Logical Equivalence

Definition

Two statements are called logically equivalent iff they have identical truth values for each possible substitution of statements for their statement variables.

Notation

The logical equivalence of statement forms $P$ and $Q$ is denoted by $P\equiv Q$

How to Show Not $\equiv$

There are two ways to show that statement forms $P$ and$Q$ are not logically equivalent:

Truth Table

Find at least one row where their truth values differ

Example

Show that $\sim (p\wedge q)$ and $\sim p\wedge \sim q$ are not logically equivalent

$p$	$q$	$\sim p$	$\sim q$	$p\wedge q$	$\sim (p\wedge q)$	$\sim p\wedge \sim q$
T	T	F	F	T	F	F
T	F	F	T	F	T	F
The last row of the table shows that they are not logically equivalent

Counter example

Find concrete statements for each of the two forms, one of which is true and the other of which is false

Example

Show that $\sim (p\wedge q)$ and $\sim p\wedge \sim q$ are not logically equivalent Let $p$ be true and $q$ be false $\sim (p\wedge q)$ Will evaluate to true $\sim p\wedge \sim q$ Will evaluate to false Therefore, they cannot be logically equivalent