# Quantification and logical relations, shorthand notation $\forall/\exists x \in M...$

I know the following shorthand: \begin{align*} \exists x \in M : P(x) & := \exists x ( x \in M \to P(x) ) \\ \forall x \in M : P(x) & := \forall x ( x \in M \to P(x) ). \end{align*} Now for me it is obvious that $$\forall x \in M : P(x) \equiv \neg \exists x \in M : \neg P(x).$$ But also it holds that $\forall x : P(x) \equiv \neg \exists x : \neg P(x)$. So applying I get \begin{align*} \forall x \in M : P(x) & \equiv \forall x ( x \in M \to P(x) ) \\ & \equiv \neg \exists x ( \neg( x \notin M \lor P(x))) \\ & \equiv \neg \exists x : ( x \in M \land \neg P(x) ) \\ \end{align*} and otherwise $$\forall x \in M : P(x) \equiv \neg \exists x \in M : \neg P(x) \equiv \neg \exists x ( x \in M \to \neg P(x) ).$$ So concluding $$\neg \exists x : ( x \in M \land \neg P(x) ) \equiv \neg \exists x ( x \in M \to \neg P(x) ).$$ or $$\exists x : ( x \in M \land \neg P(x) ) \equiv \exists x ( x \in M \to \neg P(x) ).$$ which obviously is not equivalent. But how to explain?

• Your shorthand for $\exists{x\in M}$ isn't right. You need to have $\exists{x\in M}{:}P(x)$ defined as $\exists{x}(x\in M \wedge P(x))$. Commented Sep 27, 2013 at 18:38
• ok, if so then it is wrong in my textbook and also on wikipedia: en.wikipedia.org/wiki/Quantification Commented Sep 27, 2013 at 18:46
• Not so Stefan. Look more carefully. Commented Sep 27, 2013 at 18:53
• by using this shorthand I even get more contradictions results, then $\forall x \in M : P(x) \equiv \forall x ( x \in M \land P(x) ) \equiv \neg \exists ( x \notin M \lor \neg P(x))$ and $\forall x \in M : P(x) \equiv \neg \exists x \in M : \neg P(x) \equiv \neg \exists ( x\in M \land \neg P(x))$, concluding $\exists x ( x \in M \lor \neg P(x)) \equiv \exists x (x \in M \land \neg P(x) )$ which obviously is not the same... Commented Sep 27, 2013 at 18:54
• ...Which is one reason among many why the shorthand in the first displayed line of your post is wrong.
– Did
Commented Sep 27, 2013 at 19:23

The correct definitions are schematically $$(\forall F x)G x \leftrightarrow \forall x (F x \to G x),$$ and $$(\exists F x)G x \leftrightarrow \exists x (F x \wedge G x),$$ where the variable $x$ is a free variable in $F$ and $G$.
Note carefully that the definition of universal quantification has a conditional statement in its definition, but the definition of existential quantification has a conjunction in its definition. In your specific cases the definitions would read $$\forall x[x \in M \to P(x)],$$ and $$\exists x[x \in M \wedge P(x)],$$ respectively.