A calculus of the absurd

18.3.3 The material conditional

This one can be really confusing. The basic idea that we seek to express using the “material conditional” is “if \(A\), then \(B\)”. Unfortunately natural language is often quite confusing, because when we say “if \(A\), then \(B\)” many people, not at all unreasonably, interpret this to mean that if \(A\) is false and \(B\) is true then it is not at all true to say “if \(A\), then \(B\)”. However, in mathematics “if” is a one-way street; all we are saying is that should \(A\) be true, then \(B\) is also true (for “if \(A\), then \(B\) to hold”.

  • Definition 18.3.1 The “material conditional”, i.e. the logical operator denoted by \(\implies \) has the following truth table

    \(A\) \(B\) \(A \implies B\)
    F F T
    F T T
    T F F
    T T T
  • Theorem 18.3.1 The (mathematical) natural language statement “if \(A\), then \(B\)” corresponds to \(A \implies B\) in logic.

  • Technique 18.3.1 To prove a statement in the form “if \(A\), then \(B\)” or “\(A\) if \(B\)” one must

    • 1. Assume that \(A\) is true.

    • 2. Show that therefore \(B\) is also true.

    Note that if \(A\) is false it doesn’t matter what happens, as \(A \implies B\) is true in this case.