390:; the addition of a third value in ternary logic leads to a total of 3 = 27 distinct operators on a single input value. (This may be made clear by considering all possible truth tables for an arbitrary unary operator. Given 2 possible values TF of the single Boolean input, there are four different patterns of output TT, TF, FT, FF resulting from the following unary operators acting on each value: always T, Identity, NOT, always F. Given three possible values of a ternary variable, each times three possible results of a unary operation, there are 27 different output patterns: TTT, TTU, TTF, TUT, TUU, TUF, TFT, TFU, TFF, UTT, UTU, UTF, UUT, UUU, UUF, UFT, UFU, UFF, FTT, FTU, FTF, FUT, FUU, FUF, FFT, FFU, and FFF.) Similarly, where Boolean logic has 2 = 16 distinct binary operators (operators with 2 inputs) possible, ternary logic has 3 = 19,683 such operators. Where the nontrival Boolean operators can be named (
3001:
each on a subset of the natural numbers (as occurs e.g. after completing the definitions of any two partial recursive predicates classically). Let t, f, u mean 'decidable by the algorithms (i.e. by use of only such information about Q(x) and R(x) as can be obtained by the algorithms) to be true', 'decidable by the algorithms to be false', 'undecidable by the algorithms whether true or false'. (iv) Assume a fixed state of knowledge about Q(x) and R(x) (as occurs e.g. after pursuing algorithms for each of them up to a given stage). Let t, f, u mean 'known to be true', 'known to be false', 'unknown whether true or false'.
219:. He never published it. In fact, he did not even number the three pages of notes where he defined his three-valued operators. Peirce soundly rejected the idea all propositions must be either true or false; boundary-propositions, he writes, are "at the limit between P and not P." However, as confident as he was that "Triadic Logic is universally true," he also jotted down that "All this is mighty close to nonsense." Only in 1966, when Max Fisch and Atwell Turquette began publishing what they rediscovered in his unpublished manuscripts, did Peirce's triadic ideas become widely known.
5305:
245:
observational data that a statement as to the position of a motor car can never be falsified or verified, then there may be some point to not regarding the statement as true or false, but regarding it as "middle." It is only because, in macrocosmic experience, everything that we regard as an empirically meaningful statement seems to be at least potentially verifiable or falsifiable that we prefer the convention according to which we say that every such statement is either true or false, but in many cases we don't know which.
2384:
25:
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However, it is not the case that 'middle' means "neither verified nor falsified at the present time." As we have seen, 'verified' and 'falsified' are epistemic predicates--that is to say, they are relative to the evidence at a particular time--whereas 'middle,' like 'true' and 'false' is not relative
1368:
truth value for Kleene logic is True.) However, the lack of valid formulas does not mean that it lacks valid arguments and/or inference rules. An argument is semantically valid in Kleene logic if, whenever (for any interpretation/model) all of its premises are True, the conclusion must also be True.
3000:
The strong 3-valued logic can be applied to completely defined predicates Q(x) and R(x), from which composite predicates are formed using ̅, V, &, ->, ≡ in the usual 2-valued meanings, thus, (iii) Suppose that there are fixed algorithms which decide the truth or falsity of Q(x) and of R(x),
2950:
But there is a second possible way to conceive of many-valued logics: that while a proposition, in itself, can have only two values, true or false, that is to say two responses, yes or no, it may happen that a given individual does not know the response, at least at a given moment; therefore, for
1002:
state can be thought of as neither true nor false in Kleene logic, or thought of as both true and false in Priest logic. The difference lies in the definition of tautologies. Where Kleene logic's only designated truth value is T, Priest logic's designated truth values are both T and U. In Kleene
244:
For example, if we have verified (by using a speedometer) that the velocity of a motor car is such and such, it might be impossible in such a world to verify or falsify certain statements concerning its position at that moment. If we know by reference to a physical law together with certain
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or inequality), with six trivial operators considering 0 or 1 inputs only, it is unreasonable to attempt to name all but a small fraction of the possible ternary operators. Just as in bivalent logic, where not all operators are given names and subsets of
227:
Broadly speaking, the primary motivation for research of three valued logic is to represent the truth value of a statement that cannot be represented as true or false. Łukasiewicz initially developed three valued logic for the
1363:
Kleene logic has no tautologies (valid formulas) because whenever all of the atomic components of a well-formed formula are assigned the value
Unknown, the formula itself must also have the value Unknown. (And the only
187:
is credited with first introducing additional logical truth degrees in his 1921 theory of elementary propositions. The conceptual form and basic ideas of three-valued logic were initially published by
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the individual there is a third attitude possible toward a proposition. This third attitude does not correspond to a distinct third value of yes or of no, but simply to a doubt between yes or no
2162:
where the third truth value NF (not false) has the semantics of a proposition that can be intuitionistically proven to not be false, but does not have an intuitionistic proof of correctness.
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using a three-valued logic, "it is possible that..." L is read "it is true that..." or "it is necessary that..." Finally I is read "it is unknown that..." or "it is contingent that..."
1967:
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2355:, or by explicit truth tables for its operations. In particular, conjunction and disjunction are the same as for Kleene's and Łukasiewicz's logic, while the negation is different.
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The Łukasiewicz Ł3 has the same tables for AND, OR, and NOT as the Kleene logic given above, but differs in its definition of implication in that "unknown implies unknown" is
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truth values instead of one; these are: True and Both (the analogue of
Unknown), so that LP does have tautologies but it has fewer valid inference rules).
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RM3 is a non-cartesian symmetric monoidal closed category; the product, which is left-adjoint to the implication, lacks valid projections, and has
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in the lattice of intermediate logics. In this sense it may be viewed as the "second strongest" intermediate logic after classical logic.
2800:
Cobreros, Pablo; Égré, Paul; Ripley, David; Rooij, Robert van (2 January 2014). "Foreword: Three-valued logics and their applications".
2528:
2115:
3244:
Mundici, D. The C*-Algebras of Three-Valued Logic. Logic
Colloquium ’88, Proceedings of the Colloquium held in Padova 61–77 (1989).
1015:
at any moment in time is not available. However, certain logical operations can yield an unambiguous result, even if they involve an
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89:
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are tautologies in Ł3 and also in classical logic. Not all tautologies of classical logic lift to Ł3 "as is". For example, the
61:
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As with bivalent logic, truth values in ternary logic may be represented numerically using various representations of the
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3175:. Synthesis lectures on digital circuits and systems. Vol. 12. Morgan & Claypool Publishers. pp. 41–42.
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3283:
3263:
3099:
2667:"Peirce's Deductive Logic > Peirce's Three-Valued Logic (Stanford Encyclopedia of Philosophy/Summer 2020 Edition)"
108:
4252:
75:
5007:
4751:
3576:
436:
308:, only the least-significant non-zero digit can have a value of 2, and the remaining digits have a value of 0 or 1;
236:
used a third value to represent when "a given individual does not know the response, at least at a given moment."
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4312:
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281:
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1190:{\displaystyle A\rightarrow B\ {\overset {\underset {\mathrm {def} }{}}{=}}\ {\mbox{OR}}(\ {\mbox{NOT}}(A),\ B)}
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57:
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277:, each digit has one of 3 values: −1, 0, or +1; these values may also be simplified to −, 0, +, respectively;
229:
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field content. SQL uses a common fragment of the Kleene K3 logic, restricted to AND, OR, and NOT tables.
2394:
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M is read as "it is not false that..." or in the (unsuccessful) Tarski–Łukasiewicz attempt to axiomatize
2691:
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In fact, using Łukasiewicz's implication and negation, the other usual connectives may be derived as:
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2523:
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have been introduced more recently, motivated by circuit problems rather than philosophical issues:
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It is also possible to derive a few other useful unary operators (first derived by Tarski in 1921):
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4595:
4447:
4430:
4153:
3633:
1900:
2905:
Rybaříková, Zuzana (1 May 2021). "Łukasiewicz, determinism, and the four-valued system of logic".
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284:, each digit can have a value of −1, 0, 0/1 (the value 0/1 has two different representations);
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An
Introduction to Many-Valued and Fuzzy Logic: Semantics, Algebras, and Derivation Systems
2352:
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419:
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8:
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3144:
Handbook of the
History of Logic Volume 8. The Many Valued and Nonmonotonic Turn in Logic
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446:
427:
operators are used, there may be functionally complete sets of ternary-valued operators.
368:
329:
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is True, meaning that only a proposition having this value everywhere is considered a
188:
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2998:. North-Holland Publishing Co., Amsterdam, and P. Noordhoff, Groningen. p. 336.
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1098:, then A AND B AND C... = MIN(A, B, C ...) and A OR B OR C ... = MAX(A, B, C...).
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292:
2745:"Logic : autograph manuscript notebook, November 12, 1865-November 1, 1909"
2715:"Logic : autograph manuscript notebook, November 12, 1865-November 1, 1909"
1055:
as well. In this example, because either bivalent state could be underlying the
371:
using the truth values {false, unknown, true}, and extends conventional
Boolean
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2932:
de
Finetti, Bruno (1 January 1995). "The logic of probability (translated)".
2891:
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2642:
1921:
defined above, it is possible to state tautologies that are their analogues:
450:
383:
237:
170:
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2731:
Triadic Logic is universally true. But Dyadic Logic is not aboslutely false
395:
2918:
232:
to represent the truth value of statements about the undetermined future.
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which, by adjointness, is equivalent to the projection from the product:
1851:
1395:. This section follows the presentation from Malinowski's chapter of the
442:
403:
391:
372:
154:
2383:
2376:
This logic is also known as a weak form of Kleene's three-valued logic.
2065:
A defining characteristic of RM3 is the lack of the axiom of
Weakening:
165:, and some third value. This is contrasted with the more commonly known
4138:
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It may be defined either by appending one of the two equivalent axioms
399:
3268:
2529:
Paraconsistent logic § An ideal three-valued paraconsistent logic
5290:
5193:
4246:
4163:
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4023:
3835:
3825:
3798:
3561:
3512:
3363:
352:, and a third non-integer "maybe" symbol such as ?, #, ½, or xy.
3156:
Heyting (1930). "Die formalen Regeln der intuitionistischen Logik".
2883:
2844:
2749:
hollisarchives.lib.harvard.edu/repositories/24/digital_objects/63983
2719:
hollisarchives.lib.harvard.edu/repositories/24/digital_objects/63983
2623:
2606:
1979:
The truth table for the material implication of R-mingle 3 (RM3) is
24:
5275:
5073:
4521:
4226:
3820:
3052:
2536:– an experimental Russian computer which was based on ternary logic
3094:. London, England: Penguin Books. Entry for 'three-valued logic'.
2147:
4871:
3663:
3505:
3020:. Reading, Mass.: Addison-Wesley Publishing Company. p. 190.
2469:
implements ternary logic as a means of handling comparisons with
2151:
1360:
which differs from that for Łukasiewicz logic (described below).
1373:(LP) has the same truth tables as Kleene logic, but it has two
240:
used it to represent values that cannot physically be decided:
2372:
Many-valued logic § Bochvar's internal three-valued logic
4415:
3761:
3606:
2607:"Introduction to a General Theory of Elementary Propositions"
2533:
122:
257:
that are "undecidable by algorithms whether true or false"
3517:
2422:
not(a) = (a + 1) mod (n), where (n) is the value of a logic
411:
1402:
Material implication for Łukasiewicz logic truth table is
2466:
1101:
Material implication for Kleene logic can be defined as:
407:
2963:
Putnam, Hilary (1 October 1957). "Three-valued logic".
2799:
2114:
as the monoid identity. This logic is equivalent to an
1915:
are not tautologies in Ł3. However, using the operator
1160:
1147:
3091:
The
Penguin Dictionary of Mathematics. Fourth Edition
1109:
1059:
state, and either state also yields the same result,
199:
in an axiomatic algebraic form, and also extended to
2480:
367:
This article mainly illustrates a system of ternary
3173:
Multiple valued logic: concepts and representations
49:. Unsourced material may be challenged and removed.
3171:Miller, D. Michael; Thornton, Mitchell A. (2008).
1189:
5321:
3197:Multiple-Valued Logic Synthesis and Optimization
947:
874:
801:
676:
603:
530:
2134:Many-valued logic § Gödel logics Gk and G∞
1003:logic, the knowledge of whether any particular
3258:. University of California Press. Dover 1998:
3170:
301:(trinary digit) having a value of: 0, 1, or 2;
3577:
3284:
3256:Philosophic Foundations of Quantum Mechanics
3055:, the Scientific Research Society: 490–494.
2829:"Three-Valued Logic and Future Contingents"
2742:
2712:
430:
260:
3769:
3584:
3570:
3291:
3277:
2931:
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269:. A few of the more common examples are:
2650:
2632:
2622:
109:Learn how and when to remove this message
3215:
3203:, Kluwer Academic Publishers, pp. 89-114
169:logics (such as classical sentential or
3298:
3199:, in Hassoun S. and Sasao T., editors,
3155:
2802:Journal of Applied Non-Classical Logics
2788:
449:'s "strong logic of indeterminacy" and
16:System including an indeterminate value
5322:
3591:
3142:" in Dov M. Gabbay, John Woods (eds.)
3087:
3018:The Art of Computer Programming Vol. 2
2993:
2987:
2962:
2956:
2865:
2859:
2751:. Houghton Library, Harvard University
2721:. Houghton Library, Harvard University
1706:They have the following truth tables:
3565:
3272:
3030:
3012:
2868:"The Problem of Future Contingencies"
2826:
2413:
2234:
2118:which also obeys the contrapositive.
1995:
1495:
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1213:
928:
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3140:Many-valued Logic and its Philosophy
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1380:
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950:
877:
804:
679:
606:
533:
360:, ternary values are represented by
47:adding citations to reliable sources
18:
2833:The Philosophical Quarterly (1950-)
2426:
2168:(F, false; NF, not false; T, true)
195:. These were then re-formulated by
13:
3209:
2142:, also referred as Smetanov logic
1136:
1133:
1130:
730:(−1, false; 0, unknown; +1, true)
14:
5346:
2702:from the original on Dec 6, 2023.
2594:from the original on May 3, 2023.
445:showing the logic operations for
153:systems in which there are three
5303:
3201:Logic Synthesis and Verification
2776:www.digitalpeirce.fee.unicamp.br
2769:
2743:Peirce, Charles S. (1839–1914).
2713:Peirce, Charles S. (1839–1914).
2584:"Trilean (Stanford JavaNLP API)"
2483:
2382:
2365:
2154:in 1930 as a model for studying
1968:extended contradiction principle
1397:Handbook of the History of Logic
459:(F, false; U, unknown; T, true)
437:Kleene algebra (with involution)
253:used a third value to represent
206:
23:
3189:
3164:
3149:
3132:
3121:
3108:
3081:
3070:from the original on 2019-10-30
3024:
3006:
2996:Introduction to metamathematics
2611:American Journal of Mathematics
2449:
2220:
2164:
1981:
1708:
1404:
1199:
924:MIN(MAX(A, B), NEG(MIN(A, B)))
726:
455:
282:redundant binary representation
34:needs additional citations for
3222:. Cambridge University Press.
2763:
2736:
2706:
2683:
2659:
2634:2027/uiuo.ark:/13960/t9j450f7q
2598:
2576:
1184:
1172:
1166:
1153:
1113:
1019:operand. For example, because
1:
5264:History of mathematical logic
3250:10.1016/s0049-237x(08)70262-3
2994:Kleene, Stephen Cole (1952).
2570:
2499:Binary logic (disambiguation)
2222:
2171:
2138:The logic of here and there (
1983:
1710:
1406:
1201:
733:
462:
230:problem of future contingents
222:
5189:Primitive recursive function
3128:"Beyond Propositional Logic"
2814:10.1080/11663081.2014.909631
2465:The database query language
2116:"ideal" paraconsistent logic
1974:
1063:results in all three cases.
7:
3438:Ontology (computer science)
2504:Boolean algebra (structure)
2476:
2121:
998:In these truth tables, the
10:
5351:
4253:Schröder–Bernstein theorem
3980:Monadic predicate calculus
3639:Foundations of mathematics
3331:Intuitionistic type theory
3254:Reichenbach, Hans (1944).
3034:(November–December 2001).
2458:
2419:not(a) = (a + 1) mod 3, or
2369:
2131:
2125:
1384:
1007:state secretly represents
434:
5299:
5286:Philosophy of mathematics
5235:Automated theorem proving
5217:
5112:
4944:
4837:
4689:
4406:
4382:
4360:Von Neumann–Bernays–Gödel
4305:
4199:
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4001:
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3919:
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3760:
3682:
3599:
3528:
3479:
3451:
3428:
3410:
3382:
3344:
3306:
3216:Bergmann, Merrie (2008).
2524:Homogeneity (linguistics)
2445:Dubrova and Muzio algebra
2270:
2256:
2253:
2237:
2207:
2199:
2191:
2167:
2150:G3 logic), introduced by
2031:
2017:
2014:
1998:
1836:
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1440:
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1421:
1326:
1312:
1309:
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1235:
1232:
1216:
1197:, and its truth table is
978:
964:
931:
905:
891:
858:
832:
818:
785:
765:
757:
749:
729:
707:
693:
660:
634:
620:
587:
561:
547:
514:
494:
486:
478:
458:
378:
306:skew binary number system
197:Grigore Constantin Moisil
173:) which provide only for
2872:The Philosophical Review
2866:Taylor, Richard (1957).
2248:
2245:
2242:
2186:
2183:
2009:
2006:
2003:
1901:law of non-contradiction
1857:In Łukasiewicz's Ł3 the
1810:
1805:
1765:
1760:
1720:
1715:
1509:
1506:
1503:
1432:
1429:
1426:
1304:
1301:
1298:
1227:
1224:
1221:
1070:values, are assigned to
1066:If numeric values, e.g.
942:
939:
936:
869:
866:
863:
796:
793:
790:
744:
741:
671:
668:
665:
598:
595:
592:
525:
522:
519:
473:
470:
431:Kleene and Priest logics
375:to a trivalent context.
261:Representation of values
217:many-valued logic system
203:-valued logics in 1945.
145:, sometimes abbreviated
4936:Self-verifying theories
4757:Tarski's axiomatization
3708:Tarski's undefinability
3703:incompleteness theorems
3336:Constructive set theory
3195:Dubrova, Elena (2002).
2358:HT logic is the unique
453:'s "logic of paradox".
5310:Mathematics portal
4921:Proof of impossibility
4569:propositional variable
3879:Propositional calculus
3138:Grzegorz Malinowski, "
3116:Standard Ternary Logic
3088:Nelson, David (2008).
2605:Post, Emil L. (1921).
2590:. Stanford NLP Group.
2545:Ternary numeral system
2454:
2391:This section is empty.
1943:law of excluded fourth
1887:law of excluded middle
1491:(A, B), MIN(1, 1−A+B)
1191:
289:ternary numeral system
267:ternary numeral system
247:
213:Charles Sanders Peirce
5179:Kolmogorov complexity
5132:Computably enumerable
5032:Model complete theory
4824:Principia Mathematica
3884:Propositional formula
3713:Banach–Tarski paradox
3321:Constructive analysis
2965:Philosophical Studies
2934:Philosophical Studies
2919:10.1515/sem-2019-0115
2827:Prior, A. N. (1953).
2132:Further information:
2126:Further information:
1385:Further information:
1192:
425:functionally complete
418:), and 4 variants of
242:
193:Clarence Irving Lewis
5127:Church–Turing thesis
5114:Computability theory
4323:continuum hypothesis
3841:Square of opposition
3699:Gödel's completeness
3374:Fuzzy set operations
3369:Fuzzy finite element
3316:Intuitionistic logic
3061:10.1511/2001.40.3268
2433:modulars arithmetics
2353:intuitionistic logic
2158:, is a three-valued
2156:intuitionistic logic
1107:
149:) is any of several
58:"Three-valued logic"
43:improve this article
5281:Mathematical object
5172:P versus NP problem
5137:Computable function
4931:Reverse mathematics
4857:Logical consequence
4734:primitive recursive
4729:elementary function
4502:Free/bound variable
4355:Tarski–Grothendieck
3874:Logical connectives
3804:Logical equivalence
3654:Logical consequence
3551:Non-monotonic logic
3300:Non-classical logic
2588:Stanford University
2565:The World of Null-A
2540:Strawson entailment
2231:
2180:
1992:
1492:
1415:
1287:
1286:(A, B), MAX(−A, B)
1210:
925:
852:
779:
738:
654:
581:
508:
467:
447:Stephen Cole Kleene
369:propositional logic
251:Stephen Cole Kleene
5079:Transfer principle
5042:Semantics of logic
5027:Categorical theory
5003:Non-standard model
4517:Logical connective
3644:Information theory
3593:Mathematical logic
3546:Intermediate logic
3326:Heyting arithmetic
3114:Douglas W. Jones,
3044:American Scientist
2977:10.1007/BF02304905
2946:10.1007/BF00996317
2671:plato.stanford.edu
2414:Ternary Post logic
2225:
2174:
2160:intermediate logic
2128:Intermediate logic
1986:
1486:
1409:
1281:
1209:(A, B), OR(¬A, B)
1204:
1187:
1164:
1151:
1140:
923:
850:
777:
736:
652:
579:
506:
465:
441:Below is a set of
127:three-valued logic
5335:Ternary computers
5330:Many-valued logic
5317:
5316:
5249:Abstract category
5052:Theories of truth
4862:Rule of inference
4852:Natural deduction
4833:
4832:
4378:
4377:
4083:Cartesian product
3988:
3987:
3894:Many-valued logic
3869:Boolean functions
3752:Russell's paradox
3727:diagonal argument
3624:First-order logic
3559:
3558:
3541:Inquisitive logic
3536:Dynamic semantics
3489:Three-state logic
3443:Ontology language
3229:978-0-521-88128-9
3182:978-1-59829-190-2
2690:Lane, R. (2001).
2555:Three-state logic
2519:Four-valued logic
2491:Philosophy portal
2411:
2410:
2351:to the axioms of
2302:
2301:
2298:
2297:
2219:
2218:
2215:
2214:
2063:
2062:
2059:
2058:
1848:
1847:
1844:
1843:
1799:
1798:
1754:
1753:
1563:
1562:
1559:
1558:
1482:
1481:
1387:Łukasiewicz logic
1381:Łukasiewicz logic
1358:
1357:
1354:
1353:
1277:
1276:
1180:
1163:
1158:
1150:
1145:
1141:
1128:
1127:
1121:
996:
995:
992:
991:
919:
918:
846:
845:
773:
772:
725:
724:
721:
720:
648:
647:
575:
574:
502:
501:
151:many-valued logic
119:
118:
111:
93:
5342:
5308:
5307:
5259:History of logic
5254:Category of sets
5147:Decision problem
4926:Ordinal analysis
4867:Sequent calculus
4765:Boolean algebras
4705:
4704:
4679:
4650:logical/constant
4404:
4403:
4390:
4313:Zermelo–Fraenkel
4064:Set operations:
3999:
3998:
3936:
3767:
3766:
3747:Löwenheim–Skolem
3634:Formal semantics
3586:
3579:
3572:
3563:
3562:
3494:Tri-state buffer
3293:
3286:
3279:
3270:
3269:
3240:
3238:
3236:
3204:
3193:
3187:
3186:
3168:
3162:
3161:
3153:
3147:
3146:, Elsevier, 2009
3136:
3130:
3125:
3119:
3118:, Feb. 11, 2013.
3112:
3106:
3105:
3085:
3079:
3078:
3076:
3075:
3069:
3040:
3028:
3022:
3021:
3014:Knuth, Donald E.
3010:
3004:
3003:
2991:
2985:
2984:
2982:to the evidence.
2960:
2954:
2953:
2929:
2923:
2922:
2913:(240): 129–143.
2902:
2896:
2895:
2863:
2857:
2856:
2824:
2818:
2817:
2797:
2786:
2785:
2783:
2782:
2767:
2761:
2760:
2758:
2756:
2740:
2734:
2733:
2728:
2726:
2710:
2704:
2703:
2687:
2681:
2680:
2678:
2677:
2663:
2657:
2656:
2654:
2636:
2626:
2602:
2596:
2595:
2580:
2559:tri-state buffer
2549:Balanced ternary
2509:Boolean function
2493:
2488:
2487:
2486:
2427:Modular algebras
2406:
2403:
2393:You can help by
2386:
2379:
2350:
2332:or equivalently
2331:
2232:
2224:
2221:
2181:
2173:
2165:
2113:
2104:
2083:
1993:
1985:
1982:
1965:
1940:
1920:
1914:
1898:
1884:
1874:
1859:designated value
1816:
1808:
1803:
1802:
1771:
1763:
1758:
1757:
1726:
1718:
1713:
1712:
1709:
1702:
1678:
1660:
1637:
1609:
1589:
1493:
1485:
1416:
1408:
1405:
1371:Logic of Paradox
1288:
1280:
1211:
1203:
1200:
1196:
1194:
1193:
1188:
1178:
1165:
1161:
1156:
1152:
1148:
1143:
1142:
1139:
1123:
1119:
1068:balanced ternary
926:
922:
853:
849:
780:
776:
739:
735:
727:
655:
651:
582:
578:
509:
505:
468:
464:
456:
358:ternary computer
275:balanced ternary
234:Bruno de Finetti
114:
107:
103:
100:
94:
92:
51:
27:
19:
5350:
5349:
5345:
5344:
5343:
5341:
5340:
5339:
5320:
5319:
5318:
5313:
5302:
5295:
5240:Category theory
5230:Algebraic logic
5213:
5184:Lambda calculus
5122:Church encoding
5108:
5084:Truth predicate
4940:
4906:Complete theory
4829:
4698:
4694:
4690:
4685:
4677:
4397: and
4393:
4388:
4374:
4350:New Foundations
4318:axiom of choice
4301:
4263:Gödel numbering
4203: and
4195:
4099:
3984:
3934:
3915:
3864:Boolean algebra
3850:
3814:Equiconsistency
3779:Classical logic
3756:
3737:Halting problem
3725: and
3701: and
3689: and
3688:
3683:Theorems (
3678:
3595:
3590:
3560:
3555:
3524:
3475:
3447:
3424:
3406:
3397:Relevance logic
3392:Structural rule
3378:
3354:Degree of truth
3340:
3302:
3297:
3234:
3232:
3230:
3212:
3210:Further reading
3207:
3194:
3190:
3183:
3169:
3165:
3154:
3150:
3137:
3133:
3126:
3122:
3113:
3109:
3102:
3086:
3082:
3073:
3071:
3067:
3038:
3029:
3025:
3011:
3007:
2992:
2988:
2961:
2957:
2930:
2926:
2903:
2899:
2884:10.2307/2182851
2864:
2860:
2845:10.2307/2217099
2839:(13): 317–326.
2825:
2821:
2798:
2789:
2780:
2778:
2772:"Triadic Logic"
2768:
2764:
2754:
2752:
2741:
2737:
2724:
2722:
2711:
2707:
2692:"Triadic Logic"
2688:
2684:
2675:
2673:
2665:
2664:
2660:
2624:10.2307/2370324
2603:
2599:
2582:
2581:
2577:
2573:
2514:Digital circuit
2489:
2484:
2482:
2479:
2463:
2457:
2452:
2442:Pradhan algebra
2429:
2416:
2407:
2401:
2398:
2374:
2368:
2333:
2305:
2229:
2178:
2136:
2130:
2124:
2109:
2091:
2069:
1990:
1977:
1948:
1925:
1916:
1904:
1890:
1876:
1866:
1865:. For example,
1811:
1806:
1766:
1761:
1721:
1716:
1681:
1663:
1645:
1612:
1592:
1569:
1490:
1413:
1389:
1383:
1285:
1208:
1159:
1146:
1129:
1122:
1108:
1105:
1104:
439:
433:
388:unary operators
381:
362:ternary signals
263:
225:
209:
189:Jan Łukasiewicz
115:
104:
98:
95:
52:
50:
40:
28:
17:
12:
11:
5:
5348:
5338:
5337:
5332:
5315:
5314:
5300:
5297:
5296:
5294:
5293:
5288:
5283:
5278:
5273:
5272:
5271:
5261:
5256:
5251:
5242:
5237:
5232:
5227:
5225:Abstract logic
5221:
5219:
5215:
5214:
5212:
5211:
5206:
5204:Turing machine
5201:
5196:
5191:
5186:
5181:
5176:
5175:
5174:
5169:
5164:
5159:
5154:
5144:
5142:Computable set
5139:
5134:
5129:
5124:
5118:
5116:
5110:
5109:
5107:
5106:
5101:
5096:
5091:
5086:
5081:
5076:
5071:
5070:
5069:
5064:
5059:
5049:
5044:
5039:
5037:Satisfiability
5034:
5029:
5024:
5023:
5022:
5012:
5011:
5010:
5000:
4999:
4998:
4993:
4988:
4983:
4978:
4968:
4967:
4966:
4961:
4954:Interpretation
4950:
4948:
4942:
4941:
4939:
4938:
4933:
4928:
4923:
4918:
4908:
4903:
4902:
4901:
4900:
4899:
4889:
4884:
4874:
4869:
4864:
4859:
4854:
4849:
4843:
4841:
4835:
4834:
4831:
4830:
4828:
4827:
4819:
4818:
4817:
4816:
4811:
4810:
4809:
4804:
4799:
4779:
4778:
4777:
4775:minimal axioms
4772:
4761:
4760:
4759:
4748:
4747:
4746:
4741:
4736:
4731:
4726:
4721:
4708:
4706:
4687:
4686:
4684:
4683:
4682:
4681:
4669:
4664:
4663:
4662:
4657:
4652:
4647:
4637:
4632:
4627:
4622:
4621:
4620:
4615:
4605:
4604:
4603:
4598:
4593:
4588:
4578:
4573:
4572:
4571:
4566:
4561:
4551:
4550:
4549:
4544:
4539:
4534:
4529:
4524:
4514:
4509:
4504:
4499:
4498:
4497:
4492:
4487:
4482:
4472:
4467:
4465:Formation rule
4462:
4457:
4456:
4455:
4450:
4440:
4439:
4438:
4428:
4423:
4418:
4413:
4407:
4401:
4384:Formal systems
4380:
4379:
4376:
4375:
4373:
4372:
4367:
4362:
4357:
4352:
4347:
4342:
4337:
4332:
4327:
4326:
4325:
4320:
4309:
4307:
4303:
4302:
4300:
4299:
4298:
4297:
4287:
4282:
4281:
4280:
4273:Large cardinal
4270:
4265:
4260:
4255:
4250:
4236:
4235:
4234:
4229:
4224:
4209:
4207:
4197:
4196:
4194:
4193:
4192:
4191:
4186:
4181:
4171:
4166:
4161:
4156:
4151:
4146:
4141:
4136:
4131:
4126:
4121:
4116:
4110:
4108:
4101:
4100:
4098:
4097:
4096:
4095:
4090:
4085:
4080:
4075:
4070:
4062:
4061:
4060:
4055:
4045:
4040:
4038:Extensionality
4035:
4033:Ordinal number
4030:
4020:
4015:
4014:
4013:
4002:
3996:
3990:
3989:
3986:
3985:
3983:
3982:
3977:
3972:
3967:
3962:
3957:
3952:
3951:
3950:
3940:
3939:
3938:
3925:
3923:
3917:
3916:
3914:
3913:
3912:
3911:
3906:
3901:
3891:
3886:
3881:
3876:
3871:
3866:
3860:
3858:
3852:
3851:
3849:
3848:
3843:
3838:
3833:
3828:
3823:
3818:
3817:
3816:
3806:
3801:
3796:
3791:
3786:
3781:
3775:
3773:
3764:
3758:
3757:
3755:
3754:
3749:
3744:
3739:
3734:
3729:
3717:Cantor's
3715:
3710:
3705:
3695:
3693:
3680:
3679:
3677:
3676:
3671:
3666:
3661:
3656:
3651:
3646:
3641:
3636:
3631:
3626:
3621:
3616:
3615:
3614:
3603:
3601:
3597:
3596:
3589:
3588:
3581:
3574:
3566:
3557:
3556:
3554:
3553:
3548:
3543:
3538:
3532:
3530:
3526:
3525:
3523:
3522:
3521:
3520:
3510:
3509:
3508:
3498:
3497:
3496:
3485:
3483:
3477:
3476:
3474:
3473:
3468:
3463:
3457:
3455:
3449:
3448:
3446:
3445:
3440:
3434:
3432:
3426:
3425:
3423:
3422:
3416:
3414:
3412:Paraconsistent
3408:
3407:
3405:
3404:
3399:
3394:
3388:
3386:
3380:
3379:
3377:
3376:
3371:
3366:
3361:
3356:
3350:
3348:
3342:
3341:
3339:
3338:
3333:
3328:
3323:
3318:
3312:
3310:
3308:Intuitionistic
3304:
3303:
3296:
3295:
3288:
3281:
3273:
3267:
3266:
3252:
3242:
3241:, chapters 5-9
3228:
3211:
3208:
3206:
3205:
3188:
3181:
3163:
3148:
3131:
3120:
3107:
3100:
3080:
3023:
3005:
2986:
2955:
2940:(1): 181–190.
2924:
2897:
2858:
2819:
2787:
2770:Lane, Robert.
2762:
2735:
2705:
2682:
2658:
2617:(3): 163–185.
2597:
2574:
2572:
2569:
2568:
2567:
2562:
2552:
2542:
2537:
2531:
2526:
2521:
2516:
2511:
2506:
2501:
2495:
2494:
2478:
2475:
2459:Main article:
2456:
2453:
2451:
2448:
2447:
2446:
2443:
2440:
2428:
2425:
2424:
2423:
2420:
2415:
2412:
2409:
2408:
2389:
2387:
2370:Main article:
2367:
2364:
2300:
2299:
2296:
2295:
2292:
2289:
2286:
2282:
2281:
2278:
2275:
2272:
2268:
2267:
2264:
2261:
2258:
2255:
2251:
2250:
2247:
2244:
2240:
2239:
2236:
2227:
2217:
2216:
2213:
2212:
2209:
2205:
2204:
2201:
2197:
2196:
2193:
2189:
2188:
2185:
2176:
2170:
2169:
2123:
2120:
2106:
2105:
2085:
2084:
2061:
2060:
2057:
2056:
2053:
2050:
2047:
2043:
2042:
2039:
2036:
2033:
2029:
2028:
2025:
2022:
2019:
2016:
2012:
2011:
2008:
2005:
2001:
2000:
1997:
1988:
1976:
1973:
1972:
1971:
1946:
1846:
1845:
1842:
1841:
1838:
1834:
1833:
1830:
1826:
1825:
1822:
1818:
1817:
1809:
1800:
1797:
1796:
1793:
1789:
1788:
1785:
1781:
1780:
1777:
1773:
1772:
1764:
1755:
1752:
1751:
1748:
1744:
1743:
1740:
1736:
1735:
1732:
1728:
1727:
1719:
1704:
1703:
1679:
1661:
1639:
1638:
1610:
1590:
1561:
1560:
1557:
1556:
1553:
1550:
1547:
1543:
1542:
1539:
1536:
1533:
1529:
1528:
1525:
1522:
1519:
1516:
1512:
1511:
1508:
1505:
1501:
1500:
1497:
1488:
1483:
1480:
1479:
1476:
1473:
1470:
1466:
1465:
1462:
1459:
1456:
1452:
1451:
1448:
1445:
1442:
1439:
1435:
1434:
1431:
1428:
1424:
1423:
1420:
1411:
1382:
1379:
1356:
1355:
1352:
1351:
1348:
1345:
1342:
1338:
1337:
1334:
1331:
1328:
1324:
1323:
1320:
1317:
1314:
1311:
1307:
1306:
1303:
1300:
1296:
1295:
1292:
1283:
1278:
1275:
1274:
1271:
1268:
1265:
1261:
1260:
1257:
1254:
1251:
1247:
1246:
1243:
1240:
1237:
1234:
1230:
1229:
1226:
1223:
1219:
1218:
1215:
1206:
1186:
1183:
1177:
1174:
1171:
1168:
1155:
1138:
1135:
1132:
1126:
1118:
1115:
1112:
994:
993:
990:
989:
986:
983:
980:
976:
975:
972:
969:
966:
962:
961:
958:
955:
952:
949:
945:
944:
941:
938:
934:
933:
930:
920:
917:
916:
913:
910:
907:
903:
902:
899:
896:
893:
889:
888:
885:
882:
879:
876:
872:
871:
868:
865:
861:
860:
857:
847:
844:
843:
840:
837:
834:
830:
829:
826:
823:
820:
816:
815:
812:
809:
806:
803:
799:
798:
795:
792:
788:
787:
784:
774:
771:
770:
767:
763:
762:
759:
755:
754:
751:
747:
746:
743:
732:
731:
723:
722:
719:
718:
715:
712:
709:
705:
704:
701:
698:
695:
691:
690:
687:
684:
681:
678:
674:
673:
670:
667:
663:
662:
659:
649:
646:
645:
642:
639:
636:
632:
631:
628:
625:
622:
618:
617:
614:
611:
608:
605:
601:
600:
597:
594:
590:
589:
586:
576:
573:
572:
569:
566:
563:
559:
558:
555:
552:
549:
545:
544:
541:
538:
535:
532:
528:
527:
524:
521:
517:
516:
513:
503:
500:
499:
496:
492:
491:
488:
484:
483:
480:
476:
475:
472:
461:
460:
432:
429:
380:
377:
354:
353:
342:
309:
302:
285:
278:
262:
259:
224:
221:
208:
205:
185:Emil Leon Post
117:
116:
31:
29:
22:
15:
9:
6:
4:
3:
2:
5347:
5336:
5333:
5331:
5328:
5327:
5325:
5312:
5311:
5306:
5298:
5292:
5289:
5287:
5284:
5282:
5279:
5277:
5274:
5270:
5267:
5266:
5265:
5262:
5260:
5257:
5255:
5252:
5250:
5246:
5243:
5241:
5238:
5236:
5233:
5231:
5228:
5226:
5223:
5222:
5220:
5216:
5210:
5207:
5205:
5202:
5200:
5199:Recursive set
5197:
5195:
5192:
5190:
5187:
5185:
5182:
5180:
5177:
5173:
5170:
5168:
5165:
5163:
5160:
5158:
5155:
5153:
5150:
5149:
5148:
5145:
5143:
5140:
5138:
5135:
5133:
5130:
5128:
5125:
5123:
5120:
5119:
5117:
5115:
5111:
5105:
5102:
5100:
5097:
5095:
5092:
5090:
5087:
5085:
5082:
5080:
5077:
5075:
5072:
5068:
5065:
5063:
5060:
5058:
5055:
5054:
5053:
5050:
5048:
5045:
5043:
5040:
5038:
5035:
5033:
5030:
5028:
5025:
5021:
5018:
5017:
5016:
5013:
5009:
5008:of arithmetic
5006:
5005:
5004:
5001:
4997:
4994:
4992:
4989:
4987:
4984:
4982:
4979:
4977:
4974:
4973:
4972:
4969:
4965:
4962:
4960:
4957:
4956:
4955:
4952:
4951:
4949:
4947:
4943:
4937:
4934:
4932:
4929:
4927:
4924:
4922:
4919:
4916:
4915:from ZFC
4912:
4909:
4907:
4904:
4898:
4895:
4894:
4893:
4890:
4888:
4885:
4883:
4880:
4879:
4878:
4875:
4873:
4870:
4868:
4865:
4863:
4860:
4858:
4855:
4853:
4850:
4848:
4845:
4844:
4842:
4840:
4836:
4826:
4825:
4821:
4820:
4815:
4814:non-Euclidean
4812:
4808:
4805:
4803:
4800:
4798:
4797:
4793:
4792:
4790:
4787:
4786:
4784:
4780:
4776:
4773:
4771:
4768:
4767:
4766:
4762:
4758:
4755:
4754:
4753:
4749:
4745:
4742:
4740:
4737:
4735:
4732:
4730:
4727:
4725:
4722:
4720:
4717:
4716:
4714:
4710:
4709:
4707:
4702:
4696:
4691:Example
4688:
4680:
4675:
4674:
4673:
4670:
4668:
4665:
4661:
4658:
4656:
4653:
4651:
4648:
4646:
4643:
4642:
4641:
4638:
4636:
4633:
4631:
4628:
4626:
4623:
4619:
4616:
4614:
4611:
4610:
4609:
4606:
4602:
4599:
4597:
4594:
4592:
4589:
4587:
4584:
4583:
4582:
4579:
4577:
4574:
4570:
4567:
4565:
4562:
4560:
4557:
4556:
4555:
4552:
4548:
4545:
4543:
4540:
4538:
4535:
4533:
4530:
4528:
4525:
4523:
4520:
4519:
4518:
4515:
4513:
4510:
4508:
4505:
4503:
4500:
4496:
4493:
4491:
4488:
4486:
4483:
4481:
4478:
4477:
4476:
4473:
4471:
4468:
4466:
4463:
4461:
4458:
4454:
4451:
4449:
4448:by definition
4446:
4445:
4444:
4441:
4437:
4434:
4433:
4432:
4429:
4427:
4424:
4422:
4419:
4417:
4414:
4412:
4409:
4408:
4405:
4402:
4400:
4396:
4391:
4385:
4381:
4371:
4368:
4366:
4363:
4361:
4358:
4356:
4353:
4351:
4348:
4346:
4343:
4341:
4338:
4336:
4335:Kripke–Platek
4333:
4331:
4328:
4324:
4321:
4319:
4316:
4315:
4314:
4311:
4310:
4308:
4304:
4296:
4293:
4292:
4291:
4288:
4286:
4283:
4279:
4276:
4275:
4274:
4271:
4269:
4266:
4264:
4261:
4259:
4256:
4254:
4251:
4248:
4244:
4240:
4237:
4233:
4230:
4228:
4225:
4223:
4220:
4219:
4218:
4214:
4211:
4210:
4208:
4206:
4202:
4198:
4190:
4187:
4185:
4182:
4180:
4179:constructible
4177:
4176:
4175:
4172:
4170:
4167:
4165:
4162:
4160:
4157:
4155:
4152:
4150:
4147:
4145:
4142:
4140:
4137:
4135:
4132:
4130:
4127:
4125:
4122:
4120:
4117:
4115:
4112:
4111:
4109:
4107:
4102:
4094:
4091:
4089:
4086:
4084:
4081:
4079:
4076:
4074:
4071:
4069:
4066:
4065:
4063:
4059:
4056:
4054:
4051:
4050:
4049:
4046:
4044:
4041:
4039:
4036:
4034:
4031:
4029:
4025:
4021:
4019:
4016:
4012:
4009:
4008:
4007:
4004:
4003:
4000:
3997:
3995:
3991:
3981:
3978:
3976:
3973:
3971:
3968:
3966:
3963:
3961:
3958:
3956:
3953:
3949:
3946:
3945:
3944:
3941:
3937:
3932:
3931:
3930:
3927:
3926:
3924:
3922:
3918:
3910:
3907:
3905:
3902:
3900:
3897:
3896:
3895:
3892:
3890:
3887:
3885:
3882:
3880:
3877:
3875:
3872:
3870:
3867:
3865:
3862:
3861:
3859:
3857:
3856:Propositional
3853:
3847:
3844:
3842:
3839:
3837:
3834:
3832:
3829:
3827:
3824:
3822:
3819:
3815:
3812:
3811:
3810:
3807:
3805:
3802:
3800:
3797:
3795:
3792:
3790:
3787:
3785:
3784:Logical truth
3782:
3780:
3777:
3776:
3774:
3772:
3768:
3765:
3763:
3759:
3753:
3750:
3748:
3745:
3743:
3740:
3738:
3735:
3733:
3730:
3728:
3724:
3720:
3716:
3714:
3711:
3709:
3706:
3704:
3700:
3697:
3696:
3694:
3692:
3686:
3681:
3675:
3672:
3670:
3667:
3665:
3662:
3660:
3657:
3655:
3652:
3650:
3647:
3645:
3642:
3640:
3637:
3635:
3632:
3630:
3627:
3625:
3622:
3620:
3617:
3613:
3610:
3609:
3608:
3605:
3604:
3602:
3598:
3594:
3587:
3582:
3580:
3575:
3573:
3568:
3567:
3564:
3552:
3549:
3547:
3544:
3542:
3539:
3537:
3534:
3533:
3531:
3527:
3519:
3516:
3515:
3514:
3511:
3507:
3504:
3503:
3502:
3499:
3495:
3492:
3491:
3490:
3487:
3486:
3484:
3482:
3481:Digital logic
3478:
3472:
3469:
3467:
3464:
3462:
3459:
3458:
3456:
3454:
3450:
3444:
3441:
3439:
3436:
3435:
3433:
3431:
3427:
3421:
3418:
3417:
3415:
3413:
3409:
3403:
3400:
3398:
3395:
3393:
3390:
3389:
3387:
3385:
3384:Substructural
3381:
3375:
3372:
3370:
3367:
3365:
3362:
3360:
3357:
3355:
3352:
3351:
3349:
3347:
3343:
3337:
3334:
3332:
3329:
3327:
3324:
3322:
3319:
3317:
3314:
3313:
3311:
3309:
3305:
3301:
3294:
3289:
3287:
3282:
3280:
3275:
3274:
3271:
3265:
3264:0-486-40459-5
3261:
3257:
3253:
3251:
3247:
3243:
3231:
3225:
3221:
3220:
3214:
3213:
3202:
3198:
3192:
3184:
3178:
3174:
3167:
3159:
3152:
3145:
3141:
3135:
3129:
3124:
3117:
3111:
3103:
3101:9780141920870
3097:
3093:
3092:
3084:
3066:
3062:
3058:
3054:
3050:
3046:
3045:
3037:
3033:
3027:
3019:
3015:
3009:
3002:
2997:
2990:
2983:
2978:
2974:
2970:
2966:
2959:
2952:
2947:
2943:
2939:
2935:
2928:
2920:
2916:
2912:
2908:
2901:
2893:
2889:
2885:
2881:
2877:
2873:
2869:
2862:
2854:
2850:
2846:
2842:
2838:
2834:
2830:
2823:
2815:
2811:
2808:(1–2): 1–11.
2807:
2803:
2796:
2794:
2792:
2777:
2773:
2766:
2750:
2746:
2739:
2732:
2720:
2716:
2709:
2701:
2697:
2693:
2686:
2672:
2668:
2662:
2653:
2648:
2644:
2640:
2635:
2630:
2625:
2620:
2616:
2612:
2608:
2601:
2593:
2589:
2585:
2579:
2575:
2566:
2563:
2560:
2556:
2553:
2550:
2546:
2543:
2541:
2538:
2535:
2532:
2530:
2527:
2525:
2522:
2520:
2517:
2515:
2512:
2510:
2507:
2505:
2502:
2500:
2497:
2496:
2492:
2481:
2474:
2472:
2468:
2462:
2444:
2441:
2438:
2437:
2436:
2434:
2421:
2418:
2417:
2405:
2396:
2392:
2388:
2385:
2381:
2380:
2377:
2373:
2366:Bochvar logic
2363:
2361:
2356:
2354:
2348:
2344:
2340:
2336:
2329:
2325:
2321:
2317:
2313:
2309:
2293:
2290:
2287:
2283:
2279:
2276:
2273:
2269:
2265:
2262:
2259:
2252:
2241:
2233:
2223:
2210:
2206:
2202:
2198:
2194:
2190:
2182:
2172:
2166:
2163:
2161:
2157:
2153:
2149:
2145:
2141:
2135:
2129:
2119:
2117:
2112:
2103:
2099:
2095:
2090:
2089:
2088:
2081:
2077:
2073:
2068:
2067:
2066:
2054:
2051:
2048:
2044:
2040:
2037:
2034:
2030:
2026:
2023:
2020:
2013:
2002:
1994:
1984:
1980:
1969:
1963:
1959:
1956:
1952:
1947:
1944:
1939:
1935:
1932:
1928:
1924:
1923:
1922:
1919:
1912:
1908:
1902:
1897:
1893:
1888:
1883:
1879:
1873:
1869:
1864:
1860:
1855:
1853:
1839:
1835:
1831:
1827:
1823:
1819:
1815:
1804:
1801:
1794:
1790:
1786:
1782:
1778:
1774:
1770:
1759:
1756:
1749:
1745:
1741:
1737:
1733:
1729:
1725:
1714:
1711:
1707:
1701:
1698:
1694:
1691:
1687:
1684:
1680:
1677:
1673:
1669:
1666:
1662:
1659:
1655:
1651:
1648:
1644:
1643:
1642:
1635:
1631:
1627:
1623:
1619:
1615:
1611:
1607:
1603:
1599:
1595:
1591:
1588:
1584:
1580:
1576:
1572:
1568:
1567:
1566:
1554:
1551:
1548:
1544:
1540:
1537:
1534:
1530:
1526:
1523:
1520:
1513:
1502:
1494:
1484:
1477:
1474:
1471:
1467:
1463:
1460:
1457:
1453:
1449:
1446:
1443:
1436:
1425:
1417:
1407:
1403:
1400:
1398:
1394:
1388:
1378:
1376:
1372:
1367:
1361:
1349:
1346:
1343:
1339:
1335:
1332:
1329:
1325:
1321:
1318:
1315:
1308:
1297:
1289:
1279:
1272:
1269:
1266:
1262:
1258:
1255:
1252:
1248:
1244:
1241:
1238:
1231:
1220:
1212:
1202:
1198:
1181:
1175:
1169:
1124:
1116:
1110:
1102:
1099:
1097:
1094:is less than
1093:
1089:
1086:is less than
1085:
1081:
1077:
1073:
1069:
1064:
1062:
1058:
1054:
1050:
1046:
1042:
1038:
1034:
1030:
1026:
1022:
1018:
1014:
1010:
1006:
1001:
987:
984:
981:
977:
973:
970:
967:
963:
959:
956:
953:
946:
935:
927:
921:
914:
911:
908:
904:
900:
897:
894:
890:
886:
883:
880:
873:
862:
854:
848:
841:
838:
835:
831:
827:
824:
821:
817:
813:
810:
807:
800:
789:
781:
775:
768:
764:
760:
756:
752:
748:
740:
734:
728:
716:
713:
710:
706:
702:
699:
696:
692:
688:
685:
682:
675:
664:
656:
650:
643:
640:
637:
633:
629:
626:
623:
619:
615:
612:
609:
602:
591:
583:
577:
570:
567:
564:
560:
556:
553:
550:
546:
542:
539:
536:
529:
518:
510:
504:
497:
493:
489:
485:
481:
477:
469:
463:
457:
454:
452:
451:Graham Priest
448:
444:
438:
428:
426:
421:
417:
413:
409:
405:
401:
397:
393:
389:
386:allows 2 = 4
385:
384:Boolean logic
376:
374:
370:
365:
363:
359:
351:
347:
343:
340:
336:
332:
331:
326:
322:
318:
314:
310:
307:
303:
300:
299:
294:
290:
286:
283:
279:
276:
272:
271:
270:
268:
258:
256:
252:
246:
241:
239:
238:Hilary Putnam
235:
231:
220:
218:
214:
211:Around 1910,
207:Pre-discovery
204:
202:
198:
194:
190:
186:
182:
180:
176:
172:
171:Boolean logic
168:
164:
160:
156:
152:
148:
144:
140:
136:
132:
131:trinary logic
128:
124:
113:
110:
102:
91:
88:
84:
81:
77:
74:
70:
67:
63:
60: –
59:
55:
54:Find sources:
48:
44:
38:
37:
32:This article
30:
26:
21:
20:
5301:
5099:Ultraproduct
4946:Model theory
4911:Independence
4847:Formal proof
4839:Proof theory
4822:
4795:
4752:real numbers
4724:second-order
4635:Substitution
4512:Metalanguage
4453:conservative
4426:Axiom schema
4370:Constructive
4340:Morse–Kelley
4306:Set theories
4285:Aleph number
4278:inaccessible
4184:Grothendieck
4068:intersection
3955:Higher-order
3943:Second-order
3898:
3889:Truth tables
3846:Venn diagram
3629:Formal proof
3461:Three-valued
3460:
3402:Linear logic
3255:
3233:. Retrieved
3218:
3200:
3191:
3172:
3166:
3158:Sitz. Berlin
3157:
3151:
3143:
3134:
3123:
3110:
3090:
3083:
3072:. Retrieved
3048:
3042:
3036:"Third base"
3032:Hayes, Brian
3026:
3017:
3008:
2999:
2995:
2989:
2980:
2971:(5): 73–80.
2968:
2964:
2958:
2949:
2937:
2933:
2927:
2910:
2906:
2900:
2875:
2871:
2861:
2836:
2832:
2822:
2805:
2801:
2779:. Retrieved
2775:
2765:
2753:. Retrieved
2748:
2738:
2730:
2723:. Retrieved
2718:
2708:
2695:
2685:
2674:. Retrieved
2670:
2661:
2614:
2610:
2600:
2587:
2578:
2464:
2450:Applications
2439:Cohn algebra
2430:
2399:
2395:adding to it
2390:
2375:
2357:
2346:
2342:
2338:
2334:
2327:
2323:
2319:
2315:
2311:
2307:
2303:
2143:
2139:
2137:
2110:
2107:
2101:
2097:
2093:
2086:
2079:
2075:
2071:
2064:
1978:
1961:
1957:
1954:
1950:
1937:
1933:
1930:
1926:
1917:
1910:
1906:
1895:
1891:
1881:
1877:
1871:
1867:
1856:
1849:
1813:
1768:
1723:
1705:
1699:
1696:
1692:
1689:
1685:
1682:
1675:
1671:
1667:
1664:
1657:
1653:
1649:
1646:
1640:
1633:
1629:
1625:
1621:
1617:
1613:
1605:
1601:
1597:
1593:
1586:
1582:
1578:
1574:
1570:
1564:
1401:
1396:
1392:
1390:
1374:
1365:
1362:
1359:
1103:
1100:
1095:
1091:
1087:
1083:
1079:
1075:
1071:
1065:
1060:
1056:
1052:
1048:
1044:
1040:
1039:also equals
1036:
1032:
1028:
1024:
1020:
1016:
1012:
1008:
1004:
999:
997:
443:truth tables
440:
382:
366:
355:
349:
345:
338:
334:
328:
324:
320:
319:, and 0 for
316:
312:
296:
264:
248:
243:
226:
210:
200:
183:
178:
174:
162:
158:
155:truth values
146:
142:
138:
134:
130:
126:
120:
105:
99:January 2011
96:
86:
79:
72:
65:
53:
41:Please help
36:verification
33:
5209:Type theory
5157:undecidable
5089:Truth value
4976:equivalence
4655:non-logical
4268:Enumeration
4258:Isomorphism
4205:cardinality
4189:Von Neumann
4154:Ultrafilter
4119:Uncountable
4053:equivalence
3970:Quantifiers
3960:Fixed-point
3929:First-order
3809:Consistency
3794:Proposition
3771:Traditional
3742:Lindström's
3732:Compactness
3674:Type theory
3619:Cardinality
3501:Four-valued
3471:Łukasiewicz
3466:Four-valued
3453:Many-valued
3430:Description
3420:Dialetheism
2878:(1): 1–28.
2402:August 2014
1852:modal logic
420:implication
416:equivalence
373:connectives
330:undecidable
249:Similarly,
157:indicating
5324:Categories
5020:elementary
4713:arithmetic
4581:Quantifier
4559:functional
4431:Expression
4149:Transitive
4093:identities
4078:complement
4011:hereditary
3994:Set theory
3359:Fuzzy rule
3074:2020-04-12
2781:2020-07-30
2676:2024-05-15
2571:References
2461:Null (SQL)
1899:, and the
1375:designated
1366:designated
1082:such that
851:MAX(A, B)
778:MIN(A, B)
653:XOR(A, B)
507:AND(A, B)
435:See also:
335:irrelevant
325:unknowable
255:predicates
223:Motivation
215:defined a
69:newspapers
5291:Supertask
5194:Recursion
5152:decidable
4986:saturated
4964:of models
4887:deductive
4882:axiomatic
4802:Hilbert's
4789:Euclidean
4770:canonical
4693:axiomatic
4625:Signature
4554:Predicate
4443:Extension
4365:Ackermann
4290:Operation
4169:Universal
4159:Recursive
4134:Singleton
4129:Inhabited
4114:Countable
4104:Types of
4088:power set
4058:partition
3975:Predicate
3921:Predicate
3836:Syllogism
3826:Soundness
3799:Inference
3789:Tautology
3691:paradoxes
3513:IEEE 1164
3364:Fuzzy set
3235:24 August
2907:Semiotica
2892:0031-8108
2853:0031-8094
2643:0002-9327
2431:Some 3VL
1975:RM3 logic
1863:tautology
1399:, vol 8.
1114:→
580:OR(A, B)
356:Inside a
135:trivalent
5276:Logicism
5269:timeline
5245:Concrete
5104:Validity
5074:T-schema
5067:Kripke's
5062:Tarski's
5057:semantic
5047:Strength
4996:submodel
4991:spectrum
4959:function
4807:Tarski's
4796:Elements
4783:geometry
4739:Robinson
4660:variable
4645:function
4618:spectrum
4608:Sentence
4564:variable
4507:Language
4460:Relation
4421:Automata
4411:Alphabet
4395:language
4249:-jection
4227:codomain
4213:Function
4174:Universe
4144:Infinite
4048:Relation
3831:Validity
3821:Argument
3719:theorem,
3160:. 42–56.
3065:Archived
3053:Sigma Xi
3016:(1981).
2700:Archived
2592:Archived
2477:See also
2122:HT logic
348:, 1 for
315:, 2 for
167:bivalent
5218:Related
5015:Diagram
4913: (
4892:Hilbert
4877:Systems
4872:Theorem
4750:of the
4695:systems
4475:Formula
4470:Grammar
4386: (
4330:General
4043:Forcing
4028:Element
3948:Monadic
3723:paradox
3664:Theorem
3600:General
3506:Verilog
2755:May 15,
2725:May 15,
2696:Commens
2652:2370324
2314:) → (((
2230:(A, B)
2152:Heyting
1991:(A, B)
1414:(A, B)
1092:unknown
1088:unknown
1076:unknown
1057:unknown
1051:equals
1049:unknown
1043:, then
1027:equals
1017:unknown
1005:unknown
1000:unknown
737:NEG(A)
466:NOT(A)
321:unknown
304:in the
291:, each
287:in the
280:in the
143:trilean
139:ternary
83:scholar
4981:finite
4744:Skolem
4697:
4672:Theory
4640:Symbol
4630:String
4613:atomic
4490:ground
4485:closed
4480:atomic
4436:ground
4399:syntax
4295:binary
4222:domain
4139:Finite
3904:finite
3762:Logics
3721:
3669:Theory
3529:Others
3262:
3226:
3179:
3098:
2890:
2851:
2649:
2641:
2360:coatom
2235:A → B
2146:or as
1996:A → B
1496:A → B
1419:A → B
1291:A → B
1214:A → B
1179:
1157:
1144:
1120:
1031:, and
929:A ⊕ B
856:A ∨ B
783:A ∧ B
658:A ⊕ B
585:A ∨ B
512:A ∧ B
379:Logics
344:0 for
311:1 for
129:(also
85:
78:
71:
64:
56:
4971:Model
4719:Peano
4576:Proof
4416:Arity
4345:Naive
4232:image
4164:Fuzzy
4124:Empty
4073:union
4018:Class
3659:Model
3649:Lemma
3607:Axiom
3346:Fuzzy
3068:(PDF)
3051:(6).
3039:(PDF)
2647:JSTOR
2547:(and
2534:Setun
2148:Gödel
1628:) ∧ (
1600:= ¬(¬
1369:(The
1084:false
1072:false
1037:false
1013:false
346:false
337:, or
317:false
295:is a
293:digit
179:false
163:false
141:, or
123:logic
90:JSTOR
76:books
5094:Type
4897:list
4701:list
4678:list
4667:Term
4601:rank
4495:open
4389:list
4201:Maps
4106:sets
3965:Free
3935:list
3685:list
3612:list
3518:VHDL
3260:ISBN
3237:2013
3224:ISBN
3177:ISBN
3096:ISBN
2911:2021
2888:ISSN
2849:ISSN
2757:2023
2727:2023
2639:ISSN
2471:NULL
2326:) →
2322:) →
2179:(A)
2100:) →
1875:and
1604:∨ ¬
1585:) →
1393:true
1096:true
1090:and
1080:true
1078:and
1061:true
1053:true
1045:true
1041:true
1033:true
1029:true
1025:true
1021:true
1009:true
412:XNOR
396:NAND
350:true
339:both
313:true
298:trit
191:and
177:and
175:true
159:true
125:, a
62:news
4781:of
4763:of
4711:of
4243:Sur
4217:Map
4024:Ur-
4006:Set
3246:doi
3057:doi
2973:doi
2942:doi
2915:doi
2880:doi
2841:doi
2810:doi
2629:hdl
2619:doi
2467:SQL
2455:SQL
2397:.
2341:)∨(
2337:∨(¬
2291:NF
2271:NF
2246:NF
2226:IMP
2200:NF
2187:¬A
2175:NOT
2144:SmT
2074:→ (
1989:RM3
1987:IMP
1960:∧ ¬
1953:∧ ¬
1936:∨ ¬
1909:∧ ¬
1894:∨ ¬
1695:∧ ¬
1670:= ¬
1652:= ¬
1620:= (
1577:= (
1555:+1
1549:−1
1546:+1
1541:+1
1538:+1
1527:+1
1524:+1
1521:+1
1518:−1
1510:+1
1504:−1
1487:IMP
1410:IMP
1350:+1
1344:−1
1341:+1
1336:+1
1322:+1
1319:+1
1316:+1
1313:−1
1305:+1
1299:−1
1282:IMP
1205:IMP
1162:NOT
1047:OR
1035:OR
1023:OR
1011:or
988:−1
982:+1
979:+1
960:+1
954:−1
951:−1
943:+1
937:−1
915:+1
912:+1
909:+1
906:+1
901:+1
887:+1
881:−1
878:−1
870:+1
864:−1
842:+1
836:−1
833:+1
822:−1
814:−1
811:−1
808:−1
805:−1
797:+1
791:−1
769:−1
766:+1
753:+1
750:−1
745:¬A
474:¬A
408:XOR
404:NOR
394:,
392:AND
273:in
147:3VL
121:In
45:by
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2257:F
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2249:T
2243:F
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2208:T
2203:F
2195:T
2192:F
2184:A
2177:HT
2140:HT
2096:⊗
2082:))
2078:→
2055:T
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2038:U
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2032:U
2027:T
2024:T
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2018:F
2015:A
2010:T
2007:U
2004:F
1999:B
1970:).
1949:¬(
1929:∨
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515:B
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400:OR
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