Quiz. Models of \(T_H\) [0036]

consider a structure consisting of a universe \(U\) and interpretation \(I\) as follows:

\[ U = \{A, B\} \quad I = \{\texttt {taller} \mapsto \{\langle A, A\rangle , \langle B, B\rangle \}\} \]

Are the following models of the theory \(T\) or not:

Is \(\langle U, I\rangle \) a model of \(T\)
If we change the interpretation to \[ I = \{\texttt {taller} \mapsto \{\langle A, B\rangle \}\} \] is the structure now a model of \(T\)?
If we add the following axiom \[ \forall x, y, z.\ (\texttt {taller}(x, y) \land \texttt {taller}(y, z) \to \texttt {taller}(x, z)) \] and we change the interpretation to \[ I = \{\texttt {taller} \mapsto \{\langle A, B \rangle , \langle B, C \rangle \}\} \] then is the structure a model of \(T\)?

Solution.

As a reminder we are working with the axiom:

\[ \forall x, y.\ (\texttt {taller}(x, y) \to \neg \texttt {taller}(y, x)) \]

Clearly no, as if we take \(x = A\) and \(y = A\) we have that both \(\texttt {taller}(A, A)\) and \(\texttt {taller}(A, A)\) hold, violating the axiom.
Here we are chaning or interpretation to just the relation \(\texttt {taller}(A, B)\) holding. In this case the axiom is satisfied as there are no values of \(x\) and \(y\) such that both \(\texttt {taller}(x, y)\) and \(\texttt {taller}(y, x)\) hold. So yes this is a model of \(T\).
No, as if we take \(x = A\), \(y = B\) and \(z = C\) we have that both \(\texttt {taller}(A, B)\) and \(\texttt {taller}(B, C)\) hold, but \(\texttt {taller}(A, C)\) does not hold (so not in our interpretation for the \(\texttt {taller}\) relationship), violating the axiom.