Talk:Logical connective

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It had been proposed to merge this page with Logical constant, however, the result of the discussion was keep.

Contents 1 There are unary operators, binary operators, and comparators 2 Merging boolean operators with logical operators 3 wikiproject proposed 4 Sets and Logical strength sections 5 Relative strength of operators 6 Move 7 Changes to "Arity" section (major and minor) 8 Too long? 9 Intro 10 Venn diagrams arrangement 11 "two or more well-formed formulae" 12 Truth-table 13 expressed as a relation 14 List of connectives with truth tables and Venn diagrams 15 The lede 16 Logical constant 17 moment of doubt 18 New lede

[edit] 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)

[edit] 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)

[edit] 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)

[edit] 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)

[edit] 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		↓	∨		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 ()	{ ∅ , { ∅ } , { { ∅ } } , { ∅ , { ∅ } } } - Tautology ()
{ ∅ } - NOR (↓)	{ { ∅ } , { { ∅ } } , { ∅ , { ∅ } } } - OR ()
{ { ∅ } } - Material nonimplication (⊅)	{ ∅ , { { ∅ } } , { ∅ , { ∅ } } } - Material implication (⊃)
{ ∅, { ∅ } } - Not q	{ { { ∅ } } , { ∅ , { ∅ } } } - q
{ { { ∅ } } } - Converse nonimplication (⊄)	{ ∅ , { ∅ } , { ∅ , { ∅ } } } - Converse implication (⊂)
{ ∅ , { { ∅ } } } - Not p	{ { ∅ } , { ∅ , { ∅ } } } - p
{ { ∅ } , { { ∅ } } } - Exclusive disjunction ()	{ ∅ , { ∅ , { ∅ } } } - Biconditional ()
{ ∅ , { ∅ } , { { ∅ } } } - NAND (↑ or |)	{ { ∅ , { ∅ } } } - Conjunction ()

[edit] 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)

[edit] 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)

[edit] 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)

[edit] 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)

[edit] 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)

[edit] "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)

[edit] 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

[edit] 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)

[edit] 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 n/a   Q T F P T    F   F  F    F   F	Tautology Notation Equivalent formulas Truth table Venn diagram T n/a   Q T F P T    T   T  F    T   T
Conjunction Notation Equivalent formulas Truth table Venn diagram P & Q P Q P ¬Q ¬P Q ¬P ¬Q   Q T F P T    T   F  F    F   F	Alternative denial Notation Equivalent formulas Truth table Venn diagram P ↑ Q P | Q P NAND Q P → ¬Q ¬P ← Q ¬P ¬Q   Q T F P T    F   T  F    T   T
Material nonimplication Notation Equivalent formulas Truth table Venn diagram P Q P Q P & ¬Q ¬P ↓ Q ¬P ¬Q   Q T F P T    F   T  F    F   F	Material implication Notation Equivalent formulas Truth table Venn diagram P → Q P Q P ↑ ¬Q ¬P Q ¬P ← ¬Q   Q T F P T    T   F  F    T   T
Proposition P Notation Equivalent formulas Truth table Venn diagram P n/a   Q T F P T    T   T  F    F   F	Negation of P Notation Equivalent formulas Truth table Venn diagram ¬P n/a   Q T F P T    F   F  F    T   T
Converse nonimplication Notation Equivalent formulas Truth table Venn diagram P Q P Q P ↓ ¬Q ¬P & Q ¬P ¬Q   Q T F P T    F   F  F    T   F	Converse implication Notation Equivalent formulas Truth table Venn diagram P Q P Q P ¬Q ¬P ↑ Q ¬P → ¬Q   Q T F P T    T   T  F    F   T
Proposition Q Notation Equivalent formulas Truth table Venn diagram Q n/a   Q T F P T    T   F  F    T   F	Negation of Q Notation Equivalent formulas Truth table Venn diagram ¬Q n/a   Q T F P T    F   T  F    F   T
Exclusive disjunction Notation Equivalent formulas Truth table Venn diagram P Q P Q P Q P ¬Q ¬P Q ¬P ¬Q   Q T F P T    F   T  F    T   F	Biconditional Notation Equivalent formulas Truth table Venn diagram P Q P ≡ Q P ¬Q ¬P Q ¬P ¬Q   Q T F P T    T   F  F    F   T
Disjunction Notation Equivalent formulas Truth table Venn diagram P Q P ¬Q ¬P → Q ¬P ↑ ¬Q   Q T F P T    T   T  F    T   F	Joint denial Notation Equivalent formulas Truth table Venn diagram P ↓ Q P ¬Q ¬P Q ¬P & ¬Q   Q T F P T    F   F  F    F   T

[edit] 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)

[edit] 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)

[edit] 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)

[edit] 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)