Talk:Boolean algebra (structure)/Archive 2

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This archive page covers approximately the dates from 31 October 2005 until 20 March 2007.

Post replies to the main talk page, copying or summarizing the section you are replying to if necessary.

This archive is no longer active. Please add new archivals to the latest archive on the talk page.

Paragraph from introduction

I'm not interested in following the above chit chat to discover what is at stake in the current edit war/civilised discussion about the intro to this article, however I will note that I do not find the following paragraph acceptable:

A related subject that is sometimes referred to as Boolean algebra is Boolean logic, which might be defined as what all Boolean algebras have in common. It consists of relationships among elements of a Boolean algebra that always hold, no matter which Boolean algebra one starts with. Since the algebra of logic gates and some electrical circuits is formally the same, Boolean logic is studied in engineering and computer science, as well as in mathematical logic.

This paragraph makes the following false claims:

  • That the theory described at Boolean logic captures what all Boolean algebras have in common
  • and that that theory consists of relationships among elements of a Boolean algebra that always hold, no matter which Boolean algebra one starts with

The theory at the boolean logic page gives a simplified account of the model and proof theory of Boolean inference, that is correct in the finite case but in the infinite case can diverge, so that the models are sound wrt. the notion of inference but not complete.

Furthermore, both Boolean algebra and Boolean logic are of interest to mathematical logicians: the former is not only some logically irrelevant exercise that mathematicians play with. If this paragraph is not fixed, I will delete it. (And so, I fear, find myself drawn into the above..) --- Charles Stewart 15:36, 31 October 2005 (UTC)

OK, first, in the last paragraph, when you say "Boolean algebra", do you mean "the study of Boolean algebras", or something else? What you say is true in any case; I just think we have to be extremely careful in this context to nail down the meaning of all mass-noun uses. I would prefer to completely avoid any mass-noun use of the term "Boolean algebra" on the page that talks about Boolean algebras, except when mentioning that usage as a linguistic alternative.
Now more into the substance:
  1. What's at the Boolean logic page, at least the last time I looked, really had no account, simplified or otherwise, of the model theory. Yes, there were hints from which you could reconstruct a model theory if you were so inclined, but it would be exaggerated to call it an "account". Or putting it another way, interpretation was discussed, but structures were not. So I rounded off and spoke as though that page were just about the syntactic relations. I agree that's less than an ideal solution.
  2. Given that, as far as I can tell, there is no model theory described at Boolean logic, I find it hard to interpret your claim that the model theory and the proof theory can diverge. --Trovatore 17:05, 31 October 2005 (UTC)
  3. I certainly didn't intend to imply that mathematical logicians aren't interested in Boolean algebras! (Look up the title of my dissertation--can be found through my user page's link to an external site.) Sloppy editing, possibly; I mainly wanted to remove the claim that computer scientists and engineers are interested in Boolean algebras, as I've never seen a nontrivial example of this (would love to see an example if it exists! that would be great; I'm certainly not against such applications). --Trovatore 17:38, 31 October 2005 (UTC)
For those even lazier than me, the title of Trovatore's 2003 UCLA doctoral dissertation is
  • An Inquiry into the Number of Isomorphism Classes of Boolean Algebras and the Borel Cardinality of Certain Borel Equivalence Relations
We may conclude that Trovatore would prefer not to see another Boolean algebra for, say, the next five years — but probably misses the bodies at the beach. ;-) --KSmrqT 22:32, 3 November 2005 (UTC)
Theoretical computer science today draws on some topics that even Charles Stewart might enjoy: topos theory, model theory, intuitionist logic, and more. Here's a sentence from a recent abstract:
  • Starting from a given denotational description of a computational situation, an axiomatic semantics can be derived in a quite automatic way by using the mathematical framework of Stone duality. The basic idea (due to Abramsky) is that the classic Stone representation theorem for Boolean algebras is the key to establish a correspondence (actually a duality of categories) between denotational semantics (spaces of points which are denotations of computational processes) and program logics (lattices of properties of processes).
I'd have to dig a little more to find an engineering example, but perhaps this will suffice. --KSmrqT 01:03, 2 November 2005 (UTC)
That'll do it! I withdraw my objection to saying that computer scientists use Boolean algebras. Though maybe we should emphasize theoretical computer scientists. --Trovatore 01:10, 2 November 2005 (UTC)
Computer science is theoretical computer science, and mostly consists of mathematics. That other stuff is computer programming (or hardware design), which "real" computer scientists only do for fun. ;-) Whether that's a good split or not is a different question. I notice Gauss didn't think it beneath him to apply mathematics to physics, yet still managed to prove a deep theorem or two. --KSmrqT 21:52, 3 November 2005 (UTC)
The BL page talks simply about what WP calls the algebra of sets, truth tables, Venn diagrams, etc. which are a well understood class of models of classical propositional logic for which it is not complete. It does not talk very formally about them, but they are being given as the kind of interpretation one has in mind for the logic, which can mislead and is what I object to. We know there are no nontrivial examples of BAs, unless there is such a thing as trivial, infinitely large examples... --- Charles Stewart 18:47, 31 October 2005 (UTC)
I'm afraid I'm going to need you to tell me slower what you mean by the above statement about completeness. The only interpretation I can come up with is that there are (presumably first-order) sentences that are true in every algebra of sets (or maybe every Boolean algebra?), but that are not derivable from the classical propositional calculus. What would be an example of that? Something like "there exists an atom", I guess, would be true in every algebra of sets (though not every Boolean algebra), but wouldn't be derivable; is that the kind of thing you meant? I certainly don't think we should be claiming that the existence of atoms is a part of Boolean logic; if you feel that the Boolean logic page does imply that, then maybe that's a flaw in that page. --Trovatore 20:14, 31 October 2005 (UTC)
It's the propositional theory, but where we allow infinite conjunctions and/or disjunctions. The mismatch between the two halves is described at Stone's representation theorem for Boolean algebras: it's nontrivial mathematically, and it's no problem for the BL article except for claims that the BL article is about just the same thing as the BA article but for some things only mathematicians care about. I'm a logician and not a mathematician, trust me when I say it's not true. --- Charles Stewart 20:22, 31 October 2005 (UTC)
What's the referent of "it" in the first sentence of the above paragraph? Are you saying that such infinitary formulae are part of Boolean logic? --Trovatore 20:30, 31 October 2005 (UTC)
The BL article explicitly says it is talking about the finite case. --- Charles Stewart 21:32, 31 October 2005 (UTC)
Oh, or maybe you're saying that there are infinitary formulas that hold in every BA and are thus part of "what all BAs have in common", but which are not covered by the Boolean logic article. I suppose that's a good point. Could it be addressed by saying that Boolean logic is "roughly what all BAs have in common", or maybe "all simple things that all BAs have in common"? --Trovatore 20:38, 31 October 2005 (UTC)
Better, the logic of infinite propositional formulae is a kind of logic beyond the topic treated by BL, but within the scope of BA. Roughly is a good word for indicating the kind of imprecision in the claim, though it doesn't help the reader find out what the imprecision is. --- Charles Stewart 21:32, 31 October 2005 (UTC)
So finally maybe I may see what you mean by Boolean algebra as a mass noun. As I say though, I really think we should scrupulously avoid using it as a mass noun in an article that discusses the structure. We can always say "the infinitary logic of Boolean algebras" for the concept described above.
On the "roughly" question, you're right, of course. But we aren't going to find a perfect solution and maybe that one's not too bad. I would really like to see a better definition given for "Boolean logic", whether or not it corresponds to the content of the Boolean logic article. --Trovatore 21:40, 31 October 2005 (UTC)
I agree with both points of what Trovatore said, though I think there is a question mark hanging over the BL article (principally: contributors find it difficult to figure out what goes where, but there are other issues). The infinitary logic of Boolean algebras is a good phrase, kudos. --- Charles Stewart 20:15, 1 November 2005 (UTC)
Infinitary logic is not a phrase familiar to a general audience, so please restrict use of the term to the body of the content currently under Boolean algebra. StuRat 21:10, 1 November 2005 (UTC)
I do not see why an accessible article should not introduce terms that are new to the audience, if that is what StuRat meant by his remark. --- Charles Stewart 15:20, 2 November 2005 (UTC)
One shouldn't include such terms in the intro, as that means there is a pre-req for understanding the intro. They might then follow that link, which in turn requires them to follow more links to understand it's intro, ad infinitum. Readers should be able to understand an intro immediately, without doing additional research. Also, such terms should only be included in the body when absolutely necessary. If the reader had wanted to learn about Infinitary logic, they would have presumably gone to that article in the first place. However, if a discussion of Infinitary logic is needed to understand Boolean algebra, then the body is the right place for it. StuRat 16:38, 2 November 2005 (UTC)
This all seems silly to me:
  • One shouldn't include such terms in the intro, as that means there is a pre-req for understanding the intro. - Nonsense. One can explain in more or less any terms one introduces. Nor is there any need for this explanation to appear in the introductory paragraph. --- Charles Stewart 15:13, 3 November 2005 (UTC)
You're going to explain terms in a later section after you use them ? Don't you see a slight problem with that ? They won't know what you're talking about until, and unless, they get to that description.
  • One shouldn't include such terms in the intro, as that means there is a pre-req for understanding the intro - I don't agree with the absolute necessity test in any case, but this case passes it: it is abosultely necessary to talk about the infinite case in order to indicate where the mathematical content of the BL and the BA articles diverge. --- Charles Stewart 15:13, 3 November 2005 (UTC)
That's debateable. We've survived for all this time without the term. As for not agreeing with the necessity test, that means you are in favor of using terms the average reader won't understand, in the intro, even when not necessary. Why ? StuRat 15:53, 3 November 2005 (UTC)
  • If the reader had wanted to learn about Infinitary logic, they would have presumably gone to that article in the first place. - Or they may discover that the topic is interesting by learning of this divergence. You seem awfully keen on sheperding your readers through a particular train of ideas without distractions: this is can be good in a textbook but it is rather inappropriate for an encyclopedia. --- Charles Stewart 15:13, 3 November 2005 (UTC)
The "See also" section is a good place for links to related concepts which aren't necessary in the article proper. By placing them in the article, you are forcing readers to learn those terms, if they want to understand the discussion, rather than giving them the choice. StuRat 15:53, 3 November 2005 (UTC)

Article rename

There has been a discussion on KSmrq's talk page (section named "peace") regarding renaming this article. The goal is to make it no longer a place where people looking for the Boolean logic article's content arrive in error. This would make it no longer imperative that the intro be readable by a general audience. (While all intros should technically be readable by a general audience, if no member of the general audience ever finds the article, it isn't an issue.) There have been two suggestions, both of which I support:

If either plan can garner enough support, we can solve this issue permanently and avoid any painful edit wars. StuRat 18:32, 31 October 2005 (UTC)

But Boolean lattice is certainly a less common name than Boolean algebra, so that would go against a basic policy on page titles. The second suggestion doesn't have that difficulty. I think the usual solution would mix those: Boolean algebra is ambiguous enough to have that as the dab page; and Boolean algebra (algebraic structure) or something comparable is a good-enough page name. Charles Matthews 18:52, 31 October 2005 (UTC)
Renaming this article to Boolean algebra (algebraic structure) but leaving Boolean algebra as a disambiguation page has one problem, though, I'm afraid those looking for the Boolean logic content may not be able to tell that that's the one they want. They might have been looking under the name Boolean algebra and then proceed to Boolean algebra (algebraic structure), reasoning that the title at least contains the term they are looking for. Thus, the need for an intro understandable by the general audience would remain. Either renaming this article to one which is obviously not what they want, like Boolean lattice, or filling this article with the Boolean logic content, would prevent this from happening. StuRat 19:05, 31 October 2005 (UTC)
Why not write 100 words on the disambiguation page to minimise that risk, then? If people can't or won't take trouble to read, then they get the wrong page; but no one can legislate for every eventuality. Charles Matthews 19:26, 31 October 2005 (UTC)
The problem is: it's very difficult to explain the differences to someone who isn't already familiar with both concepts. Heck, we can't even agree on how to describe the differences amongst ourselves. StuRat 20:04, 31 October 2005 (UTC)
I'm against both proposals on the grounds that the current BL article is not about BAs. --- Charles Stewart 19:21, 31 October 2005 (UTC)
Postcript: we could move the current BA article to some suitable qualification and make BA a disambig page. --- Charles Stewart 19:23, 31 October 2005 (UTC)

What about Boolean algebra theory as the disambiguated version of the mass-noun? It could have a subarticle called Boolean algebra (mathematical object) if necessary, which would hopefully make clear to anyone that this is not a theory but an object. BTW I'm now a bit lost to what exactly is Boolean logic, but I thought it was two-valued logic. Is it true that a Boolean algebra describes/models Boolean logic _for a certain number of propositions_? --MarSch 10:31, 2 November 2005 (UTC)

That name works for me. The current content of Boolean logic is really those topics from Boolean algebra which are aimed at common applications, like database queries, search page queries, and circuit design. That includes two-valued logic, but is not necessarily limited to it. Some theory is also included, but only as much as is needed for common applications. StuRat 14:37, 2 November 2005 (UTC)
(Responding to MarSch). A Boolean algebra models the propositional calculus in the following sense: If you take any tautology of the propositional calculus, and any Boolean algebra B, and substitute the letters in the tautology by elements of B, you get an expression that evaluates to the 1 of B.
I'm not sure exactly what you mean by the "for a certain number of propositions" part. Many Boolean algebras are not in any obvious way expressible as being about propositions; they're just algebraic structures that happen to satisfy the axioms of the propositional calculus in the sense above. We're interested in their structural properties, not just their logical ones. How many elements do they have? Are they atomic/atomless/free/complete/countably saturated/etc etc etc? --Trovatore 18:43, 2 November 2005 (UTC)

For the same reasons expressed above, I cannot support moving Boolean algebra to Boolean lattice.

I would support the dab model:

  1. Move Boolean algebra to Boolean algebra (lattice) or some more appropriate choice than "lattice"?
  2. Move Boolean logic to Boolean algebra (logic) or some more appropriate choice than logic? (Chalst: Do you also object to this?)
  3. Make Boolean algebra a carefully worded disambiguation page. (Making the distinction should not be all that difficult.)

Paul August 21:19, 5 December 2005 (UTC)

I would be on board for that, but would prefer Boolean algebra (algebraic structure) to Boolean algebra (lattice). Boolean algebra (mathematical structure) would also be acceptable. --Trovatore 21:30, 5 December 2005 (UTC)
I would support naming the articles Boolean algebra (abstract algebra) and Boolean algebra (logic). That's kind of parallel and consistent; the articles might well begin "In abstract algebra, a Boolean algebra is..." and "In logic, Boolean algebra is...". It shouldn't be that difficult to write a clear dab page. I wouldn't oppose Boolean algebra (algebraic structure), if that seems better to others. -GTBacchus(talk) 21:33, 5 December 2005 (UTC)
"Abstract algebra" is better than "lattice", probably, but (broken record alert) I don't like the way it mixes mass-noun and count-noun uses of "algebra". I'd like the word "structure" in there. --Trovatore 21:36, 5 December 2005 (UTC)
It certainly does mix those uses. I'll support "Algebraic structure". -GTBacchus(talk) 21:41, 5 December 2005 (UTC)
I'm fine with "Algebraic structure". Paul August 23:28, 5 December 2005 (UTC)
I'm fine with "Algebraic structure", but I'm not fine with "Logic". Boolean Algebra (logic) would be yet another article, not equivalent to the present Boolean logic. Arthur Rubin | (talk) 22:44, 6 December 2005 (UTC)


I believe the Wikibook link, although named "Boolean Algebra", belongs in the "Boolean Logic" article. Comments? Arthur Rubin (talk) 01:10, 3 November 2005 (UTC)

Yep. However it would go well with the dab notice to "Boolean logic", if somehow we could edit the template to make it clear that that's what it's being associated with, rather than the article as a whole. --Trovatore 01:13, 3 November 2005 (UTC)
Maybe that will work. It looks weird, but I think it covers the bases. -- Arthur Rubin (talk) 01:50, 3 November 2005 (UTC)
It didn't strike me as clear that the book notice was related to the dab notice, plus it's generally suggested not to have sections called "Introduction". So I hacked the template instead (in situ, not the general template of course). See what you think. --Trovatore 03:23, 3 November 2005 (UTC)
That covers it, thanks. Unless the wikibooks guru changes it again -- Arthur Rubin (talk) 19:00, 7 November 2005 (UTC)

Example in intro

I added the following text a while ago, which people seemed to like:

As a simple example, there might be three conditions that can be independently true or false. An element of the Boolean algebra might then specify exactly which ones are true; the Boolean algebra itself would be the collection of all eight possibilities, together with ways of combining them.

But it occurs to me that this passage is ambiguous. What I had in mind, when I wrote it, was a way of talking about the finite Boolean algebra with three atoms, and therefore eight elements. I now don't think that's a very natural way of reading what I wrote, which more naturally seems to be talking about the free Boolean algebra with three generators, therefore eight atoms, therefore 256 elements. Does anyone want to take a shot at cleaning it up? Any votes on whether "three atoms" or "three generators" is the more natural example to present? ---Trovatore 20:27, 10 November 2005 (UTC)

Please see the discussion at free Boolean algebra, in the "Motivation and example" section, to see what I'm talking about. There's also a Hasse diagram there that I think shows what's going on for the "propositional" interpretation. --Trovatore 20:08, 12 November 2005 (UTC)
According to my thinking below, the element of the Boolean algebra does not specify truth. This would be done by a homomorphism. The algebra itself stays agnostic on its semantics. -lethe talk 22:27, 23 November 2005 (UTC)

where is the example from propositional calculus?

Under the examples section for boolean algebra, I was hoping to see an explanation of the Boolean algebra formed of wffs made out of some propositions (possibly mod tautology, à la Lindenbaum-Tarski). I'm not sure that such a thing is a Boolean algbera (for example, are 0 and 1 usually guaranteed by propositional logic?), but it sure seems like it ought to be. Seems to me like it ought to be a free Boolean algebra, in fact, and that the example on that page is simply the language of a propositional logic with 2 propositions (mod tautology). If that is so, it seems like a very economical way to define a propositional language. And a truth assignment is simply a Boolean algebra homomorphism to {0,1}. Any comments? I'd like to add it to the article, but I'm not sure it is correct. -lethe talk 21:28, 23 November 2005 (UTC)

I went ahead and added my example anyway. If there's a problem, do what you gotta do. I have one concern though: tautologies are defined in terms of truth assignments, so I can't define a truth assignment in this way (in terms of the Lindenbaum algebra). This definition is circular. It seems to me that the truth assignment should be properly defined as a homomorphism from the free distributive lattice to {0,1}. Tautologies would then be the intersection of all the preimages of 1 (sentences which are true in every truth assignment). -lethe talk 21:54, 23 November 2005 (UTC)
I can see no objection to what you've added (except that the Lindenbaum algebra article says that it's for a specific logical theory, and you haven't explicitly said which one). But what I'm interested in is, should this be in the intro as the first example, rather than just one among many? No one has responded to my question in the section above (#Example in intro), about whether it's more natural to start with an example with three atoms, or with three generators. And unfortunately the text I wrote could conceivably be read either way. That needs to be fixed, without making the paragraph harder to read. --Trovatore 21:57, 23 November 2005 (UTC)
Oh, I see, yes. This is probably the most important example of a Boolean algebra, and should be given prominence. That's why I was surprised not to see it. And now I see that you've been looking for the same example. As for mentioning an explicit theory, I'm pretty vague on what that means. Maybe a theory is the set of sentences closed under semantic implication. or the set of sentences true in a given structure. I'm not sure how that relates to Boolean algebras. I was also nervous about the (under some mild assumptions, it becomes a boolean algebra remark. I understand the subject imperfectly, so I am afraid of adding things that are wrong. -lethe talk 22:23, 23 November 2005 (UTC)
The Lindenbaum article talks about a logical theory, which is a redlink. I assume it means something like classical propositional logic (one logical theory), or intuitionistic propositional logic (a different one). My guess is that intuitionistic propositional logic would not get you a Boolean algebra (because you wouldn't get ¬¬p=p). My concern here is not with the content, but with the link to Lindenbaum-Tarski algebra; the latter seems to be somewhat at odds with what you say in the example. Maybe if you simply explained it in words, and dropped the link to Lindenbaum algebra, it would fix the problem, granted that there is a problem. --Trovatore 22:38, 23 November 2005 (UTC)
Well, as far as I can tell, the Lindenbaum construction is entirely necessary: your algebra will not have a 1 otherwise. In fact, a whole bunch of operations will not be well-defined. -lethe talk 23:49, 23 November 2005 (UTC)
The point is that there's not really that much to the construction; you could just explain it in a sentence or two, rather than giving a link to a page that talks about something possibly more general than what you want. Something like "finite Boolean combinations of propositions, where two such combinations are considered equal if logically equivalent". --Trovatore 23:54, 23 November 2005 (UTC)

Explanation of "algebraic structure" in intro


the language you've been modifying was added in response to specific demands to make the introduction accessible, and in particular to explain what an algebraic structure is. I wouldn't have added it on my own devices. Search for "Celestianpower" in this page to see the background discussion. I don't mind tweaks to the language, but there seems to be a strong constituency to maintain something like it. --Trovatore 22:46, 23 November 2005 (UTC)

Hmmm I skimmed a lot of the discussion above between you, Celestian, StuRat, and Charles. I can see that you all have worked hard to reach a balance between the people who want concise correctness and the people who want colloquial verbosity. But the current state is, in my opinion, unacceptable. Therefore, unfortunately, a new balance will have to be struck. Yesterday I removed "a set of objects, called "elements"". That's just ridiculous, this is not the place to teach people the definitions from day 1 of set theory 101. Today it was "operation which takes one or two elements". The scare quotes are annoying, and the whole thing sounds terrible.
Here is the intro I would like:
"In mathematics, a Boolean algebra is a set with three operations which satisfy the most familiar properties of intersection, union, and complement of sets; or alternatively,the properties of conjunction, disjunction, and negation of logical propositions. The latter in fact provide a commonly encountered example of a Boolean algebra: a set of Boolean variables (variables which can take a value of true or false) and the corresponding logical operations. A Boolean algebra abstracts these notions, allowing their structure to be studied generally, and elucidating ties to other areas of mathematics. The axioms of a Boolean algebra characterize the difference between classical logic and more modern forms of logic, which are usually associated with generalizations of the Boolean algebra.
The first sentence was modelled on field (mathematics). Anyway, am I being too aggressive, rewriting an intro that's been worked over extensively? Tell me what you think. I know the abstraction removers have apparently hated this article from the beginning, but I'm afraid something or other has to be done. -lethe talk 23:44, 23 November 2005 (UTC)
I gave up on improving the intro in the face of aggressive opposition. My last compromise (incorporating bits I would not have said on my own in the penultimate paragraph) appears on the talk page here. In contrast, the present version wastes the first paragraph gently saying almost nothing, and the subsequent paragraphs are dubious. However, Celestianpower had this to say about my final attempt: "Your version, KSmrq, was (as far as non-mathematicians see it) patent nonsense and therefore was reverted." And StuRat perceived my use of notation as a "series of random characters". Concluding there was no hope of meaningful compromise with such extreme positions, I abandoned my efforts. Perhaps you will have better luck. I certainly hope so! --KSmrqT 02:05, 24 November 2005 (UTC)
I hadn't seen your version. I guess I must have stepped on some toes, coming here and complaining about an intro that has already been worked over so many times. But the intro that was left when I showed up here was just bad, no offense intended to the people who worked on it. Perhaps it was just a casualty of the wiki process. Anyway, your intro looks pretty good, so I'll see if I can use it.
I can't speak for others, but my toes are perfectly happy. I don't hold my final version as polished perfection, but a step in the right direction. Feel free to use it verbatim, or as inspiration, or not at all. Secrets to good writing include rewriting, and the guts to "murder your children". (It is common practice on DVD releases of films to show a scene that was reluctantly cut, often with a voice-over by the director rhapsodizing about the great acting. That's the ghost of a murdered child.) The secret to Wikipedia eludes me. ;-) --KSmrqT 05:41, 24 November 2005 (UTC)
Lethe, your intro isn't too bad, but I suggest a few tweaks, shown in bold:
"In mathematics, a Boolean algebra is a set with three operations which satisfy the most familiar properties of intersection, union, and complement of sets; or equivalently, the properties of conjunction, disjunction, and negation of logical propositions. The latter in fact provide a commonly encountered example of a Boolean algebra: a set of Boolean variables (variables which can take a value of "true" or "false") and the corresponding logical operations. A Boolean algebra abstracts these notions, allowing their structure to be studied generally, thus clarifying ties to other areas of mathematics. The rules of a Boolean algebra characterize the difference between classical logic and more modern forms of logic, which are usually associated with generalizations of the Boolean algebra."
Also please don't delete the explanation of the differences with Boolean logic and the brief bio link for George Boole. And, while I have no need for it personally, some of the mathematicians here will likely want to retain the "(sometimes Boolean lattice)" text. StuRat 04:05, 24 November 2005 (UTC)
I have no objections to those changes. And the ommissions were just my own sloppiness. -lethe talk 04:25, 24 November 2005 (UTC)
Rather than debate "axioms" vs "rules", I'd just leave off the last sentence (about generalizations) altogether. The phrase "more modern forms of logic" is problematic (what's really meant, I think, is weaker forms of logic, like intuitionistic or linear), but more to the point, it's likely to make people think that the content of the study of Boolean algebras is more logical than mathematical. (Here using "logic" in the strict sense, as opposed to "mathematical logic".) --Trovatore 04:15, 24 November 2005 (UTC)
That's fine with me. StuRat 04:20, 24 November 2005 (UTC)
Me too -lethe talk 04:25, 24 November 2005 (UTC)

"Free" vs "atomic" example

So I'd like to point out that Lethe's proposed first paragraph (which looks generally pretty good to me, if the accessibility party accepts it) suffers from the same ambiguity as my example. It says:
The latter in fact provide a commonly encountered example of a Boolean algebra: a set of Boolean variables (variables which can take a value of "true" or "false") and the corresponding logical operations.
OK, so say you've got three variables, p, q, and r. Does that mean that the elements of the BA are things like "p and q and ¬r", and if you "and" that with "p and ¬q and r" you get "p and ¬q and ¬r" -- that is, applying the "and" operation variable-by-variable? Or when you "and" those two things, do you just get FALSE, because (for example) one of them implies q, and the other ¬q, so they can't both be true?
The first interpretation would take p, q, and r to be atoms of the Boolean algebra, and give a BA with 8 elements. The second one (the free Boolean algebra with generators p,q,r) would have as atoms all the elements of the first BA, and would have 256 elements.
In some ways I like the 3-atom, 8-element version precisely because it's clear that it's not purely logical, because it's not free. (The 8-element BA is the smallest one that's not free.) On the other hand it's possibly a little harder to motivate in a few words. But surely we shouldn't have language that could be read as either one or the other, which is the case for both my version and Lethe's. --Trovatore 06:11, 24 November 2005 (UTC)
I don't think that level of detail needs to be discussed in the intro, put that discussion in the main body of the article. StuRat 06:18, 24 November 2005 (UTC)
I don't think the suggestion is to put that discussion into the intro; I think the suggestion is to be certain that what the intro says, at whatever level of detail, is correct, which we can talk about at length here, if that's what it takes. I agree it's a little ambiguous as it is now. Have we described a Boolean algebra with 8 elements? Or 28 elements? Or what, have we described, in that sentence? -GTBacchus(talk) 07:06, 24 November 2005 (UTC)
Ok, let me clarify my statement then:
"I don't think that level of distinction needs to be made in the intro." StuRat 06:18, 24 November 2005 (UTC)

I'm not sure I understand what the problem is. Why do we have to know exactly what algebra we're talking about? I basically modelled my intro on the intro to field (mathematics). A field has addition and multiplication and division. A field has 0 and 1. That certainly isn't enough to decide what field we're talking about. I think it's OK to say a Boolean algebra has negation disjunction and conjunction, and leave unspecified what the values of those operations are. In other words, it isn't intended to be a specific algebra, but rather a class of examples. If your p, q, and r are independent propositions, then we'll have a free algebra. If they're not independent, then it won't be (in which case, the algebra may not have 8 or 256 elements). -lethe talk 10:12, 24 November 2005 (UTC)

Agreed. StuRat 15:13, 24 November 2005 (UTC)
Ok, it's just that, when I read the current intro, I get confused, and it's not because it isn't specific, it's because it's sort of half-specific, in a way that makes me think two different things, and I find that confusing. When we say "As a simple example, there might be three conditions that can be independently true or false...," we're apparently setting up a real specific example, but there's just shy of enough information for me to get it. We're very specific in passing from 3 to 8, (and that the propositions are independent), but then silent on the next step.
In other words, we don't "have to know exactly what algebra we're talking about" if we're just being general. If we're giving an example in the intro, I think it should be one that is identifiable as itself, whatever it is. -GTBacchus(talk) 17:13, 24 November 2005 (UTC)
Well, I guess we could add some words that allude to the free algebra. Actually, just adding the words "independent" should do it. But I'm still not convinced of the necessity. I guess I don't understand what Trovatore is saying about atoms. Why would anyone assume that someone was talking about atoms? Atoms are not logically independent, at least not in the example we're looking at. As far as I can tell, the only thing special about atoms is that they are successors to 0, and they provide an easy means for calculating the size of the algebra. And what kind of an algebra would have 3 atoms? not a free one, since a free algebra has 2^n atoms. So it's probably not true that an algebra with 3 atoms has 8 total elements. -lethe talk 19:57, 24 November 2005 (UTC)
Any finite Boolean algebra (more generally, any complete atomic Boolean algebra) is completely determined by its atoms, and a Boolean algebra with n atoms has 2^n elements (an element is precisely categorized by which atoms are below it). No, atoms are not independent (in the sense that there are nontrivial relationships between them), but they could represent combinations of truth values of independent propositions, as I outlined at the start of this section). --Trovatore 23:48, 24 November 2005 (UTC)
OK, so all Boolean algebras have order 2^n, not just free algebras. I didn't realize that, since I only know about atoms what I've read here. So I guess your point is that the atoms can determine algebra, just as well as the generators can, right?lethe talk 17:07, 25 November 2005 (UTC)
That's true for complete atomic Boolean algebras, which for me more or less coincide with "trivial" Boolean algebras. All finite Boolean algebras are, of course, trivial. --Trovatore 23:50, 25 November 2005 (UTC)
So if you want to specify an algebra, you can specify its generators equally well? Sure, that makes sense. But I still don't think the intro needs to be specific about it. We have Boolean variables p, q, and r, and we have that pq and ¬r are in the algebra, but don't need to state whether they're true or false or equal to each other or anything in this intro any more than I need to state that g1g2 is equal to g1−1 in S3 in the intro of group (mathematics). Anyway, that's how I'm thinking about it now, but it could be that I'm still not understanding your concern rightly. -lethe talk 17:07, 25 November 2005 (UTC)

Proposed intro change

<<< outdenting <<<

I've been lurking, but abstaining from commenting. (Once burned, twice shy. Still, the burning happened some weeks ago.) What I'd like to suggest is that the long discussion about atoms and generators is theoretically interesting, but the intro raises such issues unnecessarily. Compare to the following wording:

In abstract algebra, a Boolean algebra is an algebraic structure (a collection of elements and operations on them obeying specific axioms) that captures essential properties of both set operations and logic operations. Specifically, it deals with the set operations of intersection, union, complement; and the logic operations of AND, OR, NOT.

For example, the logical assertion that a statement a and its negation ¬a cannot both be true,

Boolean lattice of subsets

parallels the set-theory assertion that a subset A and its complement AC have empty intersection,

Because truth values can be represented as binary numbers or as voltage levels in logic circuits, the parallel extends to these as well. Thus the theory of Boolean algebras has many practical applications in electrical engineering and computer science, as well as in mathematical logic.

A Boolean algebra is also called a Boolean lattice. The connection to lattices (special partially ordered sets) is suggested by the parallel between set inclusion, A ⊆ B, and ordering, a ≤ b. Consider the lattice of all subsets of {x,y,z}, ordered by set inclusion. This Boolean lattice is a partially ordered set in which, say, {x}  ≤ {x,y}. Any two lattice elements, say p = {x,y} and q = {y,z}, have a least upper bound, here {x,y,z}, and a greatest lower bound, here {y}. Suggestively, the least upper bound (or join or supremum) is denoted by the same symbol as logical OR, pq; and the greatest lower bound (or meet or infimum) is denoted by same symbol as logical AND, pq.

The lattice interpretation helps in generalizing to Heyting algebras, which are Boolean algebras freed from the restriction that either a statement or its negation must be true. Heyting algebras correspond to intuitionist (constructivist) logic just as Boolean algebras correspond to classical logic.

Such an introduction seems to state the heart of the topic with both specificity and generality, while avoiding questions of atoms versus generators. Like lethe, I think Boolean algebras deserve to be presented like groups or rings or fields, as serious mathematics. --KSmrqT 22:44, 26 November 2005 (UTC)

I like it, but I don't think you're going to be able to sell it. I do have a quibble: I'm still extremely skeptical that the theory of Boolean algebras, as structures, has application to electrical engineering of all things. Are you sure you aren't referring to the material at Boolean logic? --Trovatore 22:51, 26 November 2005 (UTC)
Sheesh, you didn't believe the computer science reference either, yet I convinced you. Must I do all the work? ;-)
As for selling it, let's not go looking for opposition. For those with deeper mathematical experience, the intro currently in the article falls short. If I'm counting correctly, it seems that at least I, you, MarSch, and lethe like an approach like the one I've shown, and Charles Stewart has not said (but wants the picture to be used somehow). Previously one or two people slammed the door on discussion, a violation of Wikiquette which needn't happen this time.
Anyway, whether it's adopted or not, I trust this version demonstrates to your satisfaction that issues of atoms and generators can be avoided. --KSmrqT 02:17, 27 November 2005 (UTC)
I suppose. I'm a little reluctant because it seems to me that it's the only thing that makes the discussion of finite Boolean algebras even remotely interesting. It had seemed to me that, since a finite Boolean algebra is, rather trivially, determined up to isomorphism by its size, there wasn't much to discuss about them other than their size. When I started looking at the free BAs, I realized that there is something a little bit interesting: it's obvious that permutations of the generators give rise to automorphisms, and similarly for atoms, but a priori this is a little surprising, because the atoms and the generators are related in a slightly complicated way. To put it another way, if you're just given a Hasse diagram of a Boolean algebra whose cardinality is , you know it must be free (because it's isomorphic to the free BA with n generators). But how do you pick out a set of generators? It's not quite obvious. --Trovatore 02:29, 27 November 2005 (UTC)
You've lost me. I thought you were neutral on whether the intro should address specific algebras, but that if it did it should do so unambiguously. Are you saying that the intro itself should mention atoms and generators? Are you also asking that it mention the dichotomy between finite and infinite BAs? And would you prefer that the intro discuss finding generators? --KSmrqT 03:18, 27 November 2005 (UTC)
Ah, no, not in the intro, I suppose. --Trovatore 06:55, 27 November 2005 (UTC)
I like KSmrq's suggested intro very much. I think it's a vast improvement over the current, frustrating version. This one doesn't confuse me with a half-presented example; I find it rather satisfying instead. All I see missing is a mention of George Boole. -GTBacchus(talk) 04:32, 28 November 2005 (UTC)
I like it too, though I'd like to see some mention of generalized truth values, and no mention of electrical engineering unless an example can be given. Another nice addition would be a few words to distinguish this article from Boolean logic. --Trovatore 04:38, 28 November 2005 (UTC)
By all means, include the one-sentence paragraph about George Boole. As for an electrical engineering example, would inclusion in an IEEE International Symposium suffice? --KSmrqT 06:17, 28 November 2005 (UTC)
My skepticism shifts to the proposition that this stuff is properly called "electrical engineering", even if presented under IEEE auspices. My concern here really--and this applies to the CS application as well--is that readers will assume we must be talking about Boolean logic, because that's all they can imagine would be applicable to CS and EE. That's another reason I'd like a clear statement that we're talking about what the Charleses would call the "model theory" of the situation (though it's not really what I think of as model theory). --Trovatore 06:23, 28 November 2005 (UTC)
I know that alot of work and compromise has gone into the current intro (especially by Trovatore who has done excellent work here) but I like KSmrq's intro better. Yes let's keep the bit about George Boole. And yes there may be other tweaks that might be helpful (e.g. since "Boolean lattice" redirects here I might prefer retaining the mention of it in the first sentence, like it is now). But I don't think it is necessary to hammer out all the details here and now. I would like to see KSmrq to go ahead and replace the intro with something like the above. Paul August 22:01, 4 December 2005 (UTC)
I agree (hoping that the points I raised above will be incorporated). --Trovatore 22:42, 4 December 2005 (UTC)
I have two very minor nits to pick with this intro. Most importantly, it seems to focus too much on sets and logic (that is, particular Boolean algebras) rather than the concept of Boolean algebras in general. While it is probably a good idea to mention that they are examples, I think accuracy has been sacrificed needlessly here. The second, much more minor, issue is that I would be wary of having formulae in the intro - simply because it becomes easier to assume knowledge of users that they might not have.
But, yes, definitely put this in the article ASAP. Even with these problems, it is several orders of magnitude better than the current one (which attempts to explain how to understand boolean algebras without realy explaining what they are...). LVC 03:25, 6 December 2005 (UTC)

(another) Suggested Intro

It strikes me that the current introduction is almost long enough to be a short article, and that it delves into very specific things (most of which I think too specific for an introduction). I notice similar problems with the alternate introductions proposed in this talk page. Also, I think the boolean logic article - and, by extension, the mention of it in this article - somewhat confuses the issue (despite its intentions to do otherwise). Hence, I propose the following, shortened, intro:

In mathematics, a boolean algebra (also, boolean latice) is a class of algebraic structure, named for George Boole. The study of boolean algebras (sometimes itself called "Boolean algebra") has implications across diverse areas of mathematics, including set theory and logic; and in computer science in studying logic gates. preceding unsigned comment by LVC (talk • contribs) 11:27, 4 December 2005

  1. A Boolean algebra isn't a class of algebraic structure. It's an algebraic structure, or if you want to be more specific, an algebraic structure of a certain kind.
  2. Who uses the term "Boolean algebra" to mean the study of Boolean algebras? Really, I don't think this is common, though it may happen. The mass-noun sense of "Boolean algebra" usually refers to a subject that never even mentions Boolean algebras.
  3. See Wikipedia:Lead section for what an intro should cover. I think yours is too short by that standard. (I wouldn't mind a terser standard myself, but this is what we've got.) --Trovatore 17:42, 4 December 2005 (UTC)
  1. True. How about dropping the first 'a' as in: "In mathematics, boolean algebra refers to a class of algebraic structure - structures of this class are called boolean algebras ..." ?
  2. Point taken. That sentence seems more than a little clumsy in retrospect, anyway. The mass-noun subject, as I learned it, was very similar to how I learned logic (ie, it works with the two-element boolean algebra).
  3. By my reading of that page, I still think alot of the stuff in the other intros shouldn't be in the intros (eg, a boolean algreba has a set, two binary operations, etc). I do agree that mine could do with a little beafing up, though. LVC 23:57, 4 December 2005 (UTC)
On your point 1: No, I don't like that at all. I think we have to be very careful to keep "Boolean algebra" (note the capital B) as a count noun on this page, except in a single reference explicitly disambiguating from the mass-noun usage. So "Boolean algebra" should always take a determiner or be used in the plural. I think the current wording is fine.
On point 3: No, I think this level of detail is fine. I'm in favor of something along the lines of KSmrq's intro, with the caveats I've mentioned. --Trovatore 00:05, 5 December 2005 (UTC)
I think it's necessary at a very minimum to note in the intro that a Boolean algebra is defined to be a lattice, together with two binary operations and one unary operation, satisfying certain axioms. The axioms themselves need not (and probably should not) be enumerated in the intro, but just saying that much will summarize huge amounts of information for anyone (like myself) who has worked with algebraic structures, but not necessarily with Boolean algebras. You can set it aside as a blockquote and call it a "formal" definition and reassure the reader that the terms in it will be unpacked further down the page, but it really should be there, for the mathematically savvy reader to get the information she needs quickly. -GTBacchus(talk) 00:20, 5 December 2005 (UTC)
Perhaps I should explain how discussion history has shaped intro choices. (Warning: POV ahead!) Not that long ago, the intro was brief, and targeted at students of abstract algebra. Then a cry of outrage went up, and rewrites began. The masses (mostly, one outspoken editor) had to be appeased, and even though it seems unlikely they had any interest in — nor comprehension of — the content of the article, naked phrases like algebraic structure in the intro were abhorent to them. In fact, anything that looked like mathematics, not everyday English, was vigorously rejected.
What to do? One approach is to throw away as much of the mathematical content as possible. That pleases only the non-mathematicians. Another approach is to soften the raw mathematics with explanatory phrases, like "a collection of elements and operations on them obeying specific axioms". Two words become thirteen, a dramatic increase in length; but now the non-mathematicians can follow along more easily.
I've seen a number of article intros fought over. I wish everyone could appreciate the competing interests and how difficult it is to satisfy them all. Instead, many editors write with little sensitivity for any view but their own, and write poorly at that. Attributed to Nathaniel Hawthorne, "Easy reading is damn hard writing." --KSmrqT 03:52, 5 December 2005 (UTC)
Yes I basically agree. KSmrq: By the way I think you should go ahead with your version of the intro, see my and Travatore's comments above. Paul August 05:31, 5 December 2005 (UTC)

Ok, I just re-read KSmrq's intro, and I now agree that it, while not perfect, is miles better than my suggestion (and, ofcourse, than the standing intro). So, I retract my suggestion, and I'll devote my further effort to helping improve KMSrq's. Sorry for the noise. :) LVC 09:35, 5 December 2005 (UTC)


Bravo! Three cheers! Job well done! I like that intro now. linas 05:56, 6 December 2005 (UTC)

Per request, intro replaced

Since so many people have asked, I have installed a new introduction. Please understand that I do not think of this as "my" intro, nor do I intend to "defend" it from attackers. I have basically suspended non-typo editing of Wikipedia articles themselves and confined my attention to talk pages because of an accumulation of experiences with destructive behavior. I'm convinced Wikipedia needs to incorporate a serious quality control mechanism if it expects to achieve credibility as an encyclopedia. More personally, I find it hard to justify spending professional effort putting words into an article that any idiot can instantly override. Unlike articles, talk pages don't get trashed or reverted; those who disagree with my words can ignore or argue, while those who agree or appreciate can still read them and benefit. This slightly peculiar editorial stance seems better for both me and readers until such time as Wikipedia evolves. Nevertheless, I'm moved by the expressions of approval for this revised intro, so I've made an exception. Enjoy. --KSmrqT 05:58, 6 December 2005 (UTC)

I agree, I'm getting tired of doing vandalism patrol. You may be interested in the discussions at Wikipedia talk:Stable versions linas 14:57, 6 December 2005 (UTC)

In the spirit of the new intro...

I've been thinking about this article, and of a way to write the introduction, and although KSmrg beat me to it by a few days, I'll go ahead and post what I came up with here, because I've got nothing else useful to do with it. Perhaps something out of here can be used to continue to hone the live version ever closer to wiki-perfection:

In mathematics, specifically in abstract algebra, a Boolean algebra is an algebraic structure that generalizes structures arising both in set theory and in logic. The set theoretic operations of intersection, union, and complementation follow the same algebraic rules as the logical operations of AND, OR, and NOT. A Boolean algebra is a structure on which these rules can be studied in a general context, without referring specifically to set theory or to logic. Boolean algebras are named after George Boole.

By definition, a Boolean algebra is a set, equipped with two binary operations, and with one unary operation. The operations must follow a list of axioms, given in the complete formal definition below. First, in order to better understand this definition, we consider the two examples already mentioned:

Boolean lattice of subsets
  1. For our set, we take the collection of the eight subsets of a three-element set: {{}, {x}, {y}, {z}, {x,y}, {x,z}, {y,z}, {x,y,z}}. On this set, we define the binary operations of union and intersection, and the unary operation of complementation, in the usual way. This set of subsets, and these operations, compose a Boolean algebra with eight elements.
  2. Now for our set, we choose a two element set, with binary truth values for elements: {TRUE, FALSE}. Our two binary operations are now AND and OR, and the unary operation is NOT. The usual rules, such as NOT TRUE = FALSE apply to these operations, and we thus have a very small two-element Boolean algebra, which is exploited in the Boolean logic studied in computer science.

Boolean algebras are sometimes called Boolean lattices, because every Boolean algebra also meets the definition of a lattice: a partially ordered set with two binary operations satisfying certain axioms. The axioms for a Boolean algebra allow for a partial ordering to be defined in a natural way, such that the axioms for a lattice are satisfied, so every Boolean algebra is automatically a lattice as well. Suggestively, the symbols used for logical AND ("∧"), and OR ("∨") are the same symbols used for the lattice operations of "least upper bound" (or "join" or "supremum") and "greatest lower bound" (or "meet" or "infimum").

The lattice interpretation helps in generalizing to Heyting algebras, which are Boolean algebras freed from the restriction that either a statement or its negation must be true. Heyting algebras correspond to intuitionist (constructivist) logic just as Boolean algebras correspond to classical logic.

It's a bit different from the new current version, which I shall refrain from calling KSmrg's. I think the version I'm suggesting might be aimed at a slightly more general audience (less dependence on formulas and symbols, no?), and might make the basic idea slightly more accessible, by motivating Boolean algebras as generalizations of more familiar structures, and referencing two examples in a little bit of detail. As you can see, by the end, I just transition back into the current version, not having any improvement to suggest there. If there's anything I wrote that's just dead wrong, please let me know (I was particularly stabbing in the dark about applications to comp sci). Or whatever feedback you've got, even if it's just that we should leave well enough alone. -GTBacchus(talk) 07:00, 6 December 2005 (UTC)

C Syntax Notation

C syntax is very common, should it be included?:

logical    bitwise    operation
   ||         |          or
   &&         &          and
    !         ~          not
              ^          xor

Kim Bruning 17:50, 19 May 2006 (UTC)

JA: My opine, no, this is an article on basics, and that sort of incidental is best lodged in the article on C. Jon Awbrey 17:53, 19 May 2006 (UTC)

Actually, I first picked up boolean operators in Basic, then in java (which uses mostly C syntax), then C, then etc (most programming languages taking their cue from C) . In fact, I'll wager that programmers and incidental computer users will likely actually be more familiar with the C syntax symbols than with the symbols mentioned here! Else I wouldn't have mentioned it. Kim Bruning 17:56, 19 May 2006 (UTC)

JA: And I learned my boolean algebra from reading W.S. McCulloch and C.S. Peirce, who used a binary matrix emulating dot-cross notation that led Peirce to get some poor typographer to cut him 16 different symbols for the 16 functions, but, like, who cares!? Jon Awbrey 18:12, 19 May 2006 (UTC)

if ((people.use("python") ||
   people.use("perl") ||
   people.use("c") ||
   people.use("c++") ||
   people.use("c#") ||
   people.use_other()) &&
   computer_language_users.length>=A_LOT) {

Kim Bruning 18:20, 19 May 2006 (UTC) note that python only uses the same bitwise operators, it uses words for logical operators

JA: Sorry if I'm being brusque, but I'm trying to finish up 3 or 4 things in before the weekend. The article in view is a math article, and it's kind of important in math not to confuse math notations with math objects. We try to mention 3 or 4 diff notations, but trying to mention them all is neither possible nor very helpful. And besides, there are lots of other pages that cover the peculiarities of this or that pet syntax. Jon Awbrey 18:50, 19 May 2006 (UTC)

Those C notation operators would be more at home in the Boolean logic article, which is written with applications such as computer science in mind. Specifically, I suggest adding that notation under Boolean logic#Other notation. This article, on the other hand, is written exclusively for mathematicians. StuRat 19:06, 19 May 2006 (UTC)

I would prefer to say this article is written for anyone who is interested in the fundamental mathematics of Boolean algebras. Someone who is not interested in posets, ring ideals, and topological spaces would probably be happier reading about Boolean logic. But I'm not sure Wikipedia is the right venue for being exclusive. :-) --KSmrqT 21:02, 19 May 2006 (UTC)

As usual...

Like virtually all math-related articles on Wiki, this is incomprehensible and useless to anyone who is unfamiliar with the concept being covered. Written by math geeks, for math geeks. —The preceding unsigned comment was added by (talk) 00:16, 11 December 2006 (UTC).

You say that like it's a bad thing.
Seriously, though, you probably can't understand this article unless you have some prior familiarity with algebraic structures. But that's not the fault of the article -- it's inherent to the topic. Boolean algebras are a more advanced topic (at least at the entry level) than simpler algebraic structures like groups, and you probably shouldn't try to learn about them until you've cut your teeth on the simpler ones. (Note that this is a different topic from the subject sometimes called "Boolean algebra" -- our article on that topic is at Boolean logic.) --Trovatore 02:15, 11 December 2006 (UTC)

Generalized Boolean algebras

I never heard the term before, although I suppose it might be standard. Do you have a reference? (I would add a sentence that it is defined by as an algebraic variety, as is a total operator with properties defined by equations.) — Arthur Rubin | (talk) 14:43, 27 February 2007 (UTC)

Peano 1888

Please regard Peano, Calculo geometrico, 1888, the first axiomatic version of boolean algebras de: Boolesche Algebra. 21:44, 28 February 2007 (UTC)



HOW CAN I SIMPLIFY ANY FUNCTION I SEE IT ? —The preceding unsigned comment was added by (talk) 15:05, 20 March 2007 (UTC).