006:
2003:
2000:
1980:
1977:
1974:
1971:
1968:
1965:
1962:
1952:
1940:
1937:
1934:
1914:
1911:
1908:
1905:
1902:
1899:
1896:
1876:
1873:
1870:
1867:
1864:
1861:
1858:
1831:
1828:
1825:
1822:
1802:
1799:
1796:
1776:
1773:
1772:
1771:
1759:
1756:
1753:
1750:
1730:
1727:
1724:
1721:
1707:
1706:
1695:
1692:
1689:
1686:
1666:
1663:
1660:
1657:
1642:
1639:
1626:
1623:
1620:
1617:
1614:
1611:
1608:
1602:
1598:
1594:
1590:
1586:
1583:
1579:
1559:
1556:
1553:
1550:
1547:
1544:
1541:
1538:
1535:
1532:
1526:
1522:
1518:
1514:
1510:
1507:
1503:
1500:
1480:
1477:
1474:
1471:
1468:
1465:
1462:
1459:
1456:
1453:
1448:
1444:
1440:
1437:
1434:
1431:
1416:
1415:
1404:
1401:
1398:
1395:
1392:
1389:
1386:
1381:
1377:
1373:
1370:
1367:
1347:
1344:
1341:
1338:
1335:
1332:
1329:
1326:
1323:
1320:
1315:
1311:
1307:
1304:
1301:
1298:
1284:
1283:
1271:
1268:
1265:
1262:
1259:
1256:
1253:
1248:
1244:
1240:
1237:
1234:
1214:
1211:
1208:
1205:
1202:
1199:
1196:
1193:
1190:
1187:
1182:
1178:
1174:
1171:
1168:
1165:
1135:
1132:
1128:
1125:
1120:
1116:
1113:
1110:
1106:
1103:
1099:
1096:
1076:
1073:
1070:
1067:
1064:
1044:
1024:
1004:
984:
960:
949:
948:
936:
933:
930:
910:
907:
904:
901:
898:
895:
892:
872:
869:
866:
863:
860:
857:
854:
844:
832:
829:
826:
823:
820:
817:
814:
794:
791:
788:
785:
782:
779:
776:
762:
761:
749:
746:
743:
740:
737:
734:
731:
711:
708:
705:
702:
699:
696:
693:
683:
671:
668:
665:
662:
642:
639:
636:
616:
613:
610:
607:
604:
601:
598:
578:
575:
572:
569:
566:
563:
560:
540:The rules for
537:
534:
516:
515:
474:
472:
465:
459:
456:
452:
451:
440:
437:
434:
431:
428:
425:
422:
419:
416:
413:
410:
407:
404:
401:
398:
395:
392:
389:
386:
383:
380:
377:
374:
371:
368:
365:
362:
359:
336:
333:
330:
327:
324:
321:
318:
315:
312:
309:
306:
303:
300:
297:
294:
291:
280:
279:
268:
265:
262:
259:
256:
253:
250:
247:
244:
241:
238:
235:
232:
229:
226:
223:
220:
217:
214:
211:
208:
205:
202:
199:
176:
173:
170:
167:
147:
144:
141:
138:
118:
115:
112:
109:
26:
24:
18:Matrix (logic)
14:
13:
10:
9:
6:
4:
3:
2:
4932:
4921:
4918:
4917:
4915:
4900:
4897:
4895:
4892:
4890:
4887:
4886:
4884:
4882:
4878:
4872:
4869:
4867:
4864:
4862:
4859:
4858:
4856:
4854:
4850:
4845:
4835:
4832:
4830:
4827:
4825:
4822:
4819:
4815:
4812:
4810:
4807:
4805:
4802:
4800:
4797:
4796:
4794:
4792:
4788:
4783:
4776:
4771:
4769:
4764:
4762:
4757:
4756:
4753:
4746:
4740:
4736:
4732:
4727:
4723:
4717:
4713:
4712:
4707:
4703:
4699:
4693:
4689:
4688:
4683:
4679:
4675:
4669:
4665:
4664:
4658:
4657:
4653:
4643:
4640:
4634:
4631:
4627:
4624:
4622:
4618:
4615:
4609:
4606:
4599:
4595:
4594:Skolemization
4592:
4590:
4587:
4585:
4582:
4581:
4577:
4575:
4573:
4569:
4553:
4533:
4513:
4490:
4467:
4447:
4424:
4413:
4409:
4405:
4401:
4397:
4393:
4391:
4387:
4383:
4379:
4377:
4373:
4369:
4368:proof calculi
4361:
4359:
4345:
4336:
4320:
4311:
4291:
4285:
4262:
4259:
4245:
4227:
4221:
4215:
4189:
4186:
4174:
4166:
4148:
4142:
4136:
4113:
4104:
4101:
4087:
4069:
4066:
4057:
4054:
4031:
4028:
4025:
4019:
4008:
3993:
3990:
3984:
3981:
3952:
3949:
3946:
3940:
3929:
3928:
3927:
3925:
3920:
3918:
3914:
3898:
3890:
3874:
3866:
3847:
3841:
3821:
3818:
3807:
3785:
3778:
3772:
3766:
3763:
3754:
3744:
3743:
3742:
3740:
3736:
3717:
3711:
3703:
3684:
3678:
3670:
3648:
3641:
3638:
3626:
3623:
3610:
3609:
3608:
3606:
3601:
3599:
3591:
3589:
3587:
3564:
3555:
3552:
3549:
3540:
3534:
3524:
3503:
3494:
3491:
3488:
3479:
3470:
3460:
3442:
3430:
3427:
3424:
3418:
3406:
3396:
3379:
3370:
3367:
3361:
3358:
3349:
3339:
3338:
3337:
3317:
3308:
3305:
3302:
3293:
3287:
3277:
3276:
3275:
3254:
3245:
3242:
3239:
3230:
3224:
3214:
3193:
3184:
3181:
3178:
3169:
3160:
3150:
3132:
3129:
3117:
3114:
3111:
3102:
3092:
3074:
3071:
3065:
3059:
3056:
3053:
3041:
3031:
3016:
3013:
3007:
2998:
2995:
2992:
2983:
2970:
2955:
2952:
2946:
2937:
2934:
2931:
2925:
2909:
2894:
2891:
2876:
2873:
2870:
2864:
2851:
2836:
2833:
2821:
2818:
2812:
2809:
2799:
2798:
2797:
2779:
2776:
2764:
2761:
2755:
2752:
2742:
2741:
2740:
2726:
2706:
2686:
2679:Suppose that
2674:
2672:
2670:
2666:
2662:
2658:
2654:
2651:is less than
2650:
2646:
2642:
2638:
2635:is less than
2634:
2630:
2626:
2622:
2619:is less than
2618:
2614:
2592:
2589:
2586:
2571:
2568:
2565:
2554:
2551:
2538:
2537:
2536:
2534:
2509:
2506:
2503:
2491:
2488:
2485:
2471:
2468:
2458:
2457:
2456:
2454:
2449:
2447:
2443:
2439:
2435:
2431:
2428:is less than
2427:
2424:such that if
2423:
2419:
2415:
2390:
2387:
2384:
2372:
2369:
2366:
2352:
2349:
2339:
2338:
2337:
2335:
2313:
2310:
2307:
2292:
2289:
2286:
2275:
2272:
2259:
2258:
2257:
2238:
2235:
2227:
2223:
2200:
2194:
2188:
2162:
2159:
2147:
2140:
2122:
2096:
2090:
2084:
2058:
2055:
2043:
2036:
2035:
2034:
2013:
2007:
2001:
1978:
1969:
1966:
1953:
1935:
1909:
1903:
1897:
1874:
1865:
1862:
1849:
1848:
1847:
1844:
1829:
1826:
1823:
1800:
1794:
1786:
1782:
1774:
1757:
1751:
1728:
1725:
1712:
1711:
1710:
1693:
1687:
1664:
1661:
1648:
1647:
1646:
1640:
1638:
1621:
1618:
1615:
1612:
1609:
1606:
1600:
1592:
1584:
1581:
1554:
1551:
1548:
1542:
1533:
1530:
1524:
1516:
1508:
1505:
1475:
1472:
1469:
1463:
1454:
1451:
1446:
1442:
1435:
1421:
1399:
1396:
1393:
1390:
1387:
1384:
1379:
1375:
1368:
1342:
1339:
1336:
1330:
1321:
1318:
1313:
1309:
1302:
1289:
1288:
1287:
1266:
1263:
1260:
1257:
1254:
1251:
1246:
1242:
1235:
1209:
1206:
1203:
1197:
1188:
1185:
1180:
1176:
1169:
1156:
1155:
1154:
1152:
1147:
1126:
1123:
1118:
1114:
1108:
1104:
1101:
1071:
1068:
1042:
1022:
1002:
982:
974:
973:free variable
958:
931:
905:
902:
899:
893:
870:
867:
861:
858:
845:
827:
824:
821:
815:
792:
789:
783:
780:
767:
766:
765:
744:
741:
738:
732:
709:
706:
700:
697:
684:
666:
637:
611:
608:
605:
599:
576:
573:
567:
564:
551:
550:
549:
547:
543:
535:
533:
531:
527:
523:
512:
509:
501:
491:
487:
481:
480:
475:This section
473:
469:
464:
463:
457:
455:
429:
423:
411:
405:
402:
390:
381:
375:
372:
360:
350:
349:
348:
328:
322:
313:
307:
301:
295:
289:
257:
251:
242:
236:
230:
224:
218:
212:
206:
200:
190:
189:
188:
171:
165:
142:
136:
113:
107:
99:
95:
90:
88:
84:
80:
76:
72:
68:
64:
61:, called the
60:
56:
52:
48:
44:
43:
37:
33:
19:
4870:
4782:Normal forms
4730:
4710:
4686:
4662:
4642:
4633:
4620:
4616:
4608:
4403:
4399:
4394:
4380:
4365:
4334:
4309:
4307:
3923:
3921:
3916:
3912:
3911:but no such
3888:
3864:
3863:. If each
3805:
3803:
3738:
3734:
3701:
3668:
3666:
3602:
3595:
3583:
3335:
3273:
2795:
2678:
2668:
2664:
2660:
2656:
2652:
2648:
2644:
2640:
2636:
2632:
2628:
2624:
2620:
2616:
2612:
2610:
2530:
2450:
2441:
2437:
2433:
2429:
2425:
2421:
2417:
2413:
2411:
2331:
2219:
2032:
1845:
1778:
1708:
1644:
1419:
1417:
1285:
1148:
950:
763:
539:
519:
504:
495:
484:Please help
479:verification
476:
453:
281:
91:
66:
62:
46:
39:
29:
4834:Horn clause
4621:praenectere
3891:satisfying
3867:satisfying
1781:implication
1775:Implication
546:disjunction
542:conjunction
524:formula is
522:first-order
498:August 2018
55:quantifiers
49:) if it is
42:normal form
4735:A K Peters
4654:References
2663:satisfies
2647:, because
2631:such that
2535:statement
85:useful in
4617:praenexus
4572:decidable
4554:ϕ
4534:ϕ
4511:∃
4488:∃
4468:…
4445:∀
4422:∀
4346:ϕ
4321:ψ
4292:ϕ
4289:¬
4283:∃
4263:ϕ
4257:∀
4254:¬
4228:ψ
4225:→
4222:ϕ
4213:∃
4190:ψ
4184:∃
4178:→
4175:ϕ
4149:ψ
4146:→
4143:ϕ
4134:∃
4114:ψ
4111:→
4105:ϕ
4099:∀
4070:ψ
4064:∀
4058:∨
4055:ϕ
4032:ψ
4029:∨
4026:ϕ
4017:∀
3994:ψ
3991:∨
3985:ϕ
3979:∀
3953:ψ
3950:∨
3947:ϕ
3938:∀
3899:ψ
3875:ϕ
3842:ψ
3822:ϕ
3816:∃
3773:ψ
3770:→
3767:ϕ
3761:∃
3752:∃
3712:ψ
3679:ϕ
3642:ψ
3636:∃
3633:→
3627:ϕ
3621:∃
3565:ρ
3562:→
3556:ψ
3553:∨
3550:ϕ
3538:∀
3532:∀
3504:ρ
3501:→
3495:ψ
3492:∨
3489:ϕ
3477:∀
3468:∀
3443:ρ
3440:→
3431:ψ
3428:∨
3425:ϕ
3416:∃
3404:∀
3380:ρ
3377:→
3371:ψ
3365:∃
3362:∨
3359:ϕ
3347:∀
3318:ρ
3315:→
3309:ψ
3306:∨
3303:ϕ
3291:∀
3285:∀
3255:ρ
3252:→
3246:ψ
3243:∨
3240:ϕ
3228:∀
3222:∀
3194:ρ
3191:→
3185:ψ
3182:∨
3179:ϕ
3167:∀
3158:∀
3133:ρ
3127:∀
3124:→
3118:ψ
3115:∨
3112:ϕ
3100:∀
3075:ρ
3069:∀
3066:∨
3060:ψ
3057:∨
3054:ϕ
3048:¬
3039:∀
3017:ρ
3011:∀
3008:∨
2999:ψ
2996:∨
2993:ϕ
2987:¬
2981:∀
2956:ρ
2950:∀
2947:∨
2938:ψ
2935:∨
2932:ϕ
2923:∃
2917:¬
2895:ρ
2889:∀
2886:→
2877:ψ
2874:∨
2871:ϕ
2862:∃
2837:ρ
2831:∀
2828:→
2822:ψ
2816:∃
2813:∨
2810:ϕ
2780:ρ
2774:∀
2771:→
2765:ψ
2759:∃
2756:∨
2753:ϕ
2727:ρ
2707:ψ
2687:ϕ
2581:→
2555:∈
2549:∃
2498:→
2472:∈
2466:∀
2446:tautology
2379:→
2353:∈
2347:∃
2302:→
2276:∈
2270:∀
2239:∈
2201:ψ
2198:→
2195:ϕ
2186:∀
2163:ψ
2157:∀
2151:→
2148:ϕ
2126:⊤
2120:∃
2097:ψ
2094:→
2091:ϕ
2082:∃
2059:ψ
2053:∃
2047:→
2044:ϕ
2014:ψ
2011:→
2008:ϕ
1999:∀
1979:ψ
1976:→
1970:ϕ
1964:∃
1939:⊤
1933:∃
1910:ψ
1907:→
1904:ϕ
1895:∃
1875:ψ
1872:→
1866:ϕ
1860:∀
1830:ψ
1827:∨
1824:ϕ
1821:¬
1801:ψ
1798:→
1795:ϕ
1785:rewriting
1758:ϕ
1755:¬
1749:∃
1729:ϕ
1723:∀
1720:¬
1694:ϕ
1691:¬
1685:∀
1665:ϕ
1659:∃
1656:¬
1613:∧
1578:∃
1543:∧
1502:∃
1464:∧
1433:∃
1391:∧
1366:∃
1331:∧
1300:∃
1258:∧
1233:∃
1198:∧
1167:∃
1109:ϕ
1098:∃
1072:ϕ
1066:∃
1023:ψ
983:ψ
935:⊤
929:∃
906:ψ
903:∨
900:ϕ
891:∃
871:ψ
868:∨
862:ϕ
856:∃
828:ψ
825:∧
822:ϕ
813:∃
793:ψ
790:∧
784:ϕ
778:∃
745:ψ
742:∨
739:ϕ
730:∀
710:ψ
707:∨
701:ϕ
695:∀
670:⊥
664:∀
661:¬
641:⊤
635:∃
612:ψ
609:∧
606:ϕ
597:∀
577:ψ
574:∧
568:ϕ
562:∀
548:say that
424:ρ
421:→
406:ψ
400:∃
391:∨
376:ϕ
370:∃
358:∀
323:ρ
320:→
308:ψ
302:∨
290:ϕ
252:ρ
249:→
237:ψ
231:∨
219:ϕ
210:∀
204:∃
198:∀
166:ρ
137:ψ
108:ϕ
4914:Category
4784:in logic
4578:See also
4546:, where
4374:and the
4275:implies
4205:implies
4126:implies
4047:implies
3968:implies
3586:ordering
2531:and its
1641:Negation
1597:′
1585:′
1521:′
1509:′
1127:′
1105:′
347:, while
2675:Example
2639:, then
2432:, then
51:written
40:prenex
34:of the
32:formula
4741:
4718:
4694:
4670:
4568:Tarski
4526:
4503:
4480:
4460:
4437:
2719:, and
2667:, but
2665:x<n
2228:(viz.
520:Every
158:, and
73:(e.g.
67:matrix
63:prefix
38:is in
4881:Other
4614:Latin
4600:Notes
4382:Gödel
4366:Some
2623:then
2615:, if
1709:and
1151:rings
995:; if
764:and
4739:ISBN
4716:ISBN
4692:ISBN
4668:ISBN
4388:for
4246:(5)
4167:(4)
4088:(3)
4009:(2)
3930:(1)
3924:fail
3603:The
3584:The
2590:<
2569:<
2507:<
2489:<
2388:<
2370:<
2311:<
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