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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.
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.