# Talk:Boolean algebra (structure)/Archive 1

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## Lattice?

(The following discussion occurred prior to Feb 25 2002)

Explain to me why the word "lattice" has to be mentioned in an article called "Boolean algebra" at all. For Chrissakes, Boole isn't even mentioned in the article! Sheesh! --LMS, who would have to go back to his books to set things right.

Because that's what a Boolean algebra is - a kind of lattice. Which would be more important in a page on platypuses - to mention they were discovered by sir so-and-so, or to mention they are a type of monotreme? Not that we shouldn't have both, but we aren't normally that impatient.

Well, Boole himself probably wouldn't understand the article about Boolean algebra in its present form. And it omits all sorts of totally essential information to understanding what Boolean algebra is. Have a look at this, and compare the article in its present form. I can have read Boole's formulation of Boolean algebra and understand it, without understanding our present BooleanAlgebra article. I think there's something wrong with that, particularly in an encyclopedia that attempts to explain concepts. This is, as always, MHO! --LMS

Boole would definitely understand the article once he heard the definition of Lattice. The fact remains, however, that the abstract definition given here has much less intuitive flavor for somebody not interested in abstract mathematics than a more concrete definition. Of course, that is exactly the point of the abstract definition: it is not limited to a particular example.

Most importantly, the ideas embodied in Boole's original algebra are far more limited than the abstract definition. It is important to make it clear that there are an infinite number of Boolean algebra's and that the one with just the elements 0 and 1 is only the simplest one. On the other hand, it helps beginners enormously if the simple and concrete case is explained a bit more fully.

So anyway, does my addition to the entry help make this clear?

Clearer, but not as clear as an encyclopedia article should be. The first paragraph or two should be introductory--something that someone who has had a basic course or two in logic ;-) should be able to understand, anyway. I am a fan of making difficult concepts clear, which is what is needed in an encyclopedia (and which can be difficult to do, of course).

Why not see other encyclopedia articles about B.A. online and see what they do? --LMS

## Error?

It states: An ideal I of A is called maximal if I ≠ A and if the only ideal containing I is A itself. But doesn't A always contain A implying there are no maximal ideals? I'm new to this Wikipedia thing; should I just edit this? -- anon

If you know how to fix it, then yes, just edit it. In this case, somebody else has already done that. But next time, be bold. -- Toby Bartels 13:27, 28 Sep 2003 (UTC)

## TeX or Unicode?

This page uses wiki TeX tags instead of html character entities to show OR and AND. Otherwise those characters are not rendered by my browser (IE6.)--Voodoo 15:34, 9 Apr 2004 (UTC)

The article currently has a mix of math mode ($\land$, $\lor$) and HTML entity (∧, ∨) connectives. We really should pick one or the other. I hereby propose a poll on the subject. —Ilmari Karonen 16:41:26, 2005-08-31 (UTC)

Can you see the conjunction and disjunction connectives at Table of logic symbols? Answer below:

When replying, please note your browser as OS, and whether you had to install additional Unicode fonts. Note that this poll is not about which notation we should use, although you are or course free to express your opinion on the subject. This poll is merely for deterniming the extent of browser support for Unicode logic symbols.

## Recent change of sign makes no difference

Recent change of sign at Boolean rings, ideals and filters: makes no difference, since a Boolean ring certainly has x = -x. Charles Matthews 07:02, 1 Jul 2004 (UTC)

## Rewrite?

I agree with the idea of putting something nice and simple in the first 2-3 paragraphs. With no mention of "sets" and "operators" until the later introductory paragraphs. Basically something more on the line of "Logic for kids" than "Logic for computer science graduates". -- Sjschen 15:45, 19 August 2005 (UTC)

I learnt Boolean algebra in high school, and lattices, mentions of logic and mathematical symbols never entered into it. It's not needed to understand what Boolean algebra/boolean logic is or how and where to use it. I have followed several mathematics-classes at university since then, and yes, I understand it all as it stands now, but there's still the fact that a majority of people on this planet haven't been to or even ever had the opportunity to follow university-level mathematics classes. If I knew where my high-school readers for this topic were I'd dig them out, but I don't. --Kaleissin 13:57:45, 2005-08-22 (UTC)

## Rewrite attempt

below is my attempt at rewriting stuff. my perspective is an electronic systems engineering student who also does quite a bit of computer programming in my spare time. feel free to comment and improve and hopefully we can pull something together that you can actually get you head arround without knowing about advanced mathematics.

Boolean algebra normally reffers to an algebra system in which there are only two values true and false or 0 and 1. + is usually used for OR and . or no symbol at all for for AND. The use of theese operators is an analogy with addition and multiplication that holds if you consider any nonzero value to be the same as 1. Conventionally A.B+C is interpreted as (A.B)+C not A.(B+C). This is again the same as with addition and multiplication in normal algebra. NOT is generally represented by overline. The basic identites given below can all be trivially proven by looking at every case from the truth table. There are other boolean algebras based on the normal boolean algebra but with more than two values however these are far less commonly used.

• $A.A=A$
• $A+A=A$
• $A+0=A$
• $A+1=1$
• $A.0=0$
• $A.1=1$
• $\overline\overline{A}=A$
• $A.B=\overline{\overline{A}+\overline{B}}$
• $A+B=\overline{\overline{A}.\overline{B}}$

The + and . operators are distributive that is A.(B+C) = A.B + B.B and A+B.C = (A+B).(A+C) . The first of theese is the same as with normal algebra but the second is not! Consequentyly the first is far more comfortable for most people to do than the second. For this and other reasons a sum of products (which leads to easy NAND synthsis) is more commonly used than a product of sums (which leads to easy NOR synthisys.

### De morgans theorem

De-morgans theorem states that if you invert every varible in a function swap + and . operators and invert the result you get a function equivilent to the one you started with. Repeated application of De-morgans theorem to parts of a function can be used to move all inversions to the individual variables.

(above by User:Plugwash)

That's not what a Boolean algebra is. It is not true that there are "only two values" in a Boolean algebra. There is exactly one Boolean algebra with two elements (up to isomorphism), but there are an unlimited number of Boolean algebras with more elements. Some of what is said above to be provable is usually taken as axiomatic for a BA. Josh Cherry 03:31, 30 August 2005 (UTC)

All of the stuff i listed can be proven by complete enumaration (e.g. evaluation for both A and B). Can you explain what the other boolean algebras are and how they relate to the normal two value case without needing university level maths? Plugwash 13:20, 30 August 2005 (UTC)

Well i've bitten the bullet. As noone has taken the time to rewrite the more general case in a way that is understandable without university level math knowlage, I have described the normal boolean algebra first and moved the incomprehensible math to a section towards the end. Plugwash 20:42, 30 August 2005 (UTC)

I object to this, strongly. There's no such thing as a "the normal Boolean algebra", and a BA is not defined in terms of truth and falsity, etc. What is there now is simply wrong. What was there before may have needed improvement, but it was put there by people who actually know what a Boolean algebra is. Josh Cherry 22:49, 30 August 2005 (UTC)

Sorry i had to do it this way but it was quite obvious that noone who understood the existing article (i freely admit that i can't make sense of it myself) was going to make it accessible to those without a university education in maths so the only option left was to rewrite based on what i had been taught boolean algebra was. Plugwash 23:01, 30 August 2005 (UTC)

I agree with Josh. I think we should revert to the original article then discuss what can be done to make the article more accessible. Paul August 23:56, August 30, 2005 (UTC)
I've decided to go ahead and revert this for now while we discuss what to do. Plugwash what you are trying to write about is what the article calls the "two element Boolean algebra" (see the first example). Perhaps what you want to do is write an article just about that example? Paul August 05:02, August 31, 2005 (UTC)

The existing article is so jargon infested that i can't make sense of it myself. Therefore i can't help to make the existing content more accessible. If you can do so fine but if not then i'm going to re-revert you. Just what are theese other boolean algebras actually used for anyway? Plugwash 11:51, 31 August 2005 (UTC)

Please don't do that. As Paul and I have said, what was there was based on fundamental misconceptions about the meaning of "Boolean algebra". And the "other Boolean algebras" that you keep referring to are not obscure things. The first thing that comes to my mind when I think about BAs is not the trival two-valued one but the one formed by the set of all subsets of the natural numbers. Arguably, all the interesting stuff happens in such infinite BAs. This article is not about two-valued "Boolean logic" in the context of digital electronics or Google searches. Josh Cherry 12:54, 31 August 2005 (UTC)
Unfortunately not all of mathematics can be made easily accessible, to people who lack a certain background. The article as written is correct, if perhaps somewhat difficult to understand. The article as you edited it was more accessible perhaps, but not entirely correct. For now I think the former is better. We can work on making this article more accessible, perhaps by incorporating some of you edits. But what about my suggestion about writing a different article just about the two-element Boolean algebra? As for what are these "other" Boolean algebras "used for", they are used in the study of mathematical logic, information theory, probability, universal algebra, order theory, and set theory among other things, and they are also studied in there own right. See: List of Boolean algebra topics. Paul August 15:22, August 31, 2005 (UTC)

## A modest proposal

My impression, from looking at the article and the discussion here, is that the mathematical approach taken in this article is confusing the large, nonmathematical audience who are only interested in the applications of this one algebraic structure, and who know little or nothing about abstract algebra in general. Thus, my suggestion for an quick improvement would be to move the examples section up. It's clean, well written, and provides much of what the nonmathematical audience wants from this article. The mathematical treatment can then continue in later sections.

Specifically, I suggest that the Definition and first consequences section be split up, with only the axioms in the first half, and the Examples section be placed between them. The lead section could also be made shorter, with references to new sections named History and Notation. —Ilmari Karonen 15:15:53, 2005-08-31 (UTC)

I think all these ideas have merit. Paul August 15:49, August 31, 2005 (UTC)
By the way what do you think of my idea above, that an article just about the two-element Boolean algebra be written? Paul August 15:52, August 31, 2005 (UTC)
Sure, if there's enough non-redundant content to be put there. It's probably not a good idea to shunt all the applied stuff there and leave a pure lattice-theoretic treatment here, since then nonmathematical readers wouldn't see the connection between the two. Even the general "Boolean algebra" article should be at least mostly understandable to the general public. —Ilmari Karonen 16:18:46, 2005-08-31 (UTC)
No I wouldn't remove any content from this article. And redundancy isn't really a bad thing. Since space isn't an issue and Wikipedia isn't designed to be read sequentially, some the issues that make redundancy bad in other contexts simply don't apply here. In fact on Wikipedia, judicious redundancy can actually be a good thing, since it makes us more robust, and able to recover more easily from mistakes, or vandalism. Paul August 17:26, August 31, 2005 (UTC)
I've been thinking for days how to improve this page... Since the problem seems to be that the term "boolean algebra" is ambigous (math-reading and nonmath-reading), maybe a disambiguation-tag at the beginning saying something like Are you sure you weren't looking for Boolean Logic instead? or summat. --Kaleissin 16:33:38, 2005-08-31 (UTC)
Or... a disambig-page for Boolean Algebra, moving this page to Boolean algebra (mathemathics) and making a new page, Boolean algebra (computation) for what people expect. This only points to a problem that'll only get worse. Is Wikipedia to be an encyclopedia for all, or just the unwashed masses? The latter might not want all the detail that a specialist considers essential. So, Wikipedia: The sum of all encyclopedias or Encarta with a very thorough coverage of Pokemon? Heck, if it is to be the sum of all encyclopedias it still needs thorough coverage of Pokemon, come to think of it. Bound to be a Pokemon-encyclopedia out there already. --Kaleissin 16:44:55, 2005-08-31 (UTC)
Unfortunately Boolean logic already exists, and is about a different (although historically related) concept. Also, my point (two paragraphs up) is that the "math" and "non-math" boolean algebras are the same thing, just different ways of looking at it. The article should make an attempt to show this.—Ilmari Karonen 17:15:43, 2005-08-31 (UTC)
What if, instead, we created Boolean logic (computer science) and Boolean logic (propositional calculus)? Among other things, this would allow the comp-sci guys to include pictures of nand gates and have scads of truth tables in the article, while leaving the mathematicians comfortably snuggling with Heyting algebras and field of sets the whole Category:Boolean algebra. linas 04:34, 1 September 2005 (UTC)
Besides, the multivalued boolean algebras aren't all that complicated. They're all basically equivalent to applying standard binary boolean operators to more than one bit in parallel. —Ilmari Karonen 17:15:43, 2005-08-31 (UTC)
Careful here. That's true for atomic Boolean algebras. Trovatore 20:35, 31 August 2005 (UTC)
Hm, maybe not even. It's true for atomic complete Boolean algebras. Are there atomic Boolean algebras that aren't complete? Probably, but I'm not sure off the top of my head. --Trovatore 21:54, 31 August 2005 (UTC)
Sure. An example is the set of all finite and cofinite subsets of the natural numbers (with the obvious interpretations of the Boolean operations). Josh Cherry 22:58, 31 August 2005 (UTC)
From my perspective, the interesting Boolean algebras, in practice, tend to be atomless. For example, if you're using them for forcing, either you can find a condition below which the B.a. is atomless, or the B.a. is trivial for forcing purposes. --Trovatore 20:35, 31 August 2005 (UTC)

Update: I just went ahead with the first part of this proposal, moving the Examples section up to just after the definition. The lead section still needs work; we don't need to describe the entire history of boolean algebra and all the possible notations there, only the most important bits. —Ilmari Karonen 17:25:28, 2005-08-31 (UTC)

There are only 3 notations in the intro, the first two are very common in computing and electronics respectively and the third is what we are using in the article so none of them exactly look like candiates for removal from the intro. Plugwash 20:48, 31 August 2005 (UTC)
Ugh. The AND NOT OR notation is skanky, which makes it even more clear that two articles are needed, one for electronics, and one for math. I'd really prefer to see the AND OR stuff completely removed from this article. linas 04:59, 1 September 2005 (UTC)

## New Two-element article

Plugwash has also gone ahead and and created two element boolean algebra. Paul August 17:28, August 31, 2005 (UTC)

Of course "boolean" should be capitalized. And I think it perhaps should be "two-element". That is if this article should exist at all. Paul August 17:36, August 31, 2005 (UTC)
btw the recent edits seem to have made a great improvement to the followability of this article if it continues to improve then we may even be able to get rid of the two element boolean algebra article since it doesn't seem that there is anything really specific to the two element variety after all. Plugwash 18:48, 31 August 2005 (UTC)
Per suggestion of User:Kaleissin, I strenuously recommend that the article Boolean logic (computer science) or maybe even Boolean algebra (computation) be created, with a big disambiguation marker placed on top of this current article, pointing to the new article. This will allow the computer/electronics types to take the article and run with it, while leaving the mathematicians unperturbed. In particular, two element boolean algebra could be renamed to one or the other of these. Plugwash, I encourage you to do this renaming yourself ... linas 04:50, 1 September 2005 (UTC)

## I removed some content

I removed the following content from the "Examples" section, just added by Plugwash:

• This sentence was appended at the end of the second example:
Since applying operations to sets is equivilent to applying them to the presense/absense of each element in that set individually. See also basic properties of the indicator function.
But this sentence makes no sense to me. The "operations" we are talking about are unions and intersections of sets. They aren't applied "individually" to each element of the set.
The reference to the indicator function was my contrib. And I think it is relevant, as it represants connection between boolean operations and $I_A$, which is correspondence between $\mathcal P(\mathcal S)$ and $2^\mathcal{S}$. However, I would not argue much. (Igny 22:55, 31 August 2005 (UTC))
• This sentence was added just after the new sentence above, as a new first level bullet, indicating it is a new example:
Similarly boolean algebra can be applied to bitwise operations on binary values since applying bitwise operations to binary numbers is equivilant to applying the corresponding logical operations to each bit seperately.
It seems to be referring to the previously added sentence, but I don't understand what it is trying to say.

Paul August 22:30, August 31, 2005 (UTC)

## Reformulation of the definition

I've reformatted and reformulated the definition. Note, I've dropped the requirement of Idempotency, since it follows from associativity, commutativity and absorption (idempotency was inherited from the definition of a lattice, but it is not really normally part of the definition — I've edited "lattice" to reflect that) I've also dropped boundedness, from the definition, since it follows from absorption and complements ($a \lor (a \land \lnot a)) = a \or 0$). This follows modern definitions of BAs (e.g. Just & Weese's Discovering Modern Set Theory II, and Schechter's Handbook of Analysis and Its Foundations) Paul August 16:41, September 2, 2005 (UTC)

Excellent! Now we just need to bring the rest of the article up to the same standard of quality. —Ilmari Karonen 00:23:19, 2005-09-03 (UTC)

## I've reorganized a bit more

I have reorganized the material in that used to be in the "Consequences of the definition" section. I moved some of that content, namely the other boundedness identities, that 0 and 1 are complements, de Morgan's laws and involution up to the "Definition" section. I removed the dual distributive law since it is was already one of the defining axioms. And I split the remaining content into two new sections: "Order theoretic properties" and "Principle of duality". Hope everyone approves. Paul August 05:34, September 3, 2005 (UTC)

## Proposal

It seems to me that the fundamental disagreement unearthed by Plugwash stems from the ambiguity between a Boolean algebra, count noun, which is a mathematical structure, and just plain Boolean algebra, undeclinable mass noun construed as singular, which is a notation and rules for manipulating it. While the two notions have points of commonality, they are quite distinct, and I think it's a disservice to conflate them. For example the current article is primarily about the mathematical structure, yet it includes the following passage

Specifically, Boolean algebra was an attempt to use algebraic techniques to deal with expressions in the propositional calculus.

which is about the other notion.

The two-element Boolean algebra article is sort of beside the point as an attempt to capture the mass-noun notion, because that notion is not really concerned with any particular structure, not even the two-element algebra; it's just a logical calculus.

So my proposal is:

Whaddaya think? --Trovatore 18:33, 5 September 2005 (UTC)

on one hand i'm inclined to dump my new article as this article is now understandable without a maths degree my new article is redundant in its. On the other hand i could extend my new article to be more electronics biased (boolean algebra seems far more common in electronics than computing afaict) bringing up ideas like minterms etc. btw your proposal makes no sense to me but then i'm not a mathematician Plugwash 22:59, 5 September 2005 (UTC)
What about the reader who wants to learn about using Boolean logic for searching data sets? I.e. the sort of info covered at the likes of http://library.albany.edu/internet/boolean.html. Where in wikipedia should readers find that? Nurg 06:57, 6 September 2005 (UTC)
Exactly. That's the sort of thing that should go under Boolean algebra (logical calculus), but has little to do with the mathematical structures being discussed in the current article. --Trovatore 15:02, 6 September 2005 (UTC)
I don't think Plugwash's concept is called Boolean algebra outside of circuit design. Arthur Rubin
Hm, I'm not so sure about that. I have the idea that there's a corresponding ambiguity to the one between mass-noun "algebra" meaning "the way they teach you to deal with equations in middle school" and count-noun "algebra" meaning "module with multiplication between elements". (Of course "algebra" has another mass-noun meaning, "the study of algebraic structures"; I don't think "Boolean algebra" is used much in the sense corresponding to that one.) --Trovatore 23:08, 7 September 2005 (UTC)
Perhaps it should be disambiguated from Boolean logic, instead? Arthur Rubin
That might could work. But there would need to be a prominent path there from "Boolean algebra"; I think it's clear that a fair number of people would be looking for it under that name. --Trovatore 23:08, 7 September 2005 (UTC)
Another possibility would be for Plugwash's article to be a non-technical introduction to Propositional calculus. (I haven't added a comment to the talk pages of either of those articles, yet. It's just a thought.) Arthur Rubin 18:54, 7 September 2005 (UTC)

I agree with splitting it up, but suggest that the name should indicate whether each article is basic or complex:

Boolean algebra (basic concepts)

Boolean algebra (complex theory)

I'm going to start on the basic concepts article, myself. StuRat 20:35, 14 September 2005 (UTC)

I think you're missing the point. The study of Boolean algebras is not a "more complex" version of what you call "Boolean algebra"; it's another topic entirely (though certainly with connections). A Boolean algebra is an algebraic structure, like a group or a ring, and there's no reason its article should be less technical than those. Except, of course, to accomodate those who are looking for the other notion, the one called "Boolean algebra" as a mass noun.
But those folks are simply in the wrong place; this article isn't for them. The accomodation given them should be a dab notice at the top. --Trovatore 20:57, 14 September 2005 (UTC)
I think of it as similar to the relationship between Newtonian physics and the more general rules of physics, including relativity. The Newtonian physics is a subset of the general physics, but is the far more often used version, for practical applications like figuring out how soon a train will arrive, since relativity doesn't need to be considered in most cases. BTW, I'm watching this page, so there is no need to keep putting messages on my user talk page. StuRat 21:20, 14 September 2005 (UTC)
I thought you probably thought something like that. But it's just not so. That's not what this page is about; this page is about an algebraic structure that has historical and logical connections to the notion you have in mind, but is quite a separate notion. --Trovatore 21:34, 14 September 2005 (UTC)
I agree with Travatore, that you are missing his point. This article is about a mathematical object called a "Boolean algebra". It is not about a theory, complex or basic. What you are thinking about is a different thing, and it shouldn't be called "Boolean algebra (basic concepts)". But go ahead and write your article, we can figure out what to call it later. Paul August 21:37, 14 September 2005 (UTC)
Agree with Trovatore and Paul. There's no harm in having articles on both "Boolean algebra" as a mathematical object, and "Boolean algebra" as the art of manipulating truth values and logical connectives, but they are quite distinct concepts (though related). A similar example is Commutative algebra and Commutative algebra. Or even, at a stretch, Romantic literature and Romantic literature. Dmharvey File:User dmharvey sig.png Talk 23:48, 14 September 2005 (UTC)
Agree with Trovatore and Paul and Dmharvey. Seems that StuRat wrote a nice article, except that ... well, it's got problems: either the image needs to be thrown away, as the venn diagram implicitly implies lattice (order) and partially ordered set, which the article fails to mention; or the image is to be kept, in which case the examples section needs to be discarded, since the examples are misleading. Boolean logic is a terrible example of boolean algebra. linas 00:51, 15 September 2005 (UTC)

I'm still working on it, so give me a couple of days. I intentionally didn't mention lattice (order) and partially ordered sets as those concepts are beyond the middle school/high school reader for which that article is intended. If they want to explore those concepts, they can come here, I will provide a link. StuRat 02:56, 15 September 2005 (UTC)

So it's a good-looking article, written clearly, but it seems to be rather specifically about the algebra of subsets of a given set. I'm skeptical that that's what many people understand by "Boolean algebra"; I'd like to see it sourced. In the first two pages of Google hits, only two looked like they might have considered "Boolean algebra" to be that concept, and one of those probably not (it was a book; I was guessing a bit from the TOC).
The largest single group seemed to be about the 0-1 truth-table concept, together with algebraic manipulation of variables taking those values (these tended to be about electronic circuits). Another significant group talked about Boolean search terms. And of course there were a few that actually discussed Boolean algebras as algebraic structures. --Trovatore 04:12, 15 September 2005 (UTC)
The first two pages of "Boolean logic" under Google seem to be about the same as the similar one about "[[Boolean algebra]" aka non-axiomatic propopsitional calculus. Arthur Rubin
That article will provide a basis for the 0-1 truth-table concept (it already includes one), algebraic manipulation of variables taking those values, and Boolean search terms. I will be working on expanding it and adding links for each of those subjects in the next couple of days. The only thing it will not address are the concepts in this article, but a link will be provided. StuRat 05:11, 15 September 2005 (UTC)

It's worse than I thought. I think we need to:

Arthur Rubin 00:14, 17 September 2005 (UTC)

I'll put in a voice for what is now the status quo, ie. this article stays where it is, and Boolean logic is what was User:StuRat's article. Clearly the two articles overlap heavily, being about the paradigmatic theory of algebraic logic and its applications, but it is right that Boolean algebra should be concerned with the aspects of most interest to algebraists and Boolean logic to the aspects that are key to its use in logic and engineering applications. There's some tidying up of the divide to be done, but nothing urgent, and it should wait until whatever Boole's syllogistic becomes. Two points:

• As observed by others Boolean logic refers to the Boolean algebra under a restricted class of models, namely the set-valued models. There's some importance to this in philosophical logic that probably doesn't belong in the Boolean lagebra article, namely that Boolean logic is the logic of bivalence. OTOH it might be nice to connect this discussion with Stone duality, which does demand some algebra...
• The algebra that George Boole extends Boolean rings and does not admit set-valued models. I don't know much about the models, but I 'm going to do a spot of research on them in the coming days.

I'm definitely against the disambiguation suggestion. --- Charles Stewart 14:41, 27 September 2005 (UTC)

Can you explain why you're against the disambig? Personally I think it's necessary that the article currently called Boolean logic, be named some variant of "Boolean algebra", because it seems to be what a lot of people are looking for under the name "Boolean algebra". Add to that the fact that much of Boolean logic isn't actually about logic, so not only isn't it where people will be looking for that information, it isn't where they should be looking for it either.
We do still have the problem of calling it Boolean algebra (''what''). There was a suggestion that the current Boolean logic be moved to Boolean algebra, and that Boolean algebra become Boolean algebra (mathematical structure). Obviously not my first choice, but honestly it seems less bad than anything else I can think of. --Trovatore 15:50, 27 September 2005 (UTC)
This topic, ie. what StuRat's article is about, is probably the most widely known system of logic there is, and I'm guessing far more people who are after this treatment will be thinking of "logic" than of "algebra". Already there is some treatment of logical aspects (when one starts talking about true and false, logic is not far away), and I think the article is an appropriate place for some more, particularly since non-set-valued models of Boolean algebra are generally somewhat exotic. I'm also against moving out the discussion of applications (or at least, moving out all of it), since the article is more well-rounded with it. --- Charles Stewart 16:37, 27 September 2005 (UTC)
Well, the set-valued models aren't logic at all; they're (baby) set theory. As far as the question of what people are thinking of, I think that's just not true; it seems that these topics are often taught as "Boolean algebra". "Boolean logic" in this meaning seems to be more or less a neologism. --Trovatore 16:40, 27 September 2005 (UTC)
Hold on a moment. Boolean logic is the study of certain classes of sentence schemes, which we call formula, which have the property that they have a complete and particularly simple model theory, ie. maps from the set of formula to sets of finite sets. It's logic because the maps between formulae we are interested are a plausible representation of inference, and its algebra because all of the structures involved are nice and algebraically compelling. Set-valued models might not be logic, but that's because model theory is a part of logic. Boolean logic is a very widely used term; it is the main term used in electronics and programming for these structures, and phiosophers call this Boolean logic (more or less, the logic of truth tables, see eg., Barwise & Etchemendy's Language, Proof and Logic). It is very far from being a neologism. --- Charles Stewart 18:20, 27 September 2005 (UTC)
I'm not so sure model theory is part of logic proper. It's part of mathematical logic, but most of math logic isn't really logic; it's called "logic" for historical reasons. I'll take your word on the uses of the term "Boolean logic", but I still think the subject matter is taught to a lot of high schoolers as "Boolean algebra".
Anyway I don't have a lot personally invested in what StuRat's article is called, except that it needs to be made clear to people who come looking for it under the name "Boolean algebra", that the math-structure article is not about what they're looking for, nor is it even particularly about an extension or generalization of what they're looking for. --Trovatore 18:46, 27 September 2005 (UTC)
I agree about the problematic relationship between much of mathematical logic and logic proper, but Boolean logic is an exception: it's a place where a little bit for formalism and abstraction goes a long way even for not people without much mathematical culture. I agree on the need to signpost more clearly the articles. As I said above, when I'm done with the other Boolean article, I'll do a bit with these articles. I'm puzzled about the point about generalisation: doesn't Stone's theorem say that in the finite case Boolean algebras are the structures talked about in Boolean logic? Or do you mean that the scope and manner of treatment of the articles is different?--- Charles Stewart 19:22, 27 September 2005 (UTC)
"In the finite case"? Really I don't see how there's anything in the topic treated by StuRat's article that's specifically about finite Boolean algebras. What I mean, I guess, is that the math-structure article on Boolean algebras should be seen first and foremost as something like the articles about groups or rings or lattices, rather than as about a generalization of the algebra of sets. --Trovatore 19:32, 27 September 2005 (UTC)
I'm talking about the representation theorem for BAs in terms of power set algebras: Tarski showed that in the finite case, every BA is represented by its power set, but in the infinite case that may fail, and the BA needs to be represented by a strict subalgebra. In the finite case, we're just talking about different ways of looking at the same thing, but in the infinite case (as we get in predicate logic), the algebraic view looks at things more finely than the power set view, as per Stone duality. The Field of sets article seems to be the best intro. to the concepts around here, which are definitely hairy. --- Charles Stewart 21:30, 28 September 2005 (UTC)
But for the logic part of it, the model--powerset of a finite or infinite set or something else--is irrelevant. If you want to formalize it, I suppose I mean the Π2 positive theory--whatever falls just short of letting you express the claim "there are three distinct things", which is a concept that never comes up in the topic called "Boolean algebra" when you're not thinking about mathematical structures. Boolean algebras as mathematical structures are not to be seen as generalizations or extensions of that concept, but as another thing entirely--though it helps to know the non-structure concept to understand some reasons why the structure one might be interesting. --Trovatore 05:32, 29 September 2005 (UTC)
Well, maybe not: one of the goals of proof theory is to capture argument formally, and there are non-superficial links between algebraic logic and structural proof theory. One of the goals of the book Greg Restall is writing [1] is to motivate the need for structural proof theory from ideas in the philosophy of logic: once this is accepted the wider logical interest of algebraic logic isn't too hard. --- Charles Stewart 13:53, 29 September 2005 (UTC)

## Should I do this?

I'm not very good at writing in a wikipedia style, so does anyone want to add tautologies to this for me? If it's not here in a few days then I'll stick it in, I suppose --Spankthecrumpet 00:07, 5 October 2005 (UTC)

What tautologies are you talking about? Paul August 01:12, 5 October 2005 (UTC)
Why would you want to add tautologies (redundant statements). I would think you might want to REMOVE them, if present. Or do you mean the definition of tautology: "true by virtue of its logical form alone" ? StuRat 10:39, 5 October 2005 (UTC)
He probably means "tautologies of the propositional calculus" -- things like ((p→q)&(¬q))→¬p. Those don't have much to do with Boolean algebras in the sense of this article (except that Boolean algebras are models of such tautologies). It would be a better fit at Boolean logic. But Spankthecrumpet ought to first check whether they're already there. --Trovatore 13:36, 5 October 2005 (UTC)

## Come again?

1. What is an algebraic structure?
2. How do you abstract from a truth value?
3. What is a truth value?
4. What is "set-theoretic"?
5. How are intersection, union and complement related to eachother and/or booleen algebra?

I know that in most of these cases I could find out by reading the articles linked to from here but I should not be expected to have background knowledge in those areas before understanding the article. Thank you. --Celestianpower háblame 21:55, 27 October 2005 (UTC)

I disagree. If you don't know what an algebraic structure is, then there's not much point in trying to explain to you what a Boolean algebra is, until you find out. If you want to find out, the link is right there. I think it would be terribly wasteful to re-explain it at the top of each article about each new algebraic structure. --Trovatore 22:03, 27 October 2005 (UTC)
Yes, but assuming prior knowledge is never a good practice to keep. Could you not substitute "algebraic structure" with a synonym of some kind that people like me who have little experience in this area of mathematics (or many areas of mathematics for that matter)? Do you not agree that the first sentence/paragraph of an article needs to be understandable by non-experts? --Celestianpower háblame 22:09, 27 October 2005 (UTC)
Synonym of "algebraic structure"? Can't really think of one. The definition could be substituted, but I doubt that would really help much. Or we could spin analogies, but I'm afraid that would be likely just to induce misconceptions. I doubt that it's practical in all cases to impose the understandability requirement you propose. Just maybe, for the specific case of Boolean algebras, one could get a tiny bit closer, but we have articles on far more technical topics.
I really think that the reason this keeps coming up for Boolean algebra is that people are coming here looking for a different topic taught under that name, one they expect to understand. That's the reason for the Boolean logic article. I haven't heard this sort of complaint about prewellordering or Infinity-Borel set. --Trovatore 22:19, 27 October 2005 (UTC)
Not practical for readers to understand the topic? Even a general overview? That's preposterous in my opinion. I expect to be able to read and understand the first paragraph of any Wikipedia article I come accross. Otherwise, Wikipedia fails to be a general-purpose and general readership encyclopedia. The reason for me coming here instead of either of those other topic areas is that, well, I got here first. I have little idea what booleen logic is either. Just because other articles are doing the same thing as this one doesn't mean that my argument is any less valid here. And I still have no idea what booleen logic is. And I'm sure that there are many others who feel the same. --Celestianpower háblame 22:30, 27 October 2005 (UTC)
You have to realize that there are topics that are inherently too technical for you to understand, even in broad terms, after a mere brief description. Not just "you" you; it's true for everybody, though which topics it applies to will vary from person to person. In my opinion that does not justify banning these topics from WP.
Now, given half an hour in person, I'm pretty sure I could explain to you what a Boolean algebra is. An ∞-Borel set, probably not. But in either case I don't think that half-hour discussion should go at the top of the article. There are links, and references are provided. --Trovatore 22:39, 27 October 2005 (UTC)
That's where we disagree then. Take for example the derivative. I have no idea if you know anything of it (although you probably do) but I was asked as homework to find out about the derivative. Naturally, I came to the Wikipedia article. I couldn't make head nor tail of it. Then, next lesson our teacher took 5 minutes at the start explaining it and I got it very quickly. I have no doubt that this can be done with most topics. I don't want to learn the ins and outs, I just want to know what it is and what it does. That's not too much to ask from a first paragraph. --Celestianpower háblame 22:53, 27 October 2005 (UTC)
Reproducing your teacher's five-minute explanation, complete with wording capable of substituting for his/her hand gestures and interactive responses to your questions, would run pages and pages, not a paragraph. And the concept is much simpler than lots of things on which we have important, valuable articles. --Trovatore 22:58, 27 October 2005 (UTC)
Further to what Trovatore said, this kind of trying to capture what a teacher would explain is the domain of textbooks, not encyclopedias. See WP:NOT. --- Charles Stewart 15:11, 28 October 2005 (UTC)
We don't need a five min explanation - just what rough field is this in - what is the basic significance - two sentences. If a resonably educated non-specialist can't understand the topic sentense (and I am, and I can't) then it is written for a technical dictionary not a general encyclopedia like Wikipedia. --Doc (?) 09:58, 28 October 2005 (UTC)
I agree. The opening sentence could easily be made readable just by stripping out the PhD-only language:
"In mathematics, Boolean algebra has elements which "capture the essence" of the logical operations AND, OR and NOT as well as the corresponding set operations intersection, union and complement." StuRat 14:12, 28 October 2005 (UTC)

I think the first sentence is a good intuitive explanation for mathematicians. For non-mathematicians it is apparently gibberish. Let's not point to the existence of articles on other topics which are even more difficult/abstract and for which it would be likewise more difficult to write a non-math lead. This is not constructive. Also no-one is suggesting that such articles shouldn't exist, so let's not discuss that either. On the other hand it is not feasible to include the definitions of technical terms. What we need is an intuitive lead for non-mathematicians if at all possible. I think it exists and I hope we can work together to find it. Now back to Celestianpower's questions.

1. What is "set-theoretic"?
from set theory/to do with set theory.
2. What is a truth value?
There are two truth values: "true" and "false". The values that a boolean variable can have in for example C++ or Java.
3. How do you abstract from a truth value?
It means that you allow different values. "true" and "false" are usually represented by 1 and 0 respectively. You could allow for a third value or even the full interval [0, 1].
4. How are intersection, union and complement related to eachother and/or booleen algebra?
these are very similar to the logical notions of AND, OR and NOT.

I hope this helps. --MarSch 14:32, 28 October 2005 (UTC)

I agree, although any discussion of values between true and false can be left until after the opening sentence. I also think Celestianpower wasn't so much asking for those specific answers as using them as examples of how the opening sentence is overly complex and should be simplified by removing such terms. StuRat 15:46, 28 October 2005 (UTC)

I've expanded the introduction, which seemed to me obviously too sparse. Charles Matthews 15:54, 28 October 2005 (UTC)

You expanded it, but didn't simplify it, which is what this section was discussing. For example, you retained the term "set-theoretic" instead of simpler language like "set theory". StuRat 15:57, 28 October 2005 (UTC)
Look, I am sympathetic; but that's picky. The added edits ended up with something that didn't even make sense. Charles Matthews 16:39, 28 October 2005 (UTC)
And I would say you are the one being picky about "parsing language". If you are unhappy with the way in which someone else simplifies a section, don't undo the simplification, but rather redo it so it still avoids complex terms like "set-theoretic". I am not imagining that this is needlessly complex, when I saw "set-theoretic", I wondered "what the heck is that ?", as did celestianpower and, I suspect, most of the non-PhD readers. StuRat 17:27, 28 October 2005 (UTC)

I fully expect to have my revision reverted but it's my attempt at making it simpler to understand. I still have no idea what an algebraic structure is (and the article doesn't help). --Celestianpower háblame 16:13, 28 October 2005 (UTC)

Looks good, but you could have gone a bit further by removing terms like "algebraic structure" and "Boolean lattice" from the opening paragraph. StuRat 16:35, 28 October 2005 (UTC)
Unfortunately, you were correct, your edits have been reverted to the needlessly complex language like "set-theoretic" by a PhD. Now you see the problem with PhD-controlled articles ? StuRat 16:42, 28 October 2005 (UTC)
Well, I reverted back. It is unnecessary to include such complex langauge in an introduction. As to your edit summary, mine does parse as English, as I'll find mostpeople would agree. --Celestianpower háblame 17:07, 28 October 2005 (UTC)
I hope you don't think I am the one who reverted your edits. I fully support them. If anything, I don't think they went far enough at simplification. Charles Matthews reverted your edits. StuRat 17:20, 28 October 2005 (UTC)
Yeah - I know. I reverted him not you. I feel that my introduction is simpler and a few random people on IRC seem to agree with me. --Celestianpower háblame 17:31, 28 October 2005 (UTC)
Oh sorry. It must have been lost somewhere, when I was reordering your questions. An algebraic structure is a generic name for a set on which some algebraic operation is defined. Such as addition or multiplication on the reals. It is just some mathematical object which supposedly belongs to the algebra-part of maths and not say analysis or whatever. --MarSch 16:31, 28 October 2005 (UTC)

## Revert

The revert to the version starting

In mathematics, a Boolean algebra, or Boolean lattice, is an mathematical object that can either be true or false and, as such, is required in order to handle truth tables and Venn diagrams.

is quite wrong. It is not even grammatical. A 'mathematical object' is not something can be 'true or false'. That's nonsensical. 'As such' is not literate usage. Please stop reverting to versions that fail elementary tests like making good sense. Charles Matthews 17:35, 28 October 2005 (UTC)

If it's wrong, fix it, while leaving it simple. Don't trash it all because you don't like the phrase "as such", (which incidentally had 84,400,000 Google hits from apparently "illiterate" writers). StuRat 17:41, 28 October 2005 (UTC)
Okay, my grammar wasn't perfect. I apologise. However, it did make good sense. Some random Wikipedians on IRC thought that my revision was much more helpful and easier to understand. I will change the grammar though. Thank you for pointing that out. --Celestianpower háblame 17:46, 28 October 2005 (UTC)
In fact, looking again, I see nothing wrong with my grammar. "as such" is in common usage and if you can find a better (and just as easily understandable) term for mathematical object then feel free to put it in. --Celestianpower háblame 17:52, 28 October 2005 (UTC)
Better would be like this:
In mathematics, a Boolean algebra (sometimes Boolean lattice), is a type of structure able to capture the expressive power of the idea of 'truth' of a proposition. In the style of abstract algebra, Boolean algebras consist of symbols, and operations for combining symbols. Each Boolean algebra has a symbol 1 (or true) and 0 (or false). The operations are those required in order to handle truth tables and Venn diagrams.
Charles Matthews 18:29, 28 October 2005 (UTC)
That's not too bad, but does get a bit long-winded with phrases like "able to capture the expressive power of the idea of 'truth' of a...". Also, you left out the AND/OR/NOT and UNION/INTERSECTION/COMPLEMENT language in the original text (before Celestianpower's changes). That part was the only bit which was readable to begin with. StuRat 19:22, 28 October 2005 (UTC)
I would add all that in - I thought the first sentence was the apparent stumbling block. If I could make a couple of comments: firstly, user-friendly language here is quite possible in saying what problem is being proposed for solution; but technical language really is required for saying what the solution offered is. Secondly, any programmer who has to manipulate exit conditions from a loop can understand what is being said here: some concrete condition raises a true/false flag and the point is to understand 'the rules of the game' for combining various conditions. That's what 'expressive power' is about, here. Charles Matthews 20:07, 28 October 2005 (UTC)
That's not true. I consider myself a programmer and do not understand it the way you want to say it. Plus, how many times do I have to say, "It is not necessary at all to use technical language in the first paragraph". I do not understand why my paragraph was/is so bad. It's understandable to non-experts and user-friendly and saying the things it needs to. --Celestianpower háblame 20:19, 28 October 2005 (UTC)
I said can understand it that way, not does understand it that way. Charles Matthews 20:47, 28 October 2005 (UTC)
What's the there should be no difference. If there is then we have a problem. --Celestianpower háblame 20:55, 28 October 2005 (UTC)

As it stands, I see nothing wrong with Celestianpower's version of the first paragraph and see it as a vast improvement to the original one. I disagree that technical language should be used in the first paragraph of any article and see no problem with keeping Celestianpower's version. FireFox 20:23, 28 October 2005 (UTC)

Well, to be frank, it is wrong. It is meaningless, and bad English. I respect all the confidence you have in your 15 years, but that changes nothing. Charles Matthews 20:44, 28 October 2005 (UTC)
I would rather you didn't patronise against him because he's 15. Being 15 changes nothing. An awful lot of very valid, active and mature Wikipedians (editors, admins and bureaucrats) are teenagers. That was bordering on a personal attack. --Celestianpower háblame 20:55, 28 October 2005 (UTC)
I agree. Implying that someone is incompetent due to their age is not acceptable behavior. Imagine implying that an 80 year old was incompetent due to their age, or, even worse, that someone is incompetent due to their gender or race. Any such personal attack is also committing the logical fallacy of attacking the messenger, rather than answering their arguments. StuRat 13:38, 29 October 2005 (UTC)
Come on. On the substantive point, technical language is used in the first paragraph of most mathematical articles on this site, and rightly so. There can be a reasonable objection to the exclusive use of technical language. As I have already explained, the technical and the user-friendly types of language both have a part to play. Simply trying to enforce the insertion of waffle and bluff doesn't make an encyclopedic article at all. Don't take an annoying and picky attitude towards those who have an interest in exposition. You actually do need someone who knows what the technical words mean, to write that meaning in any other terms. An introduction that isn't sound will just not last.Charles Matthews 21:14, 28 October 2005 (UTC)
I think a general in layman version paragraph should come before the TOC, and then charles matthews formal definition shoould come immediately afterwards. -- ( drini's vandalproof page ) 20:26, 28 October 2005 (UTC)
Agreed. StuRat 13:46, 29 October 2005 (UTC)
Here is some stuff to think about: Wikipedia:Manual of Style (mathematics)
It is a good idea to also have an informal introduction to the topic, without rigor, suitable for a high school student or a first-year undergraduate, as appropriate. For example,
In the case of real numbers, a continuous function corresponds to a graph that you can draw without lifting your pen from the paper, that is, without any gaps or jumps.
The informal introduction should clearly state that it is informal, and that it is only stated to introduce the formal and correct approach. If a physical or geometric analogy or diagram will help, use one: many of the readers may be non-mathematical scientists.
It is quite helpful to have a section for motivation or applications, which can illuminate the use of the mathematical idea and its connections to other areas of mathematics.

-- ( drini's vandalproof page ) 20:36, 28 October 2005 (UTC)

Yeah, well, the split into a Boolean logic page that has all the good introductory stuff, and this technical page, was wrong to do; and we see why. We see why because if people want the introduction to that page on this page, there is frankly no reason not to merge, is there? Charles Matthews 20:44, 28 October 2005 (UTC)

In my opinion tghere was no reason for the split in the first place. --Celestianpower háblame 20:55, 28 October 2005 (UTC)
The reason for the split was that earlier editers, such as Plugwash, and myself, were reverted whenever we attempted to revise this article. There were really two different, but related, issues:
1. There are two different concepts: One is called Boolean logic/Boolean algebra (no S) and is what is now covered under the articles Boolean logic. The other is call Boolean algebras (with the S) and is what is covered here.
2. Some want the material to be have a simple, easy to understand introduction and others insist that the intro be "rigorous", which conflicts with making it accessible to a general audience.
Creatng the Boolean logic article meant there is no longer a need for that material to be covered here. However, it doesn't mean that an easy to understand intro is no longer needed here, that is another issue entirely. StuRat 22:09, 28 October 2005 (UTC)
There is no more than one concept involved. The job of this article is to define it, not illustrate it. There lies the only difference. Charles Matthews 22:15, 28 October 2005 (UTC)
Your fellow mathematicians disagree with you there, as they have told me many, many times (and I believe they told you, as well). StuRat 22:17, 28 October 2005 (UTC)
In WP terms I think for myself. Charles Matthews 22:22, 28 October 2005 (UTC)
You're welcome to do so, just realize you're in the minority on that opinion, even amongst fellow mathematicians. StuRat 22:25, 28 October 2005 (UTC)

## Conditional statements

I want to make some progress on wording a way in to this article. So, someone, please give me an orthodox account of conditional statements (which are quite general enough as examples of things on which to do Boolean algebra). Charles Matthews 21:34, 28 October 2005 (UTC)

I'm not sure what you are asking here. I guess it must be worded too "rigorously" for me to understand. Are you trying to say you want some common if-then statements from real life ? For example, "if it rains, then the street will get wet". StuRat 22:37, 28 October 2005 (UTC)

Well, you could read the first sentence of that page: 'A conditional statement, in computer science, is a vital part of a programming language'. You're a programmer. I'll meet anyone half way on this. They have to be willing to meet me. Charles Matthews 06:26, 29 October 2005 (UTC)

We are willing to meet you half way. That consists of letting you have total control over the body of the article while we insist that the intro be readable by a general audience. Actually, that's far more than half way, by volume. StuRat 10:52, 29 October 2005 (UTC)

Is the first sentence of conditional statement better now? -GTBacchus 02:21, 3 November 2005 (UTC)

## Got an edit conflict

So Ksmrq was apparently working on the intro at the same time I was. His version's not bad, though I doubt StuRat's party will appreciate the ref to Heyting algebras and such, and it still claims that EE and CS use "Boolean algebra", which in context requires explanation.

Anyway, let me report my version here; maybe we can come to a synthesis. (I note that just by accident he chose the same example BA that I did--or maybe Pierce was right about "threeness"?)

In mathematics, a Boolean algebra (sometimes Boolean lattice) is a algebraic structure (that is, a set of objects, called elements, together with operations on those elements, which take one or two elements and return another element). The elements can be thought of in various ways; one of the most common is to think of them as generalized truth values. 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.
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.
The operations on a Boolean algebra are referred to as AND, OR and NOT. For the structure to be a Boolean algebra, these operations must behave as they would on the two-element Boolean algebra (the one whose only elements are TRUE and FALSE).
Boolean algebras are named after George Boole, an English mathematician at University College Cork.
I much prefer your version to KSmrq's, which gets into way too much depth for an intro section. Note that I am not driving this move to simplify the intro, Celestianpower is, although I fully support the attempt. StuRat 02:49, 29 October 2005 (UTC)
So after thinking about it a bit, I've gone ahead and put mine in (obviously KSmrq's is still there; I don't object to a synthesis but couldn't really see how to do it without making it too long). This is not intended as a hostile move against KSmrq; as I said I thought his version was pretty good.
I think the most important new element in my version is the clear distinction between this article and the Boolean logic article (and I propose that we should use this as the primary test for the division between the two articles). --Trovatore 04:38, 29 October 2005 (UTC)
Well done ! I would have made it simpler still, but this seems like a good compromise between what the "intro's must be accessible to a general audience" group and the "every statement must be rigourous" group want. StuRat 05:14, 29 October 2005 (UTC)

## new opening

Hmm, is it a general rule that editors can't keep their hands off the opening sentences, but easily ignore the body? Anyway, previous editors said that previous openings were either incorrect or incomprehensible. I have tried to write a compromise version that I hope is neither.

The fact is, Boolean algebra is an important topic for a number of different parties and for different reasons. One of my seminal college courses, a combination of abstract algebra and computer science, used the text Modern Applied Algebra by Garrett Birkhoff (mathematician) and Thomas C. Bartee (engineer); Boolean algebra is introduced on page 3. It may be tempting for an electrical engineer to dismiss the mathematicians' concerns as excessive abstraction, but that text (written in 1970!) makes it clear that a broader treatment has practical benefits. Theoretical computer science depends on such material today even more than then.

Therefore I would like to request that everyone try to edit with consideration for all the different clients of this theory, and to especially defer to the mathematicians on the more abstract elements. Even if the utility is not immediately evident in one's present endeavors, it may well become essential for later work. --KSmrqT 02:46, 29 October 2005 (UTC)

I do fully defer to the mathematicians in the body of the article, just not in the intro, which should be written for a general audience. The following items will confuse a general audience and should not be expected as pre-reqs to understand the intro:
• Set notation
• Poset
• Heyting algebra
StuRat 03:00, 29 October 2005 (UTC)
KSmrq, please see my remarks above. I think it is important that in an article that defines a Boolean algebra, we not refer to the subject as "Boolean algebra"--that's just way too confusing; it would be like referring to group theory as "group". In my proposed intro in the above section I mention that usage, because so many people are expecting it, but explain what is the (current) WP solution for disambiguating. --Trovatore 03:01, 29 October 2005 (UTC)
In the introduction it is too technical to say that Boolean algebra is the theory of free Boolean algebras, or however you put that. But cannot that thought be there? Charles Matthews 06:29, 29 October 2005 (UTC)
I have no idea what that means. StuRat 10:46, 29 October 2005 (UTC)

The nesting of these replies is confusing. I'll just continue at my original level, responding to each of the points raised. Specifically:

• StuRat, that you balk at these three items puzzles me. Every time I use any notation — set or logical or order — I say it in English first. Poset is linked, and at worst could be spelled out as "partially ordered set"; but why is it worse than set or lattice? Heyting algebra is a parenthetical remark telling the reader what they're about, not assuming prior knowledge; but if you think it's too scary (I don't), it's expendable. I'm very much in favor of discussing mathematical topics in more accessible style thoughout, not just in the opening, so we're on the same page there. I am not in favor of dumbing them down to do so, which often means working much harder at the writing. Boolean algebras are defined and used and important in connection with these other topics; we must respect that.
• Trovatore, Do you mean I should have said "… Boolean algebras have many practical applications…"? If so, I agree; sorry 'bout that. I'll comment more on your new attempt below.
• Charles Matthews, right, "theory of free Boolean algebras" is not helpful wording to folks looking for the "other" Boolean algebra. I see no way an observation at this level of abstraction can help at the disambiguation level. (But in the body, it's a fine idea.)

About the competing rewrite: Explicit mention of Boolean logic is helpful; explicit mention of the singular mass noun sense would also be helpful. Other than that, I can't say I'm a convert. It uses more words to say less, lacks the concrete examples I used, and lacks some of the links I added. One of the benefits of hypertext is that we can say "algebraic structure" without an inline definition; those who are unsure can follow the link. Obviously this can be abused, but I don't think I did so. Because I gave concrete examples, a reader can begin to get a feel for the subject even without following the link, seeing a visible parallel between the set theory and logic concepts, including the very similar notation.

Perhaps it would help if I reproduce my list of mathematical education guidelines:

• Theorem(s), Proof(s), Intuition, Example(s), Counter-example(s), Connections
• Exercises, Teaching
• Pictures, Humor

We can't cram all these into the opening, but I did include intuition, examples, connections, and pictures (in a sense, with the parallel notation). Elsewhere a suggestion was made to include a Hasse diagram (made easily with Graphviz and uploaded as SVG), but the picture might require too much explanation this early on.

I will try one more time. --KSmrqT 12:01, 29 October 2005 (UTC)

Please don't. Trovatore, Celestianpower, and I all agree that Trovatore's version is a good compromise. Let's not open up that can of worms again, please. StuRat 12:14, 29 October 2005 (UTC)
$a\land(\lnot a) = \mbox{FALSE},$
$A\cap(A^C) = \empty.$
You can't just assume a knowledge of set notation in a general audience and use it in an introduction. This is just gibberish to the average reader. StuRat 12:29, 29 October 2005 (UTC)
This is not an example of assuming knowledge of notation, which is blindingly obvious if you include the complete sentence. As I already told you: "Every time I use any notation — set or logical or order — I say it in English first." And here is the proof:
Thus, for example, the logical assertion that a statement and its negation cannot both be true,
$a\land(\lnot a) = \mbox{FALSE},$
parallels the set-theory assertion that a subset and its complement have empty intersection,
$A\cap(A^C) = \empty.$
(Emphasis added.) My only assumption is that readers can understand English; apparently that's wishful thinking. --KSmrqT 14:35, 29 October 2005 (UTC)
The point is that those symbols contribute nothing for the average reader beyond what the English statement already contributed. Imagine a fictional example:
A car engine is on fire:
$\sigma \epsilon \gamma \delta = pi^e \,$
Now, does that series of random characters help you understand anything ? No, and even if you managed to figure out what each of the symbols meant (if they actually meant something), it wouldn't tell you anything beyond "a car engine is on fire", so why waste your time ? StuRat 16:04, 29 October 2005 (UTC)
Certainly it can wait 48 hours. Charles Matthews 12:24, 29 October 2005 (UTC)
What can wait 48 hours ? StuRat 12:33, 29 October 2005 (UTC)
Looking for anything better. I'm agreeing with you here. Charles Matthews 12:42, 29 October 2005 (UTC)
Ok. I was confused since KSmrq had already made his changes to the article when you wrote that, so I would have expected past tense. I have since reverted to Trovatore's version. StuRat 12:51, 29 October 2005 (UTC)
I've spent a lot of time writing comments and then editing only to see my edits instantly obliterated, new image and all, and my views dismissed with bullshit. I'm tempted to say some very uncivil things about some very uncivil behavior. Obviously Trovatore and I have been working simultaneously, resulting in an edit conflict (very different from an edit war). Neither of us is, I believe, intentionally trying to revert the other. But others are. I cannot respect and collaborate with those who act as if my view is worthless. Where I have tried to move forward, incorporating feedback, they can only revert back, ignoring it. To name names, I'm speaking of StuRat and Celestianpower. I see comments here saying "it can wait", after my edits have been posted and reverted. Not helpful. --KSmrqT 14:35, 29 October 2005 (UTC)
Your version, KSmrq, was (as far as non-mathematicians see it) patent nonsense and therefore was reverted. It seemed you had not read any of the discussion here and had gone back to a longer version of what I came to and was horrified to see. You didn;t make any attempt at simplification whatsoever. I would also like to add that my version was reverted very quickly also. --Celestianpower háblame 14:42, 29 October 2005 (UTC)
Trovatore did initially run into an edit conflict, but then later decided to replace your version with his, on 23:34, 28 October 2005. Since 4 people (3, if you don't count Charles Matthews, who hadn't said so until after your first revert of Trovatore's work) preferred his version to yours, it was irresponsible for you to insist on reverting his edit and putting yours back in. That said, I see no reason why parts of your intro, such as the Hasse diagram, can't be moved down to the main body of the article, where the requirement to be accessible to a general audience is not as rigorously enforced. StuRat 15:43, 29 October 2005 (UTC)

## The introduction as it stands

I feel the introduction currently residing in the article is a very good compromise. Well done everyone who worked upon it and I thank you for taking my points on board. Perhaps we could try and do the same with some other complex mathematical topics. --Celestianpower háblame 08:48, 29 October 2005 (UTC)

Agreed. StuRat 10:48, 29 October 2005 (UTC)

## alternate opening

So that others can conveniently contemplate and draw from it whatever they may find of value, here's the opening as I left it:

For a basic intro to sets, Boolean operations, Venn diagrams, truth tables, and Boolean applications, see Boolean logic.
For the use of binary numbers in computer systems, please see the article binary arithmetic.

In mathematics, a Boolean algebra (equivalently, a Boolean lattice) captures essential properties of both set operations (intersection, union, complement) and logic operations (AND, OR, NOT). The parallel is explained by the fact that a subset of a set {x, y, z}, say, either does or does not select each element, and these binary choices are equivalent to truth values of TRUE or FALSE for three logical variables x, y, z. Thus, for example, the logical assertion that a statement and its negation cannot both be true,

Lattice of subsets
$a\land(\lnot a) = \mbox{FALSE},$

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

$A\cap(A^C) = \empty.$

The connection to lattice theory and partially ordered sets is suggested by the parallel between set inclusion, A ⊂ B, and ordering, a < b. Because truth values can be represented as binary numbers or as voltage levels in logic circuits, the parallel extends to these as well. Thus Boolean algebras have many practical applications in computer science and electrical engineering, as well as in mathematical logic.

The closely related subject of Boolean logic studies relationships among elements of a Boolean algebra that always hold, no matter which Boolean algebra one starts with. As used in this article, a Boolean algebra is an abstract algebraic structure, a collection of elements and operations on them obeying specific axioms; elsewhere, the term Boolean algebra (no article, always singular) may refer to the calculus of symbol manipulation based on Boolean logic.

Boolean algebras are named after George Boole, an English mathematician at University College Cork.

Enjoy. I'm outta here. --KSmrqT 14:59, 29 October 2005 (UTC)

I really like this version. And the picture. What I dislike is the weak statement that a Boolean algebra captures essential properties. It is not restrictive. A Boolean algebra captures only those properties and nothing else. I can't think of a way to reformulate ATM unfortunately. --MarSch 12:41, 31 October 2005 (UTC)
Surely the picture can be used in the article anyway. Charles Matthews 12:57, 31 October 2005 (UTC)
Yes, anywhere but the introduction