10402:
567:
4963:". Since nothing is said in the definition of contraposition with regard to the predicate of the inferred proposition, it is permissible that it could be the original subject or its contradictory, and the predicate term of the resulting type "A" proposition is again undistributed. This results in two contrapositives, one where the predicate term is distributed, and another where the predicate term is undistributed.
8642:
4971:
could be the original subject or its contradictory. This is in contradistinction to the form of the propositions of transposition, which may be material implication, or a hypothetical statement. The difference is that in its application to categorical propositions the result of contraposition is two contrapositives, each being the obvert of the other, i.e. "No non-
4889:) are both type "A" propositions). Grammatically, one cannot infer "all mortals are men" from "All men are mortal". An type "A" proposition can only be immediately inferred by conversion when both the subject and predicate are distributed, as in the inference "All bachelors are unmarried men" from "All unmarried men are bachelors".
4782:
fire or combustion is occurring. While one can infer that fire stipulates the presence of oxygen, from the presence of oxygen the converse "If there is oxygen present, then fire is present" cannot be inferred. All that can be inferred from the original proposition is that "If oxygen is not present, then there cannot be fire".
4562:, with each proposition including an antecedent and consequential term. As a matter of logical inference, to transpose or convert the terms of one proposition requires the conversion of the terms of the propositions on both sides of the biconditional relationship, meaning that transposing or converting
6629:
4970:
in which from a given categorical proposition another categorical proposition is inferred which has as its subject the contradictory of the original predicate. Since nothing is said in the definition of contraposition with regard to the predicate of the inferred proposition, it is permissible that it
4781:
An example traditionally used by logicians contrasting sufficient and necessary conditions is the statement "If there is fire, then oxygen is present". An oxygenated environment is necessary for fire or combustion, but simply because there is an oxygenated environment does not necessarily mean that
4510:
Also, notice that contraposition is a method of inference which may require the use of other rules of inference. The contrapositive is the product of the method of contraposition, with different outcomes depending upon whether the contraposition is full, or partial. The successive applications of
4341:
is obtained for all the four types (A, E, I, and O types) of traditional propositions, yielding propositions with the contradictory of the original predicate, (full) contraposition is obtained by converting the obvert of the original proposition. For "E" statements, partial contraposition can be
7542:
4546:
In the inferred proposition, the consequent is the contradictory of the antecedent in the original proposition, and the antecedent of the inferred proposition is the contradictory of the consequent of the original proposition. The symbol for material implication signifies the proposition as a
4310:
propositions, which are compounds of other propositions, e.g. "If P, then Q" (P and Q are both propositions), and their existential impact is dependent upon further propositions where quantification existence is instantiated (existential instantiation), not on the hypothetical or materially
2962:
1912:
2090:
Here, we also know that B is either true or not true. If B is not true, then A is also not true. However, it is given that A is true, so the assumption that B is not true leads to a contradiction, which means that it is not the case that B is not true. Therefore, B must be true:
3649:
4987:". The distinction between the two contrapositives is absorbed and eliminated in the principle of transposition, which presupposes the "mediate inferences" of contraposition and is also referred to as the "law of contraposition".
6449:
3028:
6875:
360:." The converse is actually the contrapositive of the inverse, and so always has the same truth value as the inverse (which as stated earlier does not always share the same truth value as that of the original proposition).
7375:
2861:
4538:) to traditional logic and categorical propositions. In this sense the use of the term "contraposition" is usually referred to by "transposition" when applied to hypothetical propositions or material implications.
3457:
720:
In practice, this equivalence can be used to make proving a statement easier. For example, if one wishes to prove that every girl in the United States (A) has brown hair (B), one can either try to directly prove
4499:). This is because the obverse of the "E" proposition is an "A" proposition which cannot be validly converted except by limitation, that is, contraposition plus a change in the quantity of the proposition from
4490:
Notice that contraposition is a valid form of immediate inference only when applied to "A" and "O" propositions. It is not valid for "I" propositions, where the obverse is an "O" proposition which has no valid
3853:
3719:
1831:
1969:
It is given that, if A is true, then B is true, and it is also given that B is not true. We can then show that A must not be true by contradiction. For if A were true, then B would have to also be true (by
3087:
1565:
7312:
6707:
4526:. In traditional logic there is more than one contrapositive inferred from each original statement. In regard to the "A" proposition this is circumvented in the symbolism of modern logic by the rule of
715:
3783:
7257:
1820:
2854:
8003:
1836:
1292:
779:
by checking that all girls without brown hair are indeed all outside the US. In particular, if one were to find at least one girl without brown hair within the US, then one would have disproved
3323:
3243:
1410:
2210:
6425:
1710:
3911:
3515:
2148:
2035:
2806:
for propositional logic, only one side of the transposition is taken as an axiom, and the other is a theorem. We describe a proof of this theorem in the system of three axioms proposed by
6277:
1148:
Strictly speaking, a contraposition can only exist in two simple conditionals. However, a contraposition may also exist in two complex, universal conditionals, if they are similar. Thus,
5137:
The previous example employed the contrapositive of a definition to prove a theorem. One can also prove a theorem by proving the contrapositive of the theorem's statement. To prove that
8050:
7865:
4813:
Necessary and sufficient conditions can be explained by analogy in terms of the concepts and the rules of immediate inference of traditional logic. In the categorical proposition "All
2085:
1196:
296:
251:
173:
7201:
7083:
6999:
405:
7789:
7716:
2666:
2467:
2415:
1964:
1056:
8082:
7586:
7344:
6366:
6052:
6020:
5962:
4346:, it can be either the original subject, or its contradictory, resulting in two contrapositives which are the obverts of one another in the "A", "O", and "E" type propositions.
1471:
7169:
6915:
5655:
3549:
3369:
3166:
3127:
2792:
1626:
809:
777:
650:
7906:
5857:
453:
348:
141:
111:
7818:
7745:
7615:
6311:
5132:
5108:
5082:
5047:
2605:
6223:
6197:
6143:
1507:
905:
6334:
1433:
7932:
7129:
7045:
6941:
6171:
6113:
6078:
5988:
5930:
5623:
5571:
5544:
5424:
5401:
5368:
5344:
5305:
5282:
2706:
2579:
2556:
2513:
2251:
1594:
835:
745:
604:
543:
498:
6754:
5884:
5804:
5717:
7644:
856:. As a result, proving or disproving either one of these statements automatically proves or disproves the other, as they are logically equivalent to each other.
7672:
7103:
7019:
6961:
6782:
6727:
5775:
5739:
5683:
5591:
3556:
2757:
2737:
2625:
2533:
2490:
2374:
2354:
2334:
2314:
2294:
2274:
1753:
1733:
1332:
1312:
1236:
1216:
1119:
1099:
1079:
1000:
976:
948:
928:
519:
474:
6624:{\displaystyle (\omega _{P{\tilde {|}}Q}^{A},\omega _{P{\tilde {|}}\lnot Q}^{A})=(\omega _{Q|P}^{A},\omega _{Q|\lnot P}^{A})\,{\widetilde {\phi \,}}\,a_{P}\,,}
4255:. In some cases, contraposition involves a change of the former's quality (i.e. affirmation or negation). For its symbolic expression in modern logic, see the
4939:". In the obversion of the original proposition to a type "E" proposition, both terms become distributed. The obverse is then converted, resulting in "No non-
2969:
4530:, or the law of contraposition. In its technical usage within the field of philosophic logic, the term "contraposition" may be limited by logicians (e.g.
8781:
7537:{\displaystyle \Pr(\lnot P\mid \lnot Q)={\frac {\Pr(\lnot Q\mid \lnot P)\,a(\lnot P)}{\Pr(\lnot Q\mid \lnot P)\,a(\lnot P)+\Pr(\lnot Q\mid P)\,a(P)}}.}
6787:
4390:
The process is completed by further obversion resulting in the 'A' type proposition that is the obverted contrapositive of the original proposition,
4360:
which presupposes that all classes have members and the existential import presumed in the form of categorical propositions, one can derive first by
5245:" instead. More often than not, this approach is preferred if the contrapositive is easier to prove than the original conditional statement itself.
4790:
The symbol for the biconditional ("↔") signifies the relationship between the propositions is both necessary and sufficient, and is verbalized as "
2957:{\displaystyle \left(\phi \to \left(\psi \rightarrow \xi \right)\right)\to \left(\left(\phi \to \psi \right)\to \left(\phi \to \xi \right)\right)}
9456:
4999:
of a statement always has the same truth value (truth or falsity) as the statement itself, it can be a powerful tool for proving mathematical
3376:
9539:
8680:
2153:
Combining the two proved statements together, we obtain the sought-after logical equivalence between a conditional and its contrapositive:
1907:{\displaystyle {\begin{aligned}\neg A\lor B\,&\,\leftrightarrow B\lor \neg A\\\,&\,\leftrightarrow \neg B\to \neg A\end{aligned}}}
3790:
3656:
3036:
2220:
Logical equivalence between two propositions means that they are true together or false together. To prove that contrapositives are
1974:). However, it is given that B is not true, so we have a contradiction. Therefore, A is not true (assuming that we are dealing with
1515:
10431:
4829:
cannot be said to be distributed, or exhausted in its expression because it is indeterminate whether every instance of a member of
7262:
9853:
6637:
4633:
The truth of the rule of transposition is dependent upon the relations of sufficient condition and necessary condition in logic.
661:
3726:
10011:
8590:
8538:
8489:
8414:
4923:, a series of immediate inferences where the rule of obversion is first applied to the original categorical proposition "All
4825:
is said to be distributed, that is, all members of its class are exhausted in its expression. Conversely, the predicate term
17:
8799:
7210:
4208:
If a statement (or its contrapositive) and the inverse (or the converse) are both true or both false, then it is known as a
3130:
1782:
9866:
9189:
2817:
7937:
1241:
4184:.) That is, having four sides is both necessary to be a quadrilateral, and alone sufficient to deem it a quadrilateral.
3254:
3174:
1356:
9871:
9861:
9598:
9451:
8804:
2159:
8795:
6374:
10007:
8519:
8360:
8299:, pp. 65–66. Also, for reference to the immediate inferences of obversion, conversion, and obversion again, see
3090:
1656:
9349:
4907:
While most authors use the terms for the same thing, some authors distinguish transposition from contraposition. In
3860:
3464:
2097:
1984:
10104:
9848:
8673:
6232:
9409:
9102:
8646:
5751:
can be given, we choose to prove this statement by contraposition. The contrapositive of the above statement is:
5003:(especially if the truth of the contrapositive is easier to establish than the truth of the statement itself). A
3246:
218:
The law of contraposition says that a conditional statement is true if, and only if, its contrapositive is true.
8843:
8602:
Brody, Bobuch A. "Glossary of
Logical Terms". Encyclopedia of Philosophy. Vol. 5-6, p. 61. Macmillan, 1973.
8008:
7823:
4222:
3330:
10365:
10067:
9830:
9825:
9650:
9071:
8755:
2046:
1151:
269:
224:
146:
10360:
10143:
10060:
9773:
9704:
9581:
8823:
4618:
of the terms of the propositions, where the violation is that the changed proposition commits the fallacy of
4610:
Otherwise, converting the terms of one proposition and not the other renders the rule invalid, violating the
4558:
The biconditional statement of the rule of transposition (↔) refers to the relation between hypothetical (→)
8257:
Brody, p. 61. Macmillan, 1973. Also, Stebbing, p.65-66, Harper, 1961, and Copi, p. 141-143, Macmillan, 1953.
7617:, i.e. in addition to assigning TRUE or FALSE we can also assign any probability to the statement. The term
7174:
4344:
nothing is said in the definition of contraposition with regard to the predicate of the inferred proposition
10285:
10111:
9797:
9431:
9030:
8400:
7050:
6966:
375:
7750:
7677:
2633:
2434:
2382:
1928:
425:. " If the negation is true, then the original proposition (and by extension the contrapositive) is false.
10426:
10163:
10158:
9768:
9507:
9436:
8765:
8666:
4898:
1020:
8376:
8152:
8055:
7553:
7317:
6339:
6025:
5993:
5935:
4062:" An object which is blue is not red, and still has color. Therefore, in this case the inverse is false.
1438:
10092:
9682:
9076:
9044:
8735:
8282:, pp. 65–66. For reference to the initial step of contraposition as obversion and conversion, see
7134:
6880:
6081:
5229:, where one infers a conditional statement from its contrapositive. In other words, the conclusion "if
5086:". This contrapositive, like the original statement, is also true. Therefore, if it can be proven that
5628:
3522:
3342:
3139:
3100:
2765:
1599:
782:
750:
623:
10382:
10331:
10228:
9726:
9687:
9164:
8809:
8479:
8838:
7870:
5823:
4681:" does not necessarily have sufficient condition. The rule of inference for sufficient condition is
432:
327:
120:
90:
10223:
10153:
9692:
9544:
9527:
9250:
8730:
5248:
Logically, the validity of proof by contrapositive can be demonstrated by the use of the following
4912:
4623:
2679:
is false)", which is the definition of a material conditional. We can then make this substitution:
7794:
7721:
7591:
6284:
4511:
conversion and obversion within the process of contraposition may be given by a variety of names.
10055:
10032:
9993:
9879:
9820:
9466:
9386:
9230:
9174:
8787:
7548:
5113:
5089:
5063:
5028:
4846:
4500:
4326:
4279:
2584:
577:
shown, if something is in A, it must be in B as well. So we can interpret "all of A is in B" as:
6202:
6176:
6122:
5015:
can also be used with contraposition, as, for example, in the proof of the irrationality of the
10345:
10072:
10050:
10017:
9910:
9756:
9741:
9714:
9665:
9549:
9484:
9309:
9275:
9270:
9144:
8975:
8952:
5519:
4619:
1641:
312:
is not at all dependent on whether or not the original proposition was true, as evidenced here.
8509:
8325:
For an explanation of the absorption of obversion and conversion as "mediate inferences" see:
4719:
Since the converse of premise (1) is not valid, all that can be stated of the relationship of
4321:
of the subject and predicate, and is valid only for the type "A" and type "O" propositions of
1480:
878:
10275:
10128:
9920:
9638:
9374:
9280:
9139:
9124:
9005:
8980:
6316:
4527:
4278:, the process of contraposition is a schema composed of several steps of inference involving
4256:
4209:
1646:
1418:
8128:
7911:
7108:
7024:
6920:
6150:
6092:
6057:
5967:
5909:
5602:
5553:
5526:
5409:
5386:
5353:
5329:
5290:
5267:
2685:
2561:
2538:
2495:
2230:
1573:
1014:
consequent of the other, and vice versa. Thus a contrapositive generally takes the form of:
814:
724:
583:
525:
480:
10248:
10210:
10087:
9891:
9731:
9655:
9633:
9461:
9419:
9318:
9285:
9149:
8937:
8848:
8102:
8005:
so that the fraction on the right-hand side of the equation above is equal to 1, and hence
6732:
5903:
5862:
5782:
5695:
5050:
5012:
4611:
4307:
4295:
4244:
4089:
2221:
747:
by checking that all girls in the United States do indeed have brown hair, or try to prove
84:
55:
51:
7620:
3644:{\displaystyle (\neg \neg p\to p)\to ((p\to \neg \neg q)\to (\neg \neg p\to \neg \neg q))}
8:
10377:
10268:
10253:
10233:
10190:
10077:
10027:
9953:
9898:
9835:
9628:
9623:
9571:
9339:
9328:
9000:
8900:
8828:
8819:
8815:
8750:
8745:
8108:
5889:
Having proved the contrapositive, we can then infer that the original statement is true.
4967:
4615:
4523:
4515:
4325:, while it is conditionally valid for "E" type propositions if a change in quantity from
4291:
4252:
4236:
2421:
1474:
1138:
4259:. Contraposition also has philosophical application distinct from the other traditional
1636:
of propositional logic. The principle was stated as a theorem of propositional logic by
10406:
10175:
10138:
10123:
10116:
10099:
9903:
9885:
9751:
9677:
9660:
9613:
9426:
9335:
9169:
9154:
9114:
9066:
9051:
9039:
8995:
8970:
8740:
8689:
7657:
7088:
7004:
6946:
6767:
6712:
5760:
5724:
5668:
5576:
5226:
4627:
4492:
4376:
4283:
4264:
4051:" This follows logically from our initial statement and, like it, it is evidently true.
2742:
2722:
2610:
2518:
2475:
2359:
2339:
2319:
2299:
2279:
2259:
2040:
We can apply the same process the other way round, starting with the assumptions that:
1738:
1718:
1317:
1297:
1221:
1201:
1104:
1084:
1064:
985:
979:
961:
933:
913:
504:
459:
114:
77:
66:
9359:
5058:
because it is a restatement of a definition. The contrapositive of this statement is "
4149:" Again, in this case, unlike the last example, the converse of the statement is true.
3023:{\displaystyle \left(\lnot \phi \to \lnot \psi \right)\to \left(\psi \to \phi \right)}
2807:
10401:
10341:
10148:
9958:
9948:
9840:
9721:
9556:
9532:
9313:
9297:
9202:
9179:
9056:
9025:
8990:
8885:
8720:
8586:
8544:
8534:
8515:
8485:
8410:
8356:
8085:
7651:
7366:
7351:
7346:
being TRUE. Hence, the subjective Bayes' theorem represents a generalization of both
6761:
5222:
4908:
4287:
4228:
1773:
1629:
555:
3913: (from (7) and (8) using the hypothetical syllogism metatheorem)
3785: (from (3) and (6) using the hypothetical syllogism metatheorem)
10355:
10350:
10243:
10200:
10022:
9983:
9978:
9963:
9789:
9746:
9643:
9441:
9391:
8965:
8927:
8200:
7204:
6440:
5016:
1637:
316:
10336:
10326:
10280:
10263:
10218:
10180:
10082:
10002:
9809:
9736:
9709:
9697:
9603:
9517:
9491:
9446:
9414:
9215:
9017:
8960:
8910:
8875:
8833:
8459:
6870:{\displaystyle (\omega _{P{\tilde {|}}Q}^{A},\omega _{P{\tilde {|}}\lnot Q}^{A})}
5020:
4084:" This statement is false because the initial statement which it negates is true.
2296:
is false. Therefore, we can reduce this proposition to the statement "False when
258:
73:
8404:
8176:
4205:
If a statement's negation is false, then the statement is true (and vice versa).
10321:
10300:
10258:
10238:
10133:
9988:
9586:
9576:
9566:
9561:
9495:
9369:
9245:
9134:
9129:
9107:
8708:
4857:" cannot be inferred by conversion from the original type "A" proposition "All
4791:
4535:
4518:
of a statement and its contrapositive as defined in traditional class logic is
4202:
If a statement's inverse is false, then its converse is false (and vice versa).
4102:
3033:(A3) already gives one of the directions of the transposition. The other side,
2803:
1975:
8548:
5110:
cannot be expressed as an irreducible fraction, then it must be the case that
4138:" In this case, unlike the last example, the inverse of the statement is true.
10420:
10295:
9973:
9480:
9265:
9255:
9225:
9210:
8880:
8235:, p. 141, Macmillan, 1953. All sources give virtually identical definitions.
5547:
5144:
4745:
4248:
4199:
If a statement's inverse is true, then its converse is true (and vice versa).
4165:
4093:
2425:
1756:
574:
551:
8652:
5182:
being positive integers with no common prime factor, and squaring to obtain
4911:
the reasoning process of transposition as a rule of inference is applied to
4495:. The contraposition of the "E" proposition is valid only with limitations (
2215:
10195:
10042:
9943:
9935:
9815:
9763:
9672:
9608:
9591:
9522:
9381:
9240:
8942:
8725:
5748:
5008:
4683:
4196:
If a statement is false, then its contrapositive is false (and vice versa).
4073:" Objects can have other colors, so the converse of our statement is false.
1971:
5312:
1145:, the other is also true, and when one is false, the other is also false.
655:
is the contrapositive of the above statement. Therefore, one can say that
10305:
10185:
9364:
9354:
9301:
8985:
8905:
8890:
8770:
8715:
5249:
5214:
4531:
4240:
4193:
If a statement is true, then its contrapositive is true (and vice versa).
4124:
1142:
309:
35:
5011:
of the contrapositive of a statement. However, indirect methods such as
9235:
9090:
9061:
8867:
5084:
cannot be expressed as an irreducible fraction, then it is not rational
4902:
4504:
4330:
4303:
4275:
1003:
70:
4136:
If a polygon is not a quadrilateral, then it does not have four sides.
4121:
If a polygon does not have four sides, then it is not a quadrilateral.
4088:
In other words, the contrapositive is logically equivalent to a given
3452:{\displaystyle (q\to \neg \neg q)\to ((p\to q)\to (p\to \neg \neg q))}
10387:
10290:
9343:
9260:
9220:
9184:
9120:
8932:
8922:
8895:
8658:
7647:
6757:
5134:
is not a rational number. The latter can be proved by contradiction.
4920:
4361:
4338:
4268:
4260:
47:
2224:, we need to understand when material implication is true or false.
10372:
10170:
9618:
9323:
8917:
6963:
can assign any subjective opinion to the statement. The case where
4318:
4164:
Since the statement and the converse are both true, it is called a
4123:" This follows logically, and as a rule, contrapositives share the
1011:
364:
6784:. The pair of derivative inverted conditional opinions is denoted
4375:
The contrapositive of the original proposition is then derived by
4158:
There is at least one quadrilateral that does not have four sides.
3848:{\displaystyle (\neg \neg p\to \neg \neg q)\to (\neg q\to \neg p)}
3714:{\displaystyle (p\to \neg \neg q)\to (\neg \neg p\to \neg \neg q)}
566:
9968:
8760:
5000:
4865:". All that can be inferred is the type "A" proposition "All non-
4322:
1633:
1347:
6709:
denotes a pair of binomial conditional opinions given by source
4349:
By example: from an original, 'A' type categorical proposition,
1825:
which can be made equivalent to its contrapositive, as follows:
1137:
to the original and is logically equivalent to it. Due to their
617:
be within A, either. This statement, which can be expressed as:
8641:
8227:. Vol. 5-6, p. 61. Macmillan, 1973. Also, Stebbing, L. Susan.
4342:
obtained by additionally making a change in quantity. Because
3082:{\displaystyle (\psi \to \phi )\to (\neg \phi \to \neg \psi )}
1767:
1560:{\displaystyle {\frac {P\to Q}{\therefore \neg Q\to \neg P}},}
9512:
8858:
8703:
8583:
Subjective Logic; A formalism for
Reasoning Under Uncertainty
5812:
This latter statement can be proven as follows: suppose that
2797:
1125:
is not the case." Using our example, this is rendered as "If
31:
8231:. Seventh edition, p.65-66. Harper, 1961, and Irving Copi's
7307:{\displaystyle \omega _{\lnot P{\widetilde {|}}\lnot Q}^{A}}
7259:
and thereby an absolute TRUE derivative conditional opinion
4271:
where equivocation varies with different proposition types.
8351:
Smith, Douglas; Eggen, Maurice; St. Andre, Richard (2001),
6702:{\displaystyle (\omega _{Q|P}^{A},\omega _{Q|\lnot P}^{A})}
6439:
represents an instance of the subjective Bayes' theorem in
5237:" is inferred by constructing a proof of the claim "if not
4837:
as a class. All that can be validly inferred is that "Some
710:{\displaystyle (A\to B)\leftrightarrow (\neg B\to \neg A).}
7207:
produces an absolute FALSE derivative conditional opinion
4243:
is inferred from another and where the former has for its
6943:, i.e. in addition to assigning TRUE or FALSE the source
5513:
2216:
More rigorous proof of the equivalence of contrapositives
1596:" appears on a line of a proof, it can be replaced with "
7171:
is absolute TRUE the subjective Bayes' theorem operator
4147:
If a polygon has four sides, then it is a quadrilateral.
4109:
If a polygon is a quadrilateral, then it has four sides.
3778:{\displaystyle (p\to q)\to (\neg \neg p\to \neg \neg q)}
1141:, stating one effectively states the other; when one is
8533:(13th ed.). New York City: McGraw-Hill Education.
4947:", maintaining distribution of both terms. The "No non-
3721: (from (4) and (5) by modus ponens)
3517: (from (1) and (2) by modus ponens)
8449:
Brody, Bobuch A. (1973). "Glossary of
Logical Terms".
7252:{\displaystyle \omega _{P{\widetilde {|}}\lnot Q}^{A}}
1815:{\displaystyle A\to B\,\leftrightarrow \,\neg A\lor B}
8478:
Copi, Irving M.; Cohen, Carl; Rodych, Victor (2016).
8058:
8011:
7940:
7914:
7873:
7826:
7797:
7753:
7724:
7680:
7660:
7623:
7594:
7556:
7378:
7320:
7265:
7213:
7177:
7137:
7111:
7091:
7085:
is an absolute FALSE opinion is equivalent to source
7053:
7027:
7007:
6969:
6949:
6923:
6883:
6790:
6770:
6735:
6715:
6640:
6452:
6377:
6342:
6319:
6287:
6235:
6205:
6179:
6153:
6125:
6095:
6060:
6028:
5996:
5970:
5938:
5912:
5865:
5826:
5820:
is odd. The product of two odd numbers is odd, hence
5785:
5763:
5727:
5698:
5671:
5631:
5605:
5579:
5556:
5529:
5412:
5389:
5356:
5332:
5293:
5270:
5149:, we can equivalently prove its contrapositive, that
5116:
5092:
5066:
5031:
4661:, as an individual or a class, materially implicates
4060:
If an object is not red, then it does not have color.
4049:
If an object does not have color, then it is not red.
3863:
3793:
3729:
3659:
3559:
3525:
3467:
3379:
3345:
3257:
3177:
3142:
3103:
3089:, is proven below, using the following lemmas proven
3039:
2972:
2864:
2820:
2768:
2745:
2725:
2688:
2636:
2613:
2587:
2564:
2541:
2521:
2498:
2478:
2437:
2385:
2362:
2342:
2322:
2302:
2282:
2262:
2233:
2162:
2100:
2049:
1987:
1931:
1834:
1785:
1741:
1721:
1659:
1602:
1576:
1518:
1483:
1441:
1421:
1359:
1320:
1300:
1244:
1224:
1204:
1154:
1107:
1087:
1067:
1023:
988:
964:
936:
916:
881:
817:
785:
753:
727:
664:
626:
586:
528:
507:
483:
462:
435:
378:
330:
272:
227:
149:
123:
93:
7131:
is FALSE. In the case when the conditional opinion
7001:
is an absolute TRUE opinion is equivalent to source
4776:
4687:, which is an argument for conditional implication:
2849:{\displaystyle \phi \to \left(\psi \to \phi \right)}
8556:Prior, Arthur Norman (1973). "Logic, Traditional".
8350:
7998:{\displaystyle \Pr(\lnot Q\mid P)=1-\Pr(Q\mid P)=0}
4892:
4743:. The rule of inference for necessary condition is
4082:
There exists a red object that does not have color.
1509:in some logical system; or as a rule of inference:
1287:{\displaystyle \forall {x}(\neg Q{x}\to \neg P{x})}
423:
Sometimes, when it is raining, I don't wear my coat
8529:Moore, Brooke Noel; Parker, Richard Burl (2020) .
8076:
8044:
7997:
7926:
7900:
7859:
7812:
7783:
7739:
7710:
7666:
7638:
7609:
7580:
7536:
7338:
7306:
7251:
7195:
7163:
7123:
7097:
7077:
7039:
7013:
6993:
6955:
6935:
6909:
6869:
6776:
6748:
6721:
6701:
6623:
6419:
6360:
6328:
6305:
6271:
6217:
6191:
6165:
6137:
6107:
6072:
6046:
6014:
5982:
5956:
5924:
5878:
5851:
5798:
5769:
5733:
5711:
5677:
5649:
5617:
5585:
5565:
5538:
5418:
5395:
5362:
5338:
5299:
5276:
5126:
5102:
5076:
5041:
3905:
3847:
3777:
3713:
3643:
3543:
3509:
3451:
3363:
3318:{\displaystyle (p\to q)\to ((q\to r)\to (p\to r))}
3317:
3238:{\displaystyle (q\to r)\to ((p\to q)\to (p\to r))}
3237:
3160:
3121:
3081:
3022:
2956:
2848:
2786:
2751:
2731:
2700:
2660:
2619:
2599:
2573:
2550:
2527:
2507:
2484:
2461:
2409:
2368:
2348:
2328:
2308:
2288:
2268:
2245:
2204:
2142:
2079:
2029:
1958:
1906:
1814:
1747:
1727:
1704:
1620:
1588:
1559:
1501:
1465:
1427:
1405:{\displaystyle (P\to Q)\vdash (\neg Q\to \neg P),}
1404:
1326:
1306:
1286:
1230:
1210:
1190:
1113:
1093:
1073:
1050:
994:
970:
942:
922:
899:
829:
803:
771:
739:
709:
644:
598:
537:
513:
492:
468:
447:
399:
342:
290:
245:
167:
135:
105:
8332:
2205:{\displaystyle (A\to B)\equiv (\neg B\to \neg A)}
10418:
8399:
8012:
7971:
7941:
7874:
7827:
7754:
7681:
7557:
7494:
7451:
7409:
7379:
6420:{\displaystyle (P\to Q)\to (\lnot Q\to \lnot P)}
4955:" is again obverted, resulting in the "All non-
4298:implies the proposition as referring to a class
1917:
1570:where the rule is that wherever an instance of "
8477:
6022:(see below), but the reverse implication, from
5625:, prove its contrapositive statement, which is
4785:
1705:{\displaystyle (P\to Q)\to (\neg Q\to \neg P),}
253:) can be compared with three other statements:
8223:Brody, Bobuch A. "Glossary of Logical Terms".
7369:which in a specific form can be expressed as:
5310:share the same truth values in all scenarios:
4547:hypothetical, or the "if–then" form, e.g. "if
3906:{\displaystyle (p\to q)\to (\neg q\to \neg p)}
3510:{\displaystyle (p\to q)\to (p\to \neg \neg q)}
2143:{\displaystyle (\neg B\to \neg A)\to (A\to B)}
2030:{\displaystyle (A\to B)\to (\neg B\to \neg A)}
8674:
4223:Categorical proposition § Contraposition
2759:, we then obtain the desired contrapositive:
1628:"; or as the statement of a truth-functional
1010:of the other only when its antecedent is the
6272:{\displaystyle (Q\to \bot )\to (P\to \bot )}
5049:is rational, then it can be expressed as an
3651: (instance of the (HS2))
3551: (instance of the (DN1))
3371: (instance of the (DN2))
308:." Unlike the contrapositive, the inverse's
65:. The contrapositive of a statement has its
62:
8528:
4653:is sufficient reason for the occurrence of
3459: (instance of the (HS1)
2336:" (i.e. "True when it is not the case that
1768:Simple proof by definition of a conditional
500:), then it can logically be concluded that
8866:
8681:
8667:
8045:{\displaystyle \Pr(\lnot P\mid \lnot Q)=1}
7860:{\displaystyle \Pr(\lnot P\mid \lnot Q)=1}
7820:being FALSE. It is then easy to see that
4990:
4673:is such that the converse proposition "If
2798:In classical propositional calculus system
61:, and an associated proof method known as
8468:
8453:. Vol. 5–6. Macmillan. p. 61ff.
8355:(5th ed.), Brooks/Cole, p. 37,
8201:"Predicates and Quantified Statements II"
7515:
7475:
7433:
7185:
6617:
6606:
6598:
6590:
5892:
5155:has a square root that is rational, then
4302:, in contrast to the conditional form of
4035:" This can be equivalently expressed as "
4003:reversal and negation of both statements
3325:- another form of Hypothetical syllogism.
2080:{\displaystyle (\neg B\to \neg A)\land A}
1881:
1876:
1856:
1851:
1799:
1795:
1191:{\displaystyle \forall {x}(P{x}\to Q{x})}
8564:
8313:
8296:
8279:
8177:"Modus ponens and modus tollens | logic"
7357:
5546:is true. Use this assumption to prove a
4541:
3949:first statement implies truth of second
3333:as a shorthand for several proof steps.
565:
561:
291:{\displaystyle \neg P\rightarrow \neg Q}
246:{\displaystyle \neg Q\rightarrow \neg P}
168:{\displaystyle \neg Q\rightarrow \neg P}
8270:. Seventh edition, p. 66. Harper, 1961.
5897:
4636:
4286:. A categorical proposition contains a
4092:statement, though not sufficient for a
4071:If an object has color, then it is red.
4037:If an object is red, then it has color.
3855: (instance of (A3))
872:when the following relationship holds:
609:It is also clear that anything that is
14:
10419:
8688:
8507:
7196:{\displaystyle {\widetilde {\phi \,}}}
5514:Difference with proof by contradiction
4714:
4251:of the original logical proposition's
8662:
8595:Blumberg, Albert E. "Logic, Modern".
8555:
8469:Copi, Irving M.; Cohen, Carl (2005).
8448:
8406:Proof in Mathematics: An Introduction
8338:
7078:{\displaystyle \omega _{Q\mid P}^{A}}
6994:{\displaystyle \omega _{Q\mid P}^{A}}
5932:cannot be proven to be equivalent to
4586:requires that the other proposition,
4311:implicative propositions themselves.
3168:- Double negation (another direction)
2671:This reads "It is not the case that (
1435:is a metalogical symbol meaning that
400:{\displaystyle \neg (P\rightarrow Q)}
8498:
8457:
8353:A Transition to Advanced Mathematics
8326:
8300:
8283:
7784:{\displaystyle \Pr(\lnot Q\mid P)=0}
7711:{\displaystyle \Pr(\lnot Q\mid P)=1}
4794:", or, according to the example "If
4317:is the simultaneous interchange and
4294:where the existential impact of the
4216:
2661:{\displaystyle \neg (R\land \neg S)}
2462:{\displaystyle \neg (\neg Q\land P)}
2410:{\displaystyle \neg (P\land \neg Q)}
1959:{\displaystyle (A\to B)\land \neg B}
859:
840:In general, for any statement where
8514:(11th ed.). Cengage Learning.
6431:
5221:, or proof by contraposition, is a
2424:can be reversed with no effect (by
1755:are propositions expressed in some
1337:
1051:{\displaystyle (\neg Q\to \neg P).}
24:
8619:Prior, A.N. "Logic, Traditional".
8616:. MacMillan, 1979, fifth edition.
8374:
8077:{\displaystyle \lnot Q\to \lnot P}
8068:
8059:
8027:
8018:
7947:
7842:
7833:
7804:
7760:
7731:
7687:
7601:
7588:generalizes the logical statement
7581:{\displaystyle \Pr(\lnot Q\mid P)}
7563:
7500:
7482:
7466:
7457:
7440:
7424:
7415:
7394:
7385:
7339:{\displaystyle \lnot Q\to \lnot P}
7330:
7321:
7291:
7271:
7236:
6917:generalizes the logical statement
6851:
6683:
6574:
6513:
6408:
6399:
6361:{\displaystyle \lnot Q\to \lnot P}
6352:
6343:
6320:
6297:
6263:
6245:
6212:
6186:
6132:
6047:{\displaystyle \lnot Q\to \lnot P}
6038:
6029:
6015:{\displaystyle \lnot Q\to \lnot P}
6006:
5997:
5957:{\displaystyle \lnot Q\to \lnot P}
5948:
5939:
5641:
5632:
5557:
5530:
5523:: Assume (for contradiction) that
5413:
5390:
5357:
5333:
5294:
5271:
5023:, the statement can be made that "
4160:" This statement is clearly false.
3894:
3885:
3836:
3827:
3812:
3809:
3800:
3797:
3766:
3763:
3754:
3751:
3702:
3699:
3690:
3687:
3672:
3669:
3629:
3626:
3617:
3614:
3599:
3596:
3566:
3563:
3529:
3526:
3498:
3495:
3437:
3434:
3392:
3389:
3355:
3352:
3331:hypothetical syllogism metatheorem
3152:
3149:
3107:
3104:
3070:
3061:
2987:
2978:
2778:
2769:
2649:
2637:
2591:
2588:
2565:
2542:
2499:
2444:
2438:
2398:
2386:
2193:
2184:
2113:
2104:
2062:
2053:
2018:
2009:
1950:
1894:
1885:
1866:
1839:
1800:
1690:
1681:
1612:
1603:
1545:
1536:
1466:{\displaystyle (\neg Q\to \neg P)}
1454:
1445:
1390:
1381:
1270:
1256:
1245:
1155:
1036:
1027:
958:." In a conditional such as this,
795:
786:
763:
754:
695:
686:
636:
627:
529:
484:
379:
282:
273:
237:
228:
159:
150:
25:
10443:
8634:
7164:{\displaystyle \omega _{Q|P}^{A}}
6910:{\displaystyle \omega _{Q|P}^{A}}
5170:equal to the rational expression
4777:Necessity and sufficiency example
4598:to be transposed or converted to
4379:to another 'E' type proposition,
1776:, the conditional is defined as:
10400:
8640:
5650:{\displaystyle \neg B\to \neg A}
4893:Distinguished from transposition
4395:All non-voters are non-residents
3544:{\displaystyle \neg \neg p\to p}
3364:{\displaystyle q\to \neg \neg q}
3161:{\displaystyle p\to \neg \neg p}
3122:{\displaystyle \neg \neg p\to p}
2787:{\displaystyle \neg Q\to \neg P}
1978:that are either true or false):
1621:{\displaystyle \neg Q\to \neg P}
1133:." This statement is said to be
804:{\displaystyle \neg B\to \neg A}
772:{\displaystyle \neg B\to \neg A}
645:{\displaystyle \neg B\to \neg A}
10432:Theorems in propositional logic
8511:A Concise Introduction to Logic
8433:
8424:
8393:
8368:
8344:
8319:
8306:
8289:
8248:, pp. 123-157, Macmillan, 1953.
8088:represents a generalization of
6371:Discharge assumption; conclude
6229:Discharge assumption; conclude
5147:, its square root is irrational
4833:as a class is also a member of
4739:is the necessary condition for
4418:Obverted (full) contraposition
4099:Similarly, take the statement "
2804:Hilbert-style deductive systems
868:is implicated by a proposition
8628:A Modern Introduction to Logic
8567:A Modern Introduction to Logic
8273:
8268:A Modern Introduction to Logic
8260:
8251:
8238:
8229:A Modern Introduction to Logic
8217:
8193:
8169:
8145:
8129:"Definition of CONTRAPOSITIVE"
8121:
8065:
8033:
8015:
7986:
7974:
7959:
7944:
7918:
7901:{\displaystyle \Pr(Q\mid P)=1}
7889:
7877:
7848:
7830:
7801:
7772:
7757:
7728:
7699:
7684:
7633:
7627:
7598:
7575:
7560:
7525:
7519:
7512:
7497:
7488:
7479:
7472:
7454:
7446:
7437:
7430:
7412:
7400:
7382:
7327:
7281:
7226:
7147:
7115:
7031:
6927:
6893:
6864:
6845:
6841:
6810:
6806:
6791:
6696:
6679:
6653:
6641:
6587:
6570:
6544:
6532:
6526:
6507:
6503:
6472:
6468:
6453:
6414:
6405:
6396:
6393:
6390:
6384:
6378:
6349:
6300:
6294:
6288:
6266:
6260:
6254:
6251:
6248:
6242:
6236:
6209:
6183:
6157:
6129:
6099:
6064:
6035:
6003:
5974:
5945:
5916:
5852:{\displaystyle x^{2}=x\cdot x}
5638:
5609:
4401:The schema of contraposition:
3917:
3900:
3891:
3882:
3879:
3876:
3870:
3864:
3842:
3833:
3824:
3821:
3818:
3806:
3794:
3772:
3760:
3748:
3745:
3742:
3736:
3730:
3708:
3696:
3684:
3681:
3678:
3666:
3660:
3638:
3635:
3623:
3611:
3608:
3605:
3593:
3587:
3584:
3581:
3578:
3572:
3560:
3535:
3504:
3492:
3486:
3483:
3480:
3474:
3468:
3446:
3443:
3431:
3425:
3422:
3419:
3413:
3407:
3404:
3401:
3398:
3386:
3380:
3349:
3329:We also use the method of the
3312:
3309:
3303:
3297:
3294:
3291:
3285:
3279:
3276:
3273:
3270:
3264:
3258:
3232:
3229:
3223:
3217:
3214:
3211:
3205:
3199:
3196:
3193:
3190:
3184:
3178:
3146:
3113:
3076:
3067:
3058:
3055:
3052:
3046:
3040:
3009:
2998:
2984:
2938:
2927:
2916:
2900:
2884:
2873:
2835:
2824:
2775:
2692:
2655:
2640:
2456:
2441:
2404:
2389:
2237:
2199:
2190:
2181:
2175:
2169:
2163:
2137:
2131:
2125:
2122:
2119:
2110:
2101:
2068:
2059:
2050:
2024:
2015:
2006:
2003:
2000:
1994:
1988:
1944:
1938:
1932:
1891:
1882:
1857:
1796:
1789:
1696:
1687:
1678:
1675:
1672:
1666:
1660:
1609:
1580:
1542:
1525:
1496:
1490:
1484:
1460:
1451:
1442:
1396:
1387:
1378:
1372:
1366:
1360:
1281:
1267:
1253:
1185:
1174:
1163:
1042:
1033:
1024:
894:
888:
882:
821:
792:
760:
731:
701:
692:
683:
680:
677:
671:
665:
633:
590:
455:is true and one is given that
448:{\displaystyle P\rightarrow Q}
439:
394:
388:
382:
343:{\displaystyle Q\rightarrow P}
334:
279:
234:
156:
136:{\displaystyle P\rightarrow Q}
127:
106:{\displaystyle P\rightarrow Q}
97:
63:§ Proof by contrapositive
13:
1:
10361:History of mathematical logic
8114:
5161:This can be shown by setting
4735:does not occur, meaning that
4170:A polygon is a quadrilateral
4107:" or equivalently expressed "
1918:Simple proof by contradiction
10286:Primitive recursive function
7813:{\displaystyle P\to \lnot Q}
7740:{\displaystyle P\to \lnot Q}
7610:{\displaystyle P\to \lnot Q}
7047:is TRUE, and the case where
6306:{\displaystyle (A\to \bot )}
4966:Contraposition is a type of
4931:"; yielding the obverse "No
4786:Relationship of propositions
4484:(O) Some non-P is not non-S
4180:is sometimes abbreviated as
4020:contradicts the implication
3985:reversal of both statements
3967:negation of both statements
545:). This is often called the
7:
8565:Stebbing, L. Susan (1961).
8508:Hurley, Patrick J. (2011).
8153:"The Law of Contraposition"
8095:
5127:{\displaystyle {\sqrt {2}}}
5103:{\displaystyle {\sqrt {2}}}
5077:{\displaystyle {\sqrt {2}}}
5042:{\displaystyle {\sqrt {2}}}
4899:Transposition (mathematics)
4769:Conclusion: Therefore, not
4384:No non-voters are residents
4369:No residents are non-voters
4168:, and can be expressed as "
4033:All red objects have color.
4026:
2600:{\displaystyle \neg \neg P}
1346:rule may be expressed as a
613:within B (the blue region)
10:
10448:
9350:Schröder–Bernstein theorem
9077:Monadic predicate calculus
8736:Foundations of mathematics
8621:Encyclopedia of Philosophy
8597:Encyclopedia of Philosophy
8575:
8558:Encyclopedia of Philosophy
8503:(5th ed.). MacMillan.
8451:Encyclopedia of Philosophy
8377:"Proofs by Contrapositive"
8225:Encyclopedia of Philosophy
8157:beisecker.faculty.unlv.edu
7547:In the equation above the
7365:represents an instance of
6877:. The conditional opinion
6218:{\displaystyle P\to \bot }
6192:{\displaystyle Q\to \bot }
6138:{\displaystyle Q\to \bot }
6082:law of the excluded middle
5660:
4896:
4727:is that in the absence of
4364:the 'E' type proposition,
4220:
522:must be also false (i.e.,
27:Mathematical logic concept
10396:
10383:Philosophy of mathematics
10332:Automated theorem proving
10314:
10209:
10041:
9934:
9786:
9503:
9479:
9457:Von Neumann–Bernays–Gödel
9402:
9296:
9200:
9098:
9089:
9016:
8951:
8857:
8779:
8696:
8630:. Cromwell Company, 1931.
8623:, Vol.5, Macmillan, 1973.
8599:, Vol.5, Macmillan, 1973.
8560:. Vol. 5. Macmillan.
7934:is TRUE. This is because
5252:, where it is shown that
5019:. By the definition of a
3336:The proof is as follows:
2607:, which is equal to just
1762:
1101:", or, more clearly, "If
8499:Copi, Irving M. (1979).
8484:. Taylor & Francis.
8458:Copi, Irving M. (1953).
8052:which is equivalent to
4913:categorical propositions
4802:'if and only if' if not
4624:affirming the consequent
4459:(O) Some S is not non-P
4354:All residents are voters
4300:with at least one member
4280:categorical propositions
4187:
2256:This is only false when
1502:{\displaystyle (P\to Q)}
900:{\displaystyle (P\to Q)}
117:: the contrapositive of
10033:Self-verifying theories
9854:Tarski's axiomatization
8805:Tarski's undefinability
8800:incompleteness theorems
8569:(7th ed.). Harper.
8181:Encyclopedia Britannica
8133:www.merriam-webster.com
7549:conditional probability
7314:which is equivalent to
6329:{\displaystyle \lnot A}
6084:or an equivalent axiom.
5597:Proof by contrapositive
5219:proof by contrapositive
5005:proof by contrapositive
4991:Proof by contrapositive
4707:Conclusion: Therefore,
4641:In the proposition "If
4435:(A) All non-P is non-S
1428:{\displaystyle \vdash }
1006:. One statement is the
411:It is not the case that
10407:Mathematics portal
10018:Proof of impossibility
9666:propositional variable
8976:Propositional calculus
8653:Improper Transposition
8078:
8046:
7999:
7928:
7927:{\displaystyle P\to Q}
7902:
7861:
7814:
7785:
7741:
7712:
7668:
7640:
7611:
7582:
7538:
7340:
7308:
7253:
7197:
7165:
7125:
7124:{\displaystyle P\to Q}
7099:
7079:
7041:
7040:{\displaystyle P\to Q}
7015:
6995:
6957:
6937:
6936:{\displaystyle P\to Q}
6911:
6871:
6778:
6750:
6723:
6703:
6625:
6429:
6421:
6362:
6330:
6307:
6273:
6219:
6193:
6167:
6166:{\displaystyle P\to Q}
6139:
6109:
6108:{\displaystyle P\to Q}
6074:
6073:{\displaystyle P\to Q}
6048:
6016:
5984:
5983:{\displaystyle P\to Q}
5958:
5926:
5925:{\displaystyle P\to Q}
5893:In nonclassical logics
5880:
5853:
5800:
5771:
5735:
5713:
5679:
5651:
5619:
5618:{\displaystyle A\to B}
5587:
5567:
5566:{\displaystyle \neg A}
5540:
5539:{\displaystyle \neg A}
5520:Proof by contradiction
5420:
5419:{\displaystyle \lnot }
5397:
5396:{\displaystyle \lnot }
5364:
5363:{\displaystyle \lnot }
5340:
5339:{\displaystyle \lnot }
5301:
5300:{\displaystyle \lnot }
5278:
5277:{\displaystyle \lnot }
5198:is a positive integer
5194:and noting that since
5151:if a positive integer
5139:if a positive integer
5128:
5104:
5078:
5043:
5013:proof by contradiction
4665:, but the relation of
4620:denying the antecedent
4415:(Full) Contraposition
4335:partial contraposition
4308:materially implicative
3907:
3849:
3779:
3715:
3645:
3545:
3511:
3453:
3365:
3319:
3247:Hypothetical syllogism
3239:
3162:
3123:
3083:
3024:
2958:
2850:
2788:
2753:
2733:
2702:
2701:{\displaystyle R\to S}
2662:
2621:
2601:
2575:
2574:{\displaystyle \neg S}
2552:
2551:{\displaystyle \neg P}
2529:
2509:
2508:{\displaystyle \neg Q}
2486:
2463:
2411:
2370:
2350:
2330:
2310:
2290:
2270:
2247:
2246:{\displaystyle P\to Q}
2206:
2144:
2081:
2031:
1960:
1908:
1816:
1749:
1729:
1706:
1622:
1590:
1589:{\displaystyle P\to Q}
1561:
1503:
1467:
1429:
1406:
1328:
1308:
1288:
1238:s," is contraposed to
1232:
1212:
1192:
1121:is not the case, then
1115:
1095:
1075:
1052:
996:
972:
944:
924:
910:This states that, "if
901:
831:
830:{\displaystyle A\to B}
805:
773:
741:
740:{\displaystyle A\to B}
711:
646:
600:
599:{\displaystyle A\to B}
570:
539:
538:{\displaystyle \neg P}
515:
494:
493:{\displaystyle \neg Q}
470:
449:
401:
344:
292:
247:
169:
137:
107:
10276:Kolmogorov complexity
10229:Computably enumerable
10129:Model complete theory
9921:Principia Mathematica
8981:Propositional formula
8810:Banach–Tarski paradox
8607:Introduction to Logic
8481:Introduction to Logic
8471:Introduction to Logic
8461:Introduction to Logic
8409:. Sydney: Kew Books.
8246:Introduction to Logic
8233:Introduction to Logic
8079:
8047:
8000:
7929:
7903:
7862:
7815:
7786:
7747:being TRUE, and that
7742:
7713:
7669:
7641:
7612:
7583:
7539:
7358:In probability theory
7341:
7309:
7254:
7198:
7166:
7126:
7100:
7080:
7042:
7016:
6996:
6958:
6938:
6912:
6872:
6779:
6751:
6749:{\displaystyle a_{P}}
6724:
6704:
6626:
6422:
6363:
6331:
6308:
6274:
6220:
6194:
6168:
6140:
6115:(initial assumption)
6110:
6087:
6075:
6049:
6017:
5985:
5959:
5927:
5881:
5879:{\displaystyle x^{2}}
5854:
5801:
5799:{\displaystyle x^{2}}
5772:
5736:
5714:
5712:{\displaystyle x^{2}}
5680:
5652:
5620:
5588:
5568:
5541:
5421:
5398:
5365:
5341:
5302:
5279:
5129:
5105:
5079:
5054:". This statement is
5044:
4649:", the occurrence of
4542:Form of transposition
4522:one of the axioms of
4407:Original proposition
4257:rule of transposition
4210:logical biconditional
4127:of their conditional.
3908:
3850:
3780:
3716:
3646:
3546:
3512:
3454:
3366:
3320:
3240:
3163:
3124:
3084:
3025:
2959:
2851:
2789:
2754:
2734:
2703:
2663:
2622:
2602:
2576:
2553:
2530:
2510:
2487:
2464:
2412:
2371:
2351:
2331:
2311:
2291:
2271:
2248:
2207:
2145:
2082:
2032:
1961:
1909:
1817:
1750:
1730:
1707:
1647:Principia Mathematica
1623:
1591:
1562:
1504:
1475:syntactic consequence
1468:
1430:
1407:
1329:
1309:
1289:
1233:
1213:
1193:
1131:Socrates is not a man
1127:Socrates is not human
1116:
1096:
1076:
1053:
997:
973:
945:
925:
902:
832:
806:
774:
742:
712:
647:
601:
569:
562:Intuitive explanation
547:law of contrapositive
540:
516:
495:
471:
450:
421:", or equivalently, "
402:
345:
293:
248:
210:I don't wear my coat,
170:
138:
108:
85:Conditional statement
52:conditional statement
18:Transposition (logic)
10224:Church–Turing thesis
10211:Computability theory
9420:continuum hypothesis
8938:Square of opposition
8796:Gödel's completeness
8649:at Wikimedia Commons
8581:Audun Jøsang, 2016,
8430:Audun Jøsang 2016:92
8381:zimmer.csufresno.edu
8266:Stebbing, L. Susan.
8103:Reductio ad absurdum
8056:
8009:
7938:
7912:
7871:
7824:
7795:
7751:
7722:
7678:
7658:
7639:{\displaystyle a(P)}
7621:
7592:
7554:
7376:
7318:
7263:
7211:
7175:
7135:
7109:
7089:
7051:
7025:
7005:
6967:
6947:
6921:
6881:
6788:
6768:
6733:
6713:
6638:
6450:
6375:
6340:
6317:
6285:
6233:
6203:
6177:
6151:
6123:
6093:
6058:
6026:
5994:
5968:
5964:. We can prove that
5936:
5910:
5904:intuitionistic logic
5898:Intuitionistic logic
5863:
5824:
5783:
5761:
5725:
5696:
5669:
5629:
5603:
5577:
5554:
5527:
5410:
5387:
5354:
5330:
5291:
5268:
5114:
5090:
5064:
5051:irreducible fraction
5029:
4847:type "A" proposition
4821:", the subject term
4637:Sufficient condition
4626:by means of illicit
4612:sufficient condition
4481:(I) Some non-P is S
4475:(I) Some S is non-P
4472:(O) Some S is not P
4031:Take the statement "
3861:
3791:
3727:
3657:
3557:
3523:
3465:
3377:
3343:
3255:
3175:
3140:
3101:
3037:
2970:
2862:
2818:
2766:
2743:
2723:
2686:
2634:
2611:
2585:
2562:
2539:
2519:
2496:
2476:
2435:
2383:
2360:
2340:
2320:
2300:
2280:
2260:
2231:
2222:logically equivalent
2160:
2098:
2047:
1985:
1929:
1832:
1783:
1739:
1719:
1657:
1600:
1574:
1516:
1481:
1439:
1419:
1357:
1318:
1298:
1242:
1222:
1202:
1152:
1105:
1085:
1065:
1021:
986:
962:
934:
914:
879:
815:
783:
751:
725:
662:
624:
584:
526:
505:
481:
460:
433:
376:
328:
306:I don't wear my coat
270:
225:
221:The contrapositive (
147:
121:
91:
56:logically equivalent
10378:Mathematical object
10269:P versus NP problem
10234:Computable function
10028:Reverse mathematics
9954:Logical consequence
9831:primitive recursive
9826:elementary function
9599:Free/bound variable
9452:Tarski–Grothendieck
8971:Logical connectives
8901:Logical equivalence
8751:Logical consequence
8609:. MacMillan, 1953.
8439:Audun Jøsang 2016:2
8403:; A. Daoud (2011).
8329:, pp. 171–174.
8109:Logical equivalence
8084:being TRUE. Hence,
7303:
7248:
7160:
7074:
6990:
6906:
6863:
6825:
6695:
6666:
6586:
6557:
6525:
6487:
5210:, a square number.
5159:is a square number.
4968:immediate inference
4715:Necessary condition
4616:necessary condition
4524:propositional logic
4516:logical equivalence
4514:The process of the
4443:(A) All S is non-P
4337:). Since the valid
4315:Full contraposition
4237:immediate inference
1976:bivalent statements
1139:logical equivalence
811:, and equivalently
10427:Mathematical logic
10176:Transfer principle
10139:Semantics of logic
10124:Categorical theory
10100:Non-standard model
9614:Logical connective
8741:Information theory
8690:Mathematical logic
8074:
8042:
7995:
7924:
7898:
7857:
7810:
7781:
7737:
7708:
7664:
7636:
7607:
7578:
7534:
7336:
7304:
7266:
7249:
7214:
7193:
7161:
7138:
7121:
7095:
7075:
7054:
7037:
7011:
6991:
6970:
6953:
6933:
6907:
6884:
6867:
6829:
6794:
6774:
6746:
6719:
6699:
6670:
6644:
6621:
6561:
6535:
6491:
6456:
6417:
6358:
6326:
6303:
6269:
6215:
6189:
6163:
6135:
6105:
6070:
6044:
6012:
5980:
5954:
5922:
5876:
5849:
5816:is not even, then
5796:
5778:is not even, then
5767:
5731:
5709:
5675:
5647:
5615:
5583:
5563:
5550:. It follows that
5536:
5416:
5393:
5360:
5336:
5297:
5274:
5124:
5100:
5074:
5039:
4432:(E) No non-P is S
4426:(E) No S is non-P
4323:Aristotelian logic
4174:it has four sides.
3903:
3845:
3775:
3711:
3641:
3541:
3507:
3449:
3361:
3315:
3235:
3158:
3119:
3079:
3020:
2954:
2846:
2784:
2749:
2729:
2698:
2658:
2617:
2597:
2571:
2548:
2525:
2505:
2482:
2459:
2420:The elements of a
2407:
2366:
2346:
2326:
2306:
2286:
2266:
2243:
2202:
2140:
2077:
2027:
1956:
1904:
1902:
1812:
1745:
1725:
1702:
1618:
1586:
1557:
1499:
1463:
1425:
1402:
1324:
1304:
1284:
1228:
1208:
1188:
1111:
1091:
1071:
1048:
992:
968:
940:
920:
897:
827:
801:
769:
737:
707:
642:
596:
571:
535:
511:
490:
466:
445:
397:
370:logical complement
340:
302:it is not raining,
288:
243:
214:it isn't raining."
206:I wear my coat" —
165:
133:
103:
10414:
10413:
10346:Abstract category
10149:Theories of truth
9959:Rule of inference
9949:Natural deduction
9930:
9929:
9475:
9474:
9180:Cartesian product
9085:
9084:
8991:Many-valued logic
8966:Boolean functions
8849:Russell's paradox
8824:diagonal argument
8721:First-order logic
8645:Media related to
8626:Stebbing, Susan.
8591:978-3-319-42337-1
8540:978-1-260-80787-5
8531:Critical thinking
8491:978-1-315-51087-3
8416:978-0-646-54509-7
7791:is equivalent to
7718:is equivalent to
7667:{\displaystyle P}
7652:prior probability
7529:
7288:
7233:
7190:
7098:{\displaystyle A}
7014:{\displaystyle A}
6956:{\displaystyle A}
6848:
6813:
6777:{\displaystyle P}
6762:prior probability
6722:{\displaystyle A}
6603:
6510:
6475:
5770:{\displaystyle x}
5734:{\displaystyle x}
5678:{\displaystyle x}
5586:{\displaystyle A}
5511:
5510:
5223:rule of inference
5145:non-square number
5122:
5098:
5072:
5037:
4909:traditional logic
4763:Premise (2): not
4488:
4487:
4276:traditional logic
4229:traditional logic
4217:Traditional logic
4024:
4023:
2752:{\displaystyle Q}
2732:{\displaystyle P}
2620:{\displaystyle P}
2528:{\displaystyle S}
2485:{\displaystyle R}
2369:{\displaystyle Q}
2349:{\displaystyle P}
2329:{\displaystyle Q}
2309:{\displaystyle P}
2289:{\displaystyle Q}
2269:{\displaystyle P}
1774:first-order logic
1748:{\displaystyle Q}
1728:{\displaystyle P}
1552:
1327:{\displaystyle P}
1307:{\displaystyle Q}
1231:{\displaystyle Q}
1211:{\displaystyle P}
1114:{\displaystyle Q}
1094:{\displaystyle P}
1074:{\displaystyle Q}
1061:That is, "If not-
995:{\displaystyle Q}
971:{\displaystyle P}
956:Socrates is human
952:Socrates is a man
943:{\displaystyle Q}
923:{\displaystyle P}
860:Formal definition
556:rule of inference
514:{\displaystyle P}
469:{\displaystyle Q}
16:(Redirected from
10439:
10405:
10404:
10356:History of logic
10351:Category of sets
10244:Decision problem
10023:Ordinal analysis
9964:Sequent calculus
9862:Boolean algebras
9802:
9801:
9776:
9747:logical/constant
9501:
9500:
9487:
9410:Zermelo–Fraenkel
9161:Set operations:
9096:
9095:
9033:
8864:
8863:
8844:Löwenheim–Skolem
8731:Formal semantics
8683:
8676:
8669:
8660:
8659:
8644:
8585:Springer, Cham,
8570:
8561:
8552:
8525:
8504:
8495:
8474:
8473:. Prentice Hall.
8465:
8454:
8440:
8437:
8431:
8428:
8422:
8420:
8397:
8391:
8390:
8388:
8387:
8372:
8366:
8365:
8348:
8342:
8336:
8330:
8323:
8317:
8310:
8304:
8293:
8287:
8277:
8271:
8264:
8258:
8255:
8249:
8242:
8236:
8221:
8215:
8214:
8212:
8211:
8205:www.csm.ornl.gov
8197:
8191:
8190:
8188:
8187:
8173:
8167:
8166:
8164:
8163:
8149:
8143:
8142:
8140:
8139:
8125:
8083:
8081:
8080:
8075:
8051:
8049:
8048:
8043:
8004:
8002:
8001:
7996:
7933:
7931:
7930:
7925:
7907:
7905:
7904:
7899:
7866:
7864:
7863:
7858:
7819:
7817:
7816:
7811:
7790:
7788:
7787:
7782:
7746:
7744:
7743:
7738:
7717:
7715:
7714:
7709:
7673:
7671:
7670:
7665:
7645:
7643:
7642:
7637:
7616:
7614:
7613:
7608:
7587:
7585:
7584:
7579:
7543:
7541:
7540:
7535:
7530:
7528:
7449:
7407:
7345:
7343:
7342:
7337:
7313:
7311:
7310:
7305:
7302:
7297:
7290:
7289:
7284:
7279:
7258:
7256:
7255:
7250:
7247:
7242:
7235:
7234:
7229:
7224:
7205:subjective logic
7202:
7200:
7199:
7194:
7192:
7191:
7186:
7180:
7170:
7168:
7167:
7162:
7159:
7154:
7150:
7130:
7128:
7127:
7122:
7104:
7102:
7101:
7096:
7084:
7082:
7081:
7076:
7073:
7068:
7046:
7044:
7043:
7038:
7020:
7018:
7017:
7012:
7000:
6998:
6997:
6992:
6989:
6984:
6962:
6960:
6959:
6954:
6942:
6940:
6939:
6934:
6916:
6914:
6913:
6908:
6905:
6900:
6896:
6876:
6874:
6873:
6868:
6862:
6857:
6850:
6849:
6844:
6839:
6824:
6819:
6815:
6814:
6809:
6804:
6783:
6781:
6780:
6775:
6755:
6753:
6752:
6747:
6745:
6744:
6729:. The parameter
6728:
6726:
6725:
6720:
6708:
6706:
6705:
6700:
6694:
6689:
6682:
6665:
6660:
6656:
6630:
6628:
6627:
6622:
6616:
6615:
6605:
6604:
6599:
6593:
6585:
6580:
6573:
6556:
6551:
6547:
6524:
6519:
6512:
6511:
6506:
6501:
6486:
6481:
6477:
6476:
6471:
6466:
6441:subjective logic
6432:Subjective logic
6426:
6424:
6423:
6418:
6367:
6365:
6364:
6359:
6335:
6333:
6332:
6327:
6312:
6310:
6309:
6304:
6278:
6276:
6275:
6270:
6224:
6222:
6221:
6216:
6198:
6196:
6195:
6190:
6172:
6170:
6169:
6164:
6144:
6142:
6141:
6136:
6114:
6112:
6111:
6106:
6079:
6077:
6076:
6071:
6053:
6051:
6050:
6045:
6021:
6019:
6018:
6013:
5989:
5987:
5986:
5981:
5963:
5961:
5960:
5955:
5931:
5929:
5928:
5923:
5906:, the statement
5885:
5883:
5882:
5877:
5875:
5874:
5858:
5856:
5855:
5850:
5836:
5835:
5805:
5803:
5802:
5797:
5795:
5794:
5776:
5774:
5773:
5768:
5740:
5738:
5737:
5732:
5718:
5716:
5715:
5710:
5708:
5707:
5684:
5682:
5681:
5676:
5656:
5654:
5653:
5648:
5624:
5622:
5621:
5616:
5592:
5590:
5589:
5584:
5572:
5570:
5569:
5564:
5545:
5543:
5542:
5537:
5425:
5423:
5422:
5417:
5402:
5400:
5399:
5394:
5369:
5367:
5366:
5361:
5345:
5343:
5342:
5337:
5313:
5306:
5304:
5303:
5298:
5283:
5281:
5280:
5275:
5169:
5168:
5133:
5131:
5130:
5125:
5123:
5118:
5109:
5107:
5106:
5101:
5099:
5094:
5083:
5081:
5080:
5075:
5073:
5068:
5048:
5046:
5045:
5040:
5038:
5033:
5017:square root of 2
4753:Premise (1): If
4691:Premise (1): If
4609:
4597:
4585:
4573:
4456:(I) Some S is P
4404:
4403:
4172:if, and only if,
4105:have four sides,
3922:
3921:
3912:
3910:
3909:
3904:
3854:
3852:
3851:
3846:
3784:
3782:
3781:
3776:
3720:
3718:
3717:
3712:
3650:
3648:
3647:
3642:
3550:
3548:
3547:
3542:
3516:
3514:
3513:
3508:
3458:
3456:
3455:
3450:
3370:
3368:
3367:
3362:
3324:
3322:
3321:
3316:
3244:
3242:
3241:
3236:
3167:
3165:
3164:
3159:
3128:
3126:
3125:
3120:
3088:
3086:
3085:
3080:
3029:
3027:
3026:
3021:
3019:
3015:
2997:
2993:
2963:
2961:
2960:
2955:
2953:
2949:
2948:
2944:
2926:
2922:
2899:
2895:
2894:
2890:
2855:
2853:
2852:
2847:
2845:
2841:
2793:
2791:
2790:
2785:
2758:
2756:
2755:
2750:
2738:
2736:
2735:
2730:
2707:
2705:
2704:
2699:
2667:
2665:
2664:
2659:
2626:
2624:
2623:
2618:
2606:
2604:
2603:
2598:
2580:
2578:
2577:
2572:
2557:
2555:
2554:
2549:
2534:
2532:
2531:
2526:
2514:
2512:
2511:
2506:
2491:
2489:
2488:
2483:
2468:
2466:
2465:
2460:
2416:
2414:
2413:
2408:
2375:
2373:
2372:
2367:
2355:
2353:
2352:
2347:
2335:
2333:
2332:
2327:
2315:
2313:
2312:
2307:
2295:
2293:
2292:
2287:
2275:
2273:
2272:
2267:
2252:
2250:
2249:
2244:
2211:
2209:
2208:
2203:
2149:
2147:
2146:
2141:
2086:
2084:
2083:
2078:
2036:
2034:
2033:
2028:
1965:
1963:
1962:
1957:
1913:
1911:
1910:
1905:
1903:
1821:
1819:
1818:
1813:
1754:
1752:
1751:
1746:
1734:
1732:
1731:
1726:
1711:
1709:
1708:
1703:
1627:
1625:
1624:
1619:
1595:
1593:
1592:
1587:
1566:
1564:
1563:
1558:
1553:
1551:
1531:
1520:
1508:
1506:
1505:
1500:
1472:
1470:
1469:
1464:
1434:
1432:
1431:
1426:
1411:
1409:
1408:
1403:
1338:Sequent notation
1333:
1331:
1330:
1325:
1313:
1311:
1310:
1305:
1293:
1291:
1290:
1285:
1280:
1266:
1252:
1237:
1235:
1234:
1229:
1217:
1215:
1214:
1209:
1197:
1195:
1194:
1189:
1184:
1173:
1162:
1120:
1118:
1117:
1112:
1100:
1098:
1097:
1092:
1080:
1078:
1077:
1072:
1057:
1055:
1054:
1049:
1001:
999:
998:
993:
977:
975:
974:
969:
949:
947:
946:
941:
929:
927:
926:
921:
906:
904:
903:
898:
836:
834:
833:
828:
810:
808:
807:
802:
778:
776:
775:
770:
746:
744:
743:
738:
716:
714:
713:
708:
651:
649:
648:
643:
605:
603:
602:
597:
544:
542:
541:
536:
520:
518:
517:
512:
499:
497:
496:
491:
477:is false (i.e.,
475:
473:
472:
467:
454:
452:
451:
446:
406:
404:
403:
398:
349:
347:
346:
341:
297:
295:
294:
289:
252:
250:
249:
244:
174:
172:
171:
166:
142:
140:
139:
134:
112:
110:
109:
104:
50:of going from a
46:, refers to the
21:
10447:
10446:
10442:
10441:
10440:
10438:
10437:
10436:
10417:
10416:
10415:
10410:
10399:
10392:
10337:Category theory
10327:Algebraic logic
10310:
10281:Lambda calculus
10219:Church encoding
10205:
10181:Truth predicate
10037:
10003:Complete theory
9926:
9795:
9791:
9787:
9782:
9774:
9494: and
9490:
9485:
9471:
9447:New Foundations
9415:axiom of choice
9398:
9360:Gödel numbering
9300: and
9292:
9196:
9081:
9031:
9012:
8961:Boolean algebra
8947:
8911:Equiconsistency
8876:Classical logic
8853:
8834:Halting problem
8822: and
8798: and
8786: and
8785:
8780:Theorems (
8775:
8692:
8687:
8655:(Fallacy Files)
8637:
8578:
8573:
8541:
8522:
8492:
8444:
8443:
8438:
8434:
8429:
8425:
8417:
8398:
8394:
8385:
8383:
8375:Cusick, Larry.
8373:
8369:
8363:
8349:
8345:
8337:
8333:
8324:
8320:
8311:
8307:
8294:
8290:
8278:
8274:
8265:
8261:
8256:
8252:
8243:
8239:
8222:
8218:
8209:
8207:
8199:
8198:
8194:
8185:
8183:
8175:
8174:
8170:
8161:
8159:
8151:
8150:
8146:
8137:
8135:
8127:
8126:
8122:
8117:
8098:
8057:
8054:
8053:
8010:
8007:
8006:
7939:
7936:
7935:
7913:
7910:
7909:
7872:
7869:
7868:
7825:
7822:
7821:
7796:
7793:
7792:
7752:
7749:
7748:
7723:
7720:
7719:
7679:
7676:
7675:
7659:
7656:
7655:
7622:
7619:
7618:
7593:
7590:
7589:
7555:
7552:
7551:
7450:
7408:
7406:
7377:
7374:
7373:
7360:
7319:
7316:
7315:
7298:
7280:
7278:
7277:
7270:
7264:
7261:
7260:
7243:
7225:
7223:
7222:
7218:
7212:
7209:
7208:
7181:
7179:
7178:
7176:
7173:
7172:
7155:
7146:
7142:
7136:
7133:
7132:
7110:
7107:
7106:
7090:
7087:
7086:
7069:
7058:
7052:
7049:
7048:
7026:
7023:
7022:
7006:
7003:
7002:
6985:
6974:
6968:
6965:
6964:
6948:
6945:
6944:
6922:
6919:
6918:
6901:
6892:
6888:
6882:
6879:
6878:
6858:
6840:
6838:
6837:
6833:
6820:
6805:
6803:
6802:
6798:
6789:
6786:
6785:
6769:
6766:
6765:
6740:
6736:
6734:
6731:
6730:
6714:
6711:
6710:
6690:
6678:
6674:
6661:
6652:
6648:
6639:
6636:
6635:
6611:
6607:
6594:
6592:
6591:
6581:
6569:
6565:
6552:
6543:
6539:
6520:
6502:
6500:
6499:
6495:
6482:
6467:
6465:
6464:
6460:
6451:
6448:
6447:
6434:
6376:
6373:
6372:
6341:
6338:
6337:
6318:
6315:
6314:
6286:
6283:
6282:
6234:
6231:
6230:
6204:
6201:
6200:
6178:
6175:
6174:
6152:
6149:
6148:
6124:
6121:
6120:
6094:
6091:
6090:
6085:
6080:, requires the
6059:
6056:
6055:
6027:
6024:
6023:
5995:
5992:
5991:
5969:
5966:
5965:
5937:
5934:
5933:
5911:
5908:
5907:
5900:
5895:
5870:
5866:
5864:
5861:
5860:
5831:
5827:
5825:
5822:
5821:
5790:
5786:
5784:
5781:
5780:
5762:
5759:
5758:
5726:
5723:
5722:
5703:
5699:
5697:
5694:
5693:
5685:be an integer.
5670:
5667:
5666:
5663:
5630:
5627:
5626:
5604:
5601:
5600:
5578:
5575:
5574:
5555:
5552:
5551:
5528:
5525:
5524:
5516:
5411:
5408:
5407:
5388:
5385:
5384:
5355:
5352:
5351:
5331:
5328:
5327:
5292:
5289:
5288:
5269:
5266:
5265:
5164:
5162:
5117:
5115:
5112:
5111:
5093:
5091:
5088:
5087:
5067:
5065:
5062:
5061:
5032:
5030:
5027:
5026:
5021:rational number
4993:
4979:" and "All non-
4905:
4895:
4788:
4779:
4717:
4639:
4599:
4587:
4575:
4563:
4544:
4423:(A) All S is P
4225:
4219:
4190:
4029:
3920:
3862:
3859:
3858:
3792:
3789:
3788:
3728:
3725:
3724:
3658:
3655:
3654:
3558:
3555:
3554:
3524:
3521:
3520:
3466:
3463:
3462:
3378:
3375:
3374:
3344:
3341:
3340:
3256:
3253:
3252:
3176:
3173:
3172:
3141:
3138:
3137:
3133:(one direction)
3131:Double negation
3102:
3099:
3098:
3038:
3035:
3034:
3005:
3001:
2977:
2973:
2971:
2968:
2967:
2934:
2930:
2912:
2908:
2907:
2903:
2880:
2876:
2869:
2865:
2863:
2860:
2859:
2831:
2827:
2819:
2816:
2815:
2808:Jan Łukasiewicz
2800:
2767:
2764:
2763:
2744:
2741:
2740:
2724:
2721:
2720:
2687:
2684:
2683:
2635:
2632:
2631:
2612:
2609:
2608:
2586:
2583:
2582:
2563:
2560:
2559:
2540:
2537:
2536:
2520:
2517:
2516:
2497:
2494:
2493:
2477:
2474:
2473:
2436:
2433:
2432:
2384:
2381:
2380:
2361:
2358:
2357:
2341:
2338:
2337:
2321:
2318:
2317:
2301:
2298:
2297:
2281:
2278:
2277:
2261:
2258:
2257:
2232:
2229:
2228:
2218:
2161:
2158:
2157:
2099:
2096:
2095:
2048:
2045:
2044:
1986:
1983:
1982:
1930:
1927:
1926:
1920:
1901:
1900:
1877:
1873:
1872:
1852:
1835:
1833:
1830:
1829:
1784:
1781:
1780:
1770:
1765:
1740:
1737:
1736:
1720:
1717:
1716:
1658:
1655:
1654:
1601:
1598:
1597:
1575:
1572:
1571:
1532:
1521:
1519:
1517:
1514:
1513:
1482:
1479:
1478:
1440:
1437:
1436:
1420:
1417:
1416:
1358:
1355:
1354:
1340:
1319:
1316:
1315:
1299:
1296:
1295:
1276:
1262:
1248:
1243:
1240:
1239:
1223:
1220:
1219:
1203:
1200:
1199:
1180:
1169:
1158:
1153:
1150:
1149:
1106:
1103:
1102:
1086:
1083:
1082:
1066:
1063:
1062:
1022:
1019:
1018:
987:
984:
983:
963:
960:
959:
935:
932:
931:
915:
912:
911:
880:
877:
876:
862:
852:always implies
816:
813:
812:
784:
781:
780:
752:
749:
748:
726:
723:
722:
663:
660:
659:
625:
622:
621:
585:
582:
581:
564:
527:
524:
523:
506:
503:
502:
482:
479:
478:
461:
458:
457:
434:
431:
430:
419:I wear my coat.
377:
374:
373:
354:I wear my coat,
329:
326:
325:
271:
268:
267:
226:
223:
222:
196:
148:
145:
144:
122:
119:
118:
92:
89:
88:
28:
23:
22:
15:
12:
11:
5:
10445:
10435:
10434:
10429:
10412:
10411:
10397:
10394:
10393:
10391:
10390:
10385:
10380:
10375:
10370:
10369:
10368:
10358:
10353:
10348:
10339:
10334:
10329:
10324:
10322:Abstract logic
10318:
10316:
10312:
10311:
10309:
10308:
10303:
10301:Turing machine
10298:
10293:
10288:
10283:
10278:
10273:
10272:
10271:
10266:
10261:
10256:
10251:
10241:
10239:Computable set
10236:
10231:
10226:
10221:
10215:
10213:
10207:
10206:
10204:
10203:
10198:
10193:
10188:
10183:
10178:
10173:
10168:
10167:
10166:
10161:
10156:
10146:
10141:
10136:
10134:Satisfiability
10131:
10126:
10121:
10120:
10119:
10109:
10108:
10107:
10097:
10096:
10095:
10090:
10085:
10080:
10075:
10065:
10064:
10063:
10058:
10051:Interpretation
10047:
10045:
10039:
10038:
10036:
10035:
10030:
10025:
10020:
10015:
10005:
10000:
9999:
9998:
9997:
9996:
9986:
9981:
9971:
9966:
9961:
9956:
9951:
9946:
9940:
9938:
9932:
9931:
9928:
9927:
9925:
9924:
9916:
9915:
9914:
9913:
9908:
9907:
9906:
9901:
9896:
9876:
9875:
9874:
9872:minimal axioms
9869:
9858:
9857:
9856:
9845:
9844:
9843:
9838:
9833:
9828:
9823:
9818:
9805:
9803:
9784:
9783:
9781:
9780:
9779:
9778:
9766:
9761:
9760:
9759:
9754:
9749:
9744:
9734:
9729:
9724:
9719:
9718:
9717:
9712:
9702:
9701:
9700:
9695:
9690:
9685:
9675:
9670:
9669:
9668:
9663:
9658:
9648:
9647:
9646:
9641:
9636:
9631:
9626:
9621:
9611:
9606:
9601:
9596:
9595:
9594:
9589:
9584:
9579:
9569:
9564:
9562:Formation rule
9559:
9554:
9553:
9552:
9547:
9537:
9536:
9535:
9525:
9520:
9515:
9510:
9504:
9498:
9481:Formal systems
9477:
9476:
9473:
9472:
9470:
9469:
9464:
9459:
9454:
9449:
9444:
9439:
9434:
9429:
9424:
9423:
9422:
9417:
9406:
9404:
9400:
9399:
9397:
9396:
9395:
9394:
9384:
9379:
9378:
9377:
9370:Large cardinal
9367:
9362:
9357:
9352:
9347:
9333:
9332:
9331:
9326:
9321:
9306:
9304:
9294:
9293:
9291:
9290:
9289:
9288:
9283:
9278:
9268:
9263:
9258:
9253:
9248:
9243:
9238:
9233:
9228:
9223:
9218:
9213:
9207:
9205:
9198:
9197:
9195:
9194:
9193:
9192:
9187:
9182:
9177:
9172:
9167:
9159:
9158:
9157:
9152:
9142:
9137:
9135:Extensionality
9132:
9130:Ordinal number
9127:
9117:
9112:
9111:
9110:
9099:
9093:
9087:
9086:
9083:
9082:
9080:
9079:
9074:
9069:
9064:
9059:
9054:
9049:
9048:
9047:
9037:
9036:
9035:
9022:
9020:
9014:
9013:
9011:
9010:
9009:
9008:
9003:
8998:
8988:
8983:
8978:
8973:
8968:
8963:
8957:
8955:
8949:
8948:
8946:
8945:
8940:
8935:
8930:
8925:
8920:
8915:
8914:
8913:
8903:
8898:
8893:
8888:
8883:
8878:
8872:
8870:
8861:
8855:
8854:
8852:
8851:
8846:
8841:
8836:
8831:
8826:
8814:Cantor's
8812:
8807:
8802:
8792:
8790:
8777:
8776:
8774:
8773:
8768:
8763:
8758:
8753:
8748:
8743:
8738:
8733:
8728:
8723:
8718:
8713:
8712:
8711:
8700:
8698:
8694:
8693:
8686:
8685:
8678:
8671:
8663:
8657:
8656:
8650:
8647:Contraposition
8636:
8635:External links
8633:
8632:
8631:
8624:
8617:
8614:Symbolic Logic
8612:Copi, Irving.
8610:
8605:Copi, Irving.
8603:
8600:
8593:
8577:
8574:
8572:
8571:
8562:
8553:
8539:
8526:
8520:
8505:
8501:Symbolic Logic
8496:
8490:
8475:
8466:
8455:
8445:
8442:
8441:
8432:
8423:
8415:
8392:
8367:
8361:
8343:
8331:
8318:
8316:, pp. 66.
8305:
8303:, p. 141.
8288:
8286:, p. 141.
8272:
8259:
8250:
8244:Irving Copi's
8237:
8216:
8192:
8168:
8144:
8119:
8118:
8116:
8113:
8112:
8111:
8106:
8097:
8094:
8090:contraposition
8086:Bayes' theorem
8073:
8070:
8067:
8064:
8061:
8041:
8038:
8035:
8032:
8029:
8026:
8023:
8020:
8017:
8014:
7994:
7991:
7988:
7985:
7982:
7979:
7976:
7973:
7970:
7967:
7964:
7961:
7958:
7955:
7952:
7949:
7946:
7943:
7923:
7920:
7917:
7897:
7894:
7891:
7888:
7885:
7882:
7879:
7876:
7856:
7853:
7850:
7847:
7844:
7841:
7838:
7835:
7832:
7829:
7809:
7806:
7803:
7800:
7780:
7777:
7774:
7771:
7768:
7765:
7762:
7759:
7756:
7736:
7733:
7730:
7727:
7707:
7704:
7701:
7698:
7695:
7692:
7689:
7686:
7683:
7674:. Assume that
7663:
7635:
7632:
7629:
7626:
7606:
7603:
7600:
7597:
7577:
7574:
7571:
7568:
7565:
7562:
7559:
7545:
7544:
7533:
7527:
7524:
7521:
7518:
7514:
7511:
7508:
7505:
7502:
7499:
7496:
7493:
7490:
7487:
7484:
7481:
7478:
7474:
7471:
7468:
7465:
7462:
7459:
7456:
7453:
7448:
7445:
7442:
7439:
7436:
7432:
7429:
7426:
7423:
7420:
7417:
7414:
7411:
7405:
7402:
7399:
7396:
7393:
7390:
7387:
7384:
7381:
7367:Bayes' theorem
7363:Contraposition
7359:
7356:
7352:Bayes' theorem
7348:contraposition
7335:
7332:
7329:
7326:
7323:
7301:
7296:
7293:
7287:
7283:
7276:
7273:
7269:
7246:
7241:
7238:
7232:
7228:
7221:
7217:
7189:
7184:
7158:
7153:
7149:
7145:
7141:
7120:
7117:
7114:
7094:
7072:
7067:
7064:
7061:
7057:
7036:
7033:
7030:
7010:
6988:
6983:
6980:
6977:
6973:
6952:
6932:
6929:
6926:
6904:
6899:
6895:
6891:
6887:
6866:
6861:
6856:
6853:
6847:
6843:
6836:
6832:
6828:
6823:
6818:
6812:
6808:
6801:
6797:
6793:
6773:
6743:
6739:
6718:
6698:
6693:
6688:
6685:
6681:
6677:
6673:
6669:
6664:
6659:
6655:
6651:
6647:
6643:
6632:
6631:
6620:
6614:
6610:
6602:
6597:
6589:
6584:
6579:
6576:
6572:
6568:
6564:
6560:
6555:
6550:
6546:
6542:
6538:
6534:
6531:
6528:
6523:
6518:
6515:
6509:
6505:
6498:
6494:
6490:
6485:
6480:
6474:
6470:
6463:
6459:
6455:
6443:expressed as:
6437:Contraposition
6433:
6430:
6416:
6413:
6410:
6407:
6404:
6401:
6398:
6395:
6392:
6389:
6386:
6383:
6380:
6369:
6368:
6357:
6354:
6351:
6348:
6345:
6325:
6322:
6302:
6299:
6296:
6293:
6290:
6279:
6268:
6265:
6262:
6259:
6256:
6253:
6250:
6247:
6244:
6241:
6238:
6227:
6226:
6225:
6214:
6211:
6208:
6188:
6185:
6182:
6162:
6159:
6156:
6134:
6131:
6128:
6104:
6101:
6098:
6069:
6066:
6063:
6043:
6040:
6037:
6034:
6031:
6011:
6008:
6005:
6002:
5999:
5979:
5976:
5973:
5953:
5950:
5947:
5944:
5941:
5921:
5918:
5915:
5899:
5896:
5894:
5891:
5873:
5869:
5848:
5845:
5842:
5839:
5834:
5830:
5810:
5809:
5793:
5789:
5766:
5745:
5744:
5730:
5720:is even, then
5706:
5702:
5674:
5662:
5659:
5646:
5643:
5640:
5637:
5634:
5614:
5611:
5608:
5582:
5562:
5559:
5535:
5532:
5515:
5512:
5509:
5508:
5505:
5502:
5499:
5496:
5493:
5489:
5488:
5485:
5482:
5479:
5476:
5473:
5469:
5468:
5465:
5462:
5459:
5456:
5453:
5449:
5448:
5445:
5442:
5439:
5436:
5433:
5429:
5428:
5415:
5392:
5381:
5372:
5359:
5348:
5335:
5324:
5319:
5296:
5273:
5121:
5097:
5071:
5036:
4997:contrapositive
4992:
4989:
4917:contraposition
4894:
4891:
4873:" (note that (
4792:if and only if
4787:
4784:
4778:
4775:
4774:
4773:
4767:
4761:
4716:
4713:
4712:
4711:
4705:
4699:
4638:
4635:
4543:
4540:
4536:Susan Stebbing
4486:
4485:
4482:
4479:
4476:
4473:
4469:
4468:
4465:
4462:
4460:
4457:
4453:
4452:
4449:
4446:
4444:
4441:
4440:(E) No S is P
4437:
4436:
4433:
4430:
4427:
4424:
4420:
4419:
4416:
4413:
4411:
4408:
4399:
4398:
4388:
4387:
4373:
4372:
4358:
4357:
4233:contraposition
4218:
4215:
4214:
4213:
4206:
4203:
4200:
4197:
4194:
4189:
4186:
4178:if and only if
4176:" (The phrase
4162:
4161:
4150:
4139:
4128:
4117:contrapositive
4103:quadrilaterals
4086:
4085:
4074:
4063:
4052:
4045:contrapositive
4028:
4025:
4022:
4021:
4018:
4009:
4005:
4004:
4001:
3991:
3990:contrapositive
3987:
3986:
3983:
3973:
3969:
3968:
3965:
3955:
3951:
3950:
3947:
3937:
3933:
3932:
3929:
3926:
3919:
3916:
3915:
3914:
3902:
3899:
3896:
3893:
3890:
3887:
3884:
3881:
3878:
3875:
3872:
3869:
3866:
3856:
3844:
3841:
3838:
3835:
3832:
3829:
3826:
3823:
3820:
3817:
3814:
3811:
3808:
3805:
3802:
3799:
3796:
3786:
3774:
3771:
3768:
3765:
3762:
3759:
3756:
3753:
3750:
3747:
3744:
3741:
3738:
3735:
3732:
3722:
3710:
3707:
3704:
3701:
3698:
3695:
3692:
3689:
3686:
3683:
3680:
3677:
3674:
3671:
3668:
3665:
3662:
3652:
3640:
3637:
3634:
3631:
3628:
3625:
3622:
3619:
3616:
3613:
3610:
3607:
3604:
3601:
3598:
3595:
3592:
3589:
3586:
3583:
3580:
3577:
3574:
3571:
3568:
3565:
3562:
3552:
3540:
3537:
3534:
3531:
3528:
3518:
3506:
3503:
3500:
3497:
3494:
3491:
3488:
3485:
3482:
3479:
3476:
3473:
3470:
3460:
3448:
3445:
3442:
3439:
3436:
3433:
3430:
3427:
3424:
3421:
3418:
3415:
3412:
3409:
3406:
3403:
3400:
3397:
3394:
3391:
3388:
3385:
3382:
3372:
3360:
3357:
3354:
3351:
3348:
3327:
3326:
3314:
3311:
3308:
3305:
3302:
3299:
3296:
3293:
3290:
3287:
3284:
3281:
3278:
3275:
3272:
3269:
3266:
3263:
3260:
3249:
3245:- one form of
3234:
3231:
3228:
3225:
3222:
3219:
3216:
3213:
3210:
3207:
3204:
3201:
3198:
3195:
3192:
3189:
3186:
3183:
3180:
3169:
3157:
3154:
3151:
3148:
3145:
3134:
3118:
3115:
3112:
3109:
3106:
3078:
3075:
3072:
3069:
3066:
3063:
3060:
3057:
3054:
3051:
3048:
3045:
3042:
3031:
3030:
3018:
3014:
3011:
3008:
3004:
3000:
2996:
2992:
2989:
2986:
2983:
2980:
2976:
2964:
2952:
2947:
2943:
2940:
2937:
2933:
2929:
2925:
2921:
2918:
2915:
2911:
2906:
2902:
2898:
2893:
2889:
2886:
2883:
2879:
2875:
2872:
2868:
2856:
2844:
2840:
2837:
2834:
2830:
2826:
2823:
2799:
2796:
2795:
2794:
2783:
2780:
2777:
2774:
2771:
2748:
2728:
2709:
2708:
2697:
2694:
2691:
2669:
2668:
2657:
2654:
2651:
2648:
2645:
2642:
2639:
2616:
2596:
2593:
2590:
2570:
2567:
2547:
2544:
2524:
2504:
2501:
2481:
2470:
2469:
2458:
2455:
2452:
2449:
2446:
2443:
2440:
2418:
2417:
2406:
2403:
2400:
2397:
2394:
2391:
2388:
2365:
2345:
2325:
2305:
2285:
2265:
2254:
2253:
2242:
2239:
2236:
2217:
2214:
2213:
2212:
2201:
2198:
2195:
2192:
2189:
2186:
2183:
2180:
2177:
2174:
2171:
2168:
2165:
2151:
2150:
2139:
2136:
2133:
2130:
2127:
2124:
2121:
2118:
2115:
2112:
2109:
2106:
2103:
2088:
2087:
2076:
2073:
2070:
2067:
2064:
2061:
2058:
2055:
2052:
2038:
2037:
2026:
2023:
2020:
2017:
2014:
2011:
2008:
2005:
2002:
1999:
1996:
1993:
1990:
1967:
1966:
1955:
1952:
1949:
1946:
1943:
1940:
1937:
1934:
1919:
1916:
1915:
1914:
1899:
1896:
1893:
1890:
1887:
1884:
1880:
1878:
1875:
1874:
1871:
1868:
1865:
1862:
1859:
1855:
1853:
1850:
1847:
1844:
1841:
1838:
1837:
1823:
1822:
1811:
1808:
1805:
1802:
1798:
1794:
1791:
1788:
1769:
1766:
1764:
1761:
1744:
1724:
1713:
1712:
1701:
1698:
1695:
1692:
1689:
1686:
1683:
1680:
1677:
1674:
1671:
1668:
1665:
1662:
1617:
1614:
1611:
1608:
1605:
1585:
1582:
1579:
1568:
1567:
1556:
1550:
1547:
1544:
1541:
1538:
1535:
1530:
1527:
1524:
1498:
1495:
1492:
1489:
1486:
1462:
1459:
1456:
1453:
1450:
1447:
1444:
1424:
1413:
1412:
1401:
1398:
1395:
1392:
1389:
1386:
1383:
1380:
1377:
1374:
1371:
1368:
1365:
1362:
1339:
1336:
1323:
1303:
1294:, or "All non-
1283:
1279:
1275:
1272:
1269:
1265:
1261:
1258:
1255:
1251:
1247:
1227:
1207:
1187:
1183:
1179:
1176:
1172:
1168:
1165:
1161:
1157:
1110:
1090:
1070:
1059:
1058:
1047:
1044:
1041:
1038:
1035:
1032:
1029:
1026:
1008:contrapositive
991:
967:
939:
919:
908:
907:
896:
893:
890:
887:
884:
864:A proposition
861:
858:
826:
823:
820:
800:
797:
794:
791:
788:
768:
765:
762:
759:
756:
736:
733:
730:
718:
717:
706:
703:
700:
697:
694:
691:
688:
685:
682:
679:
676:
673:
670:
667:
653:
652:
641:
638:
635:
632:
629:
607:
606:
595:
592:
589:
563:
560:
534:
531:
510:
489:
486:
465:
444:
441:
438:
427:
426:
407:
396:
393:
390:
387:
384:
381:
361:
350:
339:
336:
333:
313:
298:
287:
284:
281:
278:
275:
242:
239:
236:
233:
230:
202:it is raining,
178:
164:
161:
158:
155:
152:
132:
129:
126:
102:
99:
96:
59:contrapositive
40:contraposition
26:
9:
6:
4:
3:
2:
10444:
10433:
10430:
10428:
10425:
10424:
10422:
10409:
10408:
10403:
10395:
10389:
10386:
10384:
10381:
10379:
10376:
10374:
10371:
10367:
10364:
10363:
10362:
10359:
10357:
10354:
10352:
10349:
10347:
10343:
10340:
10338:
10335:
10333:
10330:
10328:
10325:
10323:
10320:
10319:
10317:
10313:
10307:
10304:
10302:
10299:
10297:
10296:Recursive set
10294:
10292:
10289:
10287:
10284:
10282:
10279:
10277:
10274:
10270:
10267:
10265:
10262:
10260:
10257:
10255:
10252:
10250:
10247:
10246:
10245:
10242:
10240:
10237:
10235:
10232:
10230:
10227:
10225:
10222:
10220:
10217:
10216:
10214:
10212:
10208:
10202:
10199:
10197:
10194:
10192:
10189:
10187:
10184:
10182:
10179:
10177:
10174:
10172:
10169:
10165:
10162:
10160:
10157:
10155:
10152:
10151:
10150:
10147:
10145:
10142:
10140:
10137:
10135:
10132:
10130:
10127:
10125:
10122:
10118:
10115:
10114:
10113:
10110:
10106:
10105:of arithmetic
10103:
10102:
10101:
10098:
10094:
10091:
10089:
10086:
10084:
10081:
10079:
10076:
10074:
10071:
10070:
10069:
10066:
10062:
10059:
10057:
10054:
10053:
10052:
10049:
10048:
10046:
10044:
10040:
10034:
10031:
10029:
10026:
10024:
10021:
10019:
10016:
10013:
10012:from ZFC
10009:
10006:
10004:
10001:
9995:
9992:
9991:
9990:
9987:
9985:
9982:
9980:
9977:
9976:
9975:
9972:
9970:
9967:
9965:
9962:
9960:
9957:
9955:
9952:
9950:
9947:
9945:
9942:
9941:
9939:
9937:
9933:
9923:
9922:
9918:
9917:
9912:
9911:non-Euclidean
9909:
9905:
9902:
9900:
9897:
9895:
9894:
9890:
9889:
9887:
9884:
9883:
9881:
9877:
9873:
9870:
9868:
9865:
9864:
9863:
9859:
9855:
9852:
9851:
9850:
9846:
9842:
9839:
9837:
9834:
9832:
9829:
9827:
9824:
9822:
9819:
9817:
9814:
9813:
9811:
9807:
9806:
9804:
9799:
9793:
9788:Example
9785:
9777:
9772:
9771:
9770:
9767:
9765:
9762:
9758:
9755:
9753:
9750:
9748:
9745:
9743:
9740:
9739:
9738:
9735:
9733:
9730:
9728:
9725:
9723:
9720:
9716:
9713:
9711:
9708:
9707:
9706:
9703:
9699:
9696:
9694:
9691:
9689:
9686:
9684:
9681:
9680:
9679:
9676:
9674:
9671:
9667:
9664:
9662:
9659:
9657:
9654:
9653:
9652:
9649:
9645:
9642:
9640:
9637:
9635:
9632:
9630:
9627:
9625:
9622:
9620:
9617:
9616:
9615:
9612:
9610:
9607:
9605:
9602:
9600:
9597:
9593:
9590:
9588:
9585:
9583:
9580:
9578:
9575:
9574:
9573:
9570:
9568:
9565:
9563:
9560:
9558:
9555:
9551:
9548:
9546:
9545:by definition
9543:
9542:
9541:
9538:
9534:
9531:
9530:
9529:
9526:
9524:
9521:
9519:
9516:
9514:
9511:
9509:
9506:
9505:
9502:
9499:
9497:
9493:
9488:
9482:
9478:
9468:
9465:
9463:
9460:
9458:
9455:
9453:
9450:
9448:
9445:
9443:
9440:
9438:
9435:
9433:
9432:Kripke–Platek
9430:
9428:
9425:
9421:
9418:
9416:
9413:
9412:
9411:
9408:
9407:
9405:
9401:
9393:
9390:
9389:
9388:
9385:
9383:
9380:
9376:
9373:
9372:
9371:
9368:
9366:
9363:
9361:
9358:
9356:
9353:
9351:
9348:
9345:
9341:
9337:
9334:
9330:
9327:
9325:
9322:
9320:
9317:
9316:
9315:
9311:
9308:
9307:
9305:
9303:
9299:
9295:
9287:
9284:
9282:
9279:
9277:
9276:constructible
9274:
9273:
9272:
9269:
9267:
9264:
9262:
9259:
9257:
9254:
9252:
9249:
9247:
9244:
9242:
9239:
9237:
9234:
9232:
9229:
9227:
9224:
9222:
9219:
9217:
9214:
9212:
9209:
9208:
9206:
9204:
9199:
9191:
9188:
9186:
9183:
9181:
9178:
9176:
9173:
9171:
9168:
9166:
9163:
9162:
9160:
9156:
9153:
9151:
9148:
9147:
9146:
9143:
9141:
9138:
9136:
9133:
9131:
9128:
9126:
9122:
9118:
9116:
9113:
9109:
9106:
9105:
9104:
9101:
9100:
9097:
9094:
9092:
9088:
9078:
9075:
9073:
9070:
9068:
9065:
9063:
9060:
9058:
9055:
9053:
9050:
9046:
9043:
9042:
9041:
9038:
9034:
9029:
9028:
9027:
9024:
9023:
9021:
9019:
9015:
9007:
9004:
9002:
8999:
8997:
8994:
8993:
8992:
8989:
8987:
8984:
8982:
8979:
8977:
8974:
8972:
8969:
8967:
8964:
8962:
8959:
8958:
8956:
8954:
8953:Propositional
8950:
8944:
8941:
8939:
8936:
8934:
8931:
8929:
8926:
8924:
8921:
8919:
8916:
8912:
8909:
8908:
8907:
8904:
8902:
8899:
8897:
8894:
8892:
8889:
8887:
8884:
8882:
8881:Logical truth
8879:
8877:
8874:
8873:
8871:
8869:
8865:
8862:
8860:
8856:
8850:
8847:
8845:
8842:
8840:
8837:
8835:
8832:
8830:
8827:
8825:
8821:
8817:
8813:
8811:
8808:
8806:
8803:
8801:
8797:
8794:
8793:
8791:
8789:
8783:
8778:
8772:
8769:
8767:
8764:
8762:
8759:
8757:
8754:
8752:
8749:
8747:
8744:
8742:
8739:
8737:
8734:
8732:
8729:
8727:
8724:
8722:
8719:
8717:
8714:
8710:
8707:
8706:
8705:
8702:
8701:
8699:
8695:
8691:
8684:
8679:
8677:
8672:
8670:
8665:
8664:
8661:
8654:
8651:
8648:
8643:
8639:
8638:
8629:
8625:
8622:
8618:
8615:
8611:
8608:
8604:
8601:
8598:
8594:
8592:
8588:
8584:
8580:
8579:
8568:
8563:
8559:
8554:
8550:
8546:
8542:
8536:
8532:
8527:
8523:
8521:9780840034175
8517:
8513:
8512:
8506:
8502:
8497:
8493:
8487:
8483:
8482:
8476:
8472:
8467:
8463:
8462:
8456:
8452:
8447:
8446:
8436:
8427:
8418:
8412:
8408:
8407:
8402:
8396:
8382:
8378:
8371:
8364:
8362:0-534-38214-2
8358:
8354:
8347:
8340:
8335:
8328:
8322:
8315:
8314:Stebbing 1961
8309:
8302:
8298:
8297:Stebbing 1961
8292:
8285:
8281:
8280:Stebbing 1961
8276:
8269:
8263:
8254:
8247:
8241:
8234:
8230:
8226:
8220:
8206:
8202:
8196:
8182:
8178:
8172:
8158:
8154:
8148:
8134:
8130:
8124:
8120:
8110:
8107:
8105:
8104:
8100:
8099:
8093:
8091:
8087:
8071:
8062:
8039:
8036:
8030:
8024:
8021:
7992:
7989:
7983:
7980:
7977:
7968:
7965:
7962:
7956:
7953:
7950:
7921:
7915:
7895:
7892:
7886:
7883:
7880:
7854:
7851:
7845:
7839:
7836:
7807:
7798:
7778:
7775:
7769:
7766:
7763:
7734:
7725:
7705:
7702:
7696:
7693:
7690:
7661:
7653:
7649:
7630:
7624:
7604:
7595:
7572:
7569:
7566:
7550:
7531:
7522:
7516:
7509:
7506:
7503:
7491:
7485:
7476:
7469:
7463:
7460:
7443:
7434:
7427:
7421:
7418:
7403:
7397:
7391:
7388:
7372:
7371:
7370:
7368:
7364:
7355:
7353:
7349:
7333:
7324:
7299:
7294:
7285:
7274:
7267:
7244:
7239:
7230:
7219:
7215:
7206:
7187:
7182:
7156:
7151:
7143:
7139:
7118:
7112:
7092:
7070:
7065:
7062:
7059:
7055:
7034:
7028:
7008:
6986:
6981:
6978:
6975:
6971:
6950:
6930:
6924:
6902:
6897:
6889:
6885:
6859:
6854:
6834:
6830:
6826:
6821:
6816:
6799:
6795:
6771:
6763:
6759:
6741:
6737:
6716:
6691:
6686:
6675:
6671:
6667:
6662:
6657:
6649:
6645:
6618:
6612:
6608:
6600:
6595:
6582:
6577:
6566:
6562:
6558:
6553:
6548:
6540:
6536:
6529:
6521:
6516:
6496:
6492:
6488:
6483:
6478:
6461:
6457:
6446:
6445:
6444:
6442:
6438:
6428:
6411:
6402:
6387:
6381:
6355:
6346:
6323:
6291:
6280:
6257:
6239:
6228:
6206:
6180:
6160:
6154:
6146:
6145:
6126:
6118:
6117:
6116:
6102:
6096:
6086:
6083:
6067:
6061:
6041:
6032:
6009:
6000:
5977:
5971:
5951:
5942:
5919:
5913:
5905:
5890:
5887:
5886:is not even.
5871:
5867:
5859:is odd. Thus
5846:
5843:
5840:
5837:
5832:
5828:
5819:
5815:
5808:
5791:
5787:
5779:
5764:
5757:
5754:
5753:
5752:
5750:
5743:
5728:
5721:
5704:
5700:
5692:
5688:
5687:
5686:
5672:
5658:
5644:
5635:
5612:
5606:
5598:
5594:
5580:
5573:is false, so
5560:
5549:
5548:contradiction
5533:
5522:
5521:
5506:
5503:
5500:
5497:
5494:
5491:
5490:
5486:
5483:
5480:
5477:
5474:
5471:
5470:
5466:
5463:
5460:
5457:
5454:
5451:
5450:
5446:
5443:
5440:
5437:
5434:
5431:
5430:
5427:
5404:
5382:
5380:
5376:
5373:
5371:
5349:
5347:
5325:
5323:
5320:
5318:
5315:
5314:
5311:
5309:
5308:
5285:
5261:
5260:
5256:
5251:
5246:
5244:
5240:
5236:
5232:
5228:
5224:
5220:
5216:
5211:
5209:
5205:
5201:
5197:
5193:
5189:
5185:
5181:
5177:
5173:
5167:
5160:
5156:
5152:
5148:
5146:
5140:
5135:
5119:
5095:
5085:
5069:
5057:
5053:
5052:
5034:
5022:
5018:
5014:
5010:
5006:
5002:
4998:
4988:
4986:
4982:
4978:
4974:
4969:
4964:
4962:
4958:
4954:
4950:
4946:
4942:
4938:
4934:
4930:
4926:
4922:
4918:
4914:
4910:
4904:
4900:
4890:
4888:
4884:
4880:
4876:
4872:
4868:
4864:
4860:
4856:
4852:
4848:
4845:". Thus, the
4844:
4840:
4836:
4832:
4828:
4824:
4820:
4816:
4811:
4809:
4805:
4801:
4797:
4793:
4783:
4772:
4768:
4766:
4762:
4760:
4756:
4752:
4751:
4750:
4748:
4747:
4746:modus tollens
4742:
4738:
4734:
4730:
4726:
4722:
4710:
4706:
4704:
4701:Premise (2):
4700:
4698:
4694:
4690:
4689:
4688:
4686:
4685:
4680:
4676:
4672:
4668:
4664:
4660:
4656:
4652:
4648:
4644:
4634:
4631:
4629:
4625:
4621:
4617:
4613:
4607:
4603:
4595:
4591:
4583:
4579:
4571:
4567:
4561:
4556:
4554:
4550:
4539:
4537:
4533:
4529:
4528:transposition
4525:
4521:
4517:
4512:
4508:
4506:
4502:
4498:
4494:
4483:
4480:
4477:
4474:
4471:
4470:
4466:
4463:
4461:
4458:
4455:
4454:
4450:
4447:
4445:
4442:
4439:
4438:
4434:
4431:
4428:
4425:
4422:
4421:
4417:
4414:
4412:
4409:
4406:
4405:
4402:
4396:
4393:
4392:
4391:
4385:
4382:
4381:
4380:
4378:
4370:
4367:
4366:
4365:
4363:
4355:
4352:
4351:
4350:
4347:
4345:
4340:
4336:
4332:
4328:
4324:
4320:
4316:
4312:
4309:
4305:
4301:
4297:
4293:
4289:
4285:
4281:
4277:
4272:
4270:
4266:
4263:processes of
4262:
4258:
4254:
4250:
4249:contradictory
4246:
4242:
4238:
4235:is a form of
4234:
4230:
4224:
4211:
4207:
4204:
4201:
4198:
4195:
4192:
4191:
4185:
4183:
4179:
4175:
4173:
4167:
4166:biconditional
4159:
4155:
4151:
4148:
4144:
4140:
4137:
4133:
4129:
4126:
4122:
4118:
4114:
4113:
4112:
4110:
4106:
4104:
4097:
4095:
4094:biconditional
4091:
4083:
4079:
4075:
4072:
4068:
4064:
4061:
4057:
4053:
4050:
4046:
4042:
4041:
4040:
4038:
4034:
4019:
4017:
4013:
4010:
4007:
4006:
4002:
4000:
3996:
3992:
3989:
3988:
3984:
3982:
3978:
3974:
3971:
3970:
3966:
3964:
3960:
3956:
3953:
3952:
3948:
3946:
3942:
3938:
3935:
3934:
3930:
3927:
3924:
3923:
3897:
3888:
3873:
3867:
3857:
3839:
3830:
3815:
3803:
3787:
3769:
3757:
3739:
3733:
3723:
3705:
3693:
3675:
3663:
3653:
3632:
3620:
3602:
3590:
3575:
3569:
3553:
3538:
3532:
3519:
3501:
3489:
3477:
3471:
3461:
3440:
3428:
3416:
3410:
3395:
3383:
3373:
3358:
3346:
3339:
3338:
3337:
3334:
3332:
3306:
3300:
3288:
3282:
3267:
3261:
3250:
3248:
3226:
3220:
3208:
3202:
3187:
3181:
3170:
3155:
3143:
3135:
3132:
3116:
3110:
3096:
3095:
3094:
3092:
3073:
3064:
3049:
3043:
3016:
3012:
3006:
3002:
2994:
2990:
2981:
2974:
2965:
2950:
2945:
2941:
2935:
2931:
2923:
2919:
2913:
2909:
2904:
2896:
2891:
2887:
2881:
2877:
2870:
2866:
2857:
2842:
2838:
2832:
2828:
2821:
2813:
2812:
2811:
2809:
2805:
2781:
2772:
2762:
2761:
2760:
2746:
2726:
2718:
2714:
2711:By reverting
2695:
2689:
2682:
2681:
2680:
2678:
2674:
2652:
2646:
2643:
2630:
2629:
2628:
2614:
2594:
2568:
2545:
2522:
2502:
2492:as equal to "
2479:
2453:
2450:
2447:
2431:
2430:
2429:
2427:
2426:commutativity
2423:
2401:
2395:
2392:
2379:
2378:
2377:
2363:
2343:
2323:
2303:
2283:
2263:
2240:
2234:
2227:
2226:
2225:
2223:
2196:
2187:
2178:
2172:
2166:
2156:
2155:
2154:
2134:
2128:
2116:
2107:
2094:
2093:
2092:
2074:
2071:
2065:
2056:
2043:
2042:
2041:
2021:
2012:
1997:
1991:
1981:
1980:
1979:
1977:
1973:
1953:
1947:
1941:
1935:
1925:
1924:
1923:
1897:
1888:
1879:
1869:
1863:
1860:
1854:
1848:
1845:
1842:
1828:
1827:
1826:
1809:
1806:
1803:
1792:
1786:
1779:
1778:
1777:
1775:
1760:
1758:
1757:formal system
1742:
1722:
1699:
1693:
1684:
1669:
1663:
1653:
1652:
1651:
1649:
1648:
1643:
1639:
1635:
1631:
1615:
1606:
1583:
1577:
1554:
1548:
1539:
1533:
1528:
1522:
1512:
1511:
1510:
1493:
1487:
1476:
1457:
1448:
1422:
1399:
1393:
1384:
1375:
1369:
1363:
1353:
1352:
1351:
1349:
1345:
1344:transposition
1335:
1321:
1301:
1277:
1273:
1263:
1259:
1249:
1225:
1205:
1181:
1177:
1170:
1166:
1159:
1146:
1144:
1140:
1136:
1132:
1128:
1124:
1108:
1088:
1068:
1045:
1039:
1030:
1017:
1016:
1015:
1013:
1009:
1005:
989:
981:
965:
957:
953:
937:
917:
891:
885:
875:
874:
873:
871:
867:
857:
855:
851:
847:
843:
838:
824:
818:
798:
789:
766:
757:
734:
728:
704:
698:
689:
674:
668:
658:
657:
656:
639:
630:
620:
619:
618:
616:
612:
593:
587:
580:
579:
578:
576:
575:Euler diagram
568:
559:
557:
554:
553:
552:modus tollens
548:
532:
521:
508:
487:
476:
463:
442:
436:
429:Note that if
424:
420:
416:
415:it is raining
412:
408:
391:
385:
371:
367:
366:
362:
359:
358:it is raining
355:
351:
337:
331:
323:
319:
318:
314:
311:
307:
303:
299:
285:
276:
265:
261:
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216:
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211:
207:
203:
199:
194:
190:
186:
182:
176:
162:
153:
130:
124:
116:
100:
94:
87:
86:
81:
79:
75:
72:
68:
64:
60:
57:
53:
49:
45:
44:transposition
41:
37:
33:
19:
10398:
10196:Ultraproduct
10043:Model theory
10008:Independence
9944:Formal proof
9936:Proof theory
9919:
9892:
9849:real numbers
9821:second-order
9732:Substitution
9609:Metalanguage
9550:conservative
9523:Axiom schema
9467:Constructive
9437:Morse–Kelley
9403:Set theories
9382:Aleph number
9375:inaccessible
9281:Grothendieck
9165:intersection
9052:Higher-order
9040:Second-order
8986:Truth tables
8943:Venn diagram
8726:Formal proof
8627:
8620:
8613:
8606:
8596:
8582:
8566:
8557:
8530:
8510:
8500:
8480:
8470:
8464:. Macmillan.
8460:
8450:
8435:
8426:
8405:
8401:Franklin, J.
8395:
8384:. Retrieved
8380:
8370:
8352:
8346:
8334:
8321:
8308:
8291:
8275:
8267:
8262:
8253:
8245:
8240:
8232:
8228:
8224:
8219:
8208:. Retrieved
8204:
8195:
8184:. Retrieved
8180:
8171:
8160:. Retrieved
8156:
8147:
8136:. Retrieved
8132:
8123:
8101:
8089:
7646:denotes the
7546:
7362:
7361:
7347:
7105:saying that
7021:saying that
6756:denotes the
6633:
6436:
6435:
6370:
6088:
5901:
5888:
5817:
5813:
5811:
5807:is not even.
5806:
5777:
5755:
5749:direct proof
5746:
5741:
5719:
5690:
5664:
5596:
5595:
5518:
5517:
5406:
5383:
5378:
5374:
5350:
5326:
5321:
5316:
5287:
5264:
5263:
5258:
5254:
5253:
5247:
5242:
5238:
5234:
5230:
5218:
5212:
5207:
5203:
5199:
5195:
5191:
5187:
5183:
5179:
5175:
5171:
5165:
5158:
5154:
5150:
5142:
5138:
5136:
5059:
5055:
5024:
5009:direct proof
5004:
4996:
4995:Because the
4994:
4984:
4980:
4976:
4972:
4965:
4960:
4956:
4952:
4948:
4944:
4940:
4936:
4932:
4928:
4924:
4916:
4906:
4886:
4882:
4878:
4874:
4870:
4866:
4862:
4858:
4854:
4850:
4842:
4838:
4834:
4830:
4826:
4822:
4818:
4814:
4812:
4807:
4803:
4799:
4795:
4789:
4780:
4770:
4764:
4758:
4754:
4744:
4740:
4736:
4732:
4728:
4724:
4720:
4718:
4708:
4702:
4696:
4692:
4684:modus ponens
4682:
4678:
4674:
4670:
4666:
4662:
4658:
4654:
4650:
4646:
4642:
4640:
4632:
4605:
4601:
4593:
4589:
4581:
4577:
4569:
4565:
4560:propositions
4559:
4557:
4552:
4548:
4545:
4519:
4513:
4509:
4497:per accidens
4496:
4489:
4400:
4394:
4389:
4383:
4374:
4368:
4359:
4353:
4348:
4343:
4334:
4314:
4313:
4304:hypothetical
4299:
4273:
4232:
4226:
4181:
4177:
4171:
4169:
4163:
4157:
4153:
4146:
4142:
4135:
4131:
4120:
4116:
4108:
4100:
4098:
4087:
4081:
4077:
4070:
4066:
4059:
4055:
4048:
4044:
4036:
4032:
4030:
4015:
4011:
3998:
3994:
3980:
3976:
3962:
3958:
3944:
3940:
3931:description
3335:
3328:
3032:
2801:
2716:
2712:
2710:
2676:
2675:is true and
2672:
2670:
2581:is equal to
2558:(from this,
2535:as equal to
2471:
2419:
2276:is true and
2255:
2219:
2152:
2089:
2039:
1972:Modus Ponens
1968:
1921:
1824:
1771:
1714:
1645:
1569:
1414:
1343:
1341:
1147:
1134:
1130:
1126:
1122:
1060:
1007:
955:
951:
909:
869:
865:
863:
853:
849:
845:
841:
839:
719:
654:
614:
610:
608:
572:
550:
546:
501:
456:
428:
422:
418:
414:
410:
369:
363:
357:
353:
321:
315:
305:
301:
263:
257:
220:
217:
213:
209:
205:
201:
197:
192:
188:
184:
180:
177:
83:
82:
58:
43:
39:
29:
10306:Type theory
10254:undecidable
10186:Truth value
10073:equivalence
9752:non-logical
9365:Enumeration
9355:Isomorphism
9302:cardinality
9286:Von Neumann
9251:Ultrafilter
9216:Uncountable
9150:equivalence
9067:Quantifiers
9057:Fixed-point
9026:First-order
8906:Consistency
8891:Proposition
8868:Traditional
8839:Lindström's
8829:Compactness
8771:Type theory
8716:Cardinality
6336:, conclude
6199:, conclude
5747:Although a
5599:: To prove
5250:truth table
5241:, then not
5215:mathematics
5202:=1 so that
4806:, then not
4532:Irving Copi
4241:proposition
4239:in which a
4125:truth value
4090:conditional
3936:implication
3918:Comparisons
2422:conjunction
1135:contraposed
1081:, then not-
950:", or, "if
310:truth value
191:, Then not
187:. — If not
36:mathematics
10421:Categories
10117:elementary
9810:arithmetic
9678:Quantifier
9656:functional
9528:Expression
9246:Transitive
9190:identities
9175:complement
9108:hereditary
9091:Set theory
8549:1122695276
8386:2019-10-26
8339:Prior 1973
8210:2019-11-26
8186:2019-11-26
8162:2019-11-26
8138:2019-11-26
8115:References
7908:i.e. when
7650:(aka. the
6760:(aka. the
5689:To prove:
4903:Set theory
4897:See also:
4628:conversion
4505:particular
4410:Obversion
4377:conversion
4331:particular
4265:conversion
4221:See also:
2719:back into
2472:We define
1314:s are non-
1198:, or "All
1004:consequent
980:antecedent
317:Conversion
71:consequent
67:antecedent
10388:Supertask
10291:Recursion
10249:decidable
10083:saturated
10061:of models
9984:deductive
9979:axiomatic
9899:Hilbert's
9886:Euclidean
9867:canonical
9790:axiomatic
9722:Signature
9651:Predicate
9540:Extension
9462:Ackermann
9387:Operation
9266:Universal
9256:Recursive
9231:Singleton
9226:Inhabited
9211:Countable
9201:Types of
9185:power set
9155:partition
9072:Predicate
9018:Predicate
8933:Syllogism
8923:Soundness
8896:Inference
8886:Tautology
8788:paradoxes
8327:Copi 1979
8301:Copi 1953
8284:Copi 1953
8069:¬
8066:→
8060:¬
8028:¬
8025:∣
8019:¬
7981:∣
7969:−
7954:∣
7948:¬
7919:→
7884:∣
7843:¬
7840:∣
7834:¬
7805:¬
7802:→
7767:∣
7761:¬
7732:¬
7729:→
7694:∣
7688:¬
7648:base rate
7602:¬
7599:→
7570:∣
7564:¬
7507:∣
7501:¬
7483:¬
7467:¬
7464:∣
7458:¬
7441:¬
7425:¬
7422:∣
7416:¬
7395:¬
7392:∣
7386:¬
7331:¬
7328:→
7322:¬
7292:¬
7286:~
7272:¬
7268:ω
7237:¬
7231:~
7216:ω
7188:~
7183:ϕ
7140:ω
7116:→
7063:∣
7056:ω
7032:→
6979:∣
6972:ω
6928:→
6886:ω
6852:¬
6846:~
6831:ω
6811:~
6796:ω
6758:base rate
6684:¬
6672:ω
6646:ω
6601:~
6596:ϕ
6575:¬
6563:ω
6537:ω
6514:¬
6508:~
6493:ω
6473:~
6458:ω
6409:¬
6406:→
6400:¬
6394:→
6385:→
6353:¬
6350:→
6344:¬
6321:¬
6298:⊥
6295:→
6264:⊥
6261:→
6252:→
6246:⊥
6243:→
6213:⊥
6210:→
6187:⊥
6184:→
6158:→
6133:⊥
6130:→
6100:→
6065:→
6039:¬
6036:→
6030:¬
6007:¬
6004:→
5998:¬
5975:→
5949:¬
5946:→
5940:¬
5917:→
5844:⋅
5642:¬
5639:→
5633:¬
5610:→
5593:is true.
5558:¬
5531:¬
5414:¬
5391:¬
5358:¬
5334:¬
5295:¬
5272:¬
4921:obversion
4501:universal
4362:obversion
4333:is made (
4327:universal
4292:predicate
4269:obversion
4261:inference
4253:predicate
3997:then not
3961:then not
3895:¬
3892:→
3886:¬
3880:→
3871:→
3837:¬
3834:→
3828:¬
3822:→
3813:¬
3810:¬
3807:→
3801:¬
3798:¬
3767:¬
3764:¬
3761:→
3755:¬
3752:¬
3746:→
3737:→
3703:¬
3700:¬
3697:→
3691:¬
3688:¬
3682:→
3673:¬
3670:¬
3667:→
3630:¬
3627:¬
3624:→
3618:¬
3615:¬
3609:→
3600:¬
3597:¬
3594:→
3582:→
3573:→
3567:¬
3564:¬
3536:→
3530:¬
3527:¬
3499:¬
3496:¬
3493:→
3484:→
3475:→
3438:¬
3435:¬
3432:→
3423:→
3414:→
3402:→
3393:¬
3390:¬
3387:→
3356:¬
3353:¬
3350:→
3304:→
3295:→
3286:→
3274:→
3265:→
3224:→
3215:→
3206:→
3194:→
3185:→
3153:¬
3150:¬
3147:→
3114:→
3108:¬
3105:¬
3074:ψ
3071:¬
3068:→
3065:ϕ
3062:¬
3056:→
3050:ϕ
3047:→
3044:ψ
3013:ϕ
3010:→
3007:ψ
2999:→
2991:ψ
2988:¬
2985:→
2982:ϕ
2979:¬
2942:ξ
2939:→
2936:ϕ
2928:→
2920:ψ
2917:→
2914:ϕ
2901:→
2888:ξ
2885:→
2882:ψ
2874:→
2871:ϕ
2839:ϕ
2836:→
2833:ψ
2825:→
2822:ϕ
2779:¬
2776:→
2770:¬
2693:→
2650:¬
2647:∧
2638:¬
2592:¬
2589:¬
2566:¬
2543:¬
2500:¬
2451:∧
2445:¬
2439:¬
2399:¬
2396:∧
2387:¬
2238:→
2194:¬
2191:→
2185:¬
2179:≡
2170:→
2132:→
2123:→
2114:¬
2111:→
2105:¬
2072:∧
2063:¬
2060:→
2054:¬
2019:¬
2016:→
2010:¬
2004:→
1995:→
1951:¬
1948:∧
1939:→
1895:¬
1892:→
1886:¬
1883:↔
1867:¬
1864:∨
1858:↔
1846:∨
1840:¬
1807:∨
1801:¬
1797:↔
1790:→
1691:¬
1688:→
1682:¬
1676:→
1667:→
1642:Whitehead
1630:tautology
1613:¬
1610:→
1604:¬
1581:→
1546:¬
1543:→
1537:¬
1534:∴
1526:→
1491:→
1455:¬
1452:→
1446:¬
1423:⊢
1391:¬
1388:→
1382:¬
1376:⊢
1367:→
1271:¬
1268:→
1257:¬
1246:∀
1175:→
1156:∀
1037:¬
1034:→
1028:¬
889:→
822:→
796:¬
793:→
787:¬
764:¬
761:→
755:¬
732:→
696:¬
693:→
687:¬
681:↔
672:→
637:¬
634:→
628:¬
591:→
549:, or the
530:¬
485:¬
440:→
389:→
380:¬
335:→
283:¬
280:→
274:¬
259:Inversion
238:¬
235:→
229:¬
160:¬
157:→
151:¬
128:→
98:→
54:into its
48:inference
10373:Logicism
10366:timeline
10342:Concrete
10201:Validity
10171:T-schema
10164:Kripke's
10159:Tarski's
10154:semantic
10144:Strength
10093:submodel
10088:spectrum
10056:function
9904:Tarski's
9893:Elements
9880:geometry
9836:Robinson
9757:variable
9742:function
9715:spectrum
9705:Sentence
9661:variable
9604:Language
9557:Relation
9518:Automata
9508:Alphabet
9492:language
9346:-jection
9324:codomain
9310:Function
9271:Universe
9241:Infinite
9145:Relation
8928:Validity
8918:Argument
8816:theorem,
8421:(p. 50).
8096:See also
6281:Turning
5990:implies
5742:is even.
5225:used in
5001:theorems
4915:through
4881:) and (¬
4493:converse
4319:negation
4154:negation
4143:converse
4078:negation
4067:converse
4027:Examples
4014:and not
4008:negation
3972:converse
2356:and not-
2316:and not-
844:implies
365:Negation
322:converse
115:formulas
74:inverted
10315:Related
10112:Diagram
10010: (
9989:Hilbert
9974:Systems
9969:Theorem
9847:of the
9792:systems
9572:Formula
9567:Grammar
9483: (
9427:General
9140:Forcing
9125:Element
9045:Monadic
8820:paradox
8761:Theorem
8697:General
8576:Sources
6119:Assume
6089:Assume
5661:Example
5233:, then
5163:√
4983:is non-
4959:is non-
4935:is non-
4869:is non-
4798:, then
4757:, then
4695:, then
4677:, then
4645:, then
4551:, then
4339:obverse
4288:subject
4284:classes
4245:subject
4132:inverse
4056:inverse
3993:if not
3957:if not
3954:inverse
2515:", and
1638:Russell
1634:theorem
1348:sequent
1129:, then
1012:negated
1002:is the
978:is the
954:, then
930:, then
573:In the
264:inverse
183:, Then
78:flipped
10078:finite
9841:Skolem
9794:
9769:Theory
9737:Symbol
9727:String
9710:atomic
9587:ground
9582:closed
9577:atomic
9533:ground
9496:syntax
9392:binary
9319:domain
9236:Finite
9001:finite
8859:Logics
8818:
8766:Theory
8589:
8547:
8537:
8518:
8488:
8413:
8359:
6634:where
5227:proofs
4296:copula
3251:(HS2)
3171:(HS1)
3136:(DN2)
3097:(DN1)
1763:Proofs
1715:where
1415:where
1218:s are
982:, and
615:cannot
10068:Model
9816:Peano
9673:Proof
9513:Arity
9442:Naive
9329:image
9261:Fuzzy
9221:Empty
9170:union
9115:Class
8756:Model
8746:Lemma
8704:Axiom
7867:when
7654:) of
6764:) of
6313:into
6147:From
5174:with
5143:is a
5007:is a
4849:"All
4467:None
4464:None
4451:None
4448:None
4188:Truth
3979:then
3943:then
1922:Let:
1473:is a
854:not A
850:not B
417:then
368:(the
356:then
320:(the
304:then
262:(the
212:then
204:then
113:. In
42:, or
32:logic
10191:Type
9994:list
9798:list
9775:list
9764:Term
9698:rank
9592:open
9486:list
9298:Maps
9203:sets
9062:Free
9032:list
8782:list
8709:list
8587:ISBN
8545:OCLC
8535:ISBN
8516:ISBN
8486:ISBN
8411:ISBN
8357:ISBN
8312:See
8295:See
7350:and
6173:and
5665:Let
5262:and
5178:and
5056:true
4919:and
4901:and
4841:are
4723:and
4614:and
4290:and
4282:and
4267:and
4247:the
4156:is "
4152:The
4145:is "
4141:The
4134:is "
4130:The
4119:is "
4115:The
4101:All
4080:is "
4076:The
4069:is "
4065:The
4058:is "
4054:The
4047:is "
4043:The
3928:form
3925:name
3091:here
2966:A3.
2858:A2.
2814:A1.
2739:and
2715:and
2376:"):
1735:and
1640:and
1342:The
1334:s."
1143:true
352:"If
300:"If
266:),
208:"If
76:and
69:and
34:and
9878:of
9860:of
9808:of
9340:Sur
9314:Map
9121:Ur-
9103:Set
7203:of
6054:to
5902:In
5756:If
5691:If
5213:In
5172:a/b
5060:If
5025:If
4975:is
4951:is
4943:is
4927:is
4885:→ ¬
4861:is
4853:is
4817:is
4810:".
4669:to
4622:or
4604:→ ¬
4592:→ ¬
4574:to
4555:".
4520:not
4503:to
4329:to
4306:or
4274:In
4227:In
4182:iff
3975:if
3939:if
2802:In
2627:):
2428:):
1772:In
1650:as
1644:in
1632:or
1477:of
611:not
413:if
372:),
324:),
200:If
179:If
143:is
30:In
10423::
10264:NP
9888::
9882::
9812::
9489:),
9344:Bi
9336:In
8543:.
8379:.
8203:.
8179:.
8155:.
8131:.
8092:.
8013:Pr
7972:Pr
7942:Pr
7875:Pr
7828:Pr
7755:Pr
7682:Pr
7558:Pr
7495:Pr
7452:Pr
7410:Pr
7380:Pr
7354:.
6427:.
5657:.
5507:T
5487:T
5467:F
5447:T
5405:→
5377:→
5286:→
5257:→
5217:,
5206:=
5186:=
4877:→
4749::
4731:,
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