What these simple rules of logical implications mean Their are many rules of logics, predicate calculus, inferences and syllogisms which haunt me always. It feels better when I find some sensible name to particular rule which also gives me intuition about the meaning of the rule. Recently I came across some simple tautologies:


$((x \rightarrow y)\wedge(x \rightarrow \neg y))\rightarrow \neg x$
$((x \rightarrow y)\vee(x \rightarrow \neg y))$


I am struggling to find intuition or reasonable interpretation of their meaning:


Interpreting  rule 1:
If a first variable $x$ implies both: second variable $y$ and its
negation $\neg y$, then why the negation of first variable $\neg x$
should always hold? Does it mean to imply that no such variable can
exist which can imply both: another variable and its negation?
Interpreting  rule 2:
Why should a first variable $x$ should always imply either (any
given) second variable or its negation?
Finally are there any formal names used to refer these rules?


I must be missing something very basic.
Contradictions can't happen. If $x$ implies a contradiction, then $x$ can't be true.
Either $y$ is true, or it isn't.  That's still the case even if I first tell you that $x$ is true.