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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:

semantics:


$[[x]]_I$ $[[y]]_I$ $[[x \Leftrightarrow y]]_I$
\(1\) \(1\) \(1\)
\(0\) \(1\) \(0\)
\(1\) \(0\) \(0\)
\(0\) \(0\) \(1\)

Notes

  • 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.

| | | | | Contributors: bookofproofs

1.Corollary: Commutativity of Equivalence


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