Definition:Iff

Definition

The logical connective iff is a convenient shorthand for if and only if.

Notation

$A$ iff $B$ is generally symbolized by means of a two-headed arrow, for example:

$A \ \leftrightarrow \ B$

$A \ \Leftrightarrow \ B $

$A \ \longleftrightarrow \ B$

$A \iff B$

The notation used on $\mathsf{Pr} \infty \mathsf{fWiki}$ is $A \iff B$.

Also known as

Some sources use the term precisely if to mean if and only if.

Also see

• Definition:Biconditional

• Results about iff can be found here.

Internationalization

Iff is translated:

In Danish:	hviss		(that is: hvis og kun hvis)
In Dutch:	desda		(that is: dan en slechts dan als)
In Finnish:	joss		(that is: jos ja vain jos)
In French:	ssi		(that is: si et seulement si)
In German:	gdw.		(that is: genau dann, wenn)
In Greek:	ανν		(that is: αν και μόνο αν)
In Hebrew:	אמ"ם		(that is: אִם וְרַק אִם)	Pronounced:  'eem v'rak 'eem
In Hindi:	यकेय		(that is: यदि व केवल यदि)	Pronounced:  yadi va keval yadi
In Icelandic:	eff		(that is: ef og aðeins ef)
In Polish:	wtw		(that is: wtedy i tylko wtedy)
In Portuguese:	sse		(that is: se e somente se)
In Spanish:	ssi		(that is: si y sólo si)

Historical Note

The use of iff to mean if and only if is believed to have originated with Paul Halmos.

Technical Note

On $\mathsf{Pr} \infty \mathsf{fWiki}$, the template {{Iff}} has been developed, which is to be used to present the phrase if and only if at any point on the page.

Sources

• 1955: John L. Kelley: General Topology ... (next): Preface
• 1959: A.H. Basson and D.J. O'Connor: Introduction to Symbolic Logic (3rd ed.) ... (previous) ... (next): $\S 2.5$: Further Logical Constants
• 1964: Donald Kalish and Richard Montague: Logic: Techniques of Formal Reasoning ... (previous) ... (next): $\text{II}$: 'AND', 'OR', 'IF AND ONLY IF': $\S 1$
• 1967: George McCarty: Topology: An Introduction with Application to Topological Groups ... (previous) ... (next): Introduction: Special Symbols
• 1975: T.S. Blyth: Set Theory and Abstract Algebra ... (previous) ... (next): $\S 1$. Sets; inclusion; intersection; union; complementation; number systems
• 1975: W.A. Sutherland: Introduction to Metric and Topological Spaces ... (previous) ... (next): Notation and Terminology
• 1982: P.M. Cohn: Algebra Volume 1 (2nd ed.) ... (previous) ... (next): Chapter $1$: Sets and mappings: $\S 1.1$: The need for logic
• 1991: Roger B. Myerson: Game Theory ... (previous) ... (next): $1.2$ Basic Concepts of Decision Theory
• 1996: H. Jerome Keisler and Joel Robbin: Mathematical Logic and Computability ... (previous) ... (next): Appendix $\text A$: Sets and Functions: $\text{A}.1$: Sets
• 1998: David Nelson: The Penguin Dictionary of Mathematics (2nd ed.) ... (previous) ... (next): iff
• 2008: David Nelson: The Penguin Dictionary of Mathematics (4th ed.) ... (previous) ... (next): iff
• 2010: Raymond M. Smullyan and Melvin Fitting: Set Theory and the Continuum Problem (revised ed.) ... (previous) ... (next): Chapter $2$: Some Basics of Class-Set Theory: $\S 2$ Transitivity and supercompleteness: Definition $2.2$ (footnote $^1$)