Talk:Law of excluded middle/Archive 1

message: How do we feel about derivations in articles? I'd quite like to add the following:
The law can be proved in propositional calculus thus:
 1    1     -(P v -P)              A (for RAA)
 2    2     P                      A (for RAA)
 2    3     P v -P                 v-I 2
 1,2  4    (P v -P) & -(P v -P)    &-I 1,3
 1    5    -P                      RAA 2,4
 1    6    P v -P                  v-I 5
 1    7    (P v -P) & -(P v -P)    &-I 1,6
      8    --(P v -P)              RAA 1,7
      9    P v -P                  DNE 8
The proof is quite sneaky - first it is assumed that (P or not-P), the law of excluded middle, is false. It is then shown that P would contradict this, so it must be the case that not-P. But not-P also contradicts the first line. So the law of excluded middle is thus proven by reductio ad absurdum.
But it's quite technical... Evercat 19:49 30 Jun 2003 (UTC)
: If you can add comments for each step which explain what is being done, I'd say go for it. -- Tarquin 19:59 30 Jun 2003 (UTC)
Your argument strikes me as circular, in that step 5 "begs the question".  You show that P leads to a contradiction (hence is false), and so assume since P is false then -P must be true.  This is exactly what we were trying to prove from the start.  Apparently your definition of RAA already entails excluding the middle. --Jdz 18:52, 27 December 2005 (UTC)
replies: text: Our basic assumption is that what we're trying to prove is false, on this basis we pose the assumption that p is true, and derived a contradition to our basic assumption, thus we are forced to conclude ~p is true.  Suppose then we are ASSUMING FURTHER that ~p is true, independent of our previous derivation, then it can be shown that p v ~p is true by disjunction intro, which again, is contradictory to our original basic assumption, thus we are forced to conclude ~~p.  We've just shown by reductio that ~p and ~~p are both true given our basic assumption, thus we have derived a contradiction from our basic assumption, thus we are forced to conclude that what we're trying to prove is true.--Macbug 07:37, 7 February 2006 (UTC)
replies: text:  Yes, but how do you justify using reductio ad absurdum?  That technique is itself a consequence of the law of excluded middle!  So using it to prove the law of excluded middle seems circular.  (Disclaimer: I am not a logician.) Jorend 14:20, 3 April 2006 (UTC)

text:  (later) I stand corrected: I am used to Metamath, in which reductio is a theorem.  Apparently that is not the way logic is usually constructed;  the example logic at propositional logic takes reductio as a built-in rule of inference.  So the proof above is valid.  I wonder if there's a shorter proof that works in the example logic.  (I added a proof to the page, but it follows Metamath's logic.  Oops.) --Jorend 18:08, 6 April 2006 (UTC)
thread_title: Derivation

message: Another note: are we sure that the law means either P is true or -P is true? See Bivalence and related laws (the bit on the sea battle) for why you might want to deny that, whilst maintaining that (P or -P) is true. Isn't the latter the correct definition of the law of excluded middle? Evercat 00:51 1 Jul 2003 (UTC)
:I may be wrong here, but I think the logical "or" does not mean "either." Specifically, in my experience with logic, "X or Y" means "X, or Y, or both", not "either X or Y, but not both." The latter would be the exclusive-or. I don't know which one applies here; probably exclusive-or. Anyway, as I understand the law of excluded middle, it simply means that something has to be either true or false; it can't be both, or partially-true, or whatever. -- Wapcaplet 01:19 1 Jul 2003 (UTC)
Well, yes, I know that or means inclusive. It doesn't affect the idea that excluded middle might hold even when bivalence does not.
Alas, I should note that I'm confused about this too - it's possible I'm wrong... but it's usually said that there is some distinction between bivalence and excluded middle. If so, you can't make both the definitions the same. :-) Evercat 01:22 1 Jul 2003 (UTC)
Ok, after a bit of reading of the Bivalence article, I think I get it. Bivalence says "it can be true or false, but not both at once." Excluded middle just says "it can only be true or false, or maybe both, but it can't be something else." -- Wapcaplet 01:27 1 Jul 2003 (UTC)
(Hm, yeah, I suppose there's no way for them both to be true, without creating a logical contradiction... but it does not appear that the law of excluded middle itself prohibits (P == not P) ) -- Wapcaplet 01:30 1 Jul 2003 (UTC)
I guess the simplest way I can put it is that, as I understand it, excluded middle says that the formula (or proposition, whatever)
: (P or -P)
is true. As I understand it, it makes no claim about what truth values are available to the proposition P itself... Bivalence does that (or, in multi-valued logics, doesn't exist)
Assuming my understanding isn't fatally flawed, which it may be. :-) Evercat 01:31 1 Jul 2003 (UTC)
:Well, I've reverted my changes. Obviously I am in no condition to edit this article :) -- Wapcaplet 01:32 1 Jul 2003 (UTC)
Well, I'm not sure I am. :-) Any logic experts? Evercat 01:33 1 Jul 2003 (UTC)
The seeming violations of the law of the excluded middle are more linguistic than logical. Many words (not only in English)have imprecise meaning, and such a word as bald can have different meanings due to subtle gradations or changing circumstances. In the most rigid sense, bald means lacking hair altogether, but it can refer to someone with complete or partial loss of head hair or to someone whose head has been shaved. Is one bald if one has no head hair yet wears a wig at the time? One could be medically bald yet sartorially rich in hair.
We use words as a sort of clothing for logical reality because formal logic, designed for reliability and simplicity, fails badly for expressing the subtle gradations of reality and consciousness, let alone the cleverness of some human efforts to deceive themselves or others. Many words have differing and at times contradictory meanings, Thus the word good can refer simultaneously to moral virtue and to technical competence. If one says "Jack is a good carpenter" then one could conceivably state that he is a good person who happens to be a carpenter by trade without reference to his performance at shaping and assembling wood products, or that one can say that Jack is a competent shaper and assembler of wood products irrespective of irrespective of his behavior on other matters. A logical fallacy arises due to differnt meanings of the same word should one  assume that technical competence implies moral goodness. Jack may have fashioned a wooden object well suited as a device for murder or torture with knowledge of its purpose, but doing so contradicts moral goodness because any deliberate contribution to the misery of others contradicts most concepts of goodness of personal conduct.
Such a subject as mathematics is structured to require rigid categories through rigid definitions. There can be no integer between five and six, and the fact that the square root of a positive integer must be an integer should  it be rational depends upon equality of any two numbers requires that those two numbers have the same properties. The law of the excluded middle implies that the square root of 2 cannot have the qualities of an integer and thus must be something other than a rational number. Human experience is more subtle than is mathematics because of ambiguities. It allows ambiguities and even oxymora (example: the reference to the 19th-century politician Stephen Douglas as the "Little Giant") to express such realities as a great orator irrespective of his short stature. Such an expression as "the integer between five and six" or "the rational square root of two" is meaningless, and any effort to assign a word to such a concept is absurd, and worthless in mathematics.--66.231.41.57 11:39, 2 February 2006 (UTC)
thread_title: Precise definition

message: In any system of logic that accepts De Morgan's rules, isn't the law of excluded middle equivalent to the law of non-contradiction? —The preceding unsigned comment was added by Bihzad (talk • contribs) 23:22, 30 May 2005 (UTC)
replies: text: Well yes, since all tautologies (validities for quantified formulas) are equivalent. If you mean to ask whether one is derivable from the other by De Morgan's, then yes again. Nortexoid 01:06, 31 May 2005 (UTC)
thread_title:  De Morgan's Rules 

message: * JA: Just giving the article and some of its hyper-associates the quick once over, I am brought to observe that considerable reserve must be exercised in comparing past and present idioms (idiomata?) in logic, even within what considers itself to be a cohortative cultural trans-mission, lest we generate travesties of a sort that were pandemic in the 1900's by way of "rendering" what our forebears understood as logic.  Just by way of example, it's not automatic that the classical "laws of thought" and the lately so-styled "principle of bivalence" (POB) even live on the same plane of comparison.  So that is one issue to which we must direct further attention.  Jon Awbrey 13:16, 2 February 2006 (UTC)
replies: text: I don't understand a word of this. Try to write in the king's english, without sounding like Miss Manners. Any concrete suggestions here or is this just rant?wvbaileyWvbailey 18:47, 3 April 2006 (UTC)
thread_title:  On being attentive 

message: Reichenbach asserts that all tautologies can be reduced to the Law of excluded middle or tertium non datur i.e. P V ~P: 
:"This is its shortest form. It is clear that we can do the same for every tautology. All tautologies have [this] same shortest form ..." (§11 Derivations, p. 52).
Reichenbach has given the following meaning with regard to so-called nomological formulas: 
:"The term 'nomological', derived from the Greek word 'nomos' meaning 'law', is chosen to express the idea that the formulas are either laws of nature or logical laws. Analytic nomological formulas are tautological formulas [e.g. always true: P V ~P, the tertium non datur, more below], or logical laws; synthetic nomological formulas are laws of nature. The term 'nomological' is therefore a generalization of term 'tautological'" (Reichenbach, p. 360?)
In our example [flying pigs], "Q = P V ~P" is called a "tautology" because it is always true-- either (1) some of the objects are flying pigs or (2) not true that some of the objects are flying pigs or (3) both statements are true. In fact Reichenbach defines "tautology" as the tertium non datur P V ~P:
:"All tautologies have the same shortest disjunctive normal form namely [P V ~P]" (cf Reichenbach p 52).
wvbaileyWvbailey 15:15, 31 May 2006 (UTC)
thread_title:  Interesting Reichenbach quotes re law of excluded middle, tie-in to "laws of nature"? 

message: :"This is clear, in the first place, if we define what the true and the false are. To say of what is that it is not, or of what is not that it is, is false, while to say of what is that it is, and of what is not that it is not, is true; so that he who says of anything that it is, or that it is not, will say either what is true or what is false; but neither what is nor what is not is said to be or not to be.-Again, the intermediate between the contradictories will be so either in the way in which grey is between black and white, 
:"...when there is an intermediate it is always observed to change into the extremes. For there is no change except to opposites and to their intermediates.
:: [Interesting quote, see below where he states "...its [the intermediate's] essence is something different". What he is saying that after a regressions to "the middle", the middle is something different, as in: grey is different from black and white.] 
:"-Again, there must be an intermediate between all contradictories, if one is not arguing merely for the sake of argument; 
:"so that it will be possible for a man to say what is neither true nor untrue, and there will be a middle between that which is and that which is not, so that there will also be a kind of change intermediate between generation and destruction.
-Again, in all classes in which the negation of an attribute involves the assertion of its contrary, even in these there will be an intermediate; 
:: [i.e. "I assert ~p" = ~p ??]
For again it will be possible to deny this intermediate with reference both to its assertion and to its negation, and this new term will be some definite thing; for its essence is something different.
::[i.e. we start with "I assert: "It is not true: 'flying pigs exist' & not-'flying pigs exist'" but this leads by regress to:
::[i.e. "I assert: "It is not true: 'flying & not-flying pigs exist' & not-'flying & not-flying pigs exist'"
::[Thus by regression the assertion settles into the middle, and Aristotle asserts "its essence is something different"] 
wvbaileyWvbailey 13:39, 2 June 2006 (UTC)
thread_title:  More of the Aristotle:7 quote 

message: Dawson asserts that "it seems to have..."(Dawson, p. 52):
:"In his Bologna address Hilbert also raised the question of syntactic completeness for formalized arithmetic [123]. Brouwer, however, took a very different view... in his Vienna lectures [footnote 8 references 10 and 14 March 1928] he went even further: In the first of them he drew a sharp distinction between "consistent" theories and "correct" ones -- an idea that seams to have suggested to Godel that even within classical mathemtics formally undecidable statements might exist [124]" (Dawson p. 52)
::"[124] According to Feigel (1969, p. 639), Brower's lecturers also galvanized Wittgenstein to resume his work in philosophy. For the texts of both lectures see Brouwer 1975, pp. 417-428 and 429-440, respectively)(Dawson p. 279)
:::"Feigel, Herbert (1969, p. 639): The Wiener Kreis in America. In Donald Fleming and Bernard Bailyn, eds., The Intellectual Migration: Europe and America, 1930-1960, 630-673. Cambridge, Mass: Harvard University Press" (Dawson p. 332)
:::"Brouwer, Luitzen E.J. 1975: Collected Works, ed. Arend heyting. Amsterdam: North-Holland Publishing Co.
This quote ends Dawson's chapter "Excursus". The next chapter is "Moment of Impact (1929-1931) which begins with the quotation of the famous antinomy: 
:::"One of themselves, even a prophet of their own, said, The Cretans are always liars .... This witness is true." -- Titus 1:12-13"(Dawson p. 53)
:"Though there is no way to pinpoint when Godel first begain work on his dissertation, internal evidence ... shows that it was sometime in 1928 or early 1929."(Dawson p. 53)  
::Although the influence is quite possible and even plausible, and in spite of others also having ventured the conjecture, to the best of my knowledge there is no directly supporting evidence, only of an entirely circumstantial nature, doing nothing else than reaffirm it is possible. We can quote but not doubleguess. --LambiamTalk 16:22, 2 June 2006 (UTC)
thread_title:  (1) Did Brouwer's "insights" trigger Godel? 

message: The fourth of Brouwer's "insights" comes from a paper of Brouwer's (1927o) where, in a footnote he asserts, the following and then he partially retracts it in 1953 (my boldface):
:"...A second argument is the way in which Hilbert seeks to settle the question ...of the reliability of the principle of excluded middle is a vicious circle; for, if we wish to provide a foundation for the correctness of this principle by means of the proof of its consistency, this implicitly presupposes the principle of the reciprocity of the complementary species and hence the principle of exluded middle itself (see Brouwer 1923c. p. 252)..." (p. 460 [see more below at (3)]
His reference to "vicious circle" is per the usage of Whitehead and Russell in PM. 
The concept of a circulum vitiosum as a logical fallacy is an old concept. Does the context suggest he refers to the usage in PM? --LambiamTalk 16:22, 2 June 2006 (UTC)
replies: text: No you are correct, the "vicious circle" is referenced to Hilbert not to PM; but I will need to get the book of Brouwer's writings to see more on this. Interesting that he picks on Hilbert incessantly, not so on Russell. Maybe he assumed that Russell was just Hilbert's shill.wvbaileyWvbailey 17:17, 2 June 2006 (UTC)
thread_title:  (2) Brouwer did fret about a "vicious circle" in the logicist formulations of Hilbert 

message: TBD
:wvbaileyWvbailey 14:23, 2 June 2006 (UTC)
replies: text: Although you can mimic this in formal logic systems, the vicious circle is in the (necessarily) somewhat informal justification as proof priciples. Just ask any mathematician, let's say Achilles, around you to justify them and you will hear Achilles say something like the following. "Assume that P or not P does not hold. Then it follows that P does not hold. In other words, we have now that not-P holds. But then the disjunction P or not-P holds, contradicting the earlier assumption. So the assumption is false, and therefore P or not-P does hold." Being a Tortoise, you now protest: "Dear friend Achilles, you have shown that the mere assumption of not-(P or not-P) is absurd and untenable. So I agree that therefore we have no choice but to conclude that not-not-(P or not-P) holds. There are a lot of 'not's in the statement. How did you manage to cut the 'not's in your conclusion down to just one occurrence?" Whereupon Achilles will deliver the following rejoinder: "Elementary, my dear Tortoise, entirely elementary. You see, what we have here is an instance of not-not-Q. Now we can greatly economize on the 'not's by a simple observation. Either Q holds, or it doesn't, as the great Aristotle already tortoise taught us. Clearly, not-not-Q tells us that the position that Q does not hold is not available to us. We have only one choice left: Q holds."  --LambiamTalk 16:22, 2 June 2006 (UTC)
thread_title:  (4) So what creates the vicious circle? LoEM or "double negation?" Can we tie what Gödel proved to either of these? 

message: comments in his intro to Brouwer's paper (1923) that the principles of finite mathematics, the origin of logic, were 
:"...applied to discourse about the physical world and then to non-finite mathematics, but in that last field there is not necessarilty a justification for each of these principles. In particular, such a justification seems to be lacking for the principle of excluded middle and that of double negation" (p. 334, 
  
Here is Brouwer on "testing" that immediately precedes his definition of LoEM:
::Within a specific finite "main system" we can always test (that is, either prove or reduce to absurdity) properties of systems, that is, test whether systems can be mapped, with prescribed correspondences between elements, into other systems; for the mapping determined by the property in question can in any case be performed in only a finite number of ways, and each of these can be undertaken by itself and pusued either to its conclusion or to a point of inhibition" (p. 335)
::[see Corrigenda (1954) where Brouwer fiddles with his definition of 'to test' (p. 341)
 
Kolmogorov (1925) seems to be both supportive and critical of the results of Brouwer. He agrees with Brouwer that "it is illegitimate to use the principle of excluded middle in the domain of transfinite arguments" but he believes he can demonstrate why "this illegitimate use has not yet led to contradictions and also why the very illegitimacy has often gone unnoticed." (p. 416). He offers a solution he calls "pseudomathematics". 
Kolmogorov openly contradicts Brouwer with his assertion (V, §1.1):
:"If a finitary conclusion is obtained with the help of arguments based on the use, even in the domain of the transfinite, of the principle of excluded middle, the conclusion is true in the usual sense."(p. 431)
He goes on to assert that:
:"The use of the principle of excluded middle never leads to a contradiction." (p. 431)
Here is Kolmogorov on "judgement":
::"In the logic of judgments the judgment is considered the ultimate element in the investigation ... it can be appraised from the point of view of truth or falsity." (p. 418)
>pseudomathematics? n{n(~(~A))-->nA} his formula 56 ??
Kolmogorov asserts that:
:"... without the help of the principle of excluded middle it is impossible to prove any proposition whose proof usually comes down to an application of the principle of transfinite induction....
:"   The proof of such propostions is often carried out without the help of the principle of transfinite induction. But all these proofs rest upon the principle of excluded middle, applied to infinite collections, or upon the principle of double negation." (p. 436) wvbaileyWvbailey 03:07, 1 June 2006 (UTC)
thread_title:  loose quotes 

message: :What counts as an acceptable proof method to intuitionists will necessarily always remain a bit vague, as these issues precede any attempts at formalization. In fact, not all intuitionists agreed or agree on all aspects; see for example the article on Ultrafinitism. Most would agree, though, that intuitionistic logic faithfully internalizes the propositional aspect of intuitionistically acceptable mathematical reasoning, and so will accept a valid theorem in that logic as being also a valid theorem schema in mathematics itself. In many ways intuitionistic type theory has gone further, extending the scope of the formal-logical approach. In general, the methods of constructive mathematics are also acceptable to intuitionists, and some people even identify intuitionism with constructivism in mathematics. 
:Intuitionists accept P V ~P for any specific proposition P whose truth or falsehood has been demonstrated. Furthermore, they accept this for classes of propositions for which an effective method is known for testing whether they hold. For example, consider the class in which P takes the form X = Y in which X and Y are arithmetic formulas involving only numerals and the familiar arithmetic operations. To test the validity of such a P, just evaluate the two sides of the equation and check whether they are the same. If such a test is not (known to be) available, usually the problem involves, so to speak, to test an infinity of cases, and no known reasoning allowing us to reduce this infinity to a finite number of effectively testable cases. A well-known example is Goldbach's conjecture. For many classes of formally infinite problems an effective (and therefore finite) test procedure is known, however, and then intuitionists will gladly assert P V ~P for these classes. What counts as effective to intuitionists, and which effective procedures count as true tests, will necessarily always remain a bit vague; see above.
:Intuitionists can look at PM derivations in two ways. One is that the logician is playing a game with self-invented rules, like peg solitaire but more complicated, and then they can look at the process and verify that the logician is not cheating. Just as for peg solitaire, this has nothing to do with notions of truth. They can also look at this game, bearing in mind that the logician claims the symbols mean something and allow the game player to express truths. Then there are many potential problems, some having to do with what counts as an acceptable definition (for example of notions like function and real number), some with the range of quantifications, some with the validity of inference rules, and some with the axioms. What these objections have in common is that intuitionists believe that the use of unreliable definitional approaches, unreliable inference rules and unreliable axioms does not offer a reliable way for examining the truth of mathematical claims. Clearly, derivations assuming LoEM or using double negative elimination are not considered kosher. Intuitionists will consider a PM derivation of a → a from a V ~a as a good example of how you can occasionally and accidentally reach correct conclusions using unreliable methods.
:The "cure" intuitionists would prescribe to the mathematical ailment of unreliability is: Use only methods whose reliability has been established. What is reliable and what not, is open to discussion, as is the question what constitutes sufficient evidence or justification. But if, in order to justify a principle, you need to resort to a circular argument, you have failed to establish reliability. Also, the argument that something has a respectable pedigree harking back to Aristotle, is believed to be sound by more than five million mathematicians, and has not failed thus far in any case where it was applied, is unacceptable.
:I'm not sure how much detail of this (assuming proper citations) should be presented in the article Law of excluded middle. If it should be presented at all, then the article Intuitionism may be a better place. --LambiamTalk 18:16, 12 June 2006 (UTC)
thread_title:  Lambiam's exposition 

message: From the article: “The proof is nonconstructive because it doesn't give specific numbers a and b that satisfy the theorem but only two separate possibilities, one of which must work. (Actually [a=$\sqrt{2}^\sqrt{2}$] is irrational but there is no known easy proof of that fact.)" (Davis p. 220)
I’m not question the fact that Davis has reported this statement and as such I’m not criticizing the accuracy of this encyclopedic article. However, I like to call attention to the apparent fallacy that this statement seems to imply. The statement seems to imply that since [$\sqrt{2}^\sqrt{2}$] is in fact irrational (with no easy way to show it), the proof is also incorrect (presumably with no easy way to show it). This implication is simply not true and the wording of the statement is misleading. To see this, go back to the original problem and read it as “prove there are irrational numbers a and b such that $a^b$ is a positive integer”. The proof is still applicable despite the fact that it is quite easy to show that [a=$\sqrt{2}^\sqrt{2}$] is not an integer. Whether [a=$\sqrt{2}^\sqrt{2}$] is an integer, a rational, or an irrational number (and how easy or difficult it is to show that fact) has nothing to do with the logic of the argument.
My only criticism of the article is that if this example is meant to show that that are problems with the non-constructive proof over the infinite, then neither the example nor Davis’s criticism of the example is applicable. In fact, there are logical problems with such “proofs”, but this example is not one of them.
—The preceding comment is by Zaquaraya (talk • contribs) 10:50, May 9, 2007 (UTC): Please sign your posts!
:Why should the fact that \sqrt{2}^{\sqrt{2}} is irrational imply that the proof (which proof exactly) is incorrect? A constructive proof of the existence of irrational a and b such that ab is rational that uses the fact that the square root of 2, raised to itself, is irrational, must contain a proof of this irrationality. The given proof does not, and can therefore not be called constructive. The known (not-easy) proof of irrationality is in fact based on the Gelfond–Schneider theorem; it is doubtful that the proof of that theorem is constructive, and it may not be easy to make it sufficiently constructive that it can be contained in another proof without losing constructivity for the larger proof.  --LambiamTalk 11:56, 9 May 2007 (UTC)
replies: text: I withdraw my criticism. I misunderstood Davis’s statement on the first reading. In fact, he is saying that not only the conjecture is true by the proof of existence that was given, but also by construction—i.e. the two irrational numbers that satisfy the conjecture are a = \sqrt{2}^{\sqrt{2}} and b = \sqrt{2}, alas with no easy way to show that a is in fact irrational. Zaquaraya 03:41, 10 May 2007 (UTC)
thread_title:  Find irrational a and b such that a^b is rational 

message: > "Every judgment is either true or false"
...or some dirty quantum-mechanicist just wrecked your nice philosophy book, because the subject of the judgement may be factually green or red or both at the same time or not even God knows what.
Therefore I think quantum mechanics should deserve a mention in this article as a sign of contradiction. Physics is applied maths, so it is a valid connection. 81.0.68.145 21:07, 15 September 2007 (UTC)
thread_title:  Schrodinger's catscrawl 

message: Is there a logical difference between the statement "A proposition MAY NOT be BOTH true and false" and "a proposition MUST be EITHER true XOR false" ? IOW, is there a logical difference between prohibiting "Socrates is BOTH mortal AND immortal" and prohibiting "Socrates is NEITHER mortal NOR immortal" ?? I'm not a logician. I'd just like a clarification here, because I'd always heard LoEM was the former and this entire article clearly states it is the latter. Thanks. roricka 2/15/09
:There certainly is a difference.  You may be interested to read the articles on dialethism and related matters —Dominus (talk) 09:13, 15 February 2009 (UTC)
thread_title:  LoEM eqivalence? 

message: The law has well been stated as "that that is is that that is not that that is not that that is". I would kinda like that included, but this is a very serious article and not sure it belongs here.
SimonTrew (talk) 17:35, 24 February 2009 (UTC)
thread_title:  Humour 

message: A question about the definition chosen in the intro, "is the principle that for any proposition, either that proposition is true, or its negation is." Why not state "either that proposition is true, or false." ? I ask because this would seem to drag things into requiring a definition of "negation"? CecilWard (talk) 17:40, 9 December 2009 (UTC)
replies: text: The statement "a proposition is either true or false" is not the Law of Excluded Middle, it is the Law of Bivalence.  The point of the Law of Excluded Middle is that a statement and its negation must always have opposite truth values.  There can never be a situation where 'P' and 'not P' are both true.  128.135.4.56 (talk) 21:54, 22 February 2010 (UTC)
thread_title:  Intro - question 

message: Are there Wikipedia articles about philosophies that do not allow for an excluded middle? (or N+1th?) Thanks. 76.202.63.148 (talk) 12:11, 29 June 2010 (UTC)
thread_title:  Non-intuitionist philosophies that do not allow for an excluded middle? 

message: "In logic, The law of excluded middle, also known as the Principle of excluded middle or excluded middle [...]"; apparently philosophers do not really appreciate the difference between a law and a principle. A 'principle' in its original meaning can be stated as a 'origin' (c.1380, "fundamental truth or proposition," from Anglo-Norm. principle, from O.Fr. principe,  from L. principium  (pl. principia ) "a beginning, first part," from princeps  (see prince). Meaning "origin, source" is attested from 1413. Sense of "general rule of conduct" is from c.1532. Used absolutely for (good or moral) principle  from 1653). Without talking about the truth or falsehood of the law of excluded middle, its formulation should be constructed by defining this middle ("neither P nor not P") and then exclude this middle. Hence, the formulation 'nothing can neither be nor not be'. This formulation is based upon the concept 'nothing' and therefore does not presuppose 'anything', that is, it is the most elementary way of formulating and can be considered the Principle of excluded middle. However, Bertrand Russell’s formulation ('everything must either be or not be') is based upon the concept 'everything' and should therefore be seen as a law instead of a principle. Also, the formulation "FOR ALL p: (p V ~p)" appears to encode a law derived from the principle of excluded middle but does not formulate the principle of excluded middle itself. A lot of philosophical confusion would most probably be avoided if this difference was pointed out more clearly. Sarahcroft (talk) 15:43, 25 August 2010 (UTC)
replies: text: Pls read the history above (where the drawings are). From Aristotle on down all are confused about what we should call the LoEM because there are many ways it has & can be stated, and there are at least two different contexts in which it can be stated: the finite-set-theoretic where we can examine each element to see if it is a p or a ~p, and the completed-infinite set-theoretic where we cannot. Starting with Aristotle, it's been called "a Law of thought", "the maxim of Excluded Middle, "an axiom of Logic", "a Rule of thought", "a principle of reasoning" etc etc. Nowadays, for the mathematician/logician, it's an "axiom of Logic" (given they choose to include it in their axiom-set, as the intuitionists are NOT inclined to do) . But for the empirically-minded philosopher it is a "Law of Nature" (i.e. a "natural law") based as it is on repeated, accumulated experience, so-called "inductive reasoning". BillWvbailey (talk) 22:29, 25 August 2010 (UTC)
replies: text: In medieval Scholasticism, the University masters were under oath to maintain that God can do absolutely anything except that... A testament to the value they placed on logic (and possibly still church doctrine?) I'd think a deity who can't manage a simple contradiction might consider a more suitable career. Work in mysterious ways smarter, not harder: p and ~p therefore whatever QED—Machine Elf 1735 (talk) 09:18, 14 November 2010 (UTC)
thread_title:  Law or Principle? 

message: There is a substantive discussion about this article (and what might be wrong or confusing with it) at Talk:Principle_of_bivalence#Musing_about_LoEM. Tijfo098 (talk) 01:07, 16 April 2011 (UTC)
thread_title:  Notice