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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]

1.Lemma: Implication as a Disjunction

2.Lemma: Negation of an Implication

3.Definition: Contrapositive


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Bibliography (further reading)

[7838] Kohar, Richard: “Basic Discrete Mathematics, Logic, Set Theory & Probability”, World Scientific, 2016

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