Talk:Law of excluded middle/Archive 2

Archive 1

Archive 2

Example seems wrong

The example given in the second paragraph seems wrong. If p = “we know a train is coming” and q = “it is dangerous to cross the tracks” then p => q is true. Wikipedia seems to be asserting that ~(p => q) states, “It is not dangerous to cross the tracks when we don’t know that a train is coming.” The statement in quotes is actually ~p => ~q, which is different from ~(p => q). I agree that it is a fallacy to conclude from p => q that ~p => ~q, but I don't think this illustrates the LoEM. What is needed is an example where p => q is true but ~(p => q) is not false. Soler97 (talk) 21:21, 7 October 2011 (UTC)

I agree that the example is opaque; frankly, when I read it, it didn't help me understand the LoEM, one iota. Maybe by cranking through the example we can come to some understanding of it. Let's start with a discussion of how we think how we derive conclusions, i.e. with modus ponens:

t & (t => ~s) | ~s

In words: We assert the truth of a premise -- that it's unsafe (~s) to cross a train tracks when the train (t) is coming: (t => ~s). Then we assert a truth, that in fact a train is coming (t). From the truth of these two occurring simultaneously in space-time: "it is true that: 'a train is coming' " & "it is true that: 'If a train is coming then it's unsafe' " we derive the truth: 'it is unsafe'.

By the way, the following is not trivial: Bertrand Russell suffered with the difficulty of it, too. We have to assert this modus ponens business as "primitive" and be very certain about what it really means: given a truth t and another truth ~s and the fact that their space-time coincidence t & ~s is also true, we can derive ~s as a truth. This is so primitive it stands outside “the logic”. This is so primitive that it was Russell’s *1.1: "*1.1 Anything implied by a true proposition is true" (Principia Mathematica, 2nd edition 1927:94, emphasis added). Russell goes on to say that "we cannot express the principle symbolically, partly because any symbolism in which p is variable only gives the hypothesis that p is true, not the fact that it is true [here he references his 1903]” (ibid).

Unfortunately, modus ponens is confusedly-expressed in the form of a "material" [? the article material implication is really confusing] implication of some sort or other, not as a logical AND. But modus ponens only works when both the t AND the ~s are truths so that their conjunction t & ~s is a truth. The truth of the terms in a modus ponens has to be formed outside the logical calculus e.g. experience, heuristics, i.e. basically by inductive reasoning (aka animal logic, e.g. see Bertrand Russell's The Problems of Philosophy on this); a modus ponens asserts nothing whatever about the other three possible cases: (no train AND safe), (no train AND not safe), (train AND not safe); it only deals with the known truths -- train t, NOT-safety of crossing the tracks ~s AND train t, given time- and space-coincidence of the two terms ~s, t.

The problem with so-called "material" [?] implication, e.g. (t => ~s) in its standard form is that it yields truth when the first term t is false, without regard to the truth of the second term ~s. The only time we can be sure of the truth of this is when t is NOT false (a truth) and in the standard logic this occurs only in the t & ~s term is true: i.e. both t and ~s are true simultaneously in space-time.

t & (t & ~s) / ~s (here I'm using / to "detach" the ~s, rather than the | of modus ponens)

In words: We assert that in fact 'a train is coming' (t), and we assert the truth of: 'a train is coming' AND 'it is unsafe to cross the tracks in space-time coincidence with the train'. From the truth of these two facts we derive the truth: 'it is unsafe to cross the tracks in space-time coincidence with the train'.

So let's go back to the example. What is it saying:

~t & (t => ~s) / ??

We cannot use traditional modus ponens (it would require this form: ~t & (~t => s) | s, for example). The fact that our thinking does not form a modus ponens directly means that we must find out what is "the truth" inside our reasoning ~t & (t => ~s). We can do this without a truth table: just convert the traditional material implication (~t --> s) to its minterms: [(~t & ~s) V (~t & s) V (t & ~s)], and again look at the assertion ~t & (~t => s) / ??

~t & [(~t & ~s) V (~t & s) V (t & ~s)] = [~t & (~t & ~s] V [~t & (~t & s)] V [~t & (t & s)] / ??

If we accept that ~t & t is void, the right term vanishes . This leaves us with the equivalence:

[~t & (t => ~s)] = [~t & (~t & ~s] V [~t & (~t & s)] / ??

This can be converted back into two modus ponens . . . if we accept the argument about the use of logical AND in its one true minterm as replacement for the vague material implication(note the vertical line indicating “detachment of the conclusion”):

[~t & (~t => ~s)] V [~t & (~t => s)] | ~s V s

In other words, we've derived that it's either safe OR it's unsafe to cross the tracks! A cautious man would be worried. Maybe there's a bomb, a booby-trap, just waiting to be triggered when we cross the tracks (in warfare this being a real possibility). The situation is "unknown" -- is it safe? Is it unsafe? Who knows from the logic? We need more real-world experience to determine the safety of the situation, i.e. if t our bomb-sniffing dogs determine that there are is no explosives under the tracks, then indeed maybe, in this instance it's safe to cross the tracks even when no train is coming.

I cranked this a couple ways, and this was the most direct derivation, but it assumes a logic that is very cautious about the use of modus ponens. Reichenbach and Russell deal with this topic. I need to go away and do some research on this. Maybe someone else has a shorter answer. Bill Wvbailey (talk) 17:23, 8 October 2011 (UTC)

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This isn't correct:

"The principle of excluded middle, along with its complement, the law of contradiction (the second of the three classic laws of thought), are correlates of the law of identity (the first of these laws). Because the principle of identity intellectually partitions the Universe into exactly two parts: "self" and "other", it creates a dichotomy wherein the two parts are "mutually exclusive" and "jointly exhaustive". The principle of contradiction is merely an expression of the mutually exclusive aspect of that dichotomy, and the principle of excluded middle is an expression of its jointly exhaustive aspect."

The principle/'law' of excluded middle [LEM] and the 'law' of contradiction [LOC] aren't correlates of the Law of Identity [LOI]. The LEM and the LOC concern the truth-functional connection between a proposition, indicative sentence or clause and its negation. The LOI concerns the alleged identity between an object and itself (or their names, depending on which version one accepts), it states nothing about the identity of a proposition and itself, nor their truth-functional connection.

Moreover, the LOI partitions nothing (or if it does then we need to see the proof), nor do the LOC or the LEM. I suspect the author of the above has confused these 'laws'/'principles' with class exclusion/inclusion operators.

See my comments about this in the discussion page of the 'Laws of Thought' article.

So, this section of the article needs to be deleted.

Rosa Lichtenstein (talk) 08:50, 21 July 2012 (UTC)

(78.234.2.195 (talk) 16:49, 13 December 2013 (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)

The Austrian mountain situation was either rain or not rain. Your inability to categorize it (and mine, for that matter), is not a failure of the Law of the Excluded Middle, but a result of an inadequate definition of the word "rain." Define "rain" sufficiently, and you will find that, according to that definition, what you experienced either was rain, or it wasn't.98.209.219.62 (talk) 15:58, 9 March 2009 (UTC)

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The notion of either being a member of or not being a member of any group already presupposes the ability to think this. It is not clear how this ability should have arisen from social membership alone. Historically, however, it would be important that the article distinguish the law of the excluded middle/third = principium exclusi tertii from more advanced formulations of it as the there is no third = tertium non datur. The former was stated by Aristotle, who however restricted it to statements about past and present as well as to elementary statements, since Aristotelian Logic lacked explicit definitions of the logical particles 'and' and 'not', whereas the t.n.d. could later be stated in the form of A or not-A. Furthermore, the section on Aristotle should concentrate on the p.e.t. and not so much on the law of contradiction (not-(P and not-P)).

For historical, but also logical, reasons, I recommend two separate articles, one on the p.e.t., and one on the t.n.d. The clear advantage of this would be historical correctness, but even more so the understanding of the reader: in the context of p.e.t. discussions of the relation to the principle of bivalence are far more clear, since classical-aristotelian logic clearly excludes a third here — in the modern contexts of t.n.d. this is different, as there are multi-valued logics that still accept the t.n.d.

-- Morton Shumway (talk) 14:13, 8 October 2009 (UTC)

Examples of what would happen if the Law of excluded middle were to fail. Aristotle used these examples in his discussion of the question: "For it is true that if a thing is a man and a not-man, evidently also it will be neither a man nor a not-man." (Metaphysics Book IV Chapter 7).

The double-negative of an "uncompleted" or expanding universe U ≡ X OR not-X (sign ≡ is logical equivalence) may produce "outliers" that are not included in the the original set X. Intuitionism posits that the mathematical universe of discourse, in particular the integers, is uncompleted. So any argument that relies on the double-negative extended over an uncompleted infinite domain such as the integers cannot be trusted; it may include new elements not in the original domain. In particular ~(~p) --> p is disallowed.

I got out my Aristotle's Metaphysics and re-read the sections re "laws of thought" (Book IV and Book IX Chapter 5 & 6). I had to reread it about half-dozen times, it's so repetitive you get lost wandering about in his words, but anyway.... He posits only one fundamental law of thought (repeating himself about 10 times, in different places): NOT(P & ~P):

"... the most certain principle of all is that regarding which it is impossible to be mistaken...then such a principle is the most certain of all; which principle this is, let us proceed to say. It is, that the same attribute cannot at the same time belong and not belong to the same subject and in the same respect.... This, then, is the most certain of all principles, since it answers to the definition above. For it is impossible for any one to believe the same thing to be and not to be, as some think Heraclitus says." (Book IV Ch 3, p. 524 in GBWW Aristotle volume I."

Okay, so what he is really doing here is rebutting the thinkers Heraclitus and Anaxagoras and Protagoras. Eventually he incorporates into his argument "no intermediate between contraries." But first after much waxing eloquent he then says something interesting:

"For it is true that if a thing is a man and a not-man, evidently also it will be neither a man nor a not-man. For to the two assertions there answer two negations, and if the former is treated as a single propostion compounded out of two, the latter also is a single proposition opposite to the former."(lines 1008, Ch4).

He is saying: M & ~M = ~(M V ~M), which is Boolean-wise correct, or perhaps ~(M & ~M) = (M V ~M). So I drew up two Venn diagrams of this, and realized that indeed his argument involves a (hypothetical) middle term ~(M V ~M): the universe of all things is the Venn rectangle, M is a circle inside the rectange, that which is the middle term (M & ~M) is an annulus around the circle, and the rest is ~M.

To me he is taking M & ~M and substituting ~M for M giving ~M & ~~M. This translates to neither a man nor a not man, with nor meaning "and not" since this is the simplest way of doing it. The neither only applies to the first M so there are no brackets as you write. 88.203.90.14 (talk) 01:37, 14 December 2013 (UTC)

Okay continuing on, in Book IV ch. 7, he finally pulls out another argument, that which what someone (not Plato, nor Aristotle, nor Aquinas, nor Locke) calls the "law of excluded middle":

"But on the other hand there cannot be an intermdiate between contradictories, but of one subject we must either affirm or deny any one predicate." (beginning of Chpt. 7)

He argues that this cannot exist by use of an infinite regress to eliminate the intermediate:

"when there is an intermediate it is always observed to change into the extemes. For there is no change except to opposites and to their intermediates... Again, the process must go on ad infinitim, and the number of realities will be not only half as great again, but even greater. 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."(beginning of Chp. 7)

And finally he finishes his argument:

"And the starting point to deal with such people [arguers such as Heraclitus et. al] is defintiion.... While the doctrine of Heraclitus, that all things are and are not, seems to make everything true, that of Anaxagoras, that there is an intermediate between the terms of a contradiction, seems to make everything false; for when things are mixed, the mixtue is neither good nor not-good, so that one cannot say anything that is true." (1012 end of Chp. 7)

He repeats this argument in Book XI Chps 5-6 (lines 1062-1064).

My old cc of the Great Books of the Western World(Britannica, 1952) has a Syntopicon in two volumes. Volume I contains "Logic", and the following authors are listed under "...the axioms of logic: the laws of thought; the principle of reasoning" (Volume 2 in the GBBW, p. 1043-1044)

Plato, Aristotle, Epictetus, Aquinas, Hobbes, Locke, Kant, Hegel, and James.

Of these I've looked at all but James. Epictetus, Hobbes are not applicable, Kant is useless; James I haven't looked at. None of these mention the "no intermediate between contraries" excepting Aristotle. I've also word hunted through De Morgan 1847, Boole 1854, Jevons 1880, Venn 1881, and Couturat 1917. Neither De Morgan nor Boole mention any of this (and I've tried every word-combination I can think of). But Venn 1881 is useful:

Venn, Symbolic Logic, 1881, "Logical statements or equations":

"But since our alternatives are collectively exhaustive as well as mutually exclusive, it is a contradiction in terms to suppose them all to vanish; this, it will be noticed, being our generalized form corresponding to the so-called Law of Excluded Middle. Suppose for instance, just for illustration, that we write down such a form as this,

Axy +Bx~y +Ox~y +D~x~y = 0,

one or more of the four factors A, B, C, D, must be supposed = 0, in order ~ avoid contradiction." (p. 237)

Jevons 1880 is also useful -- Bertrand Russell cribbed it without attribution, Jevons's 1880 presentation is identical to Russell's:

Jevons 1880 LESSON XIV. THE LAWS OF THOUGHT. p. 117

"BEFORE the reader proceeds to the lessons which treat of the most common forms of reasoning, known as the syllogism, it is desirable that he should give a careful attention to the very simple laws of thought on which all reasoning must ultimately depend. These laws describe the very simplest truths, in which all people must agree, and which at the same time apply to all notions which we can conceive. It is impossible to think correctly and avoid evident self contradiction unless we observe what are called the Three Primary Laws of Thought which may be stated as follows;

I. The Law of Identity Whatever is, is.

2. The Law of Contradiction Nothing can both be and not be.

3. The Law of Excluded Middle Everything must either be or not be.

Though these laws when thus stated may seem absurdly obvious, and were ridiculed by Locke and others on that account, I have found that students are seldom able to see at first their full meaning and importance.

It will be pointed out in Lesson XXIII, that logicians have overlooked until recent years the very simple way in which all arguments may be explained when these self-evident laws are granted; and it is not too much to say that the whole of logic will be plain to those who will constantly use these laws as the key."

In an amplification of the three laws, he continues:

"This law of excluded middle is not so evident but that plausible objections may be suggested to it. Rock, it may be urged, is not always either hard or soft, for it may be half way between, a little hard and a little soft at the same time. This objection points to a distinction which is of great logical importance, and when neglected often leads to fallacy. The law of excluded middle affirmed nothing about hard and soft, but only referred to hard and not-hard; if the reader chooses to substitute soft for not-hard he falls into a serious confusion between opposite terms and contradictory terms."

Finally, to end this, I offer up Couturat 1917, which is really interesting, especially the footnote:

r6. The Principles of Contradiction and of Excluded Middle.

"By definition, a term and its negative verify the two formulas

aa' = 0, a + a' = 1,

which represent respectively the principle of contradiction and the principle of excluded middle.¹

C. L: 1. The classes a and a' have nothing in common; in other words, no element can be at the same time both a and not-a.

2. The classes a and a' combined form the whole; In other words, every element is either a or not-a.

¹ As Mrs. LADD•FRANKLlN has truly remarked (BALDWIN, Dictionary of Philosophy and Psychology, article "Laws of Thought"), the principle of contradiction is not sufficient to define contradictories; the principle of excluded middle must be added which equally deserves the name of principle of contradiction. This is why Mrs. LADD-FRANKLIN proposes to call them respectively the principle of exclusion and the principle of exhaustion, inasmuch as, according to the first, two contradictory terms are exclusive (the one of the other); and, according to the second, they are exhaustive (of the universe of discourse)" (p. 23-24).

I haven't figured out what to do with this yet. I don't want to be accused of "original research". Whether or not "original research" includes reading original texts has come up from time to time. But the GBWW Syntopicon did most of the work re the "ancients". And word hunting can be done by a robot. Plus I haven't figured out where the name "Law of Excluded Middle" came from, nor how it got to be included in the list of "Laws". Locke 1689-1690 and Hegel 181? only recognize the first two, for example. So between Hegel 181? and Jevons 1880 the LoEM got included to make up the 3rd law. BillWvbailey (talk) 17:45, 13 October 2009 (UTC)

What you do is great. I am not sure whether you in your last remark refer to the inclusion of excluded middle in a list of Laws, or about its discussion in general. Anyway, I looked it up in Hegel 1812-16 and 1817—

Hegel in the Enzyklopädie der philosophischen Wissenschaften and the Wissenschaft der Logik contains some remarks on the 3rd law. In the Jena Logic, however, it constitutes a principle by itself. In the Science of Logic (Book Two: The Doctrine of Essence/Section One: Essence as Reflection Within Itself/Chapter 2 The Determinations of Reflection/C Contradiction/Remark 2: The Law of the Excluded Middle) there are three §§, starting with

§ 952

"The determination of opposition has also been made into a law, the so-called law of the excluded middle: something is either A or not-A; there is no third."

In the Encyclopaedia of the Philosophical Sciences in Outline (A) The Science of Logic/II. The Doctrine of Essence/(A) The Pure Categories of Reflection/(b) Difference) you can find in § 119:

"The Maxim of Excluded Middle is the maxim of the definite understanding, which would fain avoid contradiction, but in so doing falls into it. A must be either +A or -A, it says. It virtually declares in these words a third A which is neither + nor -, and which at the same time is yet invested with + and - characters."

It gets mentioned explicitly a second time in § 119, Remark 2.

-- Morton Shumway (talk) 16:08, 17 October 2009 (UTC)

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I'm happy to see someone else has some historical insight on this topic. What you found re Hegel is really interesting. I guess my questions are (1) When and how did the LoEM become "a law of thought" in its own right? i.e. who was the first "modern" to propose that it should be a "law", and why? (2)What is the exact mathematical reasoning that would demonstrate that both the LoEM is required as well as the Law of Contradiction in order to frame "logic" as we know it (this alludes to the Ladd-Franklin quote above; I'm sure it has to do with the definition of the "all" and the "none" but I'd like to see it in print). In this regard I just discovered something very interesting -- a rather deep history of the LoEM written by William Hamilton 1883 Lectures on Metaphysics & Logic Volume 2 Logic. There are other, earlier, editions with similar-sounding names, the "Volume 2 Logic" and the date 1883 is the key to the correct text. The history begins on p. 62 and includes Plato, Aristotle, "the Paripetitics", Locke, Leibniz, Schelling and Hegel (among others). I was able to download this from googlebooks (there's a cc for sale, but it's also on googlebooks for free) but I haven't had time to assimilate it (i.e. convert to text so I can search it). Bill Wvbailey (talk) 22:05, 17 October 2009 (UTC)

(1) I don't know much about that—at least in the sense of "when was it conceived of and stated as in unity with what was termed 'laws of thought'". I can say that Hegel conceives of 1. Identity, 2. Excluded Contradiction, 3. Excluded Third and 4. Sufficient Reason (Satz vom Grund) of necessary unity in regard of classical logic. My guess is that it should be alike in Leibniz. — I think it is fair to argue that already in Aristotle the principium exclusi tertii constitutes a 'law of thought' in the sense that what he is concerned with is the adequacy of thought and being (as in 'is thought capable of thinking what is?', a problem that already Parmenides is concerned with, stating that thought and being is one.). — I assume that in the context of modern, formalist logic, the question it less about 'thinking' as a human faculty, more about what is constructible as a logical calculus, which after all can be taken to grasp all of what can be thinkable in principle. However, questions can come up as to whether people 'actually' think in such-and-such way, or whether natural language complies with a proposed law of thought. I reckon that quite an amount of discussion in e.g. cognitive science, neuroscience, game theory, and all the 'computational' flavours is more or less explicitly involved with that question.

(2) Here, the problem is quite similar — a remark aside: if mathematics rest on reasoning ("mathematical reasoning"), is it then a question of math or of reasoning? – this is the point of discussions about whether it is good to mathematise logic, and whether one should only accept what is constructible —: if a calculus is reasonably possible which transcends the 'laws of thought', then any calculus able to 'demonstrate' a necessity would only do so in its own context.

In both cases I would stress the possible need for a return to ontology. But I have to say that I could not in detail work through your contribution, so I might come back to this later. — A great deal of inspiration for me at the moment comes from the works of Gotthard Günther. His PhD thesis was about Hegel's preoccupation with classical logic.

-- Morton Shumway (talk) 23:04, 17 October 2009 (UTC)

So, now I have understood that you are dealing exactly with what keeps me busy these days, namely the 'status' of the logical basic axioms. I think there is no problem as to the 'original reasearch' question - the issue has been noticed now and then, and understanding Aristotle has to deal with getting the right grasp of how the 'laws' are to be understood as such. I also think that what I wrote above already touches your concern.

I have to go now and expand later, for now there's an article on the issue: http://www2.swgc.mun.ca/animus/Articles/Volume%201/andrews.pdf

-- Morton Shumway (talk) 15:33, 19 October 2009 (UTC)

I will add one passage I found in Kant's Critique of Pure Reason (Second Division. Transcendental Dialectic / Book II. The Dialectical Inferences of Pure Reason / Chapter III. The Ideal of Pure Reason / Section 2. The Transcendental Ideal — A 571f./B 599f.) The context in Kant is that he generally problematises classical logic's consequences for thought, which we can also see here, in the distinction between Allgemeinheit (Universality) und Allheit (Totality) (The "1)" is a footnote in the text):

"But every thing, as regards its possibility, is likewise subject to the principle of complete determination, according to which if all the possible predicates of things be taken together with their contradictory opposites, then one of each pair of contradictory opposites must belong to it. This principle does not rest merely on the law of contradiction; for, besides considering each thing in its relation to the two contradictory predicates, it also considers it in its relation to the sum of all possibilities, that is, to the sum-total of all predicates of things. Presupposing this sum as being an a priori condition, it proceeds to represent everything as deriving its own possibility from the share which it possesses in this sum of all possibilities 1)".

"1) In accordance with this principle, each and every thing is therefore related to a common correlate, the sum of all possibilities. If this correlate (that is, the material for all possible predicates) should be found in the idea of some one thing, it would prove an affinity of all possible things, through identity of the ground of their complete determination. Whereas the determinability of every concept is subordinate to the universality (universalitas) of the principle of excluded middle, the determination of a thing is subordinate to the totality (universitas) or sum of all possible predicates."

-- Morton Shumway (talk) 14:51, 19 October 2009 (UTC)

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The Andrews article looks very interesting. What started out (for me) as a 1/2 hour inquiry has turned into a something huge. So to proceed will require a different tack. For just one example, I was unaware of a fourth law of thought until I read your Hegel synopsis, but now I find it also in the Hamilton 1883. And Hamilton mentions a possible 5th Law of Double Negation and its "application to the infinite" (see quote below). Then there's the business of the Hegelian "absolute" and Schelling and Hegel's "repudiation" of the "principles of Contradiction and Excluded Middle as having any appliation to the absolute"(cf Hamilton 1883:64 and very long footnote 4); this looks to be very much like the Brouwer Intuitionism issue. What I am in the process of doing now is transcribing the relevant Hamilton chapter V so that I can copy it into a wiki-article under my "name space"; in that way we (anyone) can look at it and copy it. I downloaded Hamilton as a pdf and then OCR'd it, but the result is full of OCR-errors and format issues that I have to fix line by line. Clearly this is going to take some time. I'll provide a link to that place when I create it. Bill Wvbailey (talk) 16:53, 19 October 2009 (UTC)

"In the more recent systems of philosophy, the universality and necessity of the axiom of Reason has, with other logical laws, been controverted and rejected by speculators on the absolute.⁴·

⁴[On principle of Double Negation as another law of Thought, see Fries, Logik, §41, p. 190; Calker, Denkiehre odor Logic und Dialecktik, §165, p. 453; Beneke, Lehrbuch der Logic, §64, p. 41.] (Hamilton 1883:68)

Basically put, the issue with Hegel etc., or what can be called 'transcendental logic', is that Subject/Thought/Mind (Geist) are grasped in terms of a different relation between Subject and Object, or Thought and Being. The double negation is a good example: in classical logic, not-not-A is A, but in Hegel, it is not (Thesis, Antithesis (first negation), Synthesis (second negation)). It's metaphysics (thus ontology). I think that a good introduction to the thought-being relation is the lectures on metaphysics by Adorno, but I am not sure whether there is a translation. On the classical vs. transcendental issue, I can again recommend Günther's writings, a lot of which are available online and in English. -- Morton Shumway (talk) 20:28, 19 October 2009 (UTC)

Reading your notes on Aristotle, I realised the 'ontological' background of his argument. When he talks about the 'third', that is the middle, he refers to the problem of change: the question is whether, when a thing changes in some respect, is there something between its not being this-and-this-way and its being this-and-this-way? The 'middle' would be an intermediate state, and the position he refutes is that nothing could in an infinitely small amount of time become one from the other (this issue is discussed in detail in Aristotle's "On Generation and Corruption"). -- Morton Shumway (talk) 20:45, 19 October 2009 (UTC)

Historical survey -- detailed exerpts from texts re LoEM and the "Laws of Thought"

The compilation at User:Wvbailey/Law of Excluded Middle augments the above. Here you will find either exerpts from the texts of the authors or sourcing information mostly gotten through Hamilton 1860 and The Syntopicon of the Great Books of the Western World.

• Hamilton 1860 -- General definition
• Plato, Strobaeus (exerpt of dialog referenced by Hamilton 1860)
• Aristotle (detailed quotes from above)
• Cicero (stub: see Hamilton 1860)
• Middle Ages (see Hamilton 1860)
• Thomas Aquinas
• Liebniz (stub: see Hamilton 1860)
• Hegel (stub: see Hamilton 1860)
• Hamilton 1860 (extensive exerpt)
• DeMorgan, Boole (stubs)
• Venn 1881
• Thompson 1860 (extensive exerpt)
• Jevons 1880 (extensive exerpt)
• William James (stubs)
• Couturat 1914 with Ladd-Franklin

Sources the Great Books of the Western World (Encyclopedia Britannica) and the original volumes found via Googlebooks, then converted to text by OCR, cut & pasted & then corrected by hand. Of these, I find the Ladd-Franklin quote in Couturat re notions of "mutual exclusion" and "exhaustion", also mentioned in Jevons, to be the most interesting. BillWvbailey (talk) 15:17, 28 August 2010 (UTC)

Excluded middles in Literature

Thomas Pynchon refers to "excluded middles" in The Crying of Lot 49. This has been interpreted as a political statement (i.e. moderate politics should not be excluded), but could he be referring primarily to the Law of Excluded Middles?

"The waiting above all; if not for another set of possibilities to replace those that had conditioned the land to accept any San Narciso among its most tender flesh without a reflex or a cry, then at least, at the very least, waiting for a symmetry of choices to break down, to go skew. She had heard all about excluded middles; they were bad shit, to be avoided; and how had it ever happened here, with the chances once so good for diversity? For it was now like walking among matrices of a great digital computer, the zeroes and ones twinned above, hanging like balanced mobiles right and left, ahead, thick, maybe endless. Behind the hieroglyphic streets there would either be a transcendent meaning, or only the earth."

What you are referring to is an entirely different law of excluded middles, not middle. Even though the names are similar, the subjects are not related. Feel free to create the new page for "middles". Tkuvho (talk) 21:23, 13 November 2010 (UTC)

There seems to be a principle that this article shall be within mathematical and logic science, not politics and society. In my opinion there should be a very short chapter about the application for politics and society, where the middle is not excluded, and have links to appropriate articles, such as Fallacy of the excluded middle. --BIL (talk) 09:44, 14 December 2013 (UTC)

Relationship to law of non contradiction

The law of non contradiction (LNC) is logically equivalent to the law of the excluded middle (LoEM) in Propositional Logic (see below). Doesn't that deserve mention? Isn't the article confused over this point?

Also that it's not equivalent in intuitionist logic.

LNC and LoEM are equivalent using De Morgan's law and the Double Negative law of Propositional Logic:

LNC = ~(P & ~P) = ~P V ~~P = ~P V P = P V ~P = LoEM. 88.203.90.14 (talk) 00:55, 16 December 2013 (UTC)

Importance of An inquiry into meaning and truth,chapter 20 entitled The law of excluded middle.

(84.100.243.211 (talk) 17:51, 13 December 2013 (UTC)) (84.100.243.211 (talk) 18:39, 13 December 2013 (UTC))

To my mind, the twentieth chapter entitled The law of excluded middle, constitutes a sort of climax in the celebrated An inquiry into meaning and truth. In light of Tarski and thanks to the use of the logical hexagon of the Frenchman Robert Blanché in modal logic, a lot of problems raised by Russell in his book and particularly in the twentieh chapter can be solved. Tarski said: the proposition “Snow is white” is true, if and only if snow is white. One may conclude that instead of saying the proposition p is true, one must say that the fact p is certain and symbolize the certainty of the fact p by Lp. If we are in a position to assert: ‘It snowed on Manhattan Island on the first of January in the year 1 Anno Domini’, the fact p in question must be symbolized by Lp, to be read It is a certain fact that it snowed on Manhattan Island on the first of January in the year 1 Anno Domini. If we are in a position to assert: ‘It did not snow on Manhattan Island on the first of January in the year 1 Anno Domini’, the fact not-p in question must be symbolized by L~p, to be read : It is a certain fact that it did not snow on Manhattan Island on the first of January in the year 1 Anno Domini. If we are in a state of ignorance concerning the two contradictory facts p and not-p, in other words, if we are unable to assert ‘It snowed on Manhattan Island on the first of January in the year 1 Anno Domini’ as well as ‘It did not snow on Manhattan Island on the first of January in the year 1 Anno Domini’, we experience a fact, the fact that neither p nor not-p is certain. This third fact can be symbolized by ~L~p & ~Lp, both the certainty of the fact not-p and the certainty of the fact p are excluded. I emphasize here that the third fact I mention must be given as much importance as the facts Lp and L~p we are led to consider when we are in a state of knowledge. The third fact is the fact we have to envisage when we are in a state of ignorance. It corresponds to what is called the bilateral possible. ~L~p, the non-certainty of the fact not-p is equivalent to the possibility of the fact p to be symbolized by Mp, ~Lp, the non-certainty of the fact p is equivalent to the possibibity of the fact not-p to be symbolized by M~p. There exist three situations corresponding to the case envisaged by Bertrand Russell in the chapter 20 of his An inquiry into meaning and truth and entitled The law of excluded middle. One of three things, either Lp the certainty of the fact p or L~p the certainty of the fact not-p or Mp & M~p the possibility of both p and not-p to the extent that both are non-certain. In any of the three situations, the law of excluded middle is preserved. This law can be represented thus: (l) p w not-p. The facts p and not-p are necessarily, by definition ( this is the meaning of the symbol (l) here used) contradictory. They are incompatible and they cannot be both excluded of reality.

The author of these lines thinks that the solution of the Russellian problem renders possible a consistent formula of strict implication

http://mindnewcontinent.wordpress.com/ — Preceding unsigned comment added by 84.100.243.186 (talk) 09:37, 3 January 2014 (UTC)

Opposition to the law of the excluded middle

The article does a very bad job of describing the modern constructive viewpoint. The point of not having the law of the excluded middle is not to deny it. The point is that proofs that do use the law of the excluded middle are fundamentally different than proofs that do not. From the modern viewpoint the law of the excluded middle is not that it's wrong, it's that a proof that does not make use of the law of the excluded middle is stronger than a proof that does use the law of the excluded middle. For example, a proof that does not use the law of the excluded middle gives an algorithm (via the curry-howard isomorphism). Therefore it is a bad idea to universally assume the law of the excluded middle, because that means that all proofs are automatically weaker, and do not give algorithms. It's a better idea to introduce the law of the excluded middle in each theorem that needs it. This is the modern viewpoint. However, this article presents the situation as if there is some kind of opposition between constructive and classical mathematics. There is not. The point of constructive mathematics is simply that fewer assumptions lead to stronger theorems, and that we should not make unnecessary assumptions. — Preceding unsigned comment added by 83.82.131.247 (talk) 20:00, 8 December 2014 (UTC)

Importance of the Law of the Excluded Middle and Multiple Negations of a Proposition

The law of the excluded middle is an extremely important principle in mathematics. Proof by contradiction, based on the law of the excluded middle, is used universally in mathematics. The Wikipedia article does not point this out and it should. Here are two direct quotes from Quinn:

Excluded-middle arguments are unreliable in many areas of knowledge, but absolutely essential in mathematics. Indeed we might define mathematics as the domain in which excluded middle arguments are valid. Instead of debating whether or not it is true, we should investigate the constraints it imposes on our subject.

Hilbert had proposed a precise technical meaning for “true”, namely, “provable from axioms that themselves could be shown to be consistent”. But ten years later Gödel showed that in the usual formulation of arithmetic there are statements that are impossible to contradict but not provable in Hilbert’s sense. In particular, consistency of the system could not be proved within the system. This was seen as a refutation of Hilbert’s proposal. Ironically, it had the same practical consequences because it established “impossible to contradict” as the precise mathematical meaning of “true”.

For example, the set of natural numbers N = {1, 2, 3 …} can be established as an infinite set by contradicting the proposition that it is a finite set.

This Wikipedia page defines the law of the excluded middle as:

In logic, the law of excluded middle (or the principle of excluded middle) is the third of the three classic laws of thought. It states that for any proposition, either that proposition is true, or its negation is true.

And goes on to say that

The earliest known formulation is Aristotle’s principle of non-contradiction where he says that of two contradictory propositions (i.e. where one proposition is the negation of the other) one must be true, and the other false.

The Wikipedia page “Brouwer–Hilbert controversy” adds that

Aristotelian logic is a good example – based on one’s life-experiences it just seems “logical” that an object of discourse either has a stated property (e.g. “This truck is yellow”) or it does not have that property (“This truck is not yellow”) but not both simultaneously (the Aristotelian Law of Non-Contradiction).

For something so important and truly foundational in mathematics, the definitions should be precise and clear of ambiguity. This Wikipedia page creates the following impressions:

1. For every proposition there is a single negation of the proposition. The definition says “its negation is true” and does not say one of its many possible negations is true.
2. The law of the excluded middle holds in general. It does not say that the law is true only in a very specific setting where many (perhaps, unspecified) conditions are assumed to be true.

The Aristotelian logic (example) can be easily shown to violate both the points mentioned above.

Proposition: This truck is yellow

Negation: This truck is not yellow

Law of the Excluded Middle: One of these must be true and the other must be false.

What if the truck is red and yellow? Now, the (previously incompatible) statements “this truck is yellow” and “this truck is not yellow” are both neither entirely true nor wholly false.

A second dimension of multiple colors has been introduced. However, if we look at the proposition it didn’t say anything about single or multiple colors. As stated, it should have applied to both cases. A naïve application of the principle could have resulted in false conclusions in the multiple color case.

Is the law of the excluded middle invalid?

Or, is it that we need to follow Quinn’s advice (“investigate the constraints it imposes on our subject“) and exercise extreme care in properly formulating the proposition and its negation(s)?

To properly restate the Aristotelian logic (example) one can either add an assumption or add an extra claim. For example, one can say that IF the truck is of a single color then either “this truck is yellow” or “this truck is not yellow” but not both simultaneously. But now, one must allow for the two additional negations of the proposition where the assumption is not true. The other equivalent option is to include the assumption as a second simultaneous claim:

Corrected Proposition: This truck is of a single color AND it is yellow

Now it has two simultaneous claims: (a) the truck is of a single color and (b) the truck is yellow and there are three possible negations of the proposition:

First Negation: This truck is not of a single color and it is yellow.

Second Negation: This truck is of a single color and it is not yellow.

Third Negation: This truck is not of a single color and it is not yellow.

Both reformulations are equivalent in that they both result in multiple negations of the proposition.

As soon as there is at least one underlying assumption or a second simultaneous claim, the application of the proof by contradiction argument becomes ambiguous – which one of the many resulting negations is true? The more assumptions (or additional simultaneous claims) the more negations there are. In general, with n assumptions (or n+1 simultaneous claims), there can be 2ⁿ⁺¹ – 1 negations of a proposition.

In mathematics, all underlying axioms (assumptions that are always held to be true) must be included as simultaneous claims in the properly formulated statement of any proposition. Any of the axioms need not be true and there can be multiple negations of the proposition.

This still leaves one tricky issue unresolved. Which one of the many negations does the proof by contradiction argument establish? In the very least, any argument employing proof by contradiction must explain why it chose one of the many possible negations and why it ruled out other negations.

It is accepted that Cantor’s Diagonalization Argument (CDA) does not establish in general that the unit interval is uncountable but instead establishes that IF the axiom of infinity (the set N is an infinite set) is true then the unit interval is uncountable. Some authors have stated that CDA could contradict either of the two claims – and it may be repudiating the axiom of infinity. It is certainly a viable possibility. In CDA, we pick the negation that the unit interval is uncountable over the negation that the axiom of infinity is invalid based on the principle that an axiom is held to be always true. Zukojohri (talk) 15:44, 23 April 2016 (UTC)

References

“Alternative Facts”

You might like to hint or refer to the “alternative facts” and or at Matthew 5:37 – Fritz Jörn (talk) 08:31, 25 January 2017 (UTC)

Excluded middle and principal of bivalence

This article should be preserved in argon and put in a museum, but the Law is not valid. Hilbert did the right thing because we needed to build those computers, but today we need a higher level of logic 2601:449:8200:A430:D133:A463:5DD3:3A24 (talk) 10:46, 5 September 2017 (UTC)

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 think the three laws are equivalent,law of bivalence,law of non-contradiction,law of the excluded middle.219.151.147.143 (talk) 01:53, 27 August 2010 (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

Deleted the section on use in computer science because it made no sense. In summary it argued that the Goldbach Hypothesis was decidable by LEM. FOL can't be used to demonstrate decidability in this way, because that would imply that all propositions are decidable, which is clearly false. I.e. replace GH by the continuum hypothesis or the godel sentence. (Oh, also there are no useful citations in that section to even attempt to back up the claim -- just a stackexchange conversation) --Sclv (talk) 03:54, 12 January 2018 (UTC)

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)

which Foundations for this sentence?

I am not an editor here, but in the first sentence it says: ""The law of excluded middle is logically equivalent to the law of noncontradiction by De Morgan's laws; however, no system of logic is built on just these laws, and none of these laws provide inference rules, such as modus ponens or De Morgan's laws.""; now this should depend on the choice of foundations, and is implicitly from the perspective of a non-intuitionist. — Preceding unsigned comment added by 68.36.132.247 (talk) 17:05, 19 March 2020 (UTC)

The Law of the Excluded Middle LEM Is Not a Part of the Principle of Bivalence PB

The first sentece "In logic, the law of excluded middle (or the principle of excluded middle) states that for any proposition, either that proposition is true or its negation is true." of the article wrongly (according to my understanding) states that LEM says that for each proposition A, A is true, or NOT(A) is true. In reality, this latter proposition isn't LEM, but rather a part of PB, which does indeed state that each proposition is either true or false. (Mark that falsehood is truth of negation: for every proposition, FALSE(A) = TRUE(NOT(A)).) LEM is weaker than this part of PB. LEM doesn't state that for each proposition A, we have TRUE(A) OR TRUE(NOT(A)) (this is a part of PB), but only that for each proposition A, we have TRUE(A OR NOT(A)). TRUE(A OR NOT(A)) is weaker than (TRUE(A) OR TRUE(NOT(A))) because truth doesn't necessarily distribute over disjunction; it's true that we have (TRUE(A) OR TRUE(B) => TRUE(A OR B)) for all propositions A, B, but it's not necessarily true (or even the case) that we have (TRUE(A OR B) => TRUE(A) OR TRUE(B)) for all propositions A, B. TRUE(A) is stronger than A. That A is equivalent to TRUE(A) already assumes PB. We can have a three-valued logic which isn't truth-functional in which LEM holds true (for every proposition A, we have TRUE(A OR NOT(A))) but PB breaks down because there is a proposition A whose truth-value is UNDETERMINED, so it still obeys TRUE(A OR NOT(A)), but it's neither true nor false, violating PB and in particular TRUE(A) OR TRUE(B). Kniva Keisarabani the Goth (talk) 12:23, 30 October 2020 (UTC)