(related to Lemma: Mixing-up the Sufficient and Necessary Conditions)
We want to prove that the mixing-up the sufficient and necessary conditions is a fallacy. * The fallacy can be formulated in propositional logic as $(p\Rightarrow q)\Rightarrow (q\Rightarrow p).$ * The definition of implication "$\Rightarrow$" gives us the following truth table of the function:
$[[p]]_I$ | $[[q]]_I$ | $[[p\Rightarrow q]]_I$ | $[[(q\Rightarrow p)]]_I$ |
---|---|---|---|
$0$ | $0$ | $1$ | $1$ |
$0$ | $1$ | $1$ | $0$ |
$1$ | $0$ | $0$ | $1$ |
$1$ | $1$ | $1$ | $1$ |