4274:
1789:
339:
Formulas themselves are syntactic objects. They are given meanings by interpretations. For example, in a propositional formula, each propositional variable may be interpreted as a concrete proposition, so that the overall formula expresses a relationship between these propositions. A formula need not
712:
This is, however, only a convention used to simplify the written representation of a formula. If the precedence was assumed, for example, to be left-right associative, in following order: 1. ¬ 2. ∧ 3. ∨ 4. →, then the same formula above (without parentheses)
335:
instance of formula. This distinction between the vague notion of "property" and the inductively-defined notion of well-formed formula has roots in Weyl's 1910 paper "Uber die
Definitionen der mathematischen Grundbegriffe". Thus the same formula may be written more than once, and a formula might in
650:) are assumed among the operators, making some operators more binding than others. For example, assuming the precedence (from most binding to least binding) 1. ¬ 2. → 3. ∧ 4. ∨. Then the formula
2132:
All sources supported "woof". The sources cited for "wiff", "weff", and "whiff" gave these pronunciations as alternatives to "woof". The
Gensler source gives "wood" and "woofer" as examples of how to pronounce the vowel in
330:
Although the term "formula" may be used for written marks (for instance, on a piece of paper or chalkboard), it is more precisely understood as the sequence of symbols being expressed, with the marks being a
1294:
213:
1024:
404:
1056:
1261:
1194:
1125:
778:
1647:
1613:
1582:
1548:
970:
1344:
1321:
1697:), formulas referred to any strings of symbols and among these strings, well-formed formulas were the strings that followed the formation rules of (correct) formulas.
1427:
is formed by combining atomic formulas using only logical connectives, to the exclusion of quantifiers. This is not to be confused with a formula which is not closed.
646:
A complex formula may be difficult to read, owing to, for example, the proliferation of parentheses. To alleviate this last phenomenon, precedence rules (akin to the
1234:
1165:
1096:
1076:
990:
1364:
1214:
1145:
443:
If φ and ψ are formulas, and • is any binary connective, then ( φ • ψ) is a formula. Here • could be (but is not limited to) the usual operators ∨, ∧, →, or ↔.
1841:
Formulas are a standard topic in introductory logic, and are covered by all introductory textbooks, including
Enderton (2001), Gamut (1990), and Kleene (1967)
2653:
1404:, or equivalently a formula that has no strict subformulas. The precise form of atomic formulas depends on the formal system under consideration; for
784:
of the theory at hand. This signature specifies the constant symbols, predicate symbols, and function symbols of the theory at hand, along with the
2292:
Mathematical
Learning: Report of a Conference Sponsored by the Committee on Intellective Processes Research of the Social Science Research Council
3328:
3411:
2552:
2162:
Handbook of the history of logic, (Vol 5, Logic from
Russell to Church), Tarski's logic by Keith Simmons, D. Gabbay and J. Woods Eds, p568
17:
1716:, i.e. of the concrete string representation of formulas (using this or that symbol for connectives and quantifiers, using this or that
220:
647:
1700:
Several authors simply say formula. Modern usages (especially in the context of computer science with mathematical software such as
946:
Finally, the set of formulas is defined to be the smallest set containing the set of atomic formulas such that the following holds:
3725:
2333:
2152:
327:
can be represented by sequences of formulas with certain properties, and the final formula in the sequence is what is proven.
3883:
2500:
2459:
2434:
2409:
2369:
2323:
2239:
2223:
2218:, ed. (1982), Handbook of Mathematical Logic, Studies in Logic and the Foundations of Mathematics, Amsterdam: North-Holland,
2204:
2103:
2076:
2049:
2022:
1995:
1968:
1943:
1916:
1889:
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3738:
3061:
2515:
3743:
3733:
3470:
3323:
2676:
2667:
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319:. In those contexts, a formula is a string of symbols φ for which it makes sense to ask "is φ true?", once any
3221:
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3720:
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2484:
3281:
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2163:
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2627:
781:
4232:
4015:
3932:
3645:
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3453:
2695:
117:
4318:
4157:
3983:
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3303:
2902:
1701:
87:
4313:
4035:
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3640:
3379:
3308:
2637:
2538:
2519:
2234:
Cori, Rene; Lascar, Daniel (2000), Mathematical Logic: A Course with
Exercises, Oxford University Press,
1709:
1617:
1552:
270:
172:
167:
72:
997:
365:
3964:
3554:
2948:
2916:
2607:
2315:
1375:
1029:
2290:
Allen, Layman E. (1965), "Toward
Autotelic Learning of Mathematical Logic by the WFF 'N PROOF Games",
1239:
1172:
1103:
4298:
4254:
4203:
4100:
3598:
3559:
3036:
2681:
1673:
756:
206:
192:
177:
41:
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1529:
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1705:
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292:
1712:) tend to retain of the notion of formula only the algebraic concept and to leave the question of
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3102:
3046:
2659:
1807:
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4217:
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3628:
3613:
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3421:
3356:
3181:
3147:
3142:
3016:
2847:
2824:
2260:
1744:
1622:
1557:
1461:
1409:
1401:
1326:
1303:
411:
355:
349:
332:
296:
152:
426:. The formulas will be certain expressions (that is, strings of symbols) over this alphabet.
4147:
4000:
3792:
3510:
3246:
3152:
3011:
2996:
2877:
2852:
359:
287:
A formal language can be identified with the set of formulas in the language. A formula is a
162:
2398:
1747:). The suite of games is designed to teach the principles of symbolic logic to children (in
4120:
4082:
3959:
3763:
3603:
3527:
3505:
3333:
3291:
3190:
3157:
3021:
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2720:
2469:
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1219:
1150:
1081:
1061:
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is defined recursively. Terms, informally, are expressions that represent objects from the
452:
430:
31:
2142:
W. Dean, S. Walsh, The
Prehistory of the Subsystems of Second-order Arithmetic (2016), p.6
8:
4249:
4140:
4125:
4105:
4062:
3949:
3899:
3825:
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3707:
3500:
3495:
3211:
3200:
2872:
2772:
2700:
2691:
2687:
2622:
2617:
2476:
2341:
1717:
1405:
798:
308:
238:
187:
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4278:
4047:
4010:
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2612:
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2418:
1416:, the atoms are predicate symbols together with their arguments, each argument being a
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324:
312:
234:
112:
82:
3231:
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3712:
3593:
3428:
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3185:
3169:
3074:
3051:
2928:
2897:
2862:
2757:
2592:
2496:
2455:
2430:
2405:
2396:
Hodges, Wilfrid (2001), "Classical Logic I: First-Order Logic", in Goble, Lou (ed.),
2383:
2365:
2319:
2235:
2219:
2200:
2187:
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2018:
1991:
1964:
1939:
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1817:
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316:
122:
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principle be so long that it cannot be written at all within the physical universe.
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2837:
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1774:
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1657:
1436:
51:
471:
p | q | r | s | t | u | ... (the arbitrary finite set of propositional variables)
4208:
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4052:
3954:
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2012:
1985:
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1906:
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1736:
1721:
1713:
1417:
1413:
274:
266:
242:
182:
157:
107:
2151:
First-order logic and automated theorem proving, Melvin
Fitting, Springer, 1996
4193:
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3438:
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288:
147:
127:
97:
92:
77:
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3137:
3127:
3097:
3082:
2752:
2426:
2303:
2183:
1758:
1731:
The expression "well-formed formulas" (WFF) also crept into popular culture.
1694:
1665:
320:
67:
2199:
Hodges, Wilfrid (1997), A shorter model theory, Cambridge
University Press,
4067:
3914:
3815:
3807:
3687:
3635:
3544:
3480:
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2814:
2597:
793:
619:
is a formula, because it is grammatically correct. The sequence of symbols
299:. Two key uses of formulas are in propositional logic and predicate logic.
4177:
4057:
3236:
3226:
3173:
2857:
2777:
2762:
2642:
2587:
2215:
1753:
1739:: The Game of Modern Logic", by Layman Allen, developed while he was at
3107:
2962:
2933:
2739:
791:
The definition of a formula comes in several parts. First, the set of
4259:
4162:
3215:
3132:
3092:
3056:
2992:
2804:
2794:
2767:
2530:
2068:
The New Relational Database Dictionary: Terms, Concepts, and Examples
4244:
4042:
3490:
3195:
2789:
2174:
Alonzo Church, (1944), Introduction to mathematical logic, page 49
1762:
422:
and parentheses "(" and ")", all of which are assumed to not be in
262:
1735:
is part of an esoteric pun used in the name of the academic game "
3840:
2632:
284:
is pronounced "woof", or sometimes "wiff", "weff", or "whiff".
2294:, Monographs of the Society for Research in Child Development,
340:
be interpreted, however, to be considered solely as a formula.
1468:
is a formula of a first-order language in which the variables
765:
643:
is not a formula, because it does not conform to the grammar.
3384:
2730:
2575:
2380:
Logic, Language, and Meaning, Volume 1: Introduction to Logic
785:
406:. Their definition begins with the arbitrary choice of a set
2190:(1950) , Principles of Mathematical Logic, New York: Chelsea
437:
Each propositional variable is, on its own, a formula.
1884:. Springer Science & Business Media. p. 44.
1631:
1597:
1566:
1532:
1352:
1329:
1306:
1269:
1242:
1222:
1202:
1175:
1153:
1133:
1106:
1084:
1064:
1032:
1000:
978:
955:
759:
368:
2302:
2041:
The Relational Database Dictionary, Extended Edition
1784:
1516:
2516:
Well-Formed Formula for First Order Predicate Logic
1984:Maurer, Stephen B.; Ralston, Anthony (2005-01-21).
2397:
1641:
1607:
1576:
1542:
1358:
1338:
1315:
1288:
1255:
1228:
1208:
1188:
1159:
1139:
1119:
1090:
1070:
1050:
1018:
984:
964:
772:
398:
1877:
4290:
1693:In earlier works on mathematical logic (e.g. by
808:Any constant symbol from the signature is a term
1987:Discrete Algorithmic Mathematics, Third Edition
323:in φ have been instantiated. In formal logic,
1878:Hall, Cordelia; O'Donnell, John (2013-04-17).
1728:notation, etc.) as a mere notational problem.
2546:
1983:
1680:, says that either the resulting instance of
440:If φ is a formula, then ¬φ is a formula.
214:
2481:A Concise Introduction to Mathematical Logic
2423:Gödel, Escher, Bach: An Eternal Golden Braid
2014:The Philosopher's Dictionary - Third Edition
1935:Essentials of Symbolic Logic - Third Edition
1908:Symbolic Logic: Syntax, Semantics, and Proof
587:Using this grammar, the sequence of symbols
1688:
1408:, for example, the atomic formulas are the
1381:
455:, provided the set of variables is finite:
2738:
2553:
2539:
2475:
2417:
2400:The Blackwell Guide to Philosophical Logic
1289:{\displaystyle \neg \exists x\,\neg \phi }
414:. The alphabet consists of the letters in
221:
207:
2331:
1958:
1374:is a formula starting with a sequence of
1279:
1249:
1182:
1113:
648:standard mathematical order of operations
447:This definition can also be written as a
343:
2355:
2340:. University of Michigan. Archived from
1911:. Rowman & Littlefield. p. 41.
1263:could be defined as an abbreviation for
30:For broader coverage of this topic, see
2091:
2071:. "O'Reilly Media, Inc.". p. 241.
1931:
1850:
1378:followed by a quantifier-free formula.
788:of the function and predicate symbols.
14:
4291:
2560:
2442:
2395:
2010:
2534:
2377:
2289:
1904:
1881:Discrete Mathematics Using a Computer
1456:, is a formula in which there are no
27:Syntactically correct logical formula
2358:A mathematical introduction to logic
2064:
2037:
291:object that can be given a semantic
1300:If a formula has no occurrences of
24:
2276:Allen (1965) acknowledges the pun.
1634:
1600:
1569:
1535:
1430:
1423:According to some terminology, an
1330:
1307:
1280:
1273:
1270:
1243:
1176:
1107:
1019:{\displaystyle (\phi \land \psi )}
956:
762:
745:
399:{\displaystyle (A\land (B\lor C))}
25:
4330:
2526:Well-Formed Formula at ProvenMath
2509:
2332:Ehrenberg, Rachel (Spring 2002).
1743:(he was later a professor at the
1517:Properties applicable to formulas
1051:{\displaystyle (\phi \lor \psi )}
88:Semantics (programming languages)
4272:
2038:Date, Christopher (2008-10-14).
2017:. Broadview Press. p. 323.
2011:Martin, Robert M. (2002-05-06).
1787:
1256:{\displaystyle \forall x\,\phi }
1189:{\displaystyle \forall x\,\phi }
1120:{\displaystyle \exists x\,\phi }
2485:Springer Science+Business Media
2382:, University Of Chicago Press,
2270:
2253:
2244:
2228:
2209:
2193:
2177:
2168:
2156:
2145:
2136:
2112:
2098:. Broadview Press. p. 12.
2085:
2058:
1963:. Broadview Press. p. 59.
1938:. Broadview Press. p. 14.
1684:is provable or its negation is.
859:The next step is to define the
773:{\displaystyle {\mathcal {QS}}}
750:The definition of a formula in
418:along with the symbols for the
302:
2031:
2004:
1977:
1961:A Pocket Guide to Formal Logic
1959:Laderoute, Karl (2022-10-24).
1952:
1925:
1898:
1871:
1844:
1835:
1642:{\displaystyle {\mathcal {Q}}}
1608:{\displaystyle {\mathcal {Q}}}
1577:{\displaystyle {\mathcal {Q}}}
1543:{\displaystyle {\mathcal {Q}}}
1396:is a formula that contains no
1045:
1033:
1013:
1001:
393:
390:
378:
369:
13:
1:
4233:History of mathematical logic
2283:
2092:Simpson, R. L. (1998-12-10).
1932:Simpson, R. L. (2008-03-17).
1851:Gensler, Harry (2002-09-11).
1236:is a formula (alternatively,
4158:Primitive recursive function
2360:(2nd ed.), Boston, MA:
2095:Essentials of Symbolic Logic
1484:have free occurrences, then
307:A key use of formulas is in
18:Formula (mathematical logic)
7:
1780:
1710:interactive theorem provers
906:-ary predicate symbol, and
855:are terms, is again a term.
173:Programming language theory
168:Natural language processing
10:
4335:
3222:Schröder–Bernstein theorem
2949:Monadic predicate calculus
2608:Foundations of mathematics
2483:(3rd ed.), New York:
2356:Enderton, Herbert (2001),
2316:Cambridge University Press
2065:Date, C. J. (2015-12-21).
1990:. CRC Press. p. 625.
1751:). Its name is an echo of
1434:
1385:
1376:existential quantification
965:{\displaystyle \neg \phi }
839:-ary function symbol, and
811:an expression of the form
362:, are expressions such as
347:
29:
4268:
4255:Philosophy of mathematics
4204:Automated theorem proving
4186:
4081:
3913:
3806:
3658:
3375:
3351:
3329:Von Neumann–Bernays–Gödel
3274:
3168:
3072:
2970:
2961:
2888:
2823:
2729:
2651:
2568:
2493:10.1007/978-1-4419-1221-3
2334:"He's Positively Logical"
1857:. Routledge. p. 35.
1718:parenthesizing convention
1706:automated theorem provers
1676:of the free variables of
1339:{\displaystyle \forall x}
1316:{\displaystyle \exists x}
420:propositional connectives
193:Automated theorem proving
178:Computational linguistics
1905:Agler, David W. (2013).
1828:
1689:Usage of the terminology
1556:if it is true for every
1382:Atomic and open formulas
457:
3905:Self-verifying theories
3726:Tarski's axiomatization
2677:Tarski's undefinability
2672:incompleteness theorems
2312:Computability and Logic
2044:. Apress. p. 211.
1808:Well-defined expression
1621:if it is true for some
1410:propositional variables
805:Any variable is a term.
412:propositional variables
4279:Mathematics portal
3890:Proof of impossibility
3538:propositional variable
2848:Propositional calculus
2378:Gamut, L.T.F. (1990),
1745:University of Michigan
1668:, i.e. if there is an
1643:
1609:
1578:
1544:
1360:
1340:
1317:
1290:
1257:
1230:
1210:
1190:
1161:
1141:
1121:
1092:
1072:
1052:
1020:
986:
966:
942:) is an atomic formula
774:
713:would be rewritten as
682:may be abbreviated as
400:
360:propositional formulas
356:propositional calculus
350:Propositional calculus
344:Propositional calculus
153:Propositional calculus
4148:Kolmogorov complexity
4101:Computably enumerable
4001:Model complete theory
3793:Principia Mathematica
2853:Propositional formula
2682:Banach–Tarski paradox
1854:Introduction to Logic
1644:
1610:
1579:
1545:
1361:
1341:
1318:
1291:
1258:
1231:
1229:{\displaystyle \phi }
1211:
1191:
1162:
1160:{\displaystyle \phi }
1142:
1122:
1093:
1091:{\displaystyle \psi }
1073:
1071:{\displaystyle \phi }
1053:
1021:
987:
985:{\displaystyle \phi }
967:
775:
401:
163:Mathematical notation
4096:Church–Turing thesis
4083:Computability theory
3292:continuum hypothesis
2810:Square of opposition
2668:Gödel's completeness
2477:Rautenberg, Wolfgang
2444:Kleene, Stephen Cole
2265:Fitch-style calculus
1771:The Whiffenpoof Song
1629:
1595:
1564:
1530:
1366:, then it is called
1350:
1327:
1304:
1267:
1240:
1220:
1200:
1173:
1151:
1131:
1104:
1082:
1062:
1030:
998:
976:
953:
895:is an atomic formula
757:
433:defined as follows:
366:
32:Mathematical formula
4319:Logical expressions
4250:Mathematical object
4141:P versus NP problem
4106:Computable function
3900:Reverse mathematics
3826:Logical consequence
3703:primitive recursive
3698:elementary function
3471:Free/bound variable
3324:Tarski–Grothendieck
2843:Logical connectives
2773:Logical equivalence
2623:Logical consequence
2518:- includes a short
2419:Hofstadter, Douglas
2261:propositional logic
1664:if it represents a
1656:of the language of
1406:propositional logic
1398:logical connectives
1372:existential formula
1346:, for any variable
799:domain of discourse
780:is relative to the
309:propositional logic
247:well-formed formula
239:propositional logic
188:Formal verification
103:Well-formed formula
4314:Mathematical logic
4048:Transfer principle
4011:Semantics of logic
3996:Categorical theory
3972:Non-standard model
3486:Logical connective
2613:Information theory
2562:Mathematical logic
2452:Dover Publications
2448:Mathematical logic
2259:More technically,
2188:Ackermann, Wilhelm
1639:
1605:
1574:
1540:
1356:
1336:
1313:
1286:
1253:
1226:
1216:is a variable and
1206:
1196:is a formula when
1186:
1157:
1147:is a variable and
1137:
1127:is a formula when
1117:
1088:
1068:
1058:are formulas when
1048:
1016:
982:
972:is a formula when
962:
770:
670:)) ∨ (¬
607:)) ∨ (¬
396:
273:that is part of a
235:mathematical logic
113:Regular expression
4286:
4285:
4218:Abstract category
4021:Theories of truth
3831:Rule of inference
3821:Natural deduction
3802:
3801:
3347:
3346:
3052:Cartesian product
2957:
2956:
2863:Many-valued logic
2838:Boolean functions
2721:Russell's paradox
2696:diagonal argument
2593:First-order logic
2502:978-1-4419-1220-6
2461:978-0-486-42533-7
2436:978-0-14-005579-5
2411:978-0-631-20692-7
2371:978-0-12-238452-3
2325:978-0-521-00758-0
2306:; Burgess, John;
2240:978-0-19-850048-3
2224:978-0-444-86388-1
2205:978-0-521-58713-6
2105:978-1-55111-250-3
2078:978-1-4919-5171-2
2051:978-1-4302-1042-9
2024:978-1-77048-215-9
1997:978-1-56881-166-6
1970:978-1-77048-868-7
1945:978-1-77048-495-5
1918:978-1-4422-1742-3
1891:978-1-4471-3657-6
1864:978-1-134-58880-0
1818:Glossary of logic
1803:Ground expression
1795:Philosophy portal
1507:universal closure
1359:{\displaystyle x}
1209:{\displaystyle x}
1140:{\displaystyle x}
752:first-order logic
429:The formulas are
317:first-order logic
280:The abbreviation
231:
230:
123:Ground expression
83:Semantics (logic)
16:(Redirected from
4326:
4299:Formal languages
4277:
4276:
4228:History of logic
4223:Category of sets
4116:Decision problem
3895:Ordinal analysis
3836:Sequent calculus
3734:Boolean algebras
3674:
3673:
3648:
3619:logical/constant
3373:
3372:
3359:
3282:Zermelo–Fraenkel
3033:Set operations:
2968:
2967:
2905:
2736:
2735:
2716:Löwenheim–Skolem
2603:Formal semantics
2555:
2548:
2541:
2532:
2531:
2505:
2472:
2439:
2414:
2403:
2392:
2374:
2352:
2350:
2349:
2328:
2314:(4th ed.),
2308:Jeffrey, Richard
2299:
2277:
2274:
2268:
2257:
2251:
2248:
2242:
2232:
2226:
2213:
2207:
2197:
2191:
2181:
2175:
2172:
2166:
2160:
2154:
2149:
2143:
2140:
2134:
2116:
2110:
2109:
2089:
2083:
2082:
2062:
2056:
2055:
2035:
2029:
2028:
2008:
2002:
2001:
1981:
1975:
1974:
1956:
1950:
1949:
1929:
1923:
1922:
1902:
1896:
1895:
1875:
1869:
1868:
1848:
1842:
1839:
1797:
1792:
1791:
1790:
1775:The Whiffenpoofs
1769:made popular in
1670:effective method
1648:
1646:
1645:
1640:
1638:
1637:
1614:
1612:
1611:
1606:
1604:
1603:
1583:
1581:
1580:
1575:
1573:
1572:
1549:
1547:
1546:
1541:
1539:
1538:
1504:
1483:
1458:free occurrences
1437:Sentence (logic)
1365:
1363:
1362:
1357:
1345:
1343:
1342:
1337:
1322:
1320:
1319:
1314:
1295:
1293:
1292:
1287:
1262:
1260:
1259:
1254:
1235:
1233:
1232:
1227:
1215:
1213:
1212:
1207:
1195:
1193:
1192:
1187:
1166:
1164:
1163:
1158:
1146:
1144:
1143:
1138:
1126:
1124:
1123:
1118:
1097:
1095:
1094:
1089:
1077:
1075:
1074:
1069:
1057:
1055:
1054:
1049:
1025:
1023:
1022:
1017:
991:
989:
988:
983:
971:
969:
968:
963:
922:are terms, then
779:
777:
776:
771:
769:
768:
582:
579:
576:
572:
569:
566:
562:
559:
556:
552:
549:
546:
542:
539:
536:
532:
529:
526:
522:
519:
516:
512:
509:
506:
502:
499:
496:
492:
489:
486:
483:
480:
477:
474:
470:
467:
464:
461:
453:Backus–Naur form
405:
403:
402:
397:
354:The formulas of
223:
216:
209:
52:Formal languages
37:
36:
21:
4334:
4333:
4329:
4328:
4327:
4325:
4324:
4323:
4289:
4288:
4287:
4282:
4271:
4264:
4209:Category theory
4199:Algebraic logic
4182:
4153:Lambda calculus
4091:Church encoding
4077:
4053:Truth predicate
3909:
3875:Complete theory
3798:
3667:
3663:
3659:
3654:
3646:
3366: and
3362:
3357:
3343:
3319:New Foundations
3287:axiom of choice
3270:
3232:Gödel numbering
3172: and
3164:
3068:
2953:
2903:
2884:
2833:Boolean algebra
2819:
2783:Equiconsistency
2748:Classical logic
2725:
2706:Halting problem
2694: and
2670: and
2658: and
2657:
2652:Theorems (
2647:
2564:
2559:
2512:
2503:
2462:
2437:
2412:
2390:
2372:
2347:
2345:
2326:
2286:
2281:
2280:
2275:
2271:
2258:
2254:
2249:
2245:
2233:
2229:
2214:
2210:
2198:
2194:
2182:
2178:
2173:
2169:
2161:
2157:
2150:
2146:
2141:
2137:
2117:
2113:
2106:
2090:
2086:
2079:
2063:
2059:
2052:
2036:
2032:
2025:
2009:
2005:
1998:
1982:
1978:
1971:
1957:
1953:
1946:
1930:
1926:
1919:
1903:
1899:
1892:
1876:
1872:
1865:
1849:
1845:
1840:
1836:
1831:
1813:Formal language
1793:
1788:
1786:
1783:
1767:Yale University
1749:Polish notation
1741:Yale Law School
1714:well-formedness
1691:
1672:which, given a
1633:
1632:
1630:
1627:
1626:
1599:
1598:
1596:
1593:
1592:
1568:
1567:
1565:
1562:
1561:
1534:
1533:
1531:
1528:
1527:
1519:
1502:
1496:
1489:
1481:
1475:
1469:
1439:
1433:
1431:Closed formulas
1414:predicate logic
1390:
1384:
1368:quantifier-free
1351:
1348:
1347:
1328:
1325:
1324:
1305:
1302:
1301:
1268:
1265:
1264:
1241:
1238:
1237:
1221:
1218:
1217:
1201:
1198:
1197:
1174:
1171:
1170:
1152:
1149:
1148:
1132:
1129:
1128:
1105:
1102:
1101:
1083:
1080:
1079:
1063:
1060:
1059:
1031:
1028:
1027:
999:
996:
995:
977:
974:
973:
954:
951:
950:
941:
932:
921:
912:
894:
887:
881:are terms then
880:
873:
861:atomic formulas
854:
845:
830:
821:
761:
760:
758:
755:
754:
748:
746:Predicate logic
733:∨ (¬
585:
584:
580:
577:
574:
570:
567:
564:
560:
557:
554:
550:
547:
544:
540:
537:
534:
530:
527:
524:
520:
517:
514:
510:
507:
504:
500:
497:
494:
490:
487:
484:
481:
478:
475:
472:
468:
465:
462:
459:
367:
364:
363:
352:
346:
313:predicate logic
305:
295:by means of an
275:formal language
257:, often simply
243:predicate logic
227:
198:
197:
183:Syntax analysis
158:Predicate logic
143:
142:
133:
132:
108:Automata theory
63:
62:
35:
28:
23:
22:
15:
12:
11:
5:
4332:
4322:
4321:
4316:
4311:
4309:Syntax (logic)
4306:
4301:
4284:
4283:
4269:
4266:
4265:
4263:
4262:
4257:
4252:
4247:
4242:
4241:
4240:
4230:
4225:
4220:
4211:
4206:
4201:
4196:
4194:Abstract logic
4190:
4188:
4184:
4183:
4181:
4180:
4175:
4173:Turing machine
4170:
4165:
4160:
4155:
4150:
4145:
4144:
4143:
4138:
4133:
4128:
4123:
4113:
4111:Computable set
4108:
4103:
4098:
4093:
4087:
4085:
4079:
4078:
4076:
4075:
4070:
4065:
4060:
4055:
4050:
4045:
4040:
4039:
4038:
4033:
4028:
4018:
4013:
4008:
4006:Satisfiability
4003:
3998:
3993:
3992:
3991:
3981:
3980:
3979:
3969:
3968:
3967:
3962:
3957:
3952:
3947:
3937:
3936:
3935:
3930:
3923:Interpretation
3919:
3917:
3911:
3910:
3908:
3907:
3902:
3897:
3892:
3887:
3877:
3872:
3871:
3870:
3869:
3868:
3858:
3853:
3843:
3838:
3833:
3828:
3823:
3818:
3812:
3810:
3804:
3803:
3800:
3799:
3797:
3796:
3788:
3787:
3786:
3785:
3780:
3779:
3778:
3773:
3768:
3748:
3747:
3746:
3744:minimal axioms
3741:
3730:
3729:
3728:
3717:
3716:
3715:
3710:
3705:
3700:
3695:
3690:
3677:
3675:
3656:
3655:
3653:
3652:
3651:
3650:
3638:
3633:
3632:
3631:
3626:
3621:
3616:
3606:
3601:
3596:
3591:
3590:
3589:
3584:
3574:
3573:
3572:
3567:
3562:
3557:
3547:
3542:
3541:
3540:
3535:
3530:
3520:
3519:
3518:
3513:
3508:
3503:
3498:
3493:
3483:
3478:
3473:
3468:
3467:
3466:
3461:
3456:
3451:
3441:
3436:
3434:Formation rule
3431:
3426:
3425:
3424:
3419:
3409:
3408:
3407:
3397:
3392:
3387:
3382:
3376:
3370:
3353:Formal systems
3349:
3348:
3345:
3344:
3342:
3341:
3336:
3331:
3326:
3321:
3316:
3311:
3306:
3301:
3296:
3295:
3294:
3289:
3278:
3276:
3272:
3271:
3269:
3268:
3267:
3266:
3256:
3251:
3250:
3249:
3242:Large cardinal
3239:
3234:
3229:
3224:
3219:
3205:
3204:
3203:
3198:
3193:
3178:
3176:
3166:
3165:
3163:
3162:
3161:
3160:
3155:
3150:
3140:
3135:
3130:
3125:
3120:
3115:
3110:
3105:
3100:
3095:
3090:
3085:
3079:
3077:
3070:
3069:
3067:
3066:
3065:
3064:
3059:
3054:
3049:
3044:
3039:
3031:
3030:
3029:
3024:
3014:
3009:
3007:Extensionality
3004:
3002:Ordinal number
2999:
2989:
2984:
2983:
2982:
2971:
2965:
2959:
2958:
2955:
2954:
2952:
2951:
2946:
2941:
2936:
2931:
2926:
2921:
2920:
2919:
2909:
2908:
2907:
2894:
2892:
2886:
2885:
2883:
2882:
2881:
2880:
2875:
2870:
2860:
2855:
2850:
2845:
2840:
2835:
2829:
2827:
2821:
2820:
2818:
2817:
2812:
2807:
2802:
2797:
2792:
2787:
2786:
2785:
2775:
2770:
2765:
2760:
2755:
2750:
2744:
2742:
2733:
2727:
2726:
2724:
2723:
2718:
2713:
2708:
2703:
2698:
2686:Cantor's
2684:
2679:
2674:
2664:
2662:
2649:
2648:
2646:
2645:
2640:
2635:
2630:
2625:
2620:
2615:
2610:
2605:
2600:
2595:
2590:
2585:
2584:
2583:
2572:
2570:
2566:
2565:
2558:
2557:
2550:
2543:
2535:
2529:
2528:
2523:
2511:
2510:External links
2508:
2507:
2506:
2501:
2473:
2460:
2440:
2435:
2415:
2410:
2393:
2388:
2375:
2370:
2362:Academic Press
2353:
2338:Michigan Today
2329:
2324:
2304:Boolos, George
2300:
2285:
2282:
2279:
2278:
2269:
2252:
2250:Ehrenburg 2002
2243:
2227:
2208:
2192:
2184:Hilbert, David
2176:
2167:
2155:
2144:
2135:
2131:
2130:
2127:
2124:
2121:
2111:
2104:
2084:
2077:
2057:
2050:
2030:
2023:
2003:
1996:
1976:
1969:
1951:
1944:
1924:
1917:
1897:
1890:
1870:
1863:
1843:
1833:
1832:
1830:
1827:
1826:
1825:
1820:
1815:
1810:
1805:
1799:
1798:
1782:
1779:
1702:model checkers
1690:
1687:
1686:
1685:
1650:
1636:
1623:interpretation
1602:
1591:in a language
1585:
1571:
1558:interpretation
1537:
1526:in a language
1518:
1515:
1500:
1494:
1479:
1473:
1443:closed formula
1435:Main article:
1432:
1429:
1394:atomic formula
1388:Atomic formula
1386:Main article:
1383:
1380:
1355:
1335:
1332:
1312:
1309:
1298:
1297:
1285:
1282:
1278:
1275:
1272:
1252:
1248:
1245:
1225:
1205:
1185:
1181:
1178:
1168:
1156:
1136:
1116:
1112:
1109:
1099:
1087:
1067:
1047:
1044:
1041:
1038:
1035:
1015:
1012:
1009:
1006:
1003:
993:
981:
961:
958:
944:
943:
937:
930:
917:
910:
896:
892:
885:
878:
871:
857:
856:
850:
843:
826:
819:
809:
806:
767:
764:
747:
744:
743:
742:
737:∧ ¬
710:
709:
705:∧ ¬
701:∨ ¬
680:
679:
674:∧ ¬
641:
640:
617:
616:
611:∧ ¬
458:
449:formal grammar
445:
444:
441:
438:
395:
392:
389:
386:
383:
380:
377:
374:
371:
358:, also called
348:Main article:
345:
342:
321:free variables
304:
301:
297:interpretation
261:, is a finite
249:, abbreviated
229:
228:
226:
225:
218:
211:
203:
200:
199:
196:
195:
190:
185:
180:
175:
170:
165:
160:
155:
150:
148:Formal methods
144:
140:
139:
138:
135:
134:
131:
130:
128:Atomic formula
125:
120:
115:
110:
105:
100:
98:Formation rule
95:
93:Formal grammar
90:
85:
80:
75:
70:
64:
60:
59:
58:
55:
54:
48:
47:
26:
9:
6:
4:
3:
2:
4331:
4320:
4317:
4315:
4312:
4310:
4307:
4305:
4302:
4300:
4297:
4296:
4294:
4281:
4280:
4275:
4267:
4261:
4258:
4256:
4253:
4251:
4248:
4246:
4243:
4239:
4236:
4235:
4234:
4231:
4229:
4226:
4224:
4221:
4219:
4215:
4212:
4210:
4207:
4205:
4202:
4200:
4197:
4195:
4192:
4191:
4189:
4185:
4179:
4176:
4174:
4171:
4169:
4168:Recursive set
4166:
4164:
4161:
4159:
4156:
4154:
4151:
4149:
4146:
4142:
4139:
4137:
4134:
4132:
4129:
4127:
4124:
4122:
4119:
4118:
4117:
4114:
4112:
4109:
4107:
4104:
4102:
4099:
4097:
4094:
4092:
4089:
4088:
4086:
4084:
4080:
4074:
4071:
4069:
4066:
4064:
4061:
4059:
4056:
4054:
4051:
4049:
4046:
4044:
4041:
4037:
4034:
4032:
4029:
4027:
4024:
4023:
4022:
4019:
4017:
4014:
4012:
4009:
4007:
4004:
4002:
3999:
3997:
3994:
3990:
3987:
3986:
3985:
3982:
3978:
3977:of arithmetic
3975:
3974:
3973:
3970:
3966:
3963:
3961:
3958:
3956:
3953:
3951:
3948:
3946:
3943:
3942:
3941:
3938:
3934:
3931:
3929:
3926:
3925:
3924:
3921:
3920:
3918:
3916:
3912:
3906:
3903:
3901:
3898:
3896:
3893:
3891:
3888:
3885:
3884:from ZFC
3881:
3878:
3876:
3873:
3867:
3864:
3863:
3862:
3859:
3857:
3854:
3852:
3849:
3848:
3847:
3844:
3842:
3839:
3837:
3834:
3832:
3829:
3827:
3824:
3822:
3819:
3817:
3814:
3813:
3811:
3809:
3805:
3795:
3794:
3790:
3789:
3784:
3783:non-Euclidean
3781:
3777:
3774:
3772:
3769:
3767:
3766:
3762:
3761:
3759:
3756:
3755:
3753:
3749:
3745:
3742:
3740:
3737:
3736:
3735:
3731:
3727:
3724:
3723:
3722:
3718:
3714:
3711:
3709:
3706:
3704:
3701:
3699:
3696:
3694:
3691:
3689:
3686:
3685:
3683:
3679:
3678:
3676:
3671:
3665:
3660:Example
3657:
3649:
3644:
3643:
3642:
3639:
3637:
3634:
3630:
3627:
3625:
3622:
3620:
3617:
3615:
3612:
3611:
3610:
3607:
3605:
3602:
3600:
3597:
3595:
3592:
3588:
3585:
3583:
3580:
3579:
3578:
3575:
3571:
3568:
3566:
3563:
3561:
3558:
3556:
3553:
3552:
3551:
3548:
3546:
3543:
3539:
3536:
3534:
3531:
3529:
3526:
3525:
3524:
3521:
3517:
3514:
3512:
3509:
3507:
3504:
3502:
3499:
3497:
3494:
3492:
3489:
3488:
3487:
3484:
3482:
3479:
3477:
3474:
3472:
3469:
3465:
3462:
3460:
3457:
3455:
3452:
3450:
3447:
3446:
3445:
3442:
3440:
3437:
3435:
3432:
3430:
3427:
3423:
3420:
3418:
3417:by definition
3415:
3414:
3413:
3410:
3406:
3403:
3402:
3401:
3398:
3396:
3393:
3391:
3388:
3386:
3383:
3381:
3378:
3377:
3374:
3371:
3369:
3365:
3360:
3354:
3350:
3340:
3337:
3335:
3332:
3330:
3327:
3325:
3322:
3320:
3317:
3315:
3312:
3310:
3307:
3305:
3304:Kripke–Platek
3302:
3300:
3297:
3293:
3290:
3288:
3285:
3284:
3283:
3280:
3279:
3277:
3273:
3265:
3262:
3261:
3260:
3257:
3255:
3252:
3248:
3245:
3244:
3243:
3240:
3238:
3235:
3233:
3230:
3228:
3225:
3223:
3220:
3217:
3213:
3209:
3206:
3202:
3199:
3197:
3194:
3192:
3189:
3188:
3187:
3183:
3180:
3179:
3177:
3175:
3171:
3167:
3159:
3156:
3154:
3151:
3149:
3148:constructible
3146:
3145:
3144:
3141:
3139:
3136:
3134:
3131:
3129:
3126:
3124:
3121:
3119:
3116:
3114:
3111:
3109:
3106:
3104:
3101:
3099:
3096:
3094:
3091:
3089:
3086:
3084:
3081:
3080:
3078:
3076:
3071:
3063:
3060:
3058:
3055:
3053:
3050:
3048:
3045:
3043:
3040:
3038:
3035:
3034:
3032:
3028:
3025:
3023:
3020:
3019:
3018:
3015:
3013:
3010:
3008:
3005:
3003:
3000:
2998:
2994:
2990:
2988:
2985:
2981:
2978:
2977:
2976:
2973:
2972:
2969:
2966:
2964:
2960:
2950:
2947:
2945:
2942:
2940:
2937:
2935:
2932:
2930:
2927:
2925:
2922:
2918:
2915:
2914:
2913:
2910:
2906:
2901:
2900:
2899:
2896:
2895:
2893:
2891:
2887:
2879:
2876:
2874:
2871:
2869:
2866:
2865:
2864:
2861:
2859:
2856:
2854:
2851:
2849:
2846:
2844:
2841:
2839:
2836:
2834:
2831:
2830:
2828:
2826:
2825:Propositional
2822:
2816:
2813:
2811:
2808:
2806:
2803:
2801:
2798:
2796:
2793:
2791:
2788:
2784:
2781:
2780:
2779:
2776:
2774:
2771:
2769:
2766:
2764:
2761:
2759:
2756:
2754:
2753:Logical truth
2751:
2749:
2746:
2745:
2743:
2741:
2737:
2734:
2732:
2728:
2722:
2719:
2717:
2714:
2712:
2709:
2707:
2704:
2702:
2699:
2697:
2693:
2689:
2685:
2683:
2680:
2678:
2675:
2673:
2669:
2666:
2665:
2663:
2661:
2655:
2650:
2644:
2641:
2639:
2636:
2634:
2631:
2629:
2626:
2624:
2621:
2619:
2616:
2614:
2611:
2609:
2606:
2604:
2601:
2599:
2596:
2594:
2591:
2589:
2586:
2582:
2579:
2578:
2577:
2574:
2573:
2571:
2567:
2563:
2556:
2551:
2549:
2544:
2542:
2537:
2536:
2533:
2527:
2524:
2521:
2517:
2514:
2513:
2504:
2498:
2494:
2490:
2486:
2482:
2478:
2474:
2471:
2467:
2463:
2457:
2453:
2449:
2445:
2441:
2438:
2432:
2428:
2427:Penguin Books
2424:
2420:
2416:
2413:
2407:
2404:, Blackwell,
2402:
2401:
2394:
2391:
2389:0-226-28085-3
2385:
2381:
2376:
2373:
2367:
2363:
2359:
2354:
2344:on 2009-02-08
2343:
2339:
2335:
2330:
2327:
2321:
2317:
2313:
2309:
2305:
2301:
2297:
2293:
2288:
2287:
2273:
2266:
2262:
2256:
2247:
2241:
2237:
2231:
2225:
2221:
2217:
2212:
2206:
2202:
2196:
2189:
2185:
2180:
2171:
2164:
2159:
2153:
2148:
2139:
2128:
2125:
2122:
2119:
2118:
2115:
2107:
2101:
2097:
2096:
2088:
2080:
2074:
2070:
2069:
2061:
2053:
2047:
2043:
2042:
2034:
2026:
2020:
2016:
2015:
2007:
1999:
1993:
1989:
1988:
1980:
1972:
1966:
1962:
1955:
1947:
1941:
1937:
1936:
1928:
1920:
1914:
1910:
1909:
1901:
1893:
1887:
1883:
1882:
1874:
1866:
1860:
1856:
1855:
1847:
1838:
1834:
1824:
1821:
1819:
1816:
1814:
1811:
1809:
1806:
1804:
1801:
1800:
1796:
1785:
1778:
1776:
1772:
1768:
1764:
1760:
1759:nonsense word
1756:
1755:
1750:
1746:
1742:
1738:
1734:
1729:
1727:
1723:
1719:
1715:
1711:
1707:
1703:
1698:
1696:
1683:
1679:
1675:
1671:
1667:
1666:decidable set
1663:
1659:
1655:
1651:
1624:
1620:
1619:
1590:
1586:
1559:
1555:
1554:
1525:
1521:
1520:
1514:
1512:
1508:
1503:
1493:
1487:
1482:
1472:
1467:
1463:
1459:
1455:
1451:
1449:
1444:
1438:
1428:
1426:
1421:
1419:
1415:
1411:
1407:
1403:
1399:
1395:
1389:
1379:
1377:
1373:
1369:
1353:
1333:
1310:
1283:
1276:
1250:
1246:
1223:
1203:
1183:
1179:
1169:
1167:is a formula;
1154:
1134:
1114:
1110:
1100:
1098:are formulas;
1085:
1065:
1042:
1039:
1036:
1010:
1007:
1004:
994:
979:
959:
949:
948:
947:
940:
936:
929:
925:
920:
916:
909:
905:
901:
897:
891:
884:
877:
870:
866:
865:
864:
862:
853:
849:
842:
838:
834:
829:
825:
818:
814:
810:
807:
804:
803:
802:
800:
796:
795:
789:
787:
783:
753:
740:
736:
732:
728:
724:
720:
716:
715:
714:
708:
704:
700:
696:
692:
688:
685:
684:
683:
677:
673:
669:
665:
661:
657:
653:
652:
651:
649:
644:
638:
634:
630:
626:
622:
621:
620:
614:
610:
606:
602:
598:
594:
590:
589:
588:
456:
454:
450:
442:
439:
436:
435:
434:
432:
427:
425:
421:
417:
413:
409:
387:
384:
381:
375:
372:
361:
357:
351:
341:
337:
334:
328:
326:
322:
318:
314:
310:
300:
298:
294:
290:
285:
283:
278:
276:
272:
269:from a given
268:
264:
260:
256:
252:
248:
244:
240:
236:
224:
219:
217:
212:
210:
205:
204:
202:
201:
194:
191:
189:
186:
184:
181:
179:
176:
174:
171:
169:
166:
164:
161:
159:
156:
154:
151:
149:
146:
145:
137:
136:
129:
126:
124:
121:
119:
116:
114:
111:
109:
106:
104:
101:
99:
96:
94:
91:
89:
86:
84:
81:
79:
76:
74:
71:
69:
68:Formal system
66:
65:
57:
56:
53:
50:
49:
45:
44:
39:
38:
33:
19:
4270:
4068:Ultraproduct
3915:Model theory
3880:Independence
3816:Formal proof
3808:Proof theory
3791:
3764:
3721:real numbers
3693:second-order
3604:Substitution
3481:Metalanguage
3443:
3422:conservative
3395:Axiom schema
3339:Constructive
3309:Morse–Kelley
3275:Set theories
3254:Aleph number
3247:inaccessible
3153:Grothendieck
3037:intersection
2924:Higher-order
2912:Second-order
2858:Truth tables
2815:Venn diagram
2598:Formal proof
2480:
2450:, New York:
2447:
2422:
2399:
2379:
2357:
2346:. Retrieved
2342:the original
2337:
2311:
2295:
2291:
2272:
2255:
2246:
2230:
2216:Barwise, Jon
2211:
2195:
2179:
2170:
2158:
2147:
2138:
2114:
2094:
2087:
2067:
2060:
2040:
2033:
2013:
2006:
1986:
1979:
1960:
1954:
1934:
1927:
1907:
1900:
1880:
1873:
1853:
1846:
1837:
1823:WFF 'N Proof
1770:
1752:
1737:WFF 'N PROOF
1732:
1730:
1699:
1692:
1681:
1677:
1674:substitution
1661:
1653:
1616:
1588:
1551:
1523:
1510:
1506:
1498:
1491:
1488:preceded by
1485:
1477:
1470:
1465:
1453:
1446:
1442:
1440:
1425:open formula
1424:
1422:
1393:
1391:
1371:
1367:
1299:
992:is a formula
945:
938:
934:
927:
923:
918:
914:
907:
903:
899:
889:
882:
875:
868:
858:
851:
847:
840:
836:
832:
827:
823:
816:
812:
792:
790:
749:
738:
734:
730:
729:)) → (
726:
722:
718:
711:
706:
702:
698:
694:
690:
686:
681:
675:
671:
667:
663:
659:
655:
645:
642:
636:
632:
628:
624:
618:
612:
608:
604:
600:
596:
592:
586:
446:
428:
423:
415:
407:
353:
338:
329:
306:
303:Introduction
286:
281:
279:
258:
254:
250:
246:
232:
141:Applications
102:
61:Key concepts
42:
4178:Type theory
4126:undecidable
4058:Truth value
3945:equivalence
3624:non-logical
3237:Enumeration
3227:Isomorphism
3174:cardinality
3158:Von Neumann
3123:Ultrafilter
3088:Uncountable
3022:equivalence
2939:Quantifiers
2929:Fixed-point
2898:First-order
2778:Consistency
2763:Proposition
2740:Traditional
2711:Lindström's
2701:Compactness
2643:Type theory
2588:Cardinality
1754:whiffenpoof
1618:satisfiable
1402:quantifiers
662:) ∧ (
599:) ∧ (
431:inductively
4293:Categories
3989:elementary
3682:arithmetic
3550:Quantifier
3528:functional
3400:Expression
3118:Transitive
3062:identities
3047:complement
2980:hereditary
2963:Set theory
2348:2007-08-19
2298:(1): 29–41
2284:References
2263:using the
1761:used as a
1658:arithmetic
1652:A formula
1587:A formula
1522:A formula
118:Production
4304:Metalogic
4260:Supertask
4163:Recursion
4121:decidable
3955:saturated
3933:of models
3856:deductive
3851:axiomatic
3771:Hilbert's
3758:Euclidean
3739:canonical
3662:axiomatic
3594:Signature
3523:Predicate
3412:Extension
3334:Ackermann
3259:Operation
3138:Universal
3128:Recursive
3103:Singleton
3098:Inhabited
3083:Countable
3073:Types of
3057:power set
3027:partition
2944:Predicate
2890:Predicate
2805:Syllogism
2795:Soundness
2768:Inference
2758:Tautology
2660:paradoxes
2446:(2002) ,
1662:decidable
1497:⋯ ∀
1331:∀
1308:∃
1284:ϕ
1281:¬
1274:∃
1271:¬
1251:ϕ
1244:∀
1224:ϕ
1184:ϕ
1177:∀
1155:ϕ
1115:ϕ
1108:∃
1086:ψ
1066:ϕ
1043:ψ
1040:∨
1037:ϕ
1011:ψ
1008:∧
1005:ϕ
980:ϕ
960:ϕ
957:¬
831:), where
782:signature
721:→ (
631:)→(
488:alpha set
463:alpha set
385:∨
376:∧
289:syntactic
4245:Logicism
4238:timeline
4214:Concrete
4073:Validity
4043:T-schema
4036:Kripke's
4031:Tarski's
4026:semantic
4016:Strength
3965:submodel
3960:spectrum
3928:function
3776:Tarski's
3765:Elements
3752:geometry
3708:Robinson
3629:variable
3614:function
3587:spectrum
3577:Sentence
3533:variable
3476:Language
3429:Relation
3390:Automata
3380:Alphabet
3364:language
3218:-jection
3196:codomain
3182:Function
3143:Universe
3113:Infinite
3017:Relation
2800:Validity
2790:Argument
2688:theorem,
2479:(2010),
2421:(1980),
2310:(2002),
1781:See also
1720:, using
1462:variable
1454:sentence
725:∧
697:→
693:∧
689:→
666:→
658:→
627:→
603:→
595:→
315:such as
271:alphabet
263:sequence
73:Alphabet
43:a series
40:Part of
4187:Related
3984:Diagram
3882: (
3861:Hilbert
3846:Systems
3841:Theorem
3719:of the
3664:systems
3444:Formula
3439:Grammar
3355: (
3299:General
3012:Forcing
2997:Element
2917:Monadic
2692:paradox
2633:Theorem
2569:General
2470:1950307
2133:"woof".
2129:"whiff"
1490:∀
1460:of any
1450:formula
1445:, also
786:arities
293:meaning
267:symbols
259:formula
3950:finite
3713:Skolem
3666:
3641:Theory
3609:Symbol
3599:String
3582:atomic
3459:ground
3454:closed
3449:atomic
3405:ground
3368:syntax
3264:binary
3191:domain
3108:Finite
2873:finite
2731:Logics
2690:
2638:Theory
2499:
2468:
2458:
2433:
2408:
2386:
2368:
2322:
2238:
2222:
2203:
2126:"weff"
2123:"wiff"
2120:"woof"
2102:
2075:
2048:
2021:
1994:
1967:
1942:
1915:
1888:
1861:
1722:Polish
1695:Church
1448:ground
1412:. For
902:is an
835:is an
325:proofs
78:Syntax
3940:Model
3688:Peano
3545:Proof
3385:Arity
3314:Naive
3201:image
3133:Fuzzy
3093:Empty
3042:union
2987:Class
2628:Model
2618:Lemma
2576:Axiom
2522:quiz.
1829:Notes
1763:cheer
1726:infix
1553:valid
1505:is a
1476:, …,
1464:. If
1370:. An
933:,...,
913:,...,
846:,...,
822:,...,
794:terms
563:) | (
543:) | (
523:) | (
333:token
4063:Type
3866:list
3670:list
3647:list
3636:Term
3570:rank
3464:open
3358:list
3170:Maps
3075:sets
2934:Free
2904:list
2654:list
2581:list
2520:Java
2497:ISBN
2456:ISBN
2431:ISBN
2406:ISBN
2384:ISBN
2366:ISBN
2320:ISBN
2236:ISBN
2220:ISBN
2201:ISBN
2100:ISBN
2073:ISBN
2046:ISBN
2019:ISBN
1992:ISBN
1965:ISBN
1940:ISBN
1913:ISBN
1886:ISBN
1859:ISBN
1773:and
1757:, a
1418:term
1400:nor
1078:and
1026:and
874:and
581:>
578:form
575:<
571:>
568:form
565:<
561:>
558:form
555:<
551:>
548:form
545:<
541:>
538:form
535:<
531:>
528:form
525:<
521:>
518:form
515:<
511:>
508:form
505:<
501:>
498:form
495:<
491:>
485:<
479:>
476:form
473:<
466:>
460:<
410:of
311:and
245:, a
241:and
3750:of
3732:of
3680:of
3212:Sur
3186:Map
2993:Ur-
2975:Set
2489:doi
1765:at
1733:WFF
1724:or
1660:is
1625:of
1615:is
1560:of
1550:is
1509:of
1452:or
1392:An
1323:or
898:If
867:If
654:(((
591:(((
503:| (
493:| ¬
482:::=
469:::=
451:in
282:wff
265:of
255:wff
253:or
251:WFF
233:In
4295::
4136:NP
3760::
3754::
3684::
3361:),
3216:Bi
3208:In
2495:,
2487:,
2466:MR
2464:,
2454:,
2429:,
2425:,
2364:,
2336:.
2318:,
2296:30
2186:;
1777:.
1708:,
1704:,
1513:.
1441:A
1420:.
1296:).
863:.
801:.
741:))
678:))
639:))
635:))
633:qq
623:((
615:))
583:)
277:.
237:,
46:on
4216:/
4131:P
3886:)
3672:)
3668:(
3565:∀
3560:!
3555:∃
3516:=
3511:↔
3506:→
3501:∧
3496:∨
3491:¬
3214:/
3210:/
3184:/
2995:)
2991:(
2878:∞
2868:3
2656:)
2554:e
2547:t
2540:v
2491::
2351:.
2267:.
2165:.
2108:.
2081:.
2054:.
2027:.
2000:.
1973:.
1948:.
1921:.
1894:.
1867:.
1682:A
1678:A
1654:A
1649:.
1635:Q
1601:Q
1589:A
1584:.
1570:Q
1536:Q
1524:A
1511:A
1501:n
1499:v
1495:1
1492:v
1486:A
1480:n
1478:v
1474:1
1471:v
1466:A
1354:x
1334:x
1311:x
1277:x
1247:x
1204:x
1180:x
1135:x
1111:x
1046:)
1034:(
1014:)
1002:(
939:n
935:t
931:1
928:t
926:(
924:R
919:n
915:t
911:1
908:t
904:n
900:R
893:2
890:t
888:=
886:1
883:t
879:2
876:t
872:1
869:t
852:n
848:t
844:1
841:t
837:n
833:f
828:n
824:t
820:1
817:t
815:(
813:f
766:S
763:Q
739:s
735:q
731:s
727:r
723:q
719:p
717:(
707:s
703:q
699:s
695:r
691:q
687:p
676:s
672:q
668:s
664:r
660:q
656:p
637:p
629:q
625:p
613:s
609:q
605:s
601:r
597:q
593:p
573:↔
553:→
533:∨
513:∧
424:V
416:V
408:V
394:)
391:)
388:C
382:B
379:(
373:A
370:(
222:e
215:t
208:v
34:.
20:)
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