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I already know that the below sentences are equivalent. $U$ is universal set.

$$\forall{x} \in U : q(x) \equiv \forall{x} : [x \in U \rightarrow q(x)]$$

I think that the "$\forall{x}$" in the right sentence is meaningless because it doesn't specify which set the "x" belongs to. So it should be "$\forall{x} \in U$". Is it right? If my logic is right, why do many authors often omit which set the element is in?

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    The formal specifications for the usage of universal uantifier is : $\forall x \phi(x)$, period. – 2017-02-01
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    The formula $\forall x (x \in U \to \phi(x))$ is often abbreviated as ; $(\forall x \in U) \phi(x)$. – 2017-02-01
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    In the same way, $\exists x (x \in U \land \phi(x))$ is abbreviated as $(\exists x \in U) \phi(x)$. – 2017-02-01
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    In general, $U$ is the *domain* specifying the "range" of the quantified variable: it can be any set, not necessarily the "universal set". We may have $\forall x (x \in \text {Humanity} \to \text {Mortal}(x))$ that is the same as : $\forall x ( \text {Human}(x) \to \text {Mortal}(x))$. – 2017-02-01
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    I'm worried writing "$\forall{x}$" because it is somewhat dangerous representation by Russell's paradox. There is no set of the universe. – 2017-02-01
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    Oh. I see. You mean that $x$ in "$\forall{x}$" is just a dummy notation. Am I right? – 2017-02-01
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    **NO**. $\forall x$ is $\forall x$; if you omit the $x$ how you manage $\forall x \exists y (x=y)$ ? – 2017-02-01
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    I'm very confused by the logical rigidity of it :( My question is just this: Why do people write $\forall{x} \phi(x)$ instead of $\forall{x} \in U \phi(x)$? just convention? – 2017-02-01
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    "People" write $\forall x \phi(x)$ to formalize: "All $x$ are $\phi$s". **All** means "all". – 2017-02-01

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In first order logic, sentences are defined in a first order language $\mathcal{L}$, where $x_i$ are called variables and always belong to $\mathcal{L}$. When you talk about $x$ belonging to a set, you are talking about the interpretation of $\mathcal{L}$ in some structure $\mathcal{M}$, where $x$ must belong to a specific set $M$ which itself belongs to $\mathcal{M}$.

For example, $$\forall{x}\forall{y}(x=y)$$ is a sentence in some language $\mathcal{L}$, and talking about its truth is meaningless. More precisely, it may be true or false only when it is interpreted in structures. For instance if $M$ has only one member, then $\forall{x}\forall{y}(x=y)$ is true. However, if $M$ has more than one member, it is false.

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    I didn't study first order logic. what does L mean? universal set? – 2017-02-01
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    No, M is the universal set, and L does not contain it. L contains logical symbols like $\forall$, $\lor$ , etc. and non-logical symbols, like variables, constants, function symbols, predicate symbols, etc. The point is, a sentence can be written in L, with variables $x_i$ which do not belong to a specific set. – 2017-02-01
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    Also, strictly speaking, the RHS equivalent of your sentence is the correct form in logic and LHS is its abbreviation. @cokecokecoke – 2017-02-01
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    oh, I see. Thank you. Is there any books or articles about the further study? – 2017-02-01
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    @cokecokecoke I think this book is a very good reference: https://www.amazon.com/Handbook-Mathematical-Studies-Foundations-Mathematics/dp/0444863885 – 2017-02-01