Logic: Implication
In logic, a materical conditional is a proposition of the form If the proposition \( P \) is true, then the proposition \( Q \) is also true.
A material conditional is also known as an implication or simply as a conditional proposition.
An implication is denoted as \( P \Rightarrow Q \) and vocalized as "\( P \) implies \( Q \)" or "If \( P \), then \( Q \)".
By definition, the implication \( P \Rightarrow Q \) is false only when \( P \) is true and \( Q \) is false.
Because \( P \) being true is enough to make \( Q \) true, it is also said that "\(P\) is a sufficient condition for \( Q \)".
Here's the truth table of the implication.
\( P \) |
\( Q \) |
\( P \Rightarrow Q \) |
T |
T |
T |
T |
F |
F |
F |
T |
T |
F |
F |
T |
In an implication \( P \Rightarrow Q \), the proposition \( P \) is known as the hypothesis, antecedent or premise.
The proposition \( Q \) is known as the conclusion, the consequent or the consequence.
Examples
Consider the propositions
$$ P: \text{The integer } x \text{ is a multiple of 8} $$
$$ Q: \text{The integer } x \text{ is divisible by 2} $$
It can be shown that for the integer \( x \) when \( P \) is true, then the proposition \( Q \) is also true. In this case the implication \( P \Rightarrow Q \) denotes the proposition
$$ R: \text{If the integer } x \text{ is a multiple of 8, then } x \text{ is divisible by 2} $$