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We define

$$U:=\{x;E(x)\}$$

And refer to the members of $U$ as "beings". Where $E(x): x \text{ exists.}$

Now, $\forall x\in U$, Only one of these hold:

$$1.\ E(x)\wedge P(x)\\2.\ E(x)\wedge\neg P(x)$$

Regardless of what $P(x)$ is.

Now I want to make a pure logical expression of the below paragraph which describes what I mean by the cause of an arbitrary $x$.

Each being either has its existence from itself or from some other being. If the object exists by itself, without relying on other beings, we call it a necessary being. If the object needs another object to exist, i.e. It has gotten its existence from another being, we call it contingent being. Calling that being which gives the existence "cause", A necessary being will not need a cause to exist but a contingent being needs a cause to exist and it can't exist without a cause.

$$a: x \text{ has a cause.}\implies x\text{ has only one cause.}$$ Assuming $a$, how can I define the cause relation?

$$x\to y\iff x\text{ is the cause of }y$$

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    The statement $$a: x \text{ has a cause.}\implies x\text{ has only one cause.}$$ does not really follow from what you wrote, if you ask me... – 2017-01-30
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    @5xum No I mean *assume* this. This is an additional thing. Without it, an object would accept to have multiple causes. That would make it really hard. – 2017-01-30
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    @MauroALLEGRANZA , What I mean here by a cause may not be the standard definition. I just want to translate this thing. Will I find my answer there?? – 2017-01-30
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    For a logic with an "existence" predicate, see [Free Logic](https://plato.stanford.edu/entries/logic-free/). – 2017-01-30
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    You cannot use $\to$ ("if ---, then___")... I think. Maybe you must have as primitive $x \text { is the cause of } y$. In this way : $x \text { has cause }$ is def as $\exists y (y \text { is the cause of } x)$ and $\text {Contingent}(x)$ as $\lnot \exists y (y \text { is the cause of } x)$. – 2017-01-30
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    It seems to me (a non-expert in logic) that any "definition" of x is the cause of y would need an axiomatic approach where your definition captures some of the structure of your intuition (like how points and lines are undefined in geometry). There is a statistical approach to understanding causality involving a "backdoor" criterion which might be what you want after you throw out all of the statistics. – 2017-01-30

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A not entirely standard notation for "exists exactly one" is $\exists !$. So, for example, if $A$ is a set and $P$ a property, saying

there exists exactly one element $a$ of $A$ which has the property $P$.

is like writing $$\exists!a\in A: P(a)$$

In your case, you want to say that

$$\exists x: x\text{ is the cause of }a \implies \exists! x: x\text{ is the cause of }a$$

In standard quantifiers, I would instead write

$$\exists a\in A: (P(a)\land \forall b\in A: (P(b)\implies b=a))$$

or, in your case:

$$\exists x: x\text{ is the cause of }a\implies (\forall y: y\text{ is the cause of }a\implies x=y)$$

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    You have well designed the proposition $a: x \text{ has a cause.}\implies x\text{ has only one cause.}$! but My main question was to define what a cause is for e.g. $x$, according to that paragraph. I think I haven't said what I want clearly. – 2017-01-30
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    @AHB Oh I see. Well, "to have a cause" isn't really a part of logic. Logic doesn't care what the relations "mean". So, you have to define the relation yourself. – 2017-01-30
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    I have tried to. But I doubt if it would cover all the details in the paragraph. – 2017-01-30
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Introduce a $2$-input predicate-symbol $Causes$, and $1$-input predicate-symbols $Necessary$ and $Contingent$, and axiomatize them according to what you want. It is easy to do so in first-order logic, but if you don't know how then you should really learn first-order logic first.