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


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