402:
4977:
394:
2736:
7845:
1601:
2472:
3058:, also describes rules of conversion that follow the lines of De Morgan's laws. Still, De Morgan is given credit for stating the laws in the terms of modern formal logic, and incorporating them into the language of logic. De Morgan's laws can be proved easily, and may even seem trivial. Nonetheless, these laws are helpful in making valid inferences in proofs and deductive arguments.
31:
7414:
2193:
1367:
2351:
2731:{\displaystyle {\begin{aligned}{\overline {A_{1}\land A_{2}\land \ldots \land A_{n}}}={\overline {A_{1}}}\lor {\overline {A_{2}}}\lor \ldots \lor {\overline {A_{n}}},\\{\overline {A_{1}\lor A_{2}\lor \ldots \lor A_{n}}}={\overline {A_{1}}}\land {\overline {A_{2}}}\land \ldots \land {\overline {A_{n}}}.\end{aligned}}}
1887:
1742:
1248:
3309:
Working in the opposite direction again, the second expression asserts that at least one of "not A" and "not B" must be true, or equivalently that at least one of A and B must be false. Since at least one of them must be false, then their conjunction would likewise be false. Negating said conjunction
3008:
Evaluating Search B, the search "(NOT cats)" will hit on documents that do not contain "cats", which is
Documents 2 and 4. Similarly the search "(NOT dogs)" will hit on Documents 1 and 4. Applying the AND operator to these two searches (which is Search B) will hit on the documents that are common to
3195:
Working in the opposite direction, the second expression asserts that A is false and B is false (or equivalently that "not A" and "not B" are true). Knowing this, a disjunction of A and B must be false also. The negation of said disjunction must thus be true, and the result is identical to the first
4988:
In extensions of classical propositional logic, the duality still holds (that is, to any logical operator one can always find its dual), since in the presence of the identities governing negation, one may always introduce an operator that is the De Morgan dual of another. This leads to an important
873:
731:
2013:
1596:{\displaystyle {\begin{aligned}\lnot (P_{1}\land P_{2}\land \dots \land P_{n})\leftrightarrow \lnot P_{1}\lor \lnot P_{2}\lor \ldots \lor \lnot P_{n}\\\lnot (P_{1}\lor P_{2}\lor \dots \lor P_{n})\leftrightarrow \lnot P_{1}\land \lnot P_{2}\land \ldots \land \lnot P_{n}\end{aligned}}}
1107:
992:
4645:
4052:
2976:
De Morgan's laws commonly apply to text searching using
Boolean operators AND, OR, and NOT. Consider a set of documents containing the words "cats" and "dogs". De Morgan's laws hold that these two searches will return the same set of documents:
2225:
1761:
1620:
5798:
5667:
6012:
1126:
3204:
The application of De Morgan's theorem to conjunction is very similar to its application to a disjunction both in form and rationale. Consider the following claim: "it is false that A and B are both true", which is written as:
4841:
4778:
4709:
4116:
3530:
3469:
2910:
746:
604:
2828:
6701:
6616:
6102:
6428:
2188:{\displaystyle {\begin{aligned}{\overline {\bigcap _{i\in I}A_{i}}}&\equiv \bigcup _{i\in I}{\overline {A_{i}}},\\{\overline {\bigcup _{i\in I}A_{i}}}&\equiv \bigcap _{i\in I}{\overline {A_{i}}},\end{aligned}}}
4966:
4902:
3408:
5169:
3191:
true, then the disjunction of A and B would be true, making its negation false. Presented in
English, this follows the logic that "since two things are both false, it is also false that either of them is true".
6841:
6771:
6503:
5449:
5539:
4416:
4997:: any formula is equivalent to another formula where negations only occur applied to the non-logical atoms of the formula. The existence of negation normal forms drives many applications, for example in
1006:
891:
3686:
4462:
4170:
3954:
3846:
6137:. For example, from knowing it not to be the case that both Alice and Bob showed up to their date, it does not follow who did not show up. The latter principle is equivalent to the principle of the
1614:
De Morgan's laws are normally shown in the compact form above, with the negation of the output on the left and negation of the inputs on the right. A clearer form for substitution can be stated as:
2477:
2230:
2018:
1766:
1625:
1372:
1131:
751:
609:
5894:
5852:
3005:
To evaluate Search A, clearly the search "(cats OR dogs)" will hit on
Documents 1, 2, and 3. So the negation of that search (which is Search A) will hit everything else, which is Document 4.
4506:
4211:
565:
487:
6224:
5331:
5258:
4555:
3962:
4547:
3579:
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In order for this claim to be true, either or both of A or B must be false, for if they both were true, then the conjunction of A and B would be true, making its negation false. Thus,
3182:
1359:
3301:
7181:
3908:
3800:
6300:
6267:
6171:
3611:
2346:{\displaystyle {\begin{aligned}{\overline {A\land B}}&={\overline {A}}\lor {\overline {B}},\\{\overline {A\lor B}}&={\overline {A}}\land {\overline {B}},\end{aligned}}}
6878:
3244:
6892:
In modern programming languages, due to the optimisation of compilers and interpreters, the performance differences between these options are negligible or completely absent.
3343:
3121:
2384:
1920:
1882:{\displaystyle {\begin{aligned}{\overline {A\cup B}}&={\overline {A}}\cap {\overline {B}},\\{\overline {A\cap B}}&={\overline {A}}\cup {\overline {B}},\end{aligned}}}
5069:
4246:
3875:
3767:
3738:
3712:
1737:{\displaystyle {\begin{aligned}(P\land Q)&\Longleftrightarrow \neg (\neg P\lor \neg Q),\\(P\lor Q)&\Longleftrightarrow \neg (\neg P\land \neg Q).\end{aligned}}}
8302:
6131:
6352:
6326:
4301:
4275:
2936:
2430:
2456:
1992:
1966:
5678:
1243:{\displaystyle {\begin{aligned}\neg (P\land Q)&\leftrightarrow (\neg P\lor \neg Q),\\\neg (P\lor Q)&\leftrightarrow (\neg P\land \neg Q).\\\end{aligned}}}
5550:
6526:
2958:
2404:
1940:
1291:
1271:
5933:
7109:
6948:
868:{\displaystyle {\begin{aligned}\neg (P\lor Q)&\vdash (\neg P\land \neg Q),{\text{and}}\\(\neg P\land \neg Q)&\vdash \neg (P\lor Q).\end{aligned}}}
726:{\displaystyle {\begin{aligned}\neg (P\land Q)&\vdash (\neg P\lor \neg Q),{\text{and}}\\(\neg P\lor \neg Q)&\vdash \neg (P\land Q).\end{aligned}}}
4785:
4722:
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4060:
3474:
3413:
2839:
7310:
4981:
2760:
8467:
7510:
230:
6622:
6537:
6023:
6367:
3082:
In the case of its application to a disjunction, consider the following claim: "it is false that either of A or B is true", which is written as:
7177:
4910:
4846:
3352:
7017:
6991:
7991:
7811:
6922:
5077:
3306:
Presented in
English, this follows the logic that "since it is false that two things are both true, at least one of them must be false".
8319:
6777:
6707:
6439:
1755:
In set theory, it is often stated as "union and intersection interchange under complementation", which can be formally expressed as:
1102:{\displaystyle {\frac {\neg (P\lor Q)}{\therefore \neg P\land \neg Q}}\qquad {\frac {\neg P\land \neg Q}{\therefore \neg (P\lor Q)}}}
987:{\displaystyle {\frac {\neg (P\land Q)}{\therefore \neg P\lor \neg Q}}\qquad {\frac {\neg P\lor \neg Q}{\therefore \neg (P\land Q)}}}
7213:
7164:
7113:
7083:
6966:
5373:
7717:
5460:
4375:
6234:
8297:
6902:
6107:
The converse of the last implication does not hold in pure intuitionistic logic. That is, the failure of the joint proposition
3619:
4421:
4129:
3913:
3805:
3253:
of A and B must be false (or equivalently, one or more of "not A" and "not B" must be true). This may be written directly as,
7778:
7383:
7338:
6917:
1747:
This emphasizes the need to invert both the inputs and the output, as well as change the operator when doing a substitution.
8071:
7950:
7768:
7303:
7242:
6889:
De Morgan's laws are widely used in computer engineering and digital logic for the purpose of simplifying circuit designs.
571:
401:
244:
8314:
7576:
7464:
5858:
5816:
8307:
8462:
7945:
7908:
7617:
7237:
4470:
4175:
4640:{\displaystyle \forall x{\Big (}x\in {\overline {A}}\cup {\overline {B}}\implies x\in {\overline {A\cap B}}{\Big )}}
4047:{\displaystyle \forall x{\Big (}x\in {\overline {A\cap B}}\implies x\in {\overline {A}}\cup {\overline {B}}{\Big )}}
3012:
A similar evaluation can be applied to show that the following two searches will both return
Documents 1, 2, and 4:
7591:
7581:
7484:
281:
85:
493:
415:
8457:
7996:
7888:
7876:
7871:
7632:
7622:
7373:
6907:
6179:
5266:
5193:
5014:
1969:
285:
91:
7117:
4511:
3543:
8452:
7804:
7601:
7596:
7586:
7296:
7159:. Trans. Gyula Klima. New Haven: Yale University Press, 2001. See especially Treatise 1, Chapter 7, Section 5.
5911:, the relationship of these modal operators to the quantification can be understood by setting up models using
4339:
4306:
272:
111:
98:
3140:
1305:
8416:
8334:
8209:
8161:
7975:
7898:
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7479:
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3259:
353:
276:
117:
104:
8368:
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7707:
7550:
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2747:
296:
130:
52:
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7545:
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7393:
579:
225:
206:
173:
164:
124:
7232:
6272:
6239:
6143:
4976:
3584:
7040:
6853:
3211:
3040:, which later cemented De Morgan's claim to the find. Nevertheless, a similar observation was made by
393:
8358:
8348:
8182:
8113:
8066:
8006:
7893:
7763:
7535:
7388:
5924:
5180:
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137:
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2433:
327:
323:
237:
182:
143:
4219:
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7343:
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6230:
5908:
5900:
5018:
3854:
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3717:
3691:
3029:
1995:
319:
150:
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7962:
7933:
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7834:
7438:
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7251:
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7013:
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3045:
1113:
879:
315:
263:
256:
68:
59:
45:
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7094:
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8391:
8259:
7913:
7722:
7712:
7677:
7398:
7059:
6954:
6331:
6308:
5912:
5793:{\displaystyle P(a)\lor P(b)\lor P(c)\equiv \neg (\neg P(a)\land \neg P(b)\land \neg P(c)),}
4280:
4254:
2921:
2415:
1606:
These laws generalize De Morgan’s original laws for negating conjunctions and disjunctions.
575:
359:
The complement of the intersection of two sets is the same as the union of their complements
331:
312:
5662:{\displaystyle P(a)\land P(b)\land P(c)\equiv \neg (\neg P(a)\lor \neg P(b)\lor \neg P(c))}
2441:
1977:
1951:
8290:
8228:
8046:
7866:
7758:
7662:
7319:
6847:
4998:
4990:
2988:
The corpus of documents containing "cats" or "dogs" can be represented by four documents:
1300:
provide an equivalence for negating a conjunction or disjunction involving multiple terms.
578:
and digital circuit designs. De Morgan's laws are an example of a more general concept of
300:
213:
6007:{\displaystyle \neg (P\lor Q)\,\leftrightarrow \,{\big (}(\neg P)\land (\neg Q){\big )},}
5013:
of a formula. Computer programmers use them to simplify or properly negate complicated
3310:
thus results in a true expression, and this expression is identical to the first claim.
8426:
8223:
8204:
8108:
8093:
8050:
7986:
7928:
7702:
7358:
6927:
6511:
6354:
for some arbitrary constant predicate C, meaning that the above laws are still true in
6233:. For a refined version of the failing law concerning existential statements, see the
2943:
2389:
1925:
1276:
1256:
8446:
8431:
8401:
8233:
8147:
8142:
7727:
7692:
7687:
7525:
7199:
7053:
6355:
156:
35:
7063:
17:
8381:
8376:
8194:
8123:
8081:
7940:
7844:
7697:
7672:
7667:
7520:
7254:
6508:
are tautologies even in minimal logic with negation replaced with implying a fixed
5337:
3250:
3049:
3044:, and was known to Greek and Medieval logicians. For example, in the 14th century,
3037:
383:
375:
74:
4836:{\displaystyle {\overline {A\cap B}}\subseteq {\overline {A}}\cup {\overline {B}}}
4773:{\displaystyle {\overline {A}}\cup {\overline {B}}\subseteq {\overline {A\cap B}}}
4704:{\displaystyle {\overline {A}}\cup {\overline {B}}\subseteq {\overline {A\cap B}}}
4111:{\displaystyle {\overline {A\cap B}}\subseteq {\overline {A}}\cup {\overline {B}}}
3525:{\displaystyle {\overline {A}}\cup {\overline {B}}\subseteq {\overline {A\cap B}}}
3464:{\displaystyle {\overline {A\cap B}}\subseteq {\overline {A}}\cup {\overline {B}}}
3036:. De Morgan's formulation was influenced by algebraization of logic undertaken by
2905:{\displaystyle {\overline {(A+B)}}\equiv ({\overline {A}}\cdot {\overline {B}}),}
8406:
8041:
7363:
7353:
6912:
5807:
3131:
3067:
2823:{\displaystyle {\overline {(A\cdot B)}}\equiv ({\overline {A}}+{\overline {B}})}
38:. In each case, the resultant set is the set of all points in any shade of blue.
8386:
8254:
8157:
7820:
7428:
7272:
7205:
Digital
Circuit Design for Computer Science Students: An Introductory Textbook
5002:
409:
Another form of De Morgan's law is the following as seen in the right figure.
7027:
6696:{\displaystyle (P\land Q)\,\to \,\neg {\big (}(\neg P)\lor (\neg Q){\big )},}
6611:{\displaystyle (P\lor Q)\,\to \,\neg {\big (}(\neg P)\land (\neg Q){\big )},}
6097:{\displaystyle {\big (}(\neg P)\lor (\neg Q){\big )}\,\to \,\neg (P\land Q).}
356:
of the union of two sets is the same as the intersection of their complements
8189:
8152:
8103:
8001:
7403:
7259:
6423:{\displaystyle \forall x\,\neg P(x)\,\leftrightarrow \,\neg \exists x\,P(x)}
5904:
3041:
2219:
In
Boolean algebra, similarly, this law which can be formally expressed as:
144:
7007:
6958:
6528:, while the converse of the last law does not have to be true in general.
6305:
The validity of the other three De Morgan's laws remains true if negation
322:, a 19th-century British mathematician. The rules allow the expression of
30:
7378:
4961:{\displaystyle {\overline {A\cup B}}={\overline {A}}\cap {\overline {B}}}
4897:{\displaystyle {\overline {A\cap B}}={\overline {A}}\cup {\overline {B}}}
3403:{\displaystyle {\overline {A\cap B}}={\overline {A}}\cup {\overline {B}}}
2407:
2208:
1943:
397:
The equivalency of ¬φ ∨ ¬ψ and ¬(φ ∧ ψ) is displayed in this truth table.
7055:
In Quest of
Univeral Logic: A brief overview of formal logic's evolution
7268:
5810:, relating the box ("necessarily") and diamond ("possibly") operators:
1117:
595:
138:
3032:(1806–1871), who introduced a formal version of the laws to classical
8214:
8036:
5336:
To relate these quantifier dualities to the De Morgan laws, set up a
5164:{\displaystyle {\mbox{P}}^{d}(p,q,...)=\neg P(\neg p,\neg q,\dots ).}
118:
7278:
6133:
cannot necessarily be resolved to the failure of either of the two
5179:
This duality can be generalised to quantifiers, so for example the
2204:
is some, possibly countably or uncountably infinite, indexing set.
8086:
7853:
7288:
4975:
400:
392:
29:
6836:{\displaystyle \exists x\,P(x)\,\to \,\neg \forall x\,\neg P(x),}
6766:{\displaystyle \forall x\,P(x)\,\to \,\neg \exists x\,\neg P(x),}
6498:{\displaystyle \exists x\,\neg P(x)\,\to \,\neg \forall x\,P(x).}
7413:
3048:
wrote down the words that would result by reading the laws out.
7793:
7292:
5923:
Three out of the four implications of de Morgan's laws hold in
5444:{\displaystyle \forall x\,P(x)\equiv P(a)\land P(b)\land P(c)}
2207:
In set notation, De Morgan's laws can be remembered using the
7789:
5534:{\displaystyle \exists x\,P(x)\equiv P(a)\lor P(b)\lor P(c).}
4411:{\displaystyle x\not \in {\overline {A}}\cup {\overline {B}}}
3130:
A nor B is true, then it must follow that both A is not true
4980:
De Morgan's Laws represented as a circuit with logic gates (
570:
Applications of the rules include simplification of logical
5017:. They are also often useful in computations in elementary
207:
165:
131:
86:
5806:
Then, the quantifier dualities can be extended further to
344:
The negation of "A or B" is the same as "not A and not B."
341:
The negation of "A and B" is the same as "not A or not B."
7012:. Richard Parker (10th ed.). New York: McGraw-Hill.
3681:{\displaystyle A\cap B=\{\,y\ |\ y\in A\wedge y\in B\,\}}
157:
5024:
Let one define the dual of any propositional operator P(
4457:{\displaystyle x\in {\overline {A}}\cup {\overline {B}}}
4165:{\displaystyle x\in {\overline {A}}\cup {\overline {B}}}
3949:{\displaystyle x\in {\overline {A}}\cup {\overline {B}}}
3841:{\displaystyle x\in {\overline {A}}\cup {\overline {B}}}
3066:
De Morgan's theorem may be applied to the negation of a
125:
6947:
Copi, Irving M.; Cohen, Carl; McMahon, Kenneth (2016).
238:
193:
99:
75:
5083:
5049:
5005:, and in formal logic, where it is needed to find the
6856:
6780:
6710:
6625:
6540:
6514:
6442:
6370:
6334:
6311:
6275:
6242:
6182:
6146:
6113:
6026:
5936:
5861:
5819:
5681:
5553:
5463:
5376:
5269:
5196:
5080:
5046:
4913:
4849:
4788:
4725:
4656:
4558:
4514:
4473:
4424:
4378:
4342:
4309:
4283:
4257:
4222:
4178:
4132:
4063:
3965:
3916:
3883:
3857:
3808:
3775:
3749:
3720:
3694:
3622:
3587:
3546:
3477:
3416:
3355:
3324:
3262:
3214:
3143:
3091:
2967:
is the logical NOT of what is underneath the overbar.
2946:
2924:
2842:
2763:
2475:
2444:
2418:
2392:
2365:
2228:
2016:
1980:
1954:
1928:
1901:
1764:
1623:
1370:
1308:
1279:
1259:
1129:
1009:
894:
749:
607:
496:
418:
5001:
design, where it is used to manipulate the types of
8367:
8330:
8242:
8132:
8020:
7961:
7852:
7827:
7751:
7655:
7610:
7564:
7498:
7457:
7421:
7326:
3346:
1361:, the generalized De Morgan’s Laws are as follows:
311:, are a pair of transformation rules that are both
6872:
6835:
6765:
6695:
6610:
6520:
6497:
6422:
6346:
6320:
6294:
6261:
6229:This weak form can be used as a foundation for an
6218:
6165:
6125:
6096:
6006:
5889:{\displaystyle \Diamond p\equiv \neg \Box \neg p.}
5888:
5847:{\displaystyle \Box p\equiv \neg \Diamond \neg p,}
5846:
5792:
5661:
5533:
5443:
5325:
5252:
5163:
5063:
4960:
4896:
4835:
4772:
4703:
4639:
4541:
4500:
4456:
4410:
4361:
4328:
4295:
4269:
4240:
4205:
4164:
4110:
4046:
3948:
3902:
3869:
3840:
3794:
3761:
3732:
3706:
3680:
3605:
3573:
3524:
3463:
3402:
3337:
3295:
3238:
3176:
3115:
2952:
2930:
2904:
2822:
2730:
2450:
2424:
2398:
2378:
2345:
2187:
1986:
1960:
1934:
1914:
1881:
1736:
1595:
1353:
1293:are propositions expressed in some formal system.
1285:
1265:
1242:
1101:
986:
867:
725:
559:
481:
7182:Indiana University–Purdue University Indianapolis
5803:verifying the quantifier dualities in the model.
5340:with some small number of elements in its domain
4632:
4567:
4039:
3974:
3134:B is not true, which may be written directly as:
151:
4216:Under that assumption, it must be the case that
231:
4904:; this concludes the proof of De Morgan's law.
4501:{\displaystyle x\not \in {\overline {A\cap B}}}
4206:{\displaystyle x\not \in {\overline {A\cap B}}}
3001:Document 4: Contains neither "cats" nor "dogs".
405:De Morgan's law with set subtraction operation.
200:
92:
2003:Unions and intersections of any number of sets
7805:
7304:
6685:
6651:
6600:
6566:
6361:Similarly to the above, the quantifier laws:
6063:
6029:
5996:
5962:
245:
214:
112:
105:
8:
5032:, ...) depending on elementary propositions
3675:
3635:
2998:Document 3: Contains both "cats" and "dogs".
2754:, De Morgan's laws are commonly written as:
2410:being written above the terms to be negated,
1946:being written above the terms to be negated,
560:{\displaystyle A-(B\cap C)=(A-B)\cup (A-C).}
482:{\displaystyle A-(B\cup C)=(A-B)\cap (A-C),}
7096:2000 Solved Problems in Digital Electronics
6219:{\displaystyle (\neg P)\lor \neg (\neg P).}
5326:{\displaystyle \exists x\,P(x)\equiv \neg }
5253:{\displaystyle \forall x\,P(x)\equiv \neg }
3345:to denote the complement of A, as above in
3053:
7812:
7798:
7790:
7561:
7311:
7297:
7289:
4605:
4601:
4542:{\displaystyle x\in {\overline {A\cap B}}}
4418:, in contradiction to the hypothesis that
4007:
4003:
3574:{\displaystyle x\in {\overline {A\cap B}}}
2992:Document 1: Contains only the word "cats".
337:The rules can be expressed in English as:
41:
6858:
6857:
6855:
6814:
6804:
6800:
6787:
6779:
6744:
6734:
6730:
6717:
6709:
6684:
6683:
6650:
6649:
6645:
6641:
6624:
6599:
6598:
6565:
6564:
6560:
6556:
6539:
6513:
6479:
6469:
6465:
6449:
6441:
6407:
6397:
6393:
6377:
6369:
6333:
6310:
6277:
6276:
6274:
6244:
6243:
6241:
6181:
6148:
6147:
6145:
6112:
6072:
6068:
6062:
6061:
6028:
6027:
6025:
5995:
5994:
5961:
5960:
5959:
5955:
5935:
5860:
5818:
5680:
5552:
5470:
5462:
5383:
5375:
5304:
5276:
5268:
5231:
5203:
5195:
5089:
5082:
5079:
5055:
5048:
5045:
4982:International Electrotechnical Commission
4948:
4935:
4914:
4912:
4884:
4871:
4850:
4848:
4823:
4810:
4789:
4787:
4752:
4739:
4726:
4724:
4683:
4670:
4657:
4655:
4631:
4630:
4612:
4591:
4578:
4566:
4565:
4557:
4521:
4513:
4480:
4472:
4444:
4431:
4423:
4398:
4385:
4377:
4362:{\displaystyle x\not \in {\overline {B}}}
4349:
4341:
4329:{\displaystyle x\not \in {\overline {A}}}
4316:
4308:
4282:
4256:
4221:
4185:
4177:
4152:
4139:
4131:
4098:
4085:
4064:
4062:
4038:
4037:
4027:
4014:
3985:
3973:
3972:
3964:
3936:
3923:
3915:
3890:
3882:
3856:
3828:
3815:
3807:
3782:
3774:
3748:
3719:
3693:
3674:
3645:
3638:
3621:
3586:
3553:
3545:
3504:
3491:
3478:
3476:
3451:
3438:
3417:
3415:
3390:
3377:
3356:
3354:
3325:
3323:
3261:
3213:
3142:
3090:
3009:these two searches, which is Document 4.
2945:
2923:
2886:
2873:
2843:
2841:
2807:
2794:
2764:
2762:
2710:
2704:
2684:
2678:
2664:
2658:
2643:
2624:
2611:
2604:
2586:
2580:
2560:
2554:
2540:
2534:
2519:
2500:
2487:
2480:
2476:
2474:
2443:
2417:
2391:
2366:
2364:
2326:
2313:
2288:
2271:
2258:
2233:
2229:
2227:
2167:
2161:
2149:
2126:
2110:
2103:
2085:
2079:
2067:
2044:
2028:
2021:
2017:
2015:
1979:
1953:
1927:
1902:
1900:
1862:
1849:
1824:
1807:
1794:
1769:
1765:
1763:
1624:
1622:
1583:
1561:
1545:
1526:
1507:
1494:
1474:
1452:
1436:
1417:
1398:
1385:
1371:
1369:
1345:
1326:
1313:
1307:
1278:
1258:
1130:
1128:
1055:
1010:
1008:
940:
895:
893:
803:
750:
748:
661:
608:
606:
495:
417:
3410:is completed in 2 steps by proving both
3177:{\displaystyle (\neg A)\wedge (\neg B).}
1354:{\displaystyle P_{1},P_{2},\dots ,P_{n}}
6939:
6235:lesser limited principle of omniscience
5907:observed this case, and in the case of
262:
255:
181:
58:
51:
44:
5175:Extension to predicate and modal logic
3296:{\displaystyle (\neg A)\lor (\neg B).}
3347:§ Set theory and Boolean algebra
3126:In that it has been established that
7:
6923:List of set identities and relations
4126:To prove the reverse direction, let
3903:{\displaystyle x\in {\overline {B}}}
3795:{\displaystyle x\in {\overline {A}}}
7283:Internet Encyclopedia of Philosophy
6986:(12th ed.), Cengage Learning,
4508:must not be the case, meaning that
3019:Search D: (NOT cats) OR (NOT dogs).
2984:Search B: (NOT cats) AND (NOT dogs)
2211:"break the line, change the sign".
6865:
6862:
6859:
6815:
6808:
6805:
6781:
6745:
6738:
6735:
6711:
6674:
6659:
6646:
6589:
6574:
6561:
6473:
6470:
6450:
6443:
6401:
6398:
6378:
6371:
6312:
6287:
6284:
6281:
6278:
6269:, which however is different from
6254:
6251:
6248:
6245:
6204:
6198:
6186:
6158:
6155:
6152:
6149:
6073:
6052:
6037:
5985:
5970:
5937:
5877:
5871:
5835:
5829:
5769:
5751:
5733:
5727:
5641:
5623:
5605:
5599:
5464:
5377:
5305:
5298:
5292:
5270:
5232:
5225:
5219:
5197:
5143:
5134:
5125:
4559:
3966:
3281:
3266:
3215:
3162:
3147:
3092:
1718:
1709:
1703:
1665:
1656:
1650:
1576:
1554:
1538:
1484:
1467:
1445:
1429:
1375:
1224:
1215:
1187:
1171:
1162:
1134:
1112:and expressed as truth-functional
1078:
1067:
1058:
1045:
1036:
1013:
963:
952:
943:
930:
921:
898:
840:
824:
815:
791:
782:
754:
698:
682:
673:
649:
640:
612:
370:not (A and B) = (not A) or (not B)
367:not (A or B) = (not A) and (not B)
330:purely in terms of each other via
34:De Morgan's laws represented with
25:
7114:Middle Tennessee State University
6295:{\displaystyle {\mathrm {WLPO} }}
6262:{\displaystyle {\mathrm {LLPO} }}
6166:{\displaystyle {\mathrm {WPEM} }}
3606:{\displaystyle x\not \in A\cap B}
2995:Document 2: Contains only "dogs".
7843:
7412:
7178:"Augustus De Morgan (1806–1871)"
7041:DeMorgan's [sic] Theorem
6873:{\displaystyle {\mathrm {PEM} }}
3239:{\displaystyle \neg (A\land B).}
8468:Theorems in propositional logic
6984:A Concise Introduction to Logic
6903:Conjunction/disjunction duality
4172:, and for contradiction assume
3338:{\displaystyle {\overline {A}}}
3116:{\displaystyle \neg (A\lor B).}
2379:{\displaystyle {\overline {A}}}
1915:{\displaystyle {\overline {A}}}
1054:
939:
7779:Tractatus Logico-Philosophicus
7384:Problem of multiple generality
7146:, part II, sections 32 and 33.
6918:List of Boolean algebra topics
6827:
6821:
6801:
6797:
6791:
6757:
6751:
6731:
6727:
6721:
6680:
6671:
6665:
6656:
6642:
6638:
6626:
6595:
6586:
6580:
6571:
6557:
6553:
6541:
6489:
6483:
6466:
6462:
6456:
6417:
6411:
6394:
6390:
6384:
6338:
6210:
6201:
6192:
6183:
6088:
6076:
6069:
6058:
6049:
6043:
6034:
5991:
5982:
5976:
5967:
5956:
5952:
5940:
5903:of possibility and necessity,
5784:
5781:
5775:
5763:
5757:
5745:
5739:
5730:
5721:
5715:
5706:
5700:
5691:
5685:
5656:
5653:
5647:
5635:
5629:
5617:
5611:
5602:
5593:
5587:
5578:
5572:
5563:
5557:
5525:
5519:
5510:
5504:
5495:
5489:
5480:
5474:
5438:
5432:
5423:
5417:
5408:
5402:
5393:
5387:
5320:
5317:
5311:
5295:
5286:
5280:
5247:
5244:
5238:
5222:
5213:
5207:
5155:
5131:
5119:
5095:
5064:{\displaystyle {\mbox{P}}^{d}}
4972:Generalising De Morgan duality
4602:
4004:
3646:
3287:
3278:
3272:
3263:
3230:
3218:
3168:
3159:
3153:
3144:
3107:
3095:
3016:Search C: NOT (cats AND dogs),
2896:
2870:
2858:
2846:
2817:
2791:
2779:
2767:
1724:
1706:
1700:
1693:
1681:
1671:
1653:
1647:
1640:
1628:
1535:
1532:
1487:
1426:
1423:
1378:
1230:
1212:
1209:
1202:
1190:
1177:
1159:
1156:
1149:
1137:
1093:
1081:
1028:
1016:
978:
966:
913:
901:
855:
843:
830:
812:
797:
779:
769:
757:
713:
701:
688:
670:
655:
637:
627:
615:
551:
539:
533:
521:
515:
503:
473:
461:
455:
443:
437:
425:
382:one of A or B rather than an "
1:
7769:The Principles of Mathematics
7279:Duality in Logic and Language
7064:10.13140/RG.2.2.24043.82724/1
5544:But, using De Morgan's laws,
3074:in all or part of a formula.
7465:Commutativity of conjunction
6846:but their inversion implies
4989:property of logics based on
4953:
4940:
4927:
4889:
4876:
4863:
4828:
4815:
4802:
4765:
4744:
4731:
4696:
4675:
4662:
4625:
4596:
4583:
4534:
4493:
4449:
4436:
4403:
4390:
4354:
4321:
4241:{\displaystyle x\in A\cap B}
4198:
4157:
4144:
4103:
4090:
4077:
4032:
4019:
3998:
3941:
3928:
3895:
3833:
3820:
3787:
3566:
3517:
3496:
3483:
3456:
3443:
3430:
3395:
3382:
3369:
3330:
2981:Search A: NOT (cats OR dogs)
2891:
2878:
2862:
2812:
2799:
2783:
2716:
2690:
2670:
2650:
2592:
2566:
2546:
2526:
2466:which can be generalized to
2371:
2331:
2318:
2301:
2276:
2263:
2246:
2173:
2133:
2091:
2051:
1907:
1867:
1854:
1837:
1812:
1799:
1782:
1298:generalized De Morgan’s laws
27:Pair of logical equivalences
7238:Encyclopedia of Mathematics
7006:Moore, Brooke Noel (2012).
6982:Hurley, Patrick J. (2015),
6328:is replaced by implication
4907:The other De Morgan's law,
3870:{\displaystyle x\not \in B}
3762:{\displaystyle x\not \in A}
3733:{\displaystyle x\not \in B}
3707:{\displaystyle x\not \in A}
3688:, it must be the case that
8484:
8303:von Neumann–Bernays–Gödel
7485:Monotonicity of entailment
5899:In its application to the
4993:, namely the existence of
4467:therefore, the assumption
1302:For a set of propositions
282:Existential generalization
87:Biconditional introduction
8104:One-to-one correspondence
7841:
7410:
7374:Idempotency of entailment
6908:Homogeneity (linguistics)
5927:. Specifically, we have
5040:, ... to be the operator
3200:Negation of a conjunction
3078:Negation of a disjunction
3062:Proof for Boolean algebra
3028:The laws are named after
2007:The generalized form is
7208:, Springer, p. 16,
6126:{\displaystyle P\land Q}
1120:of propositional logic:
740:rule may be written as:
273:Universal generalization
113:Disjunction introduction
100:Conjunction introduction
70:Implication introduction
7733:Willard Van Orman Quine
7132:History of Formal Logic
7052:Kashef, Arman. (2023),
6884:In computer engineering
6531:Further, one still has
5919:In intuitionistic logic
5011:disjunctive normal form
5007:conjunctive normal form
4968:, is proven similarly.
999:negation of disjunction
884:negation of conjunction
738:negation of disjunction
594:rule may be written in
592:negation of conjunction
318:. They are named after
8062:Constructible universe
7889:Constructibility (V=L)
7708:Charles Sanders Peirce
7551:Hypothetical syllogism
6874:
6837:
6767:
6697:
6612:
6522:
6499:
6424:
6348:
6347:{\displaystyle P\to C}
6322:
6321:{\displaystyle \neg P}
6296:
6263:
6220:
6167:
6127:
6098:
6008:
5890:
5848:
5794:
5663:
5535:
5445:
5327:
5254:
5185:existential quantifier
5165:
5065:
4985:
4962:
4898:
4837:
4774:
4705:
4641:
4543:
4502:
4458:
4412:
4363:
4330:
4297:
4296:{\displaystyle x\in B}
4271:
4270:{\displaystyle x\in A}
4242:
4207:
4166:
4112:
4048:
3950:
3904:
3871:
3842:
3796:
3763:
3734:
3708:
3682:
3607:
3575:
3526:
3465:
3404:
3339:
3297:
3251:one (at least) or more
3240:
3178:
3117:
3055:Summulae de Dialectica
3054:
2954:
2932:
2931:{\displaystyle \cdot }
2906:
2824:
2732:
2452:
2426:
2425:{\displaystyle \land }
2400:
2380:
2347:
2189:
1988:
1962:
1936:
1916:
1883:
1738:
1597:
1355:
1287:
1267:
1244:
1103:
988:
869:
727:
561:
483:
406:
398:
374:where "A or B" is an "
132:hypothetical syllogism
53:Propositional calculus
39:
8285:Principia Mathematica
8119:Transfinite induction
7978:(i.e. set difference)
7774:Principia Mathematica
7546:Disjunctive syllogism
7531:modus ponendo tollens
7157:Summula de Dialectica
7110:"DeMorgan's Theorems"
6959:10.4324/9781315510897
6950:Introduction to Logic
6875:
6838:
6768:
6698:
6613:
6523:
6500:
6425:
6349:
6323:
6297:
6264:
6221:
6168:
6128:
6099:
6009:
5891:
5849:
5795:
5664:
5536:
5446:
5328:
5255:
5166:
5066:
4995:negation normal forms
4979:
4963:
4899:
4838:
4775:
4706:
4642:
4544:
4503:
4459:
4413:
4364:
4331:
4298:
4272:
4243:
4208:
4167:
4113:
4049:
3951:
3905:
3872:
3843:
3797:
3764:
3735:
3709:
3683:
3608:
3576:
3527:
3466:
3405:
3340:
3298:
3241:
3179:
3118:
3070:or the negation of a
2955:
2933:
2907:
2825:
2733:
2453:
2451:{\displaystyle \lor }
2427:
2401:
2381:
2348:
2190:
1989:
1987:{\displaystyle \cup }
1963:
1961:{\displaystyle \cap }
1937:
1917:
1884:
1739:
1598:
1356:
1288:
1268:
1245:
1104:
989:
870:
728:
562:
484:
404:
396:
174:Negation introduction
167:modus ponendo tollens
33:
8359:Burali-Forti paradox
8114:Set-builder notation
8067:Continuum hypothesis
8007:Symmetric difference
7764:Function and Concept
7536:Constructive dilemma
7511:Material implication
7078:by R. L. Goodstein.
6854:
6778:
6708:
6623:
6538:
6512:
6440:
6368:
6332:
6309:
6273:
6240:
6180:
6144:
6139:weak excluded middle
6111:
6024:
5934:
5925:intuitionistic logic
5859:
5817:
5679:
5551:
5461:
5374:
5267:
5194:
5181:universal quantifier
5078:
5044:
4911:
4847:
4786:
4723:
4654:
4556:
4512:
4471:
4422:
4376:
4372:However, that means
4340:
4307:
4281:
4255:
4220:
4176:
4130:
4061:
3963:
3914:
3881:
3855:
3806:
3773:
3747:
3718:
3692:
3620:
3585:
3544:
3475:
3414:
3353:
3322:
3314:Proof for set theory
3260:
3212:
3141:
3089:
2944:
2922:
2840:
2761:
2752:computer engineering
2473:
2442:
2416:
2390:
2363:
2226:
2014:
1978:
1952:
1926:
1899:
1762:
1621:
1368:
1306:
1277:
1257:
1127:
1007:
892:
747:
605:
580:mathematical duality
494:
416:
232:Material implication
183:Rules of replacement
46:Transformation rules
18:De Morgan's law
8320:Tarski–Grothendieck
7738:Ludwig Wittgenstein
7541:Destructive dilemma
7369:Well-formed formula
7233:"Duality principle"
7142:William of Ockham,
4251:so it follows that
3034:propositional logic
2938:is the logical AND,
2460:logical disjunction
2434:logical conjunction
2386:is the negation of
1922:is the negation of
309:De Morgan's theorem
297:propositional logic
145:destructive dilemma
8463:Rules of inference
7909:Limitation of size
7683:Augustus De Morgan
7255:"de Morgan's Laws"
7252:Weisstein, Eric W.
6870:
6833:
6763:
6693:
6608:
6518:
6495:
6420:
6344:
6318:
6292:
6259:
6231:intermediate logic
6216:
6163:
6123:
6094:
6004:
5909:normal modal logic
5901:alethic modalities
5886:
5844:
5790:
5659:
5531:
5441:
5323:
5250:
5161:
5087:
5061:
5053:
5019:probability theory
5015:logical conditions
4986:
4958:
4894:
4833:
4770:
4701:
4637:
4539:
4498:
4454:
4408:
4359:
4326:
4293:
4267:
4238:
4203:
4162:
4108:
4044:
3946:
3900:
3867:
3838:
3792:
3759:
3730:
3704:
3678:
3603:
3571:
3522:
3461:
3400:
3335:
3293:
3236:
3174:
3113:
3030:Augustus De Morgan
2960:is the logical OR,
2950:
2928:
2902:
2820:
2728:
2726:
2448:
2422:
2396:
2376:
2343:
2341:
2185:
2183:
2160:
2121:
2078:
2039:
1984:
1958:
1932:
1912:
1879:
1877:
1734:
1732:
1593:
1591:
1351:
1283:
1263:
1240:
1238:
1099:
984:
865:
863:
723:
721:
557:
479:
407:
399:
320:Augustus De Morgan
316:rules of inference
264:Rules of inference
60:Rules of inference
40:
8440:
8439:
8349:Russell's paradox
8298:Zermelo–Fraenkel
8199:Dedekind-infinite
8072:Diagonal argument
7971:Cartesian product
7835:Set (mathematics)
7787:
7786:
7651:
7650:
7019:978-0-07-803828-0
7009:Critical thinking
6993:978-1-285-19654-1
6521:{\displaystyle Q}
5086:
5052:
4956:
4943:
4930:
4892:
4879:
4866:
4831:
4818:
4805:
4768:
4747:
4734:
4699:
4678:
4665:
4628:
4599:
4586:
4537:
4496:
4452:
4439:
4406:
4393:
4357:
4324:
4201:
4160:
4147:
4106:
4093:
4080:
4035:
4022:
4001:
3944:
3931:
3898:
3836:
3823:
3790:
3652:
3644:
3569:
3520:
3499:
3486:
3459:
3446:
3433:
3398:
3385:
3372:
3349:. The proof that
3333:
3187:If either A or B
3046:William of Ockham
2953:{\displaystyle +}
2894:
2881:
2865:
2815:
2802:
2786:
2719:
2693:
2673:
2653:
2595:
2569:
2549:
2529:
2399:{\displaystyle A}
2374:
2334:
2321:
2304:
2279:
2266:
2249:
2176:
2145:
2136:
2106:
2094:
2063:
2054:
2024:
1935:{\displaystyle A}
1910:
1870:
1857:
1840:
1815:
1802:
1785:
1610:Substitution form
1286:{\displaystyle Q}
1266:{\displaystyle P}
1097:
1052:
982:
937:
806:
664:
576:computer programs
293:
292:
16:(Redirected from
8475:
8458:Duality theories
8422:Bertrand Russell
8412:John von Neumann
8397:Abraham Fraenkel
8392:Richard Dedekind
8354:Suslin's problem
8265:Cantor's theorem
7982:De Morgan's laws
7847:
7814:
7807:
7800:
7791:
7723:Henry M. Sheffer
7713:Bertrand Russell
7678:Richard Dedekind
7562:
7506:De Morgan's laws
7480:Noncontradiction
7422:Classical logics
7416:
7313:
7306:
7299:
7290:
7269:de Morgan's laws
7265:
7264:
7246:
7219:
7218:
7196:
7190:
7189:
7184:. Archived from
7173:
7167:
7153:
7147:
7140:
7134:
7128:
7122:
7121:
7116:. Archived from
7106:
7100:
7092:
7086:
7073:
7067:
7066:
7049:
7043:
7038:
7032:
7031:
7003:
6997:
6996:
6979:
6973:
6972:
6944:
6879:
6877:
6876:
6871:
6869:
6868:
6842:
6840:
6839:
6834:
6772:
6770:
6769:
6764:
6702:
6700:
6699:
6694:
6689:
6688:
6655:
6654:
6617:
6615:
6614:
6609:
6604:
6603:
6570:
6569:
6527:
6525:
6524:
6519:
6504:
6502:
6501:
6496:
6429:
6427:
6426:
6421:
6353:
6351:
6350:
6345:
6327:
6325:
6324:
6319:
6301:
6299:
6298:
6293:
6291:
6290:
6268:
6266:
6265:
6260:
6258:
6257:
6225:
6223:
6222:
6217:
6172:
6170:
6169:
6164:
6162:
6161:
6132:
6130:
6129:
6124:
6103:
6101:
6100:
6095:
6067:
6066:
6033:
6032:
6013:
6011:
6010:
6005:
6000:
5999:
5966:
5965:
5913:Kripke semantics
5895:
5893:
5892:
5887:
5853:
5851:
5850:
5845:
5799:
5797:
5796:
5791:
5668:
5666:
5665:
5660:
5540:
5538:
5537:
5532:
5450:
5448:
5447:
5442:
5332:
5330:
5329:
5324:
5259:
5257:
5256:
5251:
5170:
5168:
5167:
5162:
5094:
5093:
5088:
5084:
5070:
5068:
5067:
5062:
5060:
5059:
5054:
5050:
4967:
4965:
4964:
4959:
4957:
4949:
4944:
4936:
4931:
4926:
4915:
4903:
4901:
4900:
4895:
4893:
4885:
4880:
4872:
4867:
4862:
4851:
4842:
4840:
4839:
4834:
4832:
4824:
4819:
4811:
4806:
4801:
4790:
4779:
4777:
4776:
4771:
4769:
4764:
4753:
4748:
4740:
4735:
4727:
4710:
4708:
4707:
4702:
4700:
4695:
4684:
4679:
4671:
4666:
4658:
4646:
4644:
4643:
4638:
4636:
4635:
4629:
4624:
4613:
4600:
4592:
4587:
4579:
4571:
4570:
4548:
4546:
4545:
4540:
4538:
4533:
4522:
4507:
4505:
4504:
4499:
4497:
4492:
4481:
4463:
4461:
4460:
4455:
4453:
4445:
4440:
4432:
4417:
4415:
4414:
4409:
4407:
4399:
4394:
4386:
4368:
4366:
4365:
4360:
4358:
4350:
4335:
4333:
4332:
4327:
4325:
4317:
4302:
4300:
4299:
4294:
4276:
4274:
4273:
4268:
4247:
4245:
4244:
4239:
4212:
4210:
4209:
4204:
4202:
4197:
4186:
4171:
4169:
4168:
4163:
4161:
4153:
4148:
4140:
4117:
4115:
4114:
4109:
4107:
4099:
4094:
4086:
4081:
4076:
4065:
4053:
4051:
4050:
4045:
4043:
4042:
4036:
4028:
4023:
4015:
4002:
3997:
3986:
3978:
3977:
3955:
3953:
3952:
3947:
3945:
3937:
3932:
3924:
3909:
3907:
3906:
3901:
3899:
3891:
3876:
3874:
3873:
3868:
3847:
3845:
3844:
3839:
3837:
3829:
3824:
3816:
3801:
3799:
3798:
3793:
3791:
3783:
3768:
3766:
3765:
3760:
3739:
3737:
3736:
3731:
3713:
3711:
3710:
3705:
3687:
3685:
3684:
3679:
3650:
3649:
3642:
3612:
3610:
3609:
3604:
3580:
3578:
3577:
3572:
3570:
3565:
3554:
3531:
3529:
3528:
3523:
3521:
3516:
3505:
3500:
3492:
3487:
3479:
3470:
3468:
3467:
3462:
3460:
3452:
3447:
3439:
3434:
3429:
3418:
3409:
3407:
3406:
3401:
3399:
3391:
3386:
3378:
3373:
3368:
3357:
3344:
3342:
3341:
3336:
3334:
3326:
3302:
3300:
3299:
3294:
3245:
3243:
3242:
3237:
3183:
3181:
3180:
3175:
3122:
3120:
3119:
3114:
3057:
2966:
2959:
2957:
2956:
2951:
2937:
2935:
2934:
2929:
2911:
2909:
2908:
2903:
2895:
2887:
2882:
2874:
2866:
2861:
2844:
2829:
2827:
2826:
2821:
2816:
2808:
2803:
2795:
2787:
2782:
2765:
2737:
2735:
2734:
2729:
2727:
2720:
2715:
2714:
2705:
2694:
2689:
2688:
2679:
2674:
2669:
2668:
2659:
2654:
2649:
2648:
2647:
2629:
2628:
2616:
2615:
2605:
2596:
2591:
2590:
2581:
2570:
2565:
2564:
2555:
2550:
2545:
2544:
2535:
2530:
2525:
2524:
2523:
2505:
2504:
2492:
2491:
2481:
2457:
2455:
2454:
2449:
2431:
2429:
2428:
2423:
2405:
2403:
2402:
2397:
2385:
2383:
2382:
2377:
2375:
2367:
2352:
2350:
2349:
2344:
2342:
2335:
2327:
2322:
2314:
2305:
2300:
2289:
2280:
2272:
2267:
2259:
2250:
2245:
2234:
2203:
2194:
2192:
2191:
2186:
2184:
2177:
2172:
2171:
2162:
2159:
2137:
2132:
2131:
2130:
2120:
2104:
2095:
2090:
2089:
2080:
2077:
2055:
2050:
2049:
2048:
2038:
2022:
1993:
1991:
1990:
1985:
1967:
1965:
1964:
1959:
1941:
1939:
1938:
1933:
1921:
1919:
1918:
1913:
1911:
1903:
1888:
1886:
1885:
1880:
1878:
1871:
1863:
1858:
1850:
1841:
1836:
1825:
1816:
1808:
1803:
1795:
1786:
1781:
1770:
1743:
1741:
1740:
1735:
1733:
1602:
1600:
1599:
1594:
1592:
1588:
1587:
1566:
1565:
1550:
1549:
1531:
1530:
1512:
1511:
1499:
1498:
1479:
1478:
1457:
1456:
1441:
1440:
1422:
1421:
1403:
1402:
1390:
1389:
1360:
1358:
1357:
1352:
1350:
1349:
1331:
1330:
1318:
1317:
1292:
1290:
1289:
1284:
1272:
1270:
1269:
1264:
1249:
1247:
1246:
1241:
1239:
1108:
1106:
1105:
1100:
1098:
1096:
1073:
1056:
1053:
1051:
1031:
1011:
993:
991:
990:
985:
983:
981:
958:
941:
938:
936:
916:
896:
874:
872:
871:
866:
864:
807:
804:
732:
730:
729:
724:
722:
665:
662:
566:
564:
563:
558:
488:
486:
485:
480:
307:, also known as
305:De Morgan's laws
247:
240:
233:
221:De Morgan's laws
216:
209:
202:
195:
169:
161:
153:
146:
140:
133:
127:
120:
114:
107:
101:
94:
88:
81:
71:
42:
21:
8483:
8482:
8478:
8477:
8476:
8474:
8473:
8472:
8453:Boolean algebra
8443:
8442:
8441:
8436:
8363:
8342:
8326:
8291:New Foundations
8238:
8128:
8047:Cardinal number
8030:
8016:
7957:
7848:
7839:
7823:
7818:
7788:
7783:
7759:Begriffsschrift
7747:
7743:Jan Łukasiewicz
7663:Bernard Bolzano
7647:
7618:Double negation
7606:
7577:Double negation
7560:
7494:
7470:Excluded middle
7453:
7417:
7408:
7322:
7320:Classical logic
7317:
7250:
7249:
7231:
7228:
7223:
7222:
7216:
7198:
7197:
7193:
7176:Robert H. Orr.
7175:
7174:
7170:
7154:
7150:
7141:
7137:
7129:
7125:
7108:
7107:
7103:
7093:
7089:
7076:Boolean Algebra
7074:
7070:
7051:
7050:
7046:
7039:
7035:
7020:
7005:
7004:
7000:
6994:
6981:
6980:
6976:
6969:
6946:
6945:
6941:
6936:
6899:
6886:
6852:
6851:
6848:excluded middle
6776:
6775:
6706:
6705:
6621:
6620:
6536:
6535:
6510:
6509:
6438:
6437:
6366:
6365:
6330:
6329:
6307:
6306:
6271:
6270:
6238:
6237:
6178:
6177:
6142:
6141:
6109:
6108:
6022:
6021:
5932:
5931:
5921:
5857:
5856:
5815:
5814:
5677:
5676:
5549:
5548:
5459:
5458:
5372:
5371:
5265:
5264:
5192:
5191:
5177:
5081:
5076:
5075:
5047:
5042:
5041:
4999:digital circuit
4991:classical logic
4974:
4916:
4909:
4908:
4852:
4845:
4844:
4791:
4784:
4783:
4754:
4721:
4720:
4717:
4685:
4652:
4651:
4614:
4554:
4553:
4523:
4510:
4509:
4482:
4469:
4468:
4420:
4419:
4374:
4373:
4338:
4337:
4305:
4304:
4279:
4278:
4253:
4252:
4218:
4217:
4187:
4174:
4173:
4128:
4127:
4124:
4066:
4059:
4058:
3987:
3961:
3960:
3912:
3911:
3879:
3878:
3853:
3852:
3804:
3803:
3771:
3770:
3745:
3744:
3716:
3715:
3690:
3689:
3618:
3617:
3583:
3582:
3555:
3542:
3541:
3538:
3506:
3473:
3472:
3419:
3412:
3411:
3358:
3351:
3350:
3320:
3319:
3316:
3258:
3257:
3210:
3209:
3202:
3139:
3138:
3087:
3086:
3080:
3064:
3026:
2974:
2964:
2942:
2941:
2920:
2919:
2845:
2838:
2837:
2766:
2759:
2758:
2744:
2725:
2724:
2706:
2680:
2660:
2639:
2620:
2607:
2606:
2601:
2600:
2582:
2556:
2536:
2515:
2496:
2483:
2482:
2471:
2470:
2440:
2439:
2436:operator (AND),
2414:
2413:
2388:
2387:
2361:
2360:
2340:
2339:
2306:
2290:
2285:
2284:
2251:
2235:
2224:
2223:
2217:
2215:Boolean algebra
2199:
2182:
2181:
2163:
2138:
2122:
2105:
2100:
2099:
2081:
2056:
2040:
2023:
2012:
2011:
2005:
1976:
1975:
1972:operator (AND),
1950:
1949:
1924:
1923:
1897:
1896:
1876:
1875:
1842:
1826:
1821:
1820:
1787:
1771:
1760:
1759:
1753:
1731:
1730:
1696:
1678:
1677:
1643:
1619:
1618:
1612:
1590:
1589:
1579:
1557:
1541:
1522:
1503:
1490:
1481:
1480:
1470:
1448:
1432:
1413:
1394:
1381:
1366:
1365:
1341:
1322:
1309:
1304:
1303:
1301:
1275:
1274:
1255:
1254:
1237:
1236:
1205:
1184:
1183:
1152:
1125:
1124:
1110:
1074:
1057:
1032:
1012:
1005:
1004:
995:
959:
942:
917:
897:
890:
889:
862:
861:
833:
809:
808:
772:
745:
744:
720:
719:
691:
667:
666:
630:
603:
602:
588:
586:Formal notation
492:
491:
414:
413:
390:one of A or B.
301:Boolean algebra
257:Predicate logic
251:
215:Double negation
69:
28:
23:
22:
15:
12:
11:
5:
8481:
8479:
8471:
8470:
8465:
8460:
8455:
8445:
8444:
8438:
8437:
8435:
8434:
8429:
8427:Thoralf Skolem
8424:
8419:
8414:
8409:
8404:
8399:
8394:
8389:
8384:
8379:
8373:
8371:
8365:
8364:
8362:
8361:
8356:
8351:
8345:
8343:
8341:
8340:
8337:
8331:
8328:
8327:
8325:
8324:
8323:
8322:
8317:
8312:
8311:
8310:
8295:
8294:
8293:
8281:
8280:
8279:
8268:
8267:
8262:
8257:
8252:
8246:
8244:
8240:
8239:
8237:
8236:
8231:
8226:
8221:
8212:
8207:
8202:
8192:
8187:
8186:
8185:
8180:
8175:
8165:
8155:
8150:
8145:
8139:
8137:
8130:
8129:
8127:
8126:
8121:
8116:
8111:
8109:Ordinal number
8106:
8101:
8096:
8091:
8090:
8089:
8084:
8074:
8069:
8064:
8059:
8054:
8044:
8039:
8033:
8031:
8029:
8028:
8025:
8021:
8018:
8017:
8015:
8014:
8009:
8004:
7999:
7994:
7989:
7987:Disjoint union
7984:
7979:
7973:
7967:
7965:
7959:
7958:
7956:
7955:
7954:
7953:
7948:
7937:
7936:
7934:Martin's axiom
7931:
7926:
7921:
7916:
7911:
7906:
7901:
7899:Extensionality
7896:
7891:
7886:
7885:
7884:
7879:
7874:
7864:
7858:
7856:
7850:
7849:
7842:
7840:
7838:
7837:
7831:
7829:
7825:
7824:
7819:
7817:
7816:
7809:
7802:
7794:
7785:
7784:
7782:
7781:
7776:
7771:
7766:
7761:
7755:
7753:
7749:
7748:
7746:
7745:
7740:
7735:
7730:
7725:
7720:
7718:Ernst Schröder
7715:
7710:
7705:
7703:Giuseppe Peano
7700:
7695:
7690:
7685:
7680:
7675:
7670:
7665:
7659:
7657:
7653:
7652:
7649:
7648:
7646:
7645:
7640:
7635:
7630:
7625:
7620:
7614:
7612:
7608:
7607:
7605:
7604:
7599:
7594:
7589:
7584:
7579:
7574:
7568:
7566:
7559:
7558:
7553:
7548:
7543:
7538:
7533:
7528:
7523:
7518:
7513:
7508:
7502:
7500:
7496:
7495:
7493:
7492:
7487:
7482:
7477:
7472:
7467:
7461:
7459:
7455:
7454:
7452:
7451:
7446:
7441:
7436:
7431:
7425:
7423:
7419:
7418:
7411:
7409:
7407:
7406:
7401:
7396:
7391:
7386:
7381:
7376:
7371:
7366:
7361:
7359:Truth function
7356:
7351:
7346:
7341:
7336:
7330:
7328:
7324:
7323:
7318:
7316:
7315:
7308:
7301:
7293:
7287:
7286:
7276:
7266:
7247:
7227:
7226:External links
7224:
7221:
7220:
7214:
7200:Wirth, Niklaus
7191:
7188:on 2010-07-15.
7168:
7155:Jean Buridan,
7148:
7135:
7123:
7120:on 2008-03-23.
7101:
7087:
7068:
7044:
7033:
7018:
6998:
6992:
6974:
6967:
6938:
6937:
6935:
6932:
6931:
6930:
6928:Positive logic
6925:
6920:
6915:
6910:
6905:
6898:
6895:
6894:
6893:
6890:
6885:
6882:
6867:
6864:
6861:
6844:
6843:
6832:
6829:
6826:
6823:
6820:
6817:
6813:
6810:
6807:
6803:
6799:
6796:
6793:
6790:
6786:
6783:
6773:
6762:
6759:
6756:
6753:
6750:
6747:
6743:
6740:
6737:
6733:
6729:
6726:
6723:
6720:
6716:
6713:
6703:
6692:
6687:
6682:
6679:
6676:
6673:
6670:
6667:
6664:
6661:
6658:
6653:
6648:
6644:
6640:
6637:
6634:
6631:
6628:
6618:
6607:
6602:
6597:
6594:
6591:
6588:
6585:
6582:
6579:
6576:
6573:
6568:
6563:
6559:
6555:
6552:
6549:
6546:
6543:
6517:
6506:
6505:
6494:
6491:
6488:
6485:
6482:
6478:
6475:
6472:
6468:
6464:
6461:
6458:
6455:
6452:
6448:
6445:
6431:
6430:
6419:
6416:
6413:
6410:
6406:
6403:
6400:
6396:
6392:
6389:
6386:
6383:
6380:
6376:
6373:
6343:
6340:
6337:
6317:
6314:
6289:
6286:
6283:
6280:
6256:
6253:
6250:
6247:
6227:
6226:
6215:
6212:
6209:
6206:
6203:
6200:
6197:
6194:
6191:
6188:
6185:
6160:
6157:
6154:
6151:
6122:
6119:
6116:
6105:
6104:
6093:
6090:
6087:
6084:
6081:
6078:
6075:
6071:
6065:
6060:
6057:
6054:
6051:
6048:
6045:
6042:
6039:
6036:
6031:
6015:
6014:
6003:
5998:
5993:
5990:
5987:
5984:
5981:
5978:
5975:
5972:
5969:
5964:
5958:
5954:
5951:
5948:
5945:
5942:
5939:
5920:
5917:
5897:
5896:
5885:
5882:
5879:
5876:
5873:
5870:
5867:
5864:
5854:
5843:
5840:
5837:
5834:
5831:
5828:
5825:
5822:
5801:
5800:
5789:
5786:
5783:
5780:
5777:
5774:
5771:
5768:
5765:
5762:
5759:
5756:
5753:
5750:
5747:
5744:
5741:
5738:
5735:
5732:
5729:
5726:
5723:
5720:
5717:
5714:
5711:
5708:
5705:
5702:
5699:
5696:
5693:
5690:
5687:
5684:
5670:
5669:
5658:
5655:
5652:
5649:
5646:
5643:
5640:
5637:
5634:
5631:
5628:
5625:
5622:
5619:
5616:
5613:
5610:
5607:
5604:
5601:
5598:
5595:
5592:
5589:
5586:
5583:
5580:
5577:
5574:
5571:
5568:
5565:
5562:
5559:
5556:
5542:
5541:
5530:
5527:
5524:
5521:
5518:
5515:
5512:
5509:
5506:
5503:
5500:
5497:
5494:
5491:
5488:
5485:
5482:
5479:
5476:
5473:
5469:
5466:
5452:
5451:
5440:
5437:
5434:
5431:
5428:
5425:
5422:
5419:
5416:
5413:
5410:
5407:
5404:
5401:
5398:
5395:
5392:
5389:
5386:
5382:
5379:
5365:
5364:
5334:
5333:
5322:
5319:
5316:
5313:
5310:
5307:
5303:
5300:
5297:
5294:
5291:
5288:
5285:
5282:
5279:
5275:
5272:
5261:
5260:
5249:
5246:
5243:
5240:
5237:
5234:
5230:
5227:
5224:
5221:
5218:
5215:
5212:
5209:
5206:
5202:
5199:
5176:
5173:
5172:
5171:
5160:
5157:
5154:
5151:
5148:
5145:
5142:
5139:
5136:
5133:
5130:
5127:
5124:
5121:
5118:
5115:
5112:
5109:
5106:
5103:
5100:
5097:
5092:
5058:
4973:
4970:
4955:
4952:
4947:
4942:
4939:
4934:
4929:
4925:
4922:
4919:
4891:
4888:
4883:
4878:
4875:
4870:
4865:
4861:
4858:
4855:
4830:
4827:
4822:
4817:
4814:
4809:
4804:
4800:
4797:
4794:
4767:
4763:
4760:
4757:
4751:
4746:
4743:
4738:
4733:
4730:
4716:
4713:
4698:
4694:
4691:
4688:
4682:
4677:
4674:
4669:
4664:
4661:
4634:
4627:
4623:
4620:
4617:
4611:
4608:
4604:
4598:
4595:
4590:
4585:
4582:
4577:
4574:
4569:
4564:
4561:
4536:
4532:
4529:
4526:
4520:
4517:
4495:
4491:
4488:
4485:
4479:
4476:
4451:
4448:
4443:
4438:
4435:
4430:
4427:
4405:
4402:
4397:
4392:
4389:
4384:
4381:
4356:
4353:
4348:
4345:
4323:
4320:
4315:
4312:
4292:
4289:
4286:
4266:
4263:
4260:
4237:
4234:
4231:
4228:
4225:
4200:
4196:
4193:
4190:
4184:
4181:
4159:
4156:
4151:
4146:
4143:
4138:
4135:
4123:
4120:
4105:
4102:
4097:
4092:
4089:
4084:
4079:
4075:
4072:
4069:
4041:
4034:
4031:
4026:
4021:
4018:
4013:
4010:
4006:
4000:
3996:
3993:
3990:
3984:
3981:
3976:
3971:
3968:
3943:
3940:
3935:
3930:
3927:
3922:
3919:
3897:
3894:
3889:
3886:
3866:
3863:
3860:
3851:Similarly, if
3835:
3832:
3827:
3822:
3819:
3814:
3811:
3789:
3786:
3781:
3778:
3758:
3755:
3752:
3729:
3726:
3723:
3703:
3700:
3697:
3677:
3673:
3670:
3667:
3664:
3661:
3658:
3655:
3648:
3641:
3637:
3634:
3631:
3628:
3625:
3602:
3599:
3596:
3593:
3590:
3568:
3564:
3561:
3558:
3552:
3549:
3537:
3534:
3519:
3515:
3512:
3509:
3503:
3498:
3495:
3490:
3485:
3482:
3458:
3455:
3450:
3445:
3442:
3437:
3432:
3428:
3425:
3422:
3397:
3394:
3389:
3384:
3381:
3376:
3371:
3367:
3364:
3361:
3332:
3329:
3315:
3312:
3304:
3303:
3292:
3289:
3286:
3283:
3280:
3277:
3274:
3271:
3268:
3265:
3247:
3246:
3235:
3232:
3229:
3226:
3223:
3220:
3217:
3201:
3198:
3185:
3184:
3173:
3170:
3167:
3164:
3161:
3158:
3155:
3152:
3149:
3146:
3124:
3123:
3112:
3109:
3106:
3103:
3100:
3097:
3094:
3079:
3076:
3063:
3060:
3025:
3022:
3021:
3020:
3017:
3003:
3002:
2999:
2996:
2993:
2986:
2985:
2982:
2973:
2972:Text searching
2970:
2969:
2968:
2961:
2949:
2939:
2927:
2913:
2912:
2901:
2898:
2893:
2890:
2885:
2880:
2877:
2872:
2869:
2864:
2860:
2857:
2854:
2851:
2848:
2831:
2830:
2819:
2814:
2811:
2806:
2801:
2798:
2793:
2790:
2785:
2781:
2778:
2775:
2772:
2769:
2743:
2740:
2739:
2738:
2723:
2718:
2713:
2709:
2703:
2700:
2697:
2692:
2687:
2683:
2677:
2672:
2667:
2663:
2657:
2652:
2646:
2642:
2638:
2635:
2632:
2627:
2623:
2619:
2614:
2610:
2603:
2602:
2599:
2594:
2589:
2585:
2579:
2576:
2573:
2568:
2563:
2559:
2553:
2548:
2543:
2539:
2533:
2528:
2522:
2518:
2514:
2511:
2508:
2503:
2499:
2495:
2490:
2486:
2479:
2478:
2464:
2463:
2462:operator (OR).
2447:
2437:
2421:
2411:
2395:
2373:
2370:
2354:
2353:
2338:
2333:
2330:
2325:
2320:
2317:
2312:
2309:
2307:
2303:
2299:
2296:
2293:
2287:
2286:
2283:
2278:
2275:
2270:
2265:
2262:
2257:
2254:
2252:
2248:
2244:
2241:
2238:
2232:
2231:
2216:
2213:
2196:
2195:
2180:
2175:
2170:
2166:
2158:
2155:
2152:
2148:
2144:
2141:
2139:
2135:
2129:
2125:
2119:
2116:
2113:
2109:
2102:
2101:
2098:
2093:
2088:
2084:
2076:
2073:
2070:
2066:
2062:
2059:
2057:
2053:
2047:
2043:
2037:
2034:
2031:
2027:
2020:
2019:
2004:
2001:
2000:
1999:
1998:operator (OR).
1983:
1973:
1957:
1947:
1931:
1909:
1906:
1890:
1889:
1874:
1869:
1866:
1861:
1856:
1853:
1848:
1845:
1843:
1839:
1835:
1832:
1829:
1823:
1822:
1819:
1814:
1811:
1806:
1801:
1798:
1793:
1790:
1788:
1784:
1780:
1777:
1774:
1768:
1767:
1752:
1749:
1745:
1744:
1729:
1726:
1723:
1720:
1717:
1714:
1711:
1708:
1705:
1702:
1699:
1697:
1695:
1692:
1689:
1686:
1683:
1680:
1679:
1676:
1673:
1670:
1667:
1664:
1661:
1658:
1655:
1652:
1649:
1646:
1644:
1642:
1639:
1636:
1633:
1630:
1627:
1626:
1611:
1608:
1604:
1603:
1586:
1582:
1578:
1575:
1572:
1569:
1564:
1560:
1556:
1553:
1548:
1544:
1540:
1537:
1534:
1529:
1525:
1521:
1518:
1515:
1510:
1506:
1502:
1497:
1493:
1489:
1486:
1483:
1482:
1477:
1473:
1469:
1466:
1463:
1460:
1455:
1451:
1447:
1444:
1439:
1435:
1431:
1428:
1425:
1420:
1416:
1412:
1409:
1406:
1401:
1397:
1393:
1388:
1384:
1380:
1377:
1374:
1373:
1348:
1344:
1340:
1337:
1334:
1329:
1325:
1321:
1316:
1312:
1282:
1262:
1251:
1250:
1235:
1232:
1229:
1226:
1223:
1220:
1217:
1214:
1211:
1208:
1206:
1204:
1201:
1198:
1195:
1192:
1189:
1186:
1185:
1182:
1179:
1176:
1173:
1170:
1167:
1164:
1161:
1158:
1155:
1153:
1151:
1148:
1145:
1142:
1139:
1136:
1133:
1132:
1095:
1092:
1089:
1086:
1083:
1080:
1077:
1072:
1069:
1066:
1063:
1060:
1050:
1047:
1044:
1041:
1038:
1035:
1030:
1027:
1024:
1021:
1018:
1015:
1002:
980:
977:
974:
971:
968:
965:
962:
957:
954:
951:
948:
945:
935:
932:
929:
926:
923:
920:
915:
912:
909:
906:
903:
900:
887:
876:
875:
860:
857:
854:
851:
848:
845:
842:
839:
836:
834:
832:
829:
826:
823:
820:
817:
814:
811:
810:
802:
799:
796:
793:
790:
787:
784:
781:
778:
775:
773:
771:
768:
765:
762:
759:
756:
753:
752:
734:
733:
718:
715:
712:
709:
706:
703:
700:
697:
694:
692:
690:
687:
684:
681:
678:
675:
672:
669:
668:
660:
657:
654:
651:
648:
645:
642:
639:
636:
633:
631:
629:
626:
623:
620:
617:
614:
611:
610:
587:
584:
568:
567:
556:
553:
550:
547:
544:
541:
538:
535:
532:
529:
526:
523:
520:
517:
514:
511:
508:
505:
502:
499:
489:
478:
475:
472:
469:
466:
463:
460:
457:
454:
451:
448:
445:
442:
439:
436:
433:
430:
427:
424:
421:
372:
371:
368:
361:
360:
357:
346:
345:
342:
291:
290:
289:
288:
279:
267:
266:
260:
259:
253:
252:
250:
249:
242:
235:
228:
223:
218:
211:
208:Distributivity
204:
197:
189:
186:
185:
179:
178:
177:
176:
171:
148:
135:
122:
109:
96:
83:
63:
62:
56:
55:
49:
48:
26:
24:
14:
13:
10:
9:
6:
4:
3:
2:
8480:
8469:
8466:
8464:
8461:
8459:
8456:
8454:
8451:
8450:
8448:
8433:
8432:Ernst Zermelo
8430:
8428:
8425:
8423:
8420:
8418:
8417:Willard Quine
8415:
8413:
8410:
8408:
8405:
8403:
8400:
8398:
8395:
8393:
8390:
8388:
8385:
8383:
8380:
8378:
8375:
8374:
8372:
8370:
8369:Set theorists
8366:
8360:
8357:
8355:
8352:
8350:
8347:
8346:
8344:
8338:
8336:
8333:
8332:
8329:
8321:
8318:
8316:
8315:Kripke–Platek
8313:
8309:
8306:
8305:
8304:
8301:
8300:
8299:
8296:
8292:
8289:
8288:
8287:
8286:
8282:
8278:
8275:
8274:
8273:
8270:
8269:
8266:
8263:
8261:
8258:
8256:
8253:
8251:
8248:
8247:
8245:
8241:
8235:
8232:
8230:
8227:
8225:
8222:
8220:
8218:
8213:
8211:
8208:
8206:
8203:
8200:
8196:
8193:
8191:
8188:
8184:
8181:
8179:
8176:
8174:
8171:
8170:
8169:
8166:
8163:
8159:
8156:
8154:
8151:
8149:
8146:
8144:
8141:
8140:
8138:
8135:
8131:
8125:
8122:
8120:
8117:
8115:
8112:
8110:
8107:
8105:
8102:
8100:
8097:
8095:
8092:
8088:
8085:
8083:
8080:
8079:
8078:
8075:
8073:
8070:
8068:
8065:
8063:
8060:
8058:
8055:
8052:
8048:
8045:
8043:
8040:
8038:
8035:
8034:
8032:
8026:
8023:
8022:
8019:
8013:
8010:
8008:
8005:
8003:
8000:
7998:
7995:
7993:
7990:
7988:
7985:
7983:
7980:
7977:
7974:
7972:
7969:
7968:
7966:
7964:
7960:
7952:
7951:specification
7949:
7947:
7944:
7943:
7942:
7939:
7938:
7935:
7932:
7930:
7927:
7925:
7922:
7920:
7917:
7915:
7912:
7910:
7907:
7905:
7902:
7900:
7897:
7895:
7892:
7890:
7887:
7883:
7880:
7878:
7875:
7873:
7870:
7869:
7868:
7865:
7863:
7860:
7859:
7857:
7855:
7851:
7846:
7836:
7833:
7832:
7830:
7826:
7822:
7815:
7810:
7808:
7803:
7801:
7796:
7795:
7792:
7780:
7777:
7775:
7772:
7770:
7767:
7765:
7762:
7760:
7757:
7756:
7754:
7750:
7744:
7741:
7739:
7736:
7734:
7731:
7729:
7728:Alfred Tarski
7726:
7724:
7721:
7719:
7716:
7714:
7711:
7709:
7706:
7704:
7701:
7699:
7696:
7694:
7691:
7689:
7688:Gottlob Frege
7686:
7684:
7681:
7679:
7676:
7674:
7671:
7669:
7666:
7664:
7661:
7660:
7658:
7654:
7644:
7641:
7639:
7636:
7634:
7633:Biconditional
7631:
7629:
7626:
7624:
7621:
7619:
7616:
7615:
7613:
7609:
7603:
7600:
7598:
7595:
7593:
7592:Biconditional
7590:
7588:
7585:
7583:
7580:
7578:
7575:
7573:
7570:
7569:
7567:
7563:
7557:
7554:
7552:
7549:
7547:
7544:
7542:
7539:
7537:
7534:
7532:
7529:
7527:
7526:modus tollens
7524:
7522:
7519:
7517:
7516:Transposition
7514:
7512:
7509:
7507:
7504:
7503:
7501:
7497:
7491:
7488:
7486:
7483:
7481:
7478:
7476:
7473:
7471:
7468:
7466:
7463:
7462:
7460:
7456:
7450:
7447:
7445:
7442:
7440:
7437:
7435:
7434:Propositional
7432:
7430:
7427:
7426:
7424:
7420:
7415:
7405:
7402:
7400:
7397:
7395:
7392:
7390:
7389:Associativity
7387:
7385:
7382:
7380:
7377:
7375:
7372:
7370:
7367:
7365:
7362:
7360:
7357:
7355:
7352:
7350:
7347:
7345:
7342:
7340:
7337:
7335:
7332:
7331:
7329:
7325:
7321:
7314:
7309:
7307:
7302:
7300:
7295:
7294:
7291:
7284:
7280:
7277:
7274:
7270:
7267:
7262:
7261:
7256:
7253:
7248:
7244:
7240:
7239:
7234:
7230:
7229:
7225:
7217:
7215:9783540585770
7211:
7207:
7206:
7201:
7195:
7192:
7187:
7183:
7179:
7172:
7169:
7166:
7165:0-300-08425-0
7162:
7158:
7152:
7149:
7145:
7144:Summa Logicae
7139:
7136:
7133:
7127:
7124:
7119:
7115:
7111:
7105:
7102:
7099:by S. P. Bali
7098:
7097:
7091:
7088:
7085:
7084:0-486-45894-6
7081:
7077:
7072:
7069:
7065:
7061:
7057:
7056:
7048:
7045:
7042:
7037:
7034:
7029:
7025:
7021:
7015:
7011:
7010:
7002:
6999:
6995:
6989:
6985:
6978:
6975:
6970:
6968:9781315510880
6964:
6960:
6956:
6952:
6951:
6943:
6940:
6933:
6929:
6926:
6924:
6921:
6919:
6916:
6914:
6911:
6909:
6906:
6904:
6901:
6900:
6896:
6891:
6888:
6887:
6883:
6881:
6849:
6830:
6824:
6818:
6811:
6794:
6788:
6784:
6774:
6760:
6754:
6748:
6741:
6724:
6718:
6714:
6704:
6690:
6677:
6668:
6662:
6635:
6632:
6629:
6619:
6605:
6592:
6583:
6577:
6550:
6547:
6544:
6534:
6533:
6532:
6529:
6515:
6492:
6486:
6480:
6476:
6459:
6453:
6446:
6436:
6435:
6434:
6414:
6408:
6404:
6387:
6381:
6374:
6364:
6363:
6362:
6359:
6357:
6356:minimal logic
6341:
6335:
6315:
6303:
6236:
6232:
6213:
6207:
6195:
6189:
6176:
6175:
6174:
6140:
6136:
6120:
6117:
6114:
6091:
6085:
6082:
6079:
6055:
6046:
6040:
6020:
6019:
6018:
6001:
5988:
5979:
5973:
5949:
5946:
5943:
5930:
5929:
5928:
5926:
5918:
5916:
5914:
5910:
5906:
5902:
5883:
5880:
5874:
5868:
5865:
5862:
5855:
5841:
5838:
5832:
5826:
5823:
5820:
5813:
5812:
5811:
5809:
5804:
5787:
5778:
5772:
5766:
5760:
5754:
5748:
5742:
5736:
5724:
5718:
5712:
5709:
5703:
5697:
5694:
5688:
5682:
5675:
5674:
5673:
5650:
5644:
5638:
5632:
5626:
5620:
5614:
5608:
5596:
5590:
5584:
5581:
5575:
5569:
5566:
5560:
5554:
5547:
5546:
5545:
5528:
5522:
5516:
5513:
5507:
5501:
5498:
5492:
5486:
5483:
5477:
5471:
5467:
5457:
5456:
5455:
5435:
5429:
5426:
5420:
5414:
5411:
5405:
5399:
5396:
5390:
5384:
5380:
5370:
5369:
5368:
5362:
5358:
5354:
5350:
5347:
5346:
5345:
5343:
5339:
5314:
5308:
5301:
5289:
5283:
5277:
5273:
5263:
5262:
5241:
5235:
5228:
5216:
5210:
5204:
5200:
5190:
5189:
5188:
5186:
5182:
5174:
5158:
5152:
5149:
5146:
5140:
5137:
5128:
5122:
5116:
5113:
5110:
5107:
5104:
5101:
5098:
5090:
5074:
5073:
5072:
5056:
5039:
5035:
5031:
5027:
5022:
5020:
5016:
5012:
5008:
5004:
5000:
4996:
4992:
4983:
4978:
4971:
4969:
4950:
4945:
4937:
4932:
4923:
4920:
4917:
4905:
4886:
4881:
4873:
4868:
4859:
4856:
4853:
4825:
4820:
4812:
4807:
4798:
4795:
4792:
4782:
4761:
4758:
4755:
4749:
4741:
4736:
4728:
4714:
4712:
4692:
4689:
4686:
4680:
4672:
4667:
4659:
4648:
4621:
4618:
4615:
4609:
4606:
4593:
4588:
4580:
4575:
4572:
4562:
4550:
4530:
4527:
4524:
4518:
4515:
4489:
4486:
4483:
4477:
4474:
4465:
4446:
4441:
4433:
4428:
4425:
4400:
4395:
4387:
4382:
4379:
4370:
4351:
4346:
4343:
4318:
4313:
4310:
4290:
4287:
4284:
4264:
4261:
4258:
4249:
4235:
4232:
4229:
4226:
4223:
4214:
4194:
4191:
4188:
4182:
4179:
4154:
4149:
4141:
4136:
4133:
4121:
4119:
4100:
4095:
4087:
4082:
4073:
4070:
4067:
4055:
4029:
4024:
4016:
4011:
4008:
3994:
3991:
3988:
3982:
3979:
3969:
3957:
3938:
3933:
3925:
3920:
3917:
3892:
3887:
3884:
3864:
3861:
3858:
3849:
3830:
3825:
3817:
3812:
3809:
3784:
3779:
3776:
3756:
3753:
3750:
3741:
3727:
3724:
3721:
3701:
3698:
3695:
3671:
3668:
3665:
3662:
3659:
3656:
3653:
3639:
3632:
3629:
3626:
3623:
3614:
3600:
3597:
3594:
3591:
3588:
3562:
3559:
3556:
3550:
3547:
3535:
3533:
3513:
3510:
3507:
3501:
3493:
3488:
3480:
3453:
3448:
3440:
3435:
3426:
3423:
3420:
3392:
3387:
3379:
3374:
3365:
3362:
3359:
3348:
3327:
3313:
3311:
3307:
3290:
3284:
3275:
3269:
3256:
3255:
3254:
3252:
3233:
3227:
3224:
3221:
3208:
3207:
3206:
3199:
3197:
3193:
3190:
3171:
3165:
3156:
3150:
3137:
3136:
3135:
3133:
3129:
3110:
3104:
3101:
3098:
3085:
3084:
3083:
3077:
3075:
3073:
3069:
3061:
3059:
3056:
3051:
3047:
3043:
3039:
3035:
3031:
3023:
3018:
3015:
3014:
3013:
3010:
3006:
3000:
2997:
2994:
2991:
2990:
2989:
2983:
2980:
2979:
2978:
2971:
2962:
2947:
2940:
2925:
2918:
2917:
2916:
2899:
2888:
2883:
2875:
2867:
2855:
2852:
2849:
2836:
2835:
2834:
2809:
2804:
2796:
2788:
2776:
2773:
2770:
2757:
2756:
2755:
2753:
2749:
2741:
2721:
2711:
2707:
2701:
2698:
2695:
2685:
2681:
2675:
2665:
2661:
2655:
2644:
2640:
2636:
2633:
2630:
2625:
2621:
2617:
2612:
2608:
2597:
2587:
2583:
2577:
2574:
2571:
2561:
2557:
2551:
2541:
2537:
2531:
2520:
2516:
2512:
2509:
2506:
2501:
2497:
2493:
2488:
2484:
2469:
2468:
2467:
2461:
2445:
2438:
2435:
2419:
2412:
2409:
2393:
2368:
2359:
2358:
2357:
2336:
2328:
2323:
2315:
2310:
2308:
2297:
2294:
2291:
2281:
2273:
2268:
2260:
2255:
2253:
2242:
2239:
2236:
2222:
2221:
2220:
2214:
2212:
2210:
2205:
2202:
2178:
2168:
2164:
2156:
2153:
2150:
2146:
2142:
2140:
2127:
2123:
2117:
2114:
2111:
2107:
2096:
2086:
2082:
2074:
2071:
2068:
2064:
2060:
2058:
2045:
2041:
2035:
2032:
2029:
2025:
2010:
2009:
2008:
2002:
1997:
1981:
1974:
1971:
1955:
1948:
1945:
1929:
1904:
1895:
1894:
1893:
1872:
1864:
1859:
1851:
1846:
1844:
1833:
1830:
1827:
1817:
1809:
1804:
1796:
1791:
1789:
1778:
1775:
1772:
1758:
1757:
1756:
1750:
1748:
1727:
1721:
1715:
1712:
1698:
1690:
1687:
1684:
1674:
1668:
1662:
1659:
1645:
1637:
1634:
1631:
1617:
1616:
1615:
1609:
1607:
1584:
1580:
1573:
1570:
1567:
1562:
1558:
1551:
1546:
1542:
1527:
1523:
1519:
1516:
1513:
1508:
1504:
1500:
1495:
1491:
1475:
1471:
1464:
1461:
1458:
1453:
1449:
1442:
1437:
1433:
1418:
1414:
1410:
1407:
1404:
1399:
1395:
1391:
1386:
1382:
1364:
1363:
1362:
1346:
1342:
1338:
1335:
1332:
1327:
1323:
1319:
1314:
1310:
1299:
1294:
1280:
1260:
1233:
1227:
1221:
1218:
1207:
1199:
1196:
1193:
1180:
1174:
1168:
1165:
1154:
1146:
1143:
1140:
1123:
1122:
1121:
1119:
1115:
1109:
1090:
1087:
1084:
1075:
1070:
1064:
1061:
1048:
1042:
1039:
1033:
1025:
1022:
1019:
1001:
1000:
994:
975:
972:
969:
960:
955:
949:
946:
933:
927:
924:
918:
910:
907:
904:
886:
885:
881:
858:
852:
849:
846:
837:
835:
827:
821:
818:
800:
794:
788:
785:
776:
774:
766:
763:
760:
743:
742:
741:
739:
716:
710:
707:
704:
695:
693:
685:
679:
676:
658:
652:
646:
643:
634:
632:
624:
621:
618:
601:
600:
599:
597:
593:
585:
583:
581:
577:
573:
554:
548:
545:
542:
536:
530:
527:
524:
518:
512:
509:
506:
500:
497:
490:
476:
470:
467:
464:
458:
452:
449:
446:
440:
434:
431:
428:
422:
419:
412:
411:
410:
403:
395:
391:
389:
386:" that means
385:
381:
377:
369:
366:
365:
364:
358:
355:
351:
350:
349:
343:
340:
339:
338:
335:
333:
329:
325:
321:
317:
314:
310:
306:
302:
298:
287:
286:instantiation
283:
280:
278:
277:instantiation
274:
271:
270:
269:
268:
265:
261:
258:
254:
248:
243:
241:
236:
234:
229:
227:
226:Transposition
224:
222:
219:
217:
212:
210:
205:
203:
201:Commutativity
198:
196:
194:Associativity
191:
190:
188:
187:
184:
180:
175:
172:
170:
168:
162:
160:
159:modus tollens
154:
149:
147:
141:
136:
134:
128:
123:
121:
115:
110:
108:
102:
97:
95:
89:
84:
82:
79:
76:elimination (
72:
67:
66:
65:
64:
61:
57:
54:
50:
47:
43:
37:
36:Venn diagrams
32:
19:
8382:Georg Cantor
8377:Paul Bernays
8308:Morse–Kelley
8283:
8216:
8215:Subset
8162:hereditarily
8124:Venn diagram
8082:ordered pair
7997:Intersection
7981:
7941:Axiom schema
7698:Hugh MacColl
7673:Georg Cantor
7668:George Boole
7565:Introduction
7521:modus ponens
7505:
7449:Higher-order
7444:Second-order
7394:Distribution
7354:Truth tables
7282:
7258:
7236:
7204:
7194:
7186:the original
7171:
7156:
7151:
7143:
7138:
7131:
7130:Bocheński's
7126:
7118:the original
7104:
7095:
7090:
7075:
7071:
7054:
7047:
7036:
7008:
7001:
6983:
6977:
6949:
6942:
6845:
6530:
6507:
6432:
6360:
6304:
6228:
6106:
6016:
5922:
5898:
5805:
5802:
5671:
5543:
5453:
5366:
5360:
5356:
5352:
5348:
5341:
5335:
5178:
5037:
5033:
5029:
5025:
5023:
4987:
4906:
4780:
4718:
4649:
4551:
4466:
4371:
4250:
4215:
4125:
4056:
3958:
3850:
3742:
3615:
3539:
3318:Here we use
3317:
3308:
3305:
3248:
3203:
3194:
3188:
3186:
3127:
3125:
3081:
3065:
3050:Jean Buridan
3038:George Boole
3027:
3011:
3007:
3004:
2987:
2975:
2914:
2832:
2745:
2465:
2355:
2218:
2206:
2200:
2197:
2006:
1970:intersection
1891:
1754:
1746:
1613:
1605:
1297:
1295:
1252:
1111:
1003:
998:
996:
888:
883:
877:
737:
735:
591:
589:
569:
408:
387:
384:exclusive or
379:
376:inclusive or
373:
362:
347:
336:
328:disjunctions
324:conjunctions
308:
304:
294:
284: /
275: /
220:
166:
163: /
158:
155: /
142: /
139:Constructive
129: /
116: /
103: /
90: /
78:modus ponens
77:
73: /
8407:Thomas Jech
8250:Alternative
8229:Uncountable
8183:Ultrafilter
8042:Cardinality
7946:replacement
7894:Determinacy
7643:Disjunction
7638:Conjunction
7623:Existential
7611:Elimination
7602:Disjunction
7597:Conjunction
7582:Existential
7439:First-order
7364:Truth value
7334:Quantifiers
6913:Isomorphism
5808:modal logic
5187:are duals:
5071:defined by
5003:logic gates
4303:, and thus
3072:conjunction
3068:disjunction
2742:Engineering
1114:tautologies
572:expressions
239:Exportation
126:Disjunctive
119:elimination
106:elimination
93:elimination
8447:Categories
8402:Kurt Gödel
8387:Paul Cohen
8224:Transitive
7992:Identities
7976:Complement
7963:Operations
7924:Regularity
7862:Adjunction
7821:Set theory
7693:Kurt Gödel
7556:Absorption
7458:Principles
7344:Connective
7273:PlanetMath
6934:References
5344:, such as
4984:diagrams).
4715:Conclusion
2748:electrical
1751:Set theory
598:notation:
378:" meaning
354:complement
152:Absorption
8335:Paradoxes
8255:Axiomatic
8234:Universal
8210:Singleton
8205:Recursive
8148:Countable
8143:Amorphous
8002:Power set
7919:Power set
7877:dependent
7872:countable
7628:Universal
7587:Universal
7490:Explosion
7475:Bivalence
7404:Soundness
7349:Tautology
7339:Predicate
7260:MathWorld
7243:EMS Press
7028:689858599
6816:¬
6809:∀
6806:¬
6802:→
6782:∃
6746:¬
6739:∃
6736:¬
6732:→
6712:∀
6675:¬
6669:∨
6660:¬
6647:¬
6643:→
6633:∧
6590:¬
6584:∧
6575:¬
6562:¬
6558:→
6548:∨
6474:∀
6471:¬
6467:→
6451:¬
6444:∃
6402:∃
6399:¬
6395:↔
6379:¬
6372:∀
6339:→
6313:¬
6205:¬
6199:¬
6196:∨
6187:¬
6135:conjuncts
6118:∧
6083:∧
6074:¬
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613:¬
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528:−
510:∩
501:−
468:−
459:∩
450:−
432:∪
423:−
246:Tautology
8339:Problems
8243:Theories
8219:Superset
8195:Infinite
8024:Concepts
7904:Infinity
7828:Overview
7572:Negation
7399:Validity
7379:Logicism
7202:(1995),
6897:See also
4478:∉
4383:∉
4347:∉
4314:∉
4183:∉
3862:∉
3754:∉
3725:∉
3699:∉
3616:Because
3592:∉
3581:. Then,
2408:overline
2209:mnemonic
1944:overline
1118:theorems
380:at least
332:negation
8277:General
8272:Zermelo
8178:subbase
8160: (
8099:Forcing
8077:Element
8049: (
8027:Methods
7914:Pairing
7327:General
7245:, 2001
4843:, then
4552:Hence,
3877:, then
3769:, then
3196:claim.
3128:neither
3024:History
2965:overbar
2915:where:
2458:is the
2432:is the
2356:where:
1994:is the
1968:is the
1892:where:
596:sequent
388:exactly
8168:Filter
8158:Finite
8094:Family
8037:Almost
7882:global
7867:Choice
7854:Axioms
7656:People
7212:
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4122:Part 2
3959:Thus,
3651:
3643:
3536:Part 1
2406:, the
2198:where
1942:, the
1253:where
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8190:Fuzzy
8153:Empty
8136:types
8087:tuple
8057:Class
8051:large
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7929:Union
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7499:Rules
5367:Then
5338:model
3910:, so
3802:, so
1996:union
313:valid
8173:base
7429:Term
7210:ISBN
7161:ISBN
7080:ISBN
7024:OCLC
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6988:ISBN
6963:ISBN
6433:and
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5009:and
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1296:The
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997:and
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590:The
352:The
326:and
299:and
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7271:at
7060:doi
6955:doi
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