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Given that the Philosophers believed in the beginninglessness of the world, they could not have used Burhan Al-Huduth to prove God’s existence. Instead, they utilized what is known as the Burhan Al-Imkan (Proof of Contingency):
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- If the set of all contingents is contingent, then a necessary being exists.
- The set of all contingents is contingent.
- Therefore, a necessary being exists.
Premise 1: If the set of all contingents is contingent, then a necessary being exists.
A contingent depends on an extrinsic specifier to grant it existence. And an existent that is not part of the set of all contingents, is necessary[1].
Thus, if the set of all contingents depends on an extrinsic specifier to exist, then this specifier exists necessarily.
Premise 2: The set of all contingents is contingent.
A composite depends on the existence of each of its parts to exist[2]. And if those parts exist contingently, then this composite would exist contingently as well. For that which depends on contingents to exist, is contingent a fortiori[3].
From there, we realize that the set of all contingents is nothing but a composite of contingents. A composite that would not exist if each of those contingents did not. Thus, the set of all contingents exists contingently.
Therefore, a necessary being exists.
Given the above, there exists a necessary being who the set of all contingents depends on to exist. And since “necessary being” is what is meant by “God”, God exists[4].
[1] Since an existent either accepts non-existence, or not. If so, it is a contingent. If not, it is a necessary being.
[2] For example: a house depends on the existence of its bricks to exist. If those bricks did not exist, then the house would not either.
[3] For example: if an opponent grants that the bricks that this house is made of are contingent, we say: either the house as a whole exists necessarily, or it exists contingently.
It cannot exist necessarily. This is because those bricks accept non existence by virtue of their being contingent. And the house would not exist if those bricks did not. So since it is possible for those bricks to not exist, it is possible for the house to not exist as well. And that which accepts non-existence, does not exist necessarily.
And since the house does not exist necessarily, it exists contingently.
[4] The deficiency in this proof, is its failure to identify the necessary being more clearly. An opponent could argue that this necessary being is actually a planet, or an idol, or even the matter and energy of the universe itself. Separate arguments would need to be offered to rule out all those possibilities.
On the other hand, Burhan Al-Huduth, with the aid of Burhan Huduth Al-Ajsam as a sub-argument for the second premise, more directly concludes the existence of an incorporeal and changeless necessary being, who volitionally selected existence for the world.
The above is why most of the Mutakalimun preferred Burhan Al-Huduth over Burhan Al-Imkan, even though they agreed that Burhan Al-Imkan is a valid route to proving the existence of a necessary being.
Keys to the Unseen 
I think there is a counter-example to premise 2 in your argument. How do you address brute facts for example? How would you respond to those who say that Each contingent being in the set of all contingents is self-caused?
I agree that the first premise is decisively true though.
Can you give me some examples of a contingent being?
One final question: Does this argument deal only with contingent beings like cats, oxygen, houses OR does the set of all contingent also include contingent facts like the fact that Trump was the 45th president or the contingent fact that it is raining outside? If it does also include contingent facts, how would you respond to those that grant the conclusion yet argue that the necessary being is not God but rather other necessary facts like logic or math or the necessary axioms that the mutakalimoon believe in?
Your saying: “I think there is a counter-example to premise 2 in your argument. How do you address brute facts for example?”
If by ‘brute fact’ you mean an existent whose existence was not by virtue of a specifier, then what you mean by ‘brute fact’ is what we mean by ‘necessary being’. So affirming the existence of a ‘brute fact’ is an affirmation of the necessary being’s existence, which is not contrary to this argument. If by ‘brute fact’ you mean something else, then you must elaborate.
Your saying: “Can you give me some examples of a contingent being?”
The above argument seeks only to prove that a necessary being exists. It does not seek to identify the necessary being (and this is one of its weaknesses which I mentioned in footnote 4). So you do not need examples of contingents in order for it to work. This is because an existent is either necessary or contingent. And it is established that at least one thing actually exists. So we say:
Either this one essence whose existence is established, is necessary or not.
If the opponent claims it is necessary, then he has granted the conclusion of the argument. So there is no need to go further.
If the essence whose existence is established is not necessary, then this essence is contingent. So either this existent is the only contingent in existence or there are others contingents in addition to it.
If this existent is the only contingent in existence, then whatever brought it into existence is necessary. Thus, the conclusion of the argument is entailed.
If this existent is not the only contingent in existence, then it, together with all other contingents, form the set of all contingents. And the above argument would then apply to this set. Entailing the existence of the necessary being.
In all cases, the existence of a necessary being is entailed, therefore a necessary being exists.
Your saying: “I agree that the first premise is decisively true though.”
You contradict this sentiment by asking: “How would you respond to those who say that Each contingent being in the set of all contingents is self-caused?”
How can someone who believes it possible for something to bring itself into existence, accept the first premise?
In any case, this is an obvious impossibility. For an effecter to bring its effects into existence requires the effecter to first exist. And for the effect to be brought into existence requires the effect to first not-exist (for it makes no senses to bring into existence something which already exists). So to claim that an effecter brought itself into existence requires this effecter to both exist and not-exist simultaneously. Exist in order to bring into existence. Not-exist in order to be brought into existence. This is contradictory, so it is impossible for an effecter to bring itself into existence.
Your saying: “how would you respond to those that grant the conclusion yet argue that the necessary being is not God but rather other necessary facts like logic or math or the necessary axioms that the mutakalimoon believe in?”
Was addressed above (i.e. that this argument only seeks to prove that a necessary being exists, not identify the necessary being). But to comment on the last bit when you talk about logic:
“Logic” is simply a set of standards, that when abided by, ensures that you make no mistakes in deduction. Those standards have no causal power of their own. And their reliability is not given the existence of extra-mental objects called “the laws of logic” that govern the universe (those laws are extracted abstractions). Rather, their reliability returns to the fact that a thing is itself, which is affirmed by every Mukalaf by virtue of their ability to conceptualize things. And all of logic is built upon this fact.
The same applies to maths.
As for axioms, then they are knowledge possessed by the Mukalaf, except that the Mukalaf did not infer that knowledge. And there is no doubt that all of the qualities of the contingent Mukalaf (like this Mukalaf’s knowledge) exist contingently. The necessity of axioms is not in their existence (such that they do not depend on a specifier to exist), it’s in the fact that the existence of beings with the ability to infer information, entails the existence of non-inferential knowledge that subsists within those beings. And since such beings exist (i.e. humans), then such knowledge (i.e. axioms) also exists. So the necessity is in the relation between beings with the ability to infer knowledge and axioms, not between axioms and their existence.
Analogously, when we say that the three sides of a triangle are necessary for the triangle, we do not mean that those sides exist necessarily (such that it is rationally impossible for them to not-exist). Rather, we mean that given the existence of the triangle, it is impossible for its sides to not-exist. Such that if the sides cease, so too does the triangle cease. Thus necessity here is in the relation between the triangle and its sides (i.e. if a triangle exists, then its sides must exist), not in the relation between the sides and their existence (i.e. it is not rationally impossible for the sides to not-exist, but if they do, then the triangle must also not exist).
Great answer brother. The part about the brute facts is really compelling.
However, I believe that you did skip an important question.
>>Does this argument deal only with contingent beings like cats, oxygen, houses OR does the set of all contingent also include contingent facts like the fact that Trump was the 45th president or the contingent fact that it is raining outside?
Does this argument deal with strictly beings or does it also include with contingently true propositions like the ones above. In other words, would the set of all contingents include only contingent beings or would it also include contingent propositions?
This was answered when I said that you do not need examples of contingents in order for this argument to work. So whether or not you knew the answer to this question, has no bearing on the soundness of this argument.
If you wanted an answer to this question irrespective of this argument though, then we say: what exists extra-mentally are beings and the qualities those beings are attributed with. As for relations, like “the fact that Trump was the 45th president” (which is neither a being, not an attribute that subsists within a being) then they are abstractions that the mind extracts from reality, and have no existence of their own according to the Mutakalimun. See here.
And since relations have no existence additional to that of their relata, they are excluded from relevancy (in so far as this argument is concerned).
How would you respond to those who say that the first premise is false? They say that it is possible that the explanation for the totality of contingent beings is also contingent and so lies in the set itself. This would not make the set of all contingents necessary because it relies on an explanation that is not similar to itself. A being is only necessary if its explanation is identical to itself. But the explanation for the set of all contingents here is not the set of all contingents and so is not identical to itself. Rather the explanation for the set of all contingents is one of its parts and since a part is clearly not identical to the whole then the explanation is external to the whole. How would you respond?
“How would you respond to those who say that the first premise is false? They say that it is possible that the explanation for the totality of contingent beings is also contingent and so lies in the set itself.”
This entails that the particular contingent, which you have posited to be the effecter for the set of all contingents, brought itself into existence. After all, it brought the set of all contingents into existence, while being part of this set which was brought into existence by it. This is impossible for obvious reasons, one of which being what was mentioned in the comment above.