5566:
3447:
141:, which states that every proposition is either true or false. The principle of bivalence always implies the law of excluded middle, while the converse is not always true. A commonly cited counterexample uses statements unprovable now, but provable in the future to show that the law of excluded middle may apply when the principle of bivalence fails.
2013:(Brouwer 1923 in van Heijenoort 1967:336). In general, intuitionists allow the use of the law of excluded middle when it is confined to discourse over finite collections (sets), but not when it is used in discourse over infinite sets (e.g. the natural numbers). Thus intuitionists absolutely disallow the blanket assertion: "For all propositions
441:
Let us give the name of "sense-data" to the things that are immediately known in sensation: such things as colours, sounds, smells, hardnesses, roughnesses, and so on. We shall give the name "sensation" to the experience of being immediately aware of these things … The colour itself is a sense-datum,
1936:
Davis means that "a proof that there actually are mathematic entities satisfying certain conditions would not have to provide a method to exhibit explicitly the entities in question." (p. 85). Such proofs presume the existence of a totality that is complete, a notion disallowed by intuitionists
2048:
as formalised in LP, have the law of excluded middle as a theorem, but resolve out the Liar as both true and false. In this way, the law of excluded middle is true, but because truth itself, and therefore disjunction, is not exclusive, it says next to nothing if one of the disjuncts is paradoxical,
1178:
From the first interpretation of negation, that is, the interdiction from regarding the judgment as true, it is impossible to obtain the certitude that the principle of excluded middle is true … Brouwer showed that in the case of such transfinite judgments the principle of excluded middle cannot be
1076:
The following highlights the deep mathematical and philosophic problem behind what it means to "know", and also helps elucidate what the "law" implies (i.e. what the law really means). Their difficulties with the law emerge: that they do not want to accept as true implications drawn from that which
942:
And finally constructivists … restricted mathematics to the study of concrete operations on finite or potentially (but not actually) infinite structures; completed infinite totalities … were rejected, as were indirect proof based on the Law of
Excluded Middle. Most radical among the constructivists
2809:
In a comparative analysis (pp. 43–59) of the three "-isms" (and their foremost spokesmen)—Logicism (Russell and
Whitehead), Intuitionism (Brouwer) and Formalism (Hilbert)—Kleene turns his thorough eye toward intuitionism, its "founder" Brouwer, and the intuitionists' complaints with respect to the
2780:
The original symbol as used by
Reichenbach is an upside down V, nowadays used for AND. The AND for Reichenbach is the same as that used in Principia Mathematica – a "dot" cf p. 27 where he shows a truth table where he defines "a.b". Reichenbach defines the exclusive-or on p. 35 as "the negation of
185:
It is impossible, then, that "being a man" should mean precisely "not being a man", if "man" not only signifies something about one subject but also has one significance. … And it will not be possible to be and not to be the same thing, except in virtue of an ambiguity, just as if one whom we call
882:
Kronecker insisted that there could be no existence without construction. For him, as for Paul Gordan , Hilbert's proof of the finiteness of the basis of the invariant system was simply not mathematics. Hilbert, on the other hand, throughout his life was to insist that if one can prove that the
2012:
both give examples of the law of excluded middle extended to the infinite. Hilbert's example: "the assertion that either there are only finitely many prime numbers or there are infinitely many" (quoted in Davis 2000:97); and
Brouwer's: "Every mathematical species is either finite or infinite."
221:
But
Aristotle also writes, "since it is impossible that contradictories should be at the same time true of the same thing, obviously contraries also cannot belong at the same time to the same thing" (Book IV, CH 6, p. 531). He then proposes that "there cannot be an intermediate between
1147:
Hilbert's first axiom of negation, "anything follows from the false", made its appearance only with the rise of symbolic logic, as did the first axiom of implication … while … the axiom under consideration asserts something about the consequences of something impossible: we have to accept
1726:, for example, would not accept this argument without further support for that statement. This might come in the form of a proof that the number in question is in fact irrational (or rational, as the case may be); or a finite algorithm that could determine whether the number is rational.
1713:
1061:" had "carried more weight" than "the law of excluded middle and related theorems of the propositional calculus" (Dawson p. 156). He proposed his "system Σ … and he concluded by mentioning several applications of his interpretation. Among them were a proof of the consistency with
674:
Most of these theorems—in particular ✸2.1, ✸2.11, and ✸2.14—are rejected by intuitionism. These tools are recast into another form that
Kolmogorov cites as "Hilbert's four axioms of implication" and "Hilbert's two axioms of negation" (Kolmogorov in van Heijenoort, p. 335).
424:
That is, when we judge (say) "this is red", what occurs is a relation of three terms, the mind, and "this", and "red". On the other hand, when we perceive "the redness of this", there is a relation of two terms, namely the mind and the complex object "the redness of this" (pp.
1037:
Since mathematical theorems are often proved by establishing that the negation would involve us in a contradiction, this third possibility which
Brouwer suggested would throw into question many of the mathematical statements currently
947:
The rancorous debate continued through the early 1900s into the 1920s; in 1927 Brouwer complained about "polemicizing against it in sneering tones" (Brouwer in van
Heijenoort, p. 492). But the debate was fertile: it resulted in
766:
is not exhaustive in its major terms and is therefore an inflated formula. This fact may perhaps explain why some people consider it unreasonable to write (29) with the inclusive-'or', and want to have it written with the sign of the
1093:, and in particular the principle of the reciprocity of the complementary species, that is, the principle that for every system the correctness of a property follows from the impossibility of the impossibility of this property. (335)
973:
According to
Brouwer, a statement that an object exists having a given property means that, and is only proved, when a method is known which in principle at least will enable such an object to be found or constructed
2151:. Very few mathematicians work in areas which allow for The Law of Excluded Middle to be false, as it is not compatible with the standard axiomatic system, ZFC. Namely, it is not compatible with the Axiom of Choice.
1591:
362:
2781:
the equivalence". One sign used nowadays is a circle with a + in it, i.e. ⊕ (because in binary, a ⊕ b yields modulo-2 addition – addition without carry). Other signs are ≢ (not identical to), or ≠ (not equal to).
2469:
2319:
954:(1910–1913), and that work gave a precise definition to the law of excluded middle, and all this provided an intellectual setting and the tools necessary for the mathematicians of the early 20th century:
958:
Out of the rancor, and spawned in part by it, there arose several important logical developments; Zermelo's axiomatization of set theory (1908a), that was followed two years later by the first volume of
3030:
2135:
is not an example of a statement that cannot be true or false. The law of excluded middle still holds here as the negation of this statement "This statement is not false", can be assigned true. In
2061:(tetralemma) is an ancient alternative to the law of excluded middle, which examines all four possible assignments of truth values to a proposition and its negation. It has been important in
1026:." (this was missing a closing quote) For finite sets, therefore, Brouwer accepted the principle of the excluded middle as valid. He refused to accept it for infinite sets because if the set
2376:
186:"man", and others were to call "not-man"; but the point in question is not this, whether the same thing can at the same time be and not be a man in name, but whether it can be in fact. (
386:
if it is false* … the truth-value of "p ∨ q" is truth if the truth-value of either p or q is truth, and is falsehood otherwise … that of "~ p" is the opposite of that of p …" (pp. 7–8)
1821:
1549:
2080:. Instead of a proposition's being either true or false, a proposition is either true or not able to be proved true. These two dichotomies only differ in logical systems that are not
1252:
is true by virtue of its form alone. That is, the "middle" position, that
Socrates is neither mortal nor not-mortal, is excluded by logic, and therefore either the first possibility (
1507:
1403:
2249:
3282:, reprinted in Great Books of the Western World Encyclopædia Britannica, Volume 35, 1952, p. 449 ff. This work was published by Hume in 1758 as his rewrite of his "juvenile"
2139:, such a self-referential paradox can be constructed by examining the set "the set of all sets that do not contain themselves". This set is unambiguously defined, but leads to a
690:, that is, the principle that for every system the correctness of a property follows from the impossibility of the impossibility of this property" (Brouwer, ibid, p. 335).
888:
It was his contention that nothing could be said to have mathematical existence unless it could actually be constructed with a finite number of positive integers (Reid p. 26)
1855:
1579:
1470:
1440:
1894:
1363:
1089:
On the basis of the testability just mentioned, there hold, for properties conceived within a specific finite main system, the "principle of excluded middle", that is,
818:
175:
book 3, saying that it is necessary in every case to affirm or deny, and that it is impossible that there should be anything between the two parts of a contradiction.
1927:
3202:, Cambridge at the University Press 1962 (Second Edition of 1927, reprinted). Extremely difficult because of arcane symbolism, but a must-have for serious logicians.
1335:
1159:
Hilbert's second axiom of negation expresses the principle of excluded middle. The principle is expressed here in the form in which is it used for derivations: if
1030:
is infinite, we cannot—even in principle—examine each member of the set. If, during the course of our examination, we find a member of the set with the property
390:
This is not much help. But later, in a much deeper discussion ("Definition and systematic ambiguity of Truth and Falsehood" Chapter II part III, p. 41 ff),
1782:
1762:
1308:
1288:
1065:
of the principle ~ (∀A: (A ∨ ~A)) (despite the inconsistency of the assumption ∃ A: ~ (A ∨ ~A))" (Dawson, p. 157) (no closing parenthesis had been placed)
1054:
proposed a solution: "that the negation of a universal proposition was to be understood as asserting the existence … of a counterexample" (Dawson, p. 157)
410:'red' is a sense-datum", and they "stand in relation" to one another and in relation to "I". Thus what we really mean is: "I perceive that 'This object a is red
2147:, this type of contradiction is no longer admitted. Furthermore, paradoxes of self reference can be constructed without even invoking negation at all, as in
1989:), but not in general the intuitionistic … the classical meaning, that somewhere in the completed infinite totality of the natural numbers there occurs an
1068:
The debate seemed to weaken: mathematicians, logicians and engineers continue to use the law of excluded middle (and double negation) in their daily work.
3945:
2483: – Foundational controversy in twentieth-century mathematics: an account on the formalist-intuitionist divide around the Law of the excluded middle
3232:, Littlefield, Adams & Co., Totowa, New Jersey, 1974 edition (first published 1968). Includes a wonderful essay on "The Art of drawing Inferences".
169:
propositions (i.e. where one proposition is the negation of the other) one must be true, and the other false. He also states it as a principle in the
3262:, Copernicus: Springer–Verlag New York, Inc. 1996, first published 1969. Contains a wealth of biographical information, much derived from interviews.
717:), and by the definition of implication (i.e. 1.01 p → q = ~p ∨ q) then ~p ∨ ~(~p)= p → ~(~p). QED (The derivation of 2.14 is a bit more involved.)
1041:"Taking the Principle of the Excluded Middle from the mathematician," Hilbert said, "is the same as … prohibiting the boxer the use of his fists."
883:
attributes assigned to a concept will never lead to a contradiction, the mathematical existence of the concept is thereby established (Reid p. 34)
222:
contradictories, but of one subject we must either affirm or deny any one predicate" (Book IV, CH 7, p. 531). In the context of Aristotle's
4620:
1708:{\displaystyle a^{b}=\left({\sqrt {2}}^{\sqrt {2}}\right)^{\sqrt {2}}={\sqrt {2}}^{\left({\sqrt {2}}\cdot {\sqrt {2}}\right)}={\sqrt {2}}^{2}=2}
1044:"The possible loss did not seem to bother Weyl … Brouwer's program was the coming thing, he insisted to his friends in Zürich." (Reid, p. 149)
3284:
Treatise of Human Nature: Being An attempt to introduce the experimental method of Reasoning into Moral Subjects Vol. I, Of The Understanding
2178:
and "∨" is a "max operator", then the law can be expressed in the object language by (P ∨ ~P ∨ ~~P ∨ ... ∨ ~...~P), where "~...~" represents
303:
963:, in which Russell and Whitehead showed how, via the theory of types: much of arithmetic could be developed by logicist means (Dawson p. 49)
4703:
3844:
2382:
2269:
2649:
980:"pure existence proofs have been the most important landmarks in the historical development of our science," he maintained. (Reid p. 155)
2979:
2790:
This well-known example of a non-constructive proof depending on the law of excluded middle can be found in many places, for example:
2587:
627:) (If it's true that "If this rose is red then this pig flies" then it's true that "If this pig doesn't fly then this rose isn't red.")
3343:
2123:. It is possible in logic to make well-constructed propositions that can be neither true nor false; a common example of this is the "
194:
Aristotle's assertion that "it will not be possible to be and not to be the same thing" would be written in propositional logic as ~(
394:
defines truth and falsehood in terms of a relationship between the "a" and the "b" and the "percipient". For example "This 'a' is 'b
5595:
3543:
1077:
is unverifiable (untestable, unknowable) or from the impossible or the false. (All quotes are from van Heijenoort, italics added).
5017:
1034:, the first alternative is substantiated; but if we never find such a member, the second alternative is still not substantiated.
609:) (One of the four "Principles of transposition". Similar to 1.03, 1.16 and 1.17. A very long demonstration was required here.)
800:
In line (30) the "(x)" means "for all" or "for every", a form used by Russell and Reichenbach; today the symbolism is usually
5175:
2685:
254:
Its usual form, "Every judgment is either true or false" …"(from Kolmogorov in van Heijenoort, p. 421) footnote 9: "This is
3963:
5030:
4353:
1722:
In the above argument, the assertion "this number is either rational or irrational" invokes the law of excluded middle. An
1057:
Gödel's approach to the law of excluded middle was to assert that objections against "the use of 'impredicative definitions
2897:
3316:
2928:
967:
Brouwer reduced the debate to the use of proofs designed from "negative" or "non-existence" versus "constructive" proof:
446:
Russell further described his reasoning behind his definitions of "truth" and "falsehood" in the same book (Chapter XII,
5035:
5025:
4762:
4615:
3968:
2092:. In these systems, the programmer is free to assert the law of excluded middle as a true fact, but it is not built-in
2001:), is not available to him, since he does not conceive the natural numbers as a completed totality. (Kleene 1952:49–50)
3959:
1409:
Clearly (excluded middle) this number is either rational or irrational. If it is rational, the proof is complete, and
5171:
3171:
3090:
2880:
2325:
938:
does not exist", and was thereby invoking the law of excluded middle cast into the form of the law of contradiction.
260:
4513:
3750:
5268:
5012:
3837:
3067:
3043:
2557:
859:
From the late 1800s through the 1930s, a bitter, persistent debate raged between Hilbert and his followers versus
4573:
4266:
2480:
2144:
17:
4007:
2576: – view that a proposition about the future is either necessarily true, or its negation is necessarily true
850:(For all instances of "pig" seen and unseen): ("Pig does fly" or "Pig does not fly" but not both simultaneously)
5529:
5231:
4994:
4989:
4814:
4235:
3919:
3811:
3416:
3371:
1787:
1515:
900:
from the Second International Conference in Paris in 1900) evolved from this debate (italics in the original):
155:
31:
5524:
5307:
5224:
4937:
4868:
4745:
3987:
3801:
3336:
1479:
1375:
242:
3071:, Encyclopædia Britannica, Inc., Chicago, Illinois, 1952. Cited as GB 8. 1st published, W.D. Ross (trans.),
2219:
5449:
5275:
4961:
4595:
4194:
3609:
3497:
3078:
2100:
1366:
468:
Whitehead and Russell derive some of the most powerful tools in the logician's argumentation toolkit. (In
181:
wrote that ambiguity can arise from the use of ambiguous names, but cannot exist in the facts themselves:
77:
5327:
5322:
4932:
4671:
4600:
3929:
3830:
3650:
893:
210:), through distribution of the negation in Aristotle's assertion. The former claims that no statement is
5256:
4846:
4240:
4208:
3899:
3624:
3614:
3517:
2820:
2107:
have also contested the usefulness of the law of excluded middle in the context of modern mathematics.
989:
Brouwer refused to accept the logical principle of the excluded middle, His argument was the following:
2819:
For more about the conflict between the intuitionists (e.g. Brouwer) and the formalists (Hilbert) see
1083:
offers his definition of "principle of excluded middle"; we see here also the issue of "testability":
5546:
5495:
5392:
4890:
4851:
4328:
3973:
3665:
3655:
3406:
3122:
On the significance of the principle of excluded middle in mathematics, especially in function theory
2510:
1175:
is true. Its usual form, "every judgment is either true or false" is equivalent to that given above".
255:
4002:
1263:
An example of an argument that depends on the law of excluded middle follows. We seek to prove that
202:). In modern so called classical logic, this statement is equivalent to the law of excluded middle (
5590:
5387:
5317:
4856:
4708:
4691:
4414:
3894:
3634:
3629:
3619:
3329:
2507: – Splitting of a whole into exactly two non-overlapping parts; dyadic relations and processes
1830:
1554:
1445:
1415:
1860:
5219:
5196:
5157:
5043:
4984:
4630:
4550:
4394:
4338:
3951:
3765:
3675:
3670:
3660:
3512:
3272:, Hyperion, New York, 1993. Fuzzy thinking at its finest but a good introduction to the concepts.
3062:
3054:
3038:
2499:
2494:
2127:", the statement "this statement is false", which is argued to itself be neither true nor false.
1344:
472:
formulas and propositions are identified by a leading asterisk and two numbers, such as "✸2.1".)
171:
81:
3145:, Although not directly germane, in his (1923) Brouwer uses certain words defined in this paper.
1784:
that satisfy the theorem but only two separate possibilities, one of which must work. (Actually
733:
used in his law (3). And this is the point of Reichenbach's demonstration that some believe the
5509:
5236:
5214:
5181:
5074:
4920:
4905:
4878:
4829:
4713:
4648:
4473:
4439:
4434:
4308:
4139:
4116:
3740:
3583:
3522:
3507:
3466:
3191:
2753:
2520:
2487:
567:) (Principle of double negation, part 1: if "this rose is red" is true then it's not true that
289:
138:
2870:
803:
5439:
5292:
5084:
4802:
4538:
4444:
4303:
4288:
4169:
4144:
3806:
3604:
3578:
3563:
3548:
3426:
3113:, Harvard University Press, Cambridge, Massachusetts, 1967. Reprinted with corrections, 1977.
2624:
2186:−1 disjunction signs. It is easy to check that the sentence must receive at least one of the
2009:
1973:). Under both the classical and the intuitionistic logic, by reductio ad absurdum this gives
1899:
950:
897:
463:
294:
42:
5412:
5374:
5251:
5055:
4895:
4819:
4797:
4625:
4583:
4482:
4449:
4313:
4101:
4012:
3796:
3568:
3421:
2252:
2140:
2081:
1313:
1062:
88:; however, no system of logic is built on just these laws, and none of these laws provides
8:
5541:
5432:
5417:
5397:
5354:
5241:
5191:
5117:
5062:
4999:
4792:
4787:
4735:
4503:
4492:
4164:
4064:
3992:
3983:
3979:
3914:
3909:
3770:
3573:
3538:
3401:
3106:
2573:
2263:
2213:
2160:
2085:
2077:
1239:
741:
730:
664:
281:
97:
1827:(Constructive proofs of the specific example above are not hard to produce; for example
5570:
5339:
5302:
5287:
5280:
5263:
5067:
5049:
4915:
4841:
4824:
4777:
4590:
4499:
4333:
4318:
4278:
4230:
4215:
4203:
4159:
4134:
3904:
3853:
3715:
3588:
3481:
3476:
3376:
3366:
3222:, Oxford University Press, New York, 1997 edition (first published 1912). Easy to read.
3034:
2960:
2209:
2148:
2120:
2116:
2037:
1767:
1747:
1293:
1273:
943:
were the intuitionists, led by the erstwhile topologist L. E. J. Brouwer (Dawson p. 49)
38:
4523:
3775:
2998:
2606: – Axiom used in logic and philosophy: another way of turning intuition classical
1961:), the classical mathematician may deduce a contradiction from the assumption for all
453:
5565:
5505:
5312:
5122:
5112:
5004:
4885:
4720:
4696:
4477:
4461:
4366:
4343:
4220:
4189:
4154:
4049:
3884:
3471:
3381:
3167:
3159:
3086:
2876:
2681:
2533: – Propositional calculus in which there are more than two truth values such as
2530:
2089:
2058:
1268:
1071:
872:
429:
Russell reiterated his distinction between "sense-datum" and "sensation" in his book
223:
160:
131:
89:
1216:
where ∨ means "or". The equivalence of the two forms is easily proved (p. 421)
5519:
5514:
5407:
5364:
5186:
5147:
5142:
5127:
4953:
4910:
4807:
4605:
4555:
4129:
4091:
3755:
3745:
3710:
3431:
3312:
3235:
3225:
3215:
3205:
3195:
3026:
2994:
2952:
2851:
2757:
1736:
1091:
the principle that for every system every property is either correct or impossible
864:
683:
285:
5500:
5490:
5444:
5427:
5382:
5344:
5246:
5166:
4973:
4900:
4873:
4861:
4767:
4681:
4655:
4610:
4578:
4379:
4181:
4124:
4074:
4039:
3997:
3791:
3695:
3352:
3291:
3245:
3096:
2855:
2847:
2675:
2603:
2548:
2175:
2076:
Many modern logic systems replace the law of excluded middle with the concept of
127:
85:
3162:
1952 original printing, 1971 6th printing with corrections, 10th printing 1991,
2838:
2600:
school of Buddhism, another system in which the law of excluded middle is untrue
5485:
5464:
5422:
5402:
5297:
5152:
4750:
4740:
4730:
4725:
4659:
4533:
4409:
4298:
4293:
4271:
3872:
3735:
3391:
3255:
3177:
3022:
2943:
Priest, Graham (1983). "The Logical Paradoxes and the Law of Excluded Middle".
2143:: does the set contain, as one of its elements, itself? However, in the modern
2066:
1006:." If the set is finite, it is possible—in principle—to examine each member of
3188:, Oxford University Press, Oxford, UK, 1962. Reprinted with corrections, 1975.
2761:
1823:
is irrational but there is no known easy proof of that fact.) (Davis 2000:220)
41:
notation. For a concise description of the symbols used in this notation, see
5584:
5459:
5137:
4644:
4429:
4419:
4389:
4374:
4044:
3760:
3720:
3558:
3181:
2810:
law of excluded middle as applied to arguments over the "completed infinite".
2563:
2534:
2104:
2041:
2005:
913:
To show the significance of this problem, he added the following observation:
166:
3725:
3047:, Encyclopædia Britannica, Inc., Chicago, Illinois, 1952. Cited as GB 19–20.
750:
About this issue (in admittedly very technical terms) Reichenbach observes:
5359:
5206:
5107:
5099:
4979:
4927:
4836:
4772:
4755:
4686:
4545:
4404:
4106:
3889:
3730:
3705:
3700:
3553:
2824:
2132:
2128:
2124:
2062:
2033:
1953:
existence proofs, which intuitionists do not accept. For example, to prove
1723:
868:
860:
785:
734:
726:
679:
93:
65:
5469:
5349:
4528:
4518:
4465:
4149:
4069:
4054:
3934:
3879:
3396:
3386:
2793:
2581:
2542:
2190:
2159:
Some systems of logic have different but analogous laws. For some finite
2045:
1247:
Either Socrates is mortal, or it is not the case that Socrates is mortal.
791:
420:
further defines a distinction between a "sense-datum" and a "sensation":
73:
61:
2752:
226:, this is a remarkably precise statement of the law of excluded middle,
4399:
4254:
4225:
4031:
3461:
3275:
3265:
3127:
2964:
2593:
2136:
2070:
1744:
The proof is non-constructive because it doesn't give specific numbers
697:, pp. 101–102). From the law of excluded middle (✸2.1 and ✸2.11),
490:
The proof of ✸2.1 is roughly as follows: "primitive idea" 1.08 defines
357:{\displaystyle \mathbf {*2\cdot 11} .\ \ \vdash .\ p\ \vee \thicksim p}
3212:. The William James Lectures for 1940 delivered at Harvard University.
725:
It is correct, at least for bivalent logic—i.e. it can be seen with a
693:
This principle is commonly called "the principle of double negation" (
5551:
5454:
4507:
4424:
4384:
4348:
4284:
4096:
4086:
4059:
3822:
3436:
3058:
3050:
2767:
2504:
2464:{\displaystyle (P\to (Q\lor \neg R))\to ((P\to Q)\lor (P\to \neg R))}
178:
2956:
2314:{\displaystyle \neg (P\land Q)\,\leftrightarrow \,\neg P\lor \neg Q}
241:, Aristotle seems to deny the law of excluded middle in the case of
5536:
5334:
4782:
4487:
4081:
3411:
3111:
From Frege to Gödel, A Source Book in Mathematical Logic, 1879–1931
2084:. The principle of negation as failure is used as a foundation for
2032:
Putative counterexamples to the law of excluded middle include the
998:"Suppose that A is the statement "There exists a member of the set
910:
of the consistency of the axioms of the arithmetic of real numbers.
154:
The earliest known formulation is in Aristotle's discussion of the
69:
3321:
2923:
1072:
Intuitionist definitions of the law (principle) of excluded middle
1051:
916:"If contradictory attributes be assigned to a concept, I say that
538:
is true (this is Theorem 2.08, which is proved separately), then ~
5132:
3924:
3220:
The Problems of Philosophy, With a New Introduction by John Perry
3149:
3138:
3117:
2255:, which is sometimes called the law of the weak excluded middle.
1981:). The classical logic allows this result to be transformed into
892:
The debate had a profound effect on Hilbert. Reid indicates that
277:
214:
true and false, while the latter requires that any statement is
3083:
Engines of Logic: Mathematicians and the Origin of the Computer
4676:
4022:
3867:
2566: – Type of diagrammatic notation for propositional logic
268:
112:
49:
2119:, the excluded middle has been argued to result in possible
1729:
1201:
footnote 10: "Symbolically the second form is expressed thus
1182:
footnote 9: "This is Leibniz's very simple formulation (see
2069:
as well as the ancient Greek philosophical school known as
823:. Thus an example of the expression would look like this:
3166:, North-Holland Publishing Company, Amsterdam, New York,
2872:"Proof and Knowledge in Mathematics" by Michael Detlefsen
1050:
In his lecture in 1941 at Yale and the subsequent paper,
790:
in which form it would be fully exhaustive and therefore
137:
The principle should not be confused with the semantical
3446:
367:
So just what is "truth" and "falsehood"? At the opening
2598:
Pages displaying short descriptions of redirect targets
2568:
Pages displaying short descriptions of redirect targets
2553:
Pages displaying short descriptions of redirect targets
2539:
Pages displaying short descriptions of redirect targets
2525:
Pages displaying short descriptions of redirect targets
2515:
Pages displaying short descriptions of redirect targets
2040:. Certain resolutions of these paradoxes, particularly
1719:
and 2 is certainly rational. This concludes the proof.
557:(Permutation of the assertions is allowed by axiom 1.4)
2596: – Doctrinal distinction within Tibetan Buddhism
2517:: cases where LEM appears to fail in natural language
2385:
2328:
2272:
2222:
1902:
1863:
1833:
1790:
1770:
1750:
1594:
1557:
1518:
1482:
1448:
1418:
1378:
1347:
1316:
1296:
1276:
806:
678:
Propositions ✸2.12 and ✸2.14, "double negation": The
306:
2650:"Realism – Metaphysical realism and objective truth"
2578:
Pages displaying wikidata descriptions as a fallback
1102:
s definition cites Hilbert's two axioms of negation
701:
derives principle ✸2.12 immediately. We substitute ~
688:
principle of the reciprocity of the multiple species
2490: – Pattern of reasoning in propositional logic
2795:Metamath: A Computer Language for Pure Mathematics
2463:
2370:
2313:
2243:
2203:
1921:
1888:
1849:
1815:
1776:
1756:
1707:
1573:
1543:
1501:
1464:
1434:
1397:
1357:
1329:
1302:
1282:
854:
812:
414:" and this is an undeniable-by-3rd-party "truth".
356:
3101:Logical Dilemmas, The Life and Work of Kurt Gödel
3085:, W. W. Norton & Company, NewYork, New York,
645:) (Another of the "Principles of transposition".)
581:)} (Lemma together with 2.12 used to derive 2.14)
461:From the law of excluded middle, formula ✸2.1 in
5582:
3298:, Vega, London, 2001: a reprint of a portion of
3286:first published 1739, reprinted as: David Hume,
2987:Proceedings of the American Mathematical Society
1198:" has nothing to do with the logic of judgments.
2537: – System including an indeterminate value
2523: – System including an indeterminate value
2371:{\displaystyle (P\to Q)\lor (\neg P\to \neg Q)}
1941:—for them the infinite can never be completed:
1238:then the law of excluded middle holds that the
3270:Fuzzy Thinking: The New Science of Fuzzy Logic
3103:, A.K. Peters, Wellesley, Massachusetts, 1997.
2258:This is equivalent to a few other statements:
878:Hilbert intensely disliked Kronecker's ideas:
667:the hypothesis of its own falsehood is true" (
454:Consequences of the law of excluded middle in
3838:
3337:
2850:. pp. 293–322 (Negation as a failure).
2570:: a graphical syntax for propositional logic
2545: – System for reasoning about vagueness
1896:are both easily shown to be irrational, and
794:in the narrower sense. (Reichenbach, p. 376)
120:. Another Latin designation for this law is
2700:P. T. Geach, The Law of Excluded Middle in
1010:and determine whether there is a member of
729:—that this law removes "the middle" of the
190:4.4, W. D. Ross (trans.), GBWW 8, 525–526).
4030:
3845:
3831:
3344:
3330:
3230:The Art of Philosophizing and Other Essays
3143:On the domains of definitions of functions
2977:
1258:it is not the case that Socrates is mortal
3280:An Inquiry Concerning Human Understanding
3031:Fathers of the English Dominican Province
2868:
2584: – Type of formal logic propositions
2295:
2291:
1816:{\displaystyle a={\sqrt {2}}^{\sqrt {2}}}
1730:Non-constructive proofs over the infinite
1544:{\displaystyle a={\sqrt {2}}^{\sqrt {2}}}
918:mathematically the concept does not exist
2193:(and not a value that is not one of the
2110:
906:In his second problem, had asked for a
378:. The "truth-value" of a proposition is
3154:Intuitionistic reflections on formalism
2673:
2200:Other systems reject the law entirely.
2166:, there is an analogous law called the
1502:{\displaystyle {\sqrt {2}}^{\sqrt {2}}}
1398:{\displaystyle {\sqrt {2}}^{\sqrt {2}}}
245:, in his discussion on the sea battle.
14:
5583:
3852:
3075:, Oxford University Press, Oxford, UK.
2942:
2895:
2244:{\displaystyle \neg P\lor \neg \neg P}
591:(Principle of double negation, part 2)
483:"This is the Law of excluded middle" (
433:(1912), published at the same time as
3826:
3325:
2980:"Axiom of Choice and Complementation"
2836:
2580:: the application excluded middle to
1945:In classical mathematics there occur
1929:; a proof allowed by intuitionists).
663:. It states that a proposition which
3290:, Penguin Classics, 1985. Also see:
2551: – Axioms of rational discourse
2529:Law of excluded middle is untrue in
2513: – Semantic property of plurals
371:quickly announces some definitions:
3351:
3317:Stanford Encyclopedia of Philosophy
3132:On the principle of excluded middle
2929:Internet Encyclopedia of Philosophy
2896:Priest, Graham (28 November 2010).
2869:Detlefsen, Michael (January 1992).
1740:proof disallowed by intuitionists:
1734:The above proof is an example of a
24:
2791:
2449:
2404:
2359:
2350:
2305:
2296:
2273:
2235:
2232:
2223:
807:
784:), where the symbol "⊕" signifies
25:
5607:
3306:
3210:An Inquiry Into Meaning and Truth
2999:10.1090/S0002-9939-1975-0373893-X
2154:
571:'this rose is not-red' is true".)
406:'object a' is a sense-datum" and
348:
5564:
3445:
3068:Great Books of the Western World
3044:Great Books of the Western World
2978:Diaconescu, Radu (August 1975).
2558:Limited principle of omniscience
2182:−1 negation signs and "∨ ... ∨"
934:are both shown to be true, then
398:" (e.g. "This 'object a' is 'red
317:
311:
308:
258:'s very simple formulation (see
5596:Theorems in propositional logic
3164:Introduction to Metamathematics
2971:
2936:
2915:
2889:
2862:
2830:
2813:
2803:
2800:and Davis 2000:220, footnote 2.
2784:
2204:Law of the weak excluded middle
867:. Brouwer's philosophy, called
855:Formalists versus Intuitionists
100:. The law is also known as the
3812:Tractatus Logico-Philosophicus
3417:Problem of multiple generality
3242:, Dover, New York, 1947, 1975.
2774:
2746:
2734:
2719:
2707:
2694:
2667:
2642:
2617:
2592:Non-affirming negation in the
2458:
2455:
2446:
2440:
2434:
2428:
2422:
2419:
2416:
2413:
2410:
2395:
2392:
2386:
2365:
2356:
2347:
2341:
2335:
2329:
2292:
2288:
2276:
926:Thus, Hilbert was saying: "If
720:
276:The principle was stated as a
156:principle of non-contradiction
32:fallacy of the excluded middle
13:
1:
5525:History of mathematical logic
3802:The Principles of Mathematics
3016:
2052:
1983:there exists an n such that P
1955:there exists an n such that P
1850:{\displaystyle a={\sqrt {2}}}
1574:{\displaystyle b={\sqrt {2}}}
1465:{\displaystyle b={\sqrt {2}}}
1435:{\displaystyle a={\sqrt {2}}}
740:should take the place of the
5450:Primitive recursive function
3498:Commutativity of conjunction
3200:Principia Mathematica to *56
2856:10.1007/978-1-4684-3384-5_11
2610:
2560: – Mathematical concept
2208:A particularly well-studied
1889:{\displaystyle b=\log _{2}9}
977:Hilbert naturally disagreed.
686:refer to what he calls "the
149:
126:or "no third is given". In
58:principle of excluded middle
7:
2945:The Philosophical Quarterly
2588:Mathematical constructivism
2481:Brouwer–Hilbert controversy
2474:
2145:Zermelo–Fraenkel set theory
1358:{\displaystyle {\sqrt {2}}}
1219:
659:(Called "The complement of
37:This article uses forms of
10:
5612:
4514:Schröder–Bernstein theorem
4241:Monadic predicate calculus
3900:Foundations of mathematics
3518:Monotonicity of entailment
3288:A Treatise of Human Nature
3240:Elements of Symbolic Logic
2821:Foundations of mathematics
2680:. Routledge. p. 124.
2101:L. E. J. Brouwer
1186:, IV,2). The formulation "
871:, started in earnest with
431:The Problems of Philosophy
248:
165:where he says that of two
144:
36:
29:
5560:
5547:Philosophy of mathematics
5496:Automated theorem proving
5478:
5373:
5205:
5098:
4950:
4667:
4643:
4621:Von Neumann–Bernays–Gödel
4566:
4460:
4364:
4262:
4253:
4180:
4115:
4021:
3943:
3860:
3784:
3688:
3643:
3597:
3531:
3490:
3454:
3443:
3407:Idempotency of entailment
3359:
2511:Homogeneity (linguistics)
2017:concerning infinite sets
117:principium tertii exclusi
3252:, WCB McGraw–Hill, 1997.
3186:The Development of Logic
2088:, and is widely used in
2049:or both true and false.
1509:is irrational, then let
1018:or that every member of
894:Hilbert's second problem
813:{\displaystyle \forall }
442:not a sensation. (p. 12)
68:this proposition or its
30:Not to be confused with
5197:Self-verifying theories
5018:Tarski's axiomatization
3969:Tarski's undefinability
3964:incompleteness theorems
3766:Willard Van Orman Quine
3302:starts on p. 94 ff
3063:Robert Maynard Hutchins
3039:Robert Maynard Hutchins
2654:Encyclopedia Britannica
2629:Encyclopedia Britannica
2495:Constructive set theory
2216:, which adds the axiom
2099:Mathematicians such as
1922:{\displaystyle a^{b}=3}
1369:). Consider the number
1104:
82:law of noncontradiction
5571:Mathematics portal
5182:Proof of impossibility
4830:propositional variable
4140:Propositional calculus
3741:Charles Sanders Peirce
3584:Hypothetical syllogism
3192:Alfred North Whitehead
3073:The Works of Aristotle
3037:(ed.), vols. 19–20 in
2754:Alfred North Whitehead
2674:Tomassi, Paul (1999).
2521:Law of excluded fourth
2488:Consequentia mirabilis
2465:
2372:
2315:
2245:
2003:
1923:
1890:
1851:
1825:
1817:
1778:
1758:
1709:
1575:
1545:
1503:
1466:
1436:
1399:
1359:
1331:
1304:
1284:
1156:is regarded as false …
965:
945:
890:
885:
814:
671:, pp. 103–104).)
470:Principia Mathematica,
444:
427:
388:
358:
266:
192:
139:principle of bivalence
60:states that for every
54:law of excluded middle
5440:Kolmogorov complexity
5393:Computably enumerable
5293:Model complete theory
5085:Principia Mathematica
4145:Propositional formula
3974:Banach–Tarski paradox
3807:Principia Mathematica
3579:Disjunctive syllogism
3564:modus ponendo tollens
3313:"Contradiction" entry
3148:Luitzen Egbertus Jan
3137:Luitzen Egbertus Jan
3116:Luitzen Egbertus Jan
2837:Clark, Keith (1978).
2763:Principia Mathematica
2466:
2373:
2316:
2246:
2111:In mathematical logic
2010:Luitzen E. J. Brouwer
1943:
1937:when extended to the
1924:
1891:
1852:
1818:
1779:
1759:
1742:
1710:
1576:
1546:
1504:
1467:
1437:
1400:
1360:
1332:
1330:{\displaystyle a^{b}}
1305:
1285:
1152:if the true judgment
961:Principia Mathematica
956:
951:Principia Mathematica
940:
886:
880:
815:
756:The tertium non datur
464:Principia Mathematica
456:Principia Mathematica
439:
422:
373:
359:
295:Principia Mathematica
271:Principia Mathematica
269:Bertrand Russell and
264:, IV,2)" (ibid p 421)
252:
183:
109:of the excluded third
78:three laws of thought
43:List of logic symbols
5388:Church–Turing thesis
5375:Computability theory
4584:continuum hypothesis
4102:Square of opposition
3960:Gödel's completeness
3797:Function and Concept
3569:Constructive dilemma
3544:Material implication
3061:(trans.), vol. 8 in
2840:Logic and Data Bases
2798:. footnote on p. 17.
2500:Diaconescu's theorem
2383:
2326:
2270:
2253:intuitionistic logic
2220:
2096:into these systems.
2029:" (Kleene 1952:48).
1975:not for all n, not P
1900:
1861:
1831:
1788:
1768:
1748:
1592:
1555:
1516:
1480:
1446:
1416:
1376:
1345:
1314:
1294:
1274:
1228:is the proposition:
1063:intuitionistic logic
1002:having the property
804:
661:reductio ad absurdum
514:in this rule yields
304:
158:, first proposed in
5542:Mathematical object
5433:P versus NP problem
5398:Computable function
5192:Reverse mathematics
5118:Logical consequence
4995:primitive recursive
4990:elementary function
4763:Free/bound variable
4616:Tarski–Grothendieck
4135:Logical connectives
4065:Logical equivalence
3915:Logical consequence
3771:Ludwig Wittgenstein
3574:Destructive dilemma
3402:Well-formed formula
3126:Andrei Nikolaevich
2924:"Russell's Paradox"
2898:"Paradoxical Truth"
2574:Logical determinism
2086:autoepistemic logic
2078:negation as failure
1365:is irrational (see
1256:) or its negation (
1240:logical disjunction
1233:Socrates is mortal.
1022:lacks the property
875:in the late 1800s.
448:Truth and Falsehood
282:propositional logic
76:. It is one of the
5340:Transfer principle
5303:Semantics of logic
5288:Categorical theory
5264:Non-standard model
4778:Logical connective
3905:Information theory
3854:Mathematical logic
3716:Augustus De Morgan
3296:The Vision of Hume
3107:van Heijenoort, J.
3035:Daniel J. Sullivan
2921:Kevin C. Klement,
2461:
2368:
2311:
2262:Satisfying all of
2241:
2210:intermediate logic
2121:self-contradiction
2117:mathematical logic
1919:
1886:
1847:
1813:
1774:
1754:
1705:
1571:
1541:
1499:
1462:
1432:
1395:
1355:
1327:
1300:
1280:
1269:irrational numbers
1254:Socrates is mortal
1179:considered obvious
1014:with the property
908:mathematical proof
898:Hilbert's problems
810:
709:in 2.11 to yield ~
382:if it is true and
354:
243:future contingents
5578:
5577:
5510:Abstract category
5313:Theories of truth
5123:Rule of inference
5113:Natural deduction
5094:
5093:
4639:
4638:
4344:Cartesian product
4249:
4248:
4155:Many-valued logic
4130:Boolean functions
4013:Russell's paradox
3988:diagonal argument
3885:First-order logic
3820:
3819:
3684:
3683:
3160:Stephen C. Kleene
2714:On Interpretation
2687:978-0-415-16696-6
2625:"Laws of thought"
2531:many-valued logic
2174:. If negation is
2141:Russell's paradox
2090:logic programming
1845:
1810:
1803:
1777:{\displaystyle b}
1757:{\displaystyle a}
1691:
1673:
1663:
1650:
1638:
1626:
1619:
1569:
1538:
1531:
1496:
1489:
1460:
1430:
1392:
1385:
1353:
1341:It is known that
1303:{\displaystyle b}
1283:{\displaystyle a}
1167:as well as from ~
873:Leopold Kronecker
344:
338:
329:
326:
239:On Interpretation
224:traditional logic
161:On Interpretation
123:tertium non datur
80:, along with the
16:(Redirected from
5603:
5569:
5568:
5520:History of logic
5515:Category of sets
5408:Decision problem
5187:Ordinal analysis
5128:Sequent calculus
5026:Boolean algebras
4966:
4965:
4940:
4911:logical/constant
4665:
4664:
4651:
4574:Zermelo–Fraenkel
4325:Set operations:
4260:
4259:
4197:
4028:
4027:
4008:Löwenheim–Skolem
3895:Formal semantics
3847:
3840:
3833:
3824:
3823:
3756:Henry M. Sheffer
3746:Bertrand Russell
3711:Richard Dedekind
3595:
3594:
3539:De Morgan's laws
3513:Noncontradiction
3455:Classical logics
3449:
3346:
3339:
3332:
3323:
3322:
3250:Machine Learning
3236:Hans Reichenbach
3226:Bertrand Russell
3216:Bertrand Russell
3206:Bertrand Russell
3196:Bertrand Russell
3027:Summa Theologica
3010:
3009:
3007:
3005:
2984:
2975:
2969:
2968:
2951:(131): 160–165.
2940:
2934:
2933:
2919:
2913:
2912:
2910:
2908:
2893:
2887:
2886:
2866:
2860:
2859:
2845:
2834:
2828:
2817:
2811:
2807:
2801:
2799:
2792:Megill, Norman.
2788:
2782:
2778:
2772:
2771:
2758:Bertrand Russell
2750:
2744:
2743:Γ 7, 1011b 26–27
2738:
2732:
2723:
2717:
2711:
2705:
2698:
2692:
2691:
2671:
2665:
2664:
2662:
2660:
2646:
2640:
2639:
2637:
2635:
2621:
2599:
2579:
2569:
2554:
2540:
2526:
2516:
2470:
2468:
2467:
2462:
2377:
2375:
2374:
2369:
2320:
2318:
2317:
2312:
2264:De Morgan's laws
2250:
2248:
2247:
2242:
2168:law of excluded
2131:has argued that
1947:non-constructive
1934:non-constructive
1928:
1926:
1925:
1920:
1912:
1911:
1895:
1893:
1892:
1887:
1879:
1878:
1856:
1854:
1853:
1848:
1846:
1841:
1822:
1820:
1819:
1814:
1812:
1811:
1806:
1804:
1799:
1783:
1781:
1780:
1775:
1763:
1761:
1760:
1755:
1737:non-constructive
1714:
1712:
1711:
1706:
1698:
1697:
1692:
1687:
1681:
1680:
1679:
1675:
1674:
1669:
1664:
1659:
1651:
1646:
1640:
1639:
1634:
1632:
1628:
1627:
1622:
1620:
1615:
1604:
1603:
1580:
1578:
1577:
1572:
1570:
1565:
1550:
1548:
1547:
1542:
1540:
1539:
1534:
1532:
1527:
1508:
1506:
1505:
1500:
1498:
1497:
1492:
1490:
1485:
1471:
1469:
1468:
1463:
1461:
1456:
1441:
1439:
1438:
1433:
1431:
1426:
1404:
1402:
1401:
1396:
1394:
1393:
1388:
1386:
1381:
1364:
1362:
1361:
1356:
1354:
1349:
1336:
1334:
1333:
1328:
1326:
1325:
1309:
1307:
1306:
1301:
1289:
1287:
1286:
1281:
1267:there exist two
1260:) must be true.
1224:For example, if
1060:
865:L. E. J. Brouwer
819:
817:
816:
811:
684:L. E. J. Brouwer
570:
487:, p. 101).
413:
409:
405:
402:") really means
401:
397:
363:
361:
360:
355:
342:
336:
327:
324:
320:
98:De Morgan's laws
21:
5611:
5610:
5606:
5605:
5604:
5602:
5601:
5600:
5591:Classical logic
5581:
5580:
5579:
5574:
5563:
5556:
5501:Category theory
5491:Algebraic logic
5474:
5445:Lambda calculus
5383:Church encoding
5369:
5345:Truth predicate
5201:
5167:Complete theory
5090:
4959:
4955:
4951:
4946:
4938:
4658: and
4654:
4649:
4635:
4611:New Foundations
4579:axiom of choice
4562:
4524:Gödel numbering
4464: and
4456:
4360:
4245:
4195:
4176:
4125:Boolean algebra
4111:
4075:Equiconsistency
4040:Classical logic
4017:
3998:Halting problem
3986: and
3962: and
3950: and
3949:
3944:Theorems (
3939:
3856:
3851:
3821:
3816:
3792:Begriffsschrift
3780:
3776:Jan Łukasiewicz
3696:Bernard Bolzano
3680:
3651:Double negation
3639:
3610:Double negation
3593:
3527:
3503:Excluded middle
3486:
3450:
3441:
3355:
3353:Classical logic
3350:
3309:
3292:David Applebaum
3023:Aquinas, Thomas
3019:
3014:
3013:
3003:
3001:
2982:
2976:
2972:
2957:10.2307/2218742
2941:
2937:
2922:
2920:
2916:
2906:
2904:
2894:
2890:
2883:
2867:
2863:
2848:Springer-Verlag
2843:
2835:
2831:
2818:
2814:
2808:
2804:
2789:
2785:
2779:
2775:
2751:
2747:
2739:
2735:
2724:
2720:
2712:
2708:
2699:
2695:
2688:
2672:
2668:
2658:
2656:
2648:
2647:
2643:
2633:
2631:
2623:
2622:
2618:
2613:
2597:
2577:
2567:
2552:
2549:Laws of thought
2538:
2524:
2514:
2477:
2384:
2381:
2380:
2327:
2324:
2323:
2271:
2268:
2267:
2221:
2218:
2217:
2214:De Morgan logic
2206:
2157:
2149:Curry's paradox
2113:
2055:
2038:Quine's paradox
1907:
1903:
1901:
1898:
1897:
1874:
1870:
1862:
1859:
1858:
1840:
1832:
1829:
1828:
1805:
1798:
1797:
1789:
1786:
1785:
1769:
1766:
1765:
1749:
1746:
1745:
1732:
1693:
1686:
1685:
1668:
1658:
1657:
1653:
1652:
1645:
1644:
1633:
1621:
1614:
1613:
1609:
1608:
1599:
1595:
1593:
1590:
1589:
1564:
1556:
1553:
1552:
1533:
1526:
1525:
1517:
1514:
1513:
1491:
1484:
1483:
1481:
1478:
1477:
1455:
1447:
1444:
1443:
1425:
1417:
1414:
1413:
1387:
1380:
1379:
1377:
1374:
1373:
1348:
1346:
1343:
1342:
1321:
1317:
1315:
1312:
1311:
1295:
1292:
1291:
1275:
1272:
1271:
1222:
1184:Nouveaux Essais
1074:
1058:
857:
805:
802:
801:
723:
646:
628:
610:
592:
582:
572:
568:
558:
506:. Substituting
459:
411:
407:
403:
399:
395:
307:
305:
302:
301:
274:
261:Nouveaux Essais
251:
218:true or false.
152:
147:
130:, the law is a
128:classical logic
90:inference rules
86:law of identity
46:
35:
28:
23:
22:
18:Excluded middle
15:
12:
11:
5:
5609:
5599:
5598:
5593:
5576:
5575:
5561:
5558:
5557:
5555:
5554:
5549:
5544:
5539:
5534:
5533:
5532:
5522:
5517:
5512:
5503:
5498:
5493:
5488:
5486:Abstract logic
5482:
5480:
5476:
5475:
5473:
5472:
5467:
5465:Turing machine
5462:
5457:
5452:
5447:
5442:
5437:
5436:
5435:
5430:
5425:
5420:
5415:
5405:
5403:Computable set
5400:
5395:
5390:
5385:
5379:
5377:
5371:
5370:
5368:
5367:
5362:
5357:
5352:
5347:
5342:
5337:
5332:
5331:
5330:
5325:
5320:
5310:
5305:
5300:
5298:Satisfiability
5295:
5290:
5285:
5284:
5283:
5273:
5272:
5271:
5261:
5260:
5259:
5254:
5249:
5244:
5239:
5229:
5228:
5227:
5222:
5215:Interpretation
5211:
5209:
5203:
5202:
5200:
5199:
5194:
5189:
5184:
5179:
5169:
5164:
5163:
5162:
5161:
5160:
5150:
5145:
5135:
5130:
5125:
5120:
5115:
5110:
5104:
5102:
5096:
5095:
5092:
5091:
5089:
5088:
5080:
5079:
5078:
5077:
5072:
5071:
5070:
5065:
5060:
5040:
5039:
5038:
5036:minimal axioms
5033:
5022:
5021:
5020:
5009:
5008:
5007:
5002:
4997:
4992:
4987:
4982:
4969:
4967:
4948:
4947:
4945:
4944:
4943:
4942:
4930:
4925:
4924:
4923:
4918:
4913:
4908:
4898:
4893:
4888:
4883:
4882:
4881:
4876:
4866:
4865:
4864:
4859:
4854:
4849:
4839:
4834:
4833:
4832:
4827:
4822:
4812:
4811:
4810:
4805:
4800:
4795:
4790:
4785:
4775:
4770:
4765:
4760:
4759:
4758:
4753:
4748:
4743:
4733:
4728:
4726:Formation rule
4723:
4718:
4717:
4716:
4711:
4701:
4700:
4699:
4689:
4684:
4679:
4674:
4668:
4662:
4645:Formal systems
4641:
4640:
4637:
4636:
4634:
4633:
4628:
4623:
4618:
4613:
4608:
4603:
4598:
4593:
4588:
4587:
4586:
4581:
4570:
4568:
4564:
4563:
4561:
4560:
4559:
4558:
4548:
4543:
4542:
4541:
4534:Large cardinal
4531:
4526:
4521:
4516:
4511:
4497:
4496:
4495:
4490:
4485:
4470:
4468:
4458:
4457:
4455:
4454:
4453:
4452:
4447:
4442:
4432:
4427:
4422:
4417:
4412:
4407:
4402:
4397:
4392:
4387:
4382:
4377:
4371:
4369:
4362:
4361:
4359:
4358:
4357:
4356:
4351:
4346:
4341:
4336:
4331:
4323:
4322:
4321:
4316:
4306:
4301:
4299:Extensionality
4296:
4294:Ordinal number
4291:
4281:
4276:
4275:
4274:
4263:
4257:
4251:
4250:
4247:
4246:
4244:
4243:
4238:
4233:
4228:
4223:
4218:
4213:
4212:
4211:
4201:
4200:
4199:
4186:
4184:
4178:
4177:
4175:
4174:
4173:
4172:
4167:
4162:
4152:
4147:
4142:
4137:
4132:
4127:
4121:
4119:
4113:
4112:
4110:
4109:
4104:
4099:
4094:
4089:
4084:
4079:
4078:
4077:
4067:
4062:
4057:
4052:
4047:
4042:
4036:
4034:
4025:
4019:
4018:
4016:
4015:
4010:
4005:
4000:
3995:
3990:
3978:Cantor's
3976:
3971:
3966:
3956:
3954:
3941:
3940:
3938:
3937:
3932:
3927:
3922:
3917:
3912:
3907:
3902:
3897:
3892:
3887:
3882:
3877:
3876:
3875:
3864:
3862:
3858:
3857:
3850:
3849:
3842:
3835:
3827:
3818:
3817:
3815:
3814:
3809:
3804:
3799:
3794:
3788:
3786:
3782:
3781:
3779:
3778:
3773:
3768:
3763:
3758:
3753:
3751:Ernst Schröder
3748:
3743:
3738:
3736:Giuseppe Peano
3733:
3728:
3723:
3718:
3713:
3708:
3703:
3698:
3692:
3690:
3686:
3685:
3682:
3681:
3679:
3678:
3673:
3668:
3663:
3658:
3653:
3647:
3645:
3641:
3640:
3638:
3637:
3632:
3627:
3622:
3617:
3612:
3607:
3601:
3599:
3592:
3591:
3586:
3581:
3576:
3571:
3566:
3561:
3556:
3551:
3546:
3541:
3535:
3533:
3529:
3528:
3526:
3525:
3520:
3515:
3510:
3505:
3500:
3494:
3492:
3488:
3487:
3485:
3484:
3479:
3474:
3469:
3464:
3458:
3456:
3452:
3451:
3444:
3442:
3440:
3439:
3434:
3429:
3424:
3419:
3414:
3409:
3404:
3399:
3394:
3392:Truth function
3389:
3384:
3379:
3374:
3369:
3363:
3361:
3357:
3356:
3349:
3348:
3341:
3334:
3326:
3320:
3319:
3308:
3307:External links
3305:
3304:
3303:
3273:
3263:
3256:Constance Reid
3253:
3243:
3233:
3223:
3213:
3203:
3189:
3175:
3157:
3146:
3135:
3124:
3114:
3104:
3094:
3076:
3048:
3018:
3015:
3012:
3011:
2993:(1): 176–178.
2970:
2935:
2914:
2888:
2881:
2861:
2829:
2812:
2802:
2783:
2773:
2745:
2733:
2718:
2706:
2702:Logic Matters,
2693:
2686:
2666:
2641:
2615:
2614:
2612:
2609:
2608:
2607:
2601:
2590:
2585:
2571:
2561:
2555:
2546:
2527:
2518:
2508:
2502:
2497:
2492:
2484:
2476:
2473:
2472:
2471:
2460:
2457:
2454:
2451:
2448:
2445:
2442:
2439:
2436:
2433:
2430:
2427:
2424:
2421:
2418:
2415:
2412:
2409:
2406:
2403:
2400:
2397:
2394:
2391:
2388:
2378:
2367:
2364:
2361:
2358:
2355:
2352:
2349:
2346:
2343:
2340:
2337:
2334:
2331:
2321:
2310:
2307:
2304:
2301:
2298:
2294:
2290:
2287:
2284:
2281:
2278:
2275:
2240:
2237:
2234:
2231:
2228:
2225:
2205:
2202:
2164:-valued logics
2156:
2155:Analogous laws
2153:
2125:Liar's paradox
2112:
2109:
2067:Buddhist logic
2054:
2051:
1918:
1915:
1910:
1906:
1885:
1882:
1877:
1873:
1869:
1866:
1844:
1839:
1836:
1809:
1802:
1796:
1793:
1773:
1753:
1731:
1728:
1717:
1716:
1704:
1701:
1696:
1690:
1684:
1678:
1672:
1667:
1662:
1656:
1649:
1643:
1637:
1631:
1625:
1618:
1612:
1607:
1602:
1598:
1583:
1582:
1568:
1563:
1560:
1537:
1530:
1524:
1521:
1495:
1488:
1474:
1473:
1459:
1454:
1451:
1429:
1424:
1421:
1407:
1406:
1391:
1384:
1352:
1339:
1338:
1324:
1320:
1299:
1279:
1250:
1249:
1236:
1235:
1221:
1218:
1214:
1213:
1204:
1203:
1202:
1199:
1180:
1176:
1157:
1142:
1141:
1118:
1097:
1096:
1095:
1094:
1073:
1070:
1048:
1047:
1046:
1045:
1042:
1039:
1035:
993:
992:
991:
990:
984:
983:
982:
981:
978:
975:
924:
923:
922:
921:
920:" (Reid p. 71)
914:
911:
856:
853:
852:
851:
848:
809:
798:
797:
796:
795:
788:
775:
774:
773:
772:
764:
757:
722:
719:
546:must be true.
458:
452:
353:
350:
347:
341:
335:
332:
323:
319:
316:
313:
310:
273:
267:
250:
247:
151:
148:
146:
143:
26:
9:
6:
4:
3:
2:
5608:
5597:
5594:
5592:
5589:
5588:
5586:
5573:
5572:
5567:
5559:
5553:
5550:
5548:
5545:
5543:
5540:
5538:
5535:
5531:
5528:
5527:
5526:
5523:
5521:
5518:
5516:
5513:
5511:
5507:
5504:
5502:
5499:
5497:
5494:
5492:
5489:
5487:
5484:
5483:
5481:
5477:
5471:
5468:
5466:
5463:
5461:
5460:Recursive set
5458:
5456:
5453:
5451:
5448:
5446:
5443:
5441:
5438:
5434:
5431:
5429:
5426:
5424:
5421:
5419:
5416:
5414:
5411:
5410:
5409:
5406:
5404:
5401:
5399:
5396:
5394:
5391:
5389:
5386:
5384:
5381:
5380:
5378:
5376:
5372:
5366:
5363:
5361:
5358:
5356:
5353:
5351:
5348:
5346:
5343:
5341:
5338:
5336:
5333:
5329:
5326:
5324:
5321:
5319:
5316:
5315:
5314:
5311:
5309:
5306:
5304:
5301:
5299:
5296:
5294:
5291:
5289:
5286:
5282:
5279:
5278:
5277:
5274:
5270:
5269:of arithmetic
5267:
5266:
5265:
5262:
5258:
5255:
5253:
5250:
5248:
5245:
5243:
5240:
5238:
5235:
5234:
5233:
5230:
5226:
5223:
5221:
5218:
5217:
5216:
5213:
5212:
5210:
5208:
5204:
5198:
5195:
5193:
5190:
5188:
5185:
5183:
5180:
5177:
5176:from ZFC
5173:
5170:
5168:
5165:
5159:
5156:
5155:
5154:
5151:
5149:
5146:
5144:
5141:
5140:
5139:
5136:
5134:
5131:
5129:
5126:
5124:
5121:
5119:
5116:
5114:
5111:
5109:
5106:
5105:
5103:
5101:
5097:
5087:
5086:
5082:
5081:
5076:
5075:non-Euclidean
5073:
5069:
5066:
5064:
5061:
5059:
5058:
5054:
5053:
5051:
5048:
5047:
5045:
5041:
5037:
5034:
5032:
5029:
5028:
5027:
5023:
5019:
5016:
5015:
5014:
5010:
5006:
5003:
5001:
4998:
4996:
4993:
4991:
4988:
4986:
4983:
4981:
4978:
4977:
4975:
4971:
4970:
4968:
4963:
4957:
4952:Example
4949:
4941:
4936:
4935:
4934:
4931:
4929:
4926:
4922:
4919:
4917:
4914:
4912:
4909:
4907:
4904:
4903:
4902:
4899:
4897:
4894:
4892:
4889:
4887:
4884:
4880:
4877:
4875:
4872:
4871:
4870:
4867:
4863:
4860:
4858:
4855:
4853:
4850:
4848:
4845:
4844:
4843:
4840:
4838:
4835:
4831:
4828:
4826:
4823:
4821:
4818:
4817:
4816:
4813:
4809:
4806:
4804:
4801:
4799:
4796:
4794:
4791:
4789:
4786:
4784:
4781:
4780:
4779:
4776:
4774:
4771:
4769:
4766:
4764:
4761:
4757:
4754:
4752:
4749:
4747:
4744:
4742:
4739:
4738:
4737:
4734:
4732:
4729:
4727:
4724:
4722:
4719:
4715:
4712:
4710:
4709:by definition
4707:
4706:
4705:
4702:
4698:
4695:
4694:
4693:
4690:
4688:
4685:
4683:
4680:
4678:
4675:
4673:
4670:
4669:
4666:
4663:
4661:
4657:
4652:
4646:
4642:
4632:
4629:
4627:
4624:
4622:
4619:
4617:
4614:
4612:
4609:
4607:
4604:
4602:
4599:
4597:
4596:Kripke–Platek
4594:
4592:
4589:
4585:
4582:
4580:
4577:
4576:
4575:
4572:
4571:
4569:
4565:
4557:
4554:
4553:
4552:
4549:
4547:
4544:
4540:
4537:
4536:
4535:
4532:
4530:
4527:
4525:
4522:
4520:
4517:
4515:
4512:
4509:
4505:
4501:
4498:
4494:
4491:
4489:
4486:
4484:
4481:
4480:
4479:
4475:
4472:
4471:
4469:
4467:
4463:
4459:
4451:
4448:
4446:
4443:
4441:
4440:constructible
4438:
4437:
4436:
4433:
4431:
4428:
4426:
4423:
4421:
4418:
4416:
4413:
4411:
4408:
4406:
4403:
4401:
4398:
4396:
4393:
4391:
4388:
4386:
4383:
4381:
4378:
4376:
4373:
4372:
4370:
4368:
4363:
4355:
4352:
4350:
4347:
4345:
4342:
4340:
4337:
4335:
4332:
4330:
4327:
4326:
4324:
4320:
4317:
4315:
4312:
4311:
4310:
4307:
4305:
4302:
4300:
4297:
4295:
4292:
4290:
4286:
4282:
4280:
4277:
4273:
4270:
4269:
4268:
4265:
4264:
4261:
4258:
4256:
4252:
4242:
4239:
4237:
4234:
4232:
4229:
4227:
4224:
4222:
4219:
4217:
4214:
4210:
4207:
4206:
4205:
4202:
4198:
4193:
4192:
4191:
4188:
4187:
4185:
4183:
4179:
4171:
4168:
4166:
4163:
4161:
4158:
4157:
4156:
4153:
4151:
4148:
4146:
4143:
4141:
4138:
4136:
4133:
4131:
4128:
4126:
4123:
4122:
4120:
4118:
4117:Propositional
4114:
4108:
4105:
4103:
4100:
4098:
4095:
4093:
4090:
4088:
4085:
4083:
4080:
4076:
4073:
4072:
4071:
4068:
4066:
4063:
4061:
4058:
4056:
4053:
4051:
4048:
4046:
4045:Logical truth
4043:
4041:
4038:
4037:
4035:
4033:
4029:
4026:
4024:
4020:
4014:
4011:
4009:
4006:
4004:
4001:
3999:
3996:
3994:
3991:
3989:
3985:
3981:
3977:
3975:
3972:
3970:
3967:
3965:
3961:
3958:
3957:
3955:
3953:
3947:
3942:
3936:
3933:
3931:
3928:
3926:
3923:
3921:
3918:
3916:
3913:
3911:
3908:
3906:
3903:
3901:
3898:
3896:
3893:
3891:
3888:
3886:
3883:
3881:
3878:
3874:
3871:
3870:
3869:
3866:
3865:
3863:
3859:
3855:
3848:
3843:
3841:
3836:
3834:
3829:
3828:
3825:
3813:
3810:
3808:
3805:
3803:
3800:
3798:
3795:
3793:
3790:
3789:
3787:
3783:
3777:
3774:
3772:
3769:
3767:
3764:
3762:
3761:Alfred Tarski
3759:
3757:
3754:
3752:
3749:
3747:
3744:
3742:
3739:
3737:
3734:
3732:
3729:
3727:
3724:
3722:
3721:Gottlob Frege
3719:
3717:
3714:
3712:
3709:
3707:
3704:
3702:
3699:
3697:
3694:
3693:
3691:
3687:
3677:
3674:
3672:
3669:
3667:
3666:Biconditional
3664:
3662:
3659:
3657:
3654:
3652:
3649:
3648:
3646:
3642:
3636:
3633:
3631:
3628:
3626:
3625:Biconditional
3623:
3621:
3618:
3616:
3613:
3611:
3608:
3606:
3603:
3602:
3600:
3596:
3590:
3587:
3585:
3582:
3580:
3577:
3575:
3572:
3570:
3567:
3565:
3562:
3560:
3559:modus tollens
3557:
3555:
3552:
3550:
3549:Transposition
3547:
3545:
3542:
3540:
3537:
3536:
3534:
3530:
3524:
3521:
3519:
3516:
3514:
3511:
3509:
3506:
3504:
3501:
3499:
3496:
3495:
3493:
3489:
3483:
3480:
3478:
3475:
3473:
3470:
3468:
3467:Propositional
3465:
3463:
3460:
3459:
3457:
3453:
3448:
3438:
3435:
3433:
3430:
3428:
3425:
3423:
3422:Associativity
3420:
3418:
3415:
3413:
3410:
3408:
3405:
3403:
3400:
3398:
3395:
3393:
3390:
3388:
3385:
3383:
3380:
3378:
3375:
3373:
3370:
3368:
3365:
3364:
3362:
3358:
3354:
3347:
3342:
3340:
3335:
3333:
3328:
3327:
3324:
3318:
3314:
3311:
3310:
3301:
3297:
3293:
3289:
3285:
3281:
3277:
3274:
3271:
3267:
3264:
3261:
3257:
3254:
3251:
3247:
3244:
3241:
3237:
3234:
3231:
3227:
3224:
3221:
3217:
3214:
3211:
3207:
3204:
3201:
3197:
3193:
3190:
3187:
3183:
3179:
3176:
3173:
3172:0-7204-2103-9
3169:
3165:
3161:
3158:
3155:
3151:
3147:
3144:
3140:
3136:
3133:
3129:
3125:
3123:
3119:
3115:
3112:
3108:
3105:
3102:
3098:
3095:
3092:
3091:0-393-32229-7
3088:
3084:
3080:
3077:
3074:
3070:
3069:
3064:
3060:
3056:
3052:
3049:
3046:
3045:
3040:
3036:
3032:
3028:
3024:
3021:
3020:
3000:
2996:
2992:
2988:
2981:
2974:
2966:
2962:
2958:
2954:
2950:
2946:
2939:
2931:
2930:
2925:
2918:
2903:
2899:
2892:
2884:
2882:9780415068055
2878:
2875:. Routledge.
2874:
2873:
2865:
2857:
2853:
2849:
2842:
2841:
2833:
2826:
2822:
2816:
2806:
2797:
2796:
2787:
2777:
2770:, p. 105
2769:
2765:
2764:
2759:
2755:
2749:
2742:
2737:
2731:2, 996b 26–30
2730:
2727:
2722:
2715:
2710:
2703:
2697:
2689:
2683:
2679:
2678:
2670:
2655:
2651:
2645:
2630:
2626:
2620:
2616:
2605:
2602:
2595:
2591:
2589:
2586:
2583:
2575:
2572:
2565:
2564:Logical graph
2562:
2559:
2556:
2550:
2547:
2544:
2536:
2535:ternary logic
2532:
2528:
2522:
2519:
2512:
2509:
2506:
2503:
2501:
2498:
2496:
2493:
2491:
2489:
2485:
2482:
2479:
2478:
2452:
2443:
2437:
2431:
2425:
2407:
2401:
2398:
2389:
2379:
2362:
2353:
2344:
2338:
2332:
2322:
2308:
2302:
2299:
2285:
2282:
2279:
2265:
2261:
2260:
2259:
2256:
2254:
2238:
2229:
2226:
2215:
2211:
2201:
2198:
2196:
2192:
2189:
2185:
2181:
2177:
2173:
2169:
2165:
2163:
2152:
2150:
2146:
2142:
2138:
2134:
2130:
2126:
2122:
2118:
2108:
2106:
2105:Arend Heyting
2102:
2097:
2095:
2091:
2087:
2083:
2079:
2074:
2072:
2068:
2064:
2060:
2050:
2047:
2043:
2042:Graham Priest
2039:
2035:
2030:
2028:
2024:
2020:
2016:
2011:
2007:
2006:David Hilbert
2002:
2000:
1996:
1992:
1988:
1984:
1980:
1976:
1972:
1968:
1964:
1960:
1956:
1952:
1948:
1942:
1940:
1935:
1930:
1916:
1913:
1908:
1904:
1883:
1880:
1875:
1871:
1867:
1864:
1842:
1837:
1834:
1824:
1807:
1800:
1794:
1791:
1771:
1751:
1741:
1739:
1738:
1727:
1725:
1720:
1702:
1699:
1694:
1688:
1682:
1676:
1670:
1665:
1660:
1654:
1647:
1641:
1635:
1629:
1623:
1616:
1610:
1605:
1600:
1596:
1588:
1587:
1586:
1566:
1561:
1558:
1535:
1528:
1522:
1519:
1512:
1511:
1510:
1493:
1486:
1457:
1452:
1449:
1427:
1422:
1419:
1412:
1411:
1410:
1389:
1382:
1372:
1371:
1370:
1368:
1350:
1322:
1318:
1297:
1277:
1270:
1266:
1265:
1264:
1261:
1259:
1255:
1248:
1245:
1244:
1243:
1241:
1234:
1231:
1230:
1229:
1227:
1217:
1212:
1208:
1205:
1200:
1197:
1193:
1189:
1185:
1181:
1177:
1174:
1170:
1166:
1163:follows from
1162:
1158:
1155:
1151:
1146:
1145:
1144:
1143:
1139:
1135:
1131:
1127:
1123:
1119:
1116:
1112:
1108:
1105:
1103:
1101:
1092:
1088:
1087:
1086:
1085:
1084:
1082:
1078:
1069:
1066:
1064:
1055:
1053:
1043:
1040:
1036:
1033:
1029:
1025:
1021:
1017:
1013:
1009:
1005:
1001:
997:
996:
995:
994:
988:
987:
986:
985:
979:
976:
972:
971:
970:
969:
968:
964:
962:
955:
953:
952:
944:
939:
937:
933:
929:
919:
915:
912:
909:
905:
904:
903:
902:
901:
899:
895:
889:
884:
879:
876:
874:
870:
866:
862:
849:
846:
842:
838:
834:
830:
826:
825:
824:
822:
793:
789:
787:
783:
779:
778:
777:
776:
770:
765:
762:
758:
755:
754:
753:
752:
751:
748:
746:
744:
739:
737:
732:
728:
718:
716:
712:
708:
704:
700:
696:
691:
689:
685:
681:
676:
672:
670:
666:
662:
658:
654:
650:
644:
640:
636:
632:
626:
622:
618:
614:
608:
604:
600:
596:
590:
586:
580:
576:
566:
562:
556:
552:
547:
545:
541:
537:
533:
529:
525:
521:
517:
513:
509:
505:
501:
497:
493:
488:
486:
482:
478:
473:
471:
467:
465:
457:
451:
449:
443:
438:
437:(1910–1913):
436:
432:
426:
421:
419:
415:
393:
387:
385:
381:
377:
372:
370:
365:
351:
345:
339:
333:
330:
321:
314:
299:
297:
296:
291:
287:
283:
279:
272:
265:
263:
262:
257:
246:
244:
240:
235:
233:
229:
225:
219:
217:
213:
209:
205:
201:
197:
191:
189:
182:
180:
176:
174:
173:
168:
167:contradictory
164:
162:
157:
142:
140:
135:
133:
129:
125:
124:
119:
118:
114:
110:
107:
103:
99:
95:
91:
87:
83:
79:
75:
71:
67:
63:
59:
55:
51:
44:
40:
33:
27:Logic theorem
19:
5562:
5360:Ultraproduct
5207:Model theory
5172:Independence
5108:Formal proof
5100:Proof theory
5083:
5056:
5013:real numbers
4985:second-order
4896:Substitution
4773:Metalanguage
4714:conservative
4687:Axiom schema
4631:Constructive
4601:Morse–Kelley
4567:Set theories
4546:Aleph number
4539:inaccessible
4445:Grothendieck
4329:intersection
4216:Higher-order
4204:Second-order
4150:Truth tables
4107:Venn diagram
3890:Formal proof
3731:Hugh MacColl
3706:Georg Cantor
3701:George Boole
3598:Introduction
3554:modus ponens
3502:
3482:Higher-order
3477:Second-order
3427:Distribution
3387:Truth tables
3299:
3295:
3287:
3283:
3279:
3269:
3259:
3249:
3246:Tom Mitchell
3239:
3229:
3219:
3209:
3199:
3185:
3163:
3153:
3142:
3131:
3121:
3110:
3100:
3082:
3079:Martin Davis
3072:
3066:
3042:
3002:. Retrieved
2990:
2986:
2973:
2948:
2944:
2938:
2927:
2917:
2907:10 September
2905:. Retrieved
2901:
2891:
2871:
2864:
2839:
2832:
2825:Intuitionism
2815:
2805:
2794:
2786:
2776:
2762:
2748:
2740:
2736:
2728:
2725:
2721:
2713:
2709:
2701:
2696:
2676:
2669:
2657:. Retrieved
2653:
2644:
2632:. Retrieved
2628:
2619:
2604:Peirce's law
2486:
2257:
2212:is given by
2207:
2199:
2194:
2191:truth values
2187:
2183:
2179:
2171:
2167:
2161:
2158:
2129:Arthur Prior
2114:
2098:
2093:
2075:
2063:Indian logic
2056:
2034:liar paradox
2031:
2026:
2022:
2018:
2014:
2004:
1998:
1994:
1990:
1986:
1982:
1978:
1974:
1970:
1966:
1962:
1958:
1954:
1950:
1946:
1944:
1938:
1933:
1931:
1826:
1743:
1735:
1733:
1724:intuitionist
1721:
1718:
1584:
1475:
1408:
1340:
1337:is rational.
1262:
1257:
1253:
1251:
1246:
1237:
1232:
1225:
1223:
1215:
1210:
1206:
1195:
1191:
1187:
1183:
1172:
1168:
1164:
1160:
1153:
1149:
1137:
1133:
1129:
1125:
1121:
1114:
1110:
1106:
1099:
1098:
1090:
1080:
1079:
1075:
1067:
1056:
1049:
1031:
1027:
1023:
1019:
1015:
1011:
1007:
1003:
999:
966:
960:
957:
949:
946:
941:
935:
931:
927:
925:
917:
907:
891:
887:
881:
877:
869:intuitionism
861:Hermann Weyl
858:
844:
840:
836:
832:
828:
820:
799:
786:exclusive-or
781:
768:
760:
749:
742:
735:
731:inclusive-or
727:Karnaugh map
724:
714:
710:
706:
702:
698:
694:
692:
687:
682:writings of
680:intuitionist
677:
673:
668:
665:follows from
660:
656:
652:
648:
642:
638:
634:
630:
624:
620:
616:
612:
606:
602:
598:
594:
588:
584:
578:
574:
564:
560:
554:
550:
548:
543:
539:
535:
531:
527:
523:
519:
515:
511:
507:
503:
499:
495:
491:
489:
484:
480:
476:
474:
469:
462:
460:
455:
447:
445:
440:
434:
430:
428:
423:
417:
416:
391:
389:
383:
379:
376:Truth-values
375:
374:
368:
366:
300:
293:
275:
270:
259:
253:
238:
236:
231:
227:
220:
215:
211:
207:
203:
199:
195:
193:
187:
184:
177:
170:
159:
153:
136:
122:
121:
116:
115:
108:
105:
101:
94:modus ponens
57:
53:
47:
5470:Type theory
5418:undecidable
5350:Truth value
5237:equivalence
4916:non-logical
4529:Enumeration
4519:Isomorphism
4466:cardinality
4450:Von Neumann
4415:Ultrafilter
4380:Uncountable
4314:equivalence
4231:Quantifiers
4221:Fixed-point
4190:First-order
4070:Consistency
4055:Proposition
4032:Traditional
4003:Lindström's
3993:Compactness
3935:Type theory
3880:Cardinality
3676:Disjunction
3671:Conjunction
3656:Existential
3644:Elimination
3635:Disjunction
3630:Conjunction
3615:Existential
3472:First-order
3397:Truth value
3367:Quantifiers
3152:, 1927(2),
3055:Metaphysics
2902:Opinionator
2741:Metaphysics
2726:Metaphysics
2543:fuzzy logic
2133:The Paradox
2046:dialetheism
1100:Kolmogorov'
792:nomological
721:Reichenbach
188:Metaphysics
172:Metaphysics
62:proposition
5585:Categories
5281:elementary
4974:arithmetic
4842:Quantifier
4820:functional
4692:Expression
4410:Transitive
4354:identities
4339:complement
4272:hereditary
4255:Set theory
3726:Kurt Gödel
3589:Absorption
3491:Principles
3377:Connective
3300:An Inquiry
3276:David Hume
3266:Bart Kosko
3182:Kneale, M.
3178:Kneale, W.
3128:Kolmogorov
3097:Dawson, J.
3033:(trans.),
3017:References
2594:Prasangika
2266:including
2137:set theory
2115:In modern
2071:Pyrrhonism
2053:Criticisms
1993:such that
1310:such that
1190:is either
92:, such as
84:, and the
5552:Supertask
5455:Recursion
5413:decidable
5247:saturated
5225:of models
5148:deductive
5143:axiomatic
5063:Hilbert's
5050:Euclidean
5031:canonical
4954:axiomatic
4886:Signature
4815:Predicate
4704:Extension
4626:Ackermann
4551:Operation
4430:Universal
4420:Recursive
4395:Singleton
4390:Inhabited
4375:Countable
4365:Types of
4349:power set
4319:partition
4236:Predicate
4182:Predicate
4097:Syllogism
4087:Soundness
4060:Inference
4050:Tautology
3952:paradoxes
3661:Universal
3620:Universal
3523:Explosion
3508:Bivalence
3437:Soundness
3382:Tautology
3372:Predicate
3059:W.D. Ross
3051:Aristotle
2768:Cambridge
2611:Footnotes
2505:Dichotomy
2450:¬
2447:→
2438:∨
2429:→
2417:→
2405:¬
2402:∨
2393:→
2360:¬
2357:→
2351:¬
2345:∨
2336:→
2306:¬
2303:∨
2297:¬
2293:↔
2283:∧
2274:¬
2236:¬
2233:¬
2230:∨
2224:¬
2059:Catuṣkoṭi
1881:
1666:⋅
1038:accepted.
808:∀
769:exclusive
743:inclusive
736:exclusive
629:✸2.17 ( ~
583:✸2.14 ~(~
384:falsehood
349:∼
346:∨
331:⊢
315:⋅
309:∗
290:Whitehead
179:Aristotle
150:Aristotle
132:tautology
106:principle
5537:Logicism
5530:timeline
5506:Concrete
5365:Validity
5335:T-schema
5328:Kripke's
5323:Tarski's
5318:semantic
5308:Strength
5257:submodel
5252:spectrum
5220:function
5068:Tarski's
5057:Elements
5044:geometry
5000:Robinson
4921:variable
4906:function
4879:spectrum
4869:Sentence
4825:variable
4768:Language
4721:Relation
4682:Automata
4672:Alphabet
4656:language
4510:-jection
4488:codomain
4474:Function
4435:Universe
4405:Infinite
4309:Relation
4092:Validity
4082:Argument
3980:theorem,
3605:Negation
3432:Validity
3412:Logicism
3141:, 1927,
3130:, 1925,
3120:, 1923,
3004:13 March
2760:(1910),
2659:20 March
2634:20 March
2475:See also
2094:a priori
2082:complete
1951:indirect
1939:infinite
1220:Examples
1128:) → { (~
896:(one of
647:✸2.18 (~
593:✸2.15 (~
530:. Since
237:Also in
70:negation
5479:Related
5276:Diagram
5174: (
5153:Hilbert
5138:Systems
5133:Theorem
5011:of the
4956:systems
4736:Formula
4731:Grammar
4647: (
4591:General
4304:Forcing
4289:Element
4209:Monadic
3984:paradox
3925:Theorem
3861:General
3360:General
3315:in the
3260:Hilbert
3150:Brouwer
3139:Brouwer
3118:Brouwer
3065:(ed.),
3041:(ed.),
2965:2218742
1476:But if
1194:or not-
1171:, then
1081:Brouwer
611:✸2.16 (
577:∨ ~{~(~
425:43–44).
286:Russell
278:theorem
256:Leibniz
249:Leibniz
145:History
56:or the
39:logical
5242:finite
5005:Skolem
4958:
4933:Theory
4901:Symbol
4891:String
4874:atomic
4751:ground
4746:closed
4741:atomic
4697:ground
4660:syntax
4556:binary
4483:domain
4400:Finite
4165:finite
4023:Logics
3982:
3930:Theory
3689:People
3170:
3089:
3081:2000,
2963:
2879:
2716:, c. 9
2684:
2176:cyclic
1965:, not
619:) → (~
601:) → (~
573:✸2.13
559:✸2.12
549:✸2.11
475:✸2.1 ~
343:
337:
328:
325:
216:either
66:either
52:, the
5232:Model
4980:Peano
4837:Proof
4677:Arity
4606:Naive
4493:image
4425:Fuzzy
4385:Empty
4334:union
4279:Class
3920:Model
3910:Lemma
3868:Axiom
3785:Works
3532:Rules
2983:(PDF)
2961:JSTOR
2844:(PDF)
2704:p. 74
2677:Logic
2582:modal
1585:Then
1367:proof
1052:Gödel
930:and ~
841:Flies
839:) ⊕ ~
833:Flies
780:30. (
771:-'or'
759:29. (
713:∨ ~(~
637:) → (
563:→ ~(~
380:truth
113:Latin
111:, in
50:logic
5355:Type
5158:list
4962:list
4939:list
4928:Term
4862:rank
4756:open
4650:list
4462:Maps
4367:sets
4226:Free
4196:list
3946:list
3873:list
3462:Term
3194:and
3180:and
3168:ISBN
3093:pbk.
3087:ISBN
3006:2024
2909:2023
2877:ISBN
2823:and
2682:ISBN
2661:2021
2636:2021
2541:and
2172:+1th
2103:and
2065:and
2057:The
2025:or ~
2008:and
1857:and
1764:and
1551:and
1442:and
1290:and
1136:) →
1109:→ (~
863:and
831:): (
705:for
655:) →
587:) →
510:for
298:as:
288:and
212:both
74:true
5042:of
5024:of
4972:of
4504:Sur
4478:Map
4285:Ur-
4267:Set
3057:",
3053:, "
3029:",
3025:, "
2995:doi
2953:doi
2852:doi
2251:to
2197:).
2044:'s
2036:or
1949:or
1932:By
1872:log
1209:∨ ~
845:pig
837:pig
829:pig
745:-or
738:-or
633:→ ~
623:→ ~
553:∨ ~
522:= ~
498:= ~
450:).
292:in
284:by
280:of
230:∨ ~
206:∨ ~
198:∧ ~
102:law
96:or
72:is
48:In
5587::
5428:NP
5052::
5046::
4976::
4653:),
4508:Bi
4500:In
3294:,
3278:,
3268:,
3258:,
3248:,
3238:,
3228:,
3218:,
3208:,
3198:,
3184:,
3134:,
3109:,
3099:,
2991:51
2989:.
2985:.
2959:.
2949:33
2947:.
2926:.
2900:.
2846:.
2766:,
2756:,
2652:.
2627:.
2073:.
2021::
1242::
1132:→
1124:→
1113:→
847:))
747:.
699:PM
695:PM
669:PM
651:→
641:→
615:→
605:→
597:→
542:∨
534:→
526:∨
518:→
502:∨
494:→
485:PM
479:∨
435:PM
418:PM
392:PM
369:PM
364:.
318:11
234:.
134:.
104:/
64:,
5508:/
5423:P
5178:)
4964:)
4960:(
4857:∀
4852:!
4847:∃
4808:=
4803:↔
4798:→
4793:∧
4788:∨
4783:¬
4506:/
4502:/
4476:/
4287:)
4283:(
4170:∞
4160:3
3948:)
3846:e
3839:t
3832:v
3345:e
3338:t
3331:v
3174:.
3156:,
3008:.
2997::
2967:.
2955::
2932:.
2911:.
2885:.
2858:.
2854::
2827:.
2729:B
2690:.
2663:.
2638:.
2459:)
2456:)
2453:R
2444:P
2441:(
2435:)
2432:Q
2426:P
2423:(
2420:(
2414:)
2411:)
2408:R
2399:Q
2396:(
2390:P
2387:(
2366:)
2363:Q
2354:P
2348:(
2342:)
2339:Q
2333:P
2330:(
2309:Q
2300:P
2289:)
2286:Q
2280:P
2277:(
2239:P
2227:P
2195:n
2188:n
2184:n
2180:n
2170:n
2162:n
2027:P
2023:P
2019:D
2015:P
1999:n
1997:(
1995:P
1991:n
1987:n
1985:(
1979:n
1977:(
1971:n
1969:(
1967:P
1963:n
1959:n
1957:(
1917:3
1914:=
1909:b
1905:a
1884:9
1876:2
1868:=
1865:b
1843:2
1838:=
1835:a
1808:2
1801:2
1795:=
1792:a
1772:b
1752:a
1715:,
1703:2
1700:=
1695:2
1689:2
1683:=
1677:)
1671:2
1661:2
1655:(
1648:2
1642:=
1636:2
1630:)
1624:2
1617:2
1611:(
1606:=
1601:b
1597:a
1581:.
1567:2
1562:=
1559:b
1536:2
1529:2
1523:=
1520:a
1494:2
1487:2
1472:.
1458:2
1453:=
1450:b
1428:2
1423:=
1420:a
1405:.
1390:2
1383:2
1351:2
1323:b
1319:a
1298:b
1278:a
1226:P
1211:A
1207:A
1196:B
1192:B
1188:A
1173:B
1169:A
1165:A
1161:B
1154:A
1150:B
1140:}
1138:B
1134:B
1130:A
1126:B
1122:A
1120:(
1117:)
1115:B
1111:A
1107:A
1059:'
1032:P
1028:S
1024:P
1020:S
1016:P
1012:S
1008:S
1004:P
1000:S
974:…
936:p
932:p
928:p
843:(
835:(
827:(
821:x
782:x
763:)
761:x
715:p
711:p
707:p
703:p
657:p
653:p
649:p
643:p
639:q
635:q
631:p
625:p
621:q
617:q
613:p
607:p
603:q
599:q
595:p
589:p
585:p
579:p
575:p
569:"
565:p
561:p
555:p
551:p
544:p
540:p
536:p
532:p
528:p
524:p
520:p
516:p
512:q
508:p
504:q
500:p
496:q
492:p
481:p
477:p
466:,
412:'
408:"
404:"
400:'
396:'
352:p
340:p
334:.
322:.
312:2
232:P
228:P
208:P
204:P
200:P
196:P
163:,
45:.
34:.
20:)
Text is available under the Creative Commons Attribution-ShareAlike License. Additional terms may apply.