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There are some scenarios about which I would like to get some confirmation:

  1. when defining a concept A,

    We call A, if ... [definition of concept A]

    Does "if" here mean equivalence instead of just sufficiency? Is it incorrect to replace "if" with "if and only if"?

  2. For precisely what condition is B satisfied?

    Does "precisely" mean asking for necessary and sufficient condition for B?

  3. Some other ways to say "if and only if" / "necessary and sufficient"?

Some references that summarize some standard terminologies in Mathematics such as this one?

Thanks and regards!

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"if and only if" is bidirectional; "A if and only if B" is the compact way of saying "if A, then B, AND if B, then A". – J. M. May 14 '11 at 5:47
@J.M. : I already know "if and only if" and am not asking about it per se. – Tim May 14 '11 at 5:49
@J.M. It seems to be exclusive to the statement of definitions, and it seems to be pretty well entrenched (even if I don't follow it). – Arturo Magidin May 14 '11 at 5:52
@J.M. I don't like it either; then again, I also don't like "iff", which in fact was (independently) invented by Halmos to use in definitions so they would "sound" the way the usually do, but he could still signal that they were "if and only if" statements. – Arturo Magidin May 14 '11 at 6:04
I believe John H. Conway has advocated (not-entirely-seriously?) always following "if and only if" with "then and only then" --further suggesting giving "iff" the counterpart "thenn"-- so that the two aspects of a bidirectional conditional are equally emphatic (and potentially dramatic when spoken in a lecture). I rather like the idea. – Blue May 14 '11 at 8:04

6 Answers 6

up vote 16 down vote accepted
  1. In definitions, it is very common practice to use "if" even though "if and only if" is meant. (Personally, I always use "if and only if" explicitly). So you would not have trouble finding a book that said something like

    Definition. A group $G$ is simple if $G$ is nontrivial and whenever $N\triangleleft G$, either $N=\{1\}$ or $N=G$.

    But such a definition is meant to be understood to be saying that $G$ is simple if and only if the condition is met.

  2. "Precisely" is asking for a condition that is both necessary and sufficient.

  3. "Exactly when"; "if ... then ... and conversely", among others.

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@Tim: Not that I am aware of. one recongizes common practice by participating and reading. – Arturo Magidin May 14 '11 at 6:04
@Tim: you can add "a lot" at the end of Arturo's comment. – J. M. May 14 '11 at 6:06
@Arturo I know several people who consider it circular to use "if and only if" in definitions. E.g. your example would, among other things, say "If $G$ is called simple then...", which sounds a bit nonsensical. Personally, I don't feel too strongly either way, but I tend to agree with this point of view. – Alex B. May 14 '11 at 6:15
@Alex: I never really attempted to dissuade people from using only "if" in definitions; I had an early teacher who emphasized that all definitions are, in "reality", 'if and only if' statements, so he would always say it, though he would not always write it. I like to do it just to keep emphasizing to students that we have implications going both ways: if you say A is blah, then this implies this and that; and if this and that holds, then this means A is a blah. – Arturo Magidin May 14 '11 at 20:39
@GEdgar: That's what Halmos invented it for; I don't know what Kelley used it for (it was invented independently, and Kelley's use precedes Halmos's). See my comment in response to J.M. in the question. – Arturo Magidin May 15 '11 at 0:06

It distracts the reader to use 'if and only if' in a definition, because it's so obvious that 'only if' is implied.

Definition: We say that $S$ is a snargle if all its corticles are open.

None of us would ask: "What if some of its corticles are not open? Is $S$ still a snargle?" We are not politicians, or lawyers. (I'm not, anyway.)

The phraseology 'if and only if' is best reserved to statements of lemmas, propositions, and theorems, where it plays an important role.

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Actually, there have been many times I've considered the implication in the backwards direction after reading the definition. For example, "An integer is even if it is divisible by 2." It is not unreasonable to wonder if there are even integers that are not divisible by 2. Yes, we know there aren't any, but that's not the point. Authors should be explicit in their definitions and use iff when they mean it. "An integer is even iff it is divisible by 2" is much less ambiguous and does not leave the reader questioning. – chharvey Dec 14 '13 at 23:55

The demand for economy of expression means that our shared understanding must be exploited to the fullest, and this sometimes involves importing some presumptive structures from ordinary language into the writing of mathematics materials. An important instance of this is the introductory “if”, which does not mean “if”, but rather “given that” (which is only one third as many syllables, and much less than one third of text). We see this in textbook exercises, and on tests, all the time, e.g., “If x = 3, evaluate 5x + 2.” “So,” muses the mischievous student, “if x is NOT 3, then I don’t have to bother with doing the evaluation, right?” But, in reply to the mischievous student, this really means, “Given that x = 3, evaluate 5x + 2.”

Regarding definitions, in support of conciseness, there is also operative the default of presuming maximality. That is, the stated condition is presumed to be maximal, and therefore necessary. In ordinary language, uttering non-maximal statements, intentionally or unintentionally, is highly misleading, and stomped on when detected, as in this classic exchange.:

A. “90% of Science Fiction is trash.”

B. “Of course, 90% of ANYTHING is trash.”

As far as theorems are concerned, I would prefer using something (short) that DOES distinguish aurally between “if” and “if, and only if,”. I would like to see the adoption of “fif” for this purpose, e.g., “A set M is compact fif it is closed and bounded.”

Regards, Mike Jones

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What does "fif" stand for? – Yuval Filmus May 14 '11 at 22:22
iF and only IF$\to$FIF? – Jonas Meyer May 14 '11 at 22:30
@YuvalFilmus I think it's supposed to be "iff" that you can hear. You can't hear the extra 'f', so you have to say "if and only if", but you would be able to hear "fif". And "fif" is the same in both directions, which is nice and symmetric. – msouth Apr 29 '14 at 6:24
I didn't realize there was a question what I meant by "fif". Anyway, msouth got it right. – Mike Jones Jun 21 '14 at 8:42
@msouth: Why can't "iff" be pronounced as two syllables "i-ff", with the second one stressed? Such an utterance would be concise but clearly distinct from an ordinary "if". – supercat Sep 21 '14 at 2:53

3) Necessary and sufficient condition.

  • Addapted from Wikipedia: "$p$ if and only if $q$" means that $q$ is a necessary and sufficient condition of $p$. It is the same as "$p$ if $q$" ($q$ implies $p$) and "$p$ only when $q$" (not $q$ implies not $p$).
  • From Wikitionary: "if and only if" is equivalent to; implies and is implied by; is true and false in the same cases as.
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When we write (for example) "we say that an integer is odd if it is not divisible by $2$", the statement does not explicitly deny that we might describe as odd a number that is divisible by $2$. The reason for ruling out the latter circumstance is that it would make the definition pointless. Thus, in definitions, there is an implicit message "either this definition is pointless or you may assume that it applies only if the stated conditions hold". As long as the author has the respect of the reader, the latter can be counted on to fill in the "and only if" part of the definition. If it became seen as necessary to put "and only if" in definitions, it would need to go in every definition henceforth, and any author failing to comply would have to worry about seeming wrong.

This has more to do with the conventions of discourse than with formal logic.

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instead of "A if and only if B" one can say "A is equivalent to B" But "A is equivalent to B" does not imply that the truth/falsity of A/B can be deduced from B/A.

PS: I do not recall any references for the above (if any). Any/All objections/corrections are welcome

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Actually, if A is equivalent to B does in fact mean that one can deduce the truth/falsity of A from the truth/falsity of B and the truth/falsity of B from the truth/falsity of A. Many theorems (and definitions) begin with: "The following statements are equivalent...", for which the proof (when used in a theorem, often uses transitivity proving a implies b which implies c...which implies f, which implies a..." Perhaps I'm not understanding your use of the slash between truth/false, A/B, B/A? – amWhy May 14 '11 at 17:24

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