Talk:If and only if

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This page is unintelligible to anyone who doesn't already know what it is trying to say. Simplify and make accessible. — Preceding unsigned comment added by 60.228.49.152 (talk) 21:28, 30 January 2013 (UTC)

I wholeheartedly agree. I am a CS major, and have worked as a software engineer for 14 years, and even I can't understand it. — Preceding unsigned comment added by 140.168.79.1 (talk) 06:12, 22 October 2013 (UTC)

Initial Discussion[edit]

Someone should add the triple bar to the standard symbols for "iff." But I don't know how to do so.


  1. Mary will eat pudding if it is custard.

Does this sentance need wikilinks really? Does pudding and Custard have anything at all to do with this article? I think not personally. -- 82.3.32.75 13:32, 21 Feb 2005 (UTC)


The equivalent of 'P is necessary and sufficient for Q' would be 'Q iff P' (not 'P iff Q') would it not? I've also wikilinked necessary and sufficient. - Ledge 11:18, 18 Aug 2003 (UTC)

well... since it's symmetric, it doesn't really matter that much does it? -- Tarquin 11:38, 18 Aug 2003 (UTC)

Gosh, so it is. How have I lived so long without realising that?

it should be symmetric, but the example below, which should show the difference between the equivalence and iff is not symmetric - actually the second part of the sentence (it's custart) is not even a sentence! This example is basicly wrong and it seems that the discusion of the mentioned difference is some (maybe polemic) lingual issue, but no logical nor mathematical, which (in this case) is the same. (Jester (not yet a user) 2:50, 9 Sep 2004)

"actually the second part of the sentence (it's custart) is not even a sentence!" It's == It is == subject(it) verb(is). Custard becomes a descriptive. Just thought I'd mention that.


Ark: Yes, a priest is a bachelor, at least as I understand the term. The Oxford English Dictionary says only that the man must be of marriageable age, which is arguably included in the term "man". Every American dictionary that I can find on the Net gives our original definition, possibly adding that age is irrelevant. If you have support for your definition, then I'd like to hear it; otherwise, I suggest returning the definition to what it was. OTOH, if controversy remains, we might look for a different definition to use. — Toby Bartels, Tuesday, June 18, 2002

The priest-bachelor statement is is a prime example of Imprecise language... ;-) Tarquin, Tuesday, June 18, 2002

well, to my naive surprise, this is the necessary and sufficient article. But it doesn't go into the terms necessary and sufficient...or am I missing something? Kingturtle 02:35 Apr 18, 2003 (UTC)

Well, I'm not sure. This is the iff article. It isn't clear it should go into the terms necessary and sufficient. But at the very least, necessary and sufficient are normally used in the sense of necessary condition and sufficient condition--I take it that's what you want. But the conjunction of those two is logical equivalence, which is not the same as iff (as explained in the article).

There was some confusing equivocation between use and mention here--between the biconditional, which is a connective and logical equivalence, which is a relation. I tried to clear it up, but it's a knotty topic.

I'm not sure the current version doesn't "clear it up" too much in the opposite direction. There is a distinction sometimes, but often there is not in fact a distinction, and many formal logics use a single symbol to indicate both, not the two separate symbols (single- and double-barred <->) used in this article. Delirium 18:55 12 Jun 2003 (UTC)

currently, Necessary and sufficient redirects to Iff. Kingturtle 02:46 Apr 18, 2003 (UTC)

I realized that, a bit later. I've written a brief article on it and eliminated the redirection. hope its helpful

I'm not sure I like the "iff is not equivalence" example:

Mary will eat pudding today if and only if it's custard.

I think this actually is a case of equivalence, that is being muddled by the phrasing. What we're saying is "(Mary will eat pudding today) iff (The pudding today is a custard)". Thus the logical statements "Mary will eat pudding today" and "The pudding today is a custard" are in fact equivalent: they have identical truth tables. So I still don't see the discrepancy. --Delirium 22:58 12 Jul 2003 (UTC)


I think you're right. It's bringing the meaning of the words into the matter, which is wrong -- Tarquin 10:19 13 Jul 2003 (UTC)

Regarding "if/iff" convention for defs:

I've reinserted the comment about "if" being used conventionally in math defs. I'm sorry, I've read a lot of math books, and this is a common convention. Many definitions use the terminology "if", in the sense of "If P(X), then X is called blah" or "X is said to be blah if P(X)", yet not every definition uses "iff", and all definitions are intended to be "iff", because that's what definitions are. (To counter your remark, definitions are not intended to assert equivalencies; an equivalence is usually meant to indicate a statement saying two things imply each other that has to be PROVED...definitions aren't proved, they're declared, so it doesn't make sense to say e.g. "'R is an integral domain' is equivalent to 'R is a commutative ring with identity'" because these statements aren't "equivalent" in the ordinary sense of the term, one does not PROVE they're equivalent, that simply IS the definition of an integral domain. Here are several cases where the "if" convention is used in the wikipedia itself...

  • "A prime p is called primorial or prime-factorial if it has the form p = Π(n) ± 1 for some number n" (from prime number)
  • "If a divides b and b divides a, then we say a and b are associated elements. a and b are associated if and only if there exists a unit u such that au = b." (from integral domain...notice, the first use of the word is in the sense of a definition, hence only "if" is used (although "iff" would be correct as well), but the second IS an actual theorem (result) because the equivalent condition requires proof. So, for the second statement, the meaning would change if "iff" were replaced by "if", although for the first statement it doesn't matter.
  • "In complex analysis, a function is called entire if it is defined on the whole complex plane and is holomorphic everywhere" (from entire function).

The list could go on. Revolver

There is a case for using "iff" as a definitional. There may be several different ways of defining an entity, each of which is equivalent. For example: "f is surjective if its image equals the codomain" and "f is surjective if for all t in the codomain of f there exists an element in its domain whose image is t". You can then establish that "the image of f equals the codomain of f iff for all t in the codomain of f there exists an element in its domain whose image is t". In such circumstances, to allay confusion as to which implications are in fact equivalences, it may well be worth using "iff" in the definition. In my own work I routinely use "iff" because it's rock-solidly clear. Unfortunately some of the more tedious and clothy of my intellectual mentors tended to disagree to the point of marking me down for such outrageous abuse of notation. --Matt Westwood 08:22, 20 October 2011 (UTC)

Im confused by the

  • A person is a bachelor iff that person is an unmarried but marriagable man.

example -- there could be unmarried but marriagable men (not only the priests mentioned above), for example widowers. I wouldn't think they are bachelors (are they?). If not, the (P iff Q) Q->P direction isn't true. And what about bachelor being also a term for an university diploma? Is "Tom did his B.A. well and is now a Bachelor" a correct English sentence? And what about a marriaged Tom that is a Bachelor in this sense? Would he destroy the iff above? -- till we *) 00:31, 26 Jan 2004 (UTC)


What is the pronounciation of the "iff"? Do I say "if" or "if and only if"?

I'd read "if and only if". I'm sure that's what my maths lecturers used to read, too. I guess you could say "eye eff eff". Saying "if" would be wrong. It's just a written shorthand, like using the three dots to mean therefore - you wouldn't read that as "dot dot dot". --JimmyTheWig 12:20, 31 May 2006 (UTC)
I've heard it said "ifffff..." (that is, with an extended "f" sound). However, I think the users were using it more for comic effect than anything else. "If and only if" is the sensible way to say it out loud. -- The Anome 10:35, 3 October 2007 (UTC)

Coinage of "iff" by Kelley / Halmos[edit]

The article says:

The abbreviation appeared in print for the first time in John Kelley's 1955 book General Topology.

However, the preface of the 1955 edition of General Topology says

In some cases where mathematical content requires "if and only if" and euphony demands something less I use Halmos' "iff".

which suggests that he did get it from Halmos. Now Kelley did know Halmost personally so it's possible that this was the first appearance of "iff" in print. But it seems more likely that Kelley saw it in some paper of Halmos'. I can't think of any way to pursue this any further, other than to ask Halmos. (Kelley died in 1999.) Does anyone have any other suggestions? -- Dominus 05:39, 10 May 2004 (UTC)

Possibly useful references[edit]


"Precisely if"[edit]

Does the phrase "precisely if" mean the same thing as iff? If so, it could be added to the article. Wmahan. 17:56, 2004 Aug 31 (UTC)

Yes; that is conventional usage among mathematicians (I don't know about philosophical logicians, though). Michael Hardy 20:55, 31 Aug 2004 (UTC)

Thanks. It appears to be used in logic as well (e.g. [3]), so I'll add it to the article. Wmahan. 06:34, 2004 Sep 1 (UTC)

I think the phrase "exactly when" is common also. -- Dominus 02:59, 2 Sep 2004 (UTC)

Orr?[edit]

I don't know about you, but I see "orr" and think of an imperative-logic "p' := q or r". Does anybody use "orr" for the exclusive disjunction rather than "xor"? --Damian Yerrick 08:23, 6 Sep 2004 (UTC)

I use whichever one. But I have to use "xor" in Matlab cause that is what it requires in its syntax. --GoOdCoNtEnT 01:11, 10 July 2006 (UTC)

Organization[edit]

I wrote in Talk:Mathematical jargon, in part:

Iff has two uses, imho. One is used in logic (and related fields, I suppose) to mean a binary function from a theory to a truth-value set
iff : Th x Th → {T,F}
and the other is used in arguments in any math paper or lecture. The meanings are the same, I think, but the uses are different. I think that Iff should be edited to reflect these two uses; right now it blends them. —msh210 17:03, 9 Nov 2004 (UTC)

I still think so; what do you all think?msh210 19:40, 15 Nov 2004 (UTC)

Done.msh210 18:57, 17 Nov 2004 (UTC)

"P iff Q" not equal to "P is necessary and sufficient for Q"[edit]

In my opinion, there is a little mistake in this article... I think it should be vice versa: "P iff Q" means "Q is neseccary and sufficient for P" instead of "P is necessary and sufficient for Q" isn't it?

Both are equally correct. -- Dominus 01:27, 6 Jun 2005 (UTC)
Yeah, although the suggested change does match up a little better with colloquial English usage ("P if Q" means "Q is sufficient for P", and "P only if Q" means "Q is necessary for P", so "P iff Q" means "Q is necessary and sufficient for P"). --Delirium 03:03, Jun 8, 2005 (UTC)
"P if Q" also means that P is necessary for Q, and "P only if Q" means that P is sufficient for Q. Thus, "P iff Q" means "P is necessary and sufficient for Q". I repeat, both are equally correct. -- Dominus 12:57, 8 Jun 2005 (UTC)
Perhaps you missed my phrase "with colloquial English usage". In colloquial English usage, "P if Q" does not mean "P is necessary for Q", even though this is a logical consequence. --Delirium 23:27, 6 January 2007 (UTC)
I disagree with your understanding of "colloquial English usage", but look forward to seeing your authoritative references, which I will expect around June of 2008. -- Dominus 22:02, 7 January 2007 (UTC)

Why is this page iff?[edit]

Wikipedia naming conventions states that the expanded form should be preferred unless a term is almost exclusively used as it's shorterned form (aka Scuba or Laser). So why is this page on Iff? — Ambush Commander(Talk) 21:48, 20 September 2005 (UTC)

ONLY IF instead of IFF, couldn't it be possible ?[edit]

IMHO, "ONLY IF" covers "IF", and "SUFFICIENT" includes "NECESSARY". Why doesn't one use "ONLY IF" instead of "IF AND ONLY IF", "SUFFICIENT" in the place of "NECESSARY AND SUFFICIENT" ? Seforadev 19:07, 21 November 2005 (UTC)

Convention: argue with the academics and scholars out there. I'm pretty sure there's a reason, but I don't really know (I just know that they usually use if and only if in texts). — Ambush Commander(Talk) 02:45, 22 November 2005 (UTC)
Sb said me the probable reason for those redundance was one needs REPETITION to emphasize the TWO clauses of the logical equivalence. Some other ones said NECESSARY CONDITION is for the 1st sense (=>), and SUFFICIENT CONDITION is for the 2nd (<=). I don't understand.Seforadev 02:54, 22 November 2005 (UTC)
I don't know if anyone still cares, but "only if"/"sufficient" are generally considered different from "iff"/"necessary and sufficient." For example, for natural numbers n and m, n divides m only if m is greater than n. However, the converse is not always true. n divides m is sufficent to show that m is greater than n, but it is not a necessary condition. Generally "P only if Q" can be stated "Q is a necessary condition of P" or "P => Q." "P if Q" can be stated "If P then Q," "Q is a sufficient condition for P," or "Q => P." Josh 19:02, 25 March 2006 (UTC)
Sorry to butt in (And this is 5 years later) but "P if Q" can be stated "If Q then P" but is not the same as "If P then Q". "(The grass will be wet tomorrow) if (it rains tonight)". Same as "if (it rains tonight) then (the grass will be wet tomorrow)". But not the same as "If (the grass is wet tomorrow) then (it must have rained in the night)" because the grass might be wet in the morning because of dew, for example, without it having rained (excuse the change of case and sense for linguistic accuracy). --Matt Westwood 08:31, 20 October 2011 (UTC)
I think it derives somehow from the formal English language. It is an agreed upon term among mathematicians and using "only if" would just cause confusion. --GoOdCoNtEnT 01:07, 10 July 2006 (UTC)

Absolutely! After some 20 years of pondering over this question, I suspect that mathematicians have only two (2) motives to use iff instead of only if. First, it is more chic to use an exotic expression; and second, they tend to regard an iff condition as having a mandatory character, for example: "an egg will get hard boiled iff it is cooked in boiling water for 5 or more minutes", meaning that it is mandatory to boil the egg for 5 or more minutes. This quality however, is entirely fictitious, as it is equivalent to say that an egg will get hard boiled only if it is cooked in boiling water for 5 or more minutes. Gosh, I hardly understand my own argument! Does someone else? --AVM 20:55, 22 July 2006 (UTC)

Absolutely — wrong! "Iff" means "if and only if", not "only if". — Arthur Rubin | (talk) 21:35, 22 July 2006 (UTC)
A only if B means only that A implies B. A if and only if B means that A implies B and B implies A. This is why there is a difference. "If and only if" applies only to those cases of A only if B where also B only if A. Example: a bird is a raven only if its feathers are black, but this does not exclude the possibilty of other birds with black feathers. Whereas, if a bird is a raven if its feathers are black means that any bird with black feathers is necessarily a raven. The combination of both, if and only if, would mean that every bird with black feathers is a raven, and that a bird is a raven only if its feathers are black. - Rainwarrior 07:50, 23 July 2006 (UTC)
But that's senseless - in fact NOT every bird with black feathers is a raven. So the "if and only if"-senseless-redundant-expression would mean, by your definition, that on one hand, every black bird is a raven (which is clearly wrong), AND, besides, on the other hand, a bird is only a raven when it has got black feathers.
You've missed the boat, buddy.
(1) A bird is a raven only if it is black.
(2) A bird is a raven if it is black.
(1) just says that all ravens are black. And (2) just says that all black birds are ravens. Now then:
(3) A bird is a raven if and only if (iff) it is black.
Again, (3) and (2) are clearly false. But they are certainly meaningful (think about it: if they were meaningless (viz., senseless), how would you know they were false?).
The biconditional can be understood. It's simply the conjunction of necessary and sufficient conditions.
(4) A triangle is equilateral only if it has three equal angles.
(5) A triangle is equiangular if its sides are of equal length.
(Both true, BTW.)
From (4) and (5) we get
(6) A triangle is equilateral iff it is equiangular.
Hopefully, you can see that (4),(5), and (6) don't "say" the same thing (in ::::answer to the charge of redundancy), nor are any of them "senseless".
So, reading this has convinced me that my current adviser is correct and we should just use symbolic notation to write implication when we can. But conventionally,
1. "If P then Q" or "Q if P" means P => Q.
2. "P only if Q" means P => Q
3. "P if and only if Q" means P <==> Q
One exception to rule one is the use of the "definitional" if in mathematics. "We say that a natural number n>1 is prime if it is divisible only by one and itself" really means that n is prime if and only if n > 1 and n is divisible only by one and itself. I think this is the only real exception unless someone is confused.142.151.143.163 04:09, 10 February 2007 (UTC)
Could be there a difference in the interpretation of the expression "only if" dependent on language? I am not a native speaker of English and for the expression:
(1) A bird is a raven only if it is black, I understand firstly that a black bird is a raven and secondly that a raven can be only black. The second part arises because the word "only" does not delete the expression "if it is". The word "only" is a constriction to the expression "if it is black" which prohibits the possibility to write for instance "red" instead of "black". Therefore I find that expression
"A bird is a raven only if it is black" is false as
"n divides m only if m is greater than n", for natural numbers n and m, is also false.

I think could be interesting to find out who was the first person in history, and in what language, using formally the expression "if and only if". My belief is that this expression comes from too much enthusiastic desire to emphasize the "only if" and not from the intention to do a logical conjunction between the conditionals "if" and "only if".--Amoralesm 04:11, 27 April 2007 (UTC)

"Could be there a difference in the interpretation of the expression "only if" dependent on language"" - yes, sort of, it's dependent on whether you are taking a literal linguistic interpretation, in which case "if and only if" is equivalent to "only if", or a historical mathematical interpretation, in which case "if and only if" has a conditional meaning different to its literal interpretation. Since most people will use a literal interpretation it becomes a confusing issue with two sides. It should probably be altered to reflect a stance in line with the probable linguistic interpretation.Harlequinn (talk) 11:25, 18 May 2013 (UTC)

Merge[edit]

The following discussion is closed. Please do not modify it. Subsequent comments should be made in a new section.
  • Seems reasonable. - Ekevu (talk) 17:19, 22 December 2005 (UTC)
  • Object. I think the use of iff is frequent enough across the Wikipedia that, without redirects being able to redirect to anchors (eg to Logical biconditional#If and only if), that it would be confusing for many readers. This article is long enough to support itself on its own; I think a merge is unnecessary. — OwenBlacker 17:47, 22 March 2006 (UTC)
  • Oppose merge. No need; above reasons. — goethean 17:50, 22 March 2006 (UTC)
  • Oppose merge. There is a distinction between the logical biconditional (<->) and the logical equivalence (<=>). It's a really big deal to some schools of logic (Quine, etc.), who regard their confounding on a (sub-)par with use-mention confusions. It's a deal, but not such a big deal to many in the math community, who tend to use (<=>) for both, but they have a different way of handling the distinction between assertion and contemplation that makes the symbol used less of a problem, and the fact that they save the light arrow (->) for function notation leads many to use the amphisbane arrow (<->) for a one-to-one correspondence. There is currently a mess of confusion about this in WP generally, that will eventually have to be sorted out, so I recommend keeping the articles at arm's length for the time being. Jon Awbrey 20:32, 22 March 2006 (UTC)
  • Oppose merge. It's true that logical biconditional uses iff, but iff has many applications outside of math, those of which logical biconditional doesn't have. For example, someone could say "I'll let you do that, but if and only if you do this favor for me first." The sentence wouldn't make sense if the person said: "I'll let you do that, but only if we use logical biconditional, and you do this favor for me first." To sum up my point: iff does not imply logical biconditional, although logical biconditional does imply iff. Thus, iff emcompasses too broad a meaning, and logical biconditional is a more specific thing; therefore they both deserve their own page. wickedspikes 01:00, 09 April 2006 (PST)
  • Oppose merge. A strong mention of (reference link to) the biconditional is warranted, but they aren't so indistinct that the biconditional does not deserve its own page. Rainwarrior 15:56, 19 April 2006 (UTC)
  • Comment If the articles are not merged, then the difference between iff and a logical biconditional needs to be explained in the articles. As they read now, I have a hard time seeing any difference. --PeR 07:49, 21 June 2006 (UTC)
  • Oppose merge -- agree with wickedspikes. Also, it makes more sense for any casual use of if to redirect here than to logical biconditional. See: its usage in Null set. -- AlanH (not signed in) July 18 2006
  • Oppose merge. There is a difference between iff and a logical biconditional -- iff ought to imply only a "necessary condition", but not a "sufficient condition". It is amusing how this subject, in contrast to other subjects in the field, is such a visible motive for controversy. Regards, AVM 21:20, 22 July 2006 (UTC)
Since this merge proposal has been open for two-and-a-half years without substantial support, I'm closing it iff T. :-) --tiny plastic Grey Knight 07:52, 18 June 2008 (UTC)
(Okay, I see apparently there was a stacked proposal from last year, but that's still a long time and there seemed no interest. Feel free to revert me if you disagree.) --tiny plastic Grey Knight 07:56, 18 June 2008 (UTC)

The above discussion is preserved as an archive. Please do not modify it. Subsequent comments should be made in a new section.


How it works in logic[edit]

Here is how we use double arrow ↔, i.e. iff, in logic:

1- (AB) is a shorthand symbol for [(AB) ∧ (BA)]
2- (A only if B) equals to (A → B)

Eric 06:38, 30 March 2006 (UTC)

No, (A only if B) equals (A → B); whatever follows the word "only if" is always the consequent, not the antecedent. --165.123.138.170 08:08, 10 April 2006 (UTC)
[update] You are absolutely right. It was a typos. (A → B) is obviously the correct form of (A only if B). Then I corrected my comment. Thank you for pointing it out.
Eric 22:07, 15 April 2006 (UTC)

suggested addition[edit]

I want to put in a brief note to include the formulation "just in case" which is commonly used in philosophy to mean "if and only if" even though the usual English meaning of "just in case" is "as a precaution against...", as in, e.g., "I took my umbrella just in case it started raining". This page redirects from "just in case" in the when you search for that phrase so I think it would be a useful addition. Davkal 22:22, 8 June 2006 (UTC)

A similar construction popular in mathematics is "exactly when", as in "n is the sum of two odd integers exactly when it is even". McKay 04:31, 22 June 2006 (UTC)

I have added bothDavkal 13:22, 23 June 2006 (UTC)

'In case' is used very rarely but still go ahead and add it and other synonyms for iff. --GoOdCoNtEnT 01:08, 10 July 2006 (UTC)

The difference between if and iff[edit]

If the pudding is a custard, then Madison will eat it. Does Madison eat ALL custard pudding?

Let p="pudding is a custard", q="Madison will eat it"
p→q ⇔ q ∨ (¬p) ⇔ ¬( (¬q) ∧ p )
Colloquially, IF p THEN qq OR NOT p ⇔ (NOT q AND p) is false

"If the pudding is a custard, then Madison will eat it ", hence "Madison won't eat it and it is custard" is false (she eats all custard pudding wherever they are, though she may eat non-custard ones too).

Corollary: Madison must be a very very fat lady, because she eats tons of custard pudding daily.

Proposal: Should we change the example to "If Madison eats pudding, then it is a custard" ? In this case, "it is not a custard and Madison eats it" is false: the only puddings she eats are pudding, but may exist some pudding that is not eaten by her.

Why custard? Personally, I don't know that many types of custard. I don't know how many people do. Why aren't we using fruits and vegetables as examples instead? Fruits and vegetables are a lot more globally familiar, I reckon it'd make a lot more sense. Cybersteel8 (talk) 23:43, 25 December 2008 (UTC)


Rjgodoy 16:21, 15 April 2007 (UTC)

Bachelor[edit]

Isn't a widower an "unmarried but marriageable man."? He may have previously been married, but the death of his wife means that he is now of the class "unmarried men". There is a legal difference (at the least in the eyes of the Church of England), however, between a bachelor and a widower, so the statement that "A person is a bachelor iff that person is an unmarried but marriageable man" is incorrect - the person is a bachelor ONLY IF that person is an unmarried but marriageable man. --El Pollo Diablo (Talk) 13:52, 21 January 2008 (UTC)

I looked in 3 dictionaries and got 3 different definitions of "bachelor". Therefore, we should change the example. McKay (talk) 10:50, 31 July 2008 (UTC)

contraposition/inverse[edit]

I shouldn't have reverted, but the anon edit here is subtly wrong, and the previous version is subtly correct. — Arthur Rubin (talk) 13:01, 29 July 2008 (UTC)

misconception here[edit]

It isn't relevant to the article, but rather to the contents of this talk page. What is with this misconception I'm seeing speckled with great frequency throughout this page, held by multiple posters (and undoubtedly by many many more who've never posted or thought of commenting here, as I would assume that the sample of the population who choose to write on the talk page of such an article would be more learned on the subject than average), that P only if Q is equivalent to PQ? Not only is it a misconception, it's exactly backwards! P only if Q is equivalent to if Q then P which is equivalent to QP. —Iamthedeus (talk) 03:57, 28 April 2009 (UTC)

In fact, P only if Q is equivalent to PQ because when one says P only if Q one means to say that P is true only in the case that Q is true, so P can't be true when Q is false (of course it can in the sense you can do it, but then the statement P only if Q would be false). So the interpretation "disallowed" by P only if Q is that which has P true and Q false. P can only be true when Q is as well. This is exactly what is accomplished by PQ; it is false for, as we say in logic, the interpretation where P is true and Q is false. P if Q of course means if Q then P which is QP. Agreed? – Alex Perrone (boethian), Nov 3 2010

can vs. will[edit]

Pfhorrest: you reverted a minor edit I made to the Madison-custard-pudding example. It was:

"2. Only if the pudding is a custard, will Madison eat pudding." I changed it to:
"2. Only if the pudding is a custard, can Madison eat it."

The reason you cited was "example is about her willingness to eat pudding, not her ability to do so." The reason I changed it was because, phrased as it was originally, it seemed to imply that If pudding is custard, then Madison will eat it. Of course this is not the case—it means rather that she cannot eat pudding unless it is of the custard variety, but even then, she isn't obliged to eat it if it is. The point is that, if she is to keep with the conditions set out by the statement (which is implicitly assumed by making the statement in the first place), she can eat the pudding only if it happens to be a custard.

So of course whether she chooses in the end to eat a given pudding, if it is custard, is a matter of her willingness. But that wasn't what the statement was talking about. It was saying that, if it is not a custard, then she is obliged not to eat. Hence "Only if the pudding is a custard, can Madison [even consider] eat[ing] it." —Iamthedeus (talk) 08:57, 30 April 2009 (UTC)

Hi Iamthedeus. I gathered that that was your intention, and it's good that you bring this up because I had meant to open a discussion here and then forgot about it by the time I had the time.
My concern with the usage of "can" is that we are discussing only material implication here, not strict implication (entailment) or anything involving modal logic. We don't want to say that it is possible for Madison to eat pudding only on the condition that said pudding is custard; we just want to say that Madison will, as a contingent matter of fact, eat pudding only on the condition that said pudding is a custard. In other words, given P = "The pudding is a custard" and Q = "Madison eats the pudding", we just want to say "P is implied by Q", not "P is implied by \DiamondQ".
Your usage of "obliged" in your message above seems to indicate that you might mean "can" in the sense of "may", indicating permission rather than possibility; but again, we're talking about simple truth-functional logic here, not deontic logic. We don't want to say "P is implied by \DiamondQ" for a deontic interpretation of the diamond operator either.
Strictly speaking, the use of "only if" in this sentence communicates the point you're trying to make; that all conditions where Madison eats the pudding are also conditions where the pudding is a custard. However, since the point of these examples is to clarify that that is what we mean by "only if", we can't just rely on the reader already knowing what we are trying to teach them.
I'm going to experiment around with some other, probably more verbose, ways of phrasing those examples. Let me know what you think of them here.
--Pfhorrest (talk) 09:26, 30 April 2009 (UTC)
I'm not speaking modally; I am discussing only material implication.
Respectfully, I would suggest that perhaps you do not entirely appreciate what only-if really says (not much). Keep in mind that "only if P, Q" is directly derivative of, and better stated as "if ¬P, ¬Q". The latter tells us, amongst other things, that if the ¬P term is false, then the statement can give you no information at all on the ¬Q term. This same result is what is received every time the first term in an only-if statement is true. When any given "only if P, Q" obtains (i.e. P is true), the information we are given about Q is nothing but an admission that the statement cannot tell us anything. So it is this that I mean when I say she "can", but is "not obliged to" eat the pudding. It is an epistemic possibility I speak of. We just don't know, and have no way of knowing, what she will do.
This being said, I feel that "Only if the pudding is a custard, can Madison eat it" truly expresses this nature of the only-if conditional. However, by no means do I not appreciate the stickiness of the language involved in expressing this sentence. I spent quite a while considering it until I came to what I felt was a satisfactory way of writing it. After looking at your additions, I find that the second expression of (2), "Madison will eat the pudding only if it is a custard", is pretty good—I definitely prefer it over "Only if the pudding is a custard, will Madison eat it". Nonetheless I still feel that the element of possibility that is present in my version, and lacking in the other two, is important to the meaning. What do you think?
Iamthedeus (talk) 12:38, 30 April 2009 (UTC)
I agree that the element of possibility, as you say, is present in your version, and lacking in others; but it is precisely that element of possibility which I am objecting to. That is what I meant when referring to modal logic earlier; the sentence itself is not saying anything at all about possibility, necessity, impossibility or contingency, in either alethic or epistemic senses. I follow what you mean about the epistemic implications of the sentences: from "only if P, Q" or "Q only if P", we can be certain (i.e. it is epistemically necessary) that on all conditions where "Q" is true, "P" is also true, and it it epistemically possible (we are not certain that it is false) that "Q" is false even if "P" is true. However, the preceding is a statement about the sentences "only if P, Q" and "Q only if P"; those sentences themselves do not say anything about necessity or possibility, and inserting modal language like "can" changes their meaning.
You wrote The latter tells us, amongst other things, that if the ¬P term is false, then the statement can give you no information at all on the ¬Q term. I would argue, on the above lines, that the sentence itself tells us no such thing per se, but rather, that such is true of it. It is true that we cannot deduce the truth-value of "¬Q" from a known false truth-value of "¬P"; however, "¬P→¬Q" does not itself say anything about what can be deduced from what; it simply declares a particular state of affairs to be the case.
As I suspect you are well aware, implications like these can be rendered into equivalent simpler formulae involving only conjunction, disjunction and negation (∧, ∨, & ¬; AND, OR, & NOT). "P←Q" and "Q→P" and "¬P→¬Q" are all equivalent to "P∨¬Q" and "¬(¬P∧Q)". In other words, "P←Q" (etc) is true if at least one of the following conditions is met: "P" is true, or "Q" is false. "P←Q" simply declares that "P is the case or Q is not". It is true, given knowledge of that conditional, that we can deduce certain things and not others about "P" and "Q" and their negations from each other; but the sentence itself isn't saying anything about that. If we insert modal language into the sentence, saying something like "only if P is so, can Q be so", then when rendered into a non-conditional formulation, we would get something like "P∨¬\DiamondQ" ("P or not-possibly-Q") rather than simply "P∨¬Q" ("P or not-Q").
Anyway, all that said, I think the use of epistemic modal language could be a good way to describe what the "only if" operator means, but such language should go into the analysis section below the example sentences and not in the sentences themselves. As you probably noticed, I rearranged the sentences and their analyses so that each analysis follows each sentence, to better facilitate things like that.
--Pfhorrest (talk) 06:54, 1 May 2009 (UTC)
Pfhorrest, I understand and appreciate your argument. I think that the source of our problem is the English language. The thing about "only if" is that, as it is taken to understand in English, as opposed to in logic, it actually means "if and only if". When I look at "Only if the pudding is a custard, will Madison eat it", and think of it solely as a normal English sentence, the natural meaning of it seems to be:
If the pudding is a custard, Madison will eat it.
If the pudding is not a custard, Madison will not eat it.
I think the reason for this is that the term "only if" is misleading. The fact that "if" is part of the term automatically indicates to English speakers that an if..then statement is underway, effectively conflating "if" and "only if" in the logical sense (hence why the term is taken to mean "if and only if"). A sentence in the form of "only if P, Q" indicates to English speakers, as I said above, two things: "if P, Q", because of the presence of "if", and "if ¬P, ¬Q", because of the presence of "only".
Therefore, since we are expressing the examples in English, I think that they should actually make sense as English sentences, as opposed to logic sentences. We cannot presuppose that readers understand what "only if" really means, since the very thing we are trying to do is explain that distinction. However, if the example sentence were to be as I proposed, I think that probably the first thing the description ought to do would be to stress the difference between "only if" and "if and only if" in a logical context.
Because the English speaker reads "only if P, Q" as "if P, Q, and if ¬P, ¬Q" (which is false), if we were to put it in slightly different terms, such as "only if P, ◊Q", then they would read this as "if P, ◊Q, and if ¬P, ¬◊Q" (which is true).
I do understand your objection that the modality I've been referring to is not expressed by the sentence QP, but rather is a true fact about it. My point is that expressing it as "Only if the pudding is a custard, will Madison eat it", although probably a better expression of the logical idea involved, inadvertently confuses English speakers familiar with "only if" as an English term. However, although it may not be the most technically accurate way to express the idea, "Only if the pudding is a custard, can Madison eat it" nonetheless overcomes the English speaker's misunderstanding and gets across important aspect of the logical relation and furthermore does not mislead them. Despite your objections to this way of saying it, any English speaker who reads it will immediately grasp the fundamental nature of the relationship between the two propositions. I think that this is the best way of moving forward, by giving an intuitive example, and then refining that notion with an explanation of the technical definition—rather than beginning with a technically correct, but misleading example, and then having to spend the rest of the explanation trying to fix the misconception created.
However, if you can figure out a way to express "only if P, Q" as an English sentence that fixes both problems (viz. the inaccurate modality, and English speakers' natural treatment of "only if" as "if and only if"), by all means we should change it to that. It is merely that, as yet, I have not found a way of expressing it that does this, and so I'm merely advocating the one of our current contenders whose flaw is (I believe) less severe and less detrimental to comprehension.
Iamthedeus (talk) 04:30, 2 May 2009 (UTC)
Ahh, I think I see where source of our disagreement is. I am looking at that section of this article as giving examples of "if", "only if", and "if and only if" operations in technical, logical, philosophical usage - not just colloquial English - and then subsequently explaining what those operations mean in more detail, using more colloquial English to do so. Basically, we're saying something along the lines of "In sentence (2), 'Only if the pudding is a custard, Madison will eat it', the phrase 'only if' means...". So it is very important for the example sentences to be technically precise, more so than being colloquially intuitive, because the following paragraph after each example sentence explains in more colloquial language what the sentence technically means. That was why I suggested that epistemic modal language like "can" could be quite useful in the analysis, but doesn't belong in the sentence itself. --Pfhorrest (talk) 07:53, 2 May 2009 (UTC)
So essentially it comes down to whether it would be better to start loose and intuitive, and then subsequently refine the idea towards accuracy, or to start accurate but unintuitive, and then subsequently reconcile the idea with one's intuitions, towards comprehensibility. Obviously there's no non-arbitrary way of deciding this, and although I still think the former would be best, I feel nonetheless that you've made a solid case for the latter, and you've convinced me that maybe it's not so bad as I thought it was. That being said, I won't contest it any further. Thank you for hearing my side.
Iamthedeus (talk) 09:26, 2 May 2009 (UTC)

Example on (P and Q) and R = P and (Q and R)[edit]

It's just a restatement of the definition of the associativity of "and"! 118.90.6.70 (talk) 10:10, 3 August 2009 (UTC)

Why this article needs expert attention[edit]

If and only if and Logical biconditional both begin with a sentence that identifies the topic as a logical connective between statements, which turns out to be reducible to a truth table (obviously, the same table). This is inconsistent with the result of the #Merge discussion above, where the consensus seemed to be that the articles have different topics. Melchoir (talk) 08:59, 20 February 2010 (UTC)

My impression is that even the people who wanted the articles to be separate agree that both \Leftrightarrow and \leftrightarrow are binary connectives between sentences. They just have different meanings to certain people. The clearest explanation of the difference that I see is in a different article, Material_conditional#Difference_between_material_and_logical_implication.
The people who maintain a distinction say that:
  • A \leftrightarrow B is a single formula of an object language, which could be true or false
  • A \Leftrightarrow B is a statement in the metalanguage, which asserts that A \leftrightarrow B is (logically) true.
In both cases the symbol is a binary connective; but the two have different meanings.
This sort of distinction is generally ignored in mathematical logic because it never causes a problem for us. But some philosophers are more interested in it. Since I'm not one of them, I find it very had to get up the motivation to try to edit this stuff. But at least the claims about "binary connective" seem OK to me, although the articles in general need a lot of work. — Carl (CBM · talk) 13:35, 22 February 2010 (UTC)
Thanks for the explanation! That seems like a reasonable distinction, but its implementation on Wikipedia sucks:
  1. There are no citations to the effect that "if and only if" and "logical biconditional" are the names for those two concepts. There's actually a citation to the contrary: the first footnote in Logical biconditional.
  2. There's nothing in the Logical connective article that suggests that it's meant to include a gadget that inputs two sentences in one language and outputs a sentence in a different language!
Melchoir (talk) 05:53, 23 February 2010 (UTC)
Right; without someone to explain it I don't think the two articles make very much sense. One one hand, the distinction between \leftrightarrow and \Leftrightarrow that I listed above is subtle and of interest mainly to philosophers. On the other hand, when students learn about "if and only if" in an undergraduate mathematics course, they are probably given the explanation for \leftrightarrow (mathematical logicians know but don't worry about the difference; general mathematicians won't even know it exists). So readers will bring that explanation here when they read our article if and only if. To make things worse, people commonly use the symbols \Leftrightarrow and \leftrightarrow interchangeably.
Personally, I think it would make sense to discuss the metalanguage version (\Leftrightarrow) at logical equivalence (which I just noticed), merge if and only if to logical biconditional, and rewrite the two remaining articles to be more clear. — Carl (CBM · talk) 11:21, 23 February 2010 (UTC)
I edited Logical equivalence and I think it now has a clear explanation of what's going on. — Carl (CBM · talk) 11:32, 23 February 2010 (UTC)
From citations: "if and only if (shortened iff) is a biconditional logical connective", "A sentence that is composed of two other sentences joined by "iff" is called a biconditional." I deduce that the biconditional logical connectives are just shorthands for "if and only if" phrase. Then if and only if and logical biconditional should be megred. The distinction between "logical equivalence" and "if and only if" IMHO should be discussed in logical equivalence, because, e.g. if somebody X needs to formalize some theorem in some proof checker, "if and only if" is used in theorems quite often, therefore X needs to read if and only if, but there is no splitting into object language and metalanguage, therefore "logical equivalence" makes no sense, therefore X does not need to read if and only if. --Beroal (talk) 10:15, 28 March 2011 (UTC)

Notation[edit]

Because if and only if, logical equivalence, logical equality are denoted by mostly the same symbols, I propose to move all symbols to a new article Symbols for "if and only if", "logical equivalence", "logical equality". Each of the mentioned articles should select a preferred symbol and link to the "Symbols …" article. Then each article should use only the preferred symbol. --Beroal (talk) 10:23, 28 March 2011 (UTC)

Definition via truth values and via logical connectives[edit]

p\leftrightarrow q is also defined as a propositional formula p\to q \land q\to p. E.g. "<->" in Coq is defined this way. This definition is suitable when truth values make no sense. IMO this definition also should be in the article. --Beroal (talk) 10:37, 28 March 2011 (UTC)

Mistake in Usage -- Proof[edit]

Sorry if this has been raised before in the TL;DR above.

In this sentence:

In most logical systems, one proves a statement of the form "P iff Q" by proving "if P, then Q" and "if Q, then P" (or the inverse of "if P, then Q", i.e. "if not Q, then not P").


I interpret the convoluted logic above to include the possibility that:

one proves a statement of the form "P iff Q" by proving;
"if P, then Q"
and
the inverse of "if P, then Q", i.e. "if not Q, then not P".

But this is wrong, as it won't do the job, as you see from this truth table:

\begin{array}{|cc|c||cc|c|} \hline
p & q & p \implies q & \neg q & \neg p & \neg q \implies \neg p \\
\hline
F & F & T & T & T & T \\
F & T & T & F & T & T \\
T & F & F & T & F & F \\
T & T & T & F & F & T \\
\hline
\end{array}

As you see, P \implies Q and \neg Q \implies \neg P mean exactly the same thing.

In short, to prove a "P iff Q" statement by showing P \implies Q and \neg Q \implies \neg P is a classic schoolboy blunder of a fallacy.

I suspect this can be fixed by "just" changing the offending sentence to say:

In most logical systems, one proves a statement of the form "P iff Q" by proving "if P, then Q" and "if Q, then P" (or the inverse of "if P, then Q", i.e. "if not P, then not Q")

as this is also consistent with the definition of "inverse".

However, I would also take issue with the link itself - should it not be directly to Inverse (logic)?

I'd change it myself, but I'd like someone to confirm what I'm talking about - I'm only (barely) a mathematician, and an amateur at that, and am well-known for spouting rubbish. --Matt Westwood 11:16, 11 July 2011 (UTC)

I removed the parenthetical clause. — Carl (CBM · talk) 11:36, 11 July 2011 (UTC)

Analogs[edit]

Can this be revisited? It looks as though the definitions for "orr" are being used as definitions for "iff". I'd fix it myself but I'd prefer someone who better knows what they're doing (I'm notorious for being uselessly incompetent). --Matt Westwood 08:08, 20 October 2011 (UTC)

Languages[edit]

Why is its rendering in other languages considered to be "Trivia" and "not directly relevant to the subject"? I demur. I suggest that it is completely relevant to the subject, and well worth while including.

Mind, we might like to render the structure of this section a little better. Here's an example of how it could look:

http://www.proofwiki.org/wiki/Definition:Iff

--Matt Westwood 12:15, 25 March 2012 (UTC)

These are not relevant to the concept of "if and only if" (which is the topic of the article), only to the word "iff" used to denote it (which is not). I deleted the list because lists like that, no matter how are they organized, do not add encyclopaedical value to the article. Instead, editors just add translations to their own language, which makes the article an indiscrimate collection of information. Such a list would probably be relevant to the article about the word "iff" itself should such an article exist, but not to the article about the concept. Abolen (talk) 16:30, 25 March 2012 (UTC)
Creating another page for the word "iff" would create clutter and overlap on the site without adding real value. When a person looks for information about a topic, they are generally looking for all of the information about a topic, not just the concept or just the definition. How would we determine which article is listed in the search results? The result for the concept of "iff" or the result for the definition of "iff"? I see no reason to have a distinction between a concept and a definition, since we're no longer limited by things like page numbers and paper consumption. I've found this issue on other articles as well. --Archer 70.125.154.172 (talk) 21:03, 14 April 2012 (UTC)
Of course the word iff does not deserve an article on its own. That's why we don't need such an article, and that's why we don't need its translations in the concept article. Abolen (talk) 09:51, 16 April 2012 (UTC)
Just a reminder: This article started as the article Iff. It was moved to its present name, since some wikipedians do not like abbreviations in titles, when the full name also is available.
Therefore, this is the Iff article, as well as the If and only if article; and the abbreviations in other languages are much more relevant than mere curiosities. JoergenB (talk) 23:32, 2 June 2012 (UTC)

Concept vz. terminology: Possibly better with an article split?[edit]

I find it exasperating that this is "the best article" on the important and most common usage of the concept "equivalence" in mathematics.

Of course, equivalence of mathematical statements is a special case of logical equivalence. However, that article puts the focus on the logical usage to an even higher amount than this one does. In particular, its discussion of the differences betwee logical and material equivalence in general are not to much use for a non-professional reader of a mathartice, trying to find the meaning of "equvaklence" or of "is equivalent to".

I found out the problem, when someone erroneously disambiguated a reference to equivalence to a reference to equivalence relation. Actually, the dab page Equivalence does not even mention If and only if, and lists Logical equivalence under the Logic heading, (which in itself is not quite illogical:-). I've looked around a little at the usage of various wp math related articles using "equivalence in the ordinary sense". Sometimes, there are not very useful links to the dab page; often there is no explaining link at all. My small sampling have revealed no references to if and only if in such situations; although the phrase itself often is linked.

A typical example: In Lipschitz continuity, one paragraph begins

"The inequality is (trivially) satisfied if x1 = x2. Otherwise, one can equivalently define a function to be Lipschitz continuous if and only if there exists a constant K ≥ 0 such that, for all x1x2,
\frac{d_Y(f(x_1),f(x_2))}{d_X(x_1,x_2)}\le K."

Now, this contains both the first occurrence of if and only if (which, as you can see, is linked) and of any form of the word equivalence (which is unlinked, both here and elsewhere in the article). Seemingly, we need to provide the reader with a link explaining the usage of "if and only if", but assume that (s)he understands what we mean by "equivalent definitions" without ever having any need for explanation. However, in my opinion, many first year university students of mathematics have trouble in understanding the meaning of equivalence of statements, be they definitions, parts of a theorem, or even equations, until we explain the meaning; and sometimes this still troubles them, even after we've made our best attempts at explaining.

Not even TFAE has a link to "equivalent" in its explanation; instead, actually, a beginner might come to believe that "equivalent" roughly means "of equal importance in applications". The whole list of mathematical jargon does mention and link to logical equivalence once, at the explanation of if and only if.

I'd like to have a natural target for linking equivalence and equivalent n such cases. The target page should explain the concept, in the first place, and might secondarily give various formulations for this. The explanations should start relatively simple, and contain some important common applications, like equivalence of equations. This page might be redisponated for such purposes, but that also changes its character a bit.

Hence, I'd prefer a split, into (say) Equivalence (mathematics) on the one hand, and If and only if on the other. Of course, there should be cross references between them (and to Logical equivalence), and some overlap; but I think that the emphasis should be sufficiently different for motivating both articles. JoergenB (talk) 00:51, 3 June 2012 (UTC)

"Equivalence (mathematics)" is an unlikely candidate for an article. Actually, all such equivalences are examples of an equivalence relation or its generalizations beyond the set theory (such as canonically isomorphic objects in the category theory). "Equivalence" in mathematical parlance does not specify any definite relation on a set or so – there are many cases where "up to equivalence" is used implicitly, and the appropriate wikilink has to be not a generally-nonsensical "Equivalence (mathematics)", but an article about a more concrete implementation of equivalence relation. Incnis Mrsi (talk) 08:31, 3 June 2012 (UTC)

Philosophical Interpretation[edit]

Under the "Philosophical Interpretation section, the first sentence:

"A sentence that is composed of two other sentences. . . ." Really? This is unintelligible because what it says is that a sentence is both a sentence and not a sentence.

How can a sentence be composed of two sentences? Please reword this. A sentence is either a sentence or it is not.

Dwdallam (talk) —Preceding undated comment added 21:25, 24 May 2013 (UTC)

There is nothing unintelligible about taking two sentences and composing them together via some sort of connecting word or phrase into a new, larger, third sentence.
"Bob is a bachelor" is a sentence. "He is an unmarried man" is another sentence. "Bob is a bachelor if and only if he is an unmarried man" is a sentence composed of the first two sentences plus the connecting phrase "if and only if". What's the problem? --Pfhorrest (talk) 05:39, 25 May 2013 (UTC)
A usable compromise could be "... a compound sentence formed from two simpler sentences ..." then (a) it emphasises the compound nature of the resulting sentence, and (b) it does not rely upon the implicit meaning in "composed of" that the compound sentence only contains those simpler sentences, as formed from is deliberately vaguer and so less open to quibblesome misinterpretation. --Matt Westwood 09:05, 26 May 2013 (UTC)