SYMBOLIZING
Although rather rare in colloquial speech, biconditionals are beloved by logicians because of their use in definitions. They are conditionals which “go both ways”: if p then q, and if q thenp.
This is more commonly expressed as iip if and only if q” (Remember, “if p then q” is equivalent to iip only if and “if q then p” is equivalent to iip if #.”) Where they come into their own is in any context where we are trying to be very precise about what we mean, particularly in giving definitions. Here is an example from “possible world semantics,” a branch of logic where possibility and necessity are construed in terms of being true in possible worlds:(1) Necessarily A is true in world x if and only if A is true in all worlds.[34]
Denoting iiNecessarily A is true in world ë” by N and iiA is true in all worlds” by A, this could be symbolized as (N → A) & (A → N). But it proves convenient to have a shorthand for “if and only if,” and for this we introduce our fifth statement operator,
by the following definition: 
Here the symbol =def stands for “is by definition equivalent to.” This is a Stipulative definition, so-called because we have simply stipulated how the double arrow is to be understood, rather than, say, clarifying the meaning of something previously understood. Thus statement (1) above gets symbolized as
At this point it will be useful to introduce a new term, the converse of a conditional. This is what we get from swapping its antecedent with its consequent:
The statement formed by exchanging the antecedent and consequent of a
With this terminology, the above definition says that a biconditional is equivalent to the conjunction of a conditional and its converse.
There are some synonymous expressions for “p if and only if q” The philosopher Bas van Fraassen prefers the expression “exactly if,” as in(2) World ó is physically POSSIBLE relative to world x exactly if the LAWS of x are all true in y.
Symbolized:
Logicians abbreviate the expression “if and only if’ by the made-up word “iff’; this is fine in written form, but confusing aurally, since it sounds the same as plain ‘if.’ The briefest synonymous expression I know of in English is “just in case,” which has only three syllables to the four of “exactly if.”
Another equivalent form is implicit in the example below:
(3) CHINA will sign the ban on nuclear weapons testing, but only if the US signs first.
This appears to be a conjunction of C and C only if U. But that would entail U (by Modus Ponens). So that cannot be the meaning. Further reflection shows that it must mean that China will sign if and only if the US does so first:
8.2.2