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An implication is a logical construction that essentially tells us if one condition is true, then another condition must be also true. Formally it is written
or
which would be read “ implies  ”, or “ therefore  ”, or “if  , then  ” (to name a few).
Implication is often confused for “if and only if”, or the biconditional truth function (
 ). They are not, however, the same. The implication
 is true even if only  is true. So the statement “pigs have wings, therefore it is raining today”, is true if it is indeed raining, despite the fact that the first item is false.
In fact, any implication
 is called vacuously true when  is false. By contrast,
 would be false if either  or  was by itself false (
 , or in terms of implication as
 ).
It may be useful to remember that
 only tells you that it cannot be the case that  is false while  is true;  must “follow” from  (and “false” does follow from “false”). Alternatively,
 is in fact equivalent to
The truth table for implication is therefore
| a |
b |
 |
| F |
F |
T |
| F |
T |
T |
| T |
F |
F |
| T |
T |
T |
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