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
- 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