biconditional
1 Biconditional
A biconditional
is a truth function that is true only in the case that both parameters are true or both are false.
Symbolically, the biconditional is written as
| a⇔b |
or
| a↔b |
with the latter being rare outside of formal logic. The truth table for the biconditional is
| a | b | a⇔b |
|---|---|---|
| F | F | T |
| F | T | F |
| T | F | F |
| T | T | T |
The biconditional function is often written as “iff,” meaning “if and only if.”
It gets its name from the fact that it is really two conditionals in conjunction
,
| (a→b)∧(b→a) |
This fact is important to recognize when writing a mathematical proof, as both conditionals must be proven independently.
2 Colloquial Usage
The only unambiguous way of stating a biconditional in plain English is of the form “b if a and a if b.” Slightly more formal, one would say “b implies a and a implies b.” The plain English “if” may sometimes be used as a biconditional. One must weigh context heavily.
For example, “I’ll buy you an ice cream if you pass the exam” is meant as a biconditional, since the speaker doesn’t intend a valid outcome to be buying the ice cream whether or not you pass the exam (as in a conditional). However, “it is cloudy if it is raining” is not meant as a biconditional, since it can obviously be cloudy while not raining.
| Title | biconditional |
|---|---|
| Canonical name | Biconditional |
| Date of creation | 2013-03-22 11:53:06 |
| Last modified on | 2013-03-22 11:53:06 |
| Owner | Mathprof (13753) |
| Last modified by | Mathprof (13753) |
| Numerical id | 17 |
| Author | Mathprof (13753) |
| Entry type | Definition |
| Classification | msc 03-00 |
| Synonym | iff |
| Related topic | PropositionalLogic |
| Related topic | Equivalent3 |