A confusion on baisc conditional statement. Logical falsehood entails everything as long as antecedent is not universally true? In the logic, we could assign T\F to antecedent and consequent to evaluate the conditional statement. It's easy to evaluate because it's either universally true or false.
example 1. If $4$ is an even number, then there is an infinite number of integers.
$4$ is an even number, universally true
there is an infinite number of integers, universally true.
Therefore this statement is T
Q1 What if antecedent and consequent are contingent. How should I verify the statement T or F
example 2 If tomorrow rain heavily, then the streets will be empty.
When I have not learned logic, I would say this is false. I would say
"Oh, if raining tomorrow, it's not necessarily will have no people on the street"
But now, I may say "well, tomorrow may not be raining", logical falsehood entails everything.
Q2 what kind of arguing I got here?  I felt something go wrong when dealing with a conditional statement.
When I deeply think about those, I am quite confusing. Especially when dealing with some (not logic) exam questions , like verifying the statement "If $A$ then $B$", as long as $A$ is not universally true, I always would like to claim this statement is true by logical falsehood entails everything, but this is certainly not the correct answer.
example 3 If $X$ is greater than $4$, then $X$ is divisible by $2$. (False, e.g. $7$)
Q3 When one is claiming $X$ is not necessarily divisible by 2 when X is greater than 4.  what kind of basic logic embedding in this reasoning?
I know that in the proof, one needs to find $(p\land\neg q)$ implies a contradiction to disproof the statement, but for this specific statement, I don't see a contradiction, only the existence of (p &~q) I could also argue that X is not always greater than 4, logical falsehood entails everything.
Q4 What is the contradiction in this example, and how should I convinced myself do not use logical falsehood entails everything for verifying conditional statement when antecedent is not universally true.
I've contacted my prior logic course lecturer, and he said something this required higher level of modal logic (I don't know much about this) to capture accurately, difference in possibility and necessarily, and then he did not say much more about it.
Appreciate anyone could enlighten me up by first year logic material.
 A: What is right is that the (halfway symbolic) proposition

(1) $X$ is greater than $4$ $\implies$ $X$ is divisible by $2$.

is sometimes true (which is the case, for example, when $X=3$ or $X=8$) and sometimes false (such as when $X=7$). It doesn't have a definite truth value until we decide what $X$ is.
$\Rightarrow$ doesn't care whether truths are universal, contingent, necessary, or impossible. It simply asks, "true or false?" of both antecedent and consequent and consults its truth table. If the truth value of the antecedent or consequent varies by context, then so does (potentially) the truth value of the $\Rightarrow$.
The invisible snag in your question is whether the above proposition is the same as

(2) If $X$ is greater than $4$, then $X$ is divisible by $2$.

Some introdutory textbooks in logic try hard to tell you that (1) and (2) mean exactly the same. Often they even say that the $\implies$ symbol is defined by the words "if" and "then", and then separately attempt to convince you that "if"/"then" inherently have the somewhat unusual meaning that you're trying to grasp here.
Really, though, it is a lie-to-children, and not a particularly useful one at that. The actual meaning people are trying to express when they say (2) -- even when those people are mathematicians talking about mathematics -- is not unambiguously (1). Very often it is instead something we could formalize as

(3) $\forall X$[ $X$ is greater than $4$ $\implies$ $X$ is divisible by $2$ ].

which is definitely false because the inner sentence (identical to (1), by the way) is sometimes false.
The trouble logic textbooks face is that they often introduce the quantifier $\forall X$ much later than they're explaining the "propositional" symbols $\land$, $\lor$, and $\Rightarrow$. During the part of the book where they can't mention $\forall X$, they're reduced to claiming that the English words "and", "or", "if"/"then" mean the same as $\land$, $\lor$, $\Rightarrow$, producing a horribly artificial and mutilated approximation to the meaning of natural language.
A further trouble is that there are no simple rules for when we need to insert a $\forall X$ or $\exists X$ in the logical translation of an English sentence. However, that's because the meaning of English (like most other natural languages) is fuzzy, highly context-sensitive, and sometimes downright ambiguous -- but not because the logical formalism is complicated.
Many students who go with the book's flow unfortunately leave with the opposite impression -- that it's the mathematical formalism that is complex, unpredictable and only accessible by special geniuses.
As long as you stick to propositions like (1) or (3), everything is simple and easy. If you're a native or reasonably fluent speaker of English, you will also be able to deal with (2), as long as you accept that you need to figure out from the context whether the speaker meant (1) or (3), or perhaps something subtly different from either of those. It is not determined strictly by the words in the sentence itself.
A: 
Logical falsehood entails everything as long as antecedent is not
universally true?

The basic principle of vacuous truth is given by the tautology $A \implies [\neg A \implies B]$.
Here is the truth table:

Source: https://www.erpelstolz.at/gateway/TruthTable.html
Here is a formal proof using a simplified form of natural deduction:
(Screen print from my proof checker)

Note that if the antecedent of any conditional is false (something that usually only comes up in technical proofs) then we cannot infer anything about the consequent. So, it's not, as some would have it, that anything goes in mathematics or logic.

Example 1: If 4 is an even number, then there is an infinite number of integers.



Example 2: If tomorrow (it is) rain heavily, then the streets will be empty.

Classical propositional logic can only be applied in real-world scenarios for propositions that are unambiguously either true or false at some instant in time,  usually the present. The truth value of predictions about the future are always ambiguous.

Example 3: If X is greater than 4, then X is divisible by 2.

This is contingent on the value of X, thus the truth value of the antecedent and consequent are ambiguous.
A: 
example 1 If $4$ is an even number, then there is an infinite number of integers.
$4$ is an even number, universally true; there is an infinite number of integers, universally true; therefore this statement is True.
What if antecedent and consequent are contingent. How should I verify the statement T or F

Then the conditional's truth value (by which we mean truth value in a particular context) depends on its antecedent's and consequent's combination of truth values in that interpretation. You did just this in the above analysis, and determined that the above conditional is mathematically true.
This conditional is logically contingent though.

example 2 If tomorrow rain heavily, then the streets will be empty.
When I have not learned logic, I would say this is false.

This conditional is again not a tautology, since it is alternately true and false.

But now, I may say "well, tomorrow may not be raining", logical falsehood entails everything.

Yes, so in this context/scenario, the conditional is true.
BTW, "tomorrow is not raining" is not a logical falsehood: it is true in some universe.

example 3 If $X$ is greater than $4$, then $X$ is divisible by $2$. (False, e.g. $7$)

If we are discussing $\mathbb R,$ then this conditional is not universally true; that is, $$\forall X\;(X{>}4\to2|X)\tag1$$ is false; so, it is invalid, so it is not logically true.

I could also argue that X is not always greater than 4, logical falsehood entails everything.

In fact, let's change the universe of discourse from $\mathbb R$ to $(-\infty,4],$ so that $x$ is now always not greater than $4.$ In this interpretation/context, $(1)$ becomes vacuously true.
But while $x{>}4$ is false in this context, it is true in another context, so is not logically false.

What is the contradiction in this example, and how should I convinced myself do not use logical falsehood entails everything for verifying conditional statement when antecedent is not universally true.

There is no logical falsehood and no contradiction: the antecedent is not universally false in the first interpretation, and universally false in the second interpretation.
