A conditional is an operation on two statements that creates another statement. If the first statement is true and the second statement is false then the conditional statement is false. All other combinations of truth values would lead to the conditional statement being true. The intuitive idea is that the second statement only occurs if the first one does. The conditional can be represented by a
The left side of the operator is called the hypothesis (or antecedent) and the right side is called the conclusion (or consequence)
A conditional statement that is true because the hypothesis is false is called vacuously true or true by default, this is shown in the truth table when the left argument is false.
The converse of a conditional is the statement formed by reversing the arguments. For example given the statement
The converse of "If Howard can swim across the lake, then Howard can swim to the island" is "If Howard can swim to the island, then Howard can swim across the lake."
The inverse of a conditional is the statement formed by negating the arguments. For example given the statement
The inverse of "If Howard can swim across the lake, then Howard can swim to the island" is "If Howard cannot swim across the lake, then Howard cannot swim to the island."
| T | T | T |
| T | F | F |
| F | T | T |
| F | F | T |
The conditional can be written as:
Alternatively the conditional can be written as it's contrapositive where:
This follows from the definition of a conditional where