Wikipedia talk:WikiProject Logic/Boolean algebra task force

Concerns
Here are a few concerns that inspired me to start this task force I hope that by taking a high-level view of all the articles we can find a way to organize these articles that makes sense to everyone. &mdash; Carl (CBM · talk) 14:08, 28 January 2008 (UTC)
 * The ongoing disagreements about how to organize this material have never been directly addressed with a broad enough scope to resolve them. Instead, they recur on individual talk pages, where only a few editors see them.
 * Because of the cross-disciplinary nature of these articles (spanning at least computer science, logic, and mathematics), it's important that there are articles useful to people of all backgrounds, and that these are easy to locate.

Redirects
There are lots of redirects all over the place. I have introduced a certain degree of sanity, but it's still quite bad. Anybody who is interested in the details can find them here. The page has a list of the major Boolean algebra articles, each with a convenient "What links here" link. Since this should remain only a minor distraction, I would suggest to direct any discussions on what to do with the redirects to the associated talk page. --Hans Adler (talk) 16:55, 28 January 2008 (UTC)


 * I edited Hans' page a little and transcluded it on the task force page. It may lead to ideas about better titles for articles or for topics our coverage has missed. &mdash; Carl (CBM · talk) 20:46, 28 January 2008 (UTC)

Getting started

 * ''Copied from Talk:Boolean logic

Carl's organization page has the benefits of neutrality (not hosted by the talk pages of any of the disputed articles) and the big picture (all relevant topics neatly collected and alphabetized, showing that the problem goes far beyond merely minimizing the set of articles of the form "Boolean algebra (subtopic)"). Seems to me a great forum for those who've participated in the Boolean algebra debates long enough to be thoroughly familiar with the issues (and there have been a lot of issues!). One strategy would be to start grouping the many extant articles listed on Carl's page as a step towards understanding the extent of the overlap and the appropriate connections between the groups. One candidate target might be a single introduction feeding into an unspecified number of articles on subtopics of Boolean algebra such as its structures, axiom systems (completeness), complexity theory, connections with propositional logic, close neighbors such as Heyting algebras and De Morgan algebras, etc., suitably grouped to optimally minimize the size and number of articles. The Stanford Encyclopedia Article on Boolean algebras by Don Monk is pitched at a very high level even by the standards of Boolean algebra (structure), suggesting that there is room for the latter to serve as an introduction in its own right to yet more advanced subtopics of the kind Monk addresses. --Vaughan Pratt (talk) 20:30, 28 January 2008 (UTC)


 * I think it would help to first make a list of the articles we wish we had, ignoring what already exists. Then, we can talk about how to get from where we are to the place we would like to be. I don't think we need to minimize the number of articles soley for the sake of minimizing. In some cases, it may be better to have short articles with a specific focus (Two-element Boolean algebra) and other articles with a broader scope.
 * One motivation for me getting involved here is the lack of an article that covers propositional logic in the way it is usually covered in a first logic course. This is intimately related to Boolean algebra but not covered well in any of our articles. &mdash; Carl (CBM · talk) 20:45, 28 January 2008 (UTC)
 * How is propositional logic usually covered in a first logic course? I was brought up on Hilbert systems in much the style of Propositional calculus except that the explanations weren't as obscure (or the class would have mutinied).  I refrained from trying to improve that article because there was no canonical improvement I could think of other than toning down the formalism and Greek letters---no matter what changes one makes to that article you're going to offend one school of thought or another as to the "right" presentation.  I contented myself (July 19 last) with adding a section on the equivalence with equational formulations, focusing on Boolean and Heyting algebras as the respective equational equivalents of classical and intuitionistic logic.  --Vaughan Pratt (talk) 10:35, 30 January 2008 (UTC)
 * A first course, especially at the undergrad level, would include syntax and semantics. This includes the definition of a formula, truth assignments/valuations, tautologies, and truth tables. Depending on the text and instructor, deductive systems might or might not be considered. The course would probably then go on to first-order logic. A few theorems might be proved (like an analogue of the deduction theorem, or the substitution principle for tautologies). Boolean algebras are typically not mentioned at all, nor Boolean-valued semantics, nor intuitionistic logic or Heyting algebras. In a computer-science oriented course, normal forms and resolution may be covered.
 * Of course our article on propositional logic doesn't need to stick to this list of subjects; I think Boolean-valued semantics would make a great section to tie the article into other ones. My thought at first was that I should just write an article on propositional logic, and use the existing one as a subarticle concerning deductive systems. But when I looked deeper, I realized that there is a significant issue of overlap with the Boolean algebra articles. &mdash; Carl (CBM · talk) 14:59, 30 January 2008 (UTC)
 * This seems like a great discussion for Talk:Propositional calculus where you'll be able to engage the current crop of editors for that article. While they didn't complain about my added section on equational connections they might complain about a wholesale rewriting without some prior consensus as to its form and what defects of the present article it overcomes.  I'm very much on the side of balancing syntax with semantics, but if the viewpoint there is slanted towards syntax it might be a struggle.  As Pigozzi, Blok, Font, etc. have thoroughly documented, not all logics are algebraizable.  --Vaughan Pratt (talk)  —Preceding comment was added at 12:12, 1 February 2008 (UTC)


 * I think I have failed ot explain my point. The motivation behind this page is that fragmented discussions on dozens of talk pages like Talk:Propositional calculus haven't been successful in resulting in a good arrangement of material. The propositional logic article must overlap with the logical connectives article, which in turn will overlap with the article on basic operations on a Boolean algebra (whichever article that is). That's why we need to have a higher-level plan for how to arrange the material. &mdash; Carl (CBM · talk) 15:41, 2 February 2008 (UTC)

In an ideal world
My rough idea for this project is to first make a rough list of how we wish things were organized, and then worry about how to arrange the material we have (and write more) to match that organization. Here is the beginning of a proposal. Please feel free to annotate, expand, or strike it. Let's keep the comments in this section, so the outline is easy to read.&mdash; Carl (CBM · talk) 13:35, 29 January 2008 (UTC)

Outline

 * An introductory article on boolean logic, boolean algebras, and propositional logic
 * An article on Boolean logic. Subarticles:
 * Logical connectives
 * List of topics
 * An article on Boolean algebras. Subarticles:
 * An article expounding on filters and ultrafilters
 * Boolean rings
 * Stone's representation theorem
 * The "canonically defined" fork
 * List of topics
 * An article on propositional logic Subarticles:
 * Tautology
 * Propositional formula?
 * Deductive systems for propositional logic
 * An article on normal forms of propositional formulas
 * Algebraic normal form
 * DNF and CNF
 * Zhegalkin polynomials
 * List of topics?
 * List of topics?

Choice of notation
One source of unnecessary confusion is inconsistent notation. We have the following major variants:


 * Ring-like notation ab, a+b, and a' or $$\bar a$$.
 * Logic-like notation a∧b, a∨b, and ¬a.

Ring-like notation mostly causes correct mental associations, e.g. with ideals or when teaching Boolean logic to electrical engineers. But the dual distributive law looks strange in this notation. I am also not very happy with the choice between postfix or overline notation for the complement.

Logic-like notation also causes correct mental associations, via set theory. People who are unfamiliar with set theory will be confused by the similarity of ∧ and ∨ though: they will find it hard to remember which is which. Of course, duality looks much more plausible with this notation.

From what I have seen so far, set-like notation is the standard in our pure mathematics articles, and ring-like notation is the standard in our digital engineering articles. I think logic-like notation is the modern standard in pure mathematics, so I was surprised to find ring-like notation in most newer publications I found via Google, including those on pure mathematics. Let me put it in this way:


 * 1) Digital engineering uses ring-like notation almost exclusively. Logic redundancy should continue to use it.
 * 2) Lattice theory uses logic-like notation almost exclusively. Stone space should continue to use it.

Are these observations correct? If so, what can we do about it? I would expect that the majority of our readers prefers ring-like notation and the majority of our writers prefers logic-like notation. --Hans Adler (talk) 19:37, 29 January 2008 (UTC)


 * I think all we can do is remind readers of each article what the notation means (it only takes a sentence). Each article should use the notation common to its field. &mdash; Carl (CBM · talk) 20:52, 29 January 2008 (UTC)


 * And what do we do where they intersect? E.g. I don't think it's a good idea to give both notations equal weight in the main, accessible introduction to the field, or in disjunctive normal form. People want to stick to one notation when they dive into a new subject. Superficially it looks as if the best solution for this case could be a mixture of the two such as ab, a∨b, $$\bar a$$ or a∧b, a+b, $$\bar a$$. If only one of the two dual symbols is unfamiliar, then it's very easy to remember what it means. Esthetically I prefer the first version, but unfortunately for stability theorists there are strong reasons (having to do with concatenation of tuples and implicit application of a closure operator) to read ab as a∨b in a lattice context! (This is why I always get terribly confused by the ring notation.) The second version has the problem that non-mathematicians are likely to read + as and. So this doesn't seem to work.
 * I suppose there is no general mechanism for user preferences? --Hans Adler (talk) 21:14, 29 January 2008 (UTC)


 * I don't think there is a user preference system. it is a question how to handle things like normal form articles; I confess I have always thought of them in terms of lattice operations. I see that Boolean logic currently uses that too. But the ring articles should definitely use ring notation. I hate to spend very long discussing notation, in any case. I would postpone that to an article-by-article issue. It's easy for to get wrapped up in notation and never discuss content. &mdash; Carl (CBM · talk) 21:23, 29 January 2008 (UTC)


 * In this particular case I think the ideal solution would be approximately the opposite of what we have: The very names of disjunctive normal form and conjunctive normal form indicate that they should probably be in logic-like notation. Because the duality is so obvious they should be merged. The page canonical form (Boolean algebra) should be split into sum of products and product of sums, which should treat both DNF or CNF, respectively, in the ring-like notation and in a parallel way; of course they should also treat the somewhat more special cases that are known under these names. And I suppose many applications will naturally belong on one of the two pages only. The general idea is that in such cases we should have 2 dual ring-notation versions that also cover the engineering and computer science stuff, and 1 lattice-notation version that also covers the theoretical computer science and mathematics stuff. In the intersection of lattice theory and (ring) algebra there are probably similar solutions.
 * As to the main article, we could e.g. use boolean algebra (introduction) (logic notation) and write something else more in FA style, e.g. "algebra of 0 and 1", that starts like "Suppose 0 and 1 were the only numbers, and 1+1=1. What would mathematics look like?" This article would be directed only to the general public and the very applied side.
 * Sorry for my insistence on this topic, but I think if we must live with inconsistent notation then it's important to realise this so that we can resist the natural impulse to merge the three normal form articles into one, for example. And it's important to know that some of the bigger articles will exist in a ring-notation version for one type of audience and a logic-notation version for a different type of audience. --Hans Adler (talk) 22:53, 29 January 2008 (UTC)
 * My recommendation would be to mention very early (e.g. at the beginning of Boolean algebra (introduction)) the existence of the xy+z' notation (which was the preferred notation during the 19th century and continues to be used by a number of leading exponents of the subject even today) and then spell out the preferred Wikipedia notation, the choice of which we can debate here. As a proposal to that end I would suggest 0, 1, &and;, &or;, &not;, &rarr;, &oplus; (exclusive-or), and &equiv; as the standard Boolean operators.  I have followed this convention uniformly in the Zhegalkin polynomial article (which is all about the ring basis, Zhegalkin having been the first to notice it, in 1927, strange that it took so long especially in light of Bell's 1927 much more advanced related article on modular Boolean arithmetic), even pointing out the need for &oplus; in place of + with "(In the context of Boolean algebra the addition operation mod 2, doubling as the logical operation exclusive-or, is written &oplus; in preference to + to avoid confusion with the latter's occasional use in place of &or; to denote disjunction or inclusive-or in Boolean algebra.)"   A more difficult question is whether to adopt a standard set of rules aimed at reducing the number of parentheses, for which I don't currently have any proposal other than to prefer parentheses over any unmanageably complex set of such rules.  --Vaughan Pratt (talk) 04:48, 30 January 2008 (UTC)
 * Looking at Zhegalkin polynomial just now I realize that I used xy instead of x&and;y there. Since xy has no other possible meaning in Boolean algebra, unlike x+y which could be read as either inclusive or exclusive or, I don't feel too bad about this.  It may be reasonable to allow xy and x&and;y as interchangeable, with the recommendation that xy be limited to contexts involving the ring basis, but that the ring basis stick to &oplus; in light of the potential confusion with +.  --Vaughan Pratt (talk) 05:14, 30 January 2008 (UTC)
 * As an upper bound on how complex anti-parenthesis rules should get, here is one relatively complicated proposal. Following the rule in ordinary algebra that xy+z = (xy)+z rather than x(y+z), and = is below everything, treat &and; like multiplication, &or; and &oplus; like addition, and &equiv; like =.  Insert &rarr; between &or; and &equiv;, and put &not; above everything (so &not;xy means (&not;x)y rather than &not;(xy)).  And assume right-associativity whenever it matters.  I would say reject anything more complicated than this out of hand, and probably this too as already being too complicated.  --Vaughan Pratt (talk) 05:27, 30 January 2008 (UTC)

The ring-like notation for boolean algebras seems all well and good until you want to study boolean rings. Then it becomes terribly confusing, because the addition there is different. In particular one has
 * xy = x ∧ y,
 * x + y = (x ∨ y) ∧ ¬(x ∧ y).

where the operations on the left are the boolean ring operations and the ones on the right the boolean algebra operations. For this reason it seems better (to me anyway) to always use the logic-like notation for boolean algebras. I realize, of course, that conventions in many fields are fixed. -- Fropuff (talk) 18:49, 30 January 2008 (UTC)


 * Such notational inconveniences are unavoidable. Boolean algebras can be interpreted in two non-isomorphic ways as commutative semiring structures, which is a good thing, unless you consider both in the same context and do not wisely choose your notation. In a bounded lattice top and bottom are denoted as 0 and 1, respectively, but in the lattice on the naturals (including 0) formed by the divisibility relation (so that gcd is meet) 0 is the number 1 and 1 is the number 0. We should also not forget that the ring-like notation is the original notation from the Laws of Thought. I think the best is for most articles to pick the locally most appropriate notation (corresponding to convention) while pointing out the existence of (the) other notation(s), and in one or two introductory articles to show both side-by-side. I could imagine an article devoted to notational conventions for Boolean algebra, discussing various systems, with their origins and current status (including issues as pointed out by Fropuff above), to which we could refer for readers wanting to know more. --Lambiam 06:15, 31 January 2008 (UTC)
 * I continue to support the interchangeable use of x&and;y and xy in Boolean algebra and Boolean rings, while avoiding x+y in favor of x&oplus;y as far as possible. While conventions in many fields are indeed fixed, in Boolean algebras x+y and x&or;y are both in wide use to mean inclusive-OR, whereas x+y in a ring context may mean exclusive-OR to some.  x+y should therefore be avoided because when moving from the Boolean lattice to the Boolean ring context its meaning changes from inclusive-OR to exclusive-OR, leading to confusion.  Better to stick to x&or;y for inclusive-OR and x&oplus;y for exclusive-OR uniformly across the board regardless of context. --Vaughan Pratt (talk) 12:21, 1 February 2008 (UTC)
 * In the field of logic minimization the use of + for inclusive-OR is ingrained and has been standard ever since Shannon's 1937 Master's Thesis. I don't think it is appropriate for us to use &or; instead in our articles on topics in that field; it should be sufficient if we state explicitly (which is currently not done consistently) that + is the lattice-join logical OR for which 1 + 1 = 1, and not XOR. The desire for uniformity across the board should yield here – in my opinion – for historical precedent sustained by current practice. --Lambiam 00:39, 2 February 2008 (UTC)
 * If historical precedent is the criterion, why stop at Shannon 1937 for + when you can go back to Boole 1847 (who wasn't clear in his own mind whether + denoted &or; or &oplus;, causing enormous confusion for a couple of decades) and all 19th century writers?  Peano's symbol &or;, which ironically HTML calls & or;, was not generally adopted until the 20th century when Russell and Whitehead popularized it.  It is now in almost universal use by mathematical logicians, with the notable exceptions of Tarski's school and electrical engineers such as Shannon.  Computer science splits between &or; (theory) and + (architecture), and nowadays even many computer architects have switched to &or;.  If Wikipedia were to argue priority on the basis of historical precedent its articles would be in Latin. It should suffice to mention that some writers use + for &or; and some for &oplus; and that Wikipedia avoids + altogether as a logical connective because of this ambiguity.  Peano's own motivation for replacing + by &or; was the excellent one of avoiding ambiguity in applications calling for instantiating Boolean variables with numerical formulas, e.g. x+y &le; 2 &or; 4 &le; x+y.  --Vaughan Pratt (talk) 15:54, 9 February 2008 (UTC)

One issue I have been observing
I have a concern over the several sets of multiple articles for similar concepts. I think we need to take care to include all of the appropriate bluelinks in articles, and account for different terminology used, etcetera. I keep discovering these from time to time, so I bet there are more.


 * Abstract structure --> Abstract object
 * Abstraction (mathematics) --> Abstraction
 * Model (abstract) (-->) Interpretation (logic)
 * Class (set theory) --> Class (philosophy)
 * Lemma (mathematics) --> Lemma (logic)
 * Mathematical proof --> Derivation (abstract algebra) --> Logical argument --> Derivation (logic)
 * Valuation (mathematics) --> Interpretation (logic)
 * Predicate (mathematics) --> Predicate (logic)
 * Soundness theorem --> Soundness (-->) Validity

(-->) : closely related so as to require careful attention to including different terms --> could be merged or integrated, all or in part Pontiff Greg Bard (talk) 21:22, 31 January 2008 (UTC)


 * I think that some of those are outside the scope here. The parts of propositional logic that overlap with Boolean algebras are relevant, but things like Lemma and Class aren't an issue with Boolean algebras. The subarticles about propositional logic won't overlap with algebras, either. The focus of this group is intended to be propositional logic only to the extent that it overlaps with Boolean logic and Boolean algebras. &mdash; Carl (CBM · talk) 22:30, 31 January 2008 (UTC)
 * Yes, the list includes some that are not within BAFT. It's just a list I have been collecting for a while. It's too bad about the narrow scope. This is what I mean by math-centric. I hoping that we could really do some substantial work making propositional logic accessible to people, and also a similar treatment of predicate,... I was thinking something in the way of small navigation templates, etc. Perhaps a Boolean algebra AND Prop logic task force? My goodness. Pontiff Greg Bard (talk) 23:04, 31 January 2008 (UTC)
 * I'm not looking for it to be math centric; I'd like the Boolean algebra articles to have a good balance in different areas. I don't know that there is much need for a propositional logic task force; there aren't that many articles in that particular area. &mdash; Carl (CBM · talk) 23:09, 31 January 2008 (UTC)


 * Greg, give the "my goodness" stuff a rest, would you? This task force has a specific purpose, which is somehow to find a rational solution to the unruly welter of articles with "Boolean" in the name. As they say in the movie industry, it's not a cure for cancer. As it is it will be hard enough to keep the discussions focused, without dragging in whether classes in set theory have anything to do with intensionality-vs-extensionality. --Trovatore (talk) 23:10, 31 January 2008 (UTC)


 * Hey listen, you guys are going to do whatever you want anyway. The thing is advertised as focusing on "Boolean algebra, propositional logic, logical connectives..." I am just a little surprised that oh-by-the-way it's just propositional logic as it relates to boolean algebra. That view certainly make complete sense from your point of view. I am not aware of the whole on-going boolean algebra situation (mess or otherwise). So if you have a different vision of it, and that is a common view, then fine. This task force is obvious a result of that issue. The whole thing could easily have been organized sole around propositional logic with only tangental treatment of boolean algebra, could it not? How is it that I am so flexible, and you guys are always rigid about the way you do things. I just want to share and compromise.


 * There are issues all over propositional logic in the same way. We just don't care as much about that and that's too bad. I wouldn't over-react to "my goodness" it's purely rhetorical. My goodness. Let do what we can. Pontiff Greg Bard (talk) 23:28, 31 January 2008 (UTC)


 * The issue with Boolean algebras is simply the huge spiderweb of articles on them; see the other side of this page. There is definitely some overlap with propositional logic - for example, the "main" propositional logic page, that doesn't exist, shouldn't stand in isolation to the pages on Boolean logic or logical connectives. On the other hand, the articles on predicates of various sorts are less an issue, and I don't think that Model (abstract) will overlap much.
 * It's true that it could have been organized around propositional logic with only tangential treatment of Boolean algebras. Instead, the opposite has occurred - most of the treatment is algebraic, not logical. My goal here is to try to address that from a higher vantage point. Do you have thoughts, changes, or additions on the outline higher up? &mdash; Carl (CBM · talk) 23:37, 31 January 2008 (UTC)
 * Boolean algebra (introduction) tries to remain neutral betweeen modern or abstract algebra and logic while not losing sight of Boole's original arithmetic motivation leading him away from Aristotle's syllogisms. Boolean algebra (structure) and Boolean algebra (logic) address respectively abstract algebra and logic.  Boolean algebra should not be identified with propositional calculus because they deal with respectively equations and propositions.  They are intertranslatable but modulo that their mindsets are very different: whereas the former is 0-1 neutral as in film noir the latter is like your parents, demanding the truth and deprecating the false.  --Vaughan Pratt (talk) 16:33, 9 February 2008 (UTC)
 * I have a mental distinction between propositional calculus - methods for manipulating propositional formulas - and propositional logic, which as a logic has a syntax and a semantics. Boolean algebras are the semantics, while propositional calculus is the syntax. The connection I don't see in our articles is between propositional logic and the other things. &mdash; Carl (CBM · talk) 17:28, 9 February 2008 (UTC)
 * Yes, this is a good distinction, but how broad did you have in mind? At one extreme, the simple end, if you were thinking just of Boolean algebra and classical propositional calculus, what more would you like to see about those two and their relationship (e.g. their intertranslatability) than what is already on Wikipedia?  At the other extreme, if you were thinking of "methods for manipulating propositional formulas" broadly speaking, Blok and Pigozzi's notion of algebraizable logic developed in 1989 provides an elegant framework for that concept.  The algebraizable logics are those deductive systems (propositional calculi) S whose syntactic consequence relation (the "manipulation methods") is the semantic consequence relation of a quasivariety (a class axiomatizable by conditional equations).  If such a quasivariety exists it is uniquely determined by S.  The classical and intuitionistic propositional calculi are examples of algebraizable logics, and the quasivariety they determine is in each of these two cases a variety as it happens, meaning that all axioms are unconditional equations, respectively the Boolean algebras and the Heyting algebras.  One can have propositional calculi that have no algebraic semantics axiomatizable by any set of equations, whether or not conditional; these are the non-algebraizable logics.
 * Or did you have something in between these two extremes in mind? --Vaughan Pratt (talk) 04:10, 10 February 2008 (UTC)
 * In this context I was thinking on the small end; algebraic logic is a natural generalization, but I think too far removed to be emphasized heavily in an introductory article on Boolean algebra. What I was thinking of is something like a couple paragraphs explaining how Boolean algebras can be used to give (Boolean valued) semantics for propositional logic, and how these semantics relate propositional calculus and to tautologies. This would be another way to motivate filters, as well, to show how the 2 element Boolean algebra suffices for testing propositional tautologies. Of course, most introductory books skip Boolean valued semantics altogether, but it might be nice to have a little on it in our introduction. &mdash; Carl (CBM · talk) 13:39, 10 February 2008 (UTC)
 * It is not uncommon for introductory texts at the undergraduate level (e.g. Gensler's Introduction to Logic , ISBN 978-0415226752) to define the validity of propositional arguments in terms of evaluating propositional formulas using truth tables, thus effectively using a concrete Boolean algebra without ever mentioning the term. --Lambiam 20:16, 10 February 2008 (UTC)
 * That's exactly what I'm talking about. I'm not saying we should dwell on it, but I do think it's an important connection. The thing that the undergrad books don't do, as far as I have seen, is general Boolean valued semantics in addition to two-valued semantics. &mdash; Carl (CBM · talk) 00:40, 11 February 2008 (UTC)
 * Enderton's A Mathematical Introduction to Logic (second edition, ISBN 978-0122384523) somewhat mysteriously mentions the possibility of using a complete Boolean algebra for the universe of truth values and then immediately declares: "we will confine ourselves to [two-valued logic]". I suspect the text aims at the graduate level. Mendelson's Introduction to Mathematical Logic (fourth edition, ISBN 978-0412808302) even mentions Lindenbaum algebra. I don't think we need to go as far as that in an introduction to Boolean algebra. --Lambiam 04:40, 11 February 2008 (UTC)
 * The first paragraph of Boolean algebra (introduction) says that "Boolean algebra deals with the values 0 and 1." The second paragraph sounds isomorphic to Enderton's digression, hopefully less mysteriously: it reads,  "Boolean algebra also deals with other values on which Boolean operations can be defined, such as sets or sequences of bits. However Boolean algebra is unlike many other systems of algebra in that it obeys exactly the same laws (equational properties), neither more nor fewer, no matter which of these other values are employed. Much of the subject can therefore be introduced without reference to any values besides 0 and 1. We therefore postpone to the section on Boolean algebras the treatment of these other values."  Hopefully Wikipedia is no less clear on that point than Enderton.  --Vaughan Pratt (talk) 21:20, 11 February 2008 (UTC)
 * Incidentally although I don't say so there, Example 4 of Boolean algebra (introduction) is a Lindenbaum algebra for n variables, aka the free Boolean algebra on n generators. The construction carries through to infinite n provided the unions of regions are restricted to finite unions (otherwise you just get the power set of a power set, which turns out to be the free complete atomic Boolean algebra).  --Vaughan Pratt (talk) 03:41, 12 February 2008 (UTC)

On merging Boolean algebra (introduction) and Boolean logic
As I haven't been part of all the discussions, and new readers here are even less informed than I am, could some people who have participated on those pages briefly explain the issues, and the motivation for having two separate introductory articles? &mdash; Carl (CBM · talk) 15:43, 2 February 2008 (UTC)


 * First, a little background. There are two overlapping concepts called, more or less, the same thing:


 * 1) Boolean logic, also sometimes called Boolean algebra (note the singular), refers to the application of set logic to binary values, involving AND, OR, and NOT operations performed on bits. This concept is widely used in electronics and computers, and is taught in high schools and even earlier grades.  This topic, in my opinion, can, and should, be made accessible to a general audience.


 * 2) Boolean algebras (note the plural) refers to a wider mathematic discipline, where objects can have more than just binary values. This concept is generally only taught in college to those with a major in mathematics.  This topic, in my opinion, can never be made accessible to a general audience.


 * Originally there was a single article named "Boolean algebra". That article was written for PhDs and completely incomprehensible to a general audience.  Any attempt to make that article comprehensible resulted in undos by a group of PhDs "protecting their turf".  Even one of the original founders of Wikipedia commented on how utterly incomprehensible it was (see ), way back in 2002.  To solve this problem, we split the article into "Boolean algebra", which remained aimed at PhDs, and "Boolean logic", written for the rest of us.  Unfortunately, "Boolean logic", which I started in September, 2005, was later renamed to Boolean algebra (logic), with all the material written for a general audience stripped out and replaced with more material written for a PhD audience.


 * I then restored the "Boolean logic" article in September, 2007, as it was before the changes were made. We then had two PhD level articles, Boolean algebra (structure) (the original Boolean algebra article) and Boolean algebra (logic).  Perhaps they should be merged, but I won't involve myself in that discussion.  I will, however, insist that an article remain which is aimed at a general audience (including the computer science and electronics applications).  A new article was then created by User:Vaughan Pratt, named Boolean algebra (introduction).  I don't really understand the purpose of this article.  Despite the name, it seems to include advanced material, as well. StuRat (talk) 21:51, 2 February 2008 (UTC)


 * An introductory article that includes applications is certainly warranted. It should definitely cover computer science and electronics applications. I don't see why an introductory article cannot also include brief introductions to Boolean algebras and propositional logic. I'm interested in hearing from Vaughn Pratt as well. &mdash; Carl (CBM · talk) 22:08, 2 February 2008 (UTC)


 * My question is "why should an introductory article include advanced material, when we can just refer readers to those articles via links ?". StuRat (talk) 22:33, 2 February 2008 (UTC)


 * Ordinarily, introductory articles use summary style to cover all major facets of a topic. I don't know that I agree with your division of the material into "advanced" and "not advanced". Boolean algebras are a perfectly ordinary topic for undergraduate mathematics, with numerous textbooks at the undergraduate level; they are not an esoteric area of interest only to a few researchers.
 * On the other hand, the article Boolean algebra (introduction) has its own issues, especially its narrow focus on Boolean algebras; it seems to omit applications and propositional logic. &mdash; Carl (CBM · talk) 22:55, 2 February 2008 (UTC)


 * Carl, I did not react to your outline, because I feel the situation is much too complicated for a top-down approach (at least for me). But this is a much simpler question. So here is my "In an ideal world…" reply to this specific point.
 * I think there are good reasons to have two articles on Boolean logic. Perhaps we can classify our potential readers as general public, electrical engineers, programmers, undergraduate mathematics students, and philosophical logicians. (I have excluded the theoretical computer scientists, lattice theorists, algebraists and mathematical logicians, assuming that they will all be happy with an article on Boolean algebras.)
 * An article that is good for all these groups seems to be impossible. For each group we need to give a lot of weight to the right kind of motivation, from the beginning. For the general public I would take the "What happens if all numbers except 0 and 1 disappear, and suddenly 1+1=1?" approach. Electrical engineers need digital circuits, programmers need bit twiddling idioms in various programming languages, mathematics undergraduates need sets, and philosophers need propositional logic. Drop one of these examples, and we will lose the corresponding group. Try to do all applications in one article, and we will lose them all.
 * I think from an expositorial POV the topic divides naturally into "algebra of 0 and 1 / technology" and "algebra of truth values / philosophy". It seems natural to have one article using the first approach and algebraic notation, and one using the second approach and logic notation. Both should start from zero, and explain the other approach somewhere, but not in the beginning. The first article should be the one for electrical engineers (probably by far the biggest group of readers), and the second article should be suitable for maths undergraduates and philosophers. The beginnings of both articles should be accessible to the general public, and both articles should be completely accessible to programmers (who, after all, need to switch between both POVs). But the specific target groups for the second article are (or should be) able to cope with a somewhat steeper learning curve, like what we have in Boolean algebra (introduction). So the main task is to rewrite Boolean logic so that it becomes accessible to those who are not served well by the other article. --Hans Adler (talk) 00:05, 3 February 2008 (UTC)
 * Take as the reader a member of the "general public", not being a member of any of the more special groups interested in Boolean logic (such as electrical engineers, programmers, undergraduate mathematics students, and philosophical logicians), who wishes to read up on Boolean logic purely out of intellectual curiosity. To serve this reader well, an introductory article on the topic should present
 * a concise history (Boole, Shannon);
 * the co-existence of multiple notations;
 * the existence of different interpretations, in particular truth values, sets, and binary signals;
 * the algebraic laws defining Boolean algebra; and
 * various applications, preferably illustrated by simple examples: as a logical calculus (while explaining the relationship to the propositional calculus), database query languages, logic-circuit minimization.
 * I see no reason why this can't be combined in a single introductory article. For more special audiences we have more specialized and less introductory articles that we duly refer to along the way as usual. --Lambiam 15:43, 3 February 2008 (UTC)
 * I have two concerns with this. First, I think that the reader you have in mind is a tiny minority, when compared to the vast number of engineers who need to learn about this. (Obviously this would change if the article were featured. Second, you sound as if the readers you have in mind are happy if we give them the basics with one interpretation, and then they can easily switch to any of the others. I doubt this. If you think of the drama when abstract groups and abstract topological spaces were first introduced about a hundred years ago, then it should be clear that the right way to make them understand what is going on is to present them essentially the same (mathematical) material twice, with two completely different motivations and interpretations, so that they can feel it is the same thing. --Hans Adler (talk) 16:26, 3 February 2008 (UTC)


 * Among the many things Wikipedia is not, Wikipedia is not a textbook. If engineers have a need to learn about Boolean logic at the introductory level, and they cannot put up with examples drawn from propositional logic or database queries, they should pay some money for a textbook, which can be had used already for $5. I did not propose giving one interpretation but three (truth values, sets, and binary signals). Also, whether it is morphogenetic fields or something else, many ideas that once met with dramatic and seemingly unsurmountable resistance somehow turned out to be perfectly acceptable to the next generation; abstract mathematical structures may be like that. --Lambiam 09:08, 4 February 2008 (UTC)

The discussion seems to have stalled. Some squirmishing seems to be going on instead at Talk:Boolean logic, which may be a waste of time since one user appears locked in a delusion of owning that article and being able to "fix" its shortcomings. --Lambiam 19:59, 6 February 2008 (UTC)

Sorry, I was a bit slow responding to Carl's "I'm interested in hearing from Vaughan Pratt as well." Since StuRat has put a lot of work into defending Boolean logic (far more than in following up on suggested improvements though) I figured I at least owed him the courtesy of evaluating his article to see what could usefully be merged from it into the general introduction, assuming we're in agreement that a single general introduction branching out into two or more narrowly focused articles is the way to go. I've posted my analysis of the article at Talk:Boolean logic and asked defenders of it to step forward. Absent any defenders besides its author, I'm having difficulty justifying merging anything from this article besides Boolean logic, which I've merged into Boolean algebra (introduction), which I preserved more or less as is in order that StuRat would have something of his in the introductory article.

Carl, your suggestion to include something on propositional calculus in Boolean algebra (introduction) seems a good one. That connection is documented by me at Propositional calculus. I or someone else could work up the same information into a form that would fit smoothly into Boolean algebra (introduction); while there would be a fair amount of overlap quite a bit of adjustment would be needed for a seamless adaptation.

Regarding shortage of applications in Boolean algebra (introduction), has that situation improved any since you raised that concern, and in any event what additional applications besides propositional calculus would you like to see?

Regarding StuRat's insistence that Boolean algebra (introduction) is Ph.D. level material, I rewrote much of that section today to bring it down to what any high school student who understands sets should be able to understand. I'm sure StuRat will stick to his guns that this material is incomprehensible, so I'm only interested in complaints from others that it is still too advanced for inclusion in an introduction.

At present I'm in favor of merging or deleting a large bunch of the current crop of articles while preserving more or less as is Boolean algebra (introduction), Boolean algebra (logic), and Boolean algebra (structure). The various articles treating the Boolean ring basis could be merged into the Boolean ring article. For now I don't have any proposal for Propositional calculus because it may make sense for people with that mindset to inhabit a parallel world to the algebraic one, and to have translations between those worlds in lieu of any attempt at merging. Venn diagram is fine on its own since one can get into more detail about them there, such as the results from the past decade or so about what shapes work. I'm less sold on ok with Truth tables, which seems too trivial to be its own article and would be better as a redirect to Boolean algebra (introduction). Executing on these seems a sufficiently large project as to justify holding off on looking for more such opportunities, otherwise we'll never start. --Vaughan Pratt (talk) 07:24, 7 February 2008 (UTC)


 * I broadly agree with Vaughan. But one exception is that I think each notable and separable topic, e.g. truth table, is deserving of having its own article. Paul August &#9742; 18:23, 7 February 2008 (UTC)
 * Although Vaughan has struck his remarks above about "truth table", my concern about over merging still applies. Before we merge or delete "a large bunch of the current crop of articles", let's please be sure that a consensus is reached here first. Paul August &#9742; 20:09, 7 February 2008 (UTC)
 * Indeed. I was assuming that we'd want to reach consensus on a more detailed plan before rushing in.
 * Carl, concerning your thought that Boolean algebra (introduction) could benefit from a section on propositional calculus, I made a first stab at Boolean algebra (introduction) this afternoon. Is this anything like what you had in mind?  It is still pretty rough and could use some clean-up.  --Vaughan Pratt (talk) 01:33, 8 February 2008 (UTC)
 * Also, while it's not relevant to this discussion, any fans of Boolean algebras canonically defined may be interested in my rewrite today of the first (Definition) section, aimed at making at least that section accessible to a wider audience. The approach of starting from the algebra of all finitary operations on {0,1}, as opposed to some finite basis, while certainly not standard, has a certain naturality that one gives up when introducing the material via an arbitrarily chosen basis such as the lattice or ring basis, and a fortiori an arbitrarily chosen set of equations axiomatizing the concept.  --Vaughan Pratt (talk) 16:23, 8 February 2008 (UTC)

March 2008
Now that people have had a month's break from all this, do we have any sort of consensus on how to shrink the current crop of articles on Boolean algebra? In particular what should be done about StuRat's article Boolean logic? His article on Boolean algebra was substantially revised a year ago to address its many shortcomings. This would have been progress had not StuRat disagreed that his original article had any shortcomings and reinstated the whole of the original under a new name, Boolean logic, where he is now defending it against all critics.

I gave a long laundry list of its shortcomings at Talk:Boolean_logic and StuRat responded by promising to "change the article where needed." A month later all he's changed is to replace "or" by "and" in several places, so evidently he doesn't share my concerns and considers the article fine as is. User:Paul August says above that he "broadly agrees" with my proposal to slim things down to one introduction as long as there are also separate articles on "notable and separable" aspects, which seems to be the general consensus (I'm certainly all for that).

Does anyone else have concerns about Boolean logic? And, with or without those concerns, does the article currently meet a Wikipedia need? Bear in mind Wikipedia's slogan (read the whole of the first paragraph). I fully agree with Paul that we need consensus here ("partial correctness"), but we also need some movement towards consensus ("total correctness"/termination). --Vaughan Pratt (talk) 16:04, 8 March 2008 (UTC)


 * I've said this many times already: we should not have two introductory articles to the topic, and I think that Boolean algebra (introduction) is a good start. If Mr. StuRat has problems with it because it is only accessible to "PhDs", he can help improve the article by pointing out exactly where the problems are. --Lambiam 03:42, 9 March 2008 (UTC)
 * So what role should Boolean logic play on Wikipedia, which claims to be "for a general audience?" It seems to me that the article is in violation of OWN.  Unlike other encyclopedias Wikipedia does not permit people to claim ownership of an article they wrote for Wikipedia, however attached they might be to their own work.  StuRat responded (21:42, Sept. 24, 2007) to the extreme editing his article had received earlier by reviving the whole of his original article under its present name of Boolean logic.  Is that a permissible response to editing, even when that editing was "merciless" as the guidelines put it?  If not then I suggest this revived article be dealt with in accordance with Wikipedia guidelines, whether it be deletion or whatever.  Any other recommendations?  --Vaughan Pratt (talk) 00:17, 10 March 2008 (UTC)
 * A reasonable step would be to (again, perhaps) propose a merge of the articles and see what the reactions of various editors are. I tend to agree with a merge of Boolean algebra (introduction) and Boolean logic, and agree with Lambiam's comments above. &mdash; Carl (CBM · talk) 14:16, 10 March 2008 (UTC)
 * StuRat revived his article, in its original intact form, once before after I merged his and my articles. I retained those parts of his article which I didn't object to, see Talk:Boolean_logic for the parts I did object to.  I've been hoping someone would spring to StuRat's defense and counter my objections.  Without that, if I do the merge again exactly the same material from Boolean logic will be retained.  Since it's already in Boolean algebra (introduction), this is essentially a no-op.  Please counter my objections so that we have some candidates for a real merge.  Or are people ok with a no-op merge which will leave Boolean algebra (introduction) unchanged? --Vaughan Pratt (talk) 22:10, 10 March 2008 (UTC)
 * One possible course of action in a case like this is to issue a "Request for comments". For an example, see the announcement here. --Lambiam 16:00, 11 March 2008 (UTC)
 * That example seems less critical since it's just one bit which moreover is currently set correctly ("a"). This one's a bigger one because it's a whole article we don't know what to do with.  If I were an outside observer I'd say just delete it, but as the main contributor to date to the competing article I have a nontrivial internal conflict of interest (or whatever the proper name for COI internal to Wikipedia is), so I'd feel more comfortable letting someone else step in here.  --Vaughan Pratt (talk) 05:11, 13 March 2008 (UTC)
 * The same RFC procedure can be used for any dispute where the usual discussion fails to reach consensus and lead to a resolution. A possible alternative is to take Boolean logic to Articles for deletion (nominating the page to be redirected rather than totally deleted) but that procedure is more heavy-handed. Although it is decent to reveal your role in contributing to the article, you have no COI for this article in the Wikipedia sense, which is purely external; you might have one if an issue arose over the Sun logo or the article on dynamic logic. --Lambiam 10:43, 13 March 2008 (UTC)
 * I feel an RFC would be better than an AFD, not least because the goal is to merge the articles, not delete one of them. &mdash; Carl (CBM · talk) 12:00, 13 March 2008 (UTC)
 * Although as I pointed out, operationally there will be no difference given that the merge was already done before StuRat revived his version. --Vaughan Pratt (talk) 02:43, 17 March 2008 (UTC)

Proposal for the organization of a collection of BA-related articles
Here is a proposal on how to divide the main material over various articles.
 * Introduction to Boolean algebra. An entirely elementary article, not requiring more than exit-level high-school maths background, mainly explaining the rules of the calculus (shown in action with an example of a simple yet not entirely trivial proof), and explaining its significance, both historically and in terms of its applications.
 * I'm all in favor of that. Presumably such an article would proceed at a slower pace than the Introduction to Boolean algebra that I wrote in 2008. --Vaughan Pratt (talk) 02:40, 26 March 2011 (UTC)


 * Boolean algebra. The main article on this "Calculus of Logic". In some ways similar in concept and structure to the Introduction article, but not shying away from more advanced concepts, and giving more detail. In particular, the article should offer in-depth treatments of axiomatization, completeness, and the relation to propositional logic (not necessarily in that order). The concept of Boolean algebras as algebraic structures is mentioned, of course, but mainly in their role as interpretations of the logic.
 * Except for the unexplained removal of the lead I added recently in order to fulfill WP:LEDE, the current version of Boolean algebra seems to me to fulfill those requirements reasonably well. A short section on complexity with links to the main articles in that area would round it out.  --Vaughan Pratt (talk) 02:44, 26 March 2011 (UTC)


 * Boolean logic. A redirect to Boolean algebra.
 * Now that we have more than a dab page at Boolean algebra, this is exactly the right way to deal with Boolean logic. (StuRat got a three-year lease of life on his article because this took so long.) --Vaughan Pratt (talk) 02:40, 26 March 2011 (UTC)


 * Boolean algebra (structure). The topic is clear; this can be as advanced as needed.
 * Currently this article is almost content-free, it needs a lot of work. Since Boolean algebras canonically defined contains the sort of material that should be in Boolean algebra (structure), Lambiam's understanding of it notwithstanding, I can merge some of that material in, though the "canonically" aspect of it might need to take a detour through a peer-reviewed journal before it meets Wikipedia's WP:OR guidelines.  --Vaughan Pratt (talk) 02:40, 26 March 2011 (UTC)


 * It does need a lot of work. I would like to put in stuff about, for example, atomicity, atomlessness, freeness, completeness, model-theoretic saturation.  Also natural examples of the structures, for example the regular open algebra derived from a forcing poset, and quotients by ideals (yes, I know, you don't need them if you go through the Stone stuff etc, but that is not the quick way to see them).  Can we please have this stuff earlier in the article than massive text about the definition?  --Trovatore (talk) 02:51, 26 March 2011 (UTC)
 * Goodness, I agree with Trovatore for once... --Vaughan Pratt (talk) 03:14, 26 March 2011 (UTC)
 * History of Boolean algebra. The name speaks for itself. Something we don't have now, but there is no lack of material and its inclusion is justified by the historical significance of the subject.
 * I can make a start on this, being sufficiently familiar with that area as to have given an invited talk at a conference on its history. --Vaughan Pratt (talk) 02:40, 26 March 2011 (UTC)


 * Applications of Boolean algebra. This should mainly cover the use of BA for describing and simplifying binary digital logic, and database queries. Again fairly elementary, just like the Introduction.
 * Yes! --Vaughan Pratt (talk) 02:40, 26 March 2011 (UTC)

The last two should help to keep the other articles concise. --Lambiam 17:34, 25 March 2011 (UTC)
 * Not too bad, except that:
 * I would dump the "intro" article. The "calculus" (or, as I've been calling it recently, "field of study") article can be made accessible to your target audience in its early sections.
 * Even if the early sections of a maths article are accessible to a wide audience in sone stage of its Wikilife, there is a constant creep away from that. An advantage of having a separate introductory article is that it helps, also in the future, to keep the material widely accessible. --Lambiam 17:49, 25 March 2011 (UTC)
 * I'm not deathly opposed to it, but it's not my preference. Almost everyone involved has decried the huge proliferation of articles.  I would prefer a clear two-article model, one for each of the two clear meanings of the term Boolean algebra.  (That's not an objection to the applications or history articles, which are different topics, just as there can also be articles on more specific structures, like free Boolean algebra and complete Boolean algebra.)
 * Let me clarify my position. I am not all opposed to proliferation of articles on Boolean algebra, my only complaint is with duplicative articles.  Twenty articles on twenty clearly distinct aspects of Boolean algebra would be perfectly fine by me.  On the other hand we already have comprehensive coverage of both free Boolean algebras and complete Boolean algebras; the only reason you think otherwise is that you either haven't bothered to read what we have to date or wouldn't recognize a free or complete Boolean algebra when you saw one.  --Vaughan Pratt (talk) 03:14, 26 March 2011 (UTC)
 * Intro articles to me are a bit iffy in general. In some cases they seem to be the only reasonable accommodation of the different groups needing to be satisfied, as with special relativity.  Is it really that bad in this case?  Maybe it is; I'd rather think it's not. --Trovatore (talk) 18:06, 25 March 2011 (UTC)
 * It should be clear that the "field of study" and "structure" articles are the ones that would be called Boolean algebra except for the name collision &mdash; there should be a hatnote from the field-of-study article to the structure article specifically, not to a disambig page. --Trovatore (talk) 17:38, 25 March 2011 (UTC)
 * For the hundredth time Trovatore raises his strawman argument that "Boolean algebra" should consist of exactly two articles. An article about why the left hand has exactly five fingers might be expected to have no more than two articles, but in general Wikipedia has no expectation that the content of an important subject such as Boolean algebra should limit itself to exactly two articles.  Trovatore seems to believe Wikipedia has a guideline called WP:TWO.  As I've said before I have no idea where he gets this from, but he keeps repeating it anyway.  I am convinced Trovatore is deaf.  --Vaughan Pratt (talk)
 * I have explained myself many times. By the way, my understanding of  a "strawman argument" is one where you disingenuously put a weak argument into the mouth of your opponent.  I do not see where I have done that.  --Trovatore (talk) 02:53, 26 March 2011 (UTC)
 * First, there is no consistency to your many explanations, which keep changing their rationale. Second, who besides you has proposed that there be exactly two Boolean algebra articles?  And who besides you has proposed that an article on the field of study cannot talk about Boolean algebras?  You've been disingenuously putting those weak arguments into the mouths of the other parties here.  Those are merely two examples of the strawman arguments you've been making.  --Vaughan Pratt (talk) 03:14, 26 March 2011 (UTC)
 * My arguments have been consistent since the early days of the conflict with StuRat. When I don't seem to be getting them across with one set of words, I try another, but modulo small things the argument has always been the same.
 * I do not remember whether I ever said that the field-of-study article should not talk about Boolean algebras, but I certainly have not taken that position recently. In fact I have explicitly stated the opposite.  What I have said is that it should not be about them.  I trust the distinction between an article discussing something, and being about that thing, is reasonably clear to everyone. --Trovatore (talk) 04:29, 26 March 2011 (UTC)
 * Your positions are fine by me whatever they are just so long as you don't base unilaterally reverting the contributions of others on them without discussion. If you do then we have something substantive to discuss.  Unlike you I prefer not to revert edits unilaterally but rather to raise them for discussion.  Unilaterally made edits in hotly contested areas are the stuff of edit wars.  --Vaughan Pratt (talk) 06:45, 26 March 2011 (UTC)
 * WP:BRD. The discussion takes place with the status quo ante. --Trovatore (talk) 07:52, 26 March 2011 (UTC)
 * Correct. Adding text and removing text have equal status with regards to being "unilateral". Paul August &#9742; 14:05, 26 March 2011 (UTC)
 * I've added a hatnote to Boolean algebra pointing to Boolean algebra (structure). --Lambiam 17:49, 25 March 2011 (UTC)

Can we discuss the content of the proposal, please, preferably without bickering? --Lambiam 16:26, 26 March 2011 (UTC)

My noobish take on Lambian's proposal:
 * Support for Boolean algebra and Boolean logic as proposed.
 * Support contents of Boolean algebra (structure) as proposed & amended by Trovatore, although the name might need work (I would not mind the plural), but I'd surely like to see some parts from boolean algebras canonically defined included in the "structure" article, e.g. the alternative definitions
 * Weak support for History of Boolean algebra. Add if necessary. The main Boolean algebra article should have a decent section first.
 * Weak support for Introduction as prposed (it seem too low-level); you can find that in various articles on logic gates here and what not, so it seems redundant.
 * Very weak support for applications article. Only if you really have too much of that in the main Boolean algebra article. Notable applications have their own articles anyway, so at best this would be a collection of sections with main tags.
 * Support the hatnote as best dab method, but I think the wording needs work

Did I miss anything important?

Also, we need a redirect from Switching algebra, but I'm not sure whether to sent it to two-element Boolean algebra or to boolean algebra; some undergraduate-level electronics/computing books have heard of Boolean algebras as well, and want to avoid confusion e.g.    , but others  have not. I'm surprised honestly I found so many making the distinction, I was expecting the opposite. I suspect that those who don't bother with the distinction just use Boolean algebra instead of switching algebra, so there's selection bias. Actually, out of those that mention the distinction, one makes note that they're going to use the terms interchangeably, and another notes that the distinction is seldom made. Tijfo098 (talk) 09:54, 27 March 2011 (UTC)
 * Googling the term "switching algebra" I get the impression that the count-noun use is a fairly common one, synonymous with "two-element Boolean algebra". However, it is used specifically in the context of a calculus for switching circuits as introduced in A Symbolic Analysis of Relay and Switching Circuits, Shannon's master thesis, and so the mass-noun sense of "calculus for switching circuits" (Shannon's terminology) seems quite appropriate. Perhaps we can turn the article A Symbolic Analysis of Relay and Switching Circuits into one that actually describes the calculus as introduced by Shannon and let Switching algebra redirect there. --Lambiam 10:38, 6 April 2011 (UTC)

Summarizing the above:
--Lambiam 10:41, 6 April 2011 (UTC)
 * Introduction to Boolean algebra: no strong opposition, but also no general enthusiasm. This can wait till the rest is in order and a need becomes apparent. If we don't have such an article, it is even more imperative that Boolean algebra contain an intro that is fully accessible for reders with just an exit-level high-school maths background.
 * Boolean algebra: general agreement.
 * Boolean logic: general agreement. We should perhaps wait with implementing this until Boolean algebra (and specifically the intro!) is in good shape.
 * Boolean algebra (structure): general agreement.
 * History of Boolean algebra: limited enthusiasm, but no opposition. Although this has no urgency, I think there is enough material for a full encyclopedic article.
 * Applications of Boolean algebra: reactions ranging from enthusiasm to very weak support.

More cleanup work
More Awbrey/Gregbard/StuRat-type clean-up is needed at: -- Tijfo098 (talk) 04:57, 30 March 2011 (UTC)
 * Predicate variable
 * Boolean-valued function
 * Predicate (mathematical logic)

Two-element Boolean algebra
Considering the goal to reduce duplication, will it be an improvement if the content of Two-element Boolean algebra gets merged into Boolean algebra (structure), replacing the first example there, getting instead a section on its own to which Two-element Boolean algebra then redirects? --Lambiam 11:33, 6 April 2011 (UTC)
 * It's not a bad idea, as long as the section is kept short. In my opinion there's not really a lot to say about the two-element algebra, but that doesn't always stop people from finding things to say. --Trovatore (talk) 18:30, 6 April 2011 (UTC)
 * Meh, unlike the other articles this one has a clear scope. We even have something more trivial at Boolean domain. You should ask WT:WPM too. Not many people are watching this. Tijfo098 (talk) 20:06, 6 April 2011 (UTC)
 * I left a note there myself. Tijfo098 (talk) 20:10, 6 April 2011 (UTC)
 * I think that it is Boolean domain what has to be merged with "Two-element Boolean algebra", rather than to merge "Two-element Boolean algebra" to something else. First of all, it is the same thing. And it is very notable (apart of Boolean algebra (structure)) as a prominent model of truth values. Incnis Mrsi (talk) 09:14, 26 February 2012 (UTC)


 *  Don't merge to boolean algebra, and maybe merge to boolean domain. Be very careful merging, as it is clear that Two-element Boolean algebra is written from the perspective of a middle-school student, encountering the idea for the first time. Adding any sort of college-level material to the article threatens to derail the accessibility that this article offers.  I think that there is a place for such elementary presentations of concepts in WP: we have many hundreds of articles in Category:Elementary mathematics and this could/should be one of them. linas (talk) 15:49, 21 September 2012 (UTC)

Good articlehood
Several of the articles withing the scope of WP:BATF are close to the Good Article standard. We should nominate some. &mdash; Charles Stewart (talk) 09:23, 2 May 2016 (UTC)

I propose we start with Boolean algebra (structure). &mdash; Charles Stewart (talk) 12:50, 3 May 2016 (UTC)
 * The two most obvious issues there that need fixing before nomination (to avoid a quick fail) are the lack of an inline citation for all claims, and the hidden text in "Axiomatics" (which vlolates WP:Manual of Style § Scrolling lists and collapsible content). A more minor issue is that the references are inconsistently formatted (a mix of CS1 and CS2). —David Eppstein (talk) 16:19, 3 May 2016 (UTC)
 * The lack of inline citations needs to be fixed. The good article criteria do not require conformity to the whole of WP style, just to five parts, so while it would be nice to fix the other two, it should not be necessary. &mdash; Charles Stewart (talk) 06:00, 4 May 2016 (UTC)