(related to Lemma: Denying the Antecedent of an Implication)
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 $(p\Rightarrow q)\wedge \neg p\Rightarrow \neg q.$
* The definitions of the implication "$\Rightarrow$
" and the negation "$\neg$" give us the following truth table of the function:
$[[p]]_I$ | $[[q]]_I$ | $[[p\Rightarrow q]]_I$ | $[[\neg p]]_I$ | $[[\neg q]]_I$ |
---|---|---|---|---|
$0$ | $0$ | $1$ | $1$ | $1$ |
$0$ | $1$ | $1$ | $1$ | $0$ |
$1$ | $0$ | $0$ | $0$ | $1$ |
$1$ | $1$ | $1$ | $0$ | $0$ |