# What does it take for a compound conditional statement to be false?

Jul 14, 2016

It depends upon the conjunction used to form the compound condition.

#### Explanation:

If the conjunction is and
then the statement is false if either of the components is false.
Example:
$\textcolor{w h i t e}{\text{XXX}}$Megan owns a motorcycle and Jose owns a truck.
This statement is false
1. if Megan does not own a motorcycle, or
2. if Jose does not own a truck, or
3. if Megan does not own a motorcycle and Jose does not own a truck.

If the conjunction is or
in English the result is trickier to determine since English does not normally distinguish between "inclusive or" and "exclusive or".
In either case the result is false if both of the components is false.
Example:
$\textcolor{w h i t e}{\text{XXX}}$Megan owns a motorcycle or Jose owns a truck.

This statement is false
1. if Megan does not own a motorcycle, and
2. if Jose does not own a truck.

However, what if Megan owns a motorcycle and Jose owns a truck?

In the exclusive case, one or the other must be true, but not both for the result to be true $\rightarrow$ the result is false unless exactly one of the components is false (both can not be true or the "exclusive or" is false).

Conversely, in the inclusive case, the statement is only false if both components are false.

In English when the or conjunction is used, the intent must be assumed from the context (or by directly questioning the intended meaning).