I have to find worlds that satisfy / do not satisfy the formulas
My W is the notation for my worlds U:= Universe
Now i am at the fourth and can literally not find a satisfying world but my tutors told me as a hint that there exists one
- $\psi_{1}=\forall x\left(\exists x^{\prime}\left(R\left(x, x^{\prime}\right) \wedge \neg R\left(x^{\prime}, x\right)\right)\right)$
$W_1$
$S= K^2$
$U = \mathbb{N}$
R=(x,x')| x > x'
$\alpha(x) = 2$ , $\alpha(x')=1$
$W_1 \models \varphi_1$
$W_2$
$S= K^2$\
$U = \mathbb{N}$
R=(x,x')| x = x'
$\alpha(x) = 2$ , $\alpha(x')=2$
$W_2 \not\models \varphi_1$
- $\psi_{2}=\exists x(R(x, x)) \wedge \forall x\left(\forall x^{\prime}\left(\neg R\left(x, x^{\prime}\right) \leftrightarrow R\left(x^{\prime}, x\right)\right)\right)$
$W_1$ $S= K^2$ $U = \mathbb{N}$ R=(x,x')| x = x' $\alpha(x) = 2$ , $\alpha(x')=2$ $W_1 \not\models \varphi_1$
It's a contradiction.\
- $\psi_{3}=\exists x\left(\forall x^{\prime \prime}\left(R\left(x, x^{\prime \prime}\right) \vee x=x^{\prime \prime} \vee \exists x^{\prime}\left(x=x^{\prime}\right)\right)\right)$
$W_1$
$S= K^2$
$U = \{ 1 \}$
R=(x,x'')| x = x''
$\alpha(x) = 1$ , $\alpha(x')=1$
$W_1 \models \varphi_1$
Tautology
4.$\psi_{4}=\exists x\left(\exists x^{\prime}\left(\exists x^{\prime \prime}\left(\left(x=x^{\prime}\right) \wedge \neg\left(x=x^{\prime \prime}\right) \wedge\left(R\left(x, x^{\prime}\right) \rightarrow\left(x^{\prime}=x^{\prime \prime}\right)\right)\right)\right)\right)$