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All the binary connectives we have learned so far – conjunction, disjunction and exclusive disjunction, were commutative – the order of the propositions connected by them did not matter. Now, we will learn an important binary connective which is not commutative – the implication.

## Definition: Implication

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

$\Rightarrow :=\begin{cases}L \times L & \mapsto L \\ (x,y) &\mapsto x \Rightarrow y. \end{cases}$

defined by the following truth table:

Truth Table of the Implication
$[[x]]_I$ $[[y]]_I$ $[[x \Rightarrow y]]_I$
$$1$$ $$1$$ $$1$$
$$0$$ $$1$$ $$1$$
$$1$$ $$0$$ $$0$$
$$0$$ $$0$$ $$1$$

We read the implication $x\Rightarrow y$

if $x$ then $y$”.

### Notes

• It is apparent from the truth table that the implication is not commutative – the order of the propositions connected by it counts. Therefore, both propositions have specific names – the first one is called the antecedent, the second one the consequent.
• The implication of two propositions is only false if a true antecedent implies a false consequent. This is called a contradiction. Note that a false antecedent can imply both a true and a false consequent, without creating a contradiction.
• But if a true antecedent implies a true consequent, we say it is a valid argument.
• There are more different readings of the implication $x\Rightarrow y$:
• If $x$, then $y$.
• $y$ follows from $x$.
• $x$ is sufficient for $y$.
• $y$ is necessary for $x$.
• $y$, if $x$.
• $x$, only if $y$.
• It is helpful to think about the implication as a cause and effect chain.

| | | | | Contributors: bookofproofs | References: [7838]

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