I am learning sentence logic( again) and I have an exercise which I'm not sure If I did it wrong or right:
Let $(A,\leq)$ be an totally ordered set. Translate to formal language: "Any totally ordered part of A is minorated". The way I did this:
$$(\forall M)(M\subset A \Rightarrow ((\exists x)x\in M \wedge (\exists y)(y \in M \Rightarrow x\leq y\vee y \leq x)\Rightarrow((\exists z)z \in A \wedge(\forall w)(w \in M \Rightarrow w \leq z))))$$
You can decompose your formula step by step. First define two formula $\text{TotallyOrdered}(M)$ and $\text{Minorated}(M)$ as follows \begin{align} \text{TotallyOrdered}(M)&: \forall x\ \forall y \ (x \in M \wedge y \in M) \rightarrow ((x \leqslant y) \vee (y \leqslant x)) \\ \text{Minorated}(M) &: \exists m\ \forall x\ (x \in M \rightarrow m \leqslant x) \end{align} Now the monadic second order formula $$ \forall M\ (\text{TotallyOrdered}(M) \rightarrow \text{Minorated}(M)) $$ is a formal translation of the sentence Any totally ordered part of A is minorated.