Equivalence

One of the most important connectives in logic is the equivalence which is a kind of both-sided implication.

Definition: Equivalence

Let $L$ be a set of propositions with the interpretation $I$ and the corresponding valuation function $[[]]_I$. An equivalence “$\Leftrightarrow$” is a Boolean function

\[\Leftrightarrow:=\begin{cases}L \times L & \mapsto L \\
(x,y) &\mapsto x \Leftrightarrow y\\
\end{cases}\]

defined using a conjunction of two implications:

$$x \Leftrightarrow y :=(x\Rightarrow y)\wedge (y\Rightarrow y).$$

We read the equivalence

“$x$ if, and only if $y$.”

It has the following truth table:

The equivalence of two propositions is only true if both propositions have the same truth value assigned.
Propositions, the equivalence of which is true, are called equivalent statements. Equivalent statements can be interpreted as the way of “saying the same” in two different ways.