Alternative ways to say "if and only if"?

Solution 1:

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.

Solution 2:

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.

Solution 3:

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

Solution 4:

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.

Solution 5:

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.