Logical Conditional Statements

Definition

Statements of the form $p\to q$ $p$ is called the hypothesis or antecedent $q$ is called the conclusion or consequent “if $p$, then $q$” ”$p$ only if $q$” ”$p$ is a sufficient condition for $q$” ”$q$ is a necessary condition for $p$“

Vacuously True

In the case that $p$ is false, $p\to q$ is true (vacuously true or true by default)

Implication Law

$p\to q\equiv \sim p\vee q$ $\sim (p\to q)\equiv p\wedge \sim q$ (Can be deduced from above)

Variants of Conditional Statements

Contrapositive

The contrapositive of $p\to q$ is $\sim q\to \sim p$ The contrapositive of “if p then q” is “if ~q then ~p” A conditional statement is always logically equivalent to its contrapositive $p\to q\equiv \sim q\to \sim p$ e.g. If Howard can swim across the lake, then Howard can swim to the island $\equiv$ If Howard cannot swim to the island, then Howard cannot swim across the lake

Converse

The converse of $p\to q$ is $q\to p$ The converse of “if p then q” is “if q then p” $p\to q\not\equiv q\to p$

Inverse

The inverse of $p\to q$ is $\sim p\to \sim q$ The inverse of “if p then q” is “if ~p then ~q” $p\to q/\equiv \sim p\to \sim q$

Notes

The converse of a conditional statement $\equiv$ the inverse of the logical statement (They are contrapositive) Summary:

1. conditional statement $\equiv$ contrapositive
2. converse $\equiv$ inverse
3. conditional statement $\not\equiv$ converse