I need to find a model for the following formula:
$$(\forall x \forall y \forall z.R(x,y) \wedge R(y,z) \Rightarrow R(x,z)) \wedge (\forall x\forall y.\neg R(x,y) \Leftrightarrow R(y,x))$$
So I need a underlying set and a definition of $R$. The left hand side is transitivity and it is easy to find relations satisfying that. However, I can not seem to find a formula satisfying the right hand side. The "$<$"-relation doesn't work since its negation is "$\geq$" and that is basically the problem with every example I come up with. Is there even some structure satisfying that formula?
The second part of formula is inconsistent. Instantiating $x$ and $y$ to the same element $a$ gives us $\neg R(a,a) \Leftrightarrow R(a,a)$, which is absurd.
If your logical framework allows empty models (which most don't), that would work, though.