391:
4954:
383:
2725:
7822:
1590:
2461:
3035:, 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.
20:
7391:
2182:
1356:
2340:
2720:{\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}}}
1876:
1731:
1237:
3286:
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
2985:
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
3172:
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
4965:
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
862:
720:
2002:
1585:{\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}}}
1096:
981:
4622:
4029:
2953:
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:
2214:
1750:
1609:
5775:
5644:
5989:
1115:
3181:
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:
4818:
4755:
4686:
4093:
3507:
3446:
735:
593:
2817:
6678:
6593:
6079:
2887:
6405:
2177:{\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}}}
4943:
4879:
3385:
5146:
3168:
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".
6818:
6748:
6480:
5426:
5516:
4393:
4974:: 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
995:
880:
3663:
4439:
4147:
3931:
3823:
6114:. 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
1603:
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:
2466:
2219:
2007:
1755:
1614:
1361:
1120:
740:
598:
5871:
5829:
2982:
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.
4483:
4188:
554:
476:
6201:
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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,
3159:
1348:
3278:
7158:
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6244:
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3588:
2335:{\displaystyle {\begin{aligned}{\overline {A\land B}}&={\overline {A}}\lor {\overline {B}},\\{\overline {A\lor B}}&={\overline {A}}\land {\overline {B}},\end{aligned}}}
6855:
3221:
6869:
In modern programming languages, due to the optimisation of compilers and interpreters, the performance differences between these options are negligible or completely absent.
3320:
3098:
2373:
1909:
1871:{\displaystyle {\begin{aligned}{\overline {A\cup B}}&={\overline {A}}\cap {\overline {B}},\\{\overline {A\cap B}}&={\overline {A}}\cup {\overline {B}},\end{aligned}}}
5046:
4223:
3852:
3744:
3715:
3689:
1726:{\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}}}
8279:
6108:
6329:
6303:
4278:
4252:
2913:
2419:
2445:
1981:
1955:
5655:
1232:{\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}}}
5527:
6503:
2935:
2393:
1929:
1280:
1260:
5910:
7086:
6925:
857:{\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}}}
715:{\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}}}
4762:
4699:
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4958:
2749:
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219:
6599:
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6000:
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6344:
3059:
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:
7154:
4887:
4823:
3329:
6994:
6968:
7968:
7788:
6899:
5054:
3283:
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".
8296:
6754:
6684:
6416:
1744:
In set theory, it is often stated as "union and intersection interchange under complementation", which can be formally expressed as:
1091:{\displaystyle {\frac {\neg (P\lor Q)}{\therefore \neg P\land \neg Q}}\qquad {\frac {\neg P\land \neg Q}{\therefore \neg (P\lor Q)}}}
976:{\displaystyle {\frac {\neg (P\land Q)}{\therefore \neg P\lor \neg Q}}\qquad {\frac {\neg P\lor \neg Q}{\therefore \neg (P\land Q)}}}
7190:
7141:
7090:
7060:
6943:
5350:
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5437:
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6211:
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6084:
The converse of the last implication does not hold in pure intuitionistic logic. That is, the failure of the joint proposition
3596:
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4106:
3890:
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3230:
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,
7755:
7360:
7315:
6894:
1736:
This emphasizes the need to invert both the inputs and the output, as well as change the operator when doing a substitution.
8048:
7927:
7745:
7280:
7219:
6866:
De Morgan's laws are widely used in computer engineering and digital logic for the purpose of simplifying circuit designs.
560:
390:
233:
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8284:
8439:
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7214:
4447:
4152:
4617:{\displaystyle \forall x{\Big (}x\in {\overline {A}}\cup {\overline {B}}\implies x\in {\overline {A\cap B}}{\Big )}}
4024:{\displaystyle \forall x{\Big (}x\in {\overline {A\cap B}}\implies x\in {\overline {A}}\cup {\overline {B}}{\Big )}}
2989:
A similar evaluation can be applied to show that the following two searches will both return
Documents 1, 2, and 4:
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7558:
7461:
270:
74:
482:
404:
8434:
7973:
7865:
7853:
7848:
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1958:
274:
80:
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8429:
7781:
7578:
7573:
7563:
7273:
7136:. Trans. Gyula Klima. New Haven: Yale University Press, 2001. See especially Treatise 1, Chapter 7, Section 5.
5888:, the relationship of these modal operators to the quantification can be understood by setting up models using
4316:
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261:
100:
87:
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1294:
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265:
106:
93:
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41:
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214:
195:
162:
153:
113:
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6249:
6216:
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4953:
3561:
7017:
6830:
3188:
3017:, which later cemented De Morgan's claim to the find. Nevertheless, a similar observation was made by
382:
8335:
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8159:
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7983:
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226:
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1984:
308:
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1102:
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304:
252:
245:
57:
48:
34:
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6931:
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6285:
5889:
5770:{\displaystyle P(a)\lor P(b)\lor P(c)\equiv \neg (\neg P(a)\land \neg P(b)\land \neg P(c)),}
4257:
4231:
2898:
2404:
1595:
These laws generalize De Morgan’s original laws for negating conjunctions and disjunctions.
564:
348:
The complement of the intersection of two sets is the same as the union of their complements
320:
301:
5639:{\displaystyle P(a)\land P(b)\land P(c)\equiv \neg (\neg P(a)\lor \neg P(b)\lor \neg P(c))}
2430:
1966:
1940:
8267:
8205:
8023:
7843:
7735:
7639:
7296:
6824:
4975:
4967:
2965:
The corpus of documents containing "cats" or "dogs" can be represented by four documents:
1289:
provide an equivalence for negating a conjunction or disjunction involving multiple terms.
567:
and digital circuit designs. De Morgan's laws are an example of a more general concept of
289:
202:
5984:{\displaystyle \neg (P\lor Q)\,\leftrightarrow \,{\big (}(\neg P)\land (\neg Q){\big )},}
4990:
of a formula. Computer programmers use them to simplify or properly negate complicated
3287:
thus results in a true expression, and this expression is identical to the first claim.
8403:
8200:
8181:
8085:
8070:
8027:
7963:
7905:
7679:
7335:
6904:
6488:
6331:
for some arbitrary constant predicate C, meaning that the above laws are still true in
6210:. For a refined version of the failing law concerning existential statements, see the
2920:
2378:
1914:
1265:
1245:
8423:
8408:
8378:
8210:
8124:
8119:
7704:
7669:
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7502:
7176:
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6332:
145:
24:
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8171:
8100:
8058:
7917:
7821:
7674:
7649:
7644:
7497:
7231:
6485:
are tautologies even in minimal logic with negation replaced with implying a fixed
5314:
3227:
3026:
3021:, and was known to Greek and Medieval logicians. For example, in the 14th century,
3014:
372:
364:
63:
4813:{\displaystyle {\overline {A\cap B}}\subseteq {\overline {A}}\cup {\overline {B}}}
4750:{\displaystyle {\overline {A}}\cup {\overline {B}}\subseteq {\overline {A\cap B}}}
4681:{\displaystyle {\overline {A}}\cup {\overline {B}}\subseteq {\overline {A\cap B}}}
4088:{\displaystyle {\overline {A\cap B}}\subseteq {\overline {A}}\cup {\overline {B}}}
3502:{\displaystyle {\overline {A}}\cup {\overline {B}}\subseteq {\overline {A\cap B}}}
3441:{\displaystyle {\overline {A\cap B}}\subseteq {\overline {A}}\cup {\overline {B}}}
3013:. De Morgan's formulation was influenced by algebraization of logic undertaken by
8383:
8018:
7340:
7330:
6889:
5784:
3108:
3044:
2812:{\displaystyle {\overline {(A\cdot B)}}\equiv ({\overline {A}}+{\overline {B}})}
27:. In each case, the resultant set is the set of all points in any shade of blue.
8363:
8231:
8134:
7797:
7405:
7249:
7182:
Digital
Circuit Design for Computer Science Students: An Introductory Textbook
4979:
398:
Another form of De Morgan's law is the following as seen in the right figure.
7004:
6673:{\displaystyle (P\land Q)\,\to \,\neg {\big (}(\neg P)\lor (\neg Q){\big )},}
6588:{\displaystyle (P\lor Q)\,\to \,\neg {\big (}(\neg P)\land (\neg Q){\big )},}
6074:{\displaystyle {\big (}(\neg P)\lor (\neg Q){\big )}\,\to \,\neg (P\land Q).}
2882:{\displaystyle {\overline {A+B}}\equiv {\overline {A}}\cdot {\overline {B}},}
345:
of the union of two sets is the same as the intersection of their complements
8166:
8129:
8080:
7978:
7380:
7236:
6400:{\displaystyle \forall x\,\neg P(x)\,\leftrightarrow \,\neg \exists x\,P(x)}
5881:
3018:
2208:
In
Boolean algebra, similarly, this law which can be formally expressed as:
133:
6984:
6935:
6505:, while the converse of the last law does not have to be true in general.
6282:
The validity of the other three De Morgan's laws remains true if negation
311:, a 19th-century British mathematician. The rules allow the expression of
19:
7355:
4938:{\displaystyle {\overline {A\cup B}}={\overline {A}}\cap {\overline {B}}}
4874:{\displaystyle {\overline {A\cap B}}={\overline {A}}\cup {\overline {B}}}
3380:{\displaystyle {\overline {A\cap B}}={\overline {A}}\cup {\overline {B}}}
2396:
2197:
1932:
386:
The equivalency of ¬φ ∨ ¬ψ and ¬(φ ∧ ψ) is displayed in this truth table.
7032:
In Quest of
Univeral Logic: A brief overview of formal logic's evolution
7245:
5787:, relating the box ("necessarily") and diamond ("possibly") operators:
1106:
584:
127:
3009:(1806–1871), who introduced a formal version of the laws to classical
8191:
8013:
5313:
To relate these quantifier dualities to the De Morgan laws, set up a
5141:{\displaystyle {\mbox{P}}^{d}(p,q,...)=\neg P(\neg p,\neg q,\dots ).}
107:
7255:
6110:
cannot necessarily be resolved to the failure of either of the two
5156:
This duality can be generalised to quantifiers, so for example the
2193:
is some, possibly countably or uncountably infinite, indexing set.
8063:
7830:
7265:
4952:
389:
381:
18:
6813:{\displaystyle \exists x\,P(x)\,\to \,\neg \forall x\,\neg P(x),}
6743:{\displaystyle \forall x\,P(x)\,\to \,\neg \exists x\,\neg P(x),}
6475:{\displaystyle \exists x\,\neg P(x)\,\to \,\neg \forall x\,P(x).}
7390:
3025:
wrote down the words that would result by reading the laws out.
7770:
7269:
5900:
Three out of the four implications of de Morgan's laws hold in
5421:{\displaystyle \forall x\,P(x)\equiv P(a)\land P(b)\land P(c)}
2196:
In set notation, De Morgan's laws can be remembered using the
7766:
5511:{\displaystyle \exists x\,P(x)\equiv P(a)\lor P(b)\lor P(c).}
4388:{\displaystyle x\not \in {\overline {A}}\cup {\overline {B}}}
3107:
A nor B is true, then it must follow that both A is not true
4957:
De Morgan's Laws represented as a circuit with logic gates (
559:
Applications of the rules include simplification of logical
4994:. They are also often useful in computations in elementary
196:
154:
120:
75:
5783:
Then, the quantifier dualities can be extended further to
333:
The negation of "A or B" is the same as "not A and not B."
330:
The negation of "A and B" is the same as "not A or not B."
6989:. Richard Parker (10th ed.). New York: McGraw-Hill.
3658:{\displaystyle A\cap B=\{\,y\ |\ y\in A\wedge y\in B\,\}}
146:
5001:
Let one define the dual of any propositional operator P(
4434:{\displaystyle x\in {\overline {A}}\cup {\overline {B}}}
4142:{\displaystyle x\in {\overline {A}}\cup {\overline {B}}}
3926:{\displaystyle x\in {\overline {A}}\cup {\overline {B}}}
3818:{\displaystyle x\in {\overline {A}}\cup {\overline {B}}}
3043:
De Morgan's theorem may be applied to the negation of a
114:
6924:
Copi, Irving M.; Cohen, Carl; McMahon, Kenneth (2016).
227:
182:
88:
64:
5060:
5026:
4982:, and in formal logic, where it is needed to find the
6833:
6757:
6687:
6602:
6517:
6491:
6419:
6347:
6311:
6288:
6252:
6219:
6159:
6123:
6090:
6003:
5913:
5838:
5796:
5658:
5530:
5440:
5353:
5246:
5173:
5057:
5023:
4890:
4826:
4765:
4702:
4633:
4535:
4491:
4450:
4401:
4355:
4319:
4286:
4260:
4234:
4199:
4155:
4109:
4040:
3942:
3893:
3860:
3834:
3785:
3752:
3726:
3697:
3671:
3599:
3564:
3523:
3454:
3393:
3332:
3301:
3239:
3191:
3120:
3068:
2944:
is the logical NOT of what is underneath the overbar.
2923:
2901:
2831:
2752:
2464:
2433:
2407:
2381:
2354:
2217:
2005:
1969:
1943:
1917:
1890:
1753:
1612:
1359:
1297:
1268:
1248:
1118:
998:
883:
738:
596:
485:
407:
4978:
design, where it is used to manipulate the types of
8344:
8307:
8219:
8109:
7997:
7938:
7829:
7804:
7728:
7632:
7587:
7541:
7475:
7434:
7398:
7303:
3323:
1350:, the generalized De Morgan’s Laws are as follows:
300:, are a pair of transformation rules that are both
6849:
6812:
6742:
6672:
6587:
6497:
6474:
6399:
6323:
6297:
6271:
6238:
6206:This weak form can be used as a foundation for an
6195:
6142:
6102:
6073:
5983:
5866:{\displaystyle \Diamond p\equiv \neg \Box \neg p.}
5865:
5824:{\displaystyle \Box p\equiv \neg \Diamond \neg p,}
5823:
5769:
5638:
5510:
5420:
5302:
5229:
5140:
5040:
4937:
4873:
4812:
4749:
4680:
4616:
4518:
4477:
4433:
4387:
4338:
4305:
4272:
4246:
4217:
4182:
4141:
4087:
4023:
3925:
3879:
3846:
3817:
3771:
3738:
3709:
3683:
3657:
3582:
3550:
3501:
3440:
3379:
3314:
3272:
3215:
3153:
3092:
2929:
2907:
2881:
2811:
2719:
2439:
2413:
2387:
2367:
2334:
2176:
1975:
1949:
1923:
1903:
1870:
1725:
1584:
1342:
1282:are propositions expressed in some formal system.
1274:
1254:
1231:
1090:
975:
856:
714:
548:
470:
7159:Indiana University–Purdue University Indianapolis
5780:verifying the quantifier dualities in the model.
5317:with some small number of elements in its domain
4609:
4544:
4016:
3951:
3111:B is not true, which may be written directly as:
140:
4193:Under that assumption, it must be the case that
220:
4881:; this concludes the proof of De Morgan's law.
4478:{\displaystyle x\not \in {\overline {A\cap B}}}
4183:{\displaystyle x\not \in {\overline {A\cap B}}}
2978:Document 4: Contains neither "cats" nor "dogs".
394:De Morgan's law with set subtraction operation.
189:
81:
1992:Unions and intersections of any number of sets
7782:
7281:
6662:
6628:
6577:
6543:
6338:Similarly to the above, the quantifier laws:
6040:
6006:
5973:
5939:
234:
203:
101:
94:
8:
5009:, ...) depending on elementary propositions
3652:
3612:
2975:Document 3: Contains both "cats" and "dogs".
2743:, De Morgan's laws are commonly written as:
2399:being written above the terms to be negated,
1935:being written above the terms to be negated,
549:{\displaystyle A-(B\cap C)=(A-B)\cup (A-C).}
471:{\displaystyle A-(B\cup C)=(A-B)\cap (A-C),}
7073:2000 Solved Problems in Digital Electronics
6196:{\displaystyle (\neg P)\lor \neg (\neg P).}
5303:{\displaystyle \exists x\,P(x)\equiv \neg }
5230:{\displaystyle \forall x\,P(x)\equiv \neg }
3322:to denote the complement of A, as above in
3030:
7789:
7775:
7767:
7538:
7288:
7274:
7266:
4582:
4578:
4519:{\displaystyle x\in {\overline {A\cap B}}}
4395:, in contradiction to the hypothesis that
3984:
3980:
3551:{\displaystyle x\in {\overline {A\cap B}}}
2969:Document 1: Contains only the word "cats".
326:The rules can be expressed in English as:
30:
6835:
6834:
6832:
6791:
6781:
6777:
6764:
6756:
6721:
6711:
6707:
6694:
6686:
6661:
6660:
6627:
6626:
6622:
6618:
6601:
6576:
6575:
6542:
6541:
6537:
6533:
6516:
6490:
6456:
6446:
6442:
6426:
6418:
6384:
6374:
6370:
6354:
6346:
6310:
6287:
6254:
6253:
6251:
6221:
6220:
6218:
6158:
6125:
6124:
6122:
6089:
6049:
6045:
6039:
6038:
6005:
6004:
6002:
5972:
5971:
5938:
5937:
5936:
5932:
5912:
5837:
5795:
5657:
5529:
5447:
5439:
5360:
5352:
5281:
5253:
5245:
5208:
5180:
5172:
5066:
5059:
5056:
5032:
5025:
5022:
4959:International Electrotechnical Commission
4925:
4912:
4891:
4889:
4861:
4848:
4827:
4825:
4800:
4787:
4766:
4764:
4729:
4716:
4703:
4701:
4660:
4647:
4634:
4632:
4608:
4607:
4589:
4568:
4555:
4543:
4542:
4534:
4498:
4490:
4457:
4449:
4421:
4408:
4400:
4375:
4362:
4354:
4339:{\displaystyle x\not \in {\overline {B}}}
4326:
4318:
4306:{\displaystyle x\not \in {\overline {A}}}
4293:
4285:
4259:
4233:
4198:
4162:
4154:
4129:
4116:
4108:
4075:
4062:
4041:
4039:
4015:
4014:
4004:
3991:
3962:
3950:
3949:
3941:
3913:
3900:
3892:
3867:
3859:
3833:
3805:
3792:
3784:
3759:
3751:
3725:
3696:
3670:
3651:
3622:
3615:
3598:
3563:
3530:
3522:
3481:
3468:
3455:
3453:
3428:
3415:
3394:
3392:
3367:
3354:
3333:
3331:
3302:
3300:
3238:
3190:
3119:
3067:
2986:these two searches, which is Document 4.
2922:
2900:
2866:
2853:
2832:
2830:
2796:
2783:
2753:
2751:
2699:
2693:
2673:
2667:
2653:
2647:
2632:
2613:
2600:
2593:
2575:
2569:
2549:
2543:
2529:
2523:
2508:
2489:
2476:
2469:
2465:
2463:
2432:
2406:
2380:
2355:
2353:
2315:
2302:
2277:
2260:
2247:
2222:
2218:
2216:
2156:
2150:
2138:
2115:
2099:
2092:
2074:
2068:
2056:
2033:
2017:
2010:
2006:
2004:
1968:
1942:
1916:
1891:
1889:
1851:
1838:
1813:
1796:
1783:
1758:
1754:
1752:
1613:
1611:
1572:
1550:
1534:
1515:
1496:
1483:
1463:
1441:
1425:
1406:
1387:
1374:
1360:
1358:
1334:
1315:
1302:
1296:
1267:
1247:
1119:
1117:
1044:
999:
997:
929:
884:
882:
792:
739:
737:
650:
597:
595:
484:
406:
3387:is completed in 2 steps by proving both
3154:{\displaystyle (\neg A)\wedge (\neg B).}
1343:{\displaystyle P_{1},P_{2},\dots ,P_{n}}
6916:
6212:lesser limited principle of omniscience
5884:observed this case, and in the case of
251:
244:
170:
47:
40:
33:
5152:Extension to predicate and modal logic
3273:{\displaystyle (\neg A)\lor (\neg B).}
3324:§ Set theory and Boolean algebra
3103:In that it has been established that
7:
6900:List of set identities and relations
4103:To prove the reverse direction, let
3880:{\displaystyle x\in {\overline {B}}}
3772:{\displaystyle x\in {\overline {A}}}
7260:Internet Encyclopedia of Philosophy
6963:(12th ed.), Cengage Learning,
4485:must not be the case, meaning that
2996:Search D: (NOT cats) OR (NOT dogs).
2961:Search B: (NOT cats) AND (NOT dogs)
2200:"break the line, change the sign".
6842:
6839:
6836:
6792:
6785:
6782:
6758:
6722:
6715:
6712:
6688:
6651:
6636:
6623:
6566:
6551:
6538:
6450:
6447:
6427:
6420:
6378:
6375:
6355:
6348:
6289:
6264:
6261:
6258:
6255:
6246:, which however is different from
6231:
6228:
6225:
6222:
6181:
6175:
6163:
6135:
6132:
6129:
6126:
6050:
6029:
6014:
5962:
5947:
5914:
5854:
5848:
5812:
5806:
5746:
5728:
5710:
5704:
5618:
5600:
5582:
5576:
5441:
5354:
5282:
5275:
5269:
5247:
5209:
5202:
5196:
5174:
5120:
5111:
5102:
4536:
3943:
3258:
3243:
3192:
3139:
3124:
3069:
1707:
1698:
1692:
1654:
1645:
1639:
1565:
1543:
1527:
1473:
1456:
1434:
1418:
1364:
1213:
1204:
1176:
1160:
1151:
1123:
1101:and expressed as truth-functional
1067:
1056:
1047:
1034:
1025:
1002:
952:
941:
932:
919:
910:
887:
829:
813:
804:
780:
771:
743:
687:
671:
662:
638:
629:
601:
359:not (A and B) = (not A) or (not B)
356:not (A or B) = (not A) and (not B)
319:purely in terms of each other via
23:De Morgan's laws represented with
14:
7091:Middle Tennessee State University
6272:{\displaystyle {\mathrm {WLPO} }}
6239:{\displaystyle {\mathrm {LLPO} }}
6143:{\displaystyle {\mathrm {WPEM} }}
3583:{\displaystyle x\not \in A\cap B}
2972:Document 2: Contains only "dogs".
7820:
7389:
7155:"Augustus De Morgan (1806–1871)"
7018:DeMorgan's [sic] Theorem
6850:{\displaystyle {\mathrm {PEM} }}
3216:{\displaystyle \neg (A\land B).}
8445:Theorems in propositional logic
6961:A Concise Introduction to Logic
6880:Conjunction/disjunction duality
4149:, and for contradiction assume
3315:{\displaystyle {\overline {A}}}
3093:{\displaystyle \neg (A\lor B).}
2368:{\displaystyle {\overline {A}}}
1904:{\displaystyle {\overline {A}}}
1043:
928:
7756:Tractatus Logico-Philosophicus
7361:Problem of multiple generality
7123:, part II, sections 32 and 33.
6895:List of Boolean algebra topics
6804:
6798:
6778:
6774:
6768:
6734:
6728:
6708:
6704:
6698:
6657:
6648:
6642:
6633:
6619:
6615:
6603:
6572:
6563:
6557:
6548:
6534:
6530:
6518:
6466:
6460:
6443:
6439:
6433:
6394:
6388:
6371:
6367:
6361:
6315:
6187:
6178:
6169:
6160:
6065:
6053:
6046:
6035:
6026:
6020:
6011:
5968:
5959:
5953:
5944:
5933:
5929:
5917:
5880:of possibility and necessity,
5761:
5758:
5752:
5740:
5734:
5722:
5716:
5707:
5698:
5692:
5683:
5677:
5668:
5662:
5633:
5630:
5624:
5612:
5606:
5594:
5588:
5579:
5570:
5564:
5555:
5549:
5540:
5534:
5502:
5496:
5487:
5481:
5472:
5466:
5457:
5451:
5415:
5409:
5400:
5394:
5385:
5379:
5370:
5364:
5297:
5294:
5288:
5272:
5263:
5257:
5224:
5221:
5215:
5199:
5190:
5184:
5132:
5108:
5096:
5072:
5041:{\displaystyle {\mbox{P}}^{d}}
4949:Generalising De Morgan duality
4579:
3981:
3623:
3264:
3255:
3249:
3240:
3207:
3195:
3145:
3136:
3130:
3121:
3084:
3072:
2993:Search C: NOT (cats AND dogs),
2806:
2780:
2768:
2756:
1713:
1695:
1689:
1682:
1670:
1660:
1642:
1636:
1629:
1617:
1524:
1521:
1476:
1415:
1412:
1367:
1219:
1201:
1198:
1191:
1179:
1166:
1148:
1145:
1138:
1126:
1082:
1070:
1017:
1005:
967:
955:
902:
890:
844:
832:
819:
801:
786:
768:
758:
746:
702:
690:
677:
659:
644:
626:
616:
604:
540:
528:
522:
510:
504:
492:
462:
450:
444:
432:
426:
414:
371:one of A or B rather than an "
1:
7746:The Principles of Mathematics
7256:Duality in Logic and Language
7041:10.13140/RG.2.2.24043.82724/1
5521:But, using De Morgan's laws,
3051:in all or part of a formula.
7442:Commutativity of conjunction
6823:but their inversion implies
4966:property of logics based on
4930:
4917:
4904:
4866:
4853:
4840:
4805:
4792:
4779:
4742:
4721:
4708:
4673:
4652:
4639:
4602:
4573:
4560:
4511:
4470:
4426:
4413:
4380:
4367:
4331:
4298:
4218:{\displaystyle x\in A\cap B}
4175:
4134:
4121:
4080:
4067:
4054:
4009:
3996:
3975:
3918:
3905:
3872:
3810:
3797:
3764:
3543:
3494:
3473:
3460:
3433:
3420:
3407:
3372:
3359:
3346:
3307:
2958:Search A: NOT (cats OR dogs)
2871:
2858:
2845:
2801:
2788:
2772:
2705:
2679:
2659:
2639:
2581:
2555:
2535:
2515:
2455:which can be generalized to
2360:
2320:
2307:
2290:
2265:
2252:
2235:
2162:
2122:
2080:
2040:
1896:
1856:
1843:
1826:
1801:
1788:
1771:
1287:generalized De Morgan’s laws
16:Pair of logical equivalences
7215:Encyclopedia of Mathematics
6983:Moore, Brooke Noel (2012).
6959:Hurley, Patrick J. (2015),
6305:is replaced by implication
4884:The other De Morgan's law,
3847:{\displaystyle x\not \in B}
3739:{\displaystyle x\not \in A}
3710:{\displaystyle x\not \in B}
3684:{\displaystyle x\not \in A}
3665:, it must be the case that
8461:
8280:von Neumann–Bernays–Gödel
7462:Monotonicity of entailment
5876:In its application to the
4970:, namely the existence of
4444:therefore, the assumption
1291:For a set of propositions
271:Existential generalization
76:Biconditional introduction
8081:One-to-one correspondence
7818:
7387:
7351:Idempotency of entailment
6885:Homogeneity (linguistics)
5904:. Specifically, we have
5017:, ... to be the operator
3177:Negation of a conjunction
3055:Negation of a disjunction
3039:Proof for Boolean algebra
3005:The laws are named after
1996:The generalized form is
7185:, Springer, p. 16,
6103:{\displaystyle P\land Q}
1109:of propositional logic:
729:rule may be written as:
262:Universal generalization
102:Disjunction introduction
89:Conjunction introduction
59:Implication introduction
7710:Willard Van Orman Quine
7109:History of Formal Logic
7029:Kashef, Arman. (2023),
6861:In computer engineering
6508:Further, one still has
5896:In intuitionistic logic
4988:disjunctive normal form
4984:conjunctive normal form
4945:, is proven similarly.
988:negation of disjunction
873:negation of conjunction
727:negation of disjunction
583:rule may be written in
581:negation of conjunction
307:. They are named after
8039:Constructible universe
7866:Constructibility (V=L)
7685:Charles Sanders Peirce
7528:Hypothetical syllogism
6851:
6814:
6744:
6674:
6589:
6499:
6476:
6401:
6325:
6324:{\displaystyle P\to C}
6299:
6298:{\displaystyle \neg P}
6273:
6240:
6197:
6144:
6104:
6075:
5985:
5867:
5825:
5771:
5640:
5512:
5422:
5304:
5231:
5162:existential quantifier
5142:
5042:
4962:
4939:
4875:
4814:
4751:
4682:
4618:
4520:
4479:
4435:
4389:
4340:
4307:
4274:
4273:{\displaystyle x\in B}
4248:
4247:{\displaystyle x\in A}
4219:
4184:
4143:
4089:
4025:
3927:
3881:
3848:
3819:
3773:
3740:
3711:
3685:
3659:
3584:
3552:
3503:
3442:
3381:
3316:
3274:
3228:one (at least) or more
3217:
3155:
3094:
3032:Summulae de Dialectica
3031:
2931:
2909:
2908:{\displaystyle \cdot }
2883:
2813:
2721:
2441:
2415:
2414:{\displaystyle \land }
2389:
2369:
2336:
2178:
1977:
1951:
1925:
1905:
1872:
1727:
1586:
1344:
1276:
1256:
1233:
1092:
977:
858:
716:
550:
472:
395:
387:
363:where "A or B" is an "
121:hypothetical syllogism
42:Propositional calculus
28:
8262:Principia Mathematica
8096:Transfinite induction
7955:(i.e. set difference)
7751:Principia Mathematica
7523:Disjunctive syllogism
7508:modus ponendo tollens
7134:Summula de Dialectica
7087:"DeMorgan's Theorems"
6936:10.4324/9781315510897
6927:Introduction to Logic
6852:
6815:
6745:
6675:
6590:
6500:
6477:
6402:
6326:
6300:
6274:
6241:
6198:
6145:
6105:
6076:
5986:
5868:
5826:
5772:
5641:
5513:
5423:
5305:
5232:
5143:
5043:
4972:negation normal forms
4956:
4940:
4876:
4815:
4752:
4683:
4619:
4521:
4480:
4436:
4390:
4341:
4308:
4275:
4249:
4220:
4185:
4144:
4090:
4026:
3928:
3882:
3849:
3820:
3774:
3741:
3712:
3686:
3660:
3585:
3553:
3504:
3443:
3382:
3317:
3275:
3218:
3156:
3095:
3047:or the negation of a
2932:
2910:
2884:
2814:
2722:
2442:
2440:{\displaystyle \lor }
2416:
2390:
2370:
2337:
2179:
1978:
1976:{\displaystyle \cup }
1952:
1950:{\displaystyle \cap }
1926:
1906:
1873:
1728:
1587:
1345:
1277:
1257:
1234:
1093:
978:
859:
717:
551:
473:
393:
385:
163:Negation introduction
156:modus ponendo tollens
22:
8336:Burali-Forti paradox
8091:Set-builder notation
8044:Continuum hypothesis
7984:Symmetric difference
7741:Function and Concept
7513:Constructive dilemma
7488:Material implication
7055:by R. L. Goodstein.
6831:
6755:
6685:
6600:
6515:
6489:
6417:
6345:
6309:
6286:
6250:
6217:
6157:
6121:
6116:weak excluded middle
6088:
6001:
5911:
5902:intuitionistic logic
5836:
5794:
5656:
5528:
5438:
5351:
5244:
5171:
5158:universal quantifier
5055:
5021:
4888:
4824:
4763:
4700:
4631:
4533:
4489:
4448:
4399:
4353:
4349:However, that means
4317:
4284:
4258:
4232:
4197:
4153:
4107:
4038:
3940:
3891:
3858:
3832:
3783:
3750:
3724:
3695:
3669:
3597:
3562:
3521:
3452:
3391:
3330:
3299:
3291:Proof for set theory
3237:
3189:
3118:
3066:
2921:
2899:
2829:
2750:
2741:computer engineering
2462:
2431:
2405:
2379:
2352:
2215:
2003:
1967:
1941:
1915:
1888:
1751:
1610:
1357:
1295:
1266:
1246:
1116:
996:
881:
736:
594:
569:mathematical duality
483:
405:
221:Material implication
172:Rules of replacement
35:Transformation rules
8297:Tarski–Grothendieck
7715:Ludwig Wittgenstein
7518:Destructive dilemma
7346:Well-formed formula
7210:"Duality principle"
7119:William of Ockham,
4228:so it follows that
3011:propositional logic
2915:is the logical AND,
2449:logical disjunction
2423:logical conjunction
2375:is the negation of
1911:is the negation of
298:De Morgan's theorem
286:propositional logic
134:destructive dilemma
8440:Rules of inference
7886:Limitation of size
7660:Augustus De Morgan
7232:"de Morgan's Laws"
7229:Weisstein, Eric W.
6847:
6810:
6740:
6670:
6585:
6495:
6472:
6397:
6321:
6295:
6269:
6236:
6208:intermediate logic
6193:
6140:
6100:
6071:
5981:
5886:normal modal logic
5878:alethic modalities
5863:
5821:
5767:
5636:
5508:
5418:
5300:
5227:
5138:
5064:
5038:
5030:
4996:probability theory
4992:logical conditions
4963:
4935:
4871:
4810:
4747:
4678:
4614:
4516:
4475:
4431:
4385:
4336:
4303:
4270:
4244:
4215:
4180:
4139:
4085:
4021:
3923:
3877:
3844:
3815:
3769:
3736:
3707:
3681:
3655:
3580:
3548:
3499:
3438:
3377:
3312:
3270:
3213:
3151:
3090:
3007:Augustus De Morgan
2937:is the logical OR,
2927:
2905:
2879:
2809:
2717:
2715:
2437:
2411:
2385:
2365:
2332:
2330:
2174:
2172:
2149:
2110:
2067:
2028:
1973:
1947:
1921:
1901:
1868:
1866:
1723:
1721:
1582:
1580:
1340:
1272:
1252:
1229:
1227:
1088:
973:
854:
852:
712:
710:
546:
468:
396:
388:
309:Augustus De Morgan
305:rules of inference
253:Rules of inference
49:Rules of inference
29:
8417:
8416:
8326:Russell's paradox
8275:Zermelo–Fraenkel
8176:Dedekind-infinite
8049:Diagonal argument
7948:Cartesian product
7812:Set (mathematics)
7764:
7763:
7628:
7627:
6996:978-0-07-803828-0
6986:Critical thinking
6970:978-1-285-19654-1
6498:{\displaystyle Q}
5063:
5029:
4933:
4920:
4907:
4869:
4856:
4843:
4808:
4795:
4782:
4745:
4724:
4711:
4676:
4655:
4642:
4605:
4576:
4563:
4514:
4473:
4429:
4416:
4383:
4370:
4334:
4301:
4178:
4137:
4124:
4083:
4070:
4057:
4012:
3999:
3978:
3921:
3908:
3875:
3813:
3800:
3767:
3629:
3621:
3546:
3497:
3476:
3463:
3436:
3423:
3410:
3375:
3362:
3349:
3326:. The proof that
3310:
3164:If either A or B
3023:William of Ockham
2930:{\displaystyle +}
2874:
2861:
2848:
2804:
2791:
2775:
2708:
2682:
2662:
2642:
2584:
2558:
2538:
2518:
2388:{\displaystyle A}
2363:
2323:
2310:
2293:
2268:
2255:
2238:
2165:
2134:
2125:
2095:
2083:
2052:
2043:
2013:
1924:{\displaystyle A}
1899:
1859:
1846:
1829:
1804:
1791:
1774:
1599:Substitution form
1275:{\displaystyle Q}
1255:{\displaystyle P}
1086:
1041:
971:
926:
795:
653:
565:computer programs
282:
281:
8452:
8435:Duality theories
8399:Bertrand Russell
8389:John von Neumann
8374:Abraham Fraenkel
8369:Richard Dedekind
8331:Suslin's problem
8242:Cantor's theorem
7959:De Morgan's laws
7824:
7791:
7784:
7777:
7768:
7700:Henry M. Sheffer
7690:Bertrand Russell
7655:Richard Dedekind
7539:
7483:De Morgan's laws
7457:Noncontradiction
7399:Classical logics
7393:
7290:
7283:
7276:
7267:
7246:de Morgan's laws
7242:
7241:
7223:
7196:
7195:
7173:
7167:
7166:
7161:. Archived from
7150:
7144:
7130:
7124:
7117:
7111:
7105:
7099:
7098:
7093:. Archived from
7083:
7077:
7069:
7063:
7050:
7044:
7043:
7026:
7020:
7015:
7009:
7008:
6980:
6974:
6973:
6956:
6950:
6949:
6921:
6856:
6854:
6853:
6848:
6846:
6845:
6819:
6817:
6816:
6811:
6749:
6747:
6746:
6741:
6679:
6677:
6676:
6671:
6666:
6665:
6632:
6631:
6594:
6592:
6591:
6586:
6581:
6580:
6547:
6546:
6504:
6502:
6501:
6496:
6481:
6479:
6478:
6473:
6406:
6404:
6403:
6398:
6330:
6328:
6327:
6322:
6304:
6302:
6301:
6296:
6278:
6276:
6275:
6270:
6268:
6267:
6245:
6243:
6242:
6237:
6235:
6234:
6202:
6200:
6199:
6194:
6149:
6147:
6146:
6141:
6139:
6138:
6109:
6107:
6106:
6101:
6080:
6078:
6077:
6072:
6044:
6043:
6010:
6009:
5990:
5988:
5987:
5982:
5977:
5976:
5943:
5942:
5890:Kripke semantics
5872:
5870:
5869:
5864:
5830:
5828:
5827:
5822:
5776:
5774:
5773:
5768:
5645:
5643:
5642:
5637:
5517:
5515:
5514:
5509:
5427:
5425:
5424:
5419:
5309:
5307:
5306:
5301:
5236:
5234:
5233:
5228:
5147:
5145:
5144:
5139:
5071:
5070:
5065:
5061:
5047:
5045:
5044:
5039:
5037:
5036:
5031:
5027:
4944:
4942:
4941:
4936:
4934:
4926:
4921:
4913:
4908:
4903:
4892:
4880:
4878:
4877:
4872:
4870:
4862:
4857:
4849:
4844:
4839:
4828:
4819:
4817:
4816:
4811:
4809:
4801:
4796:
4788:
4783:
4778:
4767:
4756:
4754:
4753:
4748:
4746:
4741:
4730:
4725:
4717:
4712:
4704:
4687:
4685:
4684:
4679:
4677:
4672:
4661:
4656:
4648:
4643:
4635:
4623:
4621:
4620:
4615:
4613:
4612:
4606:
4601:
4590:
4577:
4569:
4564:
4556:
4548:
4547:
4525:
4523:
4522:
4517:
4515:
4510:
4499:
4484:
4482:
4481:
4476:
4474:
4469:
4458:
4440:
4438:
4437:
4432:
4430:
4422:
4417:
4409:
4394:
4392:
4391:
4386:
4384:
4376:
4371:
4363:
4345:
4343:
4342:
4337:
4335:
4327:
4312:
4310:
4309:
4304:
4302:
4294:
4279:
4277:
4276:
4271:
4253:
4251:
4250:
4245:
4224:
4222:
4221:
4216:
4189:
4187:
4186:
4181:
4179:
4174:
4163:
4148:
4146:
4145:
4140:
4138:
4130:
4125:
4117:
4094:
4092:
4091:
4086:
4084:
4076:
4071:
4063:
4058:
4053:
4042:
4030:
4028:
4027:
4022:
4020:
4019:
4013:
4005:
4000:
3992:
3979:
3974:
3963:
3955:
3954:
3932:
3930:
3929:
3924:
3922:
3914:
3909:
3901:
3886:
3884:
3883:
3878:
3876:
3868:
3853:
3851:
3850:
3845:
3824:
3822:
3821:
3816:
3814:
3806:
3801:
3793:
3778:
3776:
3775:
3770:
3768:
3760:
3745:
3743:
3742:
3737:
3716:
3714:
3713:
3708:
3690:
3688:
3687:
3682:
3664:
3662:
3661:
3656:
3627:
3626:
3619:
3589:
3587:
3586:
3581:
3557:
3555:
3554:
3549:
3547:
3542:
3531:
3508:
3506:
3505:
3500:
3498:
3493:
3482:
3477:
3469:
3464:
3456:
3447:
3445:
3444:
3439:
3437:
3429:
3424:
3416:
3411:
3406:
3395:
3386:
3384:
3383:
3378:
3376:
3368:
3363:
3355:
3350:
3345:
3334:
3321:
3319:
3318:
3313:
3311:
3303:
3279:
3277:
3276:
3271:
3222:
3220:
3219:
3214:
3160:
3158:
3157:
3152:
3099:
3097:
3096:
3091:
3034:
2943:
2936:
2934:
2933:
2928:
2914:
2912:
2911:
2906:
2888:
2886:
2885:
2880:
2875:
2867:
2862:
2854:
2849:
2844:
2833:
2818:
2816:
2815:
2810:
2805:
2797:
2792:
2784:
2776:
2771:
2754:
2726:
2724:
2723:
2718:
2716:
2709:
2704:
2703:
2694:
2683:
2678:
2677:
2668:
2663:
2658:
2657:
2648:
2643:
2638:
2637:
2636:
2618:
2617:
2605:
2604:
2594:
2585:
2580:
2579:
2570:
2559:
2554:
2553:
2544:
2539:
2534:
2533:
2524:
2519:
2514:
2513:
2512:
2494:
2493:
2481:
2480:
2470:
2446:
2444:
2443:
2438:
2420:
2418:
2417:
2412:
2394:
2392:
2391:
2386:
2374:
2372:
2371:
2366:
2364:
2356:
2341:
2339:
2338:
2333:
2331:
2324:
2316:
2311:
2303:
2294:
2289:
2278:
2269:
2261:
2256:
2248:
2239:
2234:
2223:
2192:
2183:
2181:
2180:
2175:
2173:
2166:
2161:
2160:
2151:
2148:
2126:
2121:
2120:
2119:
2109:
2093:
2084:
2079:
2078:
2069:
2066:
2044:
2039:
2038:
2037:
2027:
2011:
1982:
1980:
1979:
1974:
1956:
1954:
1953:
1948:
1930:
1928:
1927:
1922:
1910:
1908:
1907:
1902:
1900:
1892:
1877:
1875:
1874:
1869:
1867:
1860:
1852:
1847:
1839:
1830:
1825:
1814:
1805:
1797:
1792:
1784:
1775:
1770:
1759:
1732:
1730:
1729:
1724:
1722:
1591:
1589:
1588:
1583:
1581:
1577:
1576:
1555:
1554:
1539:
1538:
1520:
1519:
1501:
1500:
1488:
1487:
1468:
1467:
1446:
1445:
1430:
1429:
1411:
1410:
1392:
1391:
1379:
1378:
1349:
1347:
1346:
1341:
1339:
1338:
1320:
1319:
1307:
1306:
1281:
1279:
1278:
1273:
1261:
1259:
1258:
1253:
1238:
1236:
1235:
1230:
1228:
1097:
1095:
1094:
1089:
1087:
1085:
1062:
1045:
1042:
1040:
1020:
1000:
982:
980:
979:
974:
972:
970:
947:
930:
927:
925:
905:
885:
863:
861:
860:
855:
853:
796:
793:
721:
719:
718:
713:
711:
654:
651:
555:
553:
552:
547:
477:
475:
474:
469:
296:, also known as
294:De Morgan's laws
236:
229:
222:
210:De Morgan's laws
205:
198:
191:
184:
158:
150:
142:
135:
129:
122:
116:
109:
103:
96:
90:
83:
77:
70:
60:
31:
8460:
8459:
8455:
8454:
8453:
8451:
8450:
8449:
8430:Boolean algebra
8420:
8419:
8418:
8413:
8340:
8319:
8303:
8268:New Foundations
8215:
8105:
8024:Cardinal number
8007:
7993:
7934:
7825:
7816:
7800:
7795:
7765:
7760:
7736:Begriffsschrift
7724:
7720:Jan Łukasiewicz
7640:Bernard Bolzano
7624:
7595:Double negation
7583:
7554:Double negation
7537:
7471:
7447:Excluded middle
7430:
7394:
7385:
7299:
7297:Classical logic
7294:
7227:
7226:
7208:
7205:
7200:
7199:
7193:
7175:
7174:
7170:
7153:Robert H. Orr.
7152:
7151:
7147:
7131:
7127:
7118:
7114:
7106:
7102:
7085:
7084:
7080:
7070:
7066:
7053:Boolean Algebra
7051:
7047:
7028:
7027:
7023:
7016:
7012:
6997:
6982:
6981:
6977:
6971:
6958:
6957:
6953:
6946:
6923:
6922:
6918:
6913:
6876:
6863:
6829:
6828:
6825:excluded middle
6753:
6752:
6683:
6682:
6598:
6597:
6513:
6512:
6487:
6486:
6415:
6414:
6343:
6342:
6307:
6306:
6284:
6283:
6248:
6247:
6215:
6214:
6155:
6154:
6119:
6118:
6086:
6085:
5999:
5998:
5909:
5908:
5898:
5834:
5833:
5792:
5791:
5654:
5653:
5526:
5525:
5436:
5435:
5349:
5348:
5242:
5241:
5169:
5168:
5154:
5058:
5053:
5052:
5024:
5019:
5018:
4976:digital circuit
4968:classical logic
4951:
4893:
4886:
4885:
4829:
4822:
4821:
4768:
4761:
4760:
4731:
4698:
4697:
4694:
4662:
4629:
4628:
4591:
4531:
4530:
4500:
4487:
4486:
4459:
4446:
4445:
4397:
4396:
4351:
4350:
4315:
4314:
4282:
4281:
4256:
4255:
4230:
4229:
4195:
4194:
4164:
4151:
4150:
4105:
4104:
4101:
4043:
4036:
4035:
3964:
3938:
3937:
3889:
3888:
3856:
3855:
3830:
3829:
3781:
3780:
3748:
3747:
3722:
3721:
3693:
3692:
3667:
3666:
3595:
3594:
3560:
3559:
3532:
3519:
3518:
3515:
3483:
3450:
3449:
3396:
3389:
3388:
3335:
3328:
3327:
3297:
3296:
3293:
3235:
3234:
3187:
3186:
3179:
3116:
3115:
3064:
3063:
3057:
3041:
3003:
2951:
2941:
2919:
2918:
2897:
2896:
2834:
2827:
2826:
2755:
2748:
2747:
2733:
2714:
2713:
2695:
2669:
2649:
2628:
2609:
2596:
2595:
2590:
2589:
2571:
2545:
2525:
2504:
2485:
2472:
2471:
2460:
2459:
2429:
2428:
2425:operator (AND),
2403:
2402:
2377:
2376:
2350:
2349:
2329:
2328:
2295:
2279:
2274:
2273:
2240:
2224:
2213:
2212:
2206:
2204:Boolean algebra
2188:
2171:
2170:
2152:
2127:
2111:
2094:
2089:
2088:
2070:
2045:
2029:
2012:
2001:
2000:
1994:
1965:
1964:
1961:operator (AND),
1939:
1938:
1913:
1912:
1886:
1885:
1865:
1864:
1831:
1815:
1810:
1809:
1776:
1760:
1749:
1748:
1742:
1720:
1719:
1685:
1667:
1666:
1632:
1608:
1607:
1601:
1579:
1578:
1568:
1546:
1530:
1511:
1492:
1479:
1470:
1469:
1459:
1437:
1421:
1402:
1383:
1370:
1355:
1354:
1330:
1311:
1298:
1293:
1292:
1290:
1264:
1263:
1244:
1243:
1226:
1225:
1194:
1173:
1172:
1141:
1114:
1113:
1099:
1063:
1046:
1021:
1001:
994:
993:
984:
948:
931:
906:
886:
879:
878:
851:
850:
822:
798:
797:
761:
734:
733:
709:
708:
680:
656:
655:
619:
592:
591:
577:
575:Formal notation
481:
480:
403:
402:
379:one of A or B.
290:Boolean algebra
246:Predicate logic
240:
204:Double negation
58:
17:
12:
11:
5:
8458:
8456:
8448:
8447:
8442:
8437:
8432:
8422:
8421:
8415:
8414:
8412:
8411:
8406:
8404:Thoralf Skolem
8401:
8396:
8391:
8386:
8381:
8376:
8371:
8366:
8361:
8356:
8350:
8348:
8342:
8341:
8339:
8338:
8333:
8328:
8322:
8320:
8318:
8317:
8314:
8308:
8305:
8304:
8302:
8301:
8300:
8299:
8294:
8289:
8288:
8287:
8272:
8271:
8270:
8258:
8257:
8256:
8245:
8244:
8239:
8234:
8229:
8223:
8221:
8217:
8216:
8214:
8213:
8208:
8203:
8198:
8189:
8184:
8179:
8169:
8164:
8163:
8162:
8157:
8152:
8142:
8132:
8127:
8122:
8116:
8114:
8107:
8106:
8104:
8103:
8098:
8093:
8088:
8086:Ordinal number
8083:
8078:
8073:
8068:
8067:
8066:
8061:
8051:
8046:
8041:
8036:
8031:
8021:
8016:
8010:
8008:
8006:
8005:
8002:
7998:
7995:
7994:
7992:
7991:
7986:
7981:
7976:
7971:
7966:
7964:Disjoint union
7961:
7956:
7950:
7944:
7942:
7936:
7935:
7933:
7932:
7931:
7930:
7925:
7914:
7913:
7911:Martin's axiom
7908:
7903:
7898:
7893:
7888:
7883:
7878:
7876:Extensionality
7873:
7868:
7863:
7862:
7861:
7856:
7851:
7841:
7835:
7833:
7827:
7826:
7819:
7817:
7815:
7814:
7808:
7806:
7802:
7801:
7796:
7794:
7793:
7786:
7779:
7771:
7762:
7761:
7759:
7758:
7753:
7748:
7743:
7738:
7732:
7730:
7726:
7725:
7723:
7722:
7717:
7712:
7707:
7702:
7697:
7695:Ernst Schröder
7692:
7687:
7682:
7680:Giuseppe Peano
7677:
7672:
7667:
7662:
7657:
7652:
7647:
7642:
7636:
7634:
7630:
7629:
7626:
7625:
7623:
7622:
7617:
7612:
7607:
7602:
7597:
7591:
7589:
7585:
7584:
7582:
7581:
7576:
7571:
7566:
7561:
7556:
7551:
7545:
7543:
7536:
7535:
7530:
7525:
7520:
7515:
7510:
7505:
7500:
7495:
7490:
7485:
7479:
7477:
7473:
7472:
7470:
7469:
7464:
7459:
7454:
7449:
7444:
7438:
7436:
7432:
7431:
7429:
7428:
7423:
7418:
7413:
7408:
7402:
7400:
7396:
7395:
7388:
7386:
7384:
7383:
7378:
7373:
7368:
7363:
7358:
7353:
7348:
7343:
7338:
7336:Truth function
7333:
7328:
7323:
7318:
7313:
7307:
7305:
7301:
7300:
7295:
7293:
7292:
7285:
7278:
7270:
7264:
7263:
7253:
7243:
7224:
7204:
7203:External links
7201:
7198:
7197:
7191:
7177:Wirth, Niklaus
7168:
7165:on 2010-07-15.
7145:
7132:Jean Buridan,
7125:
7112:
7100:
7097:on 2008-03-23.
7078:
7064:
7045:
7021:
7010:
6995:
6975:
6969:
6951:
6944:
6915:
6914:
6912:
6909:
6908:
6907:
6905:Positive logic
6902:
6897:
6892:
6887:
6882:
6875:
6872:
6871:
6870:
6867:
6862:
6859:
6844:
6841:
6838:
6821:
6820:
6809:
6806:
6803:
6800:
6797:
6794:
6790:
6787:
6784:
6780:
6776:
6773:
6770:
6767:
6763:
6760:
6750:
6739:
6736:
6733:
6730:
6727:
6724:
6720:
6717:
6714:
6710:
6706:
6703:
6700:
6697:
6693:
6690:
6680:
6669:
6664:
6659:
6656:
6653:
6650:
6647:
6644:
6641:
6638:
6635:
6630:
6625:
6621:
6617:
6614:
6611:
6608:
6605:
6595:
6584:
6579:
6574:
6571:
6568:
6565:
6562:
6559:
6556:
6553:
6550:
6545:
6540:
6536:
6532:
6529:
6526:
6523:
6520:
6494:
6483:
6482:
6471:
6468:
6465:
6462:
6459:
6455:
6452:
6449:
6445:
6441:
6438:
6435:
6432:
6429:
6425:
6422:
6408:
6407:
6396:
6393:
6390:
6387:
6383:
6380:
6377:
6373:
6369:
6366:
6363:
6360:
6357:
6353:
6350:
6320:
6317:
6314:
6294:
6291:
6266:
6263:
6260:
6257:
6233:
6230:
6227:
6224:
6204:
6203:
6192:
6189:
6186:
6183:
6180:
6177:
6174:
6171:
6168:
6165:
6162:
6137:
6134:
6131:
6128:
6099:
6096:
6093:
6082:
6081:
6070:
6067:
6064:
6061:
6058:
6055:
6052:
6048:
6042:
6037:
6034:
6031:
6028:
6025:
6022:
6019:
6016:
6013:
6008:
5992:
5991:
5980:
5975:
5970:
5967:
5964:
5961:
5958:
5955:
5952:
5949:
5946:
5941:
5935:
5931:
5928:
5925:
5922:
5919:
5916:
5897:
5894:
5874:
5873:
5862:
5859:
5856:
5853:
5850:
5847:
5844:
5841:
5831:
5820:
5817:
5814:
5811:
5808:
5805:
5802:
5799:
5778:
5777:
5766:
5763:
5760:
5757:
5754:
5751:
5748:
5745:
5742:
5739:
5736:
5733:
5730:
5727:
5724:
5721:
5718:
5715:
5712:
5709:
5706:
5703:
5700:
5697:
5694:
5691:
5688:
5685:
5682:
5679:
5676:
5673:
5670:
5667:
5664:
5661:
5647:
5646:
5635:
5632:
5629:
5626:
5623:
5620:
5617:
5614:
5611:
5608:
5605:
5602:
5599:
5596:
5593:
5590:
5587:
5584:
5581:
5578:
5575:
5572:
5569:
5566:
5563:
5560:
5557:
5554:
5551:
5548:
5545:
5542:
5539:
5536:
5533:
5519:
5518:
5507:
5504:
5501:
5498:
5495:
5492:
5489:
5486:
5483:
5480:
5477:
5474:
5471:
5468:
5465:
5462:
5459:
5456:
5453:
5450:
5446:
5443:
5429:
5428:
5417:
5414:
5411:
5408:
5405:
5402:
5399:
5396:
5393:
5390:
5387:
5384:
5381:
5378:
5375:
5372:
5369:
5366:
5363:
5359:
5356:
5342:
5341:
5311:
5310:
5299:
5296:
5293:
5290:
5287:
5284:
5280:
5277:
5274:
5271:
5268:
5265:
5262:
5259:
5256:
5252:
5249:
5238:
5237:
5226:
5223:
5220:
5217:
5214:
5211:
5207:
5204:
5201:
5198:
5195:
5192:
5189:
5186:
5183:
5179:
5176:
5153:
5150:
5149:
5148:
5137:
5134:
5131:
5128:
5125:
5122:
5119:
5116:
5113:
5110:
5107:
5104:
5101:
5098:
5095:
5092:
5089:
5086:
5083:
5080:
5077:
5074:
5069:
5035:
4950:
4947:
4932:
4929:
4924:
4919:
4916:
4911:
4906:
4902:
4899:
4896:
4868:
4865:
4860:
4855:
4852:
4847:
4842:
4838:
4835:
4832:
4807:
4804:
4799:
4794:
4791:
4786:
4781:
4777:
4774:
4771:
4744:
4740:
4737:
4734:
4728:
4723:
4720:
4715:
4710:
4707:
4693:
4690:
4675:
4671:
4668:
4665:
4659:
4654:
4651:
4646:
4641:
4638:
4611:
4604:
4600:
4597:
4594:
4588:
4585:
4581:
4575:
4572:
4567:
4562:
4559:
4554:
4551:
4546:
4541:
4538:
4513:
4509:
4506:
4503:
4497:
4494:
4472:
4468:
4465:
4462:
4456:
4453:
4428:
4425:
4420:
4415:
4412:
4407:
4404:
4382:
4379:
4374:
4369:
4366:
4361:
4358:
4333:
4330:
4325:
4322:
4300:
4297:
4292:
4289:
4269:
4266:
4263:
4243:
4240:
4237:
4214:
4211:
4208:
4205:
4202:
4177:
4173:
4170:
4167:
4161:
4158:
4136:
4133:
4128:
4123:
4120:
4115:
4112:
4100:
4097:
4082:
4079:
4074:
4069:
4066:
4061:
4056:
4052:
4049:
4046:
4018:
4011:
4008:
4003:
3998:
3995:
3990:
3987:
3983:
3977:
3973:
3970:
3967:
3961:
3958:
3953:
3948:
3945:
3920:
3917:
3912:
3907:
3904:
3899:
3896:
3874:
3871:
3866:
3863:
3843:
3840:
3837:
3828:Similarly, if
3812:
3809:
3804:
3799:
3796:
3791:
3788:
3766:
3763:
3758:
3755:
3735:
3732:
3729:
3706:
3703:
3700:
3680:
3677:
3674:
3654:
3650:
3647:
3644:
3641:
3638:
3635:
3632:
3625:
3618:
3614:
3611:
3608:
3605:
3602:
3579:
3576:
3573:
3570:
3567:
3545:
3541:
3538:
3535:
3529:
3526:
3514:
3511:
3496:
3492:
3489:
3486:
3480:
3475:
3472:
3467:
3462:
3459:
3435:
3432:
3427:
3422:
3419:
3414:
3409:
3405:
3402:
3399:
3374:
3371:
3366:
3361:
3358:
3353:
3348:
3344:
3341:
3338:
3309:
3306:
3292:
3289:
3281:
3280:
3269:
3266:
3263:
3260:
3257:
3254:
3251:
3248:
3245:
3242:
3224:
3223:
3212:
3209:
3206:
3203:
3200:
3197:
3194:
3178:
3175:
3162:
3161:
3150:
3147:
3144:
3141:
3138:
3135:
3132:
3129:
3126:
3123:
3101:
3100:
3089:
3086:
3083:
3080:
3077:
3074:
3071:
3056:
3053:
3040:
3037:
3002:
2999:
2998:
2997:
2994:
2980:
2979:
2976:
2973:
2970:
2963:
2962:
2959:
2950:
2949:Text searching
2947:
2946:
2945:
2938:
2926:
2916:
2904:
2890:
2889:
2878:
2873:
2870:
2865:
2860:
2857:
2852:
2847:
2843:
2840:
2837:
2820:
2819:
2808:
2803:
2800:
2795:
2790:
2787:
2782:
2779:
2774:
2770:
2767:
2764:
2761:
2758:
2732:
2729:
2728:
2727:
2712:
2707:
2702:
2698:
2692:
2689:
2686:
2681:
2676:
2672:
2666:
2661:
2656:
2652:
2646:
2641:
2635:
2631:
2627:
2624:
2621:
2616:
2612:
2608:
2603:
2599:
2592:
2591:
2588:
2583:
2578:
2574:
2568:
2565:
2562:
2557:
2552:
2548:
2542:
2537:
2532:
2528:
2522:
2517:
2511:
2507:
2503:
2500:
2497:
2492:
2488:
2484:
2479:
2475:
2468:
2467:
2453:
2452:
2451:operator (OR).
2436:
2426:
2410:
2400:
2384:
2362:
2359:
2343:
2342:
2327:
2322:
2319:
2314:
2309:
2306:
2301:
2298:
2296:
2292:
2288:
2285:
2282:
2276:
2275:
2272:
2267:
2264:
2259:
2254:
2251:
2246:
2243:
2241:
2237:
2233:
2230:
2227:
2221:
2220:
2205:
2202:
2185:
2184:
2169:
2164:
2159:
2155:
2147:
2144:
2141:
2137:
2133:
2130:
2128:
2124:
2118:
2114:
2108:
2105:
2102:
2098:
2091:
2090:
2087:
2082:
2077:
2073:
2065:
2062:
2059:
2055:
2051:
2048:
2046:
2042:
2036:
2032:
2026:
2023:
2020:
2016:
2009:
2008:
1993:
1990:
1989:
1988:
1987:operator (OR).
1972:
1962:
1946:
1936:
1920:
1898:
1895:
1879:
1878:
1863:
1858:
1855:
1850:
1845:
1842:
1837:
1834:
1832:
1828:
1824:
1821:
1818:
1812:
1811:
1808:
1803:
1800:
1795:
1790:
1787:
1782:
1779:
1777:
1773:
1769:
1766:
1763:
1757:
1756:
1741:
1738:
1734:
1733:
1718:
1715:
1712:
1709:
1706:
1703:
1700:
1697:
1694:
1691:
1688:
1686:
1684:
1681:
1678:
1675:
1672:
1669:
1668:
1665:
1662:
1659:
1656:
1653:
1650:
1647:
1644:
1641:
1638:
1635:
1633:
1631:
1628:
1625:
1622:
1619:
1616:
1615:
1600:
1597:
1593:
1592:
1575:
1571:
1567:
1564:
1561:
1558:
1553:
1549:
1545:
1542:
1537:
1533:
1529:
1526:
1523:
1518:
1514:
1510:
1507:
1504:
1499:
1495:
1491:
1486:
1482:
1478:
1475:
1472:
1471:
1466:
1462:
1458:
1455:
1452:
1449:
1444:
1440:
1436:
1433:
1428:
1424:
1420:
1417:
1414:
1409:
1405:
1401:
1398:
1395:
1390:
1386:
1382:
1377:
1373:
1369:
1366:
1363:
1362:
1337:
1333:
1329:
1326:
1323:
1318:
1314:
1310:
1305:
1301:
1271:
1251:
1240:
1239:
1224:
1221:
1218:
1215:
1212:
1209:
1206:
1203:
1200:
1197:
1195:
1193:
1190:
1187:
1184:
1181:
1178:
1175:
1174:
1171:
1168:
1165:
1162:
1159:
1156:
1153:
1150:
1147:
1144:
1142:
1140:
1137:
1134:
1131:
1128:
1125:
1122:
1121:
1084:
1081:
1078:
1075:
1072:
1069:
1066:
1061:
1058:
1055:
1052:
1049:
1039:
1036:
1033:
1030:
1027:
1024:
1019:
1016:
1013:
1010:
1007:
1004:
991:
969:
966:
963:
960:
957:
954:
951:
946:
943:
940:
937:
934:
924:
921:
918:
915:
912:
909:
904:
901:
898:
895:
892:
889:
876:
865:
864:
849:
846:
843:
840:
837:
834:
831:
828:
825:
823:
821:
818:
815:
812:
809:
806:
803:
800:
799:
791:
788:
785:
782:
779:
776:
773:
770:
767:
764:
762:
760:
757:
754:
751:
748:
745:
742:
741:
723:
722:
707:
704:
701:
698:
695:
692:
689:
686:
683:
681:
679:
676:
673:
670:
667:
664:
661:
658:
657:
649:
646:
643:
640:
637:
634:
631:
628:
625:
622:
620:
618:
615:
612:
609:
606:
603:
600:
599:
576:
573:
557:
556:
545:
542:
539:
536:
533:
530:
527:
524:
521:
518:
515:
512:
509:
506:
503:
500:
497:
494:
491:
488:
478:
467:
464:
461:
458:
455:
452:
449:
446:
443:
440:
437:
434:
431:
428:
425:
422:
419:
416:
413:
410:
361:
360:
357:
350:
349:
346:
335:
334:
331:
280:
279:
278:
277:
268:
256:
255:
249:
248:
242:
241:
239:
238:
231:
224:
217:
212:
207:
200:
197:Distributivity
193:
186:
178:
175:
174:
168:
167:
166:
165:
160:
137:
124:
111:
98:
85:
72:
52:
51:
45:
44:
38:
37:
15:
13:
10:
9:
6:
4:
3:
2:
8457:
8446:
8443:
8441:
8438:
8436:
8433:
8431:
8428:
8427:
8425:
8410:
8409:Ernst Zermelo
8407:
8405:
8402:
8400:
8397:
8395:
8394:Willard Quine
8392:
8390:
8387:
8385:
8382:
8380:
8377:
8375:
8372:
8370:
8367:
8365:
8362:
8360:
8357:
8355:
8352:
8351:
8349:
8347:
8346:Set theorists
8343:
8337:
8334:
8332:
8329:
8327:
8324:
8323:
8321:
8315:
8313:
8310:
8309:
8306:
8298:
8295:
8293:
8292:Kripke–Platek
8290:
8286:
8283:
8282:
8281:
8278:
8277:
8276:
8273:
8269:
8266:
8265:
8264:
8263:
8259:
8255:
8252:
8251:
8250:
8247:
8246:
8243:
8240:
8238:
8235:
8233:
8230:
8228:
8225:
8224:
8222:
8218:
8212:
8209:
8207:
8204:
8202:
8199:
8197:
8195:
8190:
8188:
8185:
8183:
8180:
8177:
8173:
8170:
8168:
8165:
8161:
8158:
8156:
8153:
8151:
8148:
8147:
8146:
8143:
8140:
8136:
8133:
8131:
8128:
8126:
8123:
8121:
8118:
8117:
8115:
8112:
8108:
8102:
8099:
8097:
8094:
8092:
8089:
8087:
8084:
8082:
8079:
8077:
8074:
8072:
8069:
8065:
8062:
8060:
8057:
8056:
8055:
8052:
8050:
8047:
8045:
8042:
8040:
8037:
8035:
8032:
8029:
8025:
8022:
8020:
8017:
8015:
8012:
8011:
8009:
8003:
8000:
7999:
7996:
7990:
7987:
7985:
7982:
7980:
7977:
7975:
7972:
7970:
7967:
7965:
7962:
7960:
7957:
7954:
7951:
7949:
7946:
7945:
7943:
7941:
7937:
7929:
7928:specification
7926:
7924:
7921:
7920:
7919:
7916:
7915:
7912:
7909:
7907:
7904:
7902:
7899:
7897:
7894:
7892:
7889:
7887:
7884:
7882:
7879:
7877:
7874:
7872:
7869:
7867:
7864:
7860:
7857:
7855:
7852:
7850:
7847:
7846:
7845:
7842:
7840:
7837:
7836:
7834:
7832:
7828:
7823:
7813:
7810:
7809:
7807:
7803:
7799:
7792:
7787:
7785:
7780:
7778:
7773:
7772:
7769:
7757:
7754:
7752:
7749:
7747:
7744:
7742:
7739:
7737:
7734:
7733:
7731:
7727:
7721:
7718:
7716:
7713:
7711:
7708:
7706:
7705:Alfred Tarski
7703:
7701:
7698:
7696:
7693:
7691:
7688:
7686:
7683:
7681:
7678:
7676:
7673:
7671:
7668:
7666:
7665:Gottlob Frege
7663:
7661:
7658:
7656:
7653:
7651:
7648:
7646:
7643:
7641:
7638:
7637:
7635:
7631:
7621:
7618:
7616:
7613:
7611:
7610:Biconditional
7608:
7606:
7603:
7601:
7598:
7596:
7593:
7592:
7590:
7586:
7580:
7577:
7575:
7572:
7570:
7569:Biconditional
7567:
7565:
7562:
7560:
7557:
7555:
7552:
7550:
7547:
7546:
7544:
7540:
7534:
7531:
7529:
7526:
7524:
7521:
7519:
7516:
7514:
7511:
7509:
7506:
7504:
7503:modus tollens
7501:
7499:
7496:
7494:
7493:Transposition
7491:
7489:
7486:
7484:
7481:
7480:
7478:
7474:
7468:
7465:
7463:
7460:
7458:
7455:
7453:
7450:
7448:
7445:
7443:
7440:
7439:
7437:
7433:
7427:
7424:
7422:
7419:
7417:
7414:
7412:
7411:Propositional
7409:
7407:
7404:
7403:
7401:
7397:
7392:
7382:
7379:
7377:
7374:
7372:
7369:
7367:
7366:Associativity
7364:
7362:
7359:
7357:
7354:
7352:
7349:
7347:
7344:
7342:
7339:
7337:
7334:
7332:
7329:
7327:
7324:
7322:
7319:
7317:
7314:
7312:
7309:
7308:
7306:
7302:
7298:
7291:
7286:
7284:
7279:
7277:
7272:
7271:
7268:
7261:
7257:
7254:
7251:
7247:
7244:
7239:
7238:
7233:
7230:
7225:
7221:
7217:
7216:
7211:
7207:
7206:
7202:
7194:
7192:9783540585770
7188:
7184:
7183:
7178:
7172:
7169:
7164:
7160:
7156:
7149:
7146:
7143:
7142:0-300-08425-0
7139:
7135:
7129:
7126:
7122:
7121:Summa Logicae
7116:
7113:
7110:
7104:
7101:
7096:
7092:
7088:
7082:
7079:
7076:by S. P. Bali
7075:
7074:
7068:
7065:
7062:
7061:0-486-45894-6
7058:
7054:
7049:
7046:
7042:
7038:
7034:
7033:
7025:
7022:
7019:
7014:
7011:
7006:
7002:
6998:
6992:
6988:
6987:
6979:
6976:
6972:
6966:
6962:
6955:
6952:
6947:
6945:9781315510880
6941:
6937:
6933:
6929:
6928:
6920:
6917:
6910:
6906:
6903:
6901:
6898:
6896:
6893:
6891:
6888:
6886:
6883:
6881:
6878:
6877:
6873:
6868:
6865:
6864:
6860:
6858:
6826:
6807:
6801:
6795:
6788:
6771:
6765:
6761:
6751:
6737:
6731:
6725:
6718:
6701:
6695:
6691:
6681:
6667:
6654:
6645:
6639:
6612:
6609:
6606:
6596:
6582:
6569:
6560:
6554:
6527:
6524:
6521:
6511:
6510:
6509:
6506:
6492:
6469:
6463:
6457:
6453:
6436:
6430:
6423:
6413:
6412:
6411:
6391:
6385:
6381:
6364:
6358:
6351:
6341:
6340:
6339:
6336:
6334:
6333:minimal logic
6318:
6312:
6292:
6280:
6213:
6209:
6190:
6184:
6172:
6166:
6153:
6152:
6151:
6117:
6113:
6097:
6094:
6091:
6068:
6062:
6059:
6056:
6032:
6023:
6017:
5997:
5996:
5995:
5978:
5965:
5956:
5950:
5926:
5923:
5920:
5907:
5906:
5905:
5903:
5895:
5893:
5891:
5887:
5883:
5879:
5860:
5857:
5851:
5845:
5842:
5839:
5832:
5818:
5815:
5809:
5803:
5800:
5797:
5790:
5789:
5788:
5786:
5781:
5764:
5755:
5749:
5743:
5737:
5731:
5725:
5719:
5713:
5701:
5695:
5689:
5686:
5680:
5674:
5671:
5665:
5659:
5652:
5651:
5650:
5627:
5621:
5615:
5609:
5603:
5597:
5591:
5585:
5573:
5567:
5561:
5558:
5552:
5546:
5543:
5537:
5531:
5524:
5523:
5522:
5505:
5499:
5493:
5490:
5484:
5478:
5475:
5469:
5463:
5460:
5454:
5448:
5444:
5434:
5433:
5432:
5412:
5406:
5403:
5397:
5391:
5388:
5382:
5376:
5373:
5367:
5361:
5357:
5347:
5346:
5345:
5339:
5335:
5331:
5327:
5324:
5323:
5322:
5320:
5316:
5291:
5285:
5278:
5266:
5260:
5254:
5250:
5240:
5239:
5218:
5212:
5205:
5193:
5187:
5181:
5177:
5167:
5166:
5165:
5163:
5159:
5151:
5135:
5129:
5126:
5123:
5117:
5114:
5105:
5099:
5093:
5090:
5087:
5084:
5081:
5078:
5075:
5067:
5051:
5050:
5049:
5033:
5016:
5012:
5008:
5004:
4999:
4997:
4993:
4989:
4985:
4981:
4977:
4973:
4969:
4960:
4955:
4948:
4946:
4927:
4922:
4914:
4909:
4900:
4897:
4894:
4882:
4863:
4858:
4850:
4845:
4836:
4833:
4830:
4802:
4797:
4789:
4784:
4775:
4772:
4769:
4759:
4738:
4735:
4732:
4726:
4718:
4713:
4705:
4691:
4689:
4669:
4666:
4663:
4657:
4649:
4644:
4636:
4625:
4598:
4595:
4592:
4586:
4583:
4570:
4565:
4557:
4552:
4549:
4539:
4527:
4507:
4504:
4501:
4495:
4492:
4466:
4463:
4460:
4454:
4451:
4442:
4423:
4418:
4410:
4405:
4402:
4377:
4372:
4364:
4359:
4356:
4347:
4328:
4323:
4320:
4295:
4290:
4287:
4267:
4264:
4261:
4241:
4238:
4235:
4226:
4212:
4209:
4206:
4203:
4200:
4191:
4171:
4168:
4165:
4159:
4156:
4131:
4126:
4118:
4113:
4110:
4098:
4096:
4077:
4072:
4064:
4059:
4050:
4047:
4044:
4032:
4006:
4001:
3993:
3988:
3985:
3971:
3968:
3965:
3959:
3956:
3946:
3934:
3915:
3910:
3902:
3897:
3894:
3869:
3864:
3861:
3841:
3838:
3835:
3826:
3807:
3802:
3794:
3789:
3786:
3761:
3756:
3753:
3733:
3730:
3727:
3718:
3704:
3701:
3698:
3678:
3675:
3672:
3648:
3645:
3642:
3639:
3636:
3633:
3630:
3616:
3609:
3606:
3603:
3600:
3591:
3577:
3574:
3571:
3568:
3565:
3539:
3536:
3533:
3527:
3524:
3512:
3510:
3490:
3487:
3484:
3478:
3470:
3465:
3457:
3430:
3425:
3417:
3412:
3403:
3400:
3397:
3369:
3364:
3356:
3351:
3342:
3339:
3336:
3325:
3304:
3290:
3288:
3284:
3267:
3261:
3252:
3246:
3233:
3232:
3231:
3229:
3210:
3204:
3201:
3198:
3185:
3184:
3183:
3176:
3174:
3170:
3167:
3148:
3142:
3133:
3127:
3114:
3113:
3112:
3110:
3106:
3087:
3081:
3078:
3075:
3062:
3061:
3060:
3054:
3052:
3050:
3046:
3038:
3036:
3033:
3028:
3024:
3020:
3016:
3012:
3008:
3000:
2995:
2992:
2991:
2990:
2987:
2983:
2977:
2974:
2971:
2968:
2967:
2966:
2960:
2957:
2956:
2955:
2948:
2939:
2924:
2917:
2902:
2895:
2894:
2893:
2876:
2868:
2863:
2855:
2850:
2841:
2838:
2835:
2825:
2824:
2823:
2798:
2793:
2785:
2777:
2765:
2762:
2759:
2746:
2745:
2744:
2742:
2738:
2730:
2710:
2700:
2696:
2690:
2687:
2684:
2674:
2670:
2664:
2654:
2650:
2644:
2633:
2629:
2625:
2622:
2619:
2614:
2610:
2606:
2601:
2597:
2586:
2576:
2572:
2566:
2563:
2560:
2550:
2546:
2540:
2530:
2526:
2520:
2509:
2505:
2501:
2498:
2495:
2490:
2486:
2482:
2477:
2473:
2458:
2457:
2456:
2450:
2434:
2427:
2424:
2408:
2401:
2398:
2382:
2357:
2348:
2347:
2346:
2325:
2317:
2312:
2304:
2299:
2297:
2286:
2283:
2280:
2270:
2262:
2257:
2249:
2244:
2242:
2231:
2228:
2225:
2211:
2210:
2209:
2203:
2201:
2199:
2194:
2191:
2167:
2157:
2153:
2145:
2142:
2139:
2135:
2131:
2129:
2116:
2112:
2106:
2103:
2100:
2096:
2085:
2075:
2071:
2063:
2060:
2057:
2053:
2049:
2047:
2034:
2030:
2024:
2021:
2018:
2014:
1999:
1998:
1997:
1991:
1986:
1970:
1963:
1960:
1944:
1937:
1934:
1918:
1893:
1884:
1883:
1882:
1861:
1853:
1848:
1840:
1835:
1833:
1822:
1819:
1816:
1806:
1798:
1793:
1785:
1780:
1778:
1767:
1764:
1761:
1747:
1746:
1745:
1739:
1737:
1716:
1710:
1704:
1701:
1687:
1679:
1676:
1673:
1663:
1657:
1651:
1648:
1634:
1626:
1623:
1620:
1606:
1605:
1604:
1598:
1596:
1573:
1569:
1562:
1559:
1556:
1551:
1547:
1540:
1535:
1531:
1516:
1512:
1508:
1505:
1502:
1497:
1493:
1489:
1484:
1480:
1464:
1460:
1453:
1450:
1447:
1442:
1438:
1431:
1426:
1422:
1407:
1403:
1399:
1396:
1393:
1388:
1384:
1380:
1375:
1371:
1353:
1352:
1351:
1335:
1331:
1327:
1324:
1321:
1316:
1312:
1308:
1303:
1299:
1288:
1283:
1269:
1249:
1222:
1216:
1210:
1207:
1196:
1188:
1185:
1182:
1169:
1163:
1157:
1154:
1143:
1135:
1132:
1129:
1112:
1111:
1110:
1108:
1104:
1098:
1079:
1076:
1073:
1064:
1059:
1053:
1050:
1037:
1031:
1028:
1022:
1014:
1011:
1008:
990:
989:
983:
964:
961:
958:
949:
944:
938:
935:
922:
916:
913:
907:
899:
896:
893:
875:
874:
870:
847:
841:
838:
835:
826:
824:
816:
810:
807:
789:
783:
777:
774:
765:
763:
755:
752:
749:
732:
731:
730:
728:
705:
699:
696:
693:
684:
682:
674:
668:
665:
647:
641:
635:
632:
623:
621:
613:
610:
607:
590:
589:
588:
586:
582:
574:
572:
570:
566:
562:
543:
537:
534:
531:
525:
519:
516:
513:
507:
501:
498:
495:
489:
486:
479:
465:
459:
456:
453:
447:
441:
438:
435:
429:
423:
420:
417:
411:
408:
401:
400:
399:
392:
384:
380:
378:
375:" that means
374:
370:
366:
358:
355:
354:
353:
347:
344:
340:
339:
338:
332:
329:
328:
327:
324:
322:
318:
314:
310:
306:
303:
299:
295:
291:
287:
276:
275:instantiation
272:
269:
267:
266:instantiation
263:
260:
259:
258:
257:
254:
250:
247:
243:
237:
232:
230:
225:
223:
218:
216:
215:Transposition
213:
211:
208:
206:
201:
199:
194:
192:
190:Commutativity
187:
185:
183:Associativity
180:
179:
177:
176:
173:
169:
164:
161:
159:
157:
151:
149:
148:modus tollens
143:
138:
136:
130:
125:
123:
117:
112:
110:
104:
99:
97:
91:
86:
84:
78:
73:
71:
68:
65:elimination (
61:
56:
55:
54:
53:
50:
46:
43:
39:
36:
32:
26:
25:Venn diagrams
21:
8359:Georg Cantor
8354:Paul Bernays
8285:Morse–Kelley
8260:
8193:
8192:Subset
8139:hereditarily
8101:Venn diagram
8059:ordered pair
7974:Intersection
7958:
7918:Axiom schema
7675:Hugh MacColl
7650:Georg Cantor
7645:George Boole
7542:Introduction
7498:modus ponens
7482:
7426:Higher-order
7421:Second-order
7371:Distribution
7331:Truth tables
7259:
7235:
7213:
7181:
7171:
7163:the original
7148:
7133:
7128:
7120:
7115:
7108:
7107:Bocheński's
7103:
7095:the original
7081:
7072:
7067:
7052:
7048:
7031:
7024:
7013:
6985:
6978:
6960:
6954:
6926:
6919:
6822:
6507:
6484:
6409:
6337:
6281:
6205:
6083:
5993:
5899:
5875:
5782:
5779:
5648:
5520:
5430:
5343:
5337:
5333:
5329:
5325:
5318:
5312:
5155:
5014:
5010:
5006:
5002:
5000:
4964:
4883:
4757:
4695:
4626:
4528:
4443:
4348:
4227:
4192:
4102:
4033:
3935:
3827:
3719:
3592:
3516:
3295:Here we use
3294:
3285:
3282:
3225:
3180:
3171:
3165:
3163:
3104:
3102:
3058:
3042:
3027:Jean Buridan
3015:George Boole
3004:
2988:
2984:
2981:
2964:
2952:
2891:
2821:
2734:
2454:
2344:
2207:
2195:
2189:
2186:
1995:
1959:intersection
1880:
1743:
1735:
1602:
1594:
1286:
1284:
1241:
1100:
992:
987:
985:
877:
872:
866:
726:
724:
580:
578:
558:
397:
376:
373:exclusive or
368:
365:inclusive or
362:
351:
336:
325:
317:disjunctions
313:conjunctions
297:
293:
283:
273: /
264: /
209:
155:
152: /
147:
144: /
131: /
128:Constructive
118: /
105: /
92: /
79: /
67:modus ponens
66:
62: /
8384:Thomas Jech
8227:Alternative
8206:Uncountable
8160:Ultrafilter
8019:Cardinality
7923:replacement
7871:Determinacy
7620:Disjunction
7615:Conjunction
7600:Existential
7588:Elimination
7579:Disjunction
7574:Conjunction
7559:Existential
7416:First-order
7341:Truth value
7311:Quantifiers
6890:Isomorphism
5785:modal logic
5164:are duals:
5048:defined by
4980:logic gates
4280:, and thus
3049:conjunction
3045:disjunction
2731:Engineering
1103:tautologies
561:expressions
228:Exportation
115:Disjunctive
108:elimination
95:elimination
82:elimination
8424:Categories
8379:Kurt Gödel
8364:Paul Cohen
8201:Transitive
7969:Identities
7953:Complement
7940:Operations
7901:Regularity
7839:Adjunction
7798:Set theory
7670:Kurt Gödel
7533:Absorption
7435:Principles
7321:Connective
7250:PlanetMath
6911:References
5321:, such as
4961:diagrams).
4692:Conclusion
2737:electrical
1740:Set theory
587:notation:
367:" meaning
343:complement
141:Absorption
8312:Paradoxes
8232:Axiomatic
8211:Universal
8187:Singleton
8182:Recursive
8125:Countable
8120:Amorphous
7979:Power set
7896:Power set
7854:dependent
7849:countable
7605:Universal
7564:Universal
7467:Explosion
7452:Bivalence
7381:Soundness
7326:Tautology
7316:Predicate
7237:MathWorld
7220:EMS Press
7005:689858599
6793:¬
6786:∀
6783:¬
6779:→
6759:∃
6723:¬
6716:∃
6713:¬
6709:→
6689:∀
6652:¬
6646:∨
6637:¬
6624:¬
6620:→
6610:∧
6567:¬
6561:∧
6552:¬
6539:¬
6535:→
6525:∨
6451:∀
6448:¬
6444:→
6428:¬
6421:∃
6379:∃
6376:¬
6372:↔
6356:¬
6349:∀
6316:→
6290:¬
6182:¬
6176:¬
6173:∨
6164:¬
6112:conjuncts
6095:∧
6060:∧
6051:¬
6047:→
6030:¬
6024:∨
6015:¬
5963:¬
5957:∧
5948:¬
5934:↔
5924:∨
5915:¬
5882:Aristotle
5855:¬
5852:◻
5849:¬
5846:≡
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490:−
457:−
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439:−
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412:−
235:Tautology
8316:Problems
8220:Theories
8196:Superset
8172:Infinite
8001:Concepts
7881:Infinity
7805:Overview
7549:Negation
7376:Validity
7356:Logicism
7179:(1995),
6874:See also
4455:∉
4360:∉
4324:∉
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3676:∉
3593:Because
3569:∉
3558:. Then,
2397:overline
2198:mnemonic
1933:overline
1107:theorems
369:at least
321:negation
8254:General
8249:Zermelo
8155:subbase
8137: (
8076:Forcing
8054:Element
8026: (
8004:Methods
7891:Pairing
7304:General
7222:, 2001
4820:, then
4529:Hence,
3854:, then
3746:, then
3173:claim.
3105:neither
3001:History
2942:overbar
2892:where:
2447:is the
2421:is the
2345:where:
1983:is the
1957:is the
1881:where:
585:sequent
377:exactly
8145:Filter
8135:Finite
8071:Family
8014:Almost
7859:global
7844:Choice
7831:Axioms
7633:People
7189:
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3513:Part 1
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8167:Fuzzy
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8034:Class
8028:large
7989:Union
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5315:model
3887:, so
3779:, so
1985:union
302:valid
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7187:ISBN
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725:The
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341:The
315:and
288:and
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