5555:
3436:
130:, 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.
2002:(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
430:
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,
1925:
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
2037:
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,
1167:
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
1065:
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
931:
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
2798:
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
2769:
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
174:
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
871:
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
2001:
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."
210:
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
1136:
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
1715:, 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.
1702:
1050:" 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
663:
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).
413:
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.
1026:
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
936:
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
755:
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
1082:, 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)
962:
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
2140:. 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.
1580:
351:
2770:
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).
2458:
2308:
943:(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:
947:
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
3019:
2124:
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
2050:(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
1015:." (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
2365:
175:"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. (
375:
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)
1810:
1538:
2069:. 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
1241:
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 (
1496:
1392:
2238:
3271:, 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"
2128:, 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
679:, 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).
877:
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)
1844:
1568:
1459:
1429:
1883:
1352:
1078:
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,
807:
164:
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.
1916:
3191:, Cambridge at the University Press 1962 (Second Edition of 1927, reprinted). Extremely difficult because of arcane symbolism, but a must-have for serious logicians.
1324:
1148:
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
1019:
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
379:
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),
1771:
1751:
1297:
1277:
1054:
of the principle ~ (∀A: (A ∨ ~A)) (despite the inconsistency of the assumption ∃ A: ~ (A ∨ ~A))" (Dawson, p. 157) (no closing parenthesis had been placed)
1043:
proposed a solution: "that the negation of a universal proposition was to be understood as asserting the existence … of a counterexample" (Dawson, p. 157)
399:'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
2136:, this type of contradiction is no longer admitted. Furthermore, paradoxes of self reference can be constructed without even invoking negation at all, as in
1978:), but not in general the intuitionistic … the classical meaning, that somewhere in the completed infinite totality of the natural numbers there occurs an
1057:
The debate seemed to weaken: mathematicians, logicians and engineers continue to use the law of excluded middle (and double negation) in their daily work.
3934:
2472: – Foundational controversy in twentieth-century mathematics: an account on the formalist-intuitionist divide around the Law of the excluded middle
3221:, Littlefield, Adams & Co., Totowa, New Jersey, 1974 edition (first published 1968). Includes a wonderful essay on "The Art of drawing Inferences".
158:
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
3251:, Copernicus: Springer–Verlag New York, Inc. 1996, first published 1969. Contains a wealth of biographical information, much derived from interviews.
706:), 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.)
1030:"Taking the Principle of the Excluded Middle from the mathematician," Hilbert said, "is the same as … prohibiting the boxer the use of his fists."
872:
attributes assigned to a concept will never lead to a contradiction, the mathematical existence of the concept is thereby established (Reid p. 34)
211:
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
4609:
1697:{\displaystyle a^{b}=\left({\sqrt {2}}^{\sqrt {2}}\right)^{\sqrt {2}}={\sqrt {2}}^{\left({\sqrt {2}}\cdot {\sqrt {2}}\right)}={\sqrt {2}}^{2}=2}
1033:"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)
3273:
Treatise of Human Nature: Being An attempt to introduce the experimental method of Reasoning into Moral Subjects Vol. I, Of The Understanding
2167:
and "∨" is a "max operator", then the law can be expressed in the object language by (P ∨ ~P ∨ ~~P ∨ ... ∨ ~...~P), where "~...~" represents
292:
952:, in which Russell and Whitehead showed how, via the theory of types: much of arithmetic could be developed by logicist means (Dawson p. 49)
4692:
3833:
2371:
2258:
2638:
969:"pure existence proofs have been the most important landmarks in the historical development of our science," he maintained. (Reid p. 155)
2968:
2779:
This well-known example of a non-constructive proof depending on the law of excluded middle can be found in many places, for example:
2576:
616:) (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.")
3332:
2112:. It is possible in logic to make well-constructed propositions that can be neither true nor false; a common example of this is the "
183:
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 ~(
383:
defines truth and falsehood in terms of a relationship between the "a" and the "b" and the "percipient". For example "This 'a' is 'b
5584:
3532:
1066:
is unverifiable (untestable, unknowable) or from the impossible or the false. (All quotes are from van Heijenoort, italics added).
5006:
1023:, the first alternative is substantiated; but if we never find such a member, the second alternative is still not substantiated.
598:) (One of the four "Principles of transposition". Similar to 1.03, 1.16 and 1.17. A very long demonstration was required here.)
789:
In line (30) the "(x)" means "for all" or "for every", a form used by Russell and Reichenbach; today the symbolism is usually
5164:
2674:
243:
Its usual form, "Every judgment is either true or false" …"(from Kolmogorov in van Heijenoort, p. 421) footnote 9: "This is
3952:
5019:
4342:
1711:
In the above argument, the assertion "this number is either rational or irrational" invokes the law of excluded middle. An
1046:
Gödel's approach to the law of excluded middle was to assert that objections against "the use of 'impredicative definitions
2886:
3305:
2917:
956:
Brouwer reduced the debate to the use of proofs designed from "negative" or "non-existence" versus "constructive" proof:
435:
Russell further described his reasoning behind his definitions of "truth" and "falsehood" in the same book (Chapter XII,
5024:
5014:
4751:
4604:
3957:
2081:. In these systems, the programmer is free to assert the law of excluded middle as a true fact, but it is not built-in
1990:), is not available to him, since he does not conceive the natural numbers as a completed totality. (Kleene 1952:49–50)
3948:
1398:
Clearly (excluded middle) this number is either rational or irrational. If it is rational, the proof is complete, and
5160:
3160:
3079:
2869:
2314:
927:
does not exist", and was thereby invoking the law of excluded middle cast into the form of the law of contradiction.
249:
4502:
3739:
5257:
5001:
3826:
3056:
3032:
2546:
848:
From the late 1800s through the 1930s, a bitter, persistent debate raged between Hilbert and his followers versus
4562:
4255:
2469:
2133:
3996:
2565: – view that a proposition about the future is either necessarily true, or its negation is necessarily true
839:(For all instances of "pig" seen and unseen): ("Pig does fly" or "Pig does not fly" but not both simultaneously)
5518:
5220:
4983:
4978:
4803:
4224:
3908:
3800:
3405:
3360:
1776:
1504:
889:
from the Second International Conference in Paris in 1900) evolved from this debate (italics in the original):
144:
20:
5513:
5296:
5213:
4926:
4857:
4734:
3976:
3790:
3325:
1468:
1364:
231:
3060:, Encyclopædia Britannica, Inc., Chicago, Illinois, 1952. Cited as GB 8. 1st published, W.D. Ross (trans.),
2208:
5438:
5264:
4950:
4584:
4183:
3598:
3486:
3067:
2089:
1355:
457:
Whitehead and Russell derive some of the most powerful tools in the logician's argumentation toolkit. (In
170:
wrote that ambiguity can arise from the use of ambiguous names, but cannot exist in the facts themselves:
66:
5316:
5311:
4921:
4660:
4589:
3918:
3819:
3639:
882:
199:), through distribution of the negation in Aristotle's assertion. The former claims that no statement is
5245:
4835:
4229:
4197:
3888:
3613:
3603:
3506:
2809:
2096:
have also contested the usefulness of the law of excluded middle in the context of modern mathematics.
978:
Brouwer refused to accept the logical principle of the excluded middle, His argument was the following:
2808:
For more about the conflict between the intuitionists (e.g. Brouwer) and the formalists (Hilbert) see
1072:
offers his definition of "principle of excluded middle"; we see here also the issue of "testability":
5535:
5484:
5381:
4879:
4840:
4317:
3962:
3654:
3644:
3395:
3111:
On the significance of the principle of excluded middle in mathematics, especially in function theory
2499:
1164:
is true. Its usual form, "every judgment is either true or false" is equivalent to that given above".
244:
3991:
1252:
An example of an argument that depends on the law of excluded middle follows. We seek to prove that
191:). In modern so called classical logic, this statement is equivalent to the law of excluded middle (
5579:
5376:
5306:
4845:
4697:
4680:
4403:
3883:
3623:
3618:
3608:
3318:
2496: – Splitting of a whole into exactly two non-overlapping parts; dyadic relations and processes
1819:
1543:
1434:
1404:
1849:
5208:
5185:
5146:
5032:
4973:
4619:
4539:
4383:
4327:
3940:
3754:
3664:
3659:
3649:
3501:
3261:, Hyperion, New York, 1993. Fuzzy thinking at its finest but a good introduction to the concepts.
3051:
3043:
3027:
2488:
2483:
2116:", the statement "this statement is false", which is argued to itself be neither true nor false.
1333:
461:
formulas and propositions are identified by a leading asterisk and two numbers, such as "✸2.1".)
160:
70:
3134:, Although not directly germane, in his (1923) Brouwer uses certain words defined in this paper.
1773:
that satisfy the theorem but only two separate possibilities, one of which must work. (Actually
722:
used in his law (3). And this is the point of Reichenbach's demonstration that some believe the
5498:
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5170:
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4909:
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3511:
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3455:
3180:
2742:
2509:
2476:
556:) (Principle of double negation, part 1: if "this rose is red" is true then it's not true that
278:
127:
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792:
5428:
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4527:
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3593:
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3537:
3415:
3102:, Harvard University Press, Cambridge, Massachusetts, 1967. Reprinted with corrections, 1977.
2613:
2175:−1 disjunction signs. It is easy to check that the sentence must receive at least one of the
1998:
1962:). Under both the classical and the intuitionistic logic, by reductio ad absurdum this gives
1888:
939:
886:
452:
283:
31:
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1051:
77:; however, no system of logic is built on just these laws, and none of these laws provides
8:
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5343:
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2149:
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270:
86:
1816:(Constructive proofs of the specific example above are not hard to produce; for example
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4123:
3893:
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3704:
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3465:
3365:
3355:
3211:, Oxford University Press, New York, 1997 edition (first published 1912). Easy to read.
3023:
2949:
2198:
2137:
2109:
2105:
2026:
1756:
1736:
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1262:
932:
were the intuitionists, led by the erstwhile topologist L. E. J. Brouwer (Dawson p. 49)
27:
4512:
3764:
2987:
2595: – Axiom used in logic and philosophy: another way of turning intuition classical
1950:), the classical mathematician may deduce a contradiction from the assumption for all
442:
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5301:
5111:
5101:
4993:
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4709:
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4178:
4143:
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3075:
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2522: – Propositional calculus in which there are more than two truth values such as
2519:
2078:
2047:
1257:
1060:
861:
418:
Russell reiterated his distinction between "sense-datum" and "sensation" in his book
212:
149:
120:
78:
1205:
where ∨ means "or". The equivalence of the two forms is easily proved (p. 421)
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the principle that for every system every property is either correct or impossible
853:
672:
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2065:
Many modern logic systems replace the law of excluded middle with the concept of
116:
74:
3151:
1952 original printing, 1971 6th printing with corrections, 10th printing 1991,
2827:
2589:
school of Buddhism, another system in which the law of excluded middle is untrue
5474:
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3724:
3380:
3244:
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3011:
2932:
Priest, Graham (1983). "The Logical Paradoxes and the Law of Excluded Middle".
2132:: does the set contain, as one of its elements, itself? However, in the modern
2055:
995:." If the set is finite, it is possible—in principle—to examine each member of
3177:, Oxford University Press, Oxford, UK, 1962. Reprinted with corrections, 1975.
2750:
1812:
is irrational but there is no known easy proof of that fact.) (Davis 2000:220)
30:
notation. For a concise description of the symbols used in this notation, see
5573:
5448:
5126:
4633:
4418:
4408:
4378:
4363:
4033:
3749:
3709:
3547:
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2799:
law of excluded middle as applied to arguments over the "completed infinite".
2552:
2523:
2093:
2030:
1994:
902:
To show the significance of this problem, he added the following observation:
155:
3714:
3036:, Encyclopædia Britannica, Inc., Chicago, Illinois, 1952. Cited as GB 19–20.
739:
About this issue (in admittedly very technical terms) Reichenbach observes:
5348:
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3694:
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2121:
2117:
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1942:
existence proofs, which intuitionists do not accept. For example, to prove
1712:
857:
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774:
723:
715:
668:
82:
54:
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5338:
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4058:
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3375:
2782:
2570:
2531:
2179:
2148:
Some systems of logic have different but analogous laws. For some finite
2034:
1236:
Either Socrates is mortal, or it is not the case that Socrates is mortal.
780:
409:
further defines a distinction between a "sense-datum" and a "sensation":
62:
50:
2741:
215:, this is a remarkably precise statement of the law of excluded middle,
4388:
4243:
4214:
4020:
3450:
3264:
3254:
3116:
2953:
2582:
2125:
2059:
1733:
The proof is non-constructive because it doesn't give specific numbers
686:, pp. 101–102). From the law of excluded middle (✸2.1 and ✸2.11),
479:
The proof of ✸2.1 is roughly as follows: "primitive idea" 1.08 defines
346:{\displaystyle \mathbf {*2\cdot 11} .\ \ \vdash .\ p\ \vee \thicksim p}
3201:. The William James Lectures for 1940 delivered at Harvard University.
714:
It is correct, at least for bivalent logic—i.e. it can be seen with a
682:
This principle is commonly called "the principle of double negation" (
5540:
5443:
4496:
4413:
4373:
4337:
4273:
4085:
4075:
4048:
3811:
3425:
3047:
3039:
2756:
2493:
2453:{\displaystyle (P\to (Q\lor \neg R))\to ((P\to Q)\lor (P\to \neg R))}
167:
2945:
2303:{\displaystyle \neg (P\land Q)\,\leftrightarrow \,\neg P\lor \neg Q}
230:, Aristotle seems to deny the law of excluded middle in the case of
5525:
5323:
4771:
4476:
4070:
3400:
3100:
From Frege to Gödel, A Source Book in Mathematical Logic, 1879–1931
2073:. The principle of negation as failure is used as a foundation for
2021:
Putative counterexamples to the law of excluded middle include the
987:"Suppose that A is the statement "There exists a member of the set
899:
of the consistency of the axioms of the arithmetic of real numbers.
143:
The earliest known formulation is in Aristotle's discussion of the
58:
3310:
2912:
1061:
Intuitionist definitions of the law (principle) of excluded middle
1040:
905:"If contradictory attributes be assigned to a concept, I say that
527:
is true (this is Theorem 2.08, which is proved separately), then ~
5121:
3913:
3209:
The Problems of Philosophy, With a New Introduction by John Perry
3138:
3127:
3106:
2244:, which is sometimes called the law of the weak excluded middle.
1970:). The classical logic allows this result to be transformed into
881:
The debate had a profound effect on Hilbert. Reid indicates that
266:
203:
true and false, while the latter requires that any statement is
3072:
Engines of Logic: Mathematicians and the Origin of the Computer
4665:
4011:
3856:
2555: – Type of diagrammatic notation for propositional logic
257:
101:
38:
2108:, the excluded middle has been argued to result in possible
1718:
1190:
footnote 10: "Symbolically the second form is expressed thus
1171:
footnote 9: "This is Leibniz's very simple formulation (see
2058:
as well as the ancient Greek philosophical school known as
812:. Thus an example of the expression would look like this:
3155:, North-Holland Publishing Company, Amsterdam, New York,
2861:"Proof and Knowledge in Mathematics" by Michael Detlefsen
1039:
In his lecture in 1941 at Yale and the subsequent paper,
779:
in which form it would be fully exhaustive and therefore
126:
The principle should not be confused with the semantical
3435:
356:
So just what is "truth" and "falsehood"? At the opening
2587:
Pages displaying short descriptions of redirect targets
2557:
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2542:
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2528:
Pages displaying short descriptions of redirect targets
2514:
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2504:
Pages displaying short descriptions of redirect targets
2029:. Certain resolutions of these paradoxes, particularly
1708:
and 2 is certainly rational. This concludes the proof.
546:(Permutation of the assertions is allowed by axiom 1.4)
2585: – Doctrinal distinction within Tibetan Buddhism
2506:: cases where LEM appears to fail in natural language
2374:
2317:
2261:
2211:
1891:
1852:
1822:
1779:
1759:
1739:
1583:
1546:
1507:
1471:
1437:
1407:
1367:
1336:
1305:
1285:
1265:
795:
667:
Propositions ✸2.12 and ✸2.14, "double negation": The
295:
2639:"Realism – Metaphysical realism and objective truth"
2567:
Pages displaying wikidata descriptions as a fallback
1091:
s definition cites Hilbert's two axioms of negation
690:
derives principle ✸2.12 immediately. We substitute ~
677:
principle of the reciprocity of the multiple species
2479: – Pattern of reasoning in propositional logic
2784:Metamath: A Computer Language for Pure Mathematics
2452:
2359:
2302:
2232:
2192:
1910:
1877:
1838:
1804:
1765:
1745:
1696:
1562:
1532:
1490:
1453:
1423:
1386:
1346:
1318:
1291:
1271:
843:
801:
403:" and this is an undeniable-by-3rd-party "truth".
345:
3090:Logical Dilemmas, The Life and Work of Kurt Gödel
3074:, W. W. Norton & Company, NewYork, New York,
634:) (Another of the "Principles of transposition".)
570:)} (Lemma together with 2.12 used to derive 2.14)
450:From the law of excluded middle, formula ✸2.1 in
5571:
3287:, Vega, London, 2001: a reprint of a portion of
3275:first published 1739, reprinted as: David Hume,
2976:Proceedings of the American Mathematical Society
1187:" has nothing to do with the logic of judgments.
2526: – System including an indeterminate value
2512: – System including an indeterminate value
2360:{\displaystyle (P\to Q)\lor (\neg P\to \neg Q)}
1930:—for them the infinite can never be completed:
1227:then the law of excluded middle holds that the
3259:Fuzzy Thinking: The New Science of Fuzzy Logic
3092:, A.K. Peters, Wellesley, Massachusetts, 1997.
2247:This is equivalent to a few other statements:
867:Hilbert intensely disliked Kronecker's ideas:
656:the hypothesis of its own falsehood is true" (
443:Consequences of the law of excluded middle in
3827:
3326:
2839:. pp. 293–322 (Negation as a failure).
2559:: a graphical syntax for propositional logic
2534: – System for reasoning about vagueness
1885:are both easily shown to be irrational, and
783:in the narrower sense. (Reichenbach, p. 376)
109:. Another Latin designation for this law is
2689:P. T. Geach, The Law of Excluded Middle in
999:and determine whether there is a member of
718:—that this law removes "the middle" of the
179:4.4, W. D. Ross (trans.), GBWW 8, 525–526).
4019:
3834:
3820:
3333:
3319:
3219:The Art of Philosophizing and Other Essays
3132:On the domains of definitions of functions
2966:
1247:it is not the case that Socrates is mortal
3269:An Inquiry Concerning Human Understanding
3020:Fathers of the English Dominican Province
2857:
2573: – Type of formal logic propositions
2284:
2280:
1805:{\displaystyle a={\sqrt {2}}^{\sqrt {2}}}
1719:Non-constructive proofs over the infinite
1533:{\displaystyle a={\sqrt {2}}^{\sqrt {2}}}
907:mathematically the concept does not exist
2182:(and not a value that is not one of the
2099:
895:In his second problem, had asked for a
367:. The "truth-value" of a proposition is
3143:Intuitionistic reflections on formalism
2662:
2189:Other systems reject the law entirely.
2155:, there is an analogous law called the
1491:{\displaystyle {\sqrt {2}}^{\sqrt {2}}}
1387:{\displaystyle {\sqrt {2}}^{\sqrt {2}}}
234:, in his discussion on the sea battle.
5572:
3841:
3064:, Oxford University Press, Oxford, UK.
2931:
2884:
2233:{\displaystyle \neg P\lor \neg \neg P}
580:(Principle of double negation, part 2)
472:"This is the Law of excluded middle" (
422:(1912), published at the same time as
3815:
3314:
2969:"Axiom of Choice and Complementation"
2825:
2569:: the application excluded middle to
1934:In classical mathematics there occur
1918:; a proof allowed by intuitionists).
652:. It states that a proposition which
3279:, Penguin Classics, 1985. Also see:
2540: – Axioms of rational discourse
2518:Law of excluded middle is untrue in
2502: – Semantic property of plurals
360:quickly announces some definitions:
3340:
3306:Stanford Encyclopedia of Philosophy
3121:On the principle of excluded middle
2918:Internet Encyclopedia of Philosophy
2885:Priest, Graham (28 November 2010).
2858:Detlefsen, Michael (January 1992).
1729:proof disallowed by intuitionists:
1723:The above proof is an example of a
13:
2780:
2438:
2393:
2348:
2339:
2294:
2285:
2262:
2224:
2221:
2212:
796:
773:), where the symbol "⊕" signifies
14:
5596:
3295:
3199:An Inquiry Into Meaning and Truth
2988:10.1090/S0002-9939-1975-0373893-X
2143:
560:'this rose is not-red' is true".)
395:'object a' is a sense-datum" and
337:
5553:
3434:
3057:Great Books of the Western World
3033:Great Books of the Western World
2967:Diaconescu, Radu (August 1975).
2547:Limited principle of omniscience
2171:−1 negation signs and "∨ ... ∨"
923:are both shown to be true, then
387:" (e.g. "This 'object a' is 'red
306:
300:
297:
247:'s very simple formulation (see
5585:Theorems in propositional logic
3153:Introduction to Metamathematics
2960:
2925:
2904:
2878:
2851:
2819:
2802:
2792:
2789:and Davis 2000:220, footnote 2.
2773:
2193:Law of the weak excluded middle
856:. Brouwer's philosophy, called
844:Formalists versus Intuitionists
89:. The law is also known as the
3801:Tractatus Logico-Philosophicus
3406:Problem of multiple generality
3231:, Dover, New York, 1947, 1975.
2763:
2735:
2723:
2708:
2696:
2683:
2656:
2631:
2606:
2581:Non-affirming negation in the
2447:
2444:
2435:
2429:
2423:
2417:
2411:
2408:
2405:
2402:
2399:
2384:
2381:
2375:
2354:
2345:
2336:
2330:
2324:
2318:
2281:
2277:
2265:
915:Thus, Hilbert was saying: "If
709:
265:The principle was stated as a
145:principle of non-contradiction
21:fallacy of the excluded middle
1:
5514:History of mathematical logic
3791:The Principles of Mathematics
3005:
2041:
1972:there exists an n such that P
1944:there exists an n such that P
1839:{\displaystyle a={\sqrt {2}}}
1563:{\displaystyle b={\sqrt {2}}}
1454:{\displaystyle b={\sqrt {2}}}
1424:{\displaystyle a={\sqrt {2}}}
729:should take the place of the
5439:Primitive recursive function
3487:Commutativity of conjunction
3189:Principia Mathematica to *56
2845:10.1007/978-1-4684-3384-5_11
2599:
2549: – Mathematical concept
2197:A particularly well-studied
1878:{\displaystyle b=\log _{2}9}
966:Hilbert naturally disagreed.
675:refer to what he calls "the
138:
115:or "no third is given". In
47:principle of excluded middle
7:
2934:The Philosophical Quarterly
2577:Mathematical constructivism
2470:Brouwer–Hilbert controversy
2463:
2134:Zermelo–Fraenkel set theory
1347:{\displaystyle {\sqrt {2}}}
1208:
648:(Called "The complement of
26:This article uses forms of
10:
5601:
4503:Schröder–Bernstein theorem
4230:Monadic predicate calculus
3889:Foundations of mathematics
3507:Monotonicity of entailment
3277:A Treatise of Human Nature
3229:Elements of Symbolic Logic
2810:Foundations of mathematics
2669:. Routledge. p. 124.
2090:L. E. J. Brouwer
1175:, IV,2). The formulation "
860:, started in earnest with
420:The Problems of Philosophy
237:
154:where he says that of two
133:
25:
18:
5549:
5536:Philosophy of mathematics
5485:Automated theorem proving
5467:
5362:
5194:
5087:
4939:
4656:
4632:
4610:Von Neumann–Bernays–Gödel
4555:
4449:
4353:
4251:
4242:
4169:
4104:
4010:
3932:
3849:
3773:
3677:
3632:
3586:
3520:
3479:
3443:
3432:
3396:Idempotency of entailment
3348:
2500:Homogeneity (linguistics)
2006:concerning infinite sets
106:principium tertii exclusi
3241:, WCB McGraw–Hill, 1997.
3175:The Development of Logic
2077:, and is widely used in
2038:or both true and false.
1498:is irrational, then let
1007:or that every member of
883:Hilbert's second problem
802:{\displaystyle \forall }
431:not a sensation. (p. 12)
57:this proposition or its
19:Not to be confused with
5186:Self-verifying theories
5007:Tarski's axiomatization
3958:Tarski's undefinability
3953:incompleteness theorems
3755:Willard Van Orman Quine
3291:starts on p. 94 ff
3052:Robert Maynard Hutchins
3028:Robert Maynard Hutchins
2643:Encyclopedia Britannica
2618:Encyclopedia Britannica
2484:Constructive set theory
2205:, which adds the axiom
2088:Mathematicians such as
1911:{\displaystyle a^{b}=3}
1358:). Consider the number
1093:
71:law of noncontradiction
5560:Mathematics portal
5171:Proof of impossibility
4819:propositional variable
4129:Propositional calculus
3730:Charles Sanders Peirce
3573:Hypothetical syllogism
3181:Alfred North Whitehead
3062:The Works of Aristotle
3026:(ed.), vols. 19–20 in
2743:Alfred North Whitehead
2663:Tomassi, Paul (1999).
2510:Law of excluded fourth
2477:Consequentia mirabilis
2454:
2361:
2304:
2234:
1992:
1912:
1879:
1840:
1814:
1806:
1767:
1747:
1698:
1564:
1534:
1492:
1455:
1425:
1388:
1348:
1320:
1293:
1273:
1145:is regarded as false …
954:
934:
879:
874:
803:
660:, pp. 103–104).)
459:Principia Mathematica,
433:
416:
377:
347:
255:
181:
128:principle of bivalence
49:states that for every
43:law of excluded middle
5429:Kolmogorov complexity
5382:Computably enumerable
5282:Model complete theory
5074:Principia Mathematica
4134:Propositional formula
3963:Banach–Tarski paradox
3796:Principia Mathematica
3568:Disjunctive syllogism
3553:modus ponendo tollens
3302:"Contradiction" entry
3137:Luitzen Egbertus Jan
3126:Luitzen Egbertus Jan
3105:Luitzen Egbertus Jan
2826:Clark, Keith (1978).
2752:Principia Mathematica
2455:
2362:
2305:
2235:
2100:In mathematical logic
1999:Luitzen E. J. Brouwer
1932:
1926:when extended to the
1913:
1880:
1841:
1807:
1768:
1748:
1731:
1699:
1565:
1535:
1493:
1456:
1426:
1389:
1349:
1321:
1319:{\displaystyle a^{b}}
1294:
1274:
1141:if the true judgment
950:Principia Mathematica
945:
940:Principia Mathematica
929:
875:
869:
804:
745:The tertium non datur
453:Principia Mathematica
445:Principia Mathematica
428:
411:
362:
348:
284:Principia Mathematica
260:Principia Mathematica
258:Bertrand Russell and
253:, IV,2)" (ibid p 421)
241:
172:
98:of the excluded third
67:three laws of thought
32:List of logic symbols
5377:Church–Turing thesis
5364:Computability theory
4573:continuum hypothesis
4091:Square of opposition
3949:Gödel's completeness
3786:Function and Concept
3558:Constructive dilemma
3533:Material implication
3050:(trans.), vol. 8 in
2829:Logic and Data Bases
2787:. footnote on p. 17.
2489:Diaconescu's theorem
2372:
2315:
2259:
2242:intuitionistic logic
2209:
2085:into these systems.
2018:" (Kleene 1952:48).
1964:not for all n, not P
1889:
1850:
1820:
1777:
1757:
1737:
1581:
1544:
1505:
1469:
1435:
1405:
1365:
1334:
1303:
1283:
1263:
1217:is the proposition:
1052:intuitionistic logic
991:having the property
793:
650:reductio ad absurdum
503:in this rule yields
293:
147:, first proposed in
5531:Mathematical object
5422:P versus NP problem
5387:Computable function
5181:Reverse mathematics
5107:Logical consequence
4984:primitive recursive
4979:elementary function
4752:Free/bound variable
4605:Tarski–Grothendieck
4124:Logical connectives
4054:Logical equivalence
3904:Logical consequence
3760:Ludwig Wittgenstein
3563:Destructive dilemma
3391:Well-formed formula
3115:Andrei Nikolaevich
2913:"Russell's Paradox"
2887:"Paradoxical Truth"
2563:Logical determinism
2075:autoepistemic logic
2067:negation as failure
1354:is irrational (see
1245:) or its negation (
1229:logical disjunction
1222:Socrates is mortal.
1011:lacks the property
864:in the late 1800s.
437:Truth and Falsehood
271:propositional logic
65:. It is one of the
5329:Transfer principle
5292:Semantics of logic
5277:Categorical theory
5253:Non-standard model
4767:Logical connective
3894:Information theory
3843:Mathematical logic
3705:Augustus De Morgan
3285:The Vision of Hume
3096:van Heijenoort, J.
3024:Daniel J. Sullivan
2910:Kevin C. Klement,
2450:
2357:
2300:
2251:Satisfying all of
2230:
2199:intermediate logic
2110:self-contradiction
2106:mathematical logic
1908:
1875:
1836:
1802:
1763:
1743:
1694:
1560:
1530:
1488:
1451:
1421:
1384:
1344:
1316:
1289:
1269:
1258:irrational numbers
1243:Socrates is mortal
1168:considered obvious
1003:with the property
897:mathematical proof
887:Hilbert's problems
799:
698:in 2.11 to yield ~
371:if it is true and
343:
232:future contingents
5567:
5566:
5499:Abstract category
5302:Theories of truth
5112:Rule of inference
5102:Natural deduction
5083:
5082:
4628:
4627:
4333:Cartesian product
4238:
4237:
4144:Many-valued logic
4119:Boolean functions
4002:Russell's paradox
3977:diagonal argument
3874:First-order logic
3809:
3808:
3673:
3672:
3149:Stephen C. Kleene
2703:On Interpretation
2676:978-0-415-16696-6
2614:"Laws of thought"
2520:many-valued logic
2163:. If negation is
2130:Russell's paradox
2079:logic programming
1834:
1799:
1792:
1766:{\displaystyle b}
1746:{\displaystyle a}
1680:
1662:
1652:
1639:
1627:
1615:
1608:
1558:
1527:
1520:
1485:
1478:
1449:
1419:
1381:
1374:
1342:
1330:It is known that
1292:{\displaystyle b}
1272:{\displaystyle a}
1156:as well as from ~
862:Leopold Kronecker
333:
327:
318:
315:
228:On Interpretation
213:traditional logic
150:On Interpretation
112:tertium non datur
69:, along with the
5592:
5558:
5557:
5509:History of logic
5504:Category of sets
5397:Decision problem
5176:Ordinal analysis
5117:Sequent calculus
5015:Boolean algebras
4955:
4954:
4929:
4900:logical/constant
4654:
4653:
4640:
4563:Zermelo–Fraenkel
4314:Set operations:
4249:
4248:
4186:
4017:
4016:
3997:Löwenheim–Skolem
3884:Formal semantics
3836:
3829:
3822:
3813:
3812:
3745:Henry M. Sheffer
3735:Bertrand Russell
3700:Richard Dedekind
3584:
3583:
3528:De Morgan's laws
3502:Noncontradiction
3444:Classical logics
3438:
3335:
3328:
3321:
3312:
3311:
3239:Machine Learning
3225:Hans Reichenbach
3215:Bertrand Russell
3205:Bertrand Russell
3195:Bertrand Russell
3185:Bertrand Russell
3016:Summa Theologica
2999:
2998:
2996:
2994:
2973:
2964:
2958:
2957:
2940:(131): 160–165.
2929:
2923:
2922:
2908:
2902:
2901:
2899:
2897:
2882:
2876:
2875:
2855:
2849:
2848:
2834:
2823:
2817:
2806:
2800:
2796:
2790:
2788:
2781:Megill, Norman.
2777:
2771:
2767:
2761:
2760:
2747:Bertrand Russell
2739:
2733:
2732:Γ 7, 1011b 26–27
2727:
2721:
2712:
2706:
2700:
2694:
2687:
2681:
2680:
2660:
2654:
2653:
2651:
2649:
2635:
2629:
2628:
2626:
2624:
2610:
2588:
2568:
2558:
2543:
2529:
2515:
2505:
2459:
2457:
2456:
2451:
2366:
2364:
2363:
2358:
2309:
2307:
2306:
2301:
2253:De Morgan's laws
2239:
2237:
2236:
2231:
2157:law of excluded
2120:has argued that
1936:non-constructive
1923:non-constructive
1917:
1915:
1914:
1909:
1901:
1900:
1884:
1882:
1881:
1876:
1868:
1867:
1845:
1843:
1842:
1837:
1835:
1830:
1811:
1809:
1808:
1803:
1801:
1800:
1795:
1793:
1788:
1772:
1770:
1769:
1764:
1752:
1750:
1749:
1744:
1726:non-constructive
1703:
1701:
1700:
1695:
1687:
1686:
1681:
1676:
1670:
1669:
1668:
1664:
1663:
1658:
1653:
1648:
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1635:
1629:
1628:
1623:
1621:
1617:
1616:
1611:
1609:
1604:
1593:
1592:
1569:
1567:
1566:
1561:
1559:
1554:
1539:
1537:
1536:
1531:
1529:
1528:
1523:
1521:
1516:
1497:
1495:
1494:
1489:
1487:
1486:
1481:
1479:
1474:
1460:
1458:
1457:
1452:
1450:
1445:
1430:
1428:
1427:
1422:
1420:
1415:
1393:
1391:
1390:
1385:
1383:
1382:
1377:
1375:
1370:
1353:
1351:
1350:
1345:
1343:
1338:
1325:
1323:
1322:
1317:
1315:
1314:
1298:
1296:
1295:
1290:
1278:
1276:
1275:
1270:
1256:there exist two
1249:) must be true.
1213:For example, if
1049:
854:L. E. J. Brouwer
808:
806:
805:
800:
673:L. E. J. Brouwer
559:
476:, p. 101).
402:
398:
394:
391:") really means
390:
386:
352:
350:
349:
344:
331:
325:
316:
313:
309:
87:De Morgan's laws
5600:
5599:
5595:
5594:
5593:
5591:
5590:
5589:
5580:Classical logic
5570:
5569:
5568:
5563:
5552:
5545:
5490:Category theory
5480:Algebraic logic
5463:
5434:Lambda calculus
5372:Church encoding
5358:
5334:Truth predicate
5190:
5156:Complete theory
5079:
4948:
4944:
4940:
4935:
4927:
4647: and
4643:
4638:
4624:
4600:New Foundations
4568:axiom of choice
4551:
4513:Gödel numbering
4453: and
4445:
4349:
4234:
4184:
4165:
4114:Boolean algebra
4100:
4064:Equiconsistency
4029:Classical logic
4006:
3987:Halting problem
3975: and
3951: and
3939: and
3938:
3933:Theorems (
3928:
3845:
3840:
3810:
3805:
3781:Begriffsschrift
3769:
3765:Jan Łukasiewicz
3685:Bernard Bolzano
3669:
3640:Double negation
3628:
3599:Double negation
3582:
3516:
3492:Excluded middle
3475:
3439:
3430:
3344:
3342:Classical logic
3339:
3298:
3281:David Applebaum
3012:Aquinas, Thomas
3008:
3003:
3002:
2992:
2990:
2971:
2965:
2961:
2946:10.2307/2218742
2930:
2926:
2911:
2909:
2905:
2895:
2893:
2883:
2879:
2872:
2856:
2852:
2837:Springer-Verlag
2832:
2824:
2820:
2807:
2803:
2797:
2793:
2778:
2774:
2768:
2764:
2740:
2736:
2728:
2724:
2713:
2709:
2701:
2697:
2688:
2684:
2677:
2661:
2657:
2647:
2645:
2637:
2636:
2632:
2622:
2620:
2612:
2611:
2607:
2602:
2586:
2566:
2556:
2541:
2538:Laws of thought
2527:
2513:
2503:
2466:
2373:
2370:
2369:
2316:
2313:
2312:
2260:
2257:
2256:
2210:
2207:
2206:
2203:De Morgan logic
2195:
2146:
2138:Curry's paradox
2102:
2044:
2027:Quine's paradox
1896:
1892:
1890:
1887:
1886:
1863:
1859:
1851:
1848:
1847:
1829:
1821:
1818:
1817:
1794:
1787:
1786:
1778:
1775:
1774:
1758:
1755:
1754:
1738:
1735:
1734:
1721:
1682:
1675:
1674:
1657:
1647:
1646:
1642:
1641:
1634:
1633:
1622:
1610:
1603:
1602:
1598:
1597:
1588:
1584:
1582:
1579:
1578:
1553:
1545:
1542:
1541:
1522:
1515:
1514:
1506:
1503:
1502:
1480:
1473:
1472:
1470:
1467:
1466:
1444:
1436:
1433:
1432:
1414:
1406:
1403:
1402:
1376:
1369:
1368:
1366:
1363:
1362:
1337:
1335:
1332:
1331:
1310:
1306:
1304:
1301:
1300:
1284:
1281:
1280:
1264:
1261:
1260:
1211:
1173:Nouveaux Essais
1063:
1047:
846:
794:
791:
790:
712:
635:
617:
599:
581:
571:
561:
557:
547:
495:. Substituting
448:
400:
396:
392:
388:
384:
296:
294:
291:
290:
263:
250:Nouveaux Essais
240:
207:true or false.
141:
136:
119:, the law is a
117:classical logic
79:inference rules
75:law of identity
35:
24:
17:
12:
11:
5:
5598:
5588:
5587:
5582:
5565:
5564:
5550:
5547:
5546:
5544:
5543:
5538:
5533:
5528:
5523:
5522:
5521:
5511:
5506:
5501:
5492:
5487:
5482:
5477:
5475:Abstract logic
5471:
5469:
5465:
5464:
5462:
5461:
5456:
5454:Turing machine
5451:
5446:
5441:
5436:
5431:
5426:
5425:
5424:
5419:
5414:
5409:
5404:
5394:
5392:Computable set
5389:
5384:
5379:
5374:
5368:
5366:
5360:
5359:
5357:
5356:
5351:
5346:
5341:
5336:
5331:
5326:
5321:
5320:
5319:
5314:
5309:
5299:
5294:
5289:
5287:Satisfiability
5284:
5279:
5274:
5273:
5272:
5262:
5261:
5260:
5250:
5249:
5248:
5243:
5238:
5233:
5228:
5218:
5217:
5216:
5211:
5204:Interpretation
5200:
5198:
5192:
5191:
5189:
5188:
5183:
5178:
5173:
5168:
5158:
5153:
5152:
5151:
5150:
5149:
5139:
5134:
5124:
5119:
5114:
5109:
5104:
5099:
5093:
5091:
5085:
5084:
5081:
5080:
5078:
5077:
5069:
5068:
5067:
5066:
5061:
5060:
5059:
5054:
5049:
5029:
5028:
5027:
5025:minimal axioms
5022:
5011:
5010:
5009:
4998:
4997:
4996:
4991:
4986:
4981:
4976:
4971:
4958:
4956:
4937:
4936:
4934:
4933:
4932:
4931:
4919:
4914:
4913:
4912:
4907:
4902:
4897:
4887:
4882:
4877:
4872:
4871:
4870:
4865:
4855:
4854:
4853:
4848:
4843:
4838:
4828:
4823:
4822:
4821:
4816:
4811:
4801:
4800:
4799:
4794:
4789:
4784:
4779:
4774:
4764:
4759:
4754:
4749:
4748:
4747:
4742:
4737:
4732:
4722:
4717:
4715:Formation rule
4712:
4707:
4706:
4705:
4700:
4690:
4689:
4688:
4678:
4673:
4668:
4663:
4657:
4651:
4634:Formal systems
4630:
4629:
4626:
4625:
4623:
4622:
4617:
4612:
4607:
4602:
4597:
4592:
4587:
4582:
4577:
4576:
4575:
4570:
4559:
4557:
4553:
4552:
4550:
4549:
4548:
4547:
4537:
4532:
4531:
4530:
4523:Large cardinal
4520:
4515:
4510:
4505:
4500:
4486:
4485:
4484:
4479:
4474:
4459:
4457:
4447:
4446:
4444:
4443:
4442:
4441:
4436:
4431:
4421:
4416:
4411:
4406:
4401:
4396:
4391:
4386:
4381:
4376:
4371:
4366:
4360:
4358:
4351:
4350:
4348:
4347:
4346:
4345:
4340:
4335:
4330:
4325:
4320:
4312:
4311:
4310:
4305:
4295:
4290:
4288:Extensionality
4285:
4283:Ordinal number
4280:
4270:
4265:
4264:
4263:
4252:
4246:
4240:
4239:
4236:
4235:
4233:
4232:
4227:
4222:
4217:
4212:
4207:
4202:
4201:
4200:
4190:
4189:
4188:
4175:
4173:
4167:
4166:
4164:
4163:
4162:
4161:
4156:
4151:
4141:
4136:
4131:
4126:
4121:
4116:
4110:
4108:
4102:
4101:
4099:
4098:
4093:
4088:
4083:
4078:
4073:
4068:
4067:
4066:
4056:
4051:
4046:
4041:
4036:
4031:
4025:
4023:
4014:
4008:
4007:
4005:
4004:
3999:
3994:
3989:
3984:
3979:
3967:Cantor's
3965:
3960:
3955:
3945:
3943:
3930:
3929:
3927:
3926:
3921:
3916:
3911:
3906:
3901:
3896:
3891:
3886:
3881:
3876:
3871:
3866:
3865:
3864:
3853:
3851:
3847:
3846:
3839:
3838:
3831:
3824:
3816:
3807:
3806:
3804:
3803:
3798:
3793:
3788:
3783:
3777:
3775:
3771:
3770:
3768:
3767:
3762:
3757:
3752:
3747:
3742:
3740:Ernst Schröder
3737:
3732:
3727:
3725:Giuseppe Peano
3722:
3717:
3712:
3707:
3702:
3697:
3692:
3687:
3681:
3679:
3675:
3674:
3671:
3670:
3668:
3667:
3662:
3657:
3652:
3647:
3642:
3636:
3634:
3630:
3629:
3627:
3626:
3621:
3616:
3611:
3606:
3601:
3596:
3590:
3588:
3581:
3580:
3575:
3570:
3565:
3560:
3555:
3550:
3545:
3540:
3535:
3530:
3524:
3522:
3518:
3517:
3515:
3514:
3509:
3504:
3499:
3494:
3489:
3483:
3481:
3477:
3476:
3474:
3473:
3468:
3463:
3458:
3453:
3447:
3445:
3441:
3440:
3433:
3431:
3429:
3428:
3423:
3418:
3413:
3408:
3403:
3398:
3393:
3388:
3383:
3381:Truth function
3378:
3373:
3368:
3363:
3358:
3352:
3350:
3346:
3345:
3338:
3337:
3330:
3323:
3315:
3309:
3308:
3297:
3296:External links
3294:
3293:
3292:
3262:
3252:
3245:Constance Reid
3242:
3232:
3222:
3212:
3202:
3192:
3178:
3164:
3146:
3135:
3124:
3113:
3103:
3093:
3083:
3065:
3037:
3007:
3004:
3001:
3000:
2982:(1): 176–178.
2959:
2924:
2903:
2877:
2870:
2850:
2818:
2801:
2791:
2772:
2762:
2734:
2722:
2707:
2695:
2691:Logic Matters,
2682:
2675:
2655:
2630:
2604:
2603:
2601:
2598:
2597:
2596:
2590:
2579:
2574:
2560:
2550:
2544:
2535:
2516:
2507:
2497:
2491:
2486:
2481:
2473:
2465:
2462:
2461:
2460:
2449:
2446:
2443:
2440:
2437:
2434:
2431:
2428:
2425:
2422:
2419:
2416:
2413:
2410:
2407:
2404:
2401:
2398:
2395:
2392:
2389:
2386:
2383:
2380:
2377:
2367:
2356:
2353:
2350:
2347:
2344:
2341:
2338:
2335:
2332:
2329:
2326:
2323:
2320:
2310:
2299:
2296:
2293:
2290:
2287:
2283:
2279:
2276:
2273:
2270:
2267:
2264:
2229:
2226:
2223:
2220:
2217:
2214:
2194:
2191:
2153:-valued logics
2145:
2144:Analogous laws
2142:
2114:Liar's paradox
2101:
2098:
2056:Buddhist logic
2043:
2040:
1907:
1904:
1899:
1895:
1874:
1871:
1866:
1862:
1858:
1855:
1833:
1828:
1825:
1798:
1791:
1785:
1782:
1762:
1742:
1720:
1717:
1706:
1705:
1693:
1690:
1685:
1679:
1673:
1667:
1661:
1656:
1651:
1645:
1638:
1632:
1626:
1620:
1614:
1607:
1601:
1596:
1591:
1587:
1572:
1571:
1557:
1552:
1549:
1526:
1519:
1513:
1510:
1484:
1477:
1463:
1462:
1448:
1443:
1440:
1418:
1413:
1410:
1396:
1395:
1380:
1373:
1341:
1328:
1327:
1313:
1309:
1288:
1268:
1239:
1238:
1225:
1224:
1210:
1207:
1203:
1202:
1193:
1192:
1191:
1188:
1169:
1165:
1146:
1131:
1130:
1107:
1086:
1085:
1084:
1083:
1062:
1059:
1037:
1036:
1035:
1034:
1031:
1028:
1024:
982:
981:
980:
979:
973:
972:
971:
970:
967:
964:
913:
912:
911:
910:
909:" (Reid p. 71)
903:
900:
845:
842:
841:
840:
837:
798:
787:
786:
785:
784:
777:
764:
763:
762:
761:
753:
746:
711:
708:
535:must be true.
447:
441:
342:
339:
336:
330:
324:
321:
312:
308:
305:
302:
299:
262:
256:
239:
236:
140:
137:
135:
132:
15:
9:
6:
4:
3:
2:
5597:
5586:
5583:
5581:
5578:
5577:
5575:
5562:
5561:
5556:
5548:
5542:
5539:
5537:
5534:
5532:
5529:
5527:
5524:
5520:
5517:
5516:
5515:
5512:
5510:
5507:
5505:
5502:
5500:
5496:
5493:
5491:
5488:
5486:
5483:
5481:
5478:
5476:
5473:
5472:
5470:
5466:
5460:
5457:
5455:
5452:
5450:
5449:Recursive set
5447:
5445:
5442:
5440:
5437:
5435:
5432:
5430:
5427:
5423:
5420:
5418:
5415:
5413:
5410:
5408:
5405:
5403:
5400:
5399:
5398:
5395:
5393:
5390:
5388:
5385:
5383:
5380:
5378:
5375:
5373:
5370:
5369:
5367:
5365:
5361:
5355:
5352:
5350:
5347:
5345:
5342:
5340:
5337:
5335:
5332:
5330:
5327:
5325:
5322:
5318:
5315:
5313:
5310:
5308:
5305:
5304:
5303:
5300:
5298:
5295:
5293:
5290:
5288:
5285:
5283:
5280:
5278:
5275:
5271:
5268:
5267:
5266:
5263:
5259:
5258:of arithmetic
5256:
5255:
5254:
5251:
5247:
5244:
5242:
5239:
5237:
5234:
5232:
5229:
5227:
5224:
5223:
5222:
5219:
5215:
5212:
5210:
5207:
5206:
5205:
5202:
5201:
5199:
5197:
5193:
5187:
5184:
5182:
5179:
5177:
5174:
5172:
5169:
5166:
5165:from ZFC
5162:
5159:
5157:
5154:
5148:
5145:
5144:
5143:
5140:
5138:
5135:
5133:
5130:
5129:
5128:
5125:
5123:
5120:
5118:
5115:
5113:
5110:
5108:
5105:
5103:
5100:
5098:
5095:
5094:
5092:
5090:
5086:
5076:
5075:
5071:
5070:
5065:
5064:non-Euclidean
5062:
5058:
5055:
5053:
5050:
5048:
5047:
5043:
5042:
5040:
5037:
5036:
5034:
5030:
5026:
5023:
5021:
5018:
5017:
5016:
5012:
5008:
5005:
5004:
5003:
4999:
4995:
4992:
4990:
4987:
4985:
4982:
4980:
4977:
4975:
4972:
4970:
4967:
4966:
4964:
4960:
4959:
4957:
4952:
4946:
4941:Example
4938:
4930:
4925:
4924:
4923:
4920:
4918:
4915:
4911:
4908:
4906:
4903:
4901:
4898:
4896:
4893:
4892:
4891:
4888:
4886:
4883:
4881:
4878:
4876:
4873:
4869:
4866:
4864:
4861:
4860:
4859:
4856:
4852:
4849:
4847:
4844:
4842:
4839:
4837:
4834:
4833:
4832:
4829:
4827:
4824:
4820:
4817:
4815:
4812:
4810:
4807:
4806:
4805:
4802:
4798:
4795:
4793:
4790:
4788:
4785:
4783:
4780:
4778:
4775:
4773:
4770:
4769:
4768:
4765:
4763:
4760:
4758:
4755:
4753:
4750:
4746:
4743:
4741:
4738:
4736:
4733:
4731:
4728:
4727:
4726:
4723:
4721:
4718:
4716:
4713:
4711:
4708:
4704:
4701:
4699:
4698:by definition
4696:
4695:
4694:
4691:
4687:
4684:
4683:
4682:
4679:
4677:
4674:
4672:
4669:
4667:
4664:
4662:
4659:
4658:
4655:
4652:
4650:
4646:
4641:
4635:
4631:
4621:
4618:
4616:
4613:
4611:
4608:
4606:
4603:
4601:
4598:
4596:
4593:
4591:
4588:
4586:
4585:Kripke–Platek
4583:
4581:
4578:
4574:
4571:
4569:
4566:
4565:
4564:
4561:
4560:
4558:
4554:
4546:
4543:
4542:
4541:
4538:
4536:
4533:
4529:
4526:
4525:
4524:
4521:
4519:
4516:
4514:
4511:
4509:
4506:
4504:
4501:
4498:
4494:
4490:
4487:
4483:
4480:
4478:
4475:
4473:
4470:
4469:
4468:
4464:
4461:
4460:
4458:
4456:
4452:
4448:
4440:
4437:
4435:
4432:
4430:
4429:constructible
4427:
4426:
4425:
4422:
4420:
4417:
4415:
4412:
4410:
4407:
4405:
4402:
4400:
4397:
4395:
4392:
4390:
4387:
4385:
4382:
4380:
4377:
4375:
4372:
4370:
4367:
4365:
4362:
4361:
4359:
4357:
4352:
4344:
4341:
4339:
4336:
4334:
4331:
4329:
4326:
4324:
4321:
4319:
4316:
4315:
4313:
4309:
4306:
4304:
4301:
4300:
4299:
4296:
4294:
4291:
4289:
4286:
4284:
4281:
4279:
4275:
4271:
4269:
4266:
4262:
4259:
4258:
4257:
4254:
4253:
4250:
4247:
4245:
4241:
4231:
4228:
4226:
4223:
4221:
4218:
4216:
4213:
4211:
4208:
4206:
4203:
4199:
4196:
4195:
4194:
4191:
4187:
4182:
4181:
4180:
4177:
4176:
4174:
4172:
4168:
4160:
4157:
4155:
4152:
4150:
4147:
4146:
4145:
4142:
4140:
4137:
4135:
4132:
4130:
4127:
4125:
4122:
4120:
4117:
4115:
4112:
4111:
4109:
4107:
4106:Propositional
4103:
4097:
4094:
4092:
4089:
4087:
4084:
4082:
4079:
4077:
4074:
4072:
4069:
4065:
4062:
4061:
4060:
4057:
4055:
4052:
4050:
4047:
4045:
4042:
4040:
4037:
4035:
4034:Logical truth
4032:
4030:
4027:
4026:
4024:
4022:
4018:
4015:
4013:
4009:
4003:
4000:
3998:
3995:
3993:
3990:
3988:
3985:
3983:
3980:
3978:
3974:
3970:
3966:
3964:
3961:
3959:
3956:
3954:
3950:
3947:
3946:
3944:
3942:
3936:
3931:
3925:
3922:
3920:
3917:
3915:
3912:
3910:
3907:
3905:
3902:
3900:
3897:
3895:
3892:
3890:
3887:
3885:
3882:
3880:
3877:
3875:
3872:
3870:
3867:
3863:
3860:
3859:
3858:
3855:
3854:
3852:
3848:
3844:
3837:
3832:
3830:
3825:
3823:
3818:
3817:
3814:
3802:
3799:
3797:
3794:
3792:
3789:
3787:
3784:
3782:
3779:
3778:
3776:
3772:
3766:
3763:
3761:
3758:
3756:
3753:
3751:
3750:Alfred Tarski
3748:
3746:
3743:
3741:
3738:
3736:
3733:
3731:
3728:
3726:
3723:
3721:
3718:
3716:
3713:
3711:
3710:Gottlob Frege
3708:
3706:
3703:
3701:
3698:
3696:
3693:
3691:
3688:
3686:
3683:
3682:
3680:
3676:
3666:
3663:
3661:
3658:
3656:
3655:Biconditional
3653:
3651:
3648:
3646:
3643:
3641:
3638:
3637:
3635:
3631:
3625:
3622:
3620:
3617:
3615:
3614:Biconditional
3612:
3610:
3607:
3605:
3602:
3600:
3597:
3595:
3592:
3591:
3589:
3585:
3579:
3576:
3574:
3571:
3569:
3566:
3564:
3561:
3559:
3556:
3554:
3551:
3549:
3548:modus tollens
3546:
3544:
3541:
3539:
3538:Transposition
3536:
3534:
3531:
3529:
3526:
3525:
3523:
3519:
3513:
3510:
3508:
3505:
3503:
3500:
3498:
3495:
3493:
3490:
3488:
3485:
3484:
3482:
3478:
3472:
3469:
3467:
3464:
3462:
3459:
3457:
3456:Propositional
3454:
3452:
3449:
3448:
3446:
3442:
3437:
3427:
3424:
3422:
3419:
3417:
3414:
3412:
3411:Associativity
3409:
3407:
3404:
3402:
3399:
3397:
3394:
3392:
3389:
3387:
3384:
3382:
3379:
3377:
3374:
3372:
3369:
3367:
3364:
3362:
3359:
3357:
3354:
3353:
3351:
3347:
3343:
3336:
3331:
3329:
3324:
3322:
3317:
3316:
3313:
3307:
3303:
3300:
3299:
3290:
3286:
3282:
3278:
3274:
3270:
3266:
3263:
3260:
3256:
3253:
3250:
3246:
3243:
3240:
3236:
3233:
3230:
3226:
3223:
3220:
3216:
3213:
3210:
3206:
3203:
3200:
3196:
3193:
3190:
3186:
3182:
3179:
3176:
3172:
3168:
3165:
3162:
3161:0-7204-2103-9
3158:
3154:
3150:
3147:
3144:
3140:
3136:
3133:
3129:
3125:
3122:
3118:
3114:
3112:
3108:
3104:
3101:
3097:
3094:
3091:
3087:
3084:
3081:
3080:0-393-32229-7
3077:
3073:
3069:
3066:
3063:
3059:
3058:
3053:
3049:
3045:
3041:
3038:
3035:
3034:
3029:
3025:
3021:
3017:
3013:
3010:
3009:
2989:
2985:
2981:
2977:
2970:
2963:
2955:
2951:
2947:
2943:
2939:
2935:
2928:
2920:
2919:
2914:
2907:
2892:
2888:
2881:
2873:
2871:9780415068055
2867:
2864:. Routledge.
2863:
2862:
2854:
2846:
2842:
2838:
2831:
2830:
2822:
2815:
2811:
2805:
2795:
2786:
2785:
2776:
2766:
2759:, p. 105
2758:
2754:
2753:
2748:
2744:
2738:
2731:
2726:
2720:2, 996b 26–30
2719:
2716:
2711:
2704:
2699:
2692:
2686:
2678:
2672:
2668:
2667:
2659:
2644:
2640:
2634:
2619:
2615:
2609:
2605:
2594:
2591:
2584:
2580:
2578:
2575:
2572:
2564:
2561:
2554:
2553:Logical graph
2551:
2548:
2545:
2539:
2536:
2533:
2525:
2524:ternary logic
2521:
2517:
2511:
2508:
2501:
2498:
2495:
2492:
2490:
2487:
2485:
2482:
2480:
2478:
2474:
2471:
2468:
2467:
2441:
2432:
2426:
2420:
2414:
2396:
2390:
2387:
2378:
2368:
2351:
2342:
2333:
2327:
2321:
2311:
2297:
2291:
2288:
2274:
2271:
2268:
2254:
2250:
2249:
2248:
2245:
2243:
2227:
2218:
2215:
2204:
2200:
2190:
2187:
2185:
2181:
2178:
2174:
2170:
2166:
2162:
2158:
2154:
2152:
2141:
2139:
2135:
2131:
2127:
2123:
2119:
2115:
2111:
2107:
2097:
2095:
2094:Arend Heyting
2091:
2086:
2084:
2080:
2076:
2072:
2068:
2063:
2061:
2057:
2053:
2049:
2039:
2036:
2032:
2031:Graham Priest
2028:
2024:
2019:
2017:
2013:
2009:
2005:
2000:
1996:
1995:David Hilbert
1991:
1989:
1985:
1981:
1977:
1973:
1969:
1965:
1961:
1957:
1953:
1949:
1945:
1941:
1937:
1931:
1929:
1924:
1919:
1905:
1902:
1897:
1893:
1872:
1869:
1864:
1860:
1856:
1853:
1831:
1826:
1823:
1813:
1796:
1789:
1783:
1780:
1760:
1740:
1730:
1728:
1727:
1716:
1714:
1709:
1691:
1688:
1683:
1677:
1671:
1665:
1659:
1654:
1649:
1643:
1636:
1630:
1624:
1618:
1612:
1605:
1599:
1594:
1589:
1585:
1577:
1576:
1575:
1555:
1550:
1547:
1524:
1517:
1511:
1508:
1501:
1500:
1499:
1482:
1475:
1446:
1441:
1438:
1416:
1411:
1408:
1401:
1400:
1399:
1378:
1371:
1361:
1360:
1359:
1357:
1339:
1311:
1307:
1286:
1266:
1259:
1255:
1254:
1253:
1250:
1248:
1244:
1237:
1234:
1233:
1232:
1230:
1223:
1220:
1219:
1218:
1216:
1206:
1201:
1197:
1194:
1189:
1186:
1182:
1178:
1174:
1170:
1166:
1163:
1159:
1155:
1152:follows from
1151:
1147:
1144:
1140:
1135:
1134:
1133:
1132:
1128:
1124:
1120:
1116:
1112:
1108:
1105:
1101:
1097:
1094:
1092:
1090:
1081:
1077:
1076:
1075:
1074:
1073:
1071:
1067:
1058:
1055:
1053:
1044:
1042:
1032:
1029:
1025:
1022:
1018:
1014:
1010:
1006:
1002:
998:
994:
990:
986:
985:
984:
983:
977:
976:
975:
974:
968:
965:
961:
960:
959:
958:
957:
953:
951:
944:
942:
941:
933:
928:
926:
922:
918:
908:
904:
901:
898:
894:
893:
892:
891:
890:
888:
884:
878:
873:
868:
865:
863:
859:
855:
851:
838:
835:
831:
827:
823:
819:
815:
814:
813:
811:
782:
778:
776:
772:
768:
767:
766:
765:
759:
754:
751:
747:
744:
743:
742:
741:
740:
737:
735:
733:
728:
726:
721:
717:
707:
705:
701:
697:
693:
689:
685:
680:
678:
674:
670:
665:
661:
659:
655:
651:
647:
643:
639:
633:
629:
625:
621:
615:
611:
607:
603:
597:
593:
589:
585:
579:
575:
569:
565:
555:
551:
545:
541:
536:
534:
530:
526:
522:
518:
514:
510:
506:
502:
498:
494:
490:
486:
482:
477:
475:
471:
467:
462:
460:
456:
454:
446:
440:
438:
432:
427:
426:(1910–1913):
425:
421:
415:
410:
408:
404:
382:
376:
374:
370:
366:
361:
359:
354:
340:
334:
328:
322:
319:
310:
303:
288:
286:
285:
280:
276:
272:
268:
261:
254:
252:
251:
246:
235:
233:
229:
224:
222:
218:
214:
208:
206:
202:
198:
194:
190:
186:
180:
178:
171:
169:
165:
163:
162:
157:
156:contradictory
153:
151:
146:
131:
129:
124:
122:
118:
114:
113:
108:
107:
103:
99:
96:
92:
88:
84:
80:
76:
72:
68:
64:
60:
56:
52:
48:
44:
40:
33:
29:
22:
16:Logic theorem
5551:
5349:Ultraproduct
5196:Model theory
5161:Independence
5097:Formal proof
5089:Proof theory
5072:
5045:
5002:real numbers
4974:second-order
4885:Substitution
4762:Metalanguage
4703:conservative
4676:Axiom schema
4620:Constructive
4590:Morse–Kelley
4556:Set theories
4535:Aleph number
4528:inaccessible
4434:Grothendieck
4318:intersection
4205:Higher-order
4193:Second-order
4139:Truth tables
4096:Venn diagram
3879:Formal proof
3720:Hugh MacColl
3695:Georg Cantor
3690:George Boole
3587:Introduction
3543:modus ponens
3491:
3471:Higher-order
3466:Second-order
3416:Distribution
3376:Truth tables
3288:
3284:
3276:
3272:
3268:
3258:
3248:
3238:
3235:Tom Mitchell
3228:
3218:
3208:
3198:
3188:
3174:
3152:
3142:
3131:
3120:
3110:
3099:
3089:
3071:
3068:Martin Davis
3061:
3055:
3031:
2991:. Retrieved
2979:
2975:
2962:
2937:
2933:
2927:
2916:
2906:
2896:10 September
2894:. Retrieved
2890:
2880:
2860:
2853:
2828:
2821:
2814:Intuitionism
2804:
2794:
2783:
2775:
2765:
2751:
2737:
2729:
2725:
2717:
2714:
2710:
2702:
2698:
2690:
2685:
2665:
2658:
2646:. Retrieved
2642:
2633:
2621:. Retrieved
2617:
2608:
2593:Peirce's law
2475:
2246:
2201:is given by
2196:
2188:
2183:
2180:truth values
2176:
2172:
2168:
2160:
2156:
2150:
2147:
2118:Arthur Prior
2103:
2087:
2082:
2064:
2052:Indian logic
2045:
2023:liar paradox
2020:
2015:
2011:
2007:
2003:
1993:
1987:
1983:
1979:
1975:
1971:
1967:
1963:
1959:
1955:
1951:
1947:
1943:
1939:
1935:
1933:
1927:
1922:
1920:
1815:
1732:
1724:
1722:
1713:intuitionist
1710:
1707:
1573:
1464:
1397:
1329:
1326:is rational.
1251:
1246:
1242:
1240:
1235:
1226:
1221:
1214:
1212:
1204:
1199:
1195:
1184:
1180:
1176:
1172:
1161:
1157:
1153:
1149:
1142:
1138:
1126:
1122:
1118:
1114:
1110:
1103:
1099:
1095:
1088:
1087:
1079:
1069:
1068:
1064:
1056:
1045:
1038:
1020:
1016:
1012:
1008:
1004:
1000:
996:
992:
988:
955:
949:
946:
938:
935:
930:
924:
920:
916:
914:
906:
896:
880:
876:
870:
866:
858:intuitionism
850:Hermann Weyl
847:
833:
829:
825:
821:
817:
809:
788:
775:exclusive-or
770:
757:
749:
738:
731:
724:
720:inclusive-or
716:Karnaugh map
713:
703:
699:
695:
691:
687:
683:
681:
676:
671:writings of
669:intuitionist
666:
662:
657:
654:follows from
649:
645:
641:
637:
631:
627:
623:
619:
613:
609:
605:
601:
595:
591:
587:
583:
577:
573:
567:
563:
553:
549:
543:
539:
537:
532:
528:
524:
520:
516:
512:
508:
504:
500:
496:
492:
488:
484:
480:
478:
473:
469:
465:
463:
458:
451:
449:
444:
436:
434:
429:
423:
419:
417:
412:
406:
405:
380:
378:
372:
368:
365:Truth-values
364:
363:
357:
355:
289:
282:
264:
259:
248:
242:
227:
225:
220:
216:
209:
204:
200:
196:
192:
188:
184:
182:
176:
173:
166:
159:
148:
142:
125:
111:
110:
105:
104:
97:
94:
90:
83:modus ponens
46:
42:
36:
5459:Type theory
5407:undecidable
5339:Truth value
5226:equivalence
4905:non-logical
4518:Enumeration
4508:Isomorphism
4455:cardinality
4439:Von Neumann
4404:Ultrafilter
4369:Uncountable
4303:equivalence
4220:Quantifiers
4210:Fixed-point
4179:First-order
4059:Consistency
4044:Proposition
4021:Traditional
3992:Lindström's
3982:Compactness
3924:Type theory
3869:Cardinality
3665:Disjunction
3660:Conjunction
3645:Existential
3633:Elimination
3624:Disjunction
3619:Conjunction
3604:Existential
3461:First-order
3386:Truth value
3356:Quantifiers
3141:, 1927(2),
3044:Metaphysics
2891:Opinionator
2730:Metaphysics
2715:Metaphysics
2532:fuzzy logic
2122:The Paradox
2035:dialetheism
1089:Kolmogorov'
781:nomological
710:Reichenbach
177:Metaphysics
161:Metaphysics
51:proposition
5574:Categories
5270:elementary
4963:arithmetic
4831:Quantifier
4809:functional
4681:Expression
4399:Transitive
4343:identities
4328:complement
4261:hereditary
4244:Set theory
3715:Kurt Gödel
3578:Absorption
3480:Principles
3366:Connective
3289:An Inquiry
3265:David Hume
3255:Bart Kosko
3171:Kneale, M.
3167:Kneale, W.
3117:Kolmogorov
3086:Dawson, J.
3022:(trans.),
3006:References
2583:Prasangika
2255:including
2126:set theory
2104:In modern
2060:Pyrrhonism
2042:Criticisms
1982:such that
1299:such that
1179:is either
81:, such as
73:, and the
5541:Supertask
5444:Recursion
5402:decidable
5236:saturated
5214:of models
5137:deductive
5132:axiomatic
5052:Hilbert's
5039:Euclidean
5020:canonical
4943:axiomatic
4875:Signature
4804:Predicate
4693:Extension
4615:Ackermann
4540:Operation
4419:Universal
4409:Recursive
4384:Singleton
4379:Inhabited
4364:Countable
4354:Types of
4338:power set
4308:partition
4225:Predicate
4171:Predicate
4086:Syllogism
4076:Soundness
4049:Inference
4039:Tautology
3941:paradoxes
3650:Universal
3609:Universal
3512:Explosion
3497:Bivalence
3426:Soundness
3371:Tautology
3361:Predicate
3048:W.D. Ross
3040:Aristotle
2757:Cambridge
2600:Footnotes
2494:Dichotomy
2439:¬
2436:→
2427:∨
2418:→
2406:→
2394:¬
2391:∨
2382:→
2349:¬
2346:→
2340:¬
2334:∨
2325:→
2295:¬
2292:∨
2286:¬
2282:↔
2272:∧
2263:¬
2225:¬
2222:¬
2219:∨
2213:¬
2048:Catuṣkoṭi
1870:
1655:⋅
1027:accepted.
797:∀
758:exclusive
732:inclusive
725:exclusive
618:✸2.17 ( ~
572:✸2.14 ~(~
373:falsehood
338:∼
335:∨
320:⊢
304:⋅
298:∗
279:Whitehead
168:Aristotle
139:Aristotle
121:tautology
95:principle
5526:Logicism
5519:timeline
5495:Concrete
5354:Validity
5324:T-schema
5317:Kripke's
5312:Tarski's
5307:semantic
5297:Strength
5246:submodel
5241:spectrum
5209:function
5057:Tarski's
5046:Elements
5033:geometry
4989:Robinson
4910:variable
4895:function
4868:spectrum
4858:Sentence
4814:variable
4757:Language
4710:Relation
4671:Automata
4661:Alphabet
4645:language
4499:-jection
4477:codomain
4463:Function
4424:Universe
4394:Infinite
4298:Relation
4081:Validity
4071:Argument
3969:theorem,
3594:Negation
3421:Validity
3401:Logicism
3130:, 1927,
3119:, 1925,
3109:, 1923,
2993:13 March
2749:(1910),
2648:20 March
2623:20 March
2464:See also
2083:a priori
2071:complete
1940:indirect
1928:infinite
1209:Examples
1117:) → { (~
885:(one of
636:✸2.18 (~
582:✸2.15 (~
519:. Since
226:Also in
59:negation
5468:Related
5265:Diagram
5163: (
5142:Hilbert
5127:Systems
5122:Theorem
5000:of the
4945:systems
4725:Formula
4720:Grammar
4636: (
4580:General
4293:Forcing
4278:Element
4198:Monadic
3973:paradox
3914:Theorem
3850:General
3349:General
3304:in the
3249:Hilbert
3139:Brouwer
3128:Brouwer
3107:Brouwer
3054:(ed.),
3030:(ed.),
2954:2218742
1465:But if
1183:or not-
1160:, then
1070:Brouwer
600:✸2.16 (
566:∨ ~{~(~
414:43–44).
275:Russell
267:theorem
245:Leibniz
238:Leibniz
134:History
45:or the
28:logical
5231:finite
4994:Skolem
4947:
4922:Theory
4890:Symbol
4880:String
4863:atomic
4740:ground
4735:closed
4730:atomic
4686:ground
4649:syntax
4545:binary
4472:domain
4389:Finite
4154:finite
4012:Logics
3971:
3919:Theory
3678:People
3159:
3078:
3070:2000,
2952:
2868:
2705:, c. 9
2673:
2165:cyclic
1954:, not
608:) → (~
590:) → (~
562:✸2.13
548:✸2.12
538:✸2.11
464:✸2.1 ~
332:
326:
317:
314:
205:either
55:either
41:, the
5221:Model
4969:Peano
4826:Proof
4666:Arity
4595:Naive
4482:image
4414:Fuzzy
4374:Empty
4323:union
4268:Class
3909:Model
3899:Lemma
3857:Axiom
3774:Works
3521:Rules
2972:(PDF)
2950:JSTOR
2833:(PDF)
2693:p. 74
2666:Logic
2571:modal
1574:Then
1356:proof
1041:Gödel
919:and ~
830:Flies
828:) ⊕ ~
822:Flies
769:30. (
760:-'or'
748:29. (
702:∨ ~(~
626:) → (
552:→ ~(~
369:truth
102:Latin
100:, in
39:logic
5344:Type
5147:list
4951:list
4928:list
4917:Term
4851:rank
4745:open
4639:list
4451:Maps
4356:sets
4215:Free
4185:list
3935:list
3862:list
3451:Term
3183:and
3169:and
3157:ISBN
3082:pbk.
3076:ISBN
2995:2024
2898:2023
2866:ISBN
2812:and
2671:ISBN
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2530:and
2161:+1th
2092:and
2054:and
2046:The
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