as i read some tutorial material on First Order Logic, i deduce that the following formula was consistent in FOL except the third one. am i right? i have doubt about the first one. any idea? thanks to all experts.
- $\bigl\{\exists y\exists x\forall z\,\bigl(C(x,y,z) \to \neg C(x,x,x)\bigr)\bigr\}$
- $\bigl\{ \forall x \bigl(A(x) \to B(x)\bigr), \forall x \bigl(A(x) \to \neg B(x)\bigr)\bigr\}$
- $\{\forall x\,A(x)\} \cup \{\neg A(t) \mid t \text{ is a term}\}$
- $\{\forall x\exists y\, B(x,y) \to \neg \exists y \forall x \,B(x,y), \exists x\,B(x,x)\}$
We construct a model for 1). The (underlying set of) the structure $M$ has one element $a$. The interpretation $C_M$ of the predicate symbol $C$ is false at $(a,a,a)$. Then the sentence 1) is true in $M$.
Similarly, we can construct models for 2) and for 4). In 4), let $M$ have two elements, $0$ and $1$. The interpretation of the predicate symbol $B$ is true at $(0,0)$, and false at the pairs $(0,1)$, $(1,0)$, and $(1,1)$. Then $\forall x\exists y B(x,y)$ is false in $M$, so the implication $\forall x\exists y B(x,y)\longrightarrow \lnot \exists y\forall x B(x,y) $ is true in $M$.