Talk:Boolean algebra (structure)/Archive 2

message: 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)
replies: text: 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)
replies: text: Maybe that will work.  It looks weird, but I think it covers the bases. -- Arthur Rubin (talk) 01:50, 3 November 2005 (UTC)
replies: text: 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)
replies: text: That covers it, thanks.  Unless the wikibooks guru changes it again -- Arthur Rubin (talk) 19:00, 7 November 2005 (UTC)
thread_title: Wikibook 

message: 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)
replies: text: 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)
replies: text: 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)
thread_title: Example in intro

message: 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)
replies: text: 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)
replies: text: 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)
replies: text: 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)

text: 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)
replies: text: 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)
replies: text: 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)
thread_title:  where is the example from propositional calculus? 

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

message: 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)
replies: text: 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)
thread_title:  Per request, intro replaced 

message: 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:
#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.
#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)
thread_title:  In the spirit of the new intro... 

message: C syntax is very common, should it be included?:
 logical    bitwise    operation
    ||         |          or
    &&         &          and
     !         ~          not
               ^          xor
Kim Bruning 17:50, 19 May 2006 (UTC)
replies: text: 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)

text: 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)
thread_title:  [[Operators in C and C++|C Syntax Notation]] 

message: 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 f\left (a,b \right) = a \setminus \left ( a \land b \right ) is a total operator with properties defined by equations.)  — Arthur Rubin |  (talk) 14:43, 27 February 2007 (UTC)
thread_title:  Generalized Boolean algebras 

message: Please regard Peano, Calculo geometrico, 1888, the first axiomatic version of boolean algebras de: Boolesche Algebra.85.216.21.153 21:44, 28 February 2007 (UTC)
thread_title:  Peano 1888 

message: HOW CAN I SIMPLIFY ANY FUNCTION I SEE IT ? —The preceding unsigned comment was added by 90.153.128.12 (talk) 15:05, 20 March 2007 (UTC)
thread_title:  I FEEL IT IS SO HARD