Talk:Boolean algebra/Archive 5 Source: en.wikipedia.org/wiki/Talk:Boolean_algebra/Archive_5

This is an archive of past discussions about Boolean algebra. Do not edit the contents of this page. If you wish to start a new discussion or revive an old one, please do so on the current talk page.

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FYI, T. Haliperin (ISBN 0444516115, Algebraical Logic 1685-1900 p. 346) considers De Morgan "The last great traditional logician" and says about his work: "Despite many insightful and forward-looking innovations the basic framework for De Morgan's logic was still the Aristotelian syllogism with its four categorical sentence forms, each with a copula connecting subject and predicate terms." However TH credits De Morgan with "his introduction of a universe [of discourse], arbitrarily specifiable, together with the removal of any distinction between a name and its contrary, either of which could be taken as the positive term." Interesting enough, De Morgan's student, Jevons, contributed to Boolean algebra by replacing Boole's (set) difference with negation along those lines. Tijfo098 (talk) 02:49, 12 April 2011 (UTC)

Could you cite a specific passage from Jevons 1864 that treats negation any differently from how De Morgan 1858 (On the Syllogism: III) treats it? --Vaughan Pratt (talk) 05:27, 12 April 2011 (UTC)

Also TH says this about De Morgan: 'Although he introduces a symbol U for "everything in the universe spoken of" and u for its contrary, denoting "nonexistence", De Morgan declines to use them in syllogistic inferences, considering them to be extreme cases which would only be of interest to mathematicians "on account of their analogy with the extreme cases which the entrance of zero and infinite magnitude oblige him to consider"'. Tijfo098 (talk) 02:52, 12 April 2011 (UTC)

Jevons is also credited (p. 367) with replacing + as xor (as Boole used/defined it) with the non-exclusive version, making it dual/symmetric to "and". Tijfo098 (talk) 02:59, 12 April 2011 (UTC)

From De Morgan's perspective all that his student Jevons had done was to change De Morgan's notation for disjunction from (A,B) to A+B. De Morgan objected strenuously to this change, perhaps in part because Boole was already using A+B with a different meaning, perhaps because De Morgan saw no reason to change his (A,B) notation. Today we write Boole's x+y as x⊕y and Jevons' A+B as A∨B (due to Russell). --Vaughan Pratt (talk) 05:27, 12 April 2011 (UTC)

However TH says on the next page that Jevons "By thus treating 0 as if it were, and yet were not, a term, Jevons fudges over the need for explaining what qualities it does have and what is meant by a combination of 0 with a genuine term." Tijfo098 (talk) 03:02, 12 April 2011 (UTC)

Perhaps a bigger complaint about Jevons is that he did not accept distributivity of disjunction over conjunction, showing that his understanding of Boolean algebra was incomplete. --Vaughan Pratt (talk) 05:27, 12 April 2011 (UTC)

Also, the next chapter in the Handbook (by Valencia) is more sympathetic to De Morgan, at least at the philosophical level by citing this: "We know that mathematicians care no more for logic than logicians for mathematics. The two eyes of exact science are mathematics and logic, the mathematical sect puts out the logical eye, the logical sect puts out the mathematical eye; each believing that it sees better with one eye than with two. [De Morgan, 1868, 71]". Tijfo098 (talk) 04:06, 12 April 2011 (UTC)

(I have to say that also applies to Wikipedia editors of today, to a certain extent.) Tijfo098 (talk) 04:07, 12 April 2011 (UTC)

I am sympathetic to Boole, De Morgan, Peirce, and Schroeder, all of whom made substantive contributions. Boole's priority and depth of understanding, even if not complete, fully justifies naming the subject for him. I would be very interested in any evidence that Jevons played any important role in the development of Boolean algebra. --Vaughan Pratt (talk) 05:27, 12 April 2011 (UTC)

"A few years after Jevons' 1864 [book], but independently of it, C. S. Peirce's first paper in logic, 1867, likewise introduced a non-exclusive sum of terms in an algebraic context. However, unlike Jevons, Peirce adheres to the extensional point-of-view of Boole's calculus. Moreover, he retains its problematic features, i.e., undetermined and uninterpretable terms, which Jevons had eliminated. It is clear from Peirce's writings that he wanted his later work—in which among other changes these problematic features no longer appear—to supersede that of this fledgling paper. " Tijfo098 (talk) 03:06, 12 April 2011 (UTC)

It's a good question what Boole himself considered to be the interpretable terms. At some gut level he seems to be aware that x+y can be treated simply as disjunction when xy = 0, where the distinction between inclusive and exclusive disjunction does not arise. This seems to be what underlies his representation of inclusive disjunction as x(1-y) + y(1-x) + xy and exclusive disjunction as x(1-y) + y(1-x). If one follows that rule consistently, all n-ary Boolean operations are expressible, for example x→y as xy + y(1-x) + (1-x)(1-y). --Vaughan Pratt (talk) 05:39, 12 April 2011 (UTC)

According to TH (p. 370-371), "Apparently Grassmann was unaware of any contemporary work in logic as he mentions only Lambert's Neues Organon of 1764 and Twesten's Logik of 1825." TH conclude with "Grassmann derives just about all the standard elementary results of Boolean algebra—not surprisingly since his algebra of concepts is just a Boolean algebra of (finitely many) atoms. Absent only is the recognition of the duality principle (but no one else had at this time), and the idea of the development of a Boolean function as a sum of constituents. If Die Begriffslehre oder Logik [1872] had appeared 25 years earlier conceivably we might all be referring to Robert-Grassmannian algebra instead of Boolean algebra. Indeed, it is a closer fit to Boolean algebra than is Boole's algebraic system." Tijfo098 (talk) 03:12, 12 April 2011 (UTC)

Robert Grassmann is a younger brother of Hermann Grassmann. The two used to collaborate a fair bit. Hermann is the inventor of linear algebra, though it had to be reinvented before his priority could be recognized: his book introducing the subject while both correct and remarkably insightful is pretty incomprehensible as written. --Vaughan Pratt (talk) 05:57, 12 April 2011 (UTC)

TH (p. 371-372): "Schroder's 1877, Der Operationskreis des Logikkalkuls, opens with the expression of surprise at the lack of attention given to Boole's remarkable achievment, that of realizing the ideal of a calculus of logic which Leibniz had propounded. Schroder was unaware of Jevons 1864 and Peirce 1867 since he cites as the only works subsequent to Boole's, two short notes (Cayley, A. J. Ellis) and the independently arrived at treatment of Grassmann's, just described. The neglect of Boole's work is attributed to its imperfections." Tijfo098 (talk) 03:16, 12 April 2011 (UTC)

TH goes on: «Unlike Jevons with his "qualities" and Grassmann with his "Begriffe", Schroder is forthrightly extensional—classes are what logic calculus is about. And, unlike Peirce, the subject is not grounded on general algebraic notions with multiple interpretations, but on clearly defined operations on classes—class union symbolized by '+', intersection by 'x' and complementation by a subscript '1' (which in his 1890 becomes a short vertical line, presumable so as not to confuse it with the numeral).» (N.b. presumably this is the origin of ¬.) «Of historical interest is Schroder's calling attention to and establishing the duality principle for logic—that to each general valid formula another is obtained on interchange of '+' with 'x' and '1' with '0'. Intimation of this principle occurs in Peirce 1867 (which Schroder had not yet seen) which called attention to the double distributivity—of multiplication over addition and addition over multiplication.» I think I'm getting close to violating copyright here, so I'll stop, but the above are sufficient to write a semi-decent early history section. Tijfo098 (talk) 03:22, 12 April 2011 (UTC)

Valencia writes (p. 477): «We shall pay attention to the theory of logic he develops in [Schroder, [1877] 1966]. This elegant booklet is the third equational logic written after Boole. The resulting system is "the algebra of logic as we know it today " [Lewis, 1918, 111]. As we shall see, Schroder defines a structure with two binary operations, multiplication and addition, a unary one, negation, and two constants, 0 an 1 that satisfy all the axioms of a Boolean algebra. However, when preparing this work Schroder was not aware of Jevon's nor of Peirce's contributions to algebraic logic.» Tijfo098 (talk) 04:21, 12 April 2011 (UTC)

And on the next page: "Indeed, it has been claimed that the main influence on his work stems not even from Boole. The roots of his equational logic, is argued in [Peckhaus, 1997], lie in the symbolic logic of Robert Grassmann and the doctrine of forms of Herman Günther Grassmann and Herman Hankel. The same thesis is defended in [Peckhaus, 1996]." So, apparently this influence is a relatively recent discovery, which probably explains why Haliperin 1986 book makes no mention of R. Grassmann, but Haliperin had changed his mind by 2004 to argue for it forcefully. Tijfo098 (talk) 04:21, 12 April 2011 (UTC)

I've found Schrőder tremendously insightful, clear, and broad in his coverage. He's definitely one of my favorite 19th century algebraic logicians. --Vaughan Pratt (talk) 06:12, 12 April 2011 (UTC)

In the earlier quote from Martin Davis, Martin omits De Morgan's contribution to turning this into a battle royal. The following is a very abridged summary of a much more complex soap opera, some of which I presented at a talk in Edinburgh in 1989 based on Peter Heath's introduction to On The Syllogism, and Other Logical Writings, an anthology in book form of De Morgan's contributions to logic.

In November 1846 De Morgan had written On the Syllogism: I. What set the battle off was Hamilton's strongly worded accusation that De Morgan had simply plagiarized his unpublished notes. De Morgan was a prickly but scrupulously honest character, a very bad combination when being so accused. De Morgan quickly persuaded Hamilton he'd gone too far, and Hamilton was ready to back down on his charge of willful plagiarism. However De Morgan demanded a full and public apology. Hamilton wasn't ready to back down that far, so De Morgan threw down the gauntlet and challenged Hamilton to a written duel in the Athenaeum, to which Hamilton agreed. The date at this point is around April 1847.

Hamilton's first shot in this duel was titled Letter to Augustus De Morgan, Esq and included all the prior correspondence. Hamilton replaced his charge of willful plagiarism with the only slightly weaker charge that De Morgan was laboring under the delusion that what he'd learned from Hamilton's notes was his own discovery. Hamilton argued that De Morgan had no way of independently figuring out any of what he'd claimed as his own prior to seeing Hamilton's notes.

De Morgan's first shot was Statement in Answer to an Assertion made by Sir William Hamilton. He argued that he had acted with complete propriety in sending Hamilton everything he was claiming as his prior to any attempt at publication, and that furthermore it contained much that was not in Hamilton's notes, for example a new syllogistic form allowing the inference from "most men have coats" and "most men have waistcoats" that some men must have both. He pointed out that Hamilton had only attempted to explain Aristotle's existing syllogisms and not to invent new ones.

Hamilton responded with a lengthy Postscript arguing that anything De Morgan might have added was based on a confused understanding of Aristotle. De Morgan replied briefly but angrily in the Athenaeum to this, Hamilton reciprocated a week later equally angrily, then at the beginning of June 1847 silence fell.

That autumn De Morgan published his book Formal Logic (originally intended to teach rigorous reasoning to his geometry students), on the very same day as Boole published his pamphlet The Mathematical Analysis of Logic while citing the battle as the inspiration for a renewal of his earlier interest in the algebra of logic. In November De Morgan sent a copy of his book to Hamilton, who returned it a week later.

This battle continued on until 1852, with De Morgan writing On the Syllogism: II in February 1850, then abated, in part because De Morgan, aware of Hamilton's failing health, appeared to find continued hostilities unchivalrous. When Hamilton died in 1856 De Morgan published a brief obituary in the Athenaeum, then returned to writing about syllogisms. On the Syllogism: III appeared in February 1858, where inter alia he introduces an explicit operation of disjunction and states De Morgan's laws. On the Syllogism: IV is dated November 1859 and gives the first treatment of relation algebra including the concept of residuation which he refers to as "Theorem K." --Vaughan Pratt (talk) 05:07, 12 April 2011 (UTC)

I see that nobody else tried to use the above notes to develop the history section insofar. I admit to having burned out of Wikipedia as a whole for a quite a while... Tijfo098 (talk) 14:31, 4 October 2012 (UTC)

This merge was a bad idea; the introductory part of the merged content is not appropriate to explain Boolean algebra.

Using analogies to arithmetics for introduction is counterproductive and not helpful. The sections "Operations" and "Laws" are full of nonsense and misleading gibberish.

• Explaining conjunction on 0 and 1 by analogy to multiplication is nonsense because Boolean Algebra in general has more than 2 elements.
• Using the semi-analogy for disjunction which even needs a trick to work is even more nonsense.
• What is "ordinary algebra"?
• Splitting the laws into those that have arithmetic analogies and those that don't is absolutely not helpful in explaining the nature of Boolean Algebra. (A hint to this in a lecture helps to prevents confusion; referring to it throughout the introduction creates confusion.)
• Speaking of "additional laws" is playing on the confusion, because Boolean Algebra is in no way an extension of arithmetics.

To be revised substantially.

— Preceding unsigned comment added by Towo (talk • contribs)

Yeah, a major revision is required. I would suggest that all laws be tabulated instead of giving explanations like that.Roshan220195 (talk) 15:01, 25 March 2012 (UTC)

This is the article on Boolean algebra as a subject. The points you're raising apply to the article Boolean algebra (structure) which treats Boolean algebras as algebraic structures, which is what you're talking about here. There is a long history behind the distinction drawn by these two articles that you will find in the archived talk pages, which you should consult before trying to merge the two articles into a single article.

To claim that Boolean algebra is not the algebra of 0 and 1 is to have failed to have read and understood Boole's original works, where he makes patently clear that he intends the subject to be the algebra of 0 and 1. Furthermore it is completely correct mathematically to say that a Boolean algebra is any model of the equational theory of 0 and 1, and is much easier to absorb than the long list of equations entailed by the definition of a Boolean algebra as any complemented distributive lattice, which is the quick way of summarizing those equations. All these points are made in this article, just not at the high speed you want to make them. --Vaughan Pratt (talk) 07:01, 14 April 2012 (UTC)

Speaking of "additional laws" is playing on the confusion, because Boolean Algebra is in no way an extension of arithmetics. This is readily contradicted both by Boole's 1854 book Laws of Thought, which emphasizes the connection with arithmetic, and the article Boolean ring, which makes that connection more precise. Just as the equational theory of commutative rings is that of the integers, so is the equational theory of Boolean rings that of the integers mod 2, noticed independently by Ivan Ivanovich Zhegalkin in 1927 and Marshall Stone in 1936. Boolean rings and Boolean algebras are essentially the same thing, in the sense that they have the same polynomials, i.e. they are intertranslatable. --Vaughan Pratt (talk) 07:39, 14 April 2012 (UTC)

S.No	Law	Expression
1	Identity	${\displaystyle {\begin{aligned}x\lor 0&=x\\x\land 1&=x\end{aligned}}}$
2	Annihilator	${\displaystyle {\begin{aligned}x\lor 1&=1\\x\land 0&=0\end{aligned}}}$
3	Idempotence	${\displaystyle {\begin{aligned}x\lor x&=x\\x\land x&=x\end{aligned}}}$
4	Involution	${\displaystyle {\begin{aligned}\lnot \lnot x&=x\end{aligned}}}$
5	Complementarity	${\displaystyle {\begin{aligned}x\lor \lnot x&=1\\x\land \lnot x&=0\end{aligned}}}$
6	Commutative	${\displaystyle {\begin{aligned}x\lor y&=y\lor x\\x\land y&=y\land x\end{aligned}}}$
7	Associative	${\displaystyle {\begin{aligned}(x\lor y)\lor z&=x\lor (y\lor z)\\(x\land y)\land z&=x\land (y\land z)\end{aligned}}}$
8	Distributive	${\displaystyle {\begin{aligned}x\land (y\lor z)&=(x\land y)\lor (x\land z)\\x\lor (y\land z)&=(x\lor y)\land (x\lor z)\end{aligned}}}$
9	Absorption	${\displaystyle {\begin{aligned}x\lor (x\land y)&=x\\x\land (x\lor y)&=x\end{aligned}}}$
10	De Morgan	${\displaystyle {\begin{aligned}\lnot (x\lor y)&=(\lnot x)\land (\lnot y)\\\lnot (x\land y)&=(\lnot x)\lor (\lnot y)\end{aligned}}}$

I would suggest that we include this table in place of monotone and non monotone laws and explain what monotone/non monotone laws are seperately. And the 'double negation' law is better known as involution law.Roshan220195 (talk) 16:26, 25 March 2012 (UTC)

First, you're at the wrong article, you want Boolean algebra (structure). Second, why do you want to oblige readers to wade through a long list of laws when exactly the same information is conveyed by defining a Boolean algebra to be a complemented distributive lattice, as done at the outset of that article? Third, "involution" is a property of negation, the law asserting negation has this property is standardly called "double negation," see e.g. http://www.britannica.com/EBchecked/topic/169990/law-of-double-negation.

I recommend reading the (very extensive!) archived talk pages over the past several years that led to the current structure before trying to reinvent this particular wheel. --Vaughan Pratt (talk) 07:22, 14 April 2012 (UTC)

There is a tag on this article suggesting to merge Boolean algebra (logic) here. I know there was a long history of POV forks on this topic, and I do not remember any of it. But looking at the two articles I am hard pressed to see a difference in topic. That makes me think the merge is a good idea. I want to leave a comment and wait for a week or so before doing it, in case I have missed something important. — Carl (CBM · talk) 11:53, 4 October 2012 (UTC)

Merge discussions tend to take a long time to complete. That one was proposed in this edit. The "Boolean algebra (logic)" article was initially written by User:StuRat and then rewritten by Vaughan Pratt [1]. I see the discussion on its talk page was mostly between the two of them. I guess we could ask them to refresh us on what all that was about, and why they found the extra article necessary. Tijfo098 (talk) 13:51, 4 October 2012 (UTC)

I agree with Carl though that the current article on Boolean algebra appears written in an introductorily enough manner so the "(logic)" one seems rather redundant at this point. (I see that Introduction to Boolean algebra was already merged/redirected to here, and it looks like I was the one who did that, although I had no recollection of it, haha.) Tijfo098 (talk) 14:02, 4 October 2012 (UTC)

I vaguely recall that there was a "master plan" agreed by consensus a couple of years ago about how to organize the articles in the BA area, but I can't seem to find it now... Tijfo098 (talk) 14:19, 4 October 2012 (UTC)

Having written over 90% of the two articles proposed for "merging," I would have no objection to merging the older one simply by (effectively) deleting it, i.e. the null merge. The newer one has pretty much everything that's in the older one.

These were the second and third of my attempts at rewriting StuRat's article, the first being Boolean algebras canonically defined (BACD). BACD was pitched at such a high level as to make a toned-down version mandatory. Judging by its talk page there seems no need to merge or delete BACD, which can be viewed as an advanced article on the topic. I would have no objections to moving BACD to a more appropriate name if there's a consensus on what it should be. --Vaughan Pratt (talk) 06:19, 7 October 2012 (UTC)

Just noticed "By introducing additional laws not listed above it becomes possible to shorten the list yet further, see Boolean algebra (logic)" in the present article. Not sure yet what this refers to, but merging Boolean algebra (logic) into the present article may need to be done more carefully than I thought. --Vaughan Pratt (talk) 06:34, 7 October 2012 (UTC)

Thanks for the comments. I am going to enact the merge this afternoon by starting with the "null merge" and seeing whether there are any specific things that should be copied from the subarticle. — Carl (CBM · talk) 16:44, 8 December 2012 (UTC)

It's old old redirect that almost everyone agrees it should be deleted. CBM re-introduces it in all pages most probably to get into conflict with other editors. -- Magioladitis (talk) 15:48, 8 December 2012 (UTC)

Although it has the same effect, otheruses4 has a more specific semantic meaning. It means we want the header to say "This article is about XXX. For YYY, see ZZZ.". So if the 'About' templates are changed later, by using the redirect here we will be able to keep the display the way we want it, by just changing the redirect to do the right thing. — Carl (CBM · talk) 15:53, 8 December 2012 (UTC)

I don't understand how you can claim that other uses, 4 or otherwise, has any kind of meaning to anyone other than the one that wrote it, and possibly you it seems. If the templates are "changed", then that either means that the display should also be changed or that it is the reason they are changing the template. Either way, the change itself would likely nullify this argument. I agree with Magio above, this seems to be nothing more than a subtle way at irritating other editors by somehow logically trying to argue nonsense. Kumioko (talk) 20:06, 8 December 2012 (UTC)

'According to Huntington the moniker "Boolean algebra" was first suggested by Sheffer in 1913.' 'Moniker' is slang: http://dictionary.reference.com/browse/moniker — Preceding unsigned comment added by 2.27.227.175 (talk) 13:03, 29 December 2012 (UTC)

I wonder if someone would please review the OR operation?

It is my understanding that x OR y = x + y, not x + y - xy. x+y-xy is the XOR function. — Preceding unsigned comment added by 208.110.205.6 (talk) 22:11, 24 January 2013 (UTC)

The IP missed the word arithmetically. All is right currently. “+” is used somewhere as an alias to “∨”, but not in the article, so 1 + 1 = 2 ☺ Incnis Mrsi (talk) 07:11, 25 January 2013 (UTC)

Wouldn't it be more appropriate to put Boolean ring in See Also? The article merely says that and, or, and not are basic operations, but clarifies this later on with "meaning that they can be taken as a basis for other Boolean operations that can be built up from them by composition." Any basis with that property, for example that of Boolean rings, would serve just as well. Most theoretical switching theorists and computer scientists would view the distinction between Boolean algebra and Boolean ring as pendantic---there are dozens of papers on the succinctness possible using different choices of basis. And viewed abstractly as an algebraic theory, meaning one containing all its polynomials as operations, there is no difference at all. --Vaughan Pratt (talk) 04:39, 10 February 2013 (UTC)

For once I have to agree with Vaughan. The difference between a Boolean algebra and a Boolean ring is fairly inessential. Even if it weren't, it's not clear why we would need to add not to be confused with Boolean ring at the top, because the names are not so similar that people are likely to arrive at the wrong article by accident. --Trovatore (talk) 09:41, 10 February 2013 (UTC)

Oh, I did forget that this is the article about the mass-noun sense of Boolean algebra, so forget the first point, but the second one still stands — we don't need to tell people not to be confused unless there's a reason to think they will be. --Trovatore (talk) 09:44, 10 February 2013 (UTC)

I agree that the difference between "Boolean algebra" (structure) and "Boolean ring" is not essential. But we have two different articles, and this is confusing for the reader. A hatnote is thus needed. I have corrected my edit by removing my hatnote and adding Boolean ring in the previous one. Nevertheless, a link to Boolean ring in the body of the article would be useful, may be in section "See also", or better in section "Boolean algebras". By the way this section should be much earlier in the article and, in any case, before section 'Concrete Boolean algebra', which is very technical. D.Lazard (talk) 11:04, 10 February 2013 (UTC)

Thanks. I really don't think we need a direct link to Boolean ring from the hatnote. Hatnotes are for disambiguation or common confusion about titles (which both don't apply to Boolean ring), not for advertising. But it's much better now and I can live with it. It has the advantage of explaining quite well what we mean by Boolean algebra (structure). Hans Adler 12:30, 10 February 2013 (UTC)

Oops, Trovatore is right, I was forgetting that Boolean algebra (structure) is the article about Boolean algebras. This article is about Boolean algebra as a subject ("mass noun" as Trovatore calls it). The counterpart would be "Boolean ring theory" (analogously to "group theory") except no one ever talks about such a thing, only Boolean rings. --Vaughan Pratt (talk) 04:03, 11 February 2013 (UTC)

I see it in Boolean algebras canonically defined#Truth tables, but there may be a better location. It certainly doesn't belong in this article. — Arthur Rubin (talk) 18:41, 28 March 2013 (UTC)

Truth function #Table of binary truth functions. Incnis Mrsi (talk) 19:26, 28 March 2013 (UTC)

He's added still more unsourced material which doesn't belong in this article. I've removed it three times, he hasn't come here to justify his additions, and I can make no sense of the justification on my talk page. — Arthur Rubin (talk) 21:52, 28 March 2013 (UTC)

The paragraph first dealing with derived operations introduces the operation sign ⊕ without any former explanation, and leaves to the reader the deducing needed to understand that it means XOR. I believe it should be fixed, although since I'm not a native English speaker and a poor writer I'd rather leave it to someone else. Thoughts? Jordissim (talk) 17:53, 5 June 2013 (UTC)

As far as I can see, the paragraph includes a detailed enough explanation and a link to the relevant article. I think that’s quite sufficient.—Emil J. 18:19, 5 June 2013 (UTC)

You're completely right, my fault. I misread the opening section. Jordissim (talk) 20:07, 5 June 2013 (UTC)

The section on Monotone laws is contradictory given that the X ∨ Y operation was previously defined to be equivalent to X + Y - XY. The section references X ∨ Y being equivalent to X + Y, without the additional term, in its separation of the two sets of laws.

Also, the Law of Distributivity of ∨ over ∧ is simply incorrect, which can be shown by setting X to be true and Y and Z false. ( X ∨ ( Y ∧ Z )) is true, but (( X ∨ Y ) ∧ ( X ∨ Z )) is false. — Preceding unsigned comment added by 128.101.13.199 (talk) 21:43, 21 September 2013 (UTC)

I guess I'm not seeing where the article says X ∨ Y is equivalent to X + Y. It only says it satisfies many of the same laws, which is true. Also the "equivalence" to x+y-xy is prefixed with "If the truth values 0 and 1 are interpreted as integers" with no mention of applicability to any other values for x and y besides truth values; certainly 7+7-7*7 is not 7, in fact x+y-xy is idempotent if and only if x is 0 or 1. Perhaps this is worth clarifying in the article.

How do you get "(( X ∨ Y ) ∧ ( X ∨ Z )) is false" when both X ∨ Y and X ∨ Z are true? Vaughan Pratt (talk) 21:04, 20 October 2013 (UTC)

Section 5.2. Digital logic gates:

The contemplations on De Morgan equivalents and the Duality Principle and what it could mean for the number of Boolean operations represented by AND and OR gates are not helpful in this section. Especially the last paragraph is quite obscure and grammatically goofed up. I suggest to remove the last 2 or 3 paragraphs here. Towopedia (talk) 09:51, 29 April 2014 (UTC)

For many readers of Wikipedia the entire article is surely not helpful. The question of which of the sixteen binary Boolean operations are representable with one AND or OR gate and inverters should be helpful for those interested in the design and analysis of Boolean circuits. The last sentence merely lists the eight such that are not so representable; I'll rephrase it, which may or may not help. Couldn't find the grammatical error, a hint please. Vaughan Pratt (talk) 16:30, 8 May 2014 (UTC)

Rephrased as promised. Let me know whether it helps. Vaughan Pratt (talk) 16:02, 9 May 2014 (UTC)

For x V y, shouldn't the intersecting circles be white, since x V y = x + y - (x ^ y)? (as in, OR does not include the product of x AND y). SquashEngineer (talk) 14:35, 2 November 2015 (UTC)

The venn diagrams are correct. XOR would have the overlapping area white. FreeFlow99 (talk) 09:31, 15 January 2016 (UTC)

There are 7 normally used boolean operators (most used are And, Or, Exclusive-Or, Not; lesser used are If and only If / XNOR, NAND, NOR) plus one I've never actually seen being used (but is mentioned in this article): Material Implication, which gives 8 in total. The lists of Order of Operations I've managed to find on the internet only contain 3 of the operators: And, Or, and Not. Even if I assume that NAND has the same precedence as AND, and NOR the same as OR, that still only covers 5 of the 8. It would be useful if a table showing the Order of Operations, for all 8 operators, for boolean algebra were included in this article. Is there an expert who can add it? FreeFlow99 (talk) 09:42, 15 January 2016 (UTC)

There seems to be an error in the "Values" paragraph confusing OR and XOR. The error continues in the "Operations" paragraph where OR is said to be: x V y = x + y - (x * y).

204.235.238.54 (talk) 19:28, 13 March 2015 (UTC)

It seems that this is you who confuses OR and XOR: It is easily to check that x + y - (x * y) is 0 if and only if both x and y are 0. This is exactly the definition of the truth table of OR. D.Lazard (talk) 20:36, 13 March 2015 (UTC)

Why does Figure 3 miss out the exclusive or? http://www.ivorcatt.org/exclusive-or.htm Ivor Catt — Preceding unsigned comment added by 31.48.249.63 (talk) 13:33, 6 June 2017 (UTC)

What is the definition of "boolean calculus" that is used in this section of the article? Jarble (talk) 16:18, 10 July 2017 (UTC)

It originally was "boolean algebra," but someone replaced it last year with "boolean calculous [sic]". I suspect that this was vandalism, though I'm not sure. Jarble (talk) 16:44, 14 July 2017 (UTC)

@Laurent Meesseman: I reverted your edit but I had another look and the result is confusing. My reasoning was based on the following:

x+y  means  x XOR y
xy   means  x AND y

  x  y   x+y  xy   x+y+xy
  -----------------------
  0  0    0   0      0
  0  1    1   0      1
  1  0    1   0      1
  1  1    0   1      1

The text in Boolean algebra#Values states that addition (x+y) is XOR and multiplication (xy) is AND. However, just below that section "Basic operations" uses + and − in the ordinary arithmetic sense where 1+1 is 2. Using those different operators it gives the rule that disjunction (OR) is x + y − xy. Bit confusing. Johnuniq (talk) 07:57, 7 May 2018 (UTC)

I just added some text to clarify what operators are intended. It's a bit clumsy but better than leaving it as a puzzle. Johnuniq (talk) 02:55, 8 May 2018 (UTC)

Maybe, but you got it wrong. When truth values are represented as numbers in GF2, they use standard GF2 arithmetic to represent the operations, not logical operations described purely in logic but with the values translated into numbers. Laurent's edit that you reverted was correct but unhelpful, because in GF2 addition and subtraction are the same thing as each other. —David Eppstein (talk) 04:02, 8 May 2018 (UTC)

Thanks for the fix. For some reason I got the last case (x=1, y=1) confused but I see that mod-2 addition works as you say, and obviously as it has to. Johnuniq (talk) 04:52, 8 May 2018 (UTC)

This article in many places makes assumption that the underlying set of elements contains only zero and one. But this is not true for boolean algebra as defined in algebra, logic, nor in the historical references that are cited upfront in this article. How can we remedy this confusion? Can we maybe factor the relevant parts that make two-element assumption to Two-element Boolean algebra? Vkuncak (talk) 12:57, 29 May 2021 (UTC)