Definition:Iff
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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
- 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 $^1$)