Talk:Logical connective

There are unary operators, binary operators, and comparators

This article and some of its relatives suffer from the lumping of operators and comparators into one big category, unfortunately called operators. You can blame computer languages on this if you wish.

In mathematical systems, there are operators and comparators. For example, in the familiar algebra of real numbers we have +, -, divide, * and others as binary operators, along with "=", "<", etc. as comparators.

In logic systems, this distinction is also made. Consider for example what comparator is used in the field ({T, F}, and, xor). It is isomorphic to the field F2.

In short, this page needs a major overhaul. —Preceding unsigned comment added by Richard B. Frost (talk • contribs) 00:32, 25 September 2007 (UTC)

@Richard B. Frost: But is not a comparator that which takes two arguments and gives a boolean? Therefore, can a logical operator not be thought of as a kind of comparator? EditorPerson53 (talk) 22:38, 6 February 2022 (UTC)

Merging boolean operators with logical operators

We have the following pairs of boolean/logical operators:

• Logical and / Conjunction
• Logical or / Disjunction

and the three-way equivalence:

• XNOR, Logical equivalence / Iff

In all of these cases I propose that what's on the left be merged into the article on the right, as is the case with Logical not and Negation at present.

Since I'm on the topic, this article could do with a lot of improving, eg. remove presumption that logic = Boolean logic, introduce slightly more high-powered mathematical analysis, such as lattice of expressiveness of sets of logical operators, and so on. ---- Charles Stewart 21:04, 11 Mar 2005 (UTC)

---

Why was this moved? Dysprosia 03:58, 28 May 2005 (UTC)

wikiproject proposed

I have proposed that this page be the centerpiece of a series of articles on the operators. Wikipedia:WikiProject_Council/Proposals#Logical_Operators

I thought the project would be too small for a formal wikiproject. There's just 16 of them. However, co-operation is needed from several disparate areas. I'd like to see:

• A) the logical, grammatical, mathematic, and computer science applications all on the same page for the same concept.
• B) similar sections including truth table, venn diagram, properties (including theorems that demonstrate the properties), symbols (preferably from all the above disciplines).
• C) several merges including If and only if, logical biconditional, and logical equality; different pages for implication; logical NOR and NOR gate; others
• D) unifying main page and nav bar.

Gregbard 05:31, 28 June 2007 (UTC)

Hmmm. Some of these in (C) are the same as binary operators, but not as ternary/poly-ary operaters. — Arthur Rubin | (talk) 20:29, 28 June 2007 (UTC)

You are correct. The operators have different qualities in different n-ary logic. They could be interpreted as a totally different thing with a different set of pages for the ternary operators, for instance. Perhaps it would be better to just include a section on the different "versions" of n-ary xor, for instance. Gregbard 20:47, 28 June 2007 (UTC)

Sets and Logical strength sections

I deleted the "Logical strength" section because I couldn't figure out what it was trying to say. I now realize that the Sets section is an arbitrary representation of the operators, which probably requires a reference, as well. If you can explain what you (Greybard) had in mind, I'll work on polishing them. — Arthur Rubin | (talk) 02:34, 29 July 2007 (UTC)

Thank you Arthur_Rubin, I'm sorry for my tone in my frustration. I think you can see that the diagram may have taken some time and effort. What I had in mind was that the diagram represents an interesting four dimensional relationship. I think it is informative on the logical connective concept. I have to go for a moment, I'll have more later. Gregbard 04:04, 29 July 2007 (UTC)

I have never seen this diagram before and have no idea how it measures logical strength, as it appears to be relatively arbitrarily labeled and oriented. Do you know a reference that discusses it? — Carl (CBM · talk) 10:52, 29 July 2007 (UTC)

Interestingly, the same diagram (Image:Logictesseract.jpg) is already on wikipedia under Hasse diagram. Along with this information, I'm looking for articles by Zellweger, Shea. There is relevant info at Finite Geometry; Lindenbaum-Tarski algebra, and maybe someday at Geometry of logic. I will keep looking. Be well, Gregbard 11:08, 1 August 2007 (UTC)

I'm quite familiar with Lindenbaum-Tarski algebras. Why not just say what you mean in the article, that the "strength" is just the ordering of this algebra (oriented with F on top), or that the figure is a Hasse diagram of the lattice? — Carl (CBM · talk) 03:41, 4 August 2007 (UTC)

I'm removing the "relative strength of operators" section. Based on this link provided by Gregbard, I figured out what is intended - that if you look at a particular 16 element sublattice of the Lindenbaum algebra of propositional logic, it gives you a way to rank the logical strength of the operators based on the partial ordering of the Lindenbaum algebra. But the link does not actually discuss that, I had to fill in the details myself. Moreover, I can't see any reason why the ratio of incoming to outgoing arrows is important - the Hasse diagram hides the transitivity of the partial order. Lacking any evidence that this method of ranking strengths is in the literature, or an important fact about the logical connectives, I'm moving the section to the talk page. — Carl (CBM · talk) 15:11, 5 August 2007 (UTC)

Relative strength of operators

{{OR|section}} The ratio of implications between operators is demonstrated by the directional lines in the tesseract The number of lines aiming away from the operator divided by the number of lines aimed toward is the ratio.

Image:Logical-connectives.gif

The relative strength of the 16 binary logical operators:

T	↑	→	~p	←	~q	$\equiv$	↓	∨	$\not \equiv$	q	⊄	p	⊅	&	F
0	1/3	1/3	1	1/3	1	1	3	1/3	1	1	3	1	3	3	∞

I'm also moving this section from the article. It's quite unclear to me what these sets are supposed to represent. It was tagged as possible OR for some time. — Carl (CBM · talk) 17:01, 28 August 2007 (UTC)

The logical operators can be expressed in terms of sets (where ∅ represents the empty set):

Set Theoretic Definitions of Logical Operators

∅ - Contradiction ($\bot$)	{ ∅ , { ∅ } , { { ∅ } } , { ∅ , { ∅ } } } - Tautology ($\top$)
{ ∅ } - NOR (↓)	{ { ∅ } , { { ∅ } } , { ∅ , { ∅ } } } - OR ($\vee$)
{ { ∅ } } - Material nonimplication (⊅)	{ ∅ , { { ∅ } } , { ∅ , { ∅ } } } - Material implication (⊃)
{ ∅, { ∅ } } - Not q	{ { { ∅ } } , { ∅ , { ∅ } } } - q
{ { { ∅ } } } - Converse nonimplication (⊄)	{ ∅ , { ∅ } , { ∅ , { ∅ } } } - Converse implication (⊂)
{ ∅ , { { ∅ } } } - Not p	{ { ∅ } , { ∅ , { ∅ } } } - p
{ { ∅ } , { { ∅ } } } - Exclusive disjunction ($\not \equiv$)	{ ∅ , { ∅ , { ∅ } } } - Biconditional ($\equiv$)
{ ∅ , { ∅ } , { { ∅ } } } - NAND (↑ or |)	{ { ∅ , { ∅ } } } - Conjunction ($\land$)

Move

This was moved (renamed) a couple weeks ago from logical operation/operator. Wondering if this was done with consensus, if connective is the best word (relation?, operation?), etc. And I also want to know if this is to be the overview article, are all linkages based in use of the term "logical operation/operator" (the convention until now, apparently) are going to be addressed. Seems like this was done out of process, and needs to be moved back, with "connective" being an alternative boldface term. Regards, -Stevertigo 02:19, 6 August 2007 (UTC)

• "Connective" seems a politically neutral term to me (as a formalist). Relation and Operator seem to have specific interpretation of the connective in mind. A specific interpretation helps in introductory examples. But hauling a specific interpretation into the terminology becomes probelematic when discussing non-classical logics. "Connective", being a purely syntatic term in English, seem appropriately agnostic to me, and I think is the right choice. Nahaj 15:38, 31 August 2007 (UTC)

Changes to "Arity" section (major and minor)

The "Arity" section currently begins:

In two-valued logic there are 4 unary operators, 16 binary operators, and 256 ternary operators. In three valued logic there are 9 unary operators, 19683 binary operators, and 7625597484987 ternary operators.

Call me crazy, but I think it should read as follows:

In two-valued logic, there are 4 unary operators, 16 binary operators, and 256 ternary operators. In three-valued logic, there are 27 unary operators, 19 683 binary operators, and 7 625 597 484 987 ternary operators.

• Mathematics: there are 27 unary operators for three-valued logic, not 9. Will someone check my math?
• Grammar
  • Punctuation: introductory phrases should end with commas.
  • Consistency: "X-valued" should use always either use a hyphen or not.
  • Number format: SI recommends spaces to format numbers into groups of three digits. Bless the SI, but this seems problematic in cases where the number reaches a line break. Some kind of thousands separator would be useful.

--75.15.135.58 06:45, 4 September 2007 (UTC)

Too long?

I don't mean to be impertenent or anything, as it is very clear that you have all spent a lot of time over this article, and care about it deeply: however, do you not think that you have perhaps taken the subject too broadly? I mean that a clear and succinct definition of a logical connective given at the beginning with examples of the main truth functional connectives would be sufficient. Once you start going beyond that, going into detail, as to the (potentially infinite) possibilities that exist for something to be a "logical connective" within a given language, then the article will be doomed to be unfinished, and, I think, you confuse the reader. Apologies if I angered anyone, I can tell you've put a lot of work into it. Wireless99 12:29, 8 September 2007 (UTC)

Intro

I have added some more examples and renderings into symbols, intended to give a better overview for the reader before he/she dives into the depths of this article. Also removed example of causal relation on the ground that such, though interesting, is not a truth-functional connective.--Philogo 13:05, 20 September 2007 (UTC)

Venn diagrams arrangement

I added a line beneath the Venn diagrams crediting the source for their arrangement, which Greg Bard mentioned above ("'Sets' and 'Logical strength' sections") in a link he titled Finite Geometry. Cullinane 11:26, 28 September 2007 (UTC)

"two or more well-formed formulae"

There's no reason to restrict to two formulae, right? Certainly, the common logical connectives are all unary or binary, but one could define a truth-functional connective to operate on three WFFs and it would still be a truth-functional connective. Shouldn't it say "one or more well-formed formulae"? Djk3 (talk) 18:45, 24 March 2008 (UTC)

Zero or more, if you want to pursue that in full generality. — Carl (CBM · talk) 19:09, 24 March 2008 (UTC)

Right, right. Djk3 (talk) 22:47, 24 March 2008 (UTC)

It's true. However everything arity greater than 2 can be expressed in terms of just binary connectives. I also think there may be a name for some of them, for instance:If P then Q, otherwise R. Pontiff Greg Bard (talk) 21:13, 24 March 2008 (UTC)

How's that? I tried to fix it so that "one or two" is no longer present, and so that it all makes sense. I don't think I changed any of the meaning, just made it clearer and neater. Djk3 (talk) 23:07, 24 March 2008 (UTC)

Truth-table

I changed the colors in the truth-table to alternating shades of white/light gray. I understand that the colors were there as an illustrative tool, but it really made the table muddy. Maybe there's another way we can present that information. Djk3 (talk) 18:48, 29 March 2008 (UTC)

I doubt very much that the readers to whom such a table may be useful can actually interpret it. In the first column, it is unclear what the arity of the symbols is. I've never seen ⌋, ⌊, ⌈ or ⌉ used as logical connectives, and they are not explained otherwise in the article. They are also not listed on Wikipedia:WikiProject Logic/Standards for notation. In the next column, why are "false" and "true" replicated four times, while P is all alone on its line? The last column must surely be totally mysterious.  --Lambiam 13:47, 30 March 2008 (UTC)

What if each row of the table was replaced with a box? Here is a very rough idea of what I am thinking of. — Carl (CBM · talk) 14:20, 30 March 2008 (UTC)

Alternative denial
Notation	Truth table	Venn diagram
P NAND Q P | Q P → ¬Q ¬P ← Q ¬P OR ¬Q	P T F Q T F T F T T

This would have the advantage of combining the Venn diagrams into the table, and should be more clear about the information being presented. — Carl (CBM · talk) 18:17, 30 March 2008 (UTC)

It is an improvement in clarity. It is a bit unfortunate that such a harsh red was used for the Venn diagram, and the colour key "red = true/included, white = false/excluded" is not the most intuitive. I'd further suggest to separate "Notation" into "Notation" and "Equivalent formulas", giving something like:

Alternative denial
Notation	Equivalent       formulas	Truth table	Venn diagram
P  NAND  Q P  |  Q	P  →  ¬Q ¬P  ←  Q ¬P  OR  ¬Q	P T F Q T     F     T   F     T     T

 --Lambiam 22:14, 30 March 2008 (UTC)

I like this idea. I could go through and change the reds in all the Venn diagrams. Do you think I should also invert the color scheme? Djk3 (talk) 00:09, 31 March 2008 (UTC)

A blue or green color for the filled in areas would be more intuitive to me - red means "no" in my mind, so it's weird to have the T cells colored red. If we want to put text on top of the color, it might need to be somewhat lighter. — Carl (CBM · talk) 01:34, 31 March 2008 (UTC)

Alternative denial
Notation	Equivalent formulas	Truth table	Venn diagram
P  NAND  Q P  |  Q	P  →  ¬Q ¬P  ←  Q ¬P  OR  ¬Q	P T F Q T     F     T   F     T     T

Is that sort of color scheme alright, or is that too flowery? Djk3 (talk) 04:02, 31 March 2008 (UTC)

It's OK for me, although you're right that the purple color does look a little flowery. I changed the colors of the rest of the box, and now I think it looks fine. — Carl (CBM · talk) 15:22, 31 March 2008 (UTC)

I think that looks good. I'll go through and change the colors for all the rest of the Venn diagrams tonight and toss them into the article in this format. Djk3 (talk) 15:55, 31 March 2008 (UTC)

It seems better to me to make a template that incorporates the code above, so that the article source itself looks like {{logicalconnective|...}} instead of being full of messy table code. I'll work on that. — Carl (CBM · talk) 15:57, 31 March 2008 (UTC)

Sure, thanks. I'll do the images in the meantime. Djk3 (talk) 16:10, 31 March 2008 (UTC)

The template is done, at Template:Logicalconnective. There is example code there that demonstrates how to fill in all the pieces. I think we can simply replace the table and Venn diagrams with 16 of those. (So if any formatting needs to be changed, we can change it once instead of 16 times.) — Carl (CBM · talk) 16:13, 31 March 2008 (UTC)

It looks fine to me. As you can see below, much lighter colours still give a quite perceptible difference with white, but the "T" is sufficiently legible as it is.  --Lambiam 16:59, 31 March 2008 (UTC)

8

9

A

B

C

D

E

F

expressed as a relation

I don't believe that pointing out that these can be expressed in different ways, for instance, as a relation, is needless complication, nor does it miss any point which is being communicated. Pontiff Greg Bard (talk) 23:01, 30 March 2008 (UTC)

The value in pointing out that it is a function is that for any input, it returns one and only one truth-value. That property isn't present in the more general relation. Djk3 (talk) 00:07, 31 March 2008 (UTC)

List of connectives with truth tables and Venn diagrams

I'm posting this here for a look-over before I put it into the main article. I spent a lot of time squinting my eyes and tipping my head doing these one after another, so they may be ripe with errors. Please check it with fresh eyes and edit as appropriate. Djk3 (talk) 01:28, 1 April 2008 (UTC)

I was afraid that might happen - I also made a version at User:CBM/Sandbox. I copied yours in. I also changed the template to fix some alignment problems. — Carl (CBM · talk) 02:14, 1 April 2008 (UTC)

Sorry if I switched the template on you while you were working on them. I thinking lining them up in two columns opposite their negations expresses what the colors in the table were expressing just as well, but it's be a little bit friendlier on the eyes. Djk3 (talk) 02:21, 1 April 2008 (UTC)

Contradiction Notation Equivalent formulas Truth table Venn diagram $\bot$ n/a   Q 0 1 P 0    {{{truthtable-00}}}   {{{truthtable-01}}}  1    {{{truthtable-10}}}   {{{truthtable-11}}}	Tautology Notation Equivalent formulas Truth table Venn diagram $T$ n/a   Q 0 1 P 0    {{{truthtable-00}}}   {{{truthtable-01}}}  1    {{{truthtable-10}}}   {{{truthtable-11}}}
Conjunction Notation Equivalent formulas Truth table Venn diagram P & Q P $\land$ Q P $\not \rightarrow$¬Q ¬P $\not \leftarrow$ Q ¬P $\downarrow$ ¬Q   Q 0 1 P 0    {{{truthtable-00}}}   {{{truthtable-01}}}  1    {{{truthtable-10}}}   {{{truthtable-11}}}	Alternative denial Notation Equivalent formulas Truth table Venn diagram P ↑ Q P | Q P NAND Q P → ¬Q ¬P ← Q ¬P $\lor$ ¬Q   Q 0 1 P 0    {{{truthtable-00}}}   {{{truthtable-01}}}  1    {{{truthtable-10}}}   {{{truthtable-11}}}
Material nonimplication Notation Equivalent formulas Truth table Venn diagram P $\not \rightarrow$ Q P $\not \supset$ Q P & ¬Q ¬P ↓ Q ¬P $\not \leftarrow$ ¬Q   Q 0 1 P 0    {{{truthtable-00}}}   {{{truthtable-01}}}  1    {{{truthtable-10}}}   {{{truthtable-11}}}	Material implication Notation Equivalent formulas Truth table Venn diagram P → Q P $\supset$ Q P ↑ ¬Q ¬P $\lor$ Q ¬P ← ¬Q   Q 0 1 P 0    {{{truthtable-00}}}   {{{truthtable-01}}}  1    {{{truthtable-10}}}   {{{truthtable-11}}}
Proposition P Notation Equivalent formulas Truth table Venn diagram P n/a   Q 0 1 P 0    {{{truthtable-00}}}   {{{truthtable-01}}}  1    {{{truthtable-10}}}   {{{truthtable-11}}}	Negation of P Notation Equivalent formulas Truth table Venn diagram ¬P n/a   Q 0 1 P 0    {{{truthtable-00}}}   {{{truthtable-01}}}  1    {{{truthtable-10}}}   {{{truthtable-11}}}
Converse nonimplication Notation Equivalent formulas Truth table Venn diagram P $\not \leftarrow$ Q P $\not \subset$ Q P ↓ ¬Q ¬P & Q ¬P $\not \rightarrow$ ¬Q   Q 0 1 P 0    {{{truthtable-00}}}   {{{truthtable-01}}}  1    {{{truthtable-10}}}   {{{truthtable-11}}}	Converse implication Notation Equivalent formulas Truth table Venn diagram P $\leftarrow$ Q P $\subset$ Q P $\lor$ ¬Q ¬P ↑ Q ¬P → ¬Q   Q 0 1 P 0    {{{truthtable-00}}}   {{{truthtable-01}}}  1    {{{truthtable-10}}}   {{{truthtable-11}}}
Proposition Q Notation Equivalent formulas Truth table Venn diagram Q n/a   Q 0 1 P 0    {{{truthtable-00}}}   {{{truthtable-01}}}  1    {{{truthtable-10}}}   {{{truthtable-11}}}	Negation of Q Notation Equivalent formulas Truth table Venn diagram ¬Q n/a   Q 0 1 P 0    {{{truthtable-00}}}   {{{truthtable-01}}}  1    {{{truthtable-10}}}   {{{truthtable-11}}}
Exclusive disjunction Notation Equivalent formulas Truth table Venn diagram P $\not \leftrightarrow$ Q P $\not \equiv$ Q P $\oplus$ Q P $\leftrightarrow$ ¬Q ¬P $\leftrightarrow$ Q ¬P $\not \leftrightarrow$ ¬Q   Q 0 1 P 0    {{{truthtable-00}}}   {{{truthtable-01}}}  1    {{{truthtable-10}}}   {{{truthtable-11}}}	Biconditional Notation Equivalent formulas Truth table Venn diagram P $\leftrightarrow$ Q P ≡ Q P $\not \leftrightarrow$ ¬Q ¬P $\not \leftrightarrow$ Q ¬P $\leftrightarrow$ ¬Q   Q 0 1 P 0    {{{truthtable-00}}}   {{{truthtable-01}}}  1    {{{truthtable-10}}}   {{{truthtable-11}}}
Disjunction Notation Equivalent formulas Truth table Venn diagram P $\lor$ Q P $\leftarrow$ ¬Q ¬P → Q ¬P ↑ ¬Q   Q 0 1 P 0    {{{truthtable-00}}}   {{{truthtable-01}}}  1    {{{truthtable-10}}}   {{{truthtable-11}}}	Joint denial Notation Equivalent formulas Truth table Venn diagram P ↓ Q P $\not \leftarrow$ ¬Q ¬P $\not \rightarrow$ Q ¬P $\wedge$ ¬Q   Q 0 1 P 0    {{{truthtable-00}}}   {{{truthtable-01}}}  1    {{{truthtable-10}}}   {{{truthtable-11}}}

The lede

EJ and I disagree about what the lede should contain. I would like to discuss that here.

A Wikipedia article, especially the first paragraph, should be readable by the average, intelligent person, who has no training in the area under discussion. Exceptions are allowed in the case of highly technical articles, Ascending chain condition for example. But logic, and logical connectives, are basic to math and computer science, and so this article should be aimed at the introductory level. For that reason, I think it is best to begin the article with the five most commonly used logical connectives (I can site a dozen books that begin that way if you want me to, but I imagine anyone else reading this could do the same), instead of leaping into the question of the infinitely many n-ary logical connectives.

Here, line by line, are the problems I had with the other introduction:

"In logic, a logical connective, also called a truth-functional connective, logical operator or propositional operator, is a logical constant which represents a syntactic operation on well-formed formulas."

This sentence could only be read by a mathematician or upper division math major, who would already know what a logical connective was. The beginner will not understand "logical constant" or "syntactic operation", and may also stumble over "well-formed formula", all concepts usually introduced after "logical connective". Also, there is no need to put all the synonyms into the first sentence, where they are stumbling blocks for the beginner. They can come later.

"The formula that results from applying a logical connective to well-formed formulas is a well-formed formula itself."

This is simply untrue, at least without considerably more discussion. What is true is that if A is a well-formed formula and B is a well-formed formula and # is a binary logical connective, then (A)#(B) is a well-formed formula. But this is too technical for the lede.

"If a logical connective is applied to sentences then the result is a compound sentence, and the truth-value of the resulting compound sentence is determined uniquely by the truth-values of the sentences to which it was applied."

Before this discussion should come a discussion of truth values. Also "applied to" is vague, and easily misunderstood.

"Consequently, a logical connective can be seen as a function which maps the truth-values of the sentences to which it is applied to either true or false."

This will strike a lay reader as meaningless and a mathematician as wrong. (A mathematician would want something like "An n-ary logical connective can be seen as a function which maps n-tuples of truth values to truth values.")

"There are infinitely many logical connectives, 22n for every arity n."

Again, a comment unnecessary for a mathematician and opaque to a non-mathematician. Since the most common logical connectives are either unary or binary, it is hardly necessary to get into n-ary connectives in the lede.

"Commonly used connectives include the binary connectives conjunction (and), disjunction (or), implication, and biconditional, the unary connective negation (not), and the nullary connectives truth and falsity."

I'm sure you can find a book that describe T and F as nullary connectives but that description does not appear in any of the textbooks or research papers I use regularly, and is in any case a construction that would only appeal to a research mathematician who already knows everything in this article. An article should be useful.

"All logical connectives can be constructed from finitely many of them, negation and conjunction for example."

After being careful about arity above, you now omit the word "binary" which is essential here. Without "binary", the "finitely" is wrong. With "binary", the word "finitely" should be replace by the word "two".

"A particular logical system will only employ some basic set connectives to construct well-formed formulas, and treat the other connectives as defined in terms of the basic ones."

And I have no problem with this sentence, if you would like to restore it to the article.

Rick Norwood (talk) 13:14, 30 May 2008 (UTC)

Agreed in general. Two-penny's worth. The opening sensne of any aricel is very imprtant. This one begins:

In logic, the five standard logical connectives are the binary connectives, "AND", "OR", "IMPLIES", and "BICONDITIONAL", which connect two logical statements, and the unary connective "NOT", which modifies one logical statement.

Step back from this. Suppose you wondered what a "trig. function" was. You turn to Wiki and it says:

The three standard trig. functions ar sin, tan and cos.

Is a reader who does not know want "trig. function" means, likely to know what sin, tan and cos are? Then how would he be any the wiser. Explanation by example only works of the examples are more familiar than the term to be explained.

It is better to give give the examples after. Eg:

Mammal: the class of verterbrate animal that bears its young live and suckles them Eg. Dog, Cow, Kangaroo. Compare other vertebrebrates: Reptile, Fish, Bird.

--Philogo 13:36, 30 May 2008 (UTC) --Philogo 13:36, 30 May 2008 (UTC)

Point taken. I'll make the change. Rick Norwood (talk) 13:58, 30 May 2008 (UTC)

Rick, I have neither the time nor desire to getting involved in a lengthy discussion, especially given your attitude that you only show the willingness to discuss after reverting to your version of the lead. The previous lead has been there for many months, and people were happy with it. Instead, I have modified your text point by point where I've seen serious issues with it. — EJ (talk) 14:15, 30 May 2008 (UTC)

Wikipedia is arrived at by consensus. An unwillingness to talk is not a good way to arrive at a consensus. I've taken the time to listen to your points and respond to them, and I'm busy, too. I've tried to address your points, and Philogo's points as well. Rick Norwood (talk) 14:18, 30 May 2008 (UTC)

I appreciate that you now edit in a constructive way, instead of reverting. I think direct editing in this way can be more productive than discussing everything first on the talk page. I am more or less happy with the current version, except for one point below, which you continue to push for reasons which I do not quite understand.

The statement "but the connectives that are always true or always false are usually omitted, leaving 14 connectives in actual use" is patently absurd. The constant for falsity is very widely used, many calculi take implication and falsity as the only connectives, for example. Also in intuitionistic logic falsity is much more often taken as basic than negation. An important reason is that without explicit constants the set of connectives is not stricly speaking functionally complete, there is no way of defining constant functions without variables. In any case, these doubts may only apply to the question of nullary connectives. There is no way in hell how constant binary functions could be excluded from the 16 binary connectives. They are binary connectives, and pragmatically speaking, they are much more useful then many other of the 16. I can't imagine any sane source which would deny constants to be binary connectives, yet include the connectives f(x,y) = x and g(x,y) = y, and distinguish between → and ←. Even if there is such a source, it is highly nonstandard, and very misleading to potential users. — EJ (talk) 14:28, 30 May 2008 (UTC)

I agree. Your recent edit is an improvement. Rick Norwood (talk) 14:50, 30 May 2008 (UTC)

I have boldy added some content at the beginning rather than discussing here first. Feel free to edit or del it if you disagree with it.--Philogo 18:57, 30 May 2008 (UTC)

Logical constant

I see that Arthur has deleted the fact that logical connectives are a type of logical constant. Does that make any sense at all? Pontiff Greg Bard (talk) 21:34, 30 May 2008 (UTC)

They're only logical constants in formal logic, which this article is not necessarily about. In any case, it's stated further down within the lead. (IMHO, it should be moved still further down, probably into another article, but that's just my opinion.) — Arthur Rubin (talk) 22:04, 30 May 2008 (UTC)

Logical constant seems to me to be seminal to the concept. I was surprised to find it buried lower down. I find the distinction you make fascinating, but I'm wondering if it isn't pretty trivial or should be an early lead distinction to cover. In formal logic a logical connective is a logical constant, in informal logic there is a sense in which logical connectives aren't logical constants? Be well, Pontiff Greg Bard (talk) 22:45, 30 May 2008 (UTC)

If this article is intended to be read by somebody who wants to know something or more about Logical connectives, then telling them that it is (if it is) a logical constant would only be informative if the term logical constant were more familiar to them than the term Logical connectives. Thus e.g. suppose a person did not know what a kangaroo was (maybe a child, or person with limited English.) Would you tell them (a) it is a marsupual? (b) an animal that hops about carrying its young in a pouch? The purpose of writing is to communicate. --Philogo 23:01, 30 May 2008 (UTC)

Then we have a basic difference of opinion on how articles should be organized. I think that we should always attempt to identify the next level of abstraction early in the lead. "A set is an abstract object." "An algorithm is a type of effective method." This is especially justified on Wikipedia since people can easily look up marsupial, for instance. Invariably, I would like to see a canonically worded account which gives some idea where in the big scheme of things the topic resides, beginning with the immediately next level of abstraction. X is a type of Y. "Oh, what's a 'Y'?" "Well just click on 'Y'." Explanations are sufficiently easy to get that we are able to go ahead and use the proper terminology, even if it stands in need of explanation itself. Pontiff Greg Bard (talk) 00:14, 31 May 2008 (UTC)

The problem with "logical constant" is that it is bad terminology in the first place. Calling quantifiers and connectives "constants" in a context where constants in a much more normal sense also play a role is incredibly bizarre, unpractical, and only serves to confuse people who are not initiated into this arcane terminology. In mathematics it's generally accepted that the subject is already hard enough and that we should do whatever we can to make it easier to understand. The term "logical constant" is obsolete; it does not contribute to making anything easier, and it can be easily replaced by "logical symbol", which has the same meaning except that it also includes the variables. There is no need to spread bad terminology all over the encyclopedia. --Hans Adler (talk) 21:00, 31 May 2008 (UTC)

moment of doubt

The article is describing logical connectives (a) as they occur in natural language (as by words like and and or, and also (b) as they occur in logic and represented by our familiar symbols. I think that is a worthy aim, but I am just wondering if the article is clear on this and not confusing to the reader, assuming as we should that they are new to this subject.--Philogo 23:12, 30 May 2008 (UTC)

As a professional logician, I can't answer questions as to clarity. As a professional philosopher, neither can Greg. Any ideas where we can go from here? — Arthur Rubin (talk) 00:11, 31 May 2008 (UTC)

Why drag me into this question like this? If there are clarifications to be made, lets be grateful to identify them and leave it at that. Pontiff Greg Bard (talk) 00:17, 31 May 2008 (UTC)

Reading the lead, I'd say that the article describes logical connectives as they occur in logic, represented by words or by symbols. The atomic statements may be in natural language or in a more symbolic form. As I see it, the differences between the following compound statements are superficial:

Socrates' dying implies that Socrates is not immortal.

Died(Socrates) IMPLIES NOT Immortal(Socrates).

Socrates died → ¬ Socrates is immortal.

Died(Socrates) → ¬ Immortal(Socrates).

 --Lambiam 08:13, 31 May 2008 (UTC)

I agree with Lambiam. The lede uses and rather than & only to improve readability by the lay reader.Rick Norwood (talk) 16:38, 31 May 2008 (UTC)

New lede

I don't like the new lede, which appears to conflate the connectives (which are symbols, living in the world of syntax) with truth functions (which are values, living in the semantic world). This may not be the intention, but the wording is very unclear. What is the antecedent to which the word "it" refers in "it is called a truth-function"? In any case, the formulation chosen is very convoluted and hard to understand, and such a heavy emphasis on truth functions is not needed or desirable. I think we should go back to earlier approaches and propose the following for the very first sentence:

In logic, a logical connective is a symbol (usually denoted as a word or a special logical symbol) that connects a number of statements to form a compound statement, whose truth value is then determined by the truth values of the individual component statements.

I think it is always good to come with an example as soon as possible, and the next sentence might be:

For example, in "x = 0 AND y = 1", "AND" is a logical connective that combines the two statements "x = 0" and "y = 1" to a single compound statement. The same connective can also be denoted by the symbols "&" and "∧", as in "x = 0 ∧ y = 1". For this example, the compound statement is true only if both component statements are.

Then I think we should list the five most common connectives, and finish off the lede with:

Other terms in use for "logical connective" are Boolean operator, logical connector, logical operation, logical operator, propositional operator, sentential connective, and truth-functional connective.

I don't see anything else that urgently needs to be put in the lede.  --Lambiam 03:16, 2 June 2008 (UTC)

Even though I wrote some of the new lede, I tend to agree with Lambiam. I was trying to preserve as much of the earlier lede as possible, and it identified the logical connective with its truth table, which is certainly one way to go. If nobody else has already fixed this, I'll take a shot later on today, working along the lines Lambiam suggests. Rick Norwood (talk) 13:38, 2 June 2008 (UTC)

Venn Diagrams

I've read that diagrams like the ones depicted in this article are actually called Johnston diagrams rather than Venn diagrams. I'd change in on the article but I can't find the table template -- thoughts?Jheiv (talk) 09:20, 1 August 2008 (UTC) Nevermind. It seems very few people call these things Johnston diagrams, I have trouble finding mention of said diagram on the web or in logic books. Jheiv (talk)`

Can you identify where you read this? The article on Johnston diagrams doesn't have any references, and I hadn't heard that term before I stumbled on our article. If my understanding is correct, for people distinguishing them a Venn diagram can solely represent a set-valued expression such as A ∩ (B ∪ C), whereas a Johnston diagrams depict propositional expressions, such as A ∧ (B ∨ C).

Boolean Operators

What happened to my article? Whoever deleted it obviously doesn't know the difference between philosophy and networking technology in search engines. I'm putting it back up, and if someone has an issue with it, I'd like a thorough explaination. Colonel Marksman (talk) 08:55, 16 November 2008 (UTC)

What are you talking about? "Boolean operator" and "Boolean operators" should be the same as "Logical connective". (If you're talking about something else, you'll have to be more specific.) — Arthur Rubin (talk) 14:50, 16 November 2008 (UTC)

To Arthur Rubin: He is clearly talking about this.

To Colonel Marksman: The term "boolean operator" clearly originates in mathematics; more exactly from the area of it in which most things are associated with George Boole. It's a synonym for logical connective, and also for boolean function, although as with all synonyms there are minor variations in usage and connotations. The very special application of this concept to search engines is at most marginally encyclopedic, and if it deserves its own article (I am not sure), then certainly the domain of application must be part of the title. Given the wider scope of the former article (quotation marks and wildcards are "orthogonal" concepts, completely unrelated to boolean operators except that in computer science they sometimes appear in similar contexts) I suggest using a title such as Search engine interfaces if you want to try again. I think an article covering the following information might be encyclopedic: the basic difference between search engines proper and web directories, the difference between "simple" (one-line) and "advanced" (web form) search interfaces. And then also differences in support for boolean operators (e.g. and, or, not), other operators (e.g. near), quotation marks, wildcards, Latin diacritics and special letters (é = è = ê = ë = e, ü = ue, ß = ss?), non-Latin alphabets, support for exact word groups (e.g. marked by quotation marks; but what does "exact" mean?), stop words, support for parentheses (e.g. (a AND b) or C vs. a AND (b OR C)) etc. --Hans Adler (talk) 20:39, 16 November 2008 (UTC)

Giant imagemap

I don't favor the giant imagemap, for the following reasons:

• It's very hard to read the image. It tries to compress too much information into too small an area.
• It's no longer possible to easily tell which row corresponds to which connective. In the table on the left, the truth values are presented in some weird order instead of being in a truth table. And the Venn diagrams are no longer displayed.
• It's completely inaccessible for users with images turned off.
• Editors do not typically move their mouse around to find out what the article is saying; they read the text to find out. Moreover, when the article is printed, any information from the image map is lost.
• The Hasse diagram conveys no useful information about logical connectives. It should simply be removed.

— Carl (CBM · talk) 21:01, 18 November 2008 (UTC)

Fully agree. And more than that I think editors should try and put information into WP in a form where it an be edited easily by others. It is supposed to be a collaborative effort, doing this sort of thing would lock out editors who might know more about the subject matter. Even if it was well designed I would still be against it. Dmcq (talk) 19:52, 19 November 2008 (UTC)

2008-11-26

Copied from the WikiProject Math talk page [1]

Hi,

what I like about the representation above is the following:

• When I touch the nibbles in the table, they are explained to me bit by bit. Please do not underestimate, that this can be helpful for people, who do not already know the subject.
  
• The tesseract shadow Hasse diagram shows all possible deductions. The Venn diagrams are helpful representations, of what the connectives actually mean or do. When I touch the odd bit connectives in the diagram, I can see that the 1 bit connectives are conjunctions and the 3 bit connectives are disjunctions (without cropping the article with text about details like this).
  
• Last but not least: I like the silver Thue-Morse sequence in the table, and the silver cube in the diagram. (These are interesting links to other regions of mathematics, and could even be mentioned in a "see also" section at the end of the article.)

The odd bit connectives in the table are ordered in the Thue-Morse sequence...

...and form a cube in the rhombic dodecahedral Hasse diagram.

Concerning accessibility:

It is true, that articles should also be accessible to blind people, and for plain text uses, may it be for wapedia or whatever. I take that very serious. But in these cases a table containing wikipedia math symbols would be not useful as well. Thus a good solution for all kinds of users is to keep the imagemap template in the article, and to add a note like this: "Here you find this information in plain text."

The lines in this table should simply look like this:

• (P AND Q) = (P material nonimplication not Q) = (not P converse nonimplication Q) = (not P NOR not Q) = 0001

The information displayed in the Hasse diagram can be shown by a simple list of conclusions like these:

• If the statement "P AND Q" is true, the statements "P", "Q" and "P XNOR Q" are alo true.

• If the statement "Q" is true, the statements "P OR Q" and "P material implication Q" are also true.

I can create this subpage Logical connectives text table, if you agree that it makes sense. I think it does.

Greetings, Boolean hexadecimal (talk) 12:19, 26 November 2008 (UTC)

I will not be able to discuss this at length over the next few days because of travel, but I will leave my thoughts here. I greatly prefer the current version of the page over the one with the imagemap and Hasse diagram.

• The current version already has the Venn diagrams and truth tables
• The truth tables in the imagemap are arranged in a linear way, which is strange because a truth table ought to be a square. The current version of the page has square truth tables. I have no idea how to read the truth tables in the imagemap (that is, which order the truth values are arranged in).
• The colors in the rows of the imagemap don't have any meaning to me. I'm sure they are there for a reason, but I don't want readers to have to think for a long time to figure out what the article is trying to tell them.
• In general, there's just too much information and not enough navigation packed into the imagemap.
• The imagemap can't be edited in any easy way, unlike the current text version of the article
• The imagemap doesn't allow readers to select their own fonts, unlike the current text version.

The Hasse diagram has a separate set of problems.

• I have never found the ordering of connectives this way to be of interest in propositional logic. I've never seen a textbook that discussed this ordering, for example.
• The diagram is upside-down compared to the corresponding lattice of the Lindenbaum algebra of propositional logic.

— Carl (CBM · talk) 13:39, 26 November 2008 (UTC)

Tables are used by blind people every day of the week. The complaint blind people have about tables is if they are used purely for formatting rather than as tables. If they are used for the purpose for which they are intended they work very well. Even tables with colspans and rowspans are good if used in a reasonable fashion. Reading the maths in wiki is done by turning off maths formatting in the preferences.

Wikipedia already has a way of finding out about things. This is by marking them as links.

Wikipedia doesn't use imagemaps or Flash or Java in a core way in the edited content. If you want to get people to adopt imagemaps as another standard I believe you start at the village pump. Otherwise the best I think one could do with it is use it as ancillary help which could be included or removed without destroying an article, the sort of thing much of the material in Wikipedia Commons does.

Have a look at Wikipedia's main pages, it doesn't use imagemaps to direct to the various subjects does it? In a place like Google's effort at an encyclopaedia where individual editors are responsible for their own patch I'm sure there wouldn't be any problem. The problem with just sticking in non-standard formats here is it hinders other editors changing the article. Anything like that should be avoided if at all possible I believe. It probably does make some articles less good but Wikipedia is supposed to be collaborative and overall it seems to work out well. In this particular case there is a reasonable alternative which is quite satisfactory for the job.

As to the funny colouring all I can say is tha sometimes more is less, it is an unnecessary distraction tha doesn't add anything wen explained. The gif for the Thue-Morse sequence seems fine though, it doesn't try to substitute for part of that article and though I'm not keen on things jumping around it does seem to be doing a competent job. Dmcq (talk) 13:56, 26 November 2008 (UTC)

External link to obscure notation

One of the external links is to a Hasse diagram similar to the one above except that it uses a strange (to me) notation in place of OR and NAND and so on. What are the curlicues and rotated question marks etc in that diagram? —Preceding unsigned comment added by 69.227.129.30 (talk) 10:33, 11 December 2008 (UTC)

These are secret free-masonic symbols. Your head will explode if you read them aloud. — Emil J. 13:29, 11 December 2008 (UTC)

It looks like a very bad case of "original research" to me. This was added by the same editor who earlier tried to push a huge table containing a similar graphic (without the "masonic" symbols) into the article [2]. This was discussed here, and I think it's safe to assume that there is a consensus that the link should be removed. Thanks. --Hans Adler (talk) 17:47, 11 December 2008 (UTC)

Karnaugh maps (2-variable k-map) are incorrectly referred to as Truth Tables

The summary says it all. To someone who knows what they are doing -- please fix.

DLA (talk) 22:58, 24 March 2009 (UTC)

They are commonly called truth tables in the literature. — Carl (CBM · talk) 00:24, 25 March 2009 (UTC)

I respectfully disagree. See the Wikipedia entries for Karnaugh Map and Truth Table, both of which treat them as distinctly different concepts. I think that this Logical Connective Wikipedia entry should be consistant with the Karnaugh Map and Truth Table Wikipedia entries. Otherwise, I think that this entry is helpful.

DLA (talk) 22:23, 31 March 2009 (UTC)

Could you explain exactly why the tables in this article should be called Karnaugh maps and not truth tables? — Carl (CBM · talk) 05:19, 1 April 2009 (UTC)

The diagrams happen to be Karnaugh maps, but that does not mean that they are not truth tables. Truth table is simply an assignment of truth values of a Boolean function to all possible combinations of truth values of its input, the layout of the table is completely irrelevant if you are going to visualize it. In particular, see truth table#Condensed truth tables for binary operators for the layout used here. As Carl pointed out, these diagrams are usually called truth tables in the literature, whereas Karnaugh maps are only referred to in very specialized contexts (integrated circuit design). — Emil J. 12:59, 1 April 2009 (UTC)

The tables in this article look like truth tables (a) to me (b) and as exampled in the article Karnaugh Map. I propose we agree any preferred terminology in the (as yet empty) section of [Wikipedia:WikiProject Logic/Standards for notation]--Philogo (talk) 13:02, 1 April 2009 (UTC) PS It Wikipedia:WikiProject Logic/Standards for notation#Terminology is no longer empty but collaborators are urgently required. (Hans?)--Philogo (talk) 22:37, 1 April 2009 (UTC)

Truth functions and Interpretation function

Would some kind ed look at and make work my non-functioning ref in the first sentence of this new section.--Philogo (talk) 22:35, 1 April 2009 (UTC)

This article uses Harvard style refs, not footnotes. So you need to add the refs at the end and insert the citation in parentheses.

There are more important problems with the text you inserted, though. The first is the sentence "Logical connective symbols can be defined by means of an interpretative function and a functionally complete set of truth-functions (Gamut 1991)." It does not makes sense to define a connective via a function - a connective is a symbol. Maybe you mean you can interpret a connective via an interpretive function?

The text also says, "Let I be an interpretative function, from sentences onto {true,false}," and then go on to say "I(~)=I(¬)=fnot", but "~" on its own is not a sentence, so it makes no sense to say I(~).

Also, I don't believe the terminology "interpretive function" is common in the literature, and I'm not sure this section needs to be in the article at all. — Carl (CBM · talk) 02:14, 2 April 2009 (UTC)

Fixed as you suggested. NB typo: I is an interpretation function not an interpretive function.--Philogo (talk) 13:37, 2 April 2009 (UTC)

ambiguity

The article appears to me to be somewhat ambigous in its use of the term "logical connective"/"truth-functional connective" A logical conenective/truth-functional connective appears to mean variously (a) a symbol such as $\neg$, $\wedge$ &c which is a truth-function (b) a symbol such as $\neg$, $\wedge$ &c when used as to represent a truth-function (c) a truth-function that can be represented by a a symbol such as $\neg$, $\wedge$ &c. (The ambiguity is perhaps akin to confusing/confounding numbers and numerals, or the plus-sign with the fucntion sum.) Eg. we should be clear when we are talking about (i) a negation sign or symbol, like $\neg$ (ii) the negation function. We should be clear whether it is the symbols or the functions whcih are propery called logical connectives/truth-functional connectives.--Philogo (talk) 14:06, 2 April 2009 (UTC)

Boolean bias

As someone coming from constructive logic and computer science, I am used to a much broader definition of "logical connective" than what is implied by this article. Although there is a brief mention of finite-valued logic in the section on arity, most of the article is heavily biased towards boolean logic. I'm especially disturbed that this bias even sneaks into the section on natural language. Of course so is a logical connective! It combines two sentences to form a new sentence, and its meaning is uniquely determined by the meaning of those subsentences—you just can't model the meaning of those sentences by boolean truth values. Perhaps much of the material in this article should be moved to a separate article on Boolean operators, and the Logical connective article could be streamlined to only discuss logical connectives in general terms? Noamz (talk) 16:21, 24 June 2009 (UTC)

Well, no. "So" depends on the extrinsic meaning of the sentences, not just their truth values, boolean or not. I don't know of an example from a constructive logic where it makes sense. Perhaps you could elaborate? — Arthur Rubin (talk) 20:25, 24 June 2009 (UTC)

Yes, that's exactly what I said: the meaning of "so" depends on the meaning of its subsentences. I did not know how broadly we are construing "truth value", which is why I used the term meaning. Backing up a bit, perhaps what I should have first suggested is that we modify the lede. Instead of, The truth value of the compound is uniquely determined by the truth values of the simpler sentences, I would just write something like, A connective is compositional when the meaning of the compound is determined by the meaning of the simpler sentences. In the usage I've been exposed to, a "logical connective" can basically be any way of combining propositions, so long as its meaning can be defined compositionally. Note that I count modal operators as logical connectives (and there are a gazillion papers on linear logic that do likewise). Noamz (talk) 03:25, 25 June 2009 (UTC)

Perhaps the article should have a small section saying there are other logics which don't follow the rules but I think this article is right to be at the straightforward propositional logic level. No need to muddle it up with things like quantum logic. Dmcq (talk) 10:30, 25 June 2009 (UTC)

I agree with Noam's suggestion. A more dramatic bias towards classical propositional logic has happened in the case of satisfiability, which used to house an article that at least tangentially mentioned the general model-theoretic definition of when a model is satisfiable [3], but now is a redirect to Boolean satisfiability problem. Fortunately, model theory does define unsatisfiable. — Charles Stewart (talk) 19:29, 26 June 2009 (UTC)

Postscript — I've created Satisfiability and validity, which is a stub that is meant to be a more informal introduction to the concepts than found in model theory. I've just realised that interpretation (logic) is meant to be just that, so I have to think about merging it. — Charles Stewart (talk) 20:35, 26 June 2009 (UTC)

This article was originally very limited, covering only truth-functional connectives in formal languages. At some point, I believe User:Philogo began to add some text about natural-language connectives. This is a good thing to add, but unfortunately it is outside the knowledge of many mathematics editors, including myself, so we can't really help with that. I believe there is a significant amount of additional material that could be added, modal connectives and other non-truth-functional connectives included. — Carl (CBM · talk) 12:32, 25 June 2009 (UTC)

Preservation properties table

Hi, I would like to offer this table for inclusion in the properties section. Any comments or modifications are welcome. Thanks Zulu Papa 5 ☆ (talk) 05:18, 1 November 2009 (UTC)

Where is this talked about in the literature please? What is its application? Thanks Dmcq (talk) 09:49, 1 November 2009 (UTC)

Good question, the table is a summary of properties that are well established in the literature and all ready in the article. I believe these properties say something about preserving validity. It serves as a summary here, how these groupings are applied requires a statement and a source. What concerns me is the “non-preserving” group as they have not been described as such in the lit. I put it as a catch all, to the others that don't have a preserving property. Also, should the first group be “True and False Preserving” or should it be “True and/or False preserving”. Maybe this should be stated differently. I can’t find a source for how the two properties combine. Zulu Papa 5 ☆ (talk) 16:21, 1 November 2009 (UTC) Zulu Papa 5 ☆ (talk) 16:25, 1 November 2009 (UTC)

Are these properties discussed in any standard textbook? I am more familiar with the mathematical logic literature, where I don't think I have seen this sort of preservation property having much interest. I guess I am asking why this information is worth including; it's clearly true, but I don't see it as having much interest. But if it is mentioned in the literature then I would not object to including it here. — Carl (CBM · talk) 11:58, 3 November 2009 (UTC)

These properties are about validity. See this source [4] Let me see if I can find an appropriate statement from this source or others. It seems like preserving validity is objectivity valid and non-preserving is subjective validation. The issue crosses over from logic to ontology and truth. Haven't found a table source or these specific statement sourced. It could be this belongs in a new separate article about the "Logical Connective Preservation Properties". Zulu Papa 5 ☆ (talk) 22:17, 3 November 2009 (UTC)

This applies [5] the "non-persistent" term (p 26). Furthermore, the application is that initial binary (true, false) requirement for a sentence changes to 3, 4 or higher order logic. These properties appear to be very relevant for the transition from zero order logic.

I am not very convinced those are relevant to an article on logical connectives; it seems like a stretch. I will look more closely tomorrow, but I would appreciate hearing why those sources actually support the table in question. — Carl (CBM · talk) 02:22, 4 November 2009 (UTC)

What's so special about these two properties, as opposed to monotonicity, affineness, self-duality, or any other properties defining a clone in Post's lattice for that matter? To me the table looks like a random selection of trivial information. — Emil J. 11:18, 4 November 2009 (UTC)

The thought in your first sentence has occurred to me as well. But the table doesn't really seem to belong there, either. Hans Adler 12:16, 4 November 2009 (UTC)

This is a tough crowd, but I expect so in binary logic (smile) LC's are about truth and false sentences, these particular properties focus on how truth is preserved. From my perspective of true and false relevance, these are the most relevant properties for validity. The other properties are interesting however, these relate to validity. Sorry to repeat myself. It seems folks dealing in the abstract don't realty care about validity however, it's a must for real applications with a purpose. The article has neglected validity. For example: a wiki sentence must be verified WP:V to a reliable source to be valid, (making it objective in place of purely subjective). Is all exactly un-sourced material in-valid? No, because there is subjective common sense about truth preservation and partial truths in the verification for validity WP:IGNORE to wiki's purpose. Without this, wiki would be a copying machine and none of us would be allowed to talk here. The truth is preserved in verification however, as these preservation properties demonstrate, it is also transformed by the applied logical connectives. Ultimate, wiki is about verified validity with logical connectives. Wikipedia, isn't about valid truths, disputes are often settled by source attribution to the cited subject and specific context. (Sorry, I can't address "a random selection of trivial information" that seems uncivil and off track for now). Thanks Zulu Papa 5 ☆ (talk) 20:21, 4 November 2009 (UTC)

Ok, I talked myself into placing this on the validity page with a short (see: validity) in this article. Thanks for the feedback. Zulu Papa 5 ☆ (talk) 16:16, 5 November 2009 (UTC)

Small changes in the table

I've slightly changed the table [6] [7], because the index elements of a matrix should be ordered lexicographically, as it is done e.g. in Karnaugh maps. So 0/false should be next to the matrice's orgin, not 1/true.

Venn diagrams in white and red are more usual, because books with two ink colors were done rather in black and red than in black and light blue. Some examples: University of Leicester; University of Illinois; The Geometry of Logic by S. H. Cullinane
These Venns (without the unnecesseary blue margin) are already used in the single articles like Logical conjunction.
Lipedia (talk) 17:15, 19 May 2010 (UTC)

Standard Notation

see Wikipedia:WikiProject Logic/Standards for notation#Symbols

It is raining if I am indoors (Q P) <- fail fix it —Preceding unsigned comment added by 85.231.122.178 (talk) 19:17, 16 February 2011 (UTC)

Non-truth-functional connectives

Shouldn't this article talk about those too? Are those not called logical? See [8] for instance. I have the impression the definitions in the lead are not really sourced. Tijfo098 (talk) 14:57, 30 March 2011 (UTC)

Non-truth-functional connectives are not usually called "logical connectives". This is the usual terminology from propositional logic, Boolean logic, etc. It is rare for such books to mention modal operators at all. — Carl (CBM · talk) 15:57, 30 March 2011 (UTC)

So what would you call them? Illogical connectives? Perhaps this article should be renamed to connective (logic) then. [9] Tijfo098 (talk) 16:03, 30 March 2011 (UTC)

Besides, your statement doesn't hold for modal logic texts [10] [11] etc. Tijfo098 (talk) 16:13, 30 March 2011 (UTC)

I don't understand why you keep linking to books. Could you just come out and say whatever argument you're trying to make? Both books you have linked seem to use the word "operator" for the modal symbols, which is the word I used above and is the word I am used to for modal operators. This article is about truth-functional logic connectives, not about modal logic, and it would be out of place to try to cover modal logic in this article.

Because they use logical connective with a different meaning that here. Tijfo098 (talk) 17:20, 30 March 2011 (UTC)

I also think that the random smattering of tags on the article isn't helpful. Just putting the POV tag at the top of the article is enough to indicate you have some issues with it. — Carl (CBM · talk) 16:22, 30 March 2011 (UTC)

Another way to say it is that this article is about the truth-functional connectives relevant to propositional logic. We can't write an article about every possible type of logical connective including modal operators, infinitary quantifiers, the operators from linear logic, and everything else, because that's not a topic that anyone actually writes about in the literature. Each logical system has its own syntax that includes particular connectives, so an article that tried to come up with a general definition of "logical connective" would be original research. On the other hand, many books (including Enderton's book in the references) cover the topic of arbitrary truth-functional logical connectives. That is an important topic for us to cover, and it's what this article is intended to discuss. — Carl (CBM · talk) 16:34, 30 March 2011 (UTC)

I think the problem is that you have adopted McGee's definition (you should have told me where it was coming from instead of complaining about the inline tags). I was able to figure this out from the SEP article on this topic:

“ McGee's strategy is to invoke semantic notions instead of modal ones: he suggests that “[a] connective is a logical connective if and only if it follows from the meaning of the connective that it is invariant under arbitrary bijections” (McGee 1996, 578) ”

But this is nowhere near the WP:COMMONNAME usage of "logical connective". And even if you want Wikipedia to follow McGee, that would mean that connective (logic) should have its separate article, not redirect here! Tijfo098 (talk) 17:20, 30 March 2011 (UTC)

I don't see the difference between "logical connective" and "connective (logic)".

The common meaning of "logical connective" is essentially "a connective in propositional logic", that is, the type of connective that corresponds to a digital logic gate. Modal logic and other nonclassical logics are very specialized topics, but many undergraduates in mathematics and computer science learn about logical connectives either in the context of Boolean logic for computing, a course in mathematical proof where they learn the mathematical meanings of 'and' and 'or', or a basic course in formal logic.

So instead of naming this article "truth-functional {0,1}-valued logical connectives" we just call it "logical connectives". That seems to match with the COMMONNAMES advice: people who are looking for modal logic will know that's what they want, while people who are interested in the connectives they just learned in their digital logic course will not expect to find modal logic. They certainly won't expect an article on the philosophical arguments about what a logical constant is, like the SEP article you quoted. That article is not about the same topic as this article. — Carl (CBM · talk) 17:41, 30 March 2011 (UTC)

Reverted again by the math department, thus page needs to be split

[12]. So, I see no option but to split this page. It's ridiculous to include two different notions in the same article and revert sourced attempts to distinguish them. Tijfo098 (talk) 23:00, 4 April 2011 (UTC)

Oppose. You appear to be trying to merge some informal semantics associated with grammatical conjunction#Coordinating conjunctions (but not actually appearing there) into this article. I don't see a reason to do so. If there were an appropriate article or section to contrast this to, I wouldn't oppose a note to that effect.

I also believe your formal semantics edits are inappropriate, due to the various mergers, such as formal theory, but I'll have to get back to that, later. I believe that article would be better split into formal semantics (logic), formal semantics (computer science) (not yet written), and formal semantics (natural language) (which apparently has some present content, at least in the article formal semantics). At least, your edits there all relate to some topic with the same name. — Arthur Rubin (talk) 23:37, 4 April 2011 (UTC)

Fortunately you don't wp:own the entire Wikipedia. Someone already created discourse connective. Try redirecting it here. And read something besides math books, it'll do you some good. Tijfo098 (talk) 23:54, 4 April 2011 (UTC)

That article seems redundant when there's already Grammatical conjunction which seems a much better title to me and is also far better developed, it has for instance ... drumroll please maestro ... references. Dmcq (talk) 00:43, 5 April 2011 (UTC)

Oppose I don't see any reason we can't have a section toward the bottom about non-truth functional connectives. I don't think we need to split it either. Please let's all just get along in the same article. Greg Bard (talk) 00:50, 5 April 2011 (UTC)

Stop the presses. Greg and I agree. (Seriously, in this instance, Greg expresses the issue better than I did. The additional comment is about non-truth-functional connectives, rather than a disguised comment on grammatical conjunction.) — Arthur Rubin (talk) 01:27, 5 April 2011 (UTC)

I am happy with the hatnote that clarifies this page is about logical connectives in classical (propositional) logic. There is no general concept of "logical connective" in an arbitrary logic that we could write about; the syntax varies so much, all we would be able to do is make a list of connectives in this logic, and this logic, and this logic. On the other hand there is a lot of philosophy discussion on arbitrary logical constants that is not really related to them being connectives (quantifiers are also important) but which should be covered in that article rather than this one. Phrases such as "whereupon" and "that is" are not really "logical" connectives, unless you think natural language is a sort of logic. — Carl (CBM · talk) 11:51, 5 April 2011 (UTC)

I am not happy with the hatnote that indeed clarifies that this page is about logical connectives in (two-valued) classical propositional logic, but that leaves no room for non two-valued interpretations of logical connectives. In particular, that leaves no room neither for discussing the use of this very same concept of logical connective in intuitionistic logic or in non-necessarily two-valued interpretations of classical logic. I can understand that many readers expects "logical connective" to be a rough synonymous of "logical operator over 0 and 1" as they maybe just learned in their digital logic course, but this is not the only way to refer to the "digital logic" meaning since for instance "two-valued logical operators" would be as relevant if not more. On the other side, I don't know other common names for what logicians call "logical connective" (i.e. a formal symbol to build new formulas by composition). What can be done to solve the problem? For instance, logical operator is a redirect to logical connective. Could it be possible to exploit the difference of meaning between operator (the connective interpreted on some domain) and connective (the symbol properly speaking) to satisfy all needs at best? --Hugo Herbelin (talk) 20:47, 17 April 2011 (UTC)

Logical operators in terms of conjunction and negation

It is possible to define logical operators in terms of only conjunction (and) and negation (not). I have added this for disjunction (or) as it is common, well known and occasionally useful to be able to do this for disjunction in particular, and shows how each operator can be formed in its most basic alphabet. Would it be useful to add it to others? 86.0.254.239 (talk) 16:15, 5 May 2011 (UTC)

Writing

The following excerpt from the article is completely incoherent. "Is some new technology (such as reversible computing, clockless logic, or quantum dots computing) "functionally complete", in that it can be used to build computers that can do all the sorts of computation that CMOS-based computers can do? If it can implement the NAND operator, only then is it functionally complete." I would correct it, but I honestly cannot make sense of this. 75.72.7.108 (talk) 18:04, 28 July 2011 (UTC)

 Done I removed the paragraph. It didn't make sense, and it didn't appear to add anything to the article. Justin W Smith ^talk/_stalk 18:49, 28 July 2011 (UTC)

Style of Venn diagrams

I propose to replace red-coloured diagrams such as with green-coloured ones. Also I think that the border should be removed from SVGs: we can use table borders. Incnis Mrsi (talk) 12:38, 9 March 2012 (UTC)

There is no rational reason to add another bunch of Venn diagrams to commons:Category:Vector Johnston diagrams. Lipedia (talk) 08:26, 24 April 2012 (UTC)

So, just change the color and upload over? Incnis Mrsi (talk) 09:16, 24 April 2012 (UTC)

I think the 'no rational reason' really means can you provide a reason for sticking in green instead of red? Why not blue of black or yellow or purple or orage or pink? Or what's particularly wrong with red? Dmcq (talk) 10:11, 24 April 2012 (UTC)

Red is inappropriate because of its negative symbolic, see numerous examples in commons:Category:SVG Anti logos. If the background remains white, I do not object against blue, azure, violet, or even black. Let Lipedia explain what means "no rational reason" himself (if he feels that further explanations are needed), and, as a homework personally to Dmcq, try to explain what would be wrong with yellow. Incnis Mrsi (talk) 10:33, 24 April 2012 (UTC)

I have never found the red very aesthetically pleasing, we had a violet shade before that was calmer. If there is a worry about having too many images on commons, you could just change the color on the images and re-upload them. — Carl (CBM · talk) 11:02, 24 April 2012 (UTC)

Resolution

So, in absence of further objections, tomorrow I proceed to following steps:

1. Download all images from commons:Category:0 and 1 circle Johnston diagrams and commons:Category:Red and white Johnston diagrams;
2. Change red to violet     by script and upload updated versions with links to this discussion;
3. Replace "Category:Red and white Johnston diagrams" with commons:Category:Violet and white Johnston diagrams;
4. Replace mentioning of red with violet in wikis, if necessary.

I do not intend to make other changes to SVGs now. Incnis Mrsi (talk) 12:13, 24 April 2012 (UTC)

This image has the color I was talking about - a light violet. The darker violet, for me, doesn't have much contrast with the black border. — Carl (CBM · talk) 13:02, 24 April 2012 (UTC)

Now I lean to creation of completely new design. Please, propose colors and proportions. My proposal is to retain orthogonally intersecting circles from Lipedia's design, with not so thick outline, filled with a color m.b. slightly lighter than proposed earlier (say, #9000FF    ) but darker than in commons:File:Set union.png, and without any foreign elements such as borders or margins. Incnis Mrsi (talk) 18:58, 24 April 2012 (UTC)

I created the original images that served this purpose. They were replaced by these red ones. The original ones I made had circles which were labeled. So I would support any replacement, however the images themselves should be labeled with circles "A" and "B" (or preferably "P" and "Q").Greg Bard (talk) 20:37, 24 April 2012 (UTC)

Please stay away from my files

Do what you want in the article, but please keep my red and white files out of this. If you want to upload something new you may overwrite the files in commons:Category:Blue and white Johnston diagrams, which are unused.

Red doesn't have any "negative" meaning when there is no green in the same context: This map doesn't show which territory is not the USA, and this one doesn't show the places to avoid when you are searching for Israel (compare this category). Here and here red is used to highlight the prime numbers. Here and here areas of Venn diagrams are highlighted. Red causes awareness, and that's also the reason for the use in anti-logos - not any symbolic meaning of this color.

Since a year there are the property sections like this one. Don't forget to delete the illustrations from them (I'm sure CBM will like that) or to upload some 3-circle diagrams in the new color. And don't forget to remove anything red from Logical biconditional. Lipedia (talk) 15:54, 24 April 2012 (UTC)

Sure, there is no rational reason to add another bunch of Venn diagrams to Commons. But since Lipedia likes his red diagrams, these diagrams should be preserved, so, I will add another bunch of diagrams. commons:Category:Blue and white Johnston diagrams does not satisfy me because of thick annoying borders. Incnis Mrsi (talk) 18:58, 24 April 2012 (UTC)

My suggestion was to overwrite these unused blue files. Of course with diagrams without the annoying border. Lipedia (talk) 19:46, 24 April 2012 (UTC)

(see comments above)-GB

only if

In the section "natural language", I removed from the table a line incorrectly stating that "only if" is the converse implication. To the contrary, "P only if Q" means the implication, P implies Q. As I am in a hurry - this implies error prone - I have not introduced a corrected version. Wlodr (talk) 14:54, 20 July 2019 (UTC)

This depends on how to write it down. “Only if P, then Q” – the converse implication P ← Q, certainly. Incnis Mrsi (talk) 15:05, 20 July 2019 (UTC)

NIMPLY / Material implication

2 slightly different symbols are used: with the slash either *after* the arrow or in the *middle* of it (this second, IMO preferable symbol appears in the "redundancy - 2 symbols" section) Confused me for a minute, anyway 2A00:23C6:B211:D100:2598:1B13:2DDC:B828 (talk) 07:52, 5 October 2020 (UTC)

About the Order of precedence section

The reference to software compilers in the Order of precedence section seems abrupt and out of context. Maybe the section should be expanded? Abd.nh (talk) 19:54, 28 October 2020 (UTC)

Doubts of the table in Section "In language", Subsection "Natural language"

I have a few doubts in the 5th row, i.e. the word "either...or...".

1, The interpretation of the word "either...or..." may be subjective. To me, this word applies to two exclusive terms which belong to the same system to categorize things. On the other hand, it is not uncommon that people use "inclusive disjunction" interpretations (esp. when it is convenient and short in natural languages), for example, "the solution is either x>0 or y>0", althou it is equivalent to "either (x>0 AND y>0), or (x>0 AND y<=0), or (x<=0 AND y>0)" if we use a categorization system using both x and y variables. Is "exclusive disjunction" more applicable than "inclusive disjunction", or vice versa? Or shall we delete this row completely since this depends on subjective interpretations of the natural language word "either...or..."?

2. The symbol "⊕" seems the logical connective "exclusive disjunction", or the logic gate "XOR". Shall it be "∨" if we go with the interpret of "either...or" as "inclusive disjunction"? Or shall we change the other columns for this row, since they ("inclusive disjuction", "⊕") are not consistent if I understand this issue correctly.

3. The 4th column, i.e. "Logical gate", shall use a term for logical gate. So the term for this row shall be "OR" (a logic gate term) if "inclusive disjunction" (a logical / boolean connective term), or "XOR" if "exclusive disjunction". It seems the term issue only occurs in the 5th row ("either...or..."), all other rows are fine.

4, This row went through a couple of revisions in history. I believe it might be due to the issue of different subjective interpretations of the word "either...or..." as described in the first doubt. This suggests that we might be better off removing the row completely. Or we shall put an extra footnote for readers about the possible subjective interpretations from the unprecise natural language word "either...or" to a precise mathematical logic concept (i.e. a logical connective). At least, we shall make the column elements in this row consistent with each other and error free. IMHO, the current edition (with the inconsistency between "inclusive disjuction" and "⊕" introduced by the revision "06:56, 16 December 2020‎", and the none logic gate term "Logical disjunction" introduced by the revision "11:54, 26 February 2021‎") may cause confusions for readers unfamiliar with this topic.

Need some help from knowledgeable Wikipedia fellows since I'm far from an expert in this topic (or I won't read this article).

Thanks for pointing this out. I've trimmed the list and added some information above it. As for exclusive disjunction, it's not so much that it's subjective in the sense of different people having different opinions. It's more that disjunction can get different interpretations in different syntactic configurations and discourse contexts. I recommend this or this if you wanna read more. Botterweg14 (talk) 23:49, 11 March 2021 (UTC)

Contradiction between definition of connective and negation being a connective

The introduction states: "A logical connective is a logical constant used to connect two or more formulas" But only one sentence later, it states: "Common connectives include negation"

This is a contradiction.

Either negation is not a connective, or a connective need not join two or more formulas.

EditorPerson53 (talk) 22:31, 6 February 2022 (UTC)

• Different people use the same terms differently. I prefer to have operator be something that ranges over 0-ary logical constants (i.e., true/false in bivalent classical logic); 1-ary operators including unary modalities and negation; 2-ary operators including binary logical connectives and bimodalities (i.e., 2-ary modal operators); and then one typically might have finite and infinite quantifiers, which generally needs some kind of set or type theory to axiomatise.

We absolutely should be consistent within each article. If WP:LOGIC had any momentum, it would be a natural place to try to reach consensus over the logic articles.

On my analysis above, connectives are usually binary, and if I want to be more general I use the language "logical operator". That differs from the scheme in this article. — Charles Stewart (talk) 01:26, 7 February 2022 (UTC)

I think EditorPerson53 has a point actually.. but instead of writing about in the talk page, just be bold and update the intro text :P · · · Omnissiahs hierophant (talk) 18:03, 7 February 2022 (UTC)