Talk:Material conditional/Archive 1

Problems with material implication

I was thinking that there should be a section about the "problems with material implication". Here's some text that I propose could be part of it:

There are several known issues concerning the standard truth-functional interpretaion of the material conditional. Such problems are sometimes referred to as the "paradoxes of material implication", though they are not quite paradoxes in the strict sense.

One such issue concerning material implication involves the truth-functional interpretation of the falsity of a conditional statement. Take, for example, the following conditional: If God exists, then the Earth is flat. If one were to think that this proposition is false (which seems reasonable), then, interpreted as a material conditional, the antecedent must be true and the consequent false—for that is the only case in which the conditional is false, according to the standard truth-functional interpretation of the material conditional. However, the antecedent is God exists. So, the standard interpretation seems to establish the existence of God from a simple false conditional in which the antecedent and the consequent are fairly unrelated. Of course, one way to respond to this would be to argue that the conditional in question is not a material conditional after all, but some other kind of conditional statement.

However, I don't know a great deal about this stuff. I thought some logic buffs might be able to create the section or help out. Any ideas? - Jaymay 18:43, 14 August 2006 (UTC)

This article is about the material conditional, as it is used in math and computer science. Some discussion of its relation to other types of conditionals is useful, but long digressions belong in the more philosophistical articles, of which there are many, as I'm sure you know. That can be handled via the requisite links in the <see also> section. Jon Awbrey 18:54, 14 August 2006 (UTC)

Jon, first, I apologize for changing the references format without discussion. I only figured that since there was no discussion going on already here that no one was really watching this article much anymore. However, by putting in the full URL to the reference that I added, I was not changing the references format, since there was no style on the article according to which "Eprint" was used. Thus, I believe you just changed the references style without discussion.

Second, I wasn't trying to be territorial by tagging this article with Wikiproject philosophy. I submit that you are being the more territorial by verbally tagging this article as restricted to math and computer science. I was trying to be pluralistic by adding philosophy-related stuff to this article, not by replacing anything with philosophy-related stuff.

Third, this article does not specify that it is about the material conditional only as it relates to math and computer science. If it is supposed to, then it should be titled "Material conditional (math and computer science)" and there should be a diambiguation article where we can distinguish a "Material conditional (philosophy)" article. There is a disambiguation article for Conditional, in which it says "Material conditional, in propositional calculus, or logical calculus in mathematics". The propositional calculus fits squarely into logic/philosophy. So, why would you say that it is restricted to "math and computer science"?

However, I think that it's a poor option anyway to create two separate articles. It's not a long article as it is. And, there is no reason that issues with the material conditional (philosophical or otherwise) should not be in this article. It need not be a "long digression" either, unless you think that any philosophically-related discussion is too long and a digression. Furthermore, while it's true that there are many philosophy articles on Wikipedia, there are none on the material conditional, except this one.

Fourth, cooperation is fun; hostility is lame. - Jaymay 22:03, 14 August 2006 (UTC)

I realize it's totally uncool to talk about one's expertise in WP, but my "commitment" here is shorter than it used to be, so maybe it will save some wasted breath to mention that I've been a student of logic in both mathematical veins and philosophical vains for 40 very odd years now, and so I'm quite familiar with all of the basic issues you mention here. But the article is titled what it is, and that is the kind of truth-functional implication that is used by mathematicians everywhere in almost (w)holy blessed ignorance of what some philosophers consider its "problematic" character. So the main aim of this article is to present the basics of material implication for the edification of those readers who came looking for that. Of course, it makes sense to make a hyper-side-long allusion to all those other issues, for which there are dedicated articles already on counterfactual conditionals, fuzzy logic, modal logic, relevance logic, and a host of others. That's the main thing. Will get to the other issues later. Jon Awbrey 01:06, 15 August 2006 (UTC)

Jon, no need to speak of your "expertise". I am well aware that the truth-functional interpretation is widely used and accepted as unproblematic. No doubt it is quite intuitive. Many philosophers recognize that. I think you misunderstood my point all along. I didnt' want to include issues with the material conditional because I thought that it has problems, in some serious sense. Fringe controversies are certainly not for an encyclopedia entry. However, issues that a large number of professionals on the subject discuss, including philosophers, are relevant to an encyclopedia entry.

However, it's really not a big deal. I just thought some people might be interested in expanding the article, since, I think, there are, from time to time, people browsing Wikipedia looking for issues surrounding material implication. But, I guess not. I'll leave it be. - Jaymay 03:58, 15 August 2006 (UTC)

Sorry if I am being brusque, but we have had some major mess-overs of related articles from both phil-logic and hard-ware folks, and since these are entry level articles my concern is not confusing initiates any more than they are likely to be already. But what I am saying about the problematique is that there are standard ways of coordinating groups of articles. In mathematical orbit, this is one of 16 on the binary connectives, and I worked a long time getting a consistent format for that group. By all means, add one or more brief sections on enrichment topics, perhaps using the {{main|...}} template under the subhead to link to the main articles on those topics. Many Regards, Jon Awbrey 04:44, 15 August 2006 (UTC)

I didn't realize you had collaborated and worked to get uniformity among various articles. I understand that you don't want all that work to be unecessarily tampered with. Besides, some issues are mentioned in the section on comparison with other conditionals. And I wasn't really planning on making the changes myself either. If anyone else what's to add info on this sort of thing, then they can.

By the way, maybe you should post something on the Talk page here warning that the article has pretty much been deemed satisfactory and that any changes should be discussed on the Talk page first. I know we're all supposed to do that first no matter what, but when the Talk page is blank or little is on it, one tends to think that no one is really working on it. - Jaymay 19:22, 15 August 2006 (UTC)

I am, of course, only expressing my personal opinions and preferences. And a note like that would probably be jes askin' fer trouble. Ha! Jon Awbrey 21:04, 15 August 2006 (UTC)

Is this right?

Subset?

Is the material conditional synonymous with the subset relation? If it is we should strongly note it or even consider merging the articles. Fresheneesz 21:44, 7 January 2007 (UTC)

Although it may be expressed in terms of sets, it is not synonymous. Gregbard 08:55, 28 June 2007 (UTC)

The material conditional $(A\rightarrow B)\Leftrightarrow (\neg A\lor B)$ is related to the entailment relation $A\Rightarrow B$ in the same way,

as the set operation $A^{c}\cup B$ is related to the subset relation $A\subseteq B$ . Lipedia (talk) 18:41, 27 July 2009 (UTC)

Proposed project

I have been working on all of the logical operators recently. I would like to see a consistent format for them. There is a wikiproject proposal for this at: Wikipedia:WikiProject_Council/Proposals#Logical_Operators. Also see Talk:Logical connective.

I would like to see the logical, grammatical, mathematical, and computer science applications of all of the operators on the single page for each of those concepts.

Gregbard 08:55, 28 June 2007 (UTC)

Material conditional vs. entailment

"The material conditional is not to be confused with the entailment relation ⊨ "

I was told by a philosophy professor they are the same. I'm a graduate student teaching logic for the first time, and I want to get it right for my students. —Preceding unsigned comment added by 24.208.177.188 (talk) 17:38, 22 April 2008 (UTC)

In Methods of Logic, Quine goes on a rant for a page or two about how we should not confuse the material conditional with implication. Truthfully, I don't understand how they're meaningfully different, but Quine says they are! Djk3 (talk) 21:10, 22 April 2008 (UTC)

They're much different. One is a relation (entailment) denoted in the metalanguage and the other is a connective (material implication) denoted in the object language. The connective takes two arguments, both of which are object-language propositions. The other takes two arguments, one of which is a set of object-language propositions and the other of which is an object-language proposition (or sometimes a set of them). For individual propositions A and B, A entails B iff the material implication from A to B is valid. (This assumes classical logic.) But it may be that for some model M, A materially implies B even though this is not true of every model (i.e. not valid), and hence A doesn't entail B. In a sense, entailment is not relative to a model, since it quantifies over all of them, while material implication is. It makes no sense to say that a material implication is true irrespective of a model unless all one means is that it is valid (and hence an entailment). Nortexoid (talk) 12:00, 23 April 2008 (UTC)

This needs to be explained in the article. Currently it asserts that they are different without giving a clue as to in what way they differ, at least not before it slips into impenetrable formal jargon. The article on entailment has the same problem.

Also, what on Earth could "the entailment relation (which is used here as a name for itself)" possibly mean? Nathanielvirgo (talk) 13:34, 26 July 2009 (UTC)

Done. The entailment tells, that the material conditional is always true, respectively never false. You need a quantifier to express the entailment.

Anyone who thinks they are the same, should ask himself if the statement $A\subseteq B$ and the set $A^{c}\cup B$ are also the same for him. Lipedia (talk) 18:56, 27 July 2009 (UTC)

The explanation of the difference between material and logical implication is difficult to understand. The problem is that the explanation of entailment is too abstract. 72.83.207.14 (talk) 14:53, 28 August 2009 (UTC)

I am addressing the comment of the anon, who tagged the entailment article, which I have reversed; it may be useful to include Peter Marcuse's comment that things might be 'true for the wrong reason'. In other words, material implication can be 'true for the wrong reason' (see the Venn diagram). BTW Peter Marcuse used to have his own wiki page which was deleted for non-notability; I disagree with that assessment. --Ancheta Wis (talk) 19:48, 28 August 2009 (UTC)

If one examines the proof in entailment#Discussion (see the Venn diagram) then one sees that material and logical implication differ in the following way:

• in logical implication, given that (A $\Rightarrow$ B), then "A → B", and also "A without B is never the case".
• in material implication, given that (A → B), then "there can be cases when B is true but A is not".

--Ancheta Wis (talk) 21:42, 29 August 2009 (UTC)

Why is it that "logical" implication, aka entailment, is not listed as a "logical" connective while material implication is? This seems very counter intuitive. Bbippy (talk) 23:23, 30 October 2011 (UTC)

Implication in natural language

I reverted a change because despite the assertion in the edit summary that the change is correct, it's not. This portion of the article is describing how implication differs between natural language and logic. --Doradus (talk) 19:45, 21 December 2008 (UTC)

I've reverted it again to its original state. We need to resolve this issue before changing the article. --Doradus (talk) 00:22, 25 January 2009 (UTC)

Summary

The summary for this article does not meet wikipedia guidelines for being understandable by the broadest possible audience. —Preceding unsigned comment added by 74.202.89.125 (talk) 19:47, 25 February 2009 (UTC)

Amen to that! It's not even understandable to this guy with a total of 4 degrees in mathematics and computer science! It comes off like so many philosophers and logicians talking to each other, and that's not what introductory paragraphs are for. Further down the article says "People often confuse material implication with logical implication," but offers no help in distinguishing between them.

Is it possible to add a real-world, everyday example or two, or at least an analogy or two?—PaulTanenbaum (talk) 22:55, 26 February 2010 (UTC)

Commutativity?

Uhm, the commutativity stated in this article confuses me, despite knowing a little math. I only know commutativity as "being allowed to swap arguments", that is, A -> B = B -> A (in this particular case. This is obviously wrong, of course, consider A=1, B=0). However, even with some calculations on paper, I cannot see how this statement helps with commutativity. Can someone explain this to me? --Tetha (talk) 10:56, 8 April 2009 (UTC)

Thanks for spotting this. This is definitely a problem, and I don't know how to fix it because I don't know a name for this property. Calling it commutativity only makes sense if you think about it as follows: Define $x\circ _{C}y$ as $(x\rightarrow (y\rightarrow C))$. Then $x\circ _{C}y\equiv y\circ _{C}x$. This is clearly too far-fetched. --Hans Adler (talk) 13:04, 8 April 2009 (UTC)

Ah. After some thinking, your post, a bit of implementation and such, I think I understand what this formula wants to say: If both A and B are antecedents for a consequence C, then it does not matter in what order the set of antecedents (in this case, A and B) are investigated. I think, calling it "commutative antecedents" or "order-invariance of multiple antecedents" would help. --Tetha (talk) 13:51, 8 April 2009 (UTC)

Contradictory example?

In "Example" it says:

Finally, if the first proposition proves false (Chris is standing at the Sydney Opera House), and the second proposition also proves false (Chris is in Sydney, Australia) then it is a false statement.

But in the truth table it is shown that F → F gives T.

Isn't there a contradiction? Dart evader (talk) 20:32, 23 June 2009 (UTC)

Does it? If Chris is not standing on the Eiffel tower then he is not in Paris. What happens if Chris is standing in the Louvre? Does that mean that he's not in Paris? - Tbsdy lives (formerly Ta bu shi da yu) ^talk 05:48, 25 June 2009 (UTC) Let me look at the example I gave and get back to you. I'll revert for now. :( Tbsdy lives (formerly Ta bu shi da yu) ^talk 05:51, 25 June 2009 (UTC)

OK, so the example is "If Chris is standing on the Eiffel tower then he is in Paris."

I actually think now that I've made a boo-boo. However... if both propositions were false, then Chris is not standing on the Eiffel tower and Chris is not in Paris. Thus it would make the statement true. Wouldn't it? - Tbsdy lives (formerly Ta bu shi da yu) ^talk 05:56, 25 June 2009 (UTC)

I suppose it would. Dart evader (talk) 14:46, 25 June 2009 (UTC)

Unintelligible!

(Yes I have put a second {{technical}} tag here.) I can be as abstract as the next guy, but this article is practically opaque. As the comments peppered throughout this talk page indicate, the article needs big help! Grad students in logic have heartburn with it. Please, somebody, at least create an introductory paragraph that gives a notion of the definition that laymen can fathom, rather than just a grab-bag of properties stitched together rather unhelpfully.—PaulTanenbaum (talk) 23:04, 26 February 2010 (UTC)

If that tag belongs anywhere, it would be on the article. -- 202.124.75.219 (talk) 01:47, 30 April 2010 (UTC)

This article effectively assumes that the reader knows as much or more about its subject matter than that which the article actually explains. In that respect it is typical of the babel of technical articles in Wikipedia. It uses letters to stand for sentences with hidden logical structure, which seems to be part of how the article ends up defining logical implication as a claim that a material conditional is "always true" (another phrase bound to leave the general reader shaking his or her head). Keep it simple. A logical or formal implication is automatically true, for instance pq logically implies p; for instance, if John's here and John's smiling, then John's smiling; if the logical structure is adequately exposed, a logically true conditional can be proven without investigation of its empirical claims (e.g. is John actually smiling?). If a claim of a logical implication turns out false, then it wasn't a logical implication. As for relevance logics and all the rest, save them for later mention, start with the ABCs. Well, maybe I'll try my hand at it when I have some time. The Tetrast (talk) 15:33, 11 April 2010 (UTC).

And this is why you SHOULDN'T be editing this article. Nortexoid (talk)

Well, maybe I should say instead that a sentence whose schema (e.g. "p → p") is valid is true in all universes, or some such, and that (logical) implication is the validity of the conditional, and is not itself called true, etc. But I stand by that which I meant, that a logical implication (at least in Quine) is not a claim of a conditional's having a valid schema, and that the article says otherwise, and neglects the common practice (according to Methods of Logic) in 1st-order logic of dealing with schemata that lay all relevant logical structure bare. On the other hand, I don't know what your complaint about my informal comment is. I also stand by my statement that the article assumes the reader's knowledge of that which is being explained. It's as if it were written by students for their professor, rather than by logicians for the general reader. Eventually, if I have time, I will do some cleanup (unless your or others do it first), and I'll source it with footnotes. The Tetrast (talk) 15:40, 12 April 2010 (UTC).

I think I see the problem here. It seems that maybe you have in mind logical implication and logical equivalence in terms of set theory or simulated set theory. I've been thinking just in terms of ordinary 1st-order logic, where (logical) implication is the validity of the conditional, and a sentence with the consistent but invalid schema $\forall$(G→H) is not taken announcing a logical implication G$\Rightarrow$H. Such terminological tangles need to be addressed without getting tangled in them for the general reader. The Tetrast (talk) 21:57, 12 April 2010 (UTC).

Duplication of content

Please see Talk:Entailment#Duplication of content. - dcljr (talk) 19:51, 15 January 2011 (UTC)

Connective versus implication

The explanation here is really confused. In contrast, here is a very simple explanation. Could someone explain how the "example" is supposed to help in understanding this distinction? Vesal (talk) 21:36, 15 January 2011 (UTC)

That seems to have originated from an earlier version of this article. I am going to very aggressively remove a lot of material which I consider nonsensical. Take the example:

$A\subseteq B\Leftrightarrow (x\in A\Rightarrow x\in B)\Leftrightarrow \forall {x}(x\in A\rightarrow x\in B)$

Is this a joke? What is the meaning of the unbound "x"?? Somewhere else I just eliminated this line, but now I see that example is supposed to show that "The relation $\Rightarrow$ can be expressed by $\rightarrow$ and the universal quantifier $\forall$." I consider this such pure and blatant nonsense that I do not want to see this back anywhere on Wikipedia unless it can be attributed to a reliable source. Vesal (talk) 21:45, 19 January 2011 (UTC)

Made clearer the differences between material implication and strict implication

I have made clear from the start there are fundamental differences between the material conditional and the strict conditional. I have also included detailed information on this difference in a new section. I have cited respectable references, with page numbers. Hopefully this addition should make the distinction clearer and easier for people new to the topic to understand. Considering the article lacked this helpful information, the addition should be greatly welcomed. Hanlon1755 (talk) 07:55, 19 December 2011 (UTC)

Reverted per WP:BRD; see Talk:Strict conditional for ongoing discussion (also Talk:Conditional statement (logic)… WP:V/WP:SYNTH/WP:NPOV).—Machine Elf ¹⁷³⁵ 16:33, 20 December 2011 (UTC)

Article inaccurate, proposal to change several parts

I dispute several parts of this article. I propose to modify the article, such that it agrees with the facts about strict conditionals. See Talk:Strict conditional for an overview of this overall discussion. Not all material conditionals can be put in "if-then" form, as this article currently suggests. "If-then" form is a type of expression reserved for only strict conditionals, not necessarily material conditionals. Furthermore, it is disputed whether or not a "material conditional" is even a type of conditional at all. What is instead the case is that all conditional statements (those that can be written in "if-then" form) are strict conditionals, which are not necessarily material conditionals. This article lacks pretty much any citations, never mind exact page numbers where this material can be found. Furthermore, it has been my expierence that some of the respected, notable, published literature on this topic is in error. Just because somebody said something about material conditionals is true doesn't necessarily mean it actually is, whether it was an "expert" or not. I want the part that material conditionals can be written in "if-then" form taken out of this article, because it isn't true. And if an entire section of this article can be about "paradoxes" or apparent "misconceptions," I propose to add to this article, at least, a sentence or two distinguishing between material conditonals and strict conditionals, and how the misconception that all material conditionals can be written in "if-then" form is not actually true. The article as currently written is very misleading and I myself am horribly a victim of it. Please aid me in these efforts to modify this article. Hanlon1755 (talk) 07:06, 23 December 2011 (UTC)

You need evidence that not all material conditionals can be written in if-then form — this means from reliable sources, not just your imagination. — Arthur Rubin (talk) 08:03, 23 December 2011 (UTC)

Every sentence I added was cited. It was either explicitly stated or was a logical consequence of what was explicitly stated in several sources. This includes the position that not all material conditionals can be written in "if-then" form. And all my sources were reliable sources. I still recommend modifying the article to improve its accuracy. Hanlon1755 (talk) 08:47, 23 December 2011 (UTC)

In mathematics, it certainly is true that every statement $A\to B$ can be stated as "A implies B" or "if A then B". (See, I just did.) The issue that some philosophers worry about is that "if A then B" could mean different things depending on the intention of the speaker. But one thing that phrase can mean is a material conditional, and every material conditional can be trivially rephrased in that way. — Carl (CBM · talk) 14:55, 23 December 2011 (UTC)

One of the sources used by Hanon1755 included Barwise who is no dummy, but for my part I don't have any of the source texts that Hanon1755 referred to. This article is about the material conditional, and only tangentially about the linguistic form "if...then...". I think that two questions are raised here:
1. The practical question is whether, in the study and application of logic, one commonly utters it "if p then q" - the general reader, coming here to learn about the material conditional in logic, deserves to be told the answer up front, i.e. in the lead, and the answer is an affirmative which we probably can source to Quine and which I seriously doubt is contradicted by any of Hanon1755's cited sources. The lead is not the place for a revisionist silence about common practice in formal, symbolic, or mathematical logic.
2. The linguistic question is, how well does "if p then q" represent "p→q"? - on which the article pretty much admits that "if p then q", as an utterance in logic, is just an imperfect convenience for "p→q". The article notes that the respective negations in everyday English do not seem equivalent to each other; i.e., the article alerts the reader to be aware of the divergence of formal logic's "it is false that if p then q" from the same form in everyday English. It would be nice if logicians pushed for use of some synthetic phrase or word like "nand not" or "nandn't" for "→" but, with few exceptions (Quine's "excl-or"), logicians resist synthetic words for connectives (whereas mathematicians seem happy to coin words for their objects and relations). The Tetrast (talk) 17:25, 23 December 2011 (UTC)