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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?

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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.

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  • $\begingroup$ I was thinking the same thing, however I assumed it had to work since its part of my homework asking for a (non-trivial, which is what excludes an empty model) model. Thank you. $\endgroup$ – Staki42 Jul 10 '16 at 16:15

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