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Welcome

Hello, welcome to Wikipedia, EJ!!

I hope you like this place and have fun editing.
We always like to meet new Wikipedians! We sure can use someone writing about logic, thanks for your first edit. Here ar some more things to do, in case you're bored. But don't feel pressed by that.


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Again, welcome!

Lady Tenar 13:12, 19 Aug 2004 (UTC)

Edit attributions

Hi EJ. Edits from 147.231.88.135 have now been reattributed to you. Regards — Kate Turner | Talk 03:01, 2004 Sep 4 (UTC)

Thanks! -- EJ 11:25, 7 Sep 2004 (UTC)

Usercat

You were listed in the Wikipedia:Wikipedians/Czech Republic page as living in or being associated with Czech Republic. As part of the Wikipedia:User categorisation project, these lists are being replaced with user categories. If you would like to add yourself to the category that is replacing the page, please visit Category:Wikipedians in Czech Republic for instructions.Rmky87 07:37, 17 August 2005 (UTC)

galois theory

Why is Gal(R/Q) trivial? Dmharvey Talk 17:40, 18 August 2005 (UTC)

Sorry, let me rephrase that. Why are there no nontrivial automorphisms of R which fix Q? Dmharvey Talk 17:40, 18 August 2005 (UTC)

R is real-closed, thus every automorphism f of R is order-preserving: if a ≤ b, there is c such that b=a+c2, which implies f(b)=f(a)+(f(c))2, thus f(a) ≤ f(b). As f fixes Q and Q is dense in R, f must fix R as well. (The assumption that f fixes Q is of course redundant, as every automorphism fixes the prime field.) -- EJ 09:42, 19 August 2005 (UTC)
Hey thanks very much, that's very cool. I'll add that to the article in a few days, I don't think it's something that everyone encountering galois theory (like me!) has seen, and otherwise its very difficult to see why the R/Q and C/Q cases so different. Dmharvey Talk 10:47, 19 August 2005 (UTC)

Regarding the Mathematician Wikipedians category

Hi, it has been suggested that Category:Mathematician Wikipedians be deleted, because it is a duplicate of the more correctly named Category:Wikipedian mathematicians. I would recomend you simply edit your userpage and add yourself to Category:Wikipedian mathematicians (or one of it's sub-categories) instead by adding this to your userpage: 

[[Category:Wikipedian mathematicians|{{subst:PAGENAME}}]]

If you disagree your can visit Wikipedia:Categories_for_deletion#Category:Mathematician_Wikipedians and vote against the deletion or just voice your opinion. --Sherool 22:15, 28 August 2005 (UTC)


list of set theory topics

Hello. If you're interested in set theory, could you help make the list of set theory topics complete? Thanks. Michael Hardy 01:34, 11 September 2005 (UTC)

Science pearls

Hi,

Please notice the above project. As a mathematician, you might be especially interested in List of publications in mathematics

I’ll appreciate any help. Thank, APH 10:04, 18 September 2005 (UTC)

Message from Randall Holmes

Dear EJ,

In the alternative set theory article, the description I had in mind was the contrast between "definite" and "indefinite", which is appropriate for the set theories they had listed, as contrasted with ZFC, but not so appropriate for New Foundations and positive set theory, which are certainly also "alternative set theories" in the broadest sense. Your edit is fine with me...

Randall Holmes 15:50, 16 December 2005 (UTC)

If you are referring to the blurb about "discrete or crisp" set theory, I somehow missed when it was added in the article, but it was misguided. "crisp" is a term used only in fuzzy set theory, and it is not meaningful in other contexts (it is not e.g. used in Hajek's theory of semisets). AFAICS "alternative set theory" does not mean much more than "not ZFC (or its simple modification)", so NF perfectly fits the bill. Anyway, I have doubts about usefulness of the alternative set theory article in its present form.
PS: the work you have done on the NF article is wonderful. -- EJ 01:19, 17 December 2005 (UTC)
It's nice to be appreciated. It ought to be wonderful, since I am one of the very few people who thinks about New Foundations (actually, mostly about NFU) professionally. On a similar note I have been contemplating adding real discussion of the axioms and motivation for the Alternative Set Theory to that article, but I note that you may be better qualified to do this. At any rate, if I jump in and do it anyway I rather imagine that my work will be edited... Randall Holmes 00:18, 21 December 2005 (UTC)
Further, is there still a community working on the Alternative Set Theory? Randall Holmes 00:18, 21 December 2005 (UTC)
Sorry for not responding earlier. You can expect your work to be edited, that's how a wiki is supposed to work, and there is nothing wrong about it. If you have an idea how to describe AST in the article, please do it. I was thinking about writing it myself, but apart from lack of time, I don't feel qualified because motivation and "philosophical" background of this theory are fundamental for its understanding. I never cared much about the motivation, and forgot what I had known about it, as I was always more interested in the formal aspects of the theory. So, I could summarize Sochor's axiom system, and write a few words about its proof theoretic properties, but that's about it. Doing just that would seriously misrepresent the theory, and do more harm than good.
Ad community: no, as far as I am aware the subject is dead since late 80's. -- EJ 17:05, 16 January 2006 (UTC)

Found You!

Please, excuse my haste and joy. You are The english:WP czech - but not referenced by language. Would you mind helping the reference Desk there ? Thanks a lot. --DLL 22:32, 21 January 2006 (UTC)

I'll take a look, but don't hold your breath, it's a quite long text. -- EJ 16:00, 22 January 2006 (UTC)

Thanks!

Thanks for your help in the traslation! Czech is a definitively difficult language for me, and Internet doesn't have many free automatic traslators. If you need some help, whatever the problem is, ask me! --COA 23:37, 22 January 2006 (UTC)

You're welcome :) -- EJ 05:36, 23 January 2006 (UTC)

Since you have taken an interest in links. Please be kind enough to vote for my new bot application to reduce overlinking of dates where they are not part of date preferences. bobblewik 20:33, 25 February 2006 (UTC)

The theory of algebraically closed fields

Thank you for the explanation and the description of the axioms. I was under the misapprehension that ACF might be able to express all theorems that someone might include in a book about algebraically closed fields, including those in a chapter on fields of characteristic zero. Would it be correct to say that the theory does not deal with the theory of fields of characteristic zero at all, since its theorems are those that are true for all algebraically closed fields? Also perhaps even expressing the concept of a field having characteristic zero in this theory would require an extension to the theory with an infinite number of axioms, or something as strong as Peano's axioms? Elroch 00:04, 16 March 2006 (UTC)

The weakness of ACF is not in missing axioms (like axioms about the characteristic), but in the poor expressive power of its language. Indeed, it is easy to extend ACF by axioms of the form 1 + 1 + 1 + ... + 1 ≠ 0, which express that the characteristic is 0; the resulting theory is usually called ACF0. Even though this sequence of extra axioms is infinite, ACF0 inherits all the nice properties of ACF (such as decidability), and it has a nice feature on its own: ACF0 is complete. This implies, for example, that every first-order sentence true in the field of complex numbers is provable in ACF0.
However, the expressive power of the language of ACF0 is extremely limited. We can express polynomial identities like y = x2 + 1, we may combine them using Boolean connectives, and we can use existential and universal quantifiers running over all elements of the field, and that's all. In fact, the quantifiers do not help here at all: ACF and ACF0 have quantifier elimination, which means that every formula can be rewritten so that it does not use any quantifiers.
Thus, almost nothing from a book about algebraically closed fields of characteristic 0 can be formulated in ACF0. We cannot talk in the theory about subfields, transcendence bases, automorphisms, and similar concepts. And, by the same token, we cannot talk about integer arithmetic. Every algebraically closed field of characteristic 0 contains an isomorphic copy of the integers, and each integer has a "name", but we cannot distinguish between integers and nonintegers by a single formula of ACF0. This means that even simple properties of the integers, like "every integer is either odd or even", cannot be formulated in the language of ACF0.
It's getting too long, so I'd better stop here. Hope this helps. -- EJ 04:13, 16 March 2006 (UTC)
Yes, thanks for the clarification. Elroch 12:09, 16 March 2006 (UTC)

Are you aware of this Wikipedia talk:Censorship ?

For myself, I would like to say that the method is not innocent. The subject is truly important : there is one talk page and twoscore people discussing auto censorship for one million (counting non active users). Will you give your advice ? --DLL 20:10, 21 March 2006 (UTC)

Caron/hacek vote

There's a vote on Talk:caron where the article should be if you're interested. +Hexagon1 (talk) 10:06, 26 March 2006 (UTC)

Thanks for the notice. Nevertheless I probably will not vote, as I don't have a clear opinion either way. -- EJ 05:05, 27 March 2006 (UTC)

Good job on Jordan curve theorem

Hi, I haven't been watching that article, but I noticed that you undid the hype about the Mizar proof. I noticed a while ago (through Usenet) that the author of the link (after the statement in the article) says quite a few misleading statements implying that somehow the Mizar proof is the first "real" proof. I didn't think to check Wikipedia and see if someone would insert this nonsense. Anyway, I just wanted to say nice job, and say that I removed the link with dubious comments; the external links already contain a link to the Mizar proof, and I think it best to avoid an overeager editor from reading the link and putting the stuff back in. --C S (Talk) 16:36, 9 April 2006 (UTC)

Thanks for appreciation, and sorry for delayed answer. I also feel uneasy about the way how some of the Mizar people present their result, discrediting the work of Veblen and other mathematicians who proved the theorem. It is cool to have a formalized proof of JCT (albeit in a ridiculously strong set theory), but that's no excuse for twisting the history. -- EJ 18:22, 16 April 2006 (UTC)

About Euclidean algorithm

Could you explain the revertion [1]? The term atomic operation is the wrong term to be used there, because a machine operation is not considered atomic even if it's a single instruction, see atomic operation. I think you mean either constant time or non-constant time operations. Second, it's more general to specify division complexity with respect to number of digits n so that both software based divisions and constant time machine divisions can be analyzed (machine divisions are always limited range arithmetics O(1) operations). Shd 17:22, 18 June 2006 (UTC)

your edit to riemann hypothesis

I think the statement as it was written originally is "almost true", in the sense that one can prove that

This is weaker than the Ω statement though, which is certainly false. I don't have a reference for this. I just think I remember seeing it somewhere (i.e. not on wikipedia :-)). Dmharvey 00:56, 25 July 2006 (UTC)

Although I note that the article Merten's conjecture contradicts my claim, so maybe I'm full of crap. Dmharvey 01:00, 25 July 2006 (UTC)
This formula is indeed an unproven conjecture, if I read the article on Merten's conjecture correctly. I also don't have any references on the subject; I simply changed the suspicious
to
because I hope that it was intended that way by the author of the original statement. -- EJ 01:29, 25 July 2006 (UTC)

Matiyasevich's theorem

Hello. Interesting edit. Fortunately, it's not nonsense (although it was after the edit I reverted this morning---someone called the deleted text "unencyclopedic", but his edit changed the meaning). I'll dig out a reference. Do you know any other sexy examples of old-fashioned narrowly-construed Aristotelian syllogisms getting so much attention from mathematicians? Michael Hardy 18:44, 7 September 2006 (UTC)

To the best of my knowledge, the syllogism embodied in the derivation of unsolvability of Hilbert's 10th problem is not getting any attention from mathematicians; I'd guess 99% of mathematicians are not actually aware it is a syllogism, or even do not know what is a syllogism. In any case, syllogisms are very common all over the place. A lot of useful theorems are universal statements like "every compact space is Baire", "every continuous function has the Darboux property", "every finite division ring is commutative", "every bounded entire function is constant", etc. These are typically applied by constructing an object which is compact (continuous, ...), and concluding that the object is a Baire space (Darboux, ...). This kind of argument is a schoolbook application of a Barbara syllogism.
So, I'm eagerly waiting for your references showing that syllogisms are rare in mathematics, and how on earth it is possible to apply a complete triviality like syllogism in a nontrivial way. -- EJ 16:58, 8 September 2006 (UTC)
"syllogisms are very common all over the place"
Yes and no. It is not "common all over the place" for mathematicians to learn and rely on Aristotle's list of valid and invalid syllogisms. As far as "getting attention" is concerned, perhaps that needs to be rephrased; it does look as if it's got enough vagueness that you're not really seeing the point. I have one specific reference in mind; I'll dig it out of the library soon. Michael Hardy 20:10, 8 September 2006 (UTC)
I indeed fail to see any point there. As for "common all over the place": no, mathematicians do not learn Aristotle's list. The uses of syllogisms I mentioned are not intentional, and are not marked explicitly as being syllogisms. Which is fine, because that's exactly the same situation as with the Hilbert's 10th problem. It would be more meaningful to consider only explicit uses of syllogisms, but under such interpretation the observation in the article makes even less sense. -- EJ 20:47, 8 September 2006 (UTC)

Doing something about the ridiculous date autoformatting/linking mess

Dear Glenford—you may be interested in putting your name to, or at least commenting on this new push to get the developers to create a parallel syntax that separates autoformatting and linking functions. IMV, it would go a long way towards fixing the untidy blueing of trivial chronological items, and would probably calm the nastiness between the anti- and pro-linking factions in the project. The proposal is to retain the existing function, to reduce the risk of objection from pro-linkers.Tony 14:57, 9 December 2006 (UTC)

TeX

Hi EJ, you replaced "TeX usually written with an uppercase X in imitation of the logo" by "usually written with an uppercase E in imitation of the logo". We are talking about the word "TeX" which has a lowercase E (in contrast with the logo). Admittedly, the previous version was not perfect; I was hesitating between mentioning "uppercase X" (implied: normal English would say Tex, but we uppercase the X) or "lowercase e" (implied: normal English would say TEX, but we lowercase the e). Maybe the "lowercase e" is better, but in any case the word is not usually written with an uppercase E so this should probably be changed (the lowercase e is probably better). Schutz 13:09, 25 January 2007 (UTC)

Sorry, I misunderstood your original intention. I didn't occur to me that anyone would want to write the name with lowercase x, so I thought the X was a mistake for E, and that it refered to the spelling TeX. Now I think it would be least confusing to omit the "usually written..." part completely: it is not necessary to repeat that the name is usually written the way it is written in the title, unless we mention some alternative spellings. -- EJ 13:33, 25 January 2007 (UTC)
Good point. I've modified the page accordingly. Schutz 13:50, 25 January 2007 (UTC)

Simplification

Hi, EJ!

We haven't met yet, but today I noticed that you made a small change in E (mathematical constant) which made me think a little bit. So let me introduce myself – I'm David Bryant, and I'm probably quite a bit older than you are (to judge from the picture on your web site – when I was younger my hair might have been even longer, but it was never so bright red!)

Anyway, you changed a formula from

into  

which is certainly correct, and in a certain sense simpler. But then I got to thinking that this is now the only instance of an improper integral in the article, and that what seems intuitively obvious to you or to me might not seem so clear to a reader with little mathematical sophistication. So I thought I'd just pop over here and ask you – is an improper integral really simpler than one with finite limits of integration? Thanks! DavidCBryant 14:16, 5 February 2007 (UTC)

Why is the integral improper? AFAICS, it is a usual convergent Lebesgue integral of a nonnegative function, there is no need to go through the limit .
Anyway, I'm notoriously bad at estimating the level of mathematical sophistication needed for understanding particular problems, so if you think the original formula is easier, do not hesitate to revert. However, I should point out that the original formula also relies on a convention which might or might not be clear to less sophisticated readers, namely for negative x. -- EJ 15:22, 5 February 2007 (UTC)

Cyclotomic polynomial bit removal from Root of Unity (old)

(Following is pasted from what I wrote in that talk page)

I've been away for a long time, so I haven't responded. But well, basically, all that I wanted to convey is the content of [2]. I'm not really sure what the best way of putting it is - the issue is that the fact that the 3rd root of unity is 1/2(1 + i * root 3) (and not some unwritedownable transcendental value, say) is not a fluke, and that for any given n, it is possible to construct some finite combination of integers, elementary operations, i, and the real kth root to give the root of unity exactly. I don't really understand what you mean by it being impossible, since well, we have examples of it being true, and an algorithm to find such radicals. (The article attributes the proof to Gauss)--Fangz 20:35, 20 June 2007 (UTC)

No need to notify me here, I have Root of unity on watchlist. -- EJ 10:08, 21 June 2007 (UTC)
We would like to invite you to join WikiProject Czech Republic, an attempt to build and maintain a large and neutral database of the Czech Republic-related articles on Wikipedia. To join, simply add your name to the members section of WikiProject Czech Republic.

≈Tulkolahten≈≈talk≈ 07:50, 24 September 2007 (UTC)

Omega Consistency

Look, I wrote the stuff about computation because the article needs to explain some context. This concept was discovered by Godel by using an explicit computation. The soundness business came later with the arithmetic heirarchy. I phrased it carefully as a rule of thumb, not as an equivalent, and made sure to point out in the next sentence that if the property P is not computable than the statements are not equivalent. I am writing in good faith. Also, it seems to me that the example in the article should be of an actual omega-consistency not sigma-whatever-soundness. The current example is a computable property.Likebox 16:56, 16 October 2007 (UTC)

Thanks again for the nice rewrite of consistency!Likebox 19:05, 17 October 2007 (UTC)

Complete theories

Hello EJ,

I am delighted to see that someone is actually noticing what I am doing and makes sure it's no nonsense. It feels safer that way. In this case, I think both version are both correct and false. The problem is that the wording of the article suggests a general context (any logic), but some of the links suggest a first-order context. What is particularly weird is that theory (mathematical logic) (general context) is a redirect to model theory, which has a clear bias for first-order logic. I think we should replace this redirect by something more neutral, and perhaps it makes sense to rewrite complete theory so it covers the general case and makes it clear that first-order is the most important case. I wouldn't have noticed this without your "fix fix". --Hans Adler (talk) 12:28, 20 November 2007 (UTC)

Well, the notion of a theory is rarely used outside of the first-order context. Furthermore, the notion of completeness discussed in the article is tailored for classical logic, it would have to be stated in a more complicated way to account for, say, intuitionistic logic. Having said that, I agree that the links are suboptimal, but the trouble is simply that the generic notions of a theory or language do not seem to currently exist in WP at all. Linking to the FO definitions is the best approximation I could think of.
In particular, notice that the formal language article discusses languages in the CS/lingustics sense (a set of strings over a finite alphabet), not in the logical sense (a vocabulary, or the set of well-formed formulas therein) which is needed for the complete theory article. The latter is not described (AFAIK) anywhere in WP (apart from the specific cases like FOL), which explains why I originally left it (intentionally) as a red link, and why I modified your change. -- EJ (talk) 13:19, 20 November 2007 (UTC)
Oh dear. There is more to be done than I thought, then.
I think the term theory is used in non-first-order cases often enough, as in Guarded fixed-point logics and the monadic theory of countable trees (Postscript), although certainly not as often as for first-order. But in the case of complete theories I couldn't find a single example. I don't know if the word complete has ever been used for them. Or for fragments of first-order. In any case I have added "first-order" to make it clear without following the links that that's what it's about. --Hans Adler (talk) 13:50, 20 November 2007 (UTC)

Mixing PNG and HTML

This is (at least) the second time that you accuse me of mixing PNG and HTML at the article concerning the fundamental theorem of algebra. And then you take what I had written, which used only HTML and convert it into something which does use both HTML and PNG images. This is how I had left it:

Another algebraic proof of the fundamental theorem can be given using Galois theory. It suffices to show that C has no proper finite field extension. Let /C be a finite extension, and without loss of generality assume that is a normal extension of R. Let be the Galois group of this extension, and let be a Sylow -group of , so that the order of is a power of , and the index of in is odd. By the fundamental theorem of Galois theory, there exists a subextension of R such that Gal(/. As R is odd, and there are no irreducible real polynomials of odd degree, we must have  R. Thus R and C are powers of . Assuming for contradiction C, the -group GalC contains a subgroup of index . Thus there exists a subextension of C of degree . However, C has no extension of degree , because every quadratic complex polynomial has a complex root, as mentioned above.

An this is how it looks now, thanks to you:

Another algebraic proof of the fundamental theorem can be given using Galois theory. It suffices to show that has no proper finite field extension. Let be a finite extension, and without loss of generality assume that is a normal extension of . Let be the Galois group of this extension, and let be a Sylow -group of , so that the order of is a power of , and the index of in is odd. By the fundamental theorem of Galois theory, there exists a subextension of such that . As is odd, and there are no irreducible real polynomials of odd degree, we must have . Thus and are powers of . Assuming for contradiction , the -group contains a subgroup of index . Thus there exists a subextension of of degree . However, has no extension of degree , because every quadratic complex polynomial has a complex root, as mentioned above.

Tell me, who is mixing PNG images with HTML?

JCSantos (talk) 09:02, 19 February 2008 (UTC)


JCSantos, you do not seem to realize that not everyone is using the same math rendering preferences as you do. Whenever you insert a <math> tag anywhere, it will always render as PNG for at least some of the users. In this edit[3] you inserted for no good reason a bunch of <math> tags (with the deceptive edit summary of "Creation of link", I should add) in a mish-mash fashion where you switch math on and off inside a single formula. Depending on the context and on the user preferences, this means that parts of the formula can be rendered as PNG, and parts are HTML, in a quite different size and font. For example, your Gal(<math>K</math>/<math>L)=H</math>, <math>[L:</math>'''R'''<math>]=[G:H]</math>, and <math>[K:</math>'''C'''<math>]>1</math> show up as
Gal(/
R
CFailed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "http://localhost:6011/en.wikipedia.org/v1/":): {\displaystyle ]>1\,\!}
(I have forced PNG here so you should see it the same under whatever preferences). I find such style absolutely insane, and I can't imagine that anybody could consider it acceptable. So, be that kind, and NEVER EVER mix <math> and non-<math> in the same formula. If you want it to show up as HTML, leave the mark-up in HTML. If you want it to show up (at least potentially) as PNG, and only then, put the whole formula in <math>...</math>. However, inline PNG does not look good anyway, so <math> embedded in text should only be used as a last resort, cf. WP:MSM.
The proper reaction to your edit should have been to revert it. However, as it was the second time you did that, I assumed that you have some reason for inserting all the PNG markup, so I fixed it to a proper <math> markup instead as a good-will gesture. That makes it indeed look suboptimal under the default preferences, but that's still better than making it look totally broken under other preference settings, as you left it. Judging from your comment here, it was a mistake, as the real reason why you inserted the <math> is apparently that you are oblivious of the consequences of your actions. -- EJ (talk) 12:31, 19 February 2008 (UTC)
Thanks to both of you for your explanations. — JCSantos (talk) 22:39, 19 February 2008 (UTC)


Move

Dear EJ: please, seek some support before moving such an important page. In addition, your move corrupted the entire archive of talk, so it had to be reverted. --Camptown (talk) 11:01, 25 February 2008 (UTC)

Quadratic Residues

Hi. I'm new here. I started adding a bit of stuff to the page, and <<< I can't get the damn monkey off !! arrgghh !!>>> next thing you know, it's rewritten. User:Virginia-American/Sandbox. It looks like you're the most recent editor of that page, so I'm giving you a heads-up; I think I've improved it, but I may have trashed your stuff on complexity without knowing I did. Are there others whom I should, by custom or courtesy, contact? Is there some way to notify the interested parties [if any] before I move it, or is that what the watch function is for?

Thanks. Virginia-American (talk) 23:32, 27 February 2008 (UTC)

Intuitionistic logic

Hello. I don't have a strong opinion one way or the other about whether to use ¬ or 0 as basic. I think 0 makes more sense for intuitionism standing on its own, and ¬ makes more sense in order to preserve the maximum degree of comparability between the intuitionist axioms and the most commonly given classical ones. If anything, I lean towards the opinion that it's better to make it clear that ¬¬F→F is the only thing added in order to get the classical classical axioms.[sic]

I think the previous axioms had been chosen for their comparability to those at Propositional calculus#Axioms. —Preceding unsigned comment added by 136.152.196.167 (talk) 04:43, 6 March 2008 (UTC)

There's universaly no such thing as "the most commonly given classical axioms" or "classical classical axioms". The classical axioms I've seen most often, for instance, are totally incompatible with intuitionistic logic because they use only →, ¬, and ∀ as primitive symbols, and use (¬A → ¬B) → (B → A) as the only additional propositional axiom on top of the intuitionistic implication fragment (i.e., MP, and the axioms currently denoted as THEN-1 and THEN-2 in the article). I've also frequently seen classical logic axiomatized by Peirce law over the intuitionistic axioms; this calculus, in fact, is remarkable in having a property which the other classical calculi lack, namely separation of the connectives: for any set S of connectives including →, the S-fragment of classical logic is axiomatized by the axioms and rules of the calculus which only use the S-connectives.
Classical logic is also often axiomatized using ⊥ as primitive; a simple calculus (due to Church IIRC) consists of those THEN-1 and THEN-2 above plus ¬¬A → A (that is, ((A → ⊥) → ⊥) → A).
Even if intuitionistic logic is axiomatized in the {→, ∧, ∨, ¬}-signature, there are simpler ways than using the (A → B) → ((A → ¬B) → ¬A) axiom. A common calculus consists of the usual positive fragment together with the axioms
(A → ¬B) → (B → ¬A),
¬A → (A → B).
Importantly, omitting the last axiom gives Johansson's minimal logic. And so on, there are countless variants of both classical and intuitionistic calculi in the literature.
The fact that adding ¬¬A → A to an intuitionistic calculus gives a classical one is independent on the choice of the calculus, so that point is moot. Furthermore, the classical calulus with ⊥ mentioned above is essentially obtained in this way from the intuitionistic one (if it is axiomatized using ∧ and ∨ as well, and if one ignores the → A axiom which becomes redundant in the classical calculus). I also do not understand why this should be regarded as a useful property. If anything, the trademark difference between classical and intuitionistic logic is the law of excluded middle, much more than the law of double negation. It would thus make more sense to demand that the classical calculus is obtained from the intuitionistic one by adding A ∨ ¬A. Needless to say, such classical calculi are rather rare (but by no means impossible; I note that Chagrov and Zakharyaschev axiomatize classical logic by A ∨ (A → ⊥) over the intuitionistic calculus with ⊥).
So, all in all, I don't think that the argument of comparison with classical logic holds any water. -- EJ (talk) 14:43, 6 March 2008 (UTC)
You're right about the law of the excluded middle being more "classical" than double negation - that's my mistake. One way or another, I guess what I'm arguing is that the relation should be made as clear as possible to whatever coice is made at Propositional calculus#Axioms, and your arguments might be valid reasons to change what's there as well.
If you think so, go ahead. However I think that the idea of keeping the axiomatizations in synch is fundamentally flawed. The world is not just intuitionistic logic. There are many other logics contained in classical logic, such as various substructural logics, fuzzy and other multivalued logics, and so on. The classical logic can be axiomatized on top of most of them by an addition of a few extra axioms (often just the law of excluded middle), hence by your argument the axiom list for classical logic should be kept as an extension of every one of them as well. The catch is that these logics are incompatible with each other, so at best you'd end up with an unwieldy list of dozens of axioms, mostly redundant when put together.
I don't think there's any harm in using different calculi as basis for intuitionistic and classical logic. In fact, as a bonus, it gives another example of a classical calculus (obtained from the intuitionistic calculus by, say, the law of double negation). -- EJ (talk) 08:49, 7 March 2008 (UTC)
Actually, I think one should try to pick whatever axiomatization is best-known for classical logic - and I know you've cast doubt on whether there's one best-known axiomatization - and have that at the article on classical logic. Then make comparability of other logics to classical logic an important consideration in choosing their axiomatizations, in a way subordinate to the presentation chosen for classical logic. That way, things are less chaotic than you suggest.
We've talked about this a lot, but as I pointed out starting out, this is one consideration only, and I'm not deadset against the changes you've made. 136.152.196.8 (talk) 20:14, 8 March 2008 (UTC)

Since Heyting algebra links to the section you've changed, can I leave it to you to make the necessary changes in order to preserve the parallelism between the two? In Heyting algebra, that involves the axiom-like definition of Heyting algebras, and then certain arguments made elsewhere in the article that refer to the definition. 136.152.196.111 (talk) 02:57, 6 March 2008 (UTC)

The "axiom-like" definition of Heyting algebras is by far the most horrible definition of Heyting algebras I've ever seen. The proper way of fixing it would be IMHO to cut it out. If there is need to axiomatize HA by equations to make it a variety, the identities in the section above it do the job in a simpler and easier to understand way. In any case, I've never seen HA being actually defined in such a way by any source, so I suspect the section may suffer from a verifiability problem. -- EJ (talk) 14:43, 6 March 2008 (UTC)
I included that because it makes the relation to intuitionistic logic clear. It separates the proof of the basic theorems in this case immediate, so in that way the fact that that is a characterization of Heyting algebras is useful to present arguments elsewhere in the article.
I agree that it's horrible as a definition, so perhaps "characterization" would be better than "definition."
If your objection on verifiability grounds is that it may not have appeared as a "definition" of Heyting algebras in that exact form, then perhaps it can still be included as a characterization. That is because in mathematics, it would be tedious to require that every minor detail of an exposition be traceable in that way, and I use the characterization elsewhere. (Proof of 1→2 and 2→1 of "Provable identities".) I believe that minor departures from what has been written elsewhere, easily verifiable for people with minimal knowledge in the field, and whose essential ideas are not novel mathematics, are acceptable. (For example, the identity 895*692=502,990 has likely never been published anywhere, but could certainly be used as an example in an article on arithmetic. If a person did not know how to multiply, could they raise objections to the example on verifiability grounds?)
If your objection is that it is not easily verifiable for people with minimal knowledge in the field, then we can talk about that separately. 136.152.196.148 (talk) 01:19, 7 March 2008 (UTC)
My objection was that I doubt anybody would seriously use it as a definition, I guess "characterization" is OK. I do not doubt that the characterization as such is correct, it is kind of obvious from the fact that HA are an equivalent algebraic semantics for the intuitionistic logic.
However, I still fail to see its usefulness. It does not simplify the relation to intuitionistic logic, it merely shifts the problem to showing that this characterization is equivalent to the usual definition of Heyting algebras, while introducing an extra unnecessary step in the middle. -- EJ (talk) 09:03, 7 March 2008 (UTC)
You're correct - it does merely shift the difficulty, but it isolates exactly what needs to be proved in order to make the relation to intuitionistic logic work. For that reason it is the definition one would give if one were setting out to define an algebra for intuitionistic logic, and I feel that is worth noting.
Since writing what I wrote last time, I've learned that this kind of definition is (probably) a special case of the notion of algebra associated with a logic introduced by H. Rasiowa in "An Algebraic Approach to Non-Classical Logics". I say "probably" because although I do not have the book at my disposal, I have learned from the internet that she defined "positive implication algebras" exactly by the conditions I've listed as EQUIV, MODUS-PONENS, THEN-1 and THEN-2, and that she also had a general definition for logics with more connectives. I'm confident that the definition I've given is just the special case of intuitionistic logic in her general definition. 136.152.196.8 (talk) 19:55, 8 March 2008 (UTC)
Intuitionistic logic is an implicative logic, yes. However, this is rather an ad hoc concept, and with all respect to Rasiowa's work, it is now very outdated. If you are interested in the general background of the connection between logics and classes of algebras, I'd suggest you have a look on the modern theory of abstract algebraic logic and the Leibniz hierarchy. Its Wikipedia article is not very informative, but the references it gives are a good place to start. In particular, the SEP article by Jansana is a nicely written introduction to the subject. -- EJ (talk) 13:29, 10 March 2008 (UTC)

Intermediate logic

On consistency, here's a quote from [4]

An intermediate propositional logic is any consistent collection of propositional formulas containing all the axioms of IPC and closed under modus ponens and substitution of arbitrary formulas for proposition letters. Each intermediate propositional logic is contained in CPC. 136.152.196.111 (talk) 03:21, 6 March 2008 (UTC)

Is it OK now? -- EJ (talk) 14:55, 6 March 2008 (UTC)

Yes, that's fine. Thanks. 136.152.196.148 (talk) 01:20, 7 March 2008 (UTC)

Lon Chaney

That also confuses me. Sorry I couldn't be of more help. Creamy3 (talk) 14:44, 18 March 2008 (UTC)

Sorry, that guy must be slightly strange to have done that. Have a good one. Creamy3 (talk) 15:32, 18 March 2008 (UTC)
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