Propositional Condition
In logic, a propositional condition (also known as a conditional statement or implication) is a compound statement of the form “if P, then Q,” denoted as P → Q.
Here, P is called the antecedent and Q the consequent.
Truth Table
The truth value of a conditional statement P → Q is defined by the following truth table:
| P | Q | P → Q |
|---|---|---|
| T | T | T |
| T | F | F |
| F | T | T |
| F | F | T |
Note: The only case where P → Q is false is when P is true and Q is false.
Key Properties
- Contrapositive:
P → Qis logically equivalent to¬Q → ¬P. - Converse: The converse of
P → QisQ → P(not logically equivalent). - Inverse: The inverse is
¬P → ¬Q(also not equivalent to the original).
Example
Consider the statement: “If it is raining, then the ground is wet.”
- Antecedent (P): It is raining.
- Consequent (Q): The ground is wet.
This does not mean that rain is the only way the ground can get wet—only that rain guarantees it.