4002:
1157:. Enderton, for example, observes that "modus ponens can produce shorter formulas from longer ones", and Russell observes that "the process of the inference cannot be reduced to symbols. Its sole record is the occurrence of ⊦q ... an inference is the dropping of a true premise; it is the dissolution of an implication".
2953:
failure. These are cases where the conditional premise describes an obligation predicated on an immoral or imprudent action, e.g., "If Doe murders his mother, he ought to do so gently," for which the dubious unconditional conclusion would be "Doe ought to gently murder his mother." It would appear to
781:
all the premises are true, then the argument is sound. For example, John might be going to work on
Wednesday. In this case, the reasoning for John's going to work (because it is Wednesday) is unsound. The argument is only sound on Tuesdays (when John goes to work), but valid on every day of the week.
2832:, the first premise is true. The second premise is also true, since starting with a set of possible authors limited to just Shakespeare and Hobbes and eliminating one of them leaves only the other. However, the conclusion is doubtful, since ruling out Shakespeare as the author of
2836:
would leave numerous possible candidates, many of them more plausible alternatives than Hobbes (if the if-thens in the inference are read as material conditionals, the conclusion comes out true simply by virtue of the false antecedent. This is one of the
2246:
1136:
in logic it must not be mistaken for a logical law; rather, it is one of the accepted mechanisms for the construction of deductive proofs that includes the "rule of definition" and the "rule of substitution".
1338:
2130:
1818:
2890:
1768:
1653:
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1208:-like structure with a single element (the "always-true") at the top and another single element (the "always-false") at the bottom. Logical equivalence becomes identity, so that when
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A justification for the "trust in inference is the belief that if the two former assertions are not in error, the final assertion is not in error". In other words: if one
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1983:
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1149:(the antecedents) and thereby not carry these antecedents forward in an ever-lengthening string of symbols; for this reason modus ponens is sometimes called the
3404:
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for a calculus says that every proof involving Cut can be transformed (generally, by a constructive method) into a proof without Cut, and hence that Cut is
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1872:, the semantics becomes more complex, the algebra may be non-Boolean, and the validity of modus ponens cannot be taken for granted.
4305:
663:, is one of the standard patterns of inference that can be applied to derive chains of conclusions that lead to the desired goal.
817:
1284:
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459:
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3459:
2241:{\displaystyle \omega _{Q\|P}^{A}=(\omega _{Q|P}^{A},\omega _{Q|\lnot P}^{A})\circledcirc \omega _{P}^{A}\,,}
4320:
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4225:
4215:
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1204:
treats every sentence as a name for an element in an ordered set. Typically, the set can be visualized as a
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remains a controversial view among logicians, but opinions vary on how the cases should be disposed of.
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2411:
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can fail for conditionals whose consequents are themselves conditionals. The following is an example:
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a second one, and the first statement or proposition is true, then the second one is also true. If
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of the conditional claim, is the case. From these two premises it can be logically concluded that
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4143:
4036:
4031:
3931:
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337:
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can be valid but nonetheless unsound if one or more premises are false; if an argument is valid
3436:
3135:
4026:
3936:
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3793:
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3623:
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Sentential
Probability Logic: Origins, Development, Current Status, and Technical Applications
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be a disjunction, as in the example given. That these kinds of cases constitute failures of
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77:
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1409:
3813:
paperback edition 1962, Cambridge at the
University Press, London UK. No ISBN, no LCCCN.
3704:
2764:
is absolute TRUE. Hence, subjective logic deduction represents a generalization of both
761:, but this has no bearing on whether any of the statements in the argument are actually
4290:
3946:
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is true. Only one line of the truth table—the first—satisfies these two conditions (
17:
4285:
4260:
4255:
1478:
1146:
654:
3677:
D. E. Over (1987). "Assumption and the
Supposed Counterexamples to Modus Ponens",
773:
argument, the premises must be true for any true instances of the conclusion. An
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he is doing exactly what he should, unconditionally, be doing. Here again,
331:
3664:
Sinnott-Armstrong, Moor, and
Fogelin (1986). "A Defense of Modus Ponens",
3966:
3025: – System of propositional logic developed by the Stoic philosophers
774:
3859:
2949:, some examples of conditional obligation also raise the possibility of
3019: – Arrangement that allows conflicting parties to coexist in peace
884:
325:
3821:
Introduction to Logic and to the
Methodology of the Deductive Sciences
996:
in classical two-valued logic can be clearly demonstrated by use of a
2806:
305:
2780:
Philosophers and linguists have identified a variety of cases where
953:
3745:
2954:
follow that if Doe is in fact gently murdering his mother, then by
3876:
3431:
Latin for the
Illiterati: Exorcizing the Ghosts of a Dead Language
762:
1655:. In the semantics for basic propositional logic, the algebra is
4001:
2962:
failure is not a popular diagnosis but is sometimes argued for.
1545:
as valid—is to say that the highest point which lies below both
3880:
3790:
Subjective Logic; A formalism for
Reasoning Under Uncertainty
2121:
represents an instance of the binomial deduction operator in
1340:. Logical implication becomes a matter of relative position:
532:
526:
512:
506:
394:
352:
318:
273:
3823:
2nd
Edition, reprinted by Dover Publications, Mineola NY.
3731:
E.g., by
Kolodny and MacFarlane (2010). "Ifs and Oughts",
952:
are statements (or propositions) in a formal language and
344:
3489:(2002). "The Development of Modus Ponens in Antiquity",
1333:{\displaystyle \neg {(P\wedge Q)}=\neg {P}\vee \neg {Q}}
312:
27:"Forward reasoning" redirects here. For other uses, see
3651:
Vann McGee (1985). "A Counterexample to Modus Ponens",
2997:
Pages displaying short descriptions of redirect targets
425:
380:
286:
262:
3622:. London: Associated University Presses. p. 203.
1481:
1281:, for instance, are equivalent (as is standard), then
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649:. The first to explicitly describe the argument form
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of the conditional claim, must be the case as well.
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3775:
A Mathematical Introduction to Logic Second Edition
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2844:The general form of McGee-type counterexamples to
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1813:{\displaystyle P\wedge (P\rightarrow Q)=P\wedge Q}
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1237:
925:
618:. Both have apparently similar but invalid forms:
216:
166:
146:
126:
106:
3746:"Fallacies | Internet Encyclopedia of Philosophy"
338:
3528:Alfred Tarski 1946:47. Also Enderton 2001:110ff.
2817:, then if Shakespeare did not do it, Hobbes did.
2068:
1963:
1928:
1887:
418:
2885:{\displaystyle P,P\rightarrow (Q\rightarrow R)}
1191:Correspondence to other mathematical frameworks
747:If today is Tuesday, then John will go to work.
387:
279:
3031: – Allegorical dialogue by Lewis Carroll
1763:{\displaystyle P\wedge (P\rightarrow Q)\leq Q}
1648:{\displaystyle P\wedge (P\rightarrow Q)\leq Q}
3892:
3811:Principia Mathematica to *56 (Second Edition)
1719:{\displaystyle P\rightarrow Q=\neg {P}\vee Q}
739:An example of an argument that fits the form
432:
401:
299:
292:
8:
3548:"Modus ponens - Encyclopedia of Mathematics"
3460:"Oxford reference: affirming the antecedent"
2732:is absolute TRUE and the antecedent opinion
2669:
2423:
2142:
39:
552:
488:
4149:
3899:
3885:
3877:
3777:, Harcourt Academic Press, Burlington MA,
2654:produces an absolute TRUE deduced opinion
1472:and is connected to it by an upward path.
926:{\displaystyle P\to Q,\;P\;\;\vdash \;\;Q}
919:
918:
914:
913:
909:
720:. The second premise is an assertion that
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3690:Bledin (2015). "Modus Ponens Defended",
3236:must always be greater than or equal to
2820:Therefore, if Shakespeare did not write
3579:
3577:
3419:
3048:The highest point that lies below both
3041:
562: 'method of putting by placing'),
449:
442:
368:
245:
238:
231:
2813:If either Shakespeare or Hobbes wrote
679:, with two premises and a conclusion:
217:{\displaystyle P\to Q,\;P\;\vdash \ Q}
3386:must always be less than or equal to
3334:must always be less than or equal to
2974:is a common misinterpretation of the
2388:. The deduced marginal opinion about
2282:denotes the subjective opinion about
1914:{\displaystyle \Pr(P\rightarrow Q)=x}
1274:{\displaystyle \neg {P}\vee \neg {Q}}
7:
2995: – Principle of classical logic
2362:generalizes the logical implication
820:between proofs and programs relates
3719:Stanford Encyclopedia of Philosophy
3516:Stanford Encyclopedia of Philosophy
2570:is an absolute FALSE opinion about
3174:
3029:What the Tortoise Said to Achilles
2685:{\displaystyle \omega _{Q\|P}^{A}}
2630:is FALSE. The deduction operator
2478:is an absolute TRUE opinion about
2439:{\displaystyle \omega _{Q\|P}^{A}}
2200:
1702:
1322:
1311:
1288:
1263:
1252:
1238:{\displaystyle \neg {(P\wedge Q)}}
1215:
1104:is also true. Therefore, whenever
608:and is closely related to another
25:
3294:will be greater than or equal to
3011: – Rule of logical inference
2839:paradoxes of material implication
2725:{\displaystyle \omega _{Q|P}^{A}}
2355:{\displaystyle \omega _{Q|P}^{A}}
1770:is then straightforward, because
1132:is one of the most commonly used
4000:
753:Therefore, John will go to work.
496:
4398:Theorems in propositional logic
3503:"Ancient Logic: Forerunners of
2757:{\displaystyle \omega _{P}^{A}}
2563:{\displaystyle \omega _{P}^{A}}
2471:{\displaystyle \omega _{P}^{A}}
2275:{\displaystyle \omega _{P}^{A}}
1845:{\displaystyle P\wedge Q\leq Q}
712:("if–then") claim, namely that
4367:Tractatus Logico-Philosophicus
3972:Problem of multiple generality
3209:{\displaystyle P\rightarrow Q}
3200:
2911:{\displaystyle Q\rightarrow R}
2902:
2879:
2873:
2867:
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2708:
2372:
2338:
2322:, and the conditional opinion
2213:
2196:
2170:
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2016:
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1584:{\displaystyle P\rightarrow Q}
1575:
1514:{\displaystyle P\rightarrow Q}
1505:
1304:
1292:
1231:
1219:
900:
191:
1:
4357:The Principles of Mathematics
3435:. London: Routledge. p.
2692:when the conditional opinion
2643:{\displaystyle \circledcirc }
1475:In this context, to say that
988:Justification via truth table
4053:Commutativity of conjunction
3618:Hailperin, Theodore (1996).
3583:Whitehead and Russell 1927:9
2828:Since Shakespeare did write
2788:, for instance, argued that
2538:is TRUE, and the case where
1865:{\displaystyle \rightarrow }
1672:{\displaystyle \rightarrow }
3848:Encyclopedia of Mathematics
3773:Herbert B. Enderton, 2001,
2918:; it is not essential that
1852:. With other treatments of
1076:we assume as premises that
818:Curry–Howard correspondence
581:. It can be summarized as "
4419:
4073:Monotonicity of entailment
1141:allows one to eliminate a
469:Existential generalization
274:Biconditional introduction
26:
3998:
3962:Idempotency of entailment
3733:The Journal of Philosophy
3692:The Journal of Philosophy
3666:The Journal of Philosophy
3653:The Journal of Philosophy
3597:. Humanities-Ebooks LLP.
3151:{\displaystyle X\wedge Y}
1985:must lie in the interval
1147:logical proof or argument
34:Rule of logical inference
2972:affirming the consequent
2776:Alleged cases of failure
2770:Law of total probability
2590:is equivalent to source
2498:is equivalent to source
1949:{\displaystyle \Pr(P)=y}
620:affirming the consequent
568:affirming the antecedent
460:Universal generalization
300:Disjunction introduction
287:Conjunction introduction
257:Implication introduction
4321:Willard Van Orman Quine
2302:as expressed by source
2029:. For the special case
1399:{\displaystyle P\leq Q}
1200:In mathematical logic,
883:rule may be written in
861:artificial intelligence
807:cut-elimination theorem
708:The first premise is a
564:implication elimination
4296:Charles Sanders Peirce
4139:Hypothetical syllogism
3803:Alfred North Whitehead
3552:encyclopediaofmath.org
3427:Stone, Jon R. (1996).
3400:
3380:
3348:
3328:
3308:
3288:
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3230:
3210:
3184:
3183:{\displaystyle \neg P}
3152:
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2381:{\displaystyle P\to Q}
2356:
2316:
2296:
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2242:
2104:
2084:
2083:{\displaystyle \Pr(Q)}
2055:
2023:
1979:
1978:{\displaystyle \Pr(Q)}
1950:
1915:
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1720:
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1446:
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1374:
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1334:
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1239:
927:
832:is a function of type
677:hypothetical syllogism
624:denying the antecedent
606:hypothetical syllogism
553:
489:
319:hypothetical syllogism
240:Propositional calculus
218:
168:
148:
128:
108:
83:Propositional calculus
4393:Latin logical phrases
4362:Principia Mathematica
4134:Disjunctive syllogism
4119:modus ponendo tollens
3401:
3381:
3379:{\displaystyle x+y-1}
3349:
3329:
3309:
3289:
3287:{\displaystyle x+y-1}
3257:
3231:
3211:
3185:
3153:
3127:
3107:
3083:
3063:
2993:Import-export (logic)
2933:
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2105:
2085:
2056:
2054:{\displaystyle x=y=1}
2024:
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1447:
1427:
1401:
1375:
1355:
1335:
1276:
1240:
1143:conditional statement
966:syntactic consequence
928:
805:is the Cut rule. The
797:In single-conclusion
361:Negation introduction
354:modus ponendo tollens
219:
169:
149:
129:
109:
4352:Function and Concept
4124:Constructive dilemma
4099:Material implication
3873:at Wolfram MathWorld
3788:Audun Jøsang, 2016,
3642:Audun Jøsang 2016:92
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3052:
2988:Condensed detachment
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1989:
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1876:Probability calculus
1856:
1824:
1774:
1730:
1687:
1681:material conditional
1663:
1615:
1595:
1569:
1549:
1541:—that is, to affirm
1525:
1499:
1479:
1456:
1436:
1410:
1406:, i.e., when either
1384:
1364:
1344:
1285:
1249:
1212:
960:symbol meaning that
894:
826:function application
675:argument is a mixed
628:Constructive dilemma
598:must also be true."
594:is true. Therefore,
554:modus ponendo ponens
419:Material implication
370:Rules of replacement
233:Transformation rules
185:
158:
154:is true. Therefore,
138:
118:
98:
18:Modus Ponendo Ponens
4326:Ludwig Wittgenstein
4129:Destructive dilemma
3957:Well-formed formula
3593:Jago, Mark (2007).
3255:{\displaystyle 1-y}
2753:
2721:
2681:
2559:
2467:
2435:
2351:
2271:
2233:
2212:
2183:
2154:
1425:{\displaystyle P=Q}
1202:algebraic semantics
1196:Algebraic semantics
1120:must also be true.
484:propositional logic
332:destructive dilemma
43:
4388:Rules of inference
4271:Augustus De Morgan
3396:
3376:
3344:
3324:
3304:
3284:
3252:
3226:
3206:
3180:
3148:
3122:
3102:
3078:
3058:
2966:Possible fallacies
2928:
2908:
2882:
2754:
2739:
2722:
2699:
2682:
2661:
2640:
2620:
2600:
2580:
2560:
2545:
2528:
2508:
2488:
2468:
2453:
2436:
2415:
2398:
2378:
2352:
2329:
2312:
2292:
2272:
2257:
2238:
2219:
2187:
2161:
2134:
2100:
2080:
2051:
2019:
1975:
1946:
1911:
1862:
1842:
1810:
1760:
1726:. Confirming that
1716:
1669:
1645:
1601:
1581:
1555:
1531:
1511:
1485:
1462:
1442:
1422:
1396:
1370:
1360:logically implies
1350:
1330:
1271:
1235:
1151:rule of detachment
923:
612:form of argument,
451:Rules of inference
247:Rules of inference
214:
179:Symbolic statement
174:must also be true.
164:
144:
124:
104:
4375:
4374:
4239:
4238:
3798:978-3-319-42337-1
3783:978-0-12-238452-3
3604:978-1-84760-041-7
3571:Enderton 2001:111
3477:Enderton 2001:110
3399:{\displaystyle x}
3347:{\displaystyle 1}
3327:{\displaystyle y}
3307:{\displaystyle 0}
3229:{\displaystyle x}
3125:{\displaystyle Y}
3105:{\displaystyle X}
3081:{\displaystyle Y}
3061:{\displaystyle X}
2931:{\displaystyle P}
2784:appears to fail.
2623:{\displaystyle P}
2603:{\displaystyle A}
2583:{\displaystyle P}
2531:{\displaystyle P}
2511:{\displaystyle A}
2491:{\displaystyle P}
2446:. The case where
2401:{\displaystyle Q}
2315:{\displaystyle A}
2295:{\displaystyle P}
2103:{\displaystyle 1}
1679:construed as the
1604:{\displaystyle Q}
1558:{\displaystyle P}
1534:{\displaystyle Q}
1465:{\displaystyle Q}
1445:{\displaystyle P}
1373:{\displaystyle Q}
1353:{\displaystyle P}
1155:law of detachment
1100:). On this line,
1070:
1069:
757:This argument is
750:Today is Tuesday.
657:. It, along with
579:rule of inference
550:), also known as
480:
479:
227:
226:
210:
167:{\displaystyle Q}
147:{\displaystyle P}
127:{\displaystyle Q}
107:{\displaystyle P}
62:Rule of inference
16:(Redirected from
4410:
4311:Henry M. Sheffer
4301:Bertrand Russell
4266:Richard Dedekind
4150:
4094:De Morgan's laws
4068:Noncontradiction
4010:Classical logics
4004:
3901:
3894:
3887:
3878:
3856:
3807:Bertrand Russell
3792:Springer, Cham,
3761:
3760:
3758:
3756:
3742:
3736:
3729:
3723:
3716:
3714:
3712:
3701:
3695:
3688:
3682:
3675:
3669:
3662:
3656:
3649:
3643:
3640:
3634:
3633:
3615:
3609:
3608:
3590:
3584:
3581:
3572:
3569:
3563:
3562:
3560:
3558:
3544:
3538:
3535:
3529:
3526:
3520:
3500:
3494:
3493:47, No. 4, 2002.
3484:
3478:
3475:
3469:
3465:Oxford Reference
3457:
3451:
3450:
3434:
3424:
3407:
3405:
3403:
3402:
3397:
3385:
3383:
3382:
3377:
3353:
3351:
3350:
3345:
3333:
3331:
3330:
3325:
3313:
3311:
3310:
3305:
3293:
3291:
3290:
3285:
3262:, and therefore
3261:
3259:
3258:
3253:
3235:
3233:
3232:
3227:
3215:
3213:
3212:
3207:
3189:
3187:
3186:
3181:
3165:
3159:
3157:
3155:
3154:
3149:
3131:
3129:
3128:
3123:
3111:
3109:
3108:
3103:
3087:
3085:
3084:
3079:
3067:
3065:
3064:
3059:
3046:
2998:
2937:
2935:
2934:
2929:
2917:
2915:
2914:
2909:
2891:
2889:
2888:
2883:
2824:, Hobbes did it.
2763:
2761:
2760:
2755:
2752:
2747:
2731:
2729:
2728:
2723:
2720:
2715:
2711:
2691:
2689:
2688:
2683:
2680:
2675:
2652:subjective logic
2649:
2647:
2646:
2641:
2629:
2627:
2626:
2621:
2609:
2607:
2606:
2601:
2589:
2587:
2586:
2581:
2569:
2567:
2566:
2561:
2558:
2553:
2537:
2535:
2534:
2529:
2517:
2515:
2514:
2509:
2497:
2495:
2494:
2489:
2477:
2475:
2474:
2469:
2466:
2461:
2445:
2443:
2442:
2437:
2434:
2429:
2407:
2405:
2404:
2399:
2387:
2385:
2384:
2379:
2361:
2359:
2358:
2353:
2350:
2345:
2341:
2321:
2319:
2318:
2313:
2301:
2299:
2298:
2293:
2281:
2279:
2278:
2273:
2270:
2265:
2247:
2245:
2244:
2239:
2232:
2227:
2211:
2206:
2199:
2182:
2177:
2173:
2153:
2148:
2123:subjective logic
2114:Subjective logic
2109:
2107:
2106:
2101:
2089:
2087:
2086:
2081:
2060:
2058:
2057:
2052:
2028:
2026:
2025:
2022:{\displaystyle }
2020:
1984:
1982:
1981:
1976:
1955:
1953:
1952:
1947:
1920:
1918:
1917:
1912:
1871:
1869:
1868:
1863:
1851:
1849:
1848:
1843:
1819:
1817:
1816:
1811:
1769:
1767:
1766:
1761:
1725:
1723:
1722:
1717:
1709:
1678:
1676:
1675:
1670:
1654:
1652:
1651:
1646:
1610:
1608:
1607:
1602:
1590:
1588:
1587:
1582:
1564:
1562:
1561:
1556:
1540:
1538:
1537:
1532:
1520:
1518:
1517:
1512:
1494:
1492:
1491:
1486:
1471:
1469:
1468:
1463:
1451:
1449:
1448:
1443:
1431:
1429:
1428:
1423:
1405:
1403:
1402:
1397:
1379:
1377:
1376:
1371:
1359:
1357:
1356:
1351:
1339:
1337:
1336:
1331:
1329:
1318:
1307:
1280:
1278:
1277:
1272:
1270:
1259:
1244:
1242:
1241:
1236:
1234:
1072:In instances of
1003:
992:The validity of
932:
930:
929:
924:
869:forward chaining
867:is often called
556:
545:
544:
541:
540:
537:
534:
531:
528:
525:
522:
518:
517:
514:
511:
508:
505:
502:
492:
434:
427:
420:
408:De Morgan's laws
403:
396:
389:
382:
356:
348:
340:
333:
327:
320:
314:
307:
301:
294:
288:
281:
275:
268:
258:
229:
223:
221:
220:
215:
208:
173:
171:
170:
165:
153:
151:
150:
145:
133:
131:
130:
125:
113:
111:
110:
105:
44:
29:Forward chaining
21:
4418:
4417:
4413:
4412:
4411:
4409:
4408:
4407:
4403:Classical logic
4378:
4377:
4376:
4371:
4347:Begriffsschrift
4335:
4331:Jan Łukasiewicz
4251:Bernard Bolzano
4235:
4206:Double negation
4194:
4165:Double negation
4148:
4082:
4058:Excluded middle
4041:
4005:
3996:
3910:
3908:Classical logic
3905:
3841:
3838:
3770:
3765:
3764:
3754:
3752:
3744:
3743:
3739:
3730:
3726:
3710:
3708:
3707:. 21 April 2010
3705:"Deontic Logic"
3703:
3702:
3698:
3689:
3685:
3676:
3672:
3663:
3659:
3650:
3646:
3641:
3637:
3630:
3617:
3616:
3612:
3605:
3592:
3591:
3587:
3582:
3575:
3570:
3566:
3556:
3554:
3546:
3545:
3541:
3536:
3532:
3527:
3523:
3501:
3497:
3487:Susanne Bobzien
3485:
3481:
3476:
3472:
3458:
3454:
3447:
3426:
3425:
3421:
3416:
3411:
3410:
3388:
3387:
3356:
3355:
3336:
3335:
3316:
3315:
3296:
3295:
3264:
3263:
3238:
3237:
3218:
3217:
3192:
3191:
3169:
3168:
3166:
3162:
3134:
3133:
3114:
3113:
3094:
3093:
3070:
3069:
3050:
3049:
3047:
3043:
3038:
2996:
2984:
2970:The fallacy of
2968:
2920:
2919:
2894:
2893:
2850:
2849:
2778:
2734:
2733:
2694:
2693:
2656:
2655:
2632:
2631:
2612:
2611:
2592:
2591:
2572:
2571:
2540:
2539:
2520:
2519:
2500:
2499:
2480:
2479:
2448:
2447:
2410:
2409:
2390:
2389:
2364:
2363:
2324:
2323:
2304:
2303:
2284:
2283:
2252:
2251:
2129:
2128:
2116:
2092:
2091:
2063:
2062:
2031:
2030:
1987:
1986:
1958:
1957:
1923:
1922:
1882:
1881:
1878:
1854:
1853:
1822:
1821:
1772:
1771:
1728:
1727:
1685:
1684:
1661:
1660:
1613:
1612:
1593:
1592:
1567:
1566:
1547:
1546:
1523:
1522:
1521:together imply
1497:
1496:
1477:
1476:
1454:
1453:
1434:
1433:
1408:
1407:
1382:
1381:
1362:
1361:
1342:
1341:
1283:
1282:
1247:
1246:
1210:
1209:
1198:
1193:
1126:
990:
892:
891:
877:
875:Formal notation
799:sequent calculi
786:argument using
669:
641:The history of
519:
499:
495:
444:Predicate logic
438:
402:Double negation
256:
183:
182:
156:
155:
136:
135:
116:
115:
96:
95:
87:
78:Classical logic
66:
35:
32:
23:
22:
15:
12:
11:
5:
4416:
4414:
4406:
4405:
4400:
4395:
4390:
4380:
4379:
4373:
4372:
4370:
4369:
4364:
4359:
4354:
4349:
4343:
4341:
4337:
4336:
4334:
4333:
4328:
4323:
4318:
4313:
4308:
4306:Ernst Schröder
4303:
4298:
4293:
4291:Giuseppe Peano
4288:
4283:
4278:
4273:
4268:
4263:
4258:
4253:
4247:
4245:
4241:
4240:
4237:
4236:
4234:
4233:
4228:
4223:
4218:
4213:
4208:
4202:
4200:
4196:
4195:
4193:
4192:
4187:
4182:
4177:
4172:
4167:
4162:
4156:
4154:
4147:
4146:
4141:
4136:
4131:
4126:
4121:
4116:
4111:
4106:
4101:
4096:
4090:
4088:
4084:
4083:
4081:
4080:
4075:
4070:
4065:
4060:
4055:
4049:
4047:
4043:
4042:
4040:
4039:
4034:
4029:
4024:
4019:
4013:
4011:
4007:
4006:
3999:
3997:
3995:
3994:
3989:
3984:
3979:
3974:
3969:
3964:
3959:
3954:
3949:
3947:Truth function
3944:
3939:
3934:
3929:
3924:
3918:
3916:
3912:
3911:
3906:
3904:
3903:
3896:
3889:
3881:
3875:
3874:
3866:
3857:
3843:"Modus ponens"
3837:
3836:External links
3834:
3833:
3832:
3814:
3800:
3786:
3769:
3766:
3763:
3762:
3737:
3724:
3696:
3683:
3670:
3657:
3644:
3635:
3628:
3610:
3603:
3585:
3573:
3564:
3539:
3537:Tarski 1946:47
3530:
3521:
3495:
3479:
3470:
3452:
3445:
3418:
3417:
3415:
3412:
3409:
3408:
3395:
3375:
3372:
3369:
3366:
3363:
3343:
3323:
3303:
3283:
3280:
3277:
3274:
3271:
3251:
3248:
3245:
3225:
3205:
3202:
3199:
3179:
3176:
3160:
3147:
3144:
3141:
3121:
3101:
3077:
3057:
3040:
3039:
3037:
3034:
3033:
3032:
3026:
3020:
3012:
3004:
2999:
2990:
2983:
2980:
2967:
2964:
2927:
2907:
2904:
2901:
2881:
2878:
2875:
2872:
2869:
2866:
2863:
2860:
2857:
2826:
2825:
2818:
2811:
2777:
2774:
2751:
2746:
2742:
2719:
2714:
2710:
2706:
2702:
2679:
2674:
2671:
2668:
2664:
2639:
2619:
2599:
2579:
2557:
2552:
2548:
2527:
2507:
2487:
2465:
2460:
2456:
2433:
2428:
2425:
2422:
2418:
2408:is denoted by
2397:
2377:
2374:
2371:
2349:
2344:
2340:
2336:
2332:
2311:
2291:
2269:
2264:
2260:
2237:
2231:
2226:
2222:
2218:
2215:
2210:
2205:
2202:
2198:
2194:
2190:
2186:
2181:
2176:
2172:
2168:
2164:
2160:
2157:
2152:
2147:
2144:
2141:
2137:
2125:expressed as:
2115:
2112:
2099:
2079:
2076:
2073:
2070:
2050:
2047:
2044:
2041:
2038:
2018:
2015:
2012:
2009:
2006:
2003:
2000:
1997:
1994:
1974:
1971:
1968:
1965:
1945:
1942:
1939:
1936:
1933:
1930:
1910:
1907:
1904:
1901:
1898:
1895:
1892:
1889:
1877:
1874:
1861:
1841:
1838:
1835:
1832:
1829:
1809:
1806:
1803:
1800:
1797:
1794:
1791:
1788:
1785:
1782:
1779:
1759:
1756:
1753:
1750:
1747:
1744:
1741:
1738:
1735:
1715:
1712:
1708:
1704:
1701:
1698:
1695:
1692:
1668:
1644:
1641:
1638:
1635:
1632:
1629:
1626:
1623:
1620:
1600:
1580:
1577:
1574:
1554:
1530:
1510:
1507:
1504:
1488:{\textstyle P}
1484:
1461:
1441:
1421:
1418:
1415:
1395:
1392:
1389:
1369:
1349:
1328:
1324:
1321:
1317:
1313:
1310:
1306:
1303:
1300:
1297:
1294:
1290:
1269:
1265:
1262:
1258:
1254:
1233:
1230:
1227:
1224:
1221:
1217:
1197:
1194:
1192:
1189:
1183:is true, then
1134:argument forms
1125:
1122:
1068:
1067:
1064:
1061:
1057:
1056:
1053:
1050:
1046:
1045:
1042:
1039:
1035:
1034:
1031:
1028:
1024:
1023:
1014:
1009:
989:
986:
982:logical system
934:
933:
922:
917:
912:
908:
905:
902:
899:
876:
873:
790:is said to be
755:
754:
751:
748:
706:
705:
698:
692:
671:The form of a
668:
665:
478:
477:
476:
475:
466:
454:
453:
447:
446:
440:
439:
437:
436:
429:
422:
415:
410:
405:
398:
395:Distributivity
391:
384:
376:
373:
372:
366:
365:
364:
363:
358:
335:
322:
309:
296:
283:
270:
250:
249:
243:
242:
236:
235:
225:
224:
213:
207:
203:
199:
196:
193:
190:
180:
176:
175:
163:
143:
123:
103:
93:
89:
88:
86:
85:
80:
74:
72:
68:
67:
65:
64:
59:
50:
48:
42:
33:
24:
14:
13:
10:
9:
6:
4:
3:
2:
4415:
4404:
4401:
4399:
4396:
4394:
4391:
4389:
4386:
4385:
4383:
4368:
4365:
4363:
4360:
4358:
4355:
4353:
4350:
4348:
4345:
4344:
4342:
4338:
4332:
4329:
4327:
4324:
4322:
4319:
4317:
4316:Alfred Tarski
4314:
4312:
4309:
4307:
4304:
4302:
4299:
4297:
4294:
4292:
4289:
4287:
4284:
4282:
4279:
4277:
4276:Gottlob Frege
4274:
4272:
4269:
4267:
4264:
4262:
4259:
4257:
4254:
4252:
4249:
4248:
4246:
4242:
4232:
4229:
4227:
4224:
4222:
4221:Biconditional
4219:
4217:
4214:
4212:
4209:
4207:
4204:
4203:
4201:
4197:
4191:
4188:
4186:
4183:
4181:
4180:Biconditional
4178:
4176:
4173:
4171:
4168:
4166:
4163:
4161:
4158:
4157:
4155:
4151:
4145:
4142:
4140:
4137:
4135:
4132:
4130:
4127:
4125:
4122:
4120:
4117:
4115:
4114:modus tollens
4112:
4110:
4107:
4105:
4104:Transposition
4102:
4100:
4097:
4095:
4092:
4091:
4089:
4085:
4079:
4076:
4074:
4071:
4069:
4066:
4064:
4061:
4059:
4056:
4054:
4051:
4050:
4048:
4044:
4038:
4035:
4033:
4030:
4028:
4025:
4023:
4022:Propositional
4020:
4018:
4015:
4014:
4012:
4008:
4003:
3993:
3990:
3988:
3985:
3983:
3980:
3978:
3977:Associativity
3975:
3973:
3970:
3968:
3965:
3963:
3960:
3958:
3955:
3953:
3950:
3948:
3945:
3943:
3940:
3938:
3935:
3933:
3930:
3928:
3925:
3923:
3920:
3919:
3917:
3913:
3909:
3902:
3897:
3895:
3890:
3888:
3883:
3882:
3879:
3872:
3871:
3867:
3865:
3861:
3858:
3854:
3850:
3849:
3844:
3840:
3839:
3835:
3830:
3829:0-486-28462-X
3826:
3822:
3818:
3817:Alfred Tarski
3815:
3812:
3808:
3804:
3801:
3799:
3795:
3791:
3787:
3784:
3780:
3776:
3772:
3771:
3767:
3751:
3747:
3741:
3738:
3735:107, 115–143.
3734:
3728:
3725:
3721:
3720:
3706:
3700:
3697:
3694:112, 462–471.
3693:
3687:
3684:
3680:
3674:
3671:
3667:
3661:
3658:
3654:
3648:
3645:
3639:
3636:
3631:
3625:
3621:
3614:
3611:
3606:
3600:
3596:
3589:
3586:
3580:
3578:
3574:
3568:
3565:
3553:
3549:
3543:
3540:
3534:
3531:
3525:
3522:
3518:
3517:
3512:
3510:
3509:Modus Tollens
3506:
3499:
3496:
3492:
3488:
3483:
3480:
3474:
3471:
3467:
3466:
3461:
3456:
3453:
3448:
3446:0-415-91775-1
3442:
3438:
3433:
3432:
3423:
3420:
3413:
3393:
3373:
3370:
3367:
3364:
3361:
3341:
3321:
3301:
3281:
3278:
3275:
3272:
3269:
3249:
3246:
3243:
3223:
3203:
3197:
3177:
3164:
3161:
3145:
3142:
3139:
3132:, denoted by
3119:
3099:
3091:
3075:
3055:
3045:
3042:
3035:
3030:
3027:
3024:
3021:
3018:
3017:
3016:Modus vivendi
3013:
3010:
3009:
3008:Modus tollens
3005:
3003:
3002:Latin phrases
3000:
2994:
2991:
2989:
2986:
2985:
2981:
2979:
2977:
2973:
2965:
2963:
2961:
2960:modus ponens'
2957:
2952:
2948:
2947:deontic logic
2943:
2941:
2925:
2905:
2899:
2892:, therefore,
2876:
2870:
2861:
2858:
2855:
2847:
2842:
2840:
2835:
2831:
2823:
2819:
2816:
2812:
2809:
2808:
2803:
2799:
2795:
2794:
2793:
2791:
2787:
2783:
2775:
2773:
2771:
2767:
2749:
2744:
2740:
2717:
2712:
2704:
2700:
2677:
2672:
2666:
2662:
2653:
2637:
2617:
2597:
2577:
2555:
2550:
2546:
2525:
2505:
2485:
2463:
2458:
2454:
2431:
2426:
2420:
2416:
2395:
2375:
2369:
2347:
2342:
2334:
2330:
2309:
2289:
2267:
2262:
2258:
2248:
2235:
2229:
2224:
2220:
2216:
2208:
2203:
2192:
2188:
2184:
2179:
2174:
2166:
2162:
2155:
2150:
2145:
2139:
2135:
2126:
2124:
2120:
2113:
2111:
2097:
2074:
2048:
2045:
2042:
2039:
2036:
2013:
2010:
2007:
2004:
2001:
1998:
1995:
1969:
1943:
1940:
1934:
1908:
1905:
1899:
1893:
1875:
1873:
1839:
1836:
1833:
1830:
1827:
1807:
1804:
1801:
1798:
1792:
1786:
1780:
1777:
1757:
1754:
1748:
1742:
1736:
1733:
1713:
1710:
1706:
1699:
1696:
1690:
1682:
1658:
1642:
1639:
1633:
1627:
1621:
1618:
1611:, i.e., that
1598:
1578:
1572:
1552:
1544:
1528:
1508:
1502:
1482:
1473:
1459:
1439:
1419:
1416:
1413:
1393:
1390:
1387:
1380:just in case
1367:
1347:
1326:
1319:
1315:
1308:
1301:
1298:
1295:
1267:
1260:
1256:
1228:
1225:
1222:
1207:
1203:
1195:
1190:
1188:
1186:
1182:
1178:
1174:
1170:
1167:
1163:
1158:
1156:
1152:
1148:
1144:
1140:
1135:
1131:
1123:
1121:
1119:
1115:
1111:
1107:
1103:
1099:
1095:
1091:
1087:
1083:
1079:
1075:
1065:
1062:
1059:
1058:
1054:
1051:
1048:
1047:
1043:
1040:
1037:
1036:
1032:
1029:
1026:
1025:
1022:
1018:
1015:
1013:
1010:
1008:
1005:
1004:
1001:
999:
995:
987:
985:
983:
979:
975:
971:
967:
963:
959:
955:
951:
947:
943:
939:
920:
915:
910:
906:
903:
897:
890:
889:
888:
886:
882:
874:
872:
870:
866:
862:
857:
855:
851:
847:
843:
839:
835:
831:
827:
823:
819:
814:
812:
808:
804:
800:
795:
793:
789:
785:
784:propositional
780:
776:
772:
768:
764:
760:
752:
749:
746:
745:
744:
742:
737:
735:
731:
727:
723:
719:
715:
711:
703:
699:
696:
693:
690:
686:
682:
681:
680:
678:
674:
666:
664:
662:
661:
660:modus tollens
656:
652:
648:
645:goes back to
644:
639:
637:
633:
629:
625:
621:
617:
616:
615:modus tollens
611:
607:
603:
599:
597:
593:
590:
587:
584:
580:
576:
575:argument form
573:
569:
565:
561:
557:
555:
549:
543:
493:
491:
485:
474:
473:instantiation
470:
467:
465:
464:instantiation
461:
458:
457:
456:
455:
452:
448:
445:
441:
435:
430:
428:
423:
421:
416:
414:
413:Transposition
411:
409:
406:
404:
399:
397:
392:
390:
388:Commutativity
385:
383:
381:Associativity
378:
377:
375:
374:
371:
367:
362:
359:
357:
355:
349:
347:
346:modus tollens
341:
336:
334:
328:
323:
321:
315:
310:
308:
302:
297:
295:
289:
284:
282:
276:
271:
269:
266:
263:elimination (
259:
254:
253:
252:
251:
248:
244:
241:
237:
234:
230:
211:
205:
201:
197:
194:
188:
181:
177:
161:
141:
121:
101:
94:
90:
84:
81:
79:
76:
75:
73:
69:
63:
60:
58:
57:argument form
55:
52:
51:
49:
45:
40:
37:
30:
19:
4286:Hugh MacColl
4261:Georg Cantor
4256:George Boole
4153:Introduction
4109:modus ponens
4108:
4037:Higher-order
4032:Second-order
3982:Distribution
3942:Truth tables
3870:Modus ponens
3868:
3860:Modus ponens
3846:
3820:
3810:
3789:
3774:
3753:. Retrieved
3749:
3740:
3732:
3727:
3717:
3709:. Retrieved
3699:
3691:
3686:
3681:47, 142–146.
3678:
3673:
3668:83, 296–300.
3665:
3660:
3655:82, 462–471.
3652:
3647:
3638:
3619:
3613:
3595:Formal Logic
3594:
3588:
3567:
3555:. Retrieved
3551:
3542:
3533:
3524:
3514:
3508:
3505:Modus Ponens
3504:
3498:
3490:
3482:
3473:
3463:
3455:
3430:
3422:
3314:. And since
3163:
3044:
3015:
3007:
2976:modus ponens
2975:
2969:
2959:
2956:modus ponens
2955:
2951:modus ponens
2950:
2944:
2940:modus ponens
2939:
2846:modus ponens
2845:
2843:
2833:
2829:
2827:
2821:
2814:
2805:
2790:modus ponens
2789:
2782:modus ponens
2781:
2779:
2766:modus ponens
2765:
2610:saying that
2518:saying that
2249:
2127:
2119:Modus ponens
2118:
2117:
1879:
1543:modus ponens
1542:
1474:
1199:
1184:
1180:
1176:
1172:
1159:
1154:
1150:
1139:Modus ponens
1138:
1130:modus ponens
1129:
1127:
1117:
1113:
1112:is true and
1109:
1105:
1101:
1097:
1093:
1089:
1085:
1084:is true and
1081:
1077:
1074:modus ponens
1073:
1071:
1020:
1016:
1011:
1006:
994:modus ponens
993:
991:
977:
973:
969:
961:
949:
945:
941:
937:
935:
887:notation as
881:modus ponens
880:
878:
865:modus ponens
864:
858:
853:
849:
845:
841:
837:
833:
829:
822:modus ponens
821:
815:
803:modus ponens
802:
796:
788:modus ponens
787:
778:
767:modus ponens
766:
756:
741:modus ponens
740:
738:
729:
721:
717:
713:
707:
701:
694:
688:
684:
673:modus ponens
672:
670:
658:
655:Theophrastus
651:modus ponens
650:
643:modus ponens
642:
640:
636:modus ponens
635:
613:
602:Modus ponens
601:
600:
595:
591:
588:
582:
567:
563:
551:
547:
490:modus ponens
487:
481:
471: /
462: /
353:
350: /
345:
342: /
329: /
326:Constructive
316: /
303: /
290: /
277: /
265:modus ponens
264:
261:
260: /
41:Modus ponens
36:
4231:Disjunction
4226:Conjunction
4211:Existential
4199:Elimination
4190:Disjunction
4185:Conjunction
4170:Existential
4027:First-order
3952:Truth value
3922:Quantifiers
3750:iep.utm.edu
3023:Stoic logic
2798:Shakespeare
2090:must equal
1591:lies below
1452:lies below
1166:proposition
998:truth table
958:metalogical
852:is of type
844:is of type
710:conditional
700:Therefore,
667:Explanation
634:version of
632:disjunctive
604:is a mixed
426:Exportation
313:Disjunctive
306:elimination
293:elimination
280:elimination
4382:Categories
4281:Kurt Gödel
4144:Absorption
4046:Principles
3932:Connective
3864:PhilPapers
3711:30 January
3629:0934223459
3414:References
2848:is simply
2786:Vann McGee
811:admissible
734:consequent
726:antecedent
558:(from
339:Absorption
4216:Universal
4175:Universal
4078:Explosion
4063:Bivalence
3992:Soundness
3937:Tautology
3927:Predicate
3853:EMS Press
3491:Phronesis
3371:−
3279:−
3247:−
3201:→
3190:implies
3175:¬
3143:∧
2903:→
2874:→
2865:→
2741:ω
2701:ω
2670:‖
2663:ω
2638:⊚
2547:ω
2455:ω
2424:‖
2417:ω
2373:→
2331:ω
2259:ω
2221:ω
2217:⊚
2201:¬
2189:ω
2163:ω
2143:‖
2136:ω
2005:−
1897:→
1860:→
1837:≤
1831:∧
1805:∧
1790:→
1781:∧
1755:≤
1746:→
1737:∧
1711:∨
1703:¬
1694:→
1667:→
1640:≤
1631:→
1622:∧
1576:→
1506:→
1391:≤
1323:¬
1320:∨
1312:¬
1299:∧
1289:¬
1264:¬
1261:∨
1253:¬
1226:∧
1216:¬
1187:is true.
1162:statement
1116:is true,
916:⊢
901:→
792:deductive
647:antiquity
572:deductive
433:Tautology
206:⊢
192:→
92:Statement
54:Deductive
4160:Negation
3987:Validity
3967:Logicism
3679:Analysis
3088:is the "
2982:See also
2768:and the
1432:or else
1175:implies
980:in some
775:argument
769:to be a
716:implies
114:implies
3915:General
3855:, 2001
3768:Sources
3755:6 March
3557:5 April
2796:Either
1956:, then
1659:, with
1657:Boolean
1206:lattice
1169:implies
1153:or the
1145:from a
885:sequent
848:, then
687:, then
630:is the
586:implies
570:, is a
4244:People
3831:(pbk).
3827:
3796:
3781:
3626:
3601:
3443:
3167:Since
2834:Hamlet
2830:Hamlet
2822:Hamlet
2815:Hamlet
2807:Hamlet
2804:wrote
2802:Hobbes
2250:where
1128:While
1124:Status
936:where
828:: if
765:; for
732:, the
724:, the
209:
4340:Works
4087:Rules
3819:1946
3809:1927
3092:" of
3036:Notes
1921:and
1179:and
964:is a
956:is a
771:sound
759:valid
610:valid
566:, or
560:Latin
71:Field
4017:Term
3825:ISBN
3805:and
3794:ISBN
3779:ISBN
3757:2020
3713:2020
3624:ISBN
3599:ISBN
3559:2018
3507:and
3441:ISBN
3112:and
3090:meet
3068:and
1820:and
1565:and
1495:and
1245:and
1092:and
972:and
944:and
879:The
840:and
816:The
763:true
653:was
622:and
577:and
47:Type
3862:at
2945:In
2841:).
2800:or
2650:of
1880:If
1164:or
968:of
859:In
850:f x
824:to
779:and
683:If
482:In
4384::
3851:,
3845:,
3748:.
3576:^
3550:.
3513:.
3462:.
3439:.
3437:60
3354:,
3216:,
2978:.
2772:.
2110:.
2069:Pr
2061:,
1964:Pr
1929:Pr
1888:Pr
1683::
1108:→
1096:→
1080:→
1066:T
1055:T
1044:F
1033:T
1019:→
1000:.
984:.
976:→
948:→
940:,
871:.
863:,
856:.
836:→
813:.
801:,
794:.
782:A
743::
638:.
626:.
589:Q.
548:MP
546:;
527:oʊ
507:oʊ
486:,
134:.
3900:e
3893:t
3886:v
3785:.
3759:.
3722:.
3715:.
3632:.
3607:.
3561:.
3519:.
3511:"
3468:.
3449:.
3406:.
3394:x
3374:1
3368:y
3365:+
3362:x
3342:1
3322:y
3302:0
3282:1
3276:y
3273:+
3270:x
3250:y
3244:1
3224:x
3204:Q
3198:P
3178:P
3158:.
3146:Y
3140:X
3120:Y
3100:X
3076:Y
3056:X
2926:P
2906:R
2900:Q
2880:)
2877:R
2871:Q
2868:(
2862:P
2859:,
2856:P
2810:.
2750:A
2745:P
2718:A
2713:P
2709:|
2705:Q
2678:A
2673:P
2667:Q
2618:P
2598:A
2578:P
2556:A
2551:P
2526:P
2506:A
2486:P
2464:A
2459:P
2432:A
2427:P
2421:Q
2396:Q
2376:Q
2370:P
2348:A
2343:P
2339:|
2335:Q
2310:A
2290:P
2268:A
2263:P
2236:,
2230:A
2225:P
2214:)
2209:A
2204:P
2197:|
2193:Q
2185:,
2180:A
2175:P
2171:|
2167:Q
2159:(
2156:=
2151:A
2146:P
2140:Q
2098:1
2078:)
2075:Q
2072:(
2049:1
2046:=
2043:y
2040:=
2037:x
2017:]
2014:x
2011:,
2008:1
2002:y
1999:+
1996:x
1993:[
1973:)
1970:Q
1967:(
1944:y
1941:=
1938:)
1935:P
1932:(
1909:x
1906:=
1903:)
1900:Q
1894:P
1891:(
1840:Q
1834:Q
1828:P
1808:Q
1802:P
1799:=
1796:)
1793:Q
1787:P
1784:(
1778:P
1758:Q
1752:)
1749:Q
1743:P
1740:(
1734:P
1714:Q
1707:P
1700:=
1697:Q
1691:P
1643:Q
1637:)
1634:Q
1628:P
1625:(
1619:P
1599:Q
1579:Q
1573:P
1553:P
1529:Q
1509:Q
1503:P
1483:P
1460:Q
1440:P
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