# Definition:Biconditional/Semantics of Biconditional

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## Definition

The concept of the biconditional has been defined such that $p \iff q$ means:

**If $p$ is true then $q$ is true, and if $q$ is true then $p$ is true.**

$p \iff q$ can be considered as a *shorthand* to replace the use of the longer and more unwieldy expression involving two conditionals and a conjunction.

If we refer to ways of expressing the conditional, we see that:

- $q \implies p$ can be interpreted as
**$p$ is true if $q$ is true**

and:

- $p \implies q$ can be interpreted as
**$p$ is true only if $q$ is true**.

Thus we arrive at the usual way of reading **$p \iff q$** which is: **$p$ is true if and only if $q$ is true.**

This can also be said as:

**The truth value of $p$ is***equivalent*to the truth value of $q$.

**$p$ is***equivalent*to $q$.

**$p$ and $q$ are***equivalent*.

**$p$ and $q$ are***coimplicant*.

**$p$ and $q$ are***logically equivalent*.

**$p$ and $q$ are***materially equivalent*.

**$p$ is true***exactly when*$q$ is true.

**$p$ is true**This is another convenient and useful (if informal) shorthand which is catching on in the mathematical community.*iff*$q$ is true.

### Necessary and Sufficient

Let:

- $p \iff q$

where $\iff$ denotes the biconditional operator.

Then it can be said that **$p$ is necessary and sufficient for $q$.**

This is a consequence of the definitions of necessary and sufficient conditions.

## Sources

- 1964: Donald Kalish and Richard Montague:
*Logic: Techniques of Formal Reasoning*... (previous) ... (next): $\text{II}$: 'AND', 'OR', 'IF AND ONLY IF': $\S 5$: Exercises, Group $\text{III}$ - 1971: Robert H. Kasriel:
*Undergraduate Topology*... (previous) ... (next): $\S1.2$: Some Remarks on the Use of the Connectives*and*,*or*,*implies*: Definition $2.1$ - 1977: Gary Chartrand:
*Introductory Graph Theory*... (previous) ... (next): Appendix $\text{A}.5$: Theorems and Proofs - 1980: D.J. O'Connor and Betty Powell:
*Elementary Logic*... (previous) ... (next): $\S \text{I}: 12$: Material Equivalence and Alternation