Talk:Boolean algebra/Archive 1

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Goal of this article

This article is intended as an introduction to Boolean algebra combining minimal prerequisites with maximal clarity. In view of the many subtopics in Boolean algebra dealing with its models, axioms, computational complexity, relationships to related systems such as propositional calculus and Heyting algebras, etc. it is expected that separate articles on those subtopics will emerge (a number already have), all of which should be able to assume this introduction as a prerequisite.

The introduction should cover the more accessible aspects of Boolean algebra at an elementary level sufficient to give a usable idea of what Boolean algebra is about, with sufficient conceptual clarity and organization to allow the material to be conveyed by the reader to a third party without reference to notes or mindlessly repeating entire sentences. (Litmus test: does the article's overall conceptualization make it any easier for the reader to explain Boolean algebra at a cocktail party?)

Every subtopic having its own article should be introduced however briefly in the introduction and related to the other subtopics, thereby providing the reader with the list of available subtopics and some preliminary insights into which if any might be sufficiently interesting as to be worth pursuing further. --Vaughan Pratt (talk) 20:58, 28 January 2008 (UTC)

Various comments

• I wasn't very happy to see yet another Boolean algebra article being started, but seeing it in its current state I am glad that Vaughan Pratt did it.
• One thing I like in particular is the way in which the section "Operations" unobtrusively lays the ground for Boolean rings.
• As to the truth tables, personally I would prefer the operations table style that StuRat is using in Boolean logic for AND and OR; it's probably least confusing to a general audience. But when you try to apply it to a unary operation such as negation it becomes awkward, so perhaps we could drop that.
• For a general audience it might be even better to replace the tables entirely by a list of 10 identities: $0\wedge 0=0$, $0\wedge 1=0$, …, $\neg 1=0$. I suppose this depends on whether we focus on explaining Boolean algebras, or whether we want to explain Boolean algebras as a first step towards abstract algebra. (Operation tables seem to play an important role for learning what abstract algebra is about.)
• This should probably become the main article and be moved to Boolean algebra.
• I would have no objections to merging Boolean algebra (structure) into this article, as long as we make sure that the article stays as focused as it is now, especially at the beginning. [Edited: doesn't make much sense after my latest changes. 00:38, 28 January 2008 (UTC)]
• This article does not discuss lattices, and I think that's good. There are so many facets that the best thing is probably to have a set of articles on Boolean algebras, Boolean lattices, Boolean rings and Stone spaces, each of which discusses one aspect, and at least one article like Boolean algebras canonically defined to discuss the connections.
• Section 3 from Boolean algebra (logic) could perhaps be considerably shortened and merged into this article.

--Hans Adler (talk) 13:16, 27 January 2008 (UTC)

• I think the sections on Venn diagrams, logic gates and laws would all profit from a short example that shows how a simple law can be applied to Venn diagrams and logic gates, and how another, slightly more complicated law (outside the scope of Venn diagrams, but we needn't talk about this) can also be applied to logic gates. --Hans Adler (talk) 14:04, 27 January 2008 (UTC) — Vaughan, I really like your excellent solution to this. Hans Adler (talk) 00:38, 28 January 2008 (UTC)

Hans, I appreciate both the positive feedback and your help with the article. Your suggestions were excellent and I think I've now managed to incorporate most of them into the article. I look forward to more good suggestions and edits from you, and from any other like-minded Wikipedians who feel like improving the shape of this article subject to its main premise that it be as introductory as possible.

I should add that despite all the flak people have been giving StuRat, without his complaints about Boolean algebra (logic) continuing to be pitched at too high a level I would not have felt the need for a more elementary account so urgently. Thanks, StuRat! --Vaughan Pratt (talk) 01:18, 28 January 2008 (UTC)

While I like Hans' suggestion of moving section 3 (on Laws) of Boolean algebra (logic) to here, there is a substantial body of material there dealing with derivations and soundness/completeness. My current feeling is to allow Boolean algebra (logic) to gradually turn into a repository for that sort of material, which is seriously technical and requires close study. The idea with Boolean algebra (introduction) is that there should be no intricate reasoning, and ideally no huge conceptual hurdles either. StuRat and I may have different thresholds there, but I've been trying to lower my threshold down to his without compromising the organization I feel even an introduction can benefit from. --Vaughan Pratt (talk) 07:27, 28 January 2008 (UTC)

That's fine with me. I will think about what to do with the material that is too applied to fit into your plan for Boolean algebra (logic). --Hans Adler (talk) 09:44, 28 January 2008 (UTC)

I just removed the following unsourced speculation from the material that came from Boolean algebra (logic): "It [i.e. the algebra of two values] has not featured prominently in law however, perhaps because mathematical methods in general have not been applied as vigorously there as in these other application areas." I think there are technical reasons why an application of logic in law would be very hard, and my own speculation on the nature of "legal set theory" would be that a good solution to that is very likely not two-valued. The earlier cross-examination and organisation membership examples seem more or less adequate however. --Hans Adler (talk) 09:44, 28 January 2008 (UTC)

Good, that makes the article shorter. However I dispute your justification for the removal: my suggested reason had little to do with two values and everything to do with the relative lack of formal mathematics in law compared e.g. to business, economics, biology, even sociology. Are you claiming formal mathematics is used in law just as much as in these other disciplines? News to me. --Vaughan Pratt (talk) 21:22, 28 January 2008 (UTC)

No, I am only doubting the word "because" here (which, I realise, is guarded by "perhaps"). My impression is that law, when done right (as it seems to be by many good academics and many higher courts), is almost as exact as mathematics, but very different. I am a bit sensitised to the question of law and mathematics because several years ago I was coaxed into reading a book on legal logic. I don't remember much of what I read, but I do remember some obvious points. I will try to explain the issue by a silly example. I am sure it's not as atypical as one might believe, since court hearings are so often about border cases of the law.

An important problem with formalising logic is the extreme context dependence of almost every word in a legal text. E.g., what does "all men" mean? Are women included? Foreigners? Slaves? Prisoners? In a text which mentions "men and women" in most places, but only "men" in connection with military service and only "women" in connection with pregnancy and birth, how about children? How about a pregnant girl? How about boys of the same age? An interesting observation is that "men and women" is probably not exactly a union of two sets, because there are always border cases. At least in Germany they are solved by the state ascribing one of the two sexes to every citizen. But if our example text also applies to foreigners, then a hermaphrodite from a country with no such rules may be meant by neither "men" nor "women", but by "men and women".

Now the very moment the legislature decides to introduce the word "hermaphrodite" somewhere in the text, you need to reevaluate the entire text. (Provided you are still interested in the silly border case.) So much for German law, which is strongly influenced by the Napoleonic code and therefore relatively straightforward. With Common Law the context dependence becomes much greater.

That's why I would speculate that a practically relevant mathematical foundation for legal logic would look radically different from what we know as logic. Perhaps it would have some similarities with quantum mechanics; perhaps a "yes or no" question to a witness is like forcing an electron to choose between two slits, and the main object of court hearings is to make wave functions collapse. So, yes, I agree with both parts of the sentence but I don't see a strong connection between them. I think I would have been more happy with something like "perhaps because today's mathematics is generally not easily applicable to law." This is also harder to interpret as blaming the jurists for the fact. But having thought all this through a bit more, I think I should have left the sentence as it was, since this was about such a marginal point. My long explanation is merely to make you understand where my puzzling behaviour came from. --Hans Adler (talk) 23:32, 28 January 2008 (UTC)

Ok, so deleting it was an improvement not only with regard to shortening the article but removing an evidently controversial claim that might point readers in a new research direction but doesn't add much to their overall comprehension of Boolean algebra. --Vaughan Pratt (talk) 03:14, 29 January 2008 (UTC)

In the section Diagrammatic representations, in the first diagram, the disjunction symbol looks like ^. (talk) —Preceding undated comment added 14:17, 15 July 2009 (UTC).

Since this is an "introduction", should there not be a least a mention of the practical use of boolean algebra? Even just the simple introductory phrase: "Boolean algebra forms the basis of digital electronics." The article does a find job with the Who, What, When and Where but says nothing about the Why. --esalkin (talk) 07:14, 29 Febuary 2010 (UTC)

Excellent suggestion ledewise, ManFromTheFuture (did you or someone else manually edit your date to a nonexistent Feb. 29 2010?). I appended a sentence to the lede, how does it look, and when should we expect your reply?  :)

Article-wise, did you not see the whole section headed "Applications"? Was that not enough "Why" for you? --Vaughan Pratt (talk) 08:21, 23 February 2010 (UTC)

What's going on???

HEY! I can't see the symbols for AND and OR in this article! What's going on?!

I can see the symbols for NOT and the other math symbols ok, but the AND and OR symbols show up as question marks in my browser. On the other hand, the AND and OR symbols on this discussion page (up and down carets) show up fine... what's going on?

Thanks for any help, thought you'd wanna know that in some browsers your symbols are getting mucked. 76.243.129.217 (talk) 09:13, 30 October 2008 (UTC)

Hmmmm, I have the same problem except my AND OR symbols show up as rectangles. Comment made on March 31 2010. —Preceding unsigned comment added by 128.206.20.54 (talk) 17:09, 31 March 2010 (UTC)

When reporting this behavior, please specify the operating system (Windows 7, Mac OS X 10.6.4, Fedora 12, Ubuntu 10.04, Debian 5.0.5, whatever) and browser (IE8, Firefox 3.6.3, Chrome 5.0.375.70, Opera 10.60, whatever) you're using. This will be very helpful in diagnosing the problem. --Vaughan Pratt (talk) 04:58, 9 July 2010 (UTC)