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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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    The Following Are Equivalent (T.F.A.E.) (Often used for more then one thing; for example, if we have $(a\iff b)\land(b\iff c)$ (which also implies $a\iff c$).)2015-03-22

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  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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    @AlexB. I don't follow why that is circular. Surely you want to say both: "If a group$G$is called simple, then these conditions hold." and "If these conditions hold, then group G is simple."2014-04-17
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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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    Context is by nature an ambiguous thing.2016-03-02
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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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    @msouth: There is a continuum of possible pronunciations ranging from a dominant stress on the first syllable, to a second syllable which is stressed and elongated to the point of ridiculousness. I don't know exactly how far it would be necessary to push the second syllable to ensure that most listeners would notice the difference in pronunciation from a normal "if", but I would expect that with enough stress many people would notice it, and with insufficient stress people would not.2014-09-22
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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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    This would be easy to do if there's only one "if". However, when the definition has multiple `if`s (e.g. multiple statements), the permutations increases exponentially, making it close to impossible for the reader to explicitly run the question "*Is the definition pointless?*" for each and every permutation. This is where ambiguity can occur.2016-03-02
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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?2011-05-14