 ?x ?y))))

&gt; So,
&gt;
&gt; 1) The &lt;=&gt; axiom above is invalid

I think you just mean that it does not adequately capture the intuitive
understanding of the "gives" relation.  In logic, "validity" and its
cognates are usually used to indicate certain properties of arguments.

&gt; and probably could be used to prove a contradiction.

Certainly it could not, at least not by itself, as it is easily shown to
be consistent.

&gt; 2) An augmented &lt;=&gt; axiom like the one above but disallowing two
&gt