A sentence that is composed of two other sentences joined by "iff" is called a biconditional. Iff joins two sentences to form a new sentence. It should not be confused with logical equivalence which is a description of a relation between two sentences. The biconditional "A iff B" uses the sentences A and B, describing a relation between the states of affairs A and B describe. By contrast "'A' is logically equivalent to 'B'" mentions the two sentences: it describes a relation between those two sentences, and not between whatever matters they describe.
The distinction is a very confusing one, and has led many a philosopher astray. Certainly it is the case that when 'A' is logically equivalent to 'B', "A iff B" is true. But the converse does not hold. For example, consider the sentence:
The corresponding logical symbols are "↔" and "⇔".
When proving the statement "P iff Q", it is equivalent to prove both the two statements "if P, then Q" and "if Q, then P".
The abbreviation appeared in print for the first time in John Kelley's 1955 book General Topology. Its invention is often credited to the mathematician Paul Halmos, but in his autobiography he states that he borrowed it from puzzlers.
In philosophy and logic, for example, "iff" is used to indicate definitions, since definitions are supposed to be universally quantified biconditionals. In mathematics, however, the word "if" is often used in definitions, rather than "iff". Here are some examples of true statements that use "iff" - true biconditionals (the first is an example of a definition):
Sometimes other words are also emphasied in the same way by repeating the last letter; for example orr for "Or and only Or" (the exclusive disjunction).