Conditional Statements

Conditional Statements, also known as implications, are a fundamental concept in formal logic. A conditional statement is typically expressed in the form "If $p$ , then $q$ ," denoted as $p \to q$ . Here, $p$ is called the antecedent (or premise), and $q$ is called the consequent (or conclusion). Arguments involving conditional statements are used to establish relationships between propositions and to derive conclusions based on given premises. Logical inferencing often relies on understanding and manipulating these conditional statements. We can express conditional statements using other logical connectives. Two statements are logically equivalent if they have the same truth value in every possible scenario. Logical equivalences can be used to simplify complex statements or to prove the validity of arguments.

Definition

Implication

Implication ( $\to$ ): Represents "If $p$ , then $q$ ". $p \to q$ means "If $p$ , then $q$ ."
- We call statements of this form conditional statements or implications, as the value of $q$ is conditional on the value of $p$ .
- If $p$ represents "It is raining" and $q$ represents "The ground is wet", then $p \to q$ represents "If it is raining, then the ground is wet."
- We say " $p$ implies $q$ ".
- $p$ only if $q$ means the same thing as $p \to q$ . We say that $q$ is a necessary condition for $p$ , and $p$ is sufficient for $q$ .
- $p \to q$ is false only if $p$ is true and $q$ is false; otherwise, it is true. This may seem counterintuitive at first, but remember that if $p$ is false, then the implication $p \to q$ is considered true regardless of the truth value of $q$ . Such statements are considered to be vacuously true. Thus, $p \to q$ is only false when a true premise leads to a false conclusion

$\begin{array}{cc|c} P & Q & P \implies Q \\ \hline \text{T} & \text{T} & \text{T} \\ \text{T} & \text{F} & \text{F} \\ \text{F} & \text{T} & \text{T} \\ \text{F} & \text{F} & \text{T} \end{array}$

The implication states that the validity of

p

and

q

can not coexist, thus

p \to q

is logically equivalent to

\neg p \lor q

.

Check your understanding

Suppose that $P\implies Q$ . What can we conclude?

Select all that apply.

Definition

Contrapositive

If we have a statement from

p \implies q

, the contrapositive is the statement

\neg q \implies \neg p

. The contrapositive is logically equivalent to the original conditional statement. This means that if

p \implies q

is true, then

\neg q \implies \neg p

is also true, and vice versa. Sometimes, the contrapositive might be easier to prove or work with than the original implication. This usage is a fundamental concept in logic and is often used in mathematical proofs and reasoning.

Definition

Converse

The converse of an implication $p \implies q$ is the statement form $q \to p$ . The converse is not logically equivalent to the original conditional statement.

Definition

Inverse

The inverse is the statement $\neg p \to \neg q$ . The inverse is also not logically equivalent to the original conditional statement

Match them up

Match each definition of $P \implies Q$ together.

Match them up

Suppose that the following is a known fact about all people: "All extroverts can engage in great conversations." What can we conclude?