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The 2012 discussion about this matter did not reveal a single instance where material implication (rule of inference) is called “material conditional” or by some other name which redirects here, or may be mistyped in a way which gets a reader to this article. At least, I do not see there any concrete direction. The [1] edit was anything else than an attempt to circumvent the due process. Incnis Mrsi (talk) 07:32, 8 May 2013 (UTC)[reply]

At present the lead section does not define "material conditional". Furthermore, it assumes some understanding of formal logic, but never actually positions "material conditional" within the study of logic. More detail, more basic explanation, and a definition of the concept would be appreciated. Cnilep (talk) 01:22, 10 May 2013 (UTC)[reply]

I've reworked the lead to try and make it clearer and more informative. I omitted the distinction between material implication and logical implication from the lead because it wasn't clearly explained and the introduction of that distinction in the lead seemed excessive and confusing. I've tried to describe the meaning of the operator in a clear manner and I've also pointed out a common confusion of beginners to formal logic in relation to that operator. I've also added two citations and extended the segment that listed logical equivalents. Your feedback would be most welcome. AnotherPseudonym (talk) 14:59, 28 May 2013 (UTC)[reply]

“Reworking” undone. I will revert on sight any edits which injects a knowledge like

(whatever a college student can derive from laws of Boolean logic), because a propositional calculus is not necessarily classical/Boolean. There is no such thing as the propositional calculus. Incnis Mrsi (talk) 07:04, 29 May 2013 (UTC)[reply]

I've reverted your revert. There may be so such thing as the propositional calculus, but removing Boolean propositional calculus form the lead would be wrong. — Arthur Rubin (talk) 03:36, 30 May 2013 (UTC)[reply]

It means that you sided with ignorance in this particular case, not more, and not less. You also pointed to some (unexplained) problems with my “ ” and other regularization of typography and some (unspecified) “other problems”, but this does not change much. I do not know who user:AnotherPseudonym is, but we know who user:Arthur Rubin is. How do you, Arthur Rubin, explain removal of Stanford Encyclopedia of Philosophy reference and its replacement with (technically broken) ones to a book written by certain Paul Teller? Incnis Mrsi (talk) 12:42, 3 June 2013 (UTC)[reply]

Using an encyclopedia to create another encyclopedia is a pointless exercise. An article should be created from primary sources not from another encyclopedia. Why not just replace the entirety of the article with a link to the Stanford Encyclopedia of Philosophy entry on the topic? There is nothing wrong with the book by Teller and it has the virtue of being online in complete form as does the other referenced book. Also it doesn't matter who I am or who Arthur Rubin is or who you are. AnotherPseudonym (talk) 11:07, 4 June 2013 (UTC)[reply]

I take your point but I think that is rather heavy handed. You could just qualify what you identified as too general. Yes a college student can derive the equivalences but the point is to provide a concise description of the operator in the lead. AnotherPseudonym (talk) 07:45, 31 May 2013 (UTC)[reply]

I have added the necessary qualification. Regarding the original lead, it was a mess. Amongst other things the original lead had redundancies, it employed terms without first defining them or linking to a definition, sometimes the term "compound" was used and other times "statment" was used, the relatively minor matter of material implication vs. logical entailment was too long, used an awful example and just confused what preceded it. In the lead it would have been sufficient to just say something like: "The material conditional is to be distinguished from logical entailment (which is usually symbolosed using [double turnstile]." The distinction can then be detailed in the body of the article. Also the failure to even mention propositional calculus -- which is the context in which someone is most likely to look up the meaning of the operator -- was an unacceptable omission. By the time someone reaches the study of paraconsistent logical systems they will likely have no need to look up what a material conditional is on Wikipedia. A novice is most likely to look up this entry in wikipedia and they will most likely have encountered the operator in the context of classical/Boolean propositional calculus. AnotherPseudonym (talk) 08:06, 31 May 2013 (UTC)[reply]

Propositional calculus has no special relevance to the topic, because the leading statement already says that “→” is a logical connective: try to think what follows from this. I do not see any point to stress the use of “→” namely in propositional calculi (not in a first-order logic or whatever). Paraconsistent logical systems also have no special relevance to the question raised and I do not realize why I should read anything about these. Which logical systems, except for Boolean-based, have the material conditional equivalent to ${\displaystyle \neg (p\land \neg q)}$? If you do not yet realize what I mean, then read logical connective #Redundancy please. Incnis Mrsi (talk) 12:42, 3 June 2013 (UTC)[reply]

I'm not arguing for the inclusion of anything about paraconsistent logical systems, I am arguing against that. If I'm not mistaken, paraconsistent logics are a species of non-Boolean logic and your contention -- if I am understanding it -- that the article lead should possess a generality that does not preclude non-Boolean logics amounts to a position that paraconsistent logics -- amongst other non-Boolean logics -- should bear on the composition of the lead. If you write a thorughly generic lead it will retain the vagueness and lack of clarity that was originally complained about. The special relevance of propositional calculus and first-order predicate logic is that they are the most likely context in which a novice will encounter the operator and will seek clarification from an encyclopedia. I believe I know what you mean but I don't agree with the completely generic form of the lead that you support. Such an article will be a useless piece of formalism. Anyone that is even aware that there are logical systems that are non-Boolean will have no need to consult a general encyclopedia regarding the material conditional. Those that are likely to consult an encyclopedia -- those encountering the material conditional in the conext of Boolean first-order logic -- will not gain anything from a generic article that takes account of non-Boolean logics in all of its descriptions. Technically what you are arguing is correct but from a pragmatic and pedagocial position it is misguided. AnotherPseudonym (talk) 06:49, 4 June 2013 (UTC)[reply]

Could you bring your pedagogical positions to wikiprojects which really need it? Wikipedia is an encyclopedia. Wikibooks is not, Wikiversity is not, but Wikipedia is! And WP:Wikipedia is not a textbook. It is a harmful misconception that Wikipedia serves to college students, not experts. There are things which experts can more likely find here than in Google, because not all structures of knowledge are detectable by text search engines. Incnis Mrsi (talk) 08:35, 4 June 2013 (UTC)[reply]

I looked up Implies in the CRC Concise Encyclopedia of Mathematics (2nd Ed.), the entry begins, "The CONNECTIVE in PROPOSITIONAL CALCULUS which has the meaning "if A is true, then B is also true."...A=>B is an abreviation for !A v B, where !A denotes NOT and v denoted OR..." pp.1466-7 So I don't think what I am proposing is unusual. I can find no merit in your argument. What sort of expert is going to need too look up this operator and will only be served with the most generic form the lead can possibly take? The lead that I replaced was a poor piece of technical writing that you'd never find anywhere other than Wikipedia. Even if your stated objective was sound the execution of the lead was like scratch paper notes that were compiled over a long period of time with lengthy gaps between the additions and no re-reading before additions. It really was an atrocious piece of technical writing by every relevant measure. I would actually excerpt that lead and use it as an example of how NOT to write a lead on a technical subject. If the previous lead looked good to you then there is no point in any discussion because your implicit writing standard doesn't even exclude redundancy and inconsistent terming which are breaches of fundamental principles of technical writing. I don't really have a great investment in the material conditional. If you want to restore a vague, inconsistent and obtuse article then go ahead. AnotherPseudonym (talk) 10:50, 4 June 2013 (UTC)[reply]

Actually, I have no objections against your current version except a minor one that I’d say “in classical logic” dropping “propositional calculus” from the lead altogether. I spend my time arguing in this talk page only because I hate when a guy like Arthur Rubin thinks that WP:BRD is a guideline for some other (minor) editors, not for himself. Incnis Mrsi (talk) 11:10, 4 June 2013 (UTC)[reply]

Done. Looking at the lead now I don't think it has lost anything significant by not explicitly referencing propositional calculus and first-order logic. AnotherPseudonym (talk) 11:52, 4 June 2013 (UTC)[reply]

In the paragraph

"In classical logic p \rightarrow q is logically equivalent to \neg(p \and \neg q) and by De Morgan's Law to \neg p \or q"

the symbols for negation, logical and and logical or are neither explained nor linked to other Wikipedia articles. In this way, the article is not understandable to the layman.--84.150.172.61 (talk) 09:06, 16 December 2013 (UTC)[reply]

I think that what it is said about monotonicity (under "Formal properties") is confusing. It is simple to see that material conditional is anti-monotonic in the first argument and monotonic in the second argument. Still, if we lift the reasoning from truth values to the inference process, then it is true that "if we know more, we cannot derive less" (in classical logic). Saying, as it is in the article, that if a→b then ∀c.(a∧c)→b doesn't mean that → is monotonic: the property is indeed true due to the anti-monotonicity of → in the first argument! (Adding "and c" to the premise can only decrease its truth value and thus increase the truth value of the whole implication, where "decrease" and "increase" refer to the total ordering of the boolean lattice ⊥ < ⊤). Is there anyone who thinks that these two levels should be clarified and kept distinct? Grace.malibran (talk) 14:01, 9 January 2014 (UTC)[reply]

"But unlike as the English construction may, the conditional statement "p→q" does not specify a causal relationship...." I doubt that. Can anybody give an example of "If p, then q" which implies causality? I can think of many examples which might give rise to a suspicion of causality, but none in which the suspicion could be considered justifiable. "If you hit the ramp going less than 50 MPH, you're not going to make it." That suggests cause and effect, but I say it doesn't imply it; it just expresses a correlation. --Marshall "Unfree" Price 208.54.85.219 (talk) 01:58, 26 May 2014 (UTC)[reply]

The current version (March 2016) reads: "However, unlike the English construction, the material conditional statement p -> q does not specify a causal relationship between p and q." This still seems specious, as the English construction does not necessarily specify a causal relationship between p and q. For example, in English I could say, "If I'm at a Fourth of July picnic, then there are going to be fireworks tonight." That does not specify any causal relationship between the two statements. (My presence at the picnic is not causing the fireworks, and the fireworks are not causing my presence at the picnic.) 74.71.76.34 (talk) 14:06, 19 March 2016 (UTC)[reply]

In your example, I think a native speaker of English would interpret that statement as the speaker claiming they'd surely launch fireworks if they were at a 4th of July picnic, especially if it had none. As in, they believe there should be fireworks so much that they would ensure that there are some. 96.60.251.105 (talk) 04:49, 31 October 2023 (UTC)[reply]

I came to this article hoping to discover whether "material conditional" was the exact opposite of "counterfactual conditional". I suspect there might be a "factual conditional", in which case, I'll have to go on another errand, seeking the opposite of "material conditional". Oh, maybe it's "immaterial conditional". Who knows? --Marshall "Unfree" Price 172.56.26.37 (talk) 02:09, 26 May 2014 (UTC)[reply]

I get it, a minimalistic style approach was taken making these diagrams, but why are they are not labeled?

http://upload.wikimedia.org/wikipedia/commons/thumb/1/1e/Venn1011.svg/440px-Venn1011.svg.png — Preceding unsigned comment added by Scire9 (talk • contribs) 20:43, 17 June 2014 (UTC)[reply]

The top of page Venn diagram of A --> B is wrong. One circle (A) should be completely inside the other (B). For example in the following image, Whale --> Mammal. http://faculty.ycp.edu/~dhovemey/fall2006/mat111/lecture/figures/whalesAreMammals.png. If it is a whale then it must be a mammal but if it is a mammal it may not be a whale. Hence whales are a subset of mammals. John Middlemas (talk) 23:05, 31 July 2014 (UTC)[reply]

I expanded the caption a little. Does that explain for you? Paradoctor (talk) 23:53, 31 July 2014 (UTC)[reply]

The average reader will wonder where is A and where is B? They are not labelled. It can be deciphered from your wording that the A is left but with effort. Also the red outside of both circles is not necessary and confusing. The white part is also irrelevant and detracts from the understanding that all of A should be inside B which is the only point anyway. All you want is a smaller red 'A' circle inside a larger white 'B' circle and white surround. The red signifies member of A which is what we have assumed. Sorry, but I think all that complication will just confuse the real meaning of A --> B. Better the whale/mammal pic. 88.203.90.14 (talk) 01:02, 1 August 2014 (UTC)[reply]

I'm beginning to wonder whether this diagram is actually helpful. It is correct, though. The idea is to represent statements through sets. A statement ${\displaystyle A}$ is true iff ${\displaystyle x\in A}$ for all ${\displaystyle x}$. Note that we're using the same name for a statement and the set representing it. ${\displaystyle A\rightarrow B}$ is false only if ${\displaystyle A}$ is true and ${\displaystyle B}$ is false. This means that the set ${\displaystyle A\rightarrow B}$ excludes exactly those ${\displaystyle x}$ for which ${\displaystyle x\in A}$ and ${\displaystyle x\notin B}$, which corresponds to the white area. Do you see why the areas outside the circles must be red? Paradoctor (talk) 01:55, 1 August 2014 (UTC)[reply]

The sentence in the intro which begins

"But unlike as the English construction may, the conditional statement "p→q" does not"

does not parse

It is trivial to prove the following using the rules of natural deduction:

${\displaystyle A\land B\implies [A\implies B]}$ (Truth table, line 1)

${\displaystyle A\land \neg B\implies \neg [A\implies B]}$ (Truth table, line 2)

${\displaystyle \neg A\implies [A\implies B]}$ (Truth table, lines 3-4)

${\displaystyle [A\implies B]\iff \neg [A\land \neg B]}$ (Often given as the definition of material implication -- not required in the above derivation of the truth table.)

It makes me wonder why so many folks believe that material implication is somehow different from the usage of implications in natural language. What's wrong with: If pigs could fly, then I'd be the King France? They should understand that anything that is true or false will follow from a falsehood.

Danchristensen (talk) 03:23, 5 January 2018 (UTC)[reply]

The introduction is appallingly bad, it begins with reference to a currently non-existent diagram. Please would someone add an appropriate simple picture to show what this means ? And shouldn't the second paragraph should be the first ?

There is a lot of this article (and a lot of argument on this page) which is quite incomprehensible to the ordinary reader. Please would all WP editors concentrate on wording articles to inform and educate those who are not familiar with specialist subjects ? Darkman101 (talk) 23:52, 6 July 2018 (UTC)[reply]

...is obviously a false statement since p and q are taken to be independent. Even worse, it is bolded in the lead. The operator, in words, should really read "if p or q is true and p is false then q is also true". Not so catchy, but more accurate and that's how the Teller source puts it. Also borne out by the article's own Venn diagram. SpinningSpark 09:05, 22 April 2020 (UTC)[reply]

There are several different forms of notation used in this page. I see the ${\displaystyle \to }$, ${\displaystyle \rightarrow }$, and → Which one should it be? Upascal (talk) 18:49, 5 March 2022 (UTC)[reply]

I think it would be best to use one of the <math> versions for the sake of forward-compatibility, since future expansions might introduce content that can't really be typset without math mode. But beyond that I have no preference. Thanks for putting in the labour by the way! Botterweg14 (talk) 19:42, 5 March 2022 (UTC)[reply]