Talk:Law of excluded middle

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Excluded middle and principal of bivalence

The first sentence of the article says "The law of the excluded middle states that a proposition is either true or false". Later on, the article says "This is not quite the same as the principle of bivalence, which states that P must be either true or false." The first one uses "is", while the second one uses "must be". What is the difference between "is" and "must be"? --Kprateek88(Talk | Contribs) 10:59, 4 November 2006 (UTC)

I don't know but I find this article absolutely confusing, if not impossible to understand! I think it should be written in more accessible language. It's as if the writer assumes we are all logicians! Monagz 22:24, 5 November 2006 (UTC)

I as one of the writers have to apologize for what is truly confusing, but in this case the apology takes the form an excuse "The poor workmen are blaming their tools". Other writers may disagree with the following (and therein lies part of the tool issue, I'm not sure you could get agreement on any of this stuff).

Why does anyone even care about "the excluded middle?" Some do. Their i:ssue really has to do with the "pigs flying" parable. The only reason anyone cares about tertium non datur is because at the bottom of it all it boils down to this parable (or, perhaps, a better-written version of it). The problem started with the guy named Kronecker who had a really bad time with the notion of "infinity". Then his case was taken up by Brouwer and his band of merry men, the so-called "intuitionists", who insisted that "an algorithm" must "produce an object". But much of mathematics, and reasoning, now relies upon the form of argument called reductio ad absurdum (I love it, myself) that do not produce objects -- at their core, they use what is known as "the double negative" together with "the existential operator FOR ALL" (often written as an upside down A but sometimes like this (x) ). But you ask, how do we get from "A V ~A" (tertium non datur) to "~~A = A" (double negative) and vise versa?? Just today I am looking through Kleene (1952) and I run into his equations #49 and #51, these two very equations, with little circles next to them meaning "not accepted by the intuitionists." To get from here to there and back is a long and winding road through the land of logic and history: not an easy trip. wvbaileyWvbailey 23:05, 5 November 2006 (UTC)

I don't think this is a good characterization of intuitionistic thinking. Suppose you take the statement "P is true" to mean not that P is true in some abstract, Platonic sense, but simply that it is possible to prove the truth of P. This is a reasonable step to take. Then not-P means that no proof of P is possible.

Similarly, "P or Q" now means that either P can be proven or Q can be proven. It is now clear why the intuitionists reject "P or not-P" in general: it's not true in general! There are values of P that cannot be proved, but for which there is no proof that a proof of P is impossible.

This is also the understanding that lies behind the intuitionistic treatment of double negation. In intuitionistic logic, P implies ~~P, but not vice versa. That is, if P can be proved, then it is impossible to produce a proof that there is no proof of P. But not vice versa: just because there is no proof that a proof of P is impossible, does not imply that P can be proved.

--Dominus 04:28, 6 November 2006 (UTC)

What I wrote may not be a good characterization, but as it follows Kleene (1952) closely (excepting the double-negative thing, which is muddled). I will leave you with a very brief quote directly from Kleene (1952).

(In the flying pig example -- All that we need to prove our asserted predicate (P V ~P) = TRUTH, one way or the other, is to demonstrate that a single instance of a flying pig exists. But if we are finding no flying pigs, to demonstrate our asserted predicate we have to examine ALL instances of pig-like creatures in this and all other universes, and we cannot do that. So the algorithm [us the intrepid searchers] that is evaluating the asserted predicate (P V ~P) =_? { t, f, u } may not ever yield "an object" (the expected output t= "TRUTH"). Since (P V ~P) is primitive recursive (cf Kleene p. 228 proof #E) this raises an interesting question re "primitive recursive functions" and an "unbounded search operator" that never succeeds. Cf Kleene p. 317 where he discusses this very issue, Chapter XII Partial Recursive Functions, §62 Chruch's thesis, etc.):

"§13. Intuitionism. In the 1880's, when the methods of Weierstrass, Dedekind and Cantor were flourishing, Kronecker argued vigorously that their fundamental definitions were only words, since they do not enable one in general to decide whether a given object satisfies the definition.

"[ a quote here from Weyl re an unjustified extension of the classical logic to infinite sets....]

"A principle of classical logic, valid in reasoning about finite sets, which Brouwer does not accept for infinite sets, is the law of the excluded middle. The law, in its general form, says for every proposition A, either A or not A. Now let A be the proposition there exists a member of the set (or domain) D having the property P. Then not A is equivalent to every member of D does not have the property P, or in other words every member of D has the property not-P. The law, applied to this A, hence gives either there exists a member of D having the property P, or every member of D has the property not-P.

"For definiteness, let us specify P to be a property such that, for any given member of D, we can determine whether that member has the property P or does not.

"Now suppose D is a finite set. Then we could examine every member of D in turn, and thus either find a member having the property P, or verify that all members have the property not-P. There might be practical difficulties, e.g. when D is a very large set having say a million members, or even for a small D when the determination whether or not a given member has the property P may be tedious. But the possibility of completing the search exists in principle. It is in this possibility which for Brouwer makes the law of the excluded middle a valid principle for reasoning with finite sets D and properties P of the kind specified.

"For an infinite set D, the situation is fundamentally different. It is no longer possible in principle to search through the entire set D.

"Moreover, in this situation the law is not saved for Brouwer by substituting, for the impossible search through all the members of the infinite set D, a mathematical solution of the problem posed. We may in some cases, i.e. for some sets D and properties P, succeed in finding a member of D having the property P; and in other cases, succeed in showing by mathematical reasoning that every member of D has the property not-P, e.g. by deducing a contradiction from the assumption that an arbitrary (i.e. unspecified member of D has the property P. ( An example for the second kind of solution is when D is the set of all the ordered pairs (m, n) of positive integers, and P is the property of a pair (m, n) that m² = 2n² . The result is then Pythagoras' discovery that sqrt(2) is irrational.) But we have no ground for affirming the possibility of obtaining either one or the other of these kinds of solutions in every case...

"Brouwer's non-acceptance of the law of the excluded middle for infinite sets D does not rest on the failure of mathematicians thus far to have solved this particular problem [Fermat's last theorem, only recently proven], or any other particular problem. To meet his objections, one would have to provide a method adequate in principle for solving not only all the outstanding unsolved mathematical problems, but any others that might ever be proposed in the future. How likely it is that such a method will be found, we leave for the time being to the reader to speculate. Later in the book we shall return to the question (§ 60) [(§ 60 "Church's theorem, the generalized Godel theorem]” (p. 47-48)

"The familiar mathematics, with its methods and logic, as developed prior to Brouwer's critque or disregarding it, we call classical; the mathematics, methods or logic which Brouwer and his school allow, we call intuitionistic. The classical includes parts which are intuitionistic and parts which are non-intuitionisic.

"The non-intuitionistic mathematics which culminated in the theories of Weierstrass, Dedekind and Canotr, and the intuitionistic mathematics of Brouwer, differ in their view of the infinite. In the former, the infinte is treated as actual or completed or extended or existential. An infinite set is regarded as existing as a completed totality, prior to or independently of any human process or generation or construction, as though it could be spread out completely for our inspection. In the latter, the infinite is treated only as potential or becoming or constructive. the recognition of this distinction, in the case of infinite magnitudes, goes back to Gauss, who in 1831 wrote, "I protest . . . against the use of an infinite magnitude as something completed, which is never permissible in mathematics."(Werke VIII p. 216).

"According to Weyl 1946, "Brouwer made it clear, as I think beyond any doubt, that there is no evidence supporting the belief in the existential character of the totality of all natural numbers . . . The sequence of numbers which grows beyond any stage already reached by passing to the next number, is a manifold of possibilities open towards infinity; it remains forever in the status of creating, but is not a closed realm of things existing in themselves. That we blindly converted one into the other is the true source of our difficulties, including the antinomies -- a source more fundamental than Russell's vicious circle principle indicated...."

"Brouwer's criticsm of the classical logic as applied to an infinite set D (say the set of the natural numbers) arises from this standpoint respecting infinity. We see this clearly by considering the meanings which the intuitionist attaches to various forms of statements.

"A generality statement all numbers n have the property P, or briefly for all n, P(n), is understood by the intuitionist as an hypothetical assertion to the effect that, if any particular natural number n were given to us, we could be sure that that number n has the property P. this is a meaning which does not require us to take into view the classical completed infinity of the natural numbers.

"Mathematical induction is an example of an intuitionistic method for proving generality propositions about the natural numbers. A proof by induction of the proposition for all n, P(n) shows that any given n would have the property P, by reasoning which uses only the numbers from 0 up to n (§7). Of course, for a particular proof by induction to be intuitionistic, also the reasonings used within its basis and induction step must be intuitionistic.

"An existence statement there exists a natural number n having the property P, or briefly there exists an n such that P(n), has its intuitionistic meaning as a partial communication (or abstract) of a statement giving a particular example of a natural number n which has the property P, or at least giving a method by which in principle one could find such an example.

"Therefore an intuitionistic proof of the proposition there exists an n such that P(n) must be constructive in the following (strict) sense. The proof actually exhibits an example of an n such that P(n), or at least indicates a method by which one could in principle find such an example.

"In classical mathematics there occur non-constructive or indirect existence proofs, which the intuitionists do not accept. For example, to prove there exits an n such that P(n), the classical mathematician may deduce a contradiction from the assumption for all n, not P(n). Under both the classical and the intuitionistic logic, by reductio ad absurdum this gives not for all n, not P(n). The classical logic allows this result to be transformed into there exiss an n such that P(n), but not (in general) the intuitionistic. Such a classical existence proof leaves us no nearer than before the proof was given to having an example of a number n such that P(n) (although sometimes we may afterwards be able to discover one by another method). The intuitionist refrains from accepting such an existence proof, because its conclusion there exists an n such that P(n) can have no meaning for him other than as a reference to an example of a numbere n such that P(n), and this example has not been produced. The classical meaning, that somewherer in the completed infinite totality of the natural numbers there occurs an n such that P(n), is not available to him, since he does not conceive the natural numbers as a completed totality."

"[etc]

"A disjunction A or B constitutes for the intuitionist an incomplete communication of a statement telling us that A holds or that B holds, or at least giving a method by which we can choose from A and B one which holds. A conjunction A and B means that both A hold and B hold. An implication A implies B (or if A, then B) expresses that B follows from A by intuitionistic reasoning, or more explicitly that one possesses a method which, from any proof of A, would procure a proof of B; and a negation not A (or A is absurd) that a contradiction B and not B follows from A by intuitionistic reasoning, or more explicitly that one possesses a method which, from any proof of A, would procure a proof of a contradiction B and not B (or of a statement already known to be absurd, such as 1 = 0). Additional comments on these intuitionistic meanings will be given in §82. See note 1 on p. 65."(italics in original, boldface added, Kleene (1952) p.46-51)

"Note 1 [added on the 6th printing, 1971]: At the top of p. 51, the seeming circularity that not B is used in explaining not A is to be avoided thus. Sameness and distinctness of two natural numbers (or of two finite sequences of symbols) are basic concepts (cf. p. 51 lines 20-24). For any B of the form m = n where m and n are natural numbers, not B shall mean that m and n are distinct. the explanation of not A in lines 5-8 then serves for any A other than of that form, by taking the B in it to be of that from. Equivalently, since the distinctness of 1 from 0 is given by intuition (so not 1 = 0 holds), not A means that one possesses a method which, from any proof of A, would procure a proof of 1 = 0 (cf. lines 8-9). [Kleene p. 65]

etc. Brouwer believes that mathematics comes from the intuition. [Kleene is quoting Brouwer here:] "There remains for mathematics 'no other source than an intuition, which places its concepts and inferences before our eyes as immediately clear'. This intuition 'is nothing other than the faculty of considering separately particular concepts and inferences which occur regularly in ordinary thinking'. The idea of the natural number series can be analyzed as resting on the possibility, first of considering an object or experience as given to us spearately from the rest of the world, second of distinguishing one such from another, and third of imagining an unlimited repetition of the second process." (p. 51)

wvbaileyWvbailey 15:14, 6 November 2006 (UTC) wvbaileyWvbailey 15:03, 7 November 2006 (UTC) wvbaileyWvbailey 21:17, 7 November 2006 (UTC)

I wrote something about the issue on the talk page of Principle of bivalence under the heading Confusing article. My suggestion for this article (Law of excluded middle) is to simplify it and remove non-essential material where possible. That definitely includes this confusing reference to the principle of bivalence.  --Lambiam^Talk 06:26, 7 November 2006 (UTC)

But I'm not sure any of us can agree ("the poor workmen blaming their tools") on what should go and what should stay. Kleene's section on intuitionism §13 is so well written that if we could copy it verbatim as far as I'm concerned I'd be done with it. Ditto the Anglin. I've typed in some more Kleene. If anyone wants me to cc this section and e-mail it to them as .pdf lemme know.

With regards to three-valued logic Kleene treats this with respect to partial recursive functions §64 that can return "u" as "undecided". I am beginning to see the tie-in with the above quote from Kleene and with the notions of "algorithm" and "total" versus "partial" recursive functions. And Kleene's "producing an object" observation. For example, here is a quote along the same lines in Minsky (1967) cautioning the reader that

"...we must always hesitate to assume that a system of equations really defines a general-recursive [total] function. We normally require auxiliary proof for this, e.g. in the form of an inductive proof that, for each argument value, the computation terminates with a unique value (Minsky 1967 p. 186)"

Sounds a bit intuitionistic/constructivistic to me. I need to read and learn and some more. wvbaileyWvbailey 15:03, 7 November 2006 (UTC)

Deletia

I am deleting the flying pigs example on the grounds that it doesn't correctly represent intuitionism at all. The intuitionist's rejection of the law of excluded middle is due to the philosophical view that math is a strictly mental activity.

This is an incorrect when applied to the LoEM, but others are confused so you're not alone. The "philosophical view" -- if there is one -- has to do with the completed infinity of Cantor, and the quote of Gauss -- see the quote from Kleene below. When restricted to finite sets (as we presume the Aristotelian logic is construed) the LoEM is fine (cf Kleene p. 46), but when our old friend the "for all" operator gets involved, and the "for all" ranges over an infinite collection, the intuitionists take exception -- use of reductio ad absurdum is a favorite target (cf page 48). How do you know "for all" is true if all you have is negative results to show for your troubles? Just because you haven't encountered a flying pig doesn't mean there isn't one. wvbaileyWvbailey 20:35, 9 January 2007 (UTC)

I don't think it would apply to real-world pigs, and I can't see that it has anything to do with a "computability" argument, as the example implies at the end. --Jorend 14:48, 9 January 2007 (UTC)

The flying pigs example represents intuitionist objects quite nicely. What may be not so good is why the example, by use of reductio ad absurdum, causes the Intuitionists to object. (cf Kleene p. 48, top of page).

Here is the quote from Kleene (1952, 1971) re what intuitionism is really fussing about. Kleene is discussing the intuitionist objection to "the completed infinity" in detail:

"The non-intuitionistic mathematics which culminated in the theories of Weierstrass, Dedekind and Cantor, and the intuitionistic mathematics of Brouwer, differ essentially in their view of the infinite. In the former, the infinite is treated as actual or completed or extended or existential. An infinite set is regarded as existing as a completed totality, prior to or independently of any human process of generation or construction, and as though it could be spread out completely for our inspection. In the later, the infinite is treated only as potential or becoming or constructive. The recognition of this distinction, in the case of infinite magnitudes, goes back to Gauss who in 1938 wrote, 'I protest . . . against the use of an infinite magnitude as something completed, which is never permissible in Mathematics' (Werke VIII p. 216.)" (p. 48, CH III A Critquie of Mathematical Reasoning, § Intuitionism).

The LoeM has to do with this very issue -- and the pigs example hints at why (in reductio, you have to produce an example to satisfy an intutionist -- its very difficult to prove non-existence "...non-constructive or indirect existence proofs, which the intuitionists do not accept" (p. 49)). Just because we didn't find a pig out there after our excursions toward the infinite doesn't mean that there isn't one out there, somewhere. This same issue comes up with the partial recursive functions, undecidability, etc. Probably the example needs to be reworked to illustrate better why reductio is usually disallowed by intuitionists. wvbaileyWvbailey 20:35, 9 January 2007 (UTC)`

I concur with Jorend. What is special about finite sets is that if we have a testable property, the finiteness guarantees we have an effective procedure for testing that property for all members. In general, no effective procedure is known. But what is finite? Let S stand for the set of numbers n that have the property of being a counterexample to Goldbach's conjecture, while no number less than n has that property. S has at most one element, and so would be considered finite by all "classical" mathematicians. Now take the testable property of a number that it can be written in the form k! for some natural number k. There is no obvious way of determining the truth or falsehood of the proposition that every element of the finite set S satisfies this testable property. The solution to this apparent paradox is that, although intuitionists agree that S cannot have more than one element, they do not consider this set as (being known to be) finite. Further, if x is a real number, intuitionists will in general not accept the statement that x = 0 or x ≠ 0 as being true without a proof taking account of the definition of x. It is not a priori clear that this has to do with the non-finiteness of a set. While it is true that classical mathematics and intuitionism differ essentially in their view of the infinite, this difference stems from different deeper, underlying views of what mathematics is about. The latter cannot be reduced to the former. The example is further unnecessarily confusing by bringing in aspects that are related to practical problems: how could anyone actually look under a rock on a planet in an unknown universe? Of course we'll never "know" then whether the claim was true or not. Aristotle, Russell and Hilbert would have agreed, but the claim in this case is not a mathematical statement.  --Lambiam^Talk 23:52, 9 January 2007 (UTC)

(I figured a Platonist would weigh in.) I was the guy who added the quote re intuitioniosm that appears at the Foundations of Mathematics that apparently Jorend is quoting. (Not that I necessarily believe it is a truth, but rather it just reflects an author's published POV). Problem is: what does that quote have to do with the intuitionist objections to 'the completed infinity'? I have no faith that we can go from that quote to an understanding of why intuitionists and finitists of all shapes and sizes dislike 'completed infinities'. One man's meat (completed infinities) is another man's poison (completed infinities).

Wiki cares only about what the literature reflects: e.g. immediately above I present wiki with a quote from Kleene. In the same section Kleene later brings in a "philosophic" aspect to the discussion: "Quoting from Heyting 1934, 'According to Brouwer, mathematics is identical with the exact part of our thinking.... no science, in particular not philosophy or logic, can be a presupposition for mathematics. It would be circular to apply any philosphical or logical principles as means of proof, since such mathematical conceptions are already presupposed in the formulation of such principles.' There remains for mathematics 'no other source than an intuition...' (Kleene p. 51).

But what does this have to do with completed infinities and the LoEM? As I wrote, the pig example is flawed (and so it can go away) because it doesn't neatly express the reason why the LoEM offends the intuitionists. It does address the notion of infinities, completed and otherwise; what it misses is the (perceived) differences between ~(A & ~A) versus A V ~A [also problems with ~~A = A . See formulas *50 and *51^o in Kleene p. 119].

A discussion of "What is 'the infinite' " (what a black hole that is... ) and Goldbach's conjecture would be fun (I've been wondering why I should accept the "existence" of ε₀) but is beside the point. Haven't we been around this before at Intuitionism?

For those who dare adventure into desperate climes, in van Heijenoort (1967, 1976 3rd edition) appears Brouwer's own article "On the significance of the princple of excluded middle in mathematics, especially in function theory" (p. 335, reprint of 1923, with commentary.) The commentary before all the following articles is especially valuable. Kolmogorov's article "On the principle of excluded middle" appears on p. 415. More Brouwer appears on p. 439. Weyl appears on p. 481.

Here is a quote from Kolmogorov:

"...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." (p. 436).

There is more re transfinite induction in the introduction to the Weyl paper. wvbaileyWvbailey 03:13, 10 January 2007 (UTC)

Who are you calling a Platonist? Hopefully not me. Already in his Ph.D. thesis Brouwer stresses from the start that mathematics is a free creation of the human mind that comes forth from an a priori given ur-intuition, and his aim is to show that this is independent of so-called laws of logic. He was quite explicit about this. Brouwer had no problem with infinities per se; he accepted the infinite ordinal ω as being the set of all natural numbers, and the continuum. What he did not accept was definition by set comprehension in which there is no method for constructing all elements by repeating some construction principle. With regard to the Law of Excluded Middle, the issue is really quite simple. To Brouwer, to assert the validity of "P or not P" for some proposition P means the assertion that there is a method to determine whether P is true or false. Therefore, the Law of Excluded Middle, which asserts that for all P the disjunction P or not P is valid, is equivalent with the assertion that for all propositions P there is a method to determine whether P is true or false. There is no reason to assume, however, that the latter is the case. Therefore this law is an unreliable logical principle. Examples to show the problems in determining whether an arbitrary proposition holds or not are most easily found when P involves a quantification over an infinite set. But that is not the reason why Brouwer rejects the LoEM. As I already wrote, there are other examples, like x = 0 or x ≠ 0, or the question whether a (single) given ordinal number is finite or infinite. We can simply refer to well-known unsolved problems (taking the role of P), like P=NP?, the Generalized Riemann Hypothesis, or – a problem that is understandable with only an elementary background in maths - Goldbach's conjecture.  --Lambiam^Talk 05:46, 10 January 2007 (UTC)

You and I will not agree on "the reason": I'm saying his "philosophy" (hunch? gut feel?) -- that there is no philosphy to be applied, only a mysterious a priori "intuition" -- (what a philosphy!: "there is no philosphy...") was a mere a posteriori explanation for why he disallowed the following: "For all propositions P: P V ~P over infinite sets D." This is like me pulling a "philosphy" out of thin air to explain why I dislike yellow cars -- "Suddenly on 10 January 2007 I, Bill, became aware of an priori knowledge that phases of the moon cause accidents involving yellow cars." Without any evidence to support my "philosphy" -- and why on earth would I even concoct such a notion? -- it is just a silly belief based on blind faith. A scientist or mathematician demands examples. At least Brouwer (1923) goes further and gives us examples of why he dislikes yellow cars:

"The following two fundamental properties, which follow from the principle of excluded middle, have been of basic significance for this incorrect "logical" mathematics of infinity ("logical" because it makes use of the principle of excluded middle), especially for the theory of real functions (developed mainly by the Paris school):

1. The points of the continuum form an ordered point species;

2. every mathematical species is either finite or infinite" (p. 335, van Heinjenoort)

Here is the reason why he had a "philosphy" -- the years are 1905-1908 or thereabouts: he observes the paradoxes of Russell etc and hears/reads the discussions around the questions of Hilbert; Cantor's work is available but seems peculiar; Gauss, Kummer and Kronecker objected to completed infinities -- Kronecker virtually destroys Cantor's career, Brouwer discusses and argues with his cronies over drinks, he gets a hunch, he experiments and tests, and finally he can produce examples that support his hunch. Only then does he feel the need to concoct "a philosophy". Like Darwin -- first he observed, then wondered and questioned why, then formulated a hunch, then assembled evidence to support his hunch (theory) from the observations/experiments.

I think you and I might agree on this: as you note and Kleene amplifies, the LoEM objection is a sort of "meta-issue" having to do with asserting "for all" with regards to propositions about infinite sets, not an objection to the sets themselves. If we assert the "For all propositions P about sets D: P V ~P" means that to quote Kleene:

"Brouwer's non-acceptance of the law of the excluded middle for infinite sets D does not rest on the failure of mathematicians thus far to have solved this particular problem [his example is Fermat's Last theorem], or any other particular problem. To meet his objection one would have to provide a method adequate in principle for solving not only all the outstanding unsolved mathematical problems, but any others that might ever be proposed in the future."

If I understand this correctly this objection is a "meta-objection" and falls into the province of 2nd order logic. Is my understanding correct here? wvbaileyWvbailey 19:41, 10 January 2007 (UTC)

Regardless of the above discussion, I still think the text about flying pigs was more perplexing than helpful. The article is more enlightening without it. I think it would be better still if most of the remaining discussion of intuitionism vs. formalism were moved to Intuitionism or yet a third article, perhaps "Intuitionistic logic". --Jorend 22:01, 10 January 2007 (UTC)

The pigs are gone. I've wondered the same thing about the intuitionism part. It seems long-winded. But part of it is necessary in order to present a background for why the LoEM is of any interest at all. Basically, if the intuitionists hadn't harped on about the LoEM there'd probably be no article, or an article about 10 lines long. I came to wikipedia about this time last year, wondering what the fuss was about "the LoEM" -- and what "the LoEM" is -- after reading about Godel and Hilbert and Russell and some oddball named Brouwer:

"the polemical tone of his writing, as well as his combative personality, led him into conflict with many of his peers, including Hilbert and Karl Menger" (Dawson p. 321)

The wikipedia article then didn't help so I did some research and ran into this snarl of conflict in "the foundations of mathematics" that ran rampant through the 1st half of the 20th C -- the LoEM was at the center of it (hence all the articles in van Heijenoort addressing the issue, the defensive tone in Godel's writing w.r.t the LoEM, etc, etc.). So I believe the article should reflect its historical importance in this regard. (And the form ~~P = P is another one that gets beat around the face and neck too).

I'm in favour of a drastic prune-back of the article, as I have stated several times before (now buried in the Archives). Let us keep only what is both essential and clear. But I don't think that we should move material that is less clear to other articles.

As to wvbailey's earlier comment, quite possibly Brouwer did not arrive overnight at the complete form of his views on the foundations of mathematics. It took him a couple of years to develop his Ph.D. Thesis, much of which went into studying the literature. The speculation about the process by which he arrived there seems baseless. Brouwer's position is that mathematics is independent both of any external experience and of logical laws. I don't understand what you mean in this context by "experiments and tests", "evidence", and "theory".

What I am saying that I cannot see how Brouwer's "philosphy" had any operative power with regards to what Brouwer truly did as a mathematician. And I don't believe for a nanosecond that his mathematics sprung whole-formed from his mind without any experience (as you said: he read the literature. Literature dwells in libraries as objects independent of Brouwer -- it is "evidence", "theory", etc.) Here is an example of "tests" interacting with "evidence", "theory" etc: I have been studying Ulam's problem. The first thing I did was build a model on a spreadsheet (took about 10 minutes) and ran the thing down to N = 2^15 and out about 250 steps (computer chokes about there). A quick read of some literature mentioned the idea of a particular sequence "terminating" -- i.e. a number Ns "terminates" when, at step S, it is less than the starting number N. I modified my spreadsheet to illustrate this. Very quickly I noted that the Ns < N fall into patterns. I counted them up. Lo and behold, such patterns repeat every 2^S -- and with some work I determined the equivalence classes are of the form N = C*2^S + K, where K is the first instance of its appearance. (Someone discovered at least part of this back in the '60's). Then I proved it for myself, derived a bunch of stuff, and now I can tell you why a particular constant K appears when it does. That is what I mean by an "experiment". What I did not do was wake up one morning with a random thought -- "Lo and behold equivalence classes will emerge if, when a number is odd I multiply it by 3 and add one and if even I divide by two, and those eq-classes are of the form N = C*2^S + K, and the K are 3 when S=4, 11 and 23 when S=5, etc....". Rather, I determined this out of my immediate and past experiences as I interacted with a machine, out of my training, my observations etc. The spreadsheet model -- (I built a abstract counter-machine model on the spreadsheet too) -- is an object with behaviors that I studied, just like a biologist studies the behaviors of a type of animal.

This is getting again beyond what is relevant to the article. I am just trying to tell what Brouwer is saying; I don't think we need to speculate about the influence on his mathematical activities – although it is not a secret that his proof of the topological invariance of dimension is not intuitionistically acceptable. By the independence of experience I'm sure he refers to people like Kant and Russell, who maintained that our mathematical intuition of the mathematical abstractions used to model space and time was the result of experiencing actual space and time, whereas Brouwer maintained – right or wrong – that to have a mathematical intuition about a mathematical system it is never necessary to have experience of any actual system "behaving" like that mathematical system but existing independently from it and operating outside the realm of mathematics. I don't know whether he ever said anything about whether such experience might be helpful – it may certainly be helpful as inspiration for formulating conjectures – but that is different from being necessary.  --Lambiam^Talk 01:00, 12 January 2007 (UTC)

In this regard: You sound like you've researched Brouwer. Where or what biographies etc. can you point us toward to help us understand this man? What I've read about him paints him as an abrasive person no one would want to be around.

There is the biography by van Dalen mentioned in the article on Brouwer. Unfortunately, I don't have access to a library, and it is beyond my budget to order it – it is quite expensive. From what I remember reading about the person, long time ago, my impression was more of a strange man than of an abrasive one. Although I met several people who knew him personally, including Heyting and Beth, they never mentioned anything in this connection.  --Lambiam^Talk 01:00, 12 January 2007 (UTC)

Concerning the last question. Brouwer's position is not a mathematical proposition: it is about mathematics, and as such external to mathematics. If second-order logic is a mathematical system, then Brouwer's objections do not fall in its province. Brouwer's objection to the belief that formal logic can provide the foundation of mathematics cannot be expressed in formal logic.  --Lambiam^Talk 23:11, 10 January 2007 (UTC)

(Are you sure about that last statement? You're saying this statment cannot be Godelized and cannot be used e.g. to see if it creates an antinomy, or it cannot be put to the diagonal method?)

In a sense his objection is rather similar to the objection Wittgenstein had to certain approaches to philosophy. Paraphrasing: "You can play with words and symbols in formal logic, but what does it have to do with mathematics and mathematical truth? To the extent I can see a meaning in them, I see no reason for believing that the rules for playing with them are valid in the mathematical domain; the justifications offered will not wash." I am quite sure this objection cannot be formalized without losing its essence.

My last question is on topic: it was not addressing Brouwer's "philosophy" (aka opinion, position ...) but rather is a more general, straight-up logic question: does a statement such as "For all sets D, given any proposition P about D: P V ~P" belong in 2nd order logic? Is it an axiom? What is it? Just an opinion? If one asserts the contrary: 'It is not a truth that "for all sets D, given any proposition P about D: P V ~P' " again, what is this? An axiom? Suppose the specified sets D are finite? i.e. "For all finite sets D, given any proposition P about D: P V ~P ". Can we prove this is a truth? What axioms lay beneath the assertions? (As I write this I am bothered by the possibility that we could godelize some of these assertions and make antinomies out of them). wvbaileyWvbailey 21:12, 11 January 2007 (UTC)

Systems I can think of that are powerful enough to express that statement are Girard's System F, and Martin-Löf's Intuitionistic type theory. The first is clearly second-order, but not a logical system: there are no proof rules. The second is a logical system, but does not fit the usual definition of being n-th order, and while one can argue it is higher-order, it is not clear that you need that for the formalization of tertium exclusum. However, we can add the Law as an axiom schema to first-order propositional calculus, meaning that for every well-formed proposition P the disjunction P ∨ ¬P is an axiom – so there are infinitely many axioms, but all are first-order. And you can consider the statement, and try to prove it mathematically, not in the formal system, that for every P the disjunction P ∨ ¬P is a theorem that follows from the axioms. A logical system may or may not have that property. To formalize it does not require the use of higher-order constructs.  --Lambiam^Talk 01:39, 12 January 2007 (UTC)

Thanks, good stuff. I found something very interesting, by accident as luck would have it, in:

Martin Davis, 2000, Engines of Logic, W. W. Norton, London, ISBN 0-393-32229-7 pbk.

This is on task, because it contains a couple gorgeous quotes re the LoEM. And you will love the quote re Wittgenstein. It both supports your belief, as expressed in his thesis, that his math flowed from his "philosophy" but it also supports vise versa -- it describes at the influences of the times. The chapter is called "Hilbert to the Rescue" and the subchapter is "Kronecker's Ghost" (p. 91ff):

Kronecker's Ghost:

"The misgivings many mathematicians felt about Cantor's transfinite, and indeed about the entire direction of foundational research, came to a head with Bertrand Russell's making known the contradiction he had found in what seemed to be straightforward reasoning. As we have seen, Frege simply gave up on his life's work when he received a letter containing Russell's paradox....

... wherein begin two pages of background history re Frege, Dedekind, Peano, Hilbert and Poincare, Russell and Principia Mathematica ...

"While Bertrand Russell labored to find a logical basis for the full breadth of classical mathematics while avoiding the paradoxes, a brilliant young Dutch mathematician, L. E. J. Brouwer had convinced himself that much of it was fatally flawed and needed to be discarded. Brouwer's doctoral dissertation of 1907 showed such hostility to Cantor's transfinite and to much of contemporary mathematical practice that one might have thought him possessed by Kronecker's spirit. In 1905, Brouwer had taken time from his mathematical pursuits to publish a short book, Life, Art and Mysticism, drenched in romantic pessimism. After portraying life in this 'sad word,' as an illusion, this morose young man concluded with:

... here ensues a long quote of Brouwer's nihilistic rant ...

"Despite his praise for the life of self-abnegation, Brouwer embarked on a self-righteous campaign to reconstruct mathematical practice from the ground up so as to satisfy his philosphical convictions. Although he could easily have chosen a conventional mathematical topic, he was determined instread to write his doctoral dissertation on the foundations of mathematics.¹⁷ His adviser reluctantly agreed, but appalled by his prize student's insistence on injecting his strange and irrelevant ideas into his dissertation, he wrote:

" 'I have again considered whether I could accept Chapter II as it stands, but honestly, Brouwer, I cannot. I find it all interwoven with some kind of pessimism and mystical attitude to life which is not mathemamtics, nor has anything to do with te foundations of mathematics.'¹⁸

"For Brouwer, mathematics exists in the consciousness of the mathematician and is ultimately derived from time [Davis's italics] as the "mathematical Primordial Intuition." The real mathematics is in the mathematician's intuition and not in its expression in language. Far from mathematics being logic (as Frege and Russell had maintained), logic itself is derived from mathematics. For Brouwer, Cantor's belief that he had found different sizes of infinity was nonsense and his continuum problem was a triviality. Hilbert was mistaken in claiming that consistency is all that is need for mathematical existence. On the contrary:

' to exist [Brouwer's italics] in mathematics means; to be constructed by intuition; and the question whether a certain language is consistent, is not only unimportant in itself, it is also not a test for mathematical existence.'

"Echoing Kronecker's call for construction as the only valid method for establishing existence in mathematics, Brouwer went further and denounced the use of a fundamental law of logic,Aristotle's law of the exluded middle [Davis's italics, boldface added] (which simply asserts that any proposition is either true or false) when applied to infinite sets²⁰. For Brouwer, some propositions can neither be said to be true or to be false; thee are propositions for which no method is currently known by means of which this can be decided one way or the other. Hilbert's original proof of Gordan's conjecture used the law of the excluded middle [boldface added] in the way mathematicians usually do: he showed that denying the conjecture would lead to a contradiction. To Brouwer such a proof was unacceptable.

"After completing his dissertation, Brouwer made a conscious decision to temporarily keep his contentions ideas under wraps ... [but] ...After obtaining a regular academic appointment... [Davis notes, with the help of Hilbert]... Brouwer felt free to return to his revolutionary project which he was now calling intuitionism.

...wherein Weyl deserts his mentor Hilbert and turns to the dark side ... "he was hooked"

"...in an address delivered in 1922, Hilbert responded to his former student's [Weyl's] desertion as if to treason:

" 'What Weyl and Brouwer are doing amounts in essence to taking the path once laid out by Kronecker: they seek to provide a foundation for mathematics by pitching overboard whatever discomforts them and declaring an embargo a la Kroneker. But this would mean dismembering and mutilating our science, and, should we follow such reformers, we would run the risk of losing a large part of our most valued treasures. Weyl and Brouwer outlaw the general notion of irrational number, of function, even of number-theoretic function, Cantor's [ordinal] [Davis's brackets] numbers of higher number classes, etc. The theorem that among infinitely many natural numbers there is always a least, and even the logical law of the excluded middle [boldface added], e.g. in the assertion that either there are only finitely many prime numbers or there are infinitely many; these are examples of forbidden theorems and modes of inference. I believe that impotent as Kronecker was to abolish irrational numbers (Weyl and Brouwer do permit us to retain a torso), no less impotent will their efforst prove today. No! Brouwer's [program] [Davis's brackets] is not as Weyl thinks, the revolution, but only a repetition of an attempted putsch with old methods, that in its day was undertaken with greater verve yet failed utterly. Especially today, when the state power is thourough armed and fortified by the work of Frege, Dedekind, and canotr, these efforts are foredoomed to failure.22"

"²⁰In the example given of a nonconstructive proof, the law of the excluded middle [boldface added] is used in the assertion "q must be either rational or irrational."

"²¹Weyl was particularly upset by the use of so-called impredicative defintions in the work of Cantor and Dedkind. Something is defined impredicatively if the defintion is in terms of a set which the item being defined is a member. From the point of view of a philosophy in which mathematical objects are constructed a bit at a time, such a definition is seen as being objectionable because the set in question cannot have been constructed before one of its elements. The contrary phiosophical view that mathematical objects are pre-existing and definitions mererely signle them out (like the charaterization: Mathilda is the tallest person in the room) rather than construct them is called Platonism and was unacceptable to Weyl."

The sub-chapter ends with a discussion of WWI and Hilbert.

The above quote is echoed, in a slightly different translation and in summary by Reid (1996) p. 155-156.

With regards to the LoEM I believe we can safely say that Brouwer disallowed the use of it for infinite sets. But from the above, his attitude toward infinity remains murky. [Somewhere I will find a quote re Brouwer's muddled philosphy... I just don't know where? Reid? Goldstein?] Above you wrote that he allowed ω. But my understanding of ω is it is just one of many ω's that we can construct at will. And I cannot follow at all how Brouwer got from romantic pessimism and a belief that "logic itself is derived from mathematics" to "Cantor's belief that he had found different sizes of infinity was nonsense and his continuum problem was a triviality". There is an interesting sort of thread running through this about the impredicative definitions and time... i.e. Brouwer's example mentioned in van Heijenoort:

"In 1948 he introduced an infinitely proceeding sequence whose definition depends upon whether a certain mathematical problem has, or has not, been solved at a certain time [van h. gives the example here] p. 334-335."

Dawson (1997) has a great quote re the influence of Brouwer on Godel:

"Brouwer, however, took a very different view. For an intuitionist there was no reasaon to expect of every formula that either it or its negation should admit of a constructive proof ... and he went even further: In the first of [his lectures] he drew a sharp distinction between "consistent" theories and "correct" ones -- an idea that seems to have suggested to Godel that even within classical mathematics formally undecidable statements might exist¹²⁴.

[124] According to Feigl (1969, p. 639), Brouwer's lectures also galvanized Wittgenstein to resume his work in phiosophy. For the texts of both lectures see Bourwer 1975, pp. 417-428 and 429-440, respectively.

I have more stuff, but will have to look it up. wvbaileyWvbailey 18:07, 12 January 2007 (UTC)

Here's a footnote in Goldstein (2005) Incompleteness: The Proof and Paradox of Kurt Godel that gets at (opines ... is her opinion a truth? ) how an intuitionist gets from "intuitionism" as a philosophy to disavowal of infinite (as opposed to unbounded) numbers:

⁶ "The intuitionists were the most severe of all when it came to the question of acceptable methods of proof. Mathematical proofs were to be limited, according to the intuitionist, to "constructive proofs,', i.e., those that employed concrete operations on finite or "potentially" (but not actually) infinite structures. Reference to completed infinite structures were forbidden, as were indirect proofs making reference to the law of the excluded middle [boldface added].... "Intuitionism," by the way, might seem like a misleading name, considering the way we have been speaking of intuitions up until now, as just the sort of appeals to objective mathematical truths that formlists and intuitionists meant to eliminate. The intuitionists claimed that their finitary constructions were actually mental constructions, and in fact the only sort of mathematical constructions that we, being finite, could actually perform. So they were claiming that their strictures on mathematical proofs actually corresponded to human psychology."

Question: is this a true reading of intuitionism? Because if it is, it goes a long way to explaining their "philosophy." I read this in the sense of Turing (1936), if more tape is needed for the computation, we just add it on. Thus the length of tape is unbounded -- no upper limit defined -- and yet not "infinite" in extent. wvbaileyWvbailey 19:18, 12 January 2007 (UTC)

ChristopherMathsEdwards 23:34, 25 April 2007 (UTC)

Hello, this is my first ever Wiki edit so bear with me. I have decided to have a go at altering this article with a few paragraphs on Michael Dummett's interpretation of Intuitionsism over the next few weeks, his thought explain more clearly why one might adopt the (usually) more restrictive forms of mathematical reasoning that Browser etc. mooted, and hopefully this will be relevant to those who want to learn about the subject. Just to test the water, have any of the contributors of the talk page above read any of Dummett's work on Intuitionism?

No, I have not read any of Dummett; the name's vaguely familiar; he's a philosopher isn't he? Perhaps add a new section with his point of view as a 'characterization' -- I did this with the algorithm page (and eventually had to create an algorithm characterizations sub-article because there are so many different 'charaterizations'). Such an approach could be expanded to include Kleene (1952), etc. as a 'characterization' as well. This would help the reader to realize that "intuitionism" has been and is subject to various interpretations, that it is not "cut and dried". wvbaileyWvbailey 16:10, 26 April 2007 (UTC)

Find irrational a and b such that a^b is rational

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 a^b 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.  --Lambiam^Talk 11:56, 9 May 2007 (UTC)

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)

A historical perspective

First of all, the article should mention that LEM is also known as "The law of the excluded third". reference: http://time-binding.org/akml/akmls/35-taylor.pdf page 21, and other references you can easily find on the web.

This is a very old idea, please research it historically.

From the beginning it was a social, religious rule. You are either a member of the christian community or you are not, there is no third option, there is no middle option. You are either a reborn or you are not.

You are either a member of the bully gang in this schoolyard or you are a bully victim, no other option is allowed. You either submit to the god or you do not submit, no other option is allowed.

This old social traditional rule became a religious rule, and the theologians tried for a long time to use logic to control the language and the minds of the people.

In modern times some people do not know anything about the history and the background of this rule, and they are trying to make it into a purely theoretical law in a logical model.

I was standing in the Austrian mountains, surrounded by billions of small drops of water. The drops did not fall to the ground, they just hang there in the air.

As long as I stood still I was completely dry. When I moved forward my face became wet.

According to the law of the excluded middle it either rains or it doesn't rain. No middle possibility is allowed or possible. But that situation in the Austrian mountains shows that there can be other possibilities, which cannot be defined as rain or no rain.

Remember the difference between theoretical ideas and the reality, logic as we see it in the modern world is purely theoretical, but the background of logical ideas is based in reality, in a tribal or religious tradition.

Many problems in modern science and the formal sciences come from the historical heritage which is incorporated in these areas. (Roger J)

If you can find reliable sources treating this that will serve as a reference, you can make the additions yourself. Without such sources, the foregoing is what we call "original research" and can not be used.  --Lambiam^Talk 07:36, 20 May 2007 (UTC)

Aristotle's quotes and Reichenbach's tertium non datur argument about the proposed use of the exclusive- rather than inclusive-or touches on the problem. If Roger J really wants to pursue this I suggest he get a cc of Reichenbach (it's in paper-back print and not expensive). It is not an easy read. Probably the philosophy behind the whole thing evolved not from the need to coerce the unfaithful, but rather from the human desire to comprehend "space-time simultaneity" -- Kafka can be a student OR a beetle but not a student AND a beetle simultaneously in space and time.

"By exploring issues arising from spatiotemporal locality, we will run into many (though not all!) of the most fundamental and historically significant questions in the philosophy of physics...."(Marc Lang, 2002, An Introduction to the Philosophy of Physics: Locality, Fields, Energy and Mass, Blackwell Publishing, Oxford UK).

If Roger J or anyone else knows of any good books along this line and the spillover into tertium non datur I'd be interested in knowing about them. wvbaileyWvbailey 20:17, 20 May 2007 (UTC)

Argument map representation

I am thinking of adding a visual representation of the simple deductive schema of excluded middle to this page. Image:Anonymous Lefty excluded middle.png. It is a real example of an informal argument, see here. Any thoughts? - Grumpyyoungman01 00:29, 20 May 2007 (UTC)

It is a good example of a fallacious argument, but what would it be doing here? Do you really want to point out the hidden assumptions in the example making it an informal fallacy (such as that the reduction in danger, if real, is due to the leadership of these man; or more egregious, that only these men can protect us against evil if it hasn't been subdued)? This is not the place to bring in political convictions. But if the fallacy is not exposed, it may be even more confusing; readers would think it is a formal fallacy due to the application of the Law. Since, in the example, the application of LEM is just the capstone on a crumbling construction, and the fallacy is in that construction, it is not clear what this would illustrate. It is not truly a "real example", since it is clearly presented in irony by Mr. Sear. In choosing examples, it is best if they do need no explanation and don't draw on the political nonce; sooner than you think, younger readers will say: "Who the **** is Tony Blair?"  --Lambiam^Talk 06:36, 20 May 2007 (UTC)

Thanks Lambiam, one thing I am not sure of is on which article to put this image if it were appropriate. This does seem like the wrong place and maybe nobody has as yet created the fallacy(argument) page. You should also be aware of the NPOV dispute about the image on its talk page. - Grumpyyoungman01 11:59, 20 May 2007 (UTC)

Since you ask, I don't think it would be appropriate anywhere (except perhaps, if Mr. Sear's blog was notable enough to merit an article, to explain the point made to the ironically challenged – but even then I would have my doubts). I don't really see what fallacy or whatever the diagram is supposed to illustrate, but nevertheless, whatever the fallacy may be, I'm fairly convinced this diagram is not a good illustration of it.  --Lambiam^Talk 12:50, 20 May 2007 (UTC)

I better correct something here, I obviously wasn't paying attention. It is not a fallacy, but may appear that way because the premises are not backed up by other premises (the whole argument is not fully fleshed out). I have the argument form written down as a simple deductive schema of excluded middle. Does this argument pattern have its own article? If not I will start it and use the image keeping the Ps and Qs and ditch the other stuff. In summary this is not supposed to represent a false dilemma but a true one. - Grumpyyoungman01 23:21, 20 May 2007 (UTC)

Yes, mind your Ps and Qs. Is this particular form of dilemma notable? You can't just start a new article on an argument pattern and invent your own name for it. The topic of an article has to enjoy a certain notability as such, the first sign of which for this would be a name commonly understood to mean this specific argument pattern. This form of the dilemma is found in that leading to the implicit conclusion of the common saying "Damned if you do, damned if you don't". (Take P = "do" and Q = "damned".) (In Wikipedia, Damned if you do, damned if you don't redirects to Catch-22 (logic), an article – whose own logic is severely impaired – that displays a quite different argument as being "Catch-22".) This form of the dilemma is also the pattern of the nonconstructive proof of the theorem "There exist irrational numbers a and b such that a^b is rational" also found in this article. And it is given in symbolic form in the article on Philosophical skepticism as the form of Thomas Reid's argument against skepticism. There Reid's argument is called "a dilemma, like this: if P, then Q; if not-P, then Q; either P or not-P; therefore, in either case, Q." I don't know a name for this particular form, and I'd think that if there was a common name for it, one of the editors at the Philosophical skepticism article, and before at Skepticism, would have known and supplied that name in the now almost six years this text has been with us.[1]

What I think you could do, if you feel up to it, is replace the current redirect page at Damned if you do, damned if you don't by a somewhat decent, if still stubby, proper article, and add a Rationale diagram as illustration. If you choose to do that, then you should (in my opinion) substitute P = "Damned" and Q = "Do" in the diagram at Image:Anonymous Lefty excluded middle.png, and, moreover, leave out the first of the three bottom boxes, since "P or not P" is not an assumption of the argument.  --Lambiam^Talk 05:07, 21 May 2007 (UTC)

Done!, although it needs copyediting and references, not quite quality yet. Thanks for your input. - Grumpyyoungman01 10:10, 21 May 2007 (UTC)

Schrodinger's catscrawl

> "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)