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 (
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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 Only in 1966, when Max Fisch and Atwell Turquette began publishing what they rediscovered in his unpublished manuscripts, did Peirce's triadic ideas become widely known.
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.
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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.
2995:
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),
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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
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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
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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
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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..."
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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.
2795:
Cobreros, Pablo; Égré, Paul; Ripley, David; Rooij, Robert van (2 January 2014). "Foreword: Three-valued logics and their applications".
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Mundici, D. The C*-Algebras of Three-Valued Logic. Logic
Colloquium ’88, Proceedings of the Colloquium held in Padova 61–77 (1989).
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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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are tautologies in Ł3 and also in classical logic. Not all tautologies of classical logic lift to Ł3 "as is". For example, the
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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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3170:. Synthesis lectures on digital circuits and systems. Vol. 12. Morgan & Claypool Publishers. pp. 41–42.
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1967:
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2662:"Peirce's Deductive Logic > Peirce's Three-Valued Logic (Stanford Encyclopedia of Philosophy/Summer 2020 Edition)"
75:
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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;
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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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1190:{\displaystyle A\rightarrow B\ {\overset {\underset {\mathrm {def} }{}}{=}}\ {\mbox{OR}}(\ {\mbox{NOT}}(A),\ B)}
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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;
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field content. SQL uses a common fragment of the Kleene K3 logic, restricted to AND, OR, and NOT tables.
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M is read as "it is not false that..." or in the (unsuccessful) Tarski–Łukasiewicz attempt to axiomatize
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In fact, using Łukasiewicz's implication and negation, the other usual connectives may be derived as:
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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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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
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Handbook of the
History of Logic Volume 8. The Many Valued and Nonmonotonic Turn in Logic
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operators are used, there may be functionally complete sets of ternary-valued operators.
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is True, meaning that only a proposition having this value everywhere is considered a
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2993:. 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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2740:"Logic : autograph manuscript notebook, November 12, 1865-November 1, 1909"
2710:"Logic : autograph manuscript notebook, November 12, 1865-November 1, 1909"
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as well. In this example, because either bivalent state could be underlying the
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372:
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using the truth values {false, unknown, true}, and extends conventional
Boolean
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399:
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de
Finetti, Bruno (1 January 1995). "The logic of probability (translated)".
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defined above, it is possible to state tautologies that are their analogues:
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Triadic Logic is universally true. But Dyadic Logic is not aboslutely false
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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:
1395:. This section follows the presentation from Malinowski's chapter of the
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
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It may be defined either by appending one of the two equivalent axioms
2151:
3263:
2529:
Paraconsistent logic § An ideal three-valued paraconsistent logic
3556:
352:, and a third non-integer "maybe" symbol such as ?, #, ½, or xy.
3151:
Heyting (1930). "Die formalen Regeln der intuitionistischen Logik".
2878:
2839:
2744:
hollisarchives.lib.harvard.edu/repositories/24/digital_objects/63983
2714:
hollisarchives.lib.harvard.edu/repositories/24/digital_objects/63983
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2601:
1979:
The truth table for the material implication of R-mingle 3 (RM3) is
2536:– an experimental Russian computer which was based on ternary logic
4410:
3756:
3601:
3089:. London, England: Penguin Books. Entry for 'three-valued logic'.
2533:
122:
3015:. Reading, Mass.: Addison-Wesley Publishing Company. p. 190.
2469:
implements ternary logic as a means of handling comparisons with
1360:
which differs from that for Łukasiewicz logic (described below).
3512:
411:
3560:
3267:
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:
2466:
2378:
2372:
Many-valued logic § Bochvar's internal three-valued logic
407:
18:
2602:"Introduction to a General Theory of Elementary Propositions"
2284:
2045:
1468:
1263:
257:
that are "undecidable by algorithms whether true or false"
2422:
not(a) = (a + 1) mod (n), where (n) is the value of a logic
679:
606:
533:
1402:
Material implication for Łukasiewicz logic truth table is
1101:
Material implication for Kleene logic can be defined as:
2958:
Putnam, Hilary (1 October 1957). "Three-valued logic".
2394:
2114:
as the monoid identity. This logic is equivalent to an
1915:
are not tautologies in Ł3. However, using the operator
1160:
1147:
3086:
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
367:
This article mainly illustrates a system of ternary
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Multiple valued logic: concepts and representations
49:. Unsourced material may be challenged and removed.
3166:Miller, D. Michael; Thornton, Mitchell A. (2008).
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3192:Multiple-Valued Logic Synthesis and Optimization
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2134:Many-valued logic § Gödel logics Gk and G∞
2003:
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1003:logic, the knowledge of whether any particular
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3253:. University of California Press. Dover 1998:
301:(trinary digit) having a value of: 0, 1, or 2;
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8:
3251:Philosophic Foundations of Quantum Mechanics
3050:, the Scientific Research Society: 490–494.
2824:"Three-Valued Logic and Future Contingents"
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269:. A few of the more common examples are:
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109:Learn how and when to remove this message
3198:, Kluwer Academic Publishers, pp. 89-114
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169:logics (such as classical sentential or
3194:, in Hassoun S. and Sasao T., editors,
2797:Journal of Applied Non-Classical Logics
2570:
449:'s "strong logic of indeterminacy" and
16:System including an indeterminate value
3137:" in Dov M. Gabbay, John Woods (eds.)
3013:The Art of Computer Programming Vol. 2
2746:. Houghton Library, Harvard University
2716:. Houghton Library, Harvard University
1706:They have the following truth tables:
2863:"The Problem of Future Contingencies"
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2118:which also obeys the contrapositive.
1995:
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3135:Many-valued Logic and its Philosophy
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360:, ternary values are represented by
47:adding citations to reliable sources
2828:The Philosophical Quarterly (1950-)
2168:(F, false; NF, not false; T, true)
195:. These were then re-formulated by
2142:, also referred as Smetanov logic
1136:
1133:
1130:
730:(−1, false; 0, unknown; +1, true)
14:
2697:from the original on Dec 6, 2023.
2589:from the original on May 3, 2023.
445:showing the logic operations for
153:systems in which there are three
5298:
3196:Logic Synthesis and Verification
2771:www.digitalpeirce.fee.unicamp.br
2738:Peirce, Charles S. (1839–1914).
2708:Peirce, Charles S. (1839–1914).
2579:"Trilean (Stanford JavaNLP API)"
2483:
2382:
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
23:
3065:from the original on 2019-10-30
2991:Introduction to metamathematics
2606:American Journal of Mathematics
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
3217:. Cambridge University Press.
2629:2027/uiuo.ark:/13960/t9j450f7q
1184:
1172:
1166:
1153:
1113:
1019:operand. For example, because
1:
5259:History of mathematical logic
3245:10.1016/s0049-237x(08)70262-3
2989:Kleene, Stephen Cole (1952).
2499:Binary logic (disambiguation)
2222:
2171:
2138:The logic of here and there (
1983:
1710:
1406:
1201:
733:
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230:problem of future contingents
5184:Primitive recursive function
3123:"Beyond Propositional Logic"
2809:10.1080/11663081.2014.909631
2465:The database query language
2116:"ideal" paraconsistent logic
1063:results in all three cases.
3433:Ontology (computer science)
2504:Boolean algebra (structure)
998:In these truth tables, the
5346:
4248:Schröder–Bernstein theorem
3975:Monadic predicate calculus
3634:Foundations of mathematics
3326:Intuitionistic type theory
3249:Reichenbach, Hans (1944).
3029:(November–December 2001).
2458:
2419:not(a) = (a + 1) mod 3, or
2369:
2131:
2125:
1384:
1007:state secretly represents
434:
5294:
5281:Philosophy of mathematics
5230:Automated theorem proving
4401:
4355:Von Neumann–Bernays–Gödel
3996:
3211:Bergmann, Merrie (2008).
2524:Homogeneity (linguistics)
2445:Dubrova and Muzio algebra
2270:
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2150:G3 logic), introduced by
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1197:, and its truth table is
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693:
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306:skew binary number system
197:Grigore Constantin Moisil
173:) which provide only for
2867:The Philosophical Review
2861:Taylor, Richard (1957).
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2186:
2183:
2006:
1901:law of non-contradiction
1857:In Łukasiewicz's Ł3 the
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1066:If numeric values, e.g.
942:
939:
936:
869:
866:
863:
796:
793:
790:
744:
741:
668:
595:
522:
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
4931:Self-verifying theories
4752:Tarski's axiomatization
3703:Tarski's undefinability
3698:incompleteness theorems
3331:Constructive set theory
3190:Dubrova, Elena (2002).
2358:HT logic is the unique
453:'s "logic of paradox".
5305:Mathematics portal
4916:Proof of impossibility
4564:propositional variable
3874:Propositional calculus
3133:Grzegorz Malinowski, "
3111:Standard Ternary Logic
3083:Nelson, David (2008).
2600:Post, Emil L. (1921).
2585:. Stanford NLP Group.
2540:Ternary numeral system
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
5174:Kolmogorov complexity
5127:Computably enumerable
5027:Model complete theory
4819:Principia Mathematica
3879:Propositional formula
3708:Banach–Tarski paradox
3316:Constructive analysis
2960:Philosophical Studies
2929:Philosophical Studies
2914:10.1515/sem-2019-0115
2822:Prior, A. N. (1953).
2132:Further information:
2126:Further information:
1385:Further information:
1192:
425:functionally complete
418:), and 4 variants of
193:Clarence Irving Lewis
5122:Church–Turing thesis
5109:Computability theory
4318:continuum hypothesis
3836:Square of opposition
3694:Gödel's completeness
3369:Fuzzy set operations
3364:Fuzzy finite element
3311:Intuitionistic logic
3056:10.1511/2001.40.3268
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
5276:Mathematical object
5167:P versus NP problem
5132:Computable function
4926:Reverse mathematics
4852:Logical consequence
4729:primitive recursive
4724:elementary function
4497:Free/bound variable
4350:Tarski–Grothendieck
3869:Logical connectives
3799:Logical equivalence
3649:Logical consequence
3546:Non-monotonic logic
3295:Non-classical logic
2583:Stanford University
2560:The World of Null-A
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
5074:Transfer principle
5037:Semantics of logic
5022:Categorical theory
4998:Non-standard model
4512:Logical connective
3639:Information theory
3588:Mathematical logic
3541:Intermediate logic
3321:Heyting arithmetic
3109:Douglas W. Jones,
3039:American Scientist
2972:10.1007/BF02304905
2941:10.1007/BF00996317
2666: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
5330:Ternary computers
5325:Many-valued logic
5312:
5311:
5244:Abstract category
5047:Theories of truth
4857:Rule of inference
4847:Natural deduction
4828:
4827:
4373:
4372:
4078:Cartesian product
3983:
3982:
3889:Many-valued logic
3864:Boolean functions
3747:Russell's paradox
3722:diagonal argument
3619:First-order logic
3554:
3553:
3536:Inquisitive logic
3531:Dynamic semantics
3484:Three-state logic
3438:Ontology language
3224:978-0-521-88128-9
3177:978-1-59829-190-2
2685:Lane, R. (2001).
2550: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:
5337:
5303:
5302:
5254:History of logic
5249:Category of sets
5142:Decision problem
4921:Ordinal analysis
4862:Sequent calculus
4760:Boolean algebras
4700:
4699:
4674:
4645:logical/constant
4399:
4385:
4308:Zermelo–Fraenkel
4059:Set operations:
3994:
3931:
3762:
3742:Löwenheim–Skolem
3629:Formal semantics
3581:
3574:
3567:
3558:
3489:Tri-state buffer
3288:
3281:
3274:
3265:
3235:
3233:
3231:
3199:
3188:
3182:
3181:
3163:
3157:
3156:
3148:
3142:
3141:, Elsevier, 2009
3131:
3125:
3120:
3114:
3113:, Feb. 11, 2013.
3107:
3101:
3100:
3080:
3074:
3073:
3071:
3070:
3064:
3035:
3023:
3017:
3016:
3009:Knuth, Donald E.
3005:
2999:
2998:
2986:
2980:
2979:
2977:to the evidence.
2955:
2949:
2948:
2924:
2918:
2917:
2908:(240): 129–143.
2897:
2891:
2890:
2858:
2852:
2851:
2819:
2813:
2812:
2792:
2781:
2780:
2778:
2777:
2762:
2756:
2755:
2753:
2751:
2735:
2729:
2728:
2723:
2721:
2705:
2699:
2698:
2682:
2676:
2675:
2673:
2672:
2658:
2652:
2651:
2649:
2631:
2621:
2597:
2591:
2590:
2575:
2554:tri-state buffer
2544: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:
2221:
2181:
2165:
2113:
2104:
2083:
1993:
1982:
1965:
1940:
1920:
1914:
1898:
1884:
1874:
1859:designated value
1816:
1808:
1803:
1771:
1763:
1758:
1726:
1718:
1713:
1709:
1702:
1678:
1660:
1637:
1609:
1589:
1493:
1416:
1405:
1371:Logic of Paradox
1288:
1211:
1200:
1196:
1194:
1193:
1188:
1178:
1165:
1161:
1156:
1152:
1148:
1143:
1142:
1139:
1123:
1119:
1068:balanced ternary
926:
853:
780:
739:
727:
655:
582:
509:
468:
456:
358:ternary computer
275:balanced ternary
234:Bruno de Finetti
114:
107:
103:
100:
94:
92:
51:
27:
19:
5345:
5344:
5340:
5339:
5338:
5336:
5335:
5334:
5315:
5314:
5313:
5308:
5297:
5290:
5235:Category theory
5225:Algebraic logic
5208:
5179:Lambda calculus
5117:Church encoding
5103:
5079:Truth predicate
4935:
4901:Complete theory
4824:
4693:
4689:
4685:
4680:
4672:
4392: and
4388:
4383:
4369:
4345:New Foundations
4313:axiom of choice
4296:
4258:Gödel numbering
4198: and
4190:
4094:
3979:
3929:
3910:
3859:Boolean algebra
3845:
3809:Equiconsistency
3774:Classical logic
3751:
3732:Halting problem
3720: and
3696: and
3684: and
3683:
3678:Theorems (
3673:
3590:
3585:
3555:
3550:
3519:
3470:
3442:
3419:
3401:
3392:Relevance logic
3387:Structural rule
3373:
3349:Degree of truth
3335:
3297:
3292:
3229:
3227:
3225:
3210:
3207:
3205:Further reading
3202:
3189:
3185:
3178:
3165:
3164:
3160:
3150:
3149:
3145:
3132:
3128:
3121:
3117:
3108:
3104:
3097:
3082:
3081:
3077:
3068:
3066:
3062:
3033:
3025:
3024:
3020:
3007:
3006:
3002:
2988:
2987:
2983:
2957:
2956:
2952:
2926:
2925:
2921:
2899:
2898:
2894:
2879:10.2307/2182851
2860:
2859:
2855:
2840:10.2307/2217099
2834:(13): 317–326.
2821:
2820:
2816:
2794:
2793:
2784:
2775:
2773:
2767:"Triadic Logic"
2764:
2763:
2759:
2749:
2747:
2737:
2736:
2732:
2719:
2717:
2707:
2706:
2702:
2687:"Triadic Logic"
2684:
2683:
2679:
2670:
2668:
2660:
2659:
2655:
2619:10.2307/2370324
2599:
2598:
2594:
2577:
2576:
2572:
2568:
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:
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:
5343:
5341:
5333:
5332:
5327:
5317:
5316:
5310:
5309:
5295:
5292:
5291:
5289:
5288:
5283:
5278:
5273:
5268:
5267:
5266:
5256:
5251:
5246:
5237:
5232:
5227:
5222:
5220:Abstract logic
5216:
5214:
5210:
5209:
5207:
5206:
5201:
5199:Turing machine
5196:
5191:
5186:
5181:
5176:
5171:
5170:
5169:
5164:
5159:
5154:
5149:
5139:
5137:Computable set
5134:
5129:
5124:
5119:
5113:
5111:
5105:
5104:
5102:
5101:
5096:
5091:
5086:
5081:
5076:
5071:
5066:
5065:
5064:
5059:
5054:
5044:
5039:
5034:
5032:Satisfiability
5029:
5024:
5019:
5018:
5017:
5007:
5006:
5005:
4995:
4994:
4993:
4988:
4983:
4978:
4973:
4963:
4962:
4961:
4956:
4949:Interpretation
4945:
4943:
4937:
4936:
4934:
4933:
4928:
4923:
4918:
4913:
4903:
4898:
4897:
4896:
4895:
4894:
4884:
4879:
4869:
4864:
4859:
4854:
4849:
4844:
4838:
4836:
4830:
4829:
4826:
4825:
4823:
4822:
4814:
4813:
4812:
4811:
4806:
4805:
4804:
4799:
4794:
4774:
4773:
4772:
4770:minimal axioms
4767:
4756:
4755:
4754:
4743:
4742:
4741:
4736:
4731:
4726:
4721:
4716:
4703:
4701:
4682:
4681:
4679:
4678:
4677:
4676:
4664:
4659:
4658:
4657:
4652:
4647:
4642:
4632:
4627:
4622:
4617:
4616:
4615:
4610:
4600:
4599:
4598:
4593:
4588:
4583:
4573:
4568:
4567:
4566:
4561:
4556:
4546:
4545:
4544:
4539:
4534:
4529:
4524:
4519:
4509:
4504:
4499:
4494:
4493:
4492:
4487:
4482:
4477:
4467:
4462:
4460:Formation rule
4457:
4452:
4451:
4450:
4445:
4435:
4434:
4433:
4423:
4418:
4413:
4408:
4402:
4396:
4379:Formal systems
4375:
4374:
4371:
4370:
4368:
4367:
4362:
4357:
4352:
4347:
4342:
4337:
4332:
4327:
4322:
4321:
4320:
4315:
4304:
4302:
4298:
4297:
4295:
4294:
4293:
4292:
4282:
4277:
4276:
4275:
4268:Large cardinal
4265:
4260:
4255:
4250:
4245:
4231:
4230:
4229:
4224:
4219:
4204:
4202:
4192:
4191:
4189:
4188:
4187:
4186:
4181:
4176:
4166:
4161:
4156:
4151:
4146:
4141:
4136:
4131:
4126:
4121:
4116:
4111:
4105:
4103:
4096:
4095:
4093:
4092:
4091:
4090:
4085:
4080:
4075:
4070:
4065:
4057:
4056:
4055:
4050:
4040:
4035:
4033:Extensionality
4030:
4028:Ordinal number
4025:
4015:
4010:
4009:
4008:
3997:
3991:
3985:
3984:
3981:
3980:
3978:
3977:
3972:
3967:
3962:
3957:
3952:
3947:
3946:
3945:
3935:
3934:
3933:
3920:
3918:
3912:
3911:
3909:
3908:
3907:
3906:
3901:
3896:
3886:
3881:
3876:
3871:
3866:
3861:
3855:
3853:
3847:
3846:
3844:
3843:
3838:
3833:
3828:
3823:
3818:
3813:
3812:
3811:
3801:
3796:
3791:
3786:
3781:
3776:
3770:
3768:
3759:
3753:
3752:
3750:
3749:
3744:
3739:
3734:
3729:
3724:
3712:Cantor's
3710:
3705:
3700:
3690:
3688:
3675:
3674:
3672:
3671:
3666:
3661:
3656:
3651:
3646:
3641:
3636:
3631:
3626:
3621:
3616:
3611:
3610:
3609:
3598:
3596:
3592:
3591:
3586:
3584:
3583:
3576:
3569:
3561:
3552:
3551:
3549:
3548:
3543:
3538:
3533:
3527:
3525:
3521:
3520:
3518:
3517:
3516:
3515:
3505:
3504:
3503:
3493:
3492:
3491:
3480:
3478:
3472:
3471:
3469:
3468:
3463:
3458:
3452:
3450:
3444:
3443:
3441:
3440:
3435:
3429:
3427:
3421:
3420:
3418:
3417:
3411:
3409:
3407:Paraconsistent
3403:
3402:
3400:
3399:
3394:
3389:
3383:
3381:
3375:
3374:
3372:
3371:
3366:
3361:
3356:
3351:
3345:
3343:
3337:
3336:
3334:
3333:
3328:
3323:
3318:
3313:
3307:
3305:
3303:Intuitionistic
3299:
3298:
3293:
3291:
3290:
3283:
3276:
3268:
3262:
3261:
3247:
3237:
3236:, chapters 5-9
3223:
3206:
3203:
3201:
3200:
3183:
3176:
3158:
3143:
3126:
3115:
3102:
3095:
3075:
3018:
3000:
2981:
2950:
2935:(1): 181–190.
2919:
2892:
2853:
2814:
2782:
2765:Lane, Robert.
2757:
2730:
2700:
2677:
2653:
2612:(3): 163–185.
2592:
2569:
2567:
2564:
2563:
2562:
2557:
2547:
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:
13:
10:
9:
6:
4:
3:
2:
5342:
5331:
5328:
5326:
5323:
5322:
5320:
5307:
5306:
5301:
5293:
5287:
5284:
5282:
5279:
5277:
5274:
5272:
5269:
5265:
5262:
5261:
5260:
5257:
5255:
5252:
5250:
5247:
5245:
5241:
5238:
5236:
5233:
5231:
5228:
5226:
5223:
5221:
5218:
5217:
5215:
5211:
5205:
5202:
5200:
5197:
5195:
5194:Recursive set
5192:
5190:
5187:
5185:
5182:
5180:
5177:
5175:
5172:
5168:
5165:
5163:
5160:
5158:
5155:
5153:
5150:
5148:
5145:
5144:
5143:
5140:
5138:
5135:
5133:
5130:
5128:
5125:
5123:
5120:
5118:
5115:
5114:
5112:
5110:
5106:
5100:
5097:
5095:
5092:
5090:
5087:
5085:
5082:
5080:
5077:
5075:
5072:
5070:
5067:
5063:
5060:
5058:
5055:
5053:
5050:
5049:
5048:
5045:
5043:
5040:
5038:
5035:
5033:
5030:
5028:
5025:
5023:
5020:
5016:
5013:
5012:
5011:
5008:
5004:
5003:of arithmetic
5001:
5000:
4999:
4996:
4992:
4989:
4987:
4984:
4982:
4979:
4977:
4974:
4972:
4969:
4968:
4967:
4964:
4960:
4957:
4955:
4952:
4951:
4950:
4947:
4946:
4944:
4942:
4938:
4932:
4929:
4927:
4924:
4922:
4919:
4917:
4914:
4911:
4910:from ZFC
4907:
4904:
4902:
4899:
4893:
4890:
4889:
4888:
4885:
4883:
4880:
4878:
4875:
4874:
4873:
4870:
4868:
4865:
4863:
4860:
4858:
4855:
4853:
4850:
4848:
4845:
4843:
4840:
4839:
4837:
4835:
4831:
4821:
4820:
4816:
4815:
4810:
4809:non-Euclidean
4807:
4803:
4800:
4798:
4795:
4793:
4792:
4788:
4787:
4785:
4782:
4781:
4779:
4775:
4771:
4768:
4766:
4763:
4762:
4761:
4757:
4753:
4750:
4749:
4748:
4744:
4740:
4737:
4735:
4732:
4730:
4727:
4725:
4722:
4720:
4717:
4715:
4712:
4711:
4709:
4705:
4704:
4702:
4697:
4691:
4686:Example
4683:
4675:
4670:
4669:
4668:
4665:
4663:
4660:
4656:
4653:
4651:
4648:
4646:
4643:
4641:
4638:
4637:
4636:
4633:
4631:
4628:
4626:
4623:
4621:
4618:
4614:
4611:
4609:
4606:
4605:
4604:
4601:
4597:
4594:
4592:
4589:
4587:
4584:
4582:
4579:
4578:
4577:
4574:
4572:
4569:
4565:
4562:
4560:
4557:
4555:
4552:
4551:
4550:
4547:
4543:
4540:
4538:
4535:
4533:
4530:
4528:
4525:
4523:
4520:
4518:
4515:
4514:
4513:
4510:
4508:
4505:
4503:
4500:
4498:
4495:
4491:
4488:
4486:
4483:
4481:
4478:
4476:
4473:
4472:
4471:
4468:
4466:
4463:
4461:
4458:
4456:
4453:
4449:
4446:
4444:
4443:by definition
4441:
4440:
4439:
4436:
4432:
4429:
4428:
4427:
4424:
4422:
4419:
4417:
4414:
4412:
4409:
4407:
4404:
4403:
4400:
4397:
4395:
4391:
4386:
4380:
4376:
4366:
4363:
4361:
4358:
4356:
4353:
4351:
4348:
4346:
4343:
4341:
4338:
4336:
4333:
4331:
4330:Kripke–Platek
4328:
4326:
4323:
4319:
4316:
4314:
4311:
4310:
4309:
4306:
4305:
4303:
4299:
4291:
4288:
4287:
4286:
4283:
4281:
4278:
4274:
4271:
4270:
4269:
4266:
4264:
4261:
4259:
4256:
4254:
4251:
4249:
4246:
4243:
4239:
4235:
4232:
4228:
4225:
4223:
4220:
4218:
4215:
4214:
4213:
4209:
4206:
4205:
4203:
4201:
4197:
4193:
4185:
4182:
4180:
4177:
4175:
4174:constructible
4172:
4171:
4170:
4167:
4165:
4162:
4160:
4157:
4155:
4152:
4150:
4147:
4145:
4142:
4140:
4137:
4135:
4132:
4130:
4127:
4125:
4122:
4120:
4117:
4115:
4112:
4110:
4107:
4106:
4104:
4102:
4097:
4089:
4086:
4084:
4081:
4079:
4076:
4074:
4071:
4069:
4066:
4064:
4061:
4060:
4058:
4054:
4051:
4049:
4046:
4045:
4044:
4041:
4039:
4036:
4034:
4031:
4029:
4026:
4024:
4020:
4016:
4014:
4011:
4007:
4004:
4003:
4002:
3999:
3998:
3995:
3992:
3990:
3986:
3976:
3973:
3971:
3968:
3966:
3963:
3961:
3958:
3956:
3953:
3951:
3948:
3944:
3941:
3940:
3939:
3936:
3932:
3927:
3926:
3925:
3922:
3921:
3919:
3917:
3913:
3905:
3902:
3900:
3897:
3895:
3892:
3891:
3890:
3887:
3885:
3882:
3880:
3877:
3875:
3872:
3870:
3867:
3865:
3862:
3860:
3857:
3856:
3854:
3852:
3851:Propositional
3848:
3842:
3839:
3837:
3834:
3832:
3829:
3827:
3824:
3822:
3819:
3817:
3814:
3810:
3807:
3806:
3805:
3802:
3800:
3797:
3795:
3792:
3790:
3787:
3785:
3782:
3780:
3779:Logical truth
3777:
3775:
3772:
3771:
3769:
3767:
3763:
3760:
3758:
3754:
3748:
3745:
3743:
3740:
3738:
3735:
3733:
3730:
3728:
3725:
3723:
3719:
3715:
3711:
3709:
3706:
3704:
3701:
3699:
3695:
3692:
3691:
3689:
3687:
3681:
3676:
3670:
3667:
3665:
3662:
3660:
3657:
3655:
3652:
3650:
3647:
3645:
3642:
3640:
3637:
3635:
3632:
3630:
3627:
3625:
3622:
3620:
3617:
3615:
3612:
3608:
3605:
3604:
3603:
3600:
3599:
3597:
3593:
3589:
3582:
3577:
3575:
3570:
3568:
3563:
3562:
3559:
3547:
3544:
3542:
3539:
3537:
3534:
3532:
3529:
3528:
3526:
3522:
3514:
3511:
3510:
3509:
3506:
3502:
3499:
3498:
3497:
3494:
3490:
3487:
3486:
3485:
3482:
3481:
3479:
3477:
3476:Digital logic
3473:
3467:
3464:
3462:
3459:
3457:
3454:
3453:
3451:
3449:
3445:
3439:
3436:
3434:
3431:
3430:
3428:
3426:
3422:
3416:
3413:
3412:
3410:
3408:
3404:
3398:
3395:
3393:
3390:
3388:
3385:
3384:
3382:
3380:
3379:Substructural
3376:
3370:
3367:
3365:
3362:
3360:
3357:
3355:
3352:
3350:
3347:
3346:
3344:
3342:
3338:
3332:
3329:
3327:
3324:
3322:
3319:
3317:
3314:
3312:
3309:
3308:
3306:
3304:
3300:
3296:
3289:
3284:
3282:
3277:
3275:
3270:
3269:
3266:
3260:
3259:0-486-40459-5
3256:
3252:
3248:
3246:
3242:
3238:
3226:
3220:
3216:
3215:
3209:
3208:
3204:
3197:
3193:
3187:
3184:
3179:
3173:
3169:
3162:
3159:
3154:
3147:
3144:
3140:
3136:
3130:
3127:
3124:
3119:
3116:
3112:
3106:
3103:
3098:
3096:9780141920870
3092:
3088:
3087:
3079:
3076:
3061:
3057:
3053:
3049:
3045:
3041:
3040:
3032:
3028:
3022:
3019:
3014:
3010:
3004:
3001:
2997:
2992:
2985:
2982:
2978:
2973:
2969:
2965:
2961:
2954:
2951:
2947:
2942:
2938:
2934:
2930:
2923:
2920:
2915:
2911:
2907:
2903:
2896:
2893:
2888:
2884:
2880:
2876:
2872:
2868:
2864:
2857:
2854:
2849:
2845:
2841:
2837:
2833:
2829:
2825:
2818:
2815:
2810:
2806:
2803:(1–2): 1–11.
2802:
2798:
2791:
2789:
2787:
2783:
2772:
2768:
2761:
2758:
2745:
2741:
2734:
2731:
2727:
2715:
2711:
2704:
2701:
2696:
2692:
2688:
2681:
2678:
2667:
2663:
2657:
2654:
2648:
2643:
2639:
2635:
2630:
2625:
2620:
2615:
2611:
2607:
2603:
2596:
2593:
2588:
2584:
2580:
2574:
2571:
2565:
2561:
2558:
2555:
2551:
2548:
2545:
2541:
2538:
2535:
2532:
2530:
2527:
2525:
2522:
2520:
2517:
2515:
2512:
2510:
2507:
2505:
2502:
2500:
2497:
2496:
2492:
2481:
2476:
2474:
2472:
2468:
2462:
2454:
2449:
2444:
2441:
2438:
2437:
2436:
2434:
2426:
2421:
2418:
2417:
2413:
2405:
2396:
2392:
2388:
2385:
2381:
2380:
2377:
2373:
2366:Bochvar logic
2365:
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:
2121:
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:
1974:
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:
1380:
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:
430:
428:
426:
421:
417:
413:
409:
405:
401:
397:
393:
389:
386:allows 2 = 4
385:
384:Boolean logic
378:
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:
260:
258:
256:
252:
246:
241:
239:
238:Hilary Putnam
235:
231:
222:
220:
218:
214:
211:Around 1910,
207:Pre-discovery
206:
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:
5296:
5094:Ultraproduct
4941:Model theory
4906:Independence
4842:Formal proof
4834:Proof theory
4817:
4790:
4747:real numbers
4719:second-order
4630:Substitution
4507:Metalanguage
4448:conservative
4421:Axiom schema
4365:Constructive
4335:Morse–Kelley
4301:Set theories
4280:Aleph number
4273:inaccessible
4179:Grothendieck
4063:intersection
3950:Higher-order
3938:Second-order
3893:
3884:Truth tables
3841:Venn diagram
3624:Formal proof
3456:Three-valued
3455:
3397:Linear logic
3250:
3228:. Retrieved
3213:
3195:
3186:
3167:
3161:
3153:Sitz. Berlin
3152:
3146:
3138:
3129:
3118:
3105:
3085:
3078:
3067:. Retrieved
3043:
3037:
3031:"Third base"
3027:Hayes, Brian
3021:
3012:
3003:
2994:
2990:
2984:
2975:
2966:(5): 73–80.
2963:
2959:
2953:
2944:
2932:
2928:
2922:
2905:
2901:
2895:
2870:
2866:
2856:
2831:
2827:
2817:
2800:
2796:
2774:. Retrieved
2770:
2760:
2748:. Retrieved
2743:
2733:
2725:
2718:. Retrieved
2713:
2703:
2690:
2680:
2669:. Retrieved
2665:
2656:
2609:
2605:
2595:
2582:
2573:
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:
5204:Type theory
5152:undecidable
5084:Truth value
4971:equivalence
4650:non-logical
4263:Enumeration
4253:Isomorphism
4200:cardinality
4184:Von Neumann
4149:Ultrafilter
4114:Uncountable
4048:equivalence
3965:Quantifiers
3955:Fixed-point
3924:First-order
3804:Consistency
3789:Proposition
3766:Traditional
3737:Lindström's
3727:Compactness
3669:Type theory
3614:Cardinality
3496:Four-valued
3466:Łukasiewicz
3461:Four-valued
3448:Many-valued
3425:Description
3415:Dialetheism
2873:(1): 1–28.
2402:August 2014
1852:modal logic
420:implication
416:equivalence
373:connectives
330:undecidable
249:Similarly,
157:indicating
5319:Categories
5015:elementary
4708:arithmetic
4576:Quantifier
4554:functional
4426:Expression
4144:Transitive
4088:identities
4073:complement
4006:hereditary
3989:Set theory
3354:Fuzzy rule
3069:2020-04-12
2776:2020-07-30
2671:2024-05-15
2566: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
5286:Supertask
5189:Recursion
5147:decidable
4981:saturated
4959:of models
4882:deductive
4877:axiomatic
4797:Hilbert's
4784:Euclidean
4765:canonical
4688:axiomatic
4620:Signature
4549:Predicate
4438:Extension
4360:Ackermann
4285:Operation
4164:Universal
4154:Recursive
4129:Singleton
4124:Inhabited
4109:Countable
4099:Types of
4083:power set
4053:partition
3970:Predicate
3916:Predicate
3831:Syllogism
3821:Soundness
3794:Inference
3784:Tautology
3686:paradoxes
3508:IEEE 1164
3359:Fuzzy set
3230:24 August
2902:Semiotica
2887:0031-8108
2848:0031-8094
2638:0002-9327
2431:Some 3VL
1975:RM3 logic
1863:tautology
1399:, vol 8.
1114:→
580:OR(A, B)
356:Inside a
135:trivalent
5271:Logicism
5264:timeline
5240:Concrete
5099:Validity
5069:T-schema
5062:Kripke's
5057:Tarski's
5052:semantic
5042:Strength
4991:submodel
4986:spectrum
4954:function
4802:Tarski's
4791:Elements
4778:geometry
4734:Robinson
4655:variable
4640:function
4613:spectrum
4603:Sentence
4559:variable
4502:Language
4455:Relation
4416:Automata
4406:Alphabet
4390:language
4244:-jection
4222:codomain
4208:Function
4169:Universe
4139:Infinite
4043:Relation
3826:Validity
3816:Argument
3714:theorem,
3155:. 42–56.
3060:Archived
3048:Sigma Xi
3011:(1981).
2695:Archived
2587:Archived
2477:See also
2122:HT logic
348:, 1 for
315:, 2 for
167:bivalent
5213:Related
5010:Diagram
4908: (
4887:Hilbert
4872:Systems
4867:Theorem
4745:of the
4690:systems
4470:Formula
4465:Grammar
4381: (
4325:General
4038:Forcing
4023:Element
3943:Monadic
3718:paradox
3659:Theorem
3595:General
3501:Verilog
2750:May 15,
2720:May 15,
2691:Commens
2647:2370324
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
4976:finite
4739:Skolem
4692:
4667:Theory
4635:Symbol
4625:String
4608:atomic
4485:ground
4480:closed
4475:atomic
4431:ground
4394:syntax
4290:binary
4217:domain
4134:Finite
3899:finite
3757:Logics
3716:
3664:Theory
3524:Others
3257:
3221:
3174:
3093:
2885:
2846:
2644:
2636:
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:
4966:Model
4714:Peano
4571:Proof
4411:Arity
4340:Naive
4227:image
4159:Fuzzy
4119:Empty
4068:union
4013:Class
3654:Model
3644:Lemma
3602:Axiom
3341:Fuzzy
3063:(PDF)
3046:(6).
3034:(PDF)
2642:JSTOR
2542:(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
5089:Type
4892:list
4696:list
4673:list
4662:Term
4596:rank
4490:open
4384:list
4196:Maps
4101:sets
3960:Free
3930:list
3680:list
3607:list
3513:VHDL
3255:ISBN
3232:2013
3219:ISBN
3172:ISBN
3091:ISBN
2906:2021
2883:ISSN
2844:ISSN
2752:2023
2722:2023
2634: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
4776:of
4758:of
4706:of
4238:Sur
4212:Map
4019:Ur-
4001:Set
3241:doi
3052:doi
2968:doi
2937:doi
2910:doi
2875:doi
2836:doi
2805:doi
2624:hdl
2614: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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5162:NP
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4242:Bi
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3058:.
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2257:F
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2249:T
2243:F
2238:B
2228:HT
2211:F
2208:T
2203:F
2195:T
2192:F
2184:A
2177:HT
2140:HT
2096:⊗
2082:))
2078:→
2055:T
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2049:F
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2038:U
2035:F
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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1889:,
1880:↔
1870:→
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400:OR
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