THE APEIRON ONTOLOGICAL ARGUMENT
1. All that exists in space and time must be a possible quantity. Otherwise it would be partly quantifiable and partly unquantifiable, which is absurd.
2. All possible quantities are measurable by numbers.
3. Infinity is not measurable by any number.
4. Therefore, infinity cannot exist in space and time.
1. All numbers are bounded, but the set of all numbers is unbounded.
2. Unboundness is greater than boundness, as an unbounded quantity includes a greater number of elements than a bounded quantity. Thus, if infinity is unbounded and numbers are bounded, then infinity is greater than numbers.
3. Therefore, infinity is greater than numbers in a non-numerical sense.
4. On the other hand, we conceded that infinity cannot exist in space and time.
5. We also concede that infinity is possible, since it is non-contradictory.
6. All that exists is possible, given that impossible entities cannot exist. Therefore, all that is possible can exist.
7. If something can exist, it can exist either by itself or by another.
8. If infinity could exist by another, there would be two beings existing simultaneously as cause and effect. The plurality of beings would entail space, and the coming into existence of infinity would imply time. This contradicts premise 4 and must be rejected.
9. Therefore, infinity can exist by itself, which -in accordance to premise 3- is a non-numerical property greater than the numerical property of not existing by itself.
10. If something can exist by itself, it must exist by itself.
11. Infinity can exist by itself.
12. Therefore, infinity must exist by itself.
13. God is the being that exists maximally by itself outside of space and time.
14. Infinity exists maximally by itself outside of space and time, being identical to God.
15. Therefore, God exists.
THE HYLOMORPHIC CONTINGENCY ARGUMENT
Matter cannot exist without form, as it is limited by form, which determines what matter can and cannot be. Formless matter would be unlimited and could be absolutely everything, which means it could have two opposite predicates at the same time, i.e., being existent and non-existent, mobile and immobile, caused and uncaused, etc. This extreme violates the principle of non-contradiction and must be rejected, as it is impossible.
If matter cannot exist without form, it follows that if matter is necessary, then form is necessary.
However, if the form ceases to be, it will follow that the form is contingent.
Now, the form ceases to be in order to become another form. Therefore, the form is contingent.
A necessary entity cannot depend on any contingent entity.
As a result of the above, it is false to assert that a necessary entity cannot exist without a contingent entity.
Consequently, it is false to affirm that matter, being necessary, cannot exist without form, which is contingent. Therefore, since matter cannot exist without form and form is contingent, it follows that matter is contingent.
On the other hand, we know that no entity can be composed of entities with predicates opposite to the first entity.
It follows that if the universe is composed of matter and form and both are contingent, then the universe must be contingent.
Now, the universe is composed of matter and form, both being contingent. Therefore the universe is contingent.
Consequently, the universe's existence presupposes a non-material, non-spatial and non-temporal necessary being, which is God.
THE OVERRIDING TRUTH ARGUMENT
"No truth is necessary" cannot be a necessary truth, since it is part of the set of truths of which non-necessity is predicated. Nor can it be a contingent truth, since this would entail that "Some truth is necessary" is also a contingent truth. However, if there are necessary truths they cannot depend on a contingent truth. Therefore, it must be concluded that "No truth is necessary" is neither a necessary truth nor a contingent truth, but a falsehood.
If "No truth is necessary" is a false statement and consequently "Some truth is necessary" is a true statement, suppose "Reality exists" is a necessary truth. In this case, some truth and reality will be necessary. This will lead to the fact that the truths that we call necessary can exist by themselves without reality and that, on the other hand, the reality that we call necessary can exist by itself without the necessary truths. However, both scenarios are incompossible and mutually exclusive, since in the scenario in which truth can exist without reality we assume that reality may not exist and is therefore contingent, while in the scenario in which reality can exist without the truth we assume that the truth may not exist and is therefore contingent. And since it is impossible for anything to be necessary and contingent, it follows that truth can only be necessary if reality is not, and reality can only be necessary if truth is not. Yet we have conceded that "Some truth is necessary" is a true statement. Therefore, "Reality exists" is not a necessary truth.
THE HERCULEAN ONTOLOGICAL ARGUMENT
1. Everything real is true.
2. There is no infinite regress in dependence relations. Consequently, there is no infinite regress in causal relations nor in noetic relations, so there is a first cause and a first truth.
3. The first truth is the principle of non-contradiction. Since it does not depend on any other truth and is not in space or time, it is absolutely unlimited.
4. To exist and to act are equivalent.
5. If something is absolutely unlimited, it has no limits in its action other than those imposed by the principle of non-contradiction.
6. If there were a first cause that was real and did not correspond to a truth that was true, it would contradict premise 1.
7. If the first cause corresponded to a truth different from the first truth, then the truth that corresponded to the first cause would depend on another truth that did not correspond to the first cause. This situation would imply that the first truth cannot cause anything and that it is limited, which contradicts the thesis that the first truth has no limits, stated in premise 3, and the thesis that if something is absolutely unlimited, it has no limits in its action other than those imposed by the principle of non-contradiction, stated in premise 5.
8. Therefore, the first cause must correspond to the first truth.
9. The first truth is immaterial (the principle of non-contradiction), as established by premise 3.
10. Therefore, the first cause is immaterial and is the principle of non-contradiction.
11. The principle of non-contradiction and God are indistinguishable.
12. Therefore, God is the first truth and the first cause.
Proof of premise 1:
1.1. For every entity x, x is contradictory if and only if there exist predicates p and q such that x has predicate p, x has predicate q, and p and q are opposite.
1.2. If a subject bearing two opposite predicates is true, then it would be both true and not true, which leads to an absurd conclusion. Therefore, a subject bearing two opposite predicates is not true.
1.3. For every entity x, x is real if and only if x exists and is in space and time.
1.4. If it is not true that the real is true, then the real is a subject bearing two opposite predicates and exists and does not exist in space and time, which leads to an absurd conclusion. Therefore, the real is true.
Proof of premise 2:
Proof 1:
2.1.1. All multiplicity can be reduced to unity.
2.1.2. Unity cannot be reduced to multiplicity.
2.1.3. Therefore, there is no infinite regress in numerical dependence relations.
Proof 2:
I.
2.2.1.1. Equal producers create equal products.
2.2.1.2. So if a cause B, being distinct from its consequent C, were equal to its antecedent A, we would be faced with two equal producers, A and B, creating two unequal products, B and C respectively, which is absurd.
2.2.1.3. Therefore, the effect is not equal to the cause.
II.
2.2.2.1. For all entities x, there is no causal relationship between x and itself.
2.2.2.2. For all entities x, there is no causal relationship between nothing and x.
2.2.2.3. For all entities A and B, if there is a causal relationship between A and B, then all elements of B proceed from A.
2.2.2.4. Therefore, there is no relationship of superiority between the effect and the cause.
III.
2.2.3.1. Therefore, there is a relationship of superiority between the cause and the effect.
IV.
2.2.4.1. Let's assume that everything has a cause (= there is an infinite regress in the dependency chain of causes).
2.2.4.2. If the previous premise is true, the whole (the sum of all entities) has a cause.
2.2.4.3. The cause is superior to the effect.
2.2.4.4. Any part is inferior to the whole (i.e., less inclusive, less representative, and less general).
2.2.4.5. If the cause of the whole belonged to the whole as its part, it would be superior and inferior to the whole, which is absurd. Otherwise, the cause of the whole would be limited by something subsequent to it, implying that it would be unlimited (and therefore not a part of the whole) before producing its effect.
2.2.4.6. Therefore, the cause of the whole does not belong to the whole as its part, and the whole is not the whole, which is absurd.
2.2.4.7. Therefore, the first premise stating that everything has a cause is false. Hence, there is no infinite regress in the dependency chain of causes.
V. A fortiori.
2.2.5.1. The cause is never subsequent to the effect.
If the cause were subsequent to the effect, the effect would be prior to the cause and without the cause, which goes against its own notion.
2.2.5.2. The part is always subsequent to the whole.
Given two unique elements, namely, the part and the whole, if the part were prior to the whole, it follows that the part would be the whole and the potential whole would be nothing when the potential part is an actual whole. Likewise, such nothing would become the whole when the past whole becomes part of the new whole. Now, this is absurd, as the notions of part and whole, like those of whole and nothing, are not mutually convertible.
The part is not simultaneous with the whole either, as the whole is arrived at either by composition or immediately. If by composition, we find the paradox that nothing becomes everything and the whole becomes part, which must be excluded. If immediately, the part can only be distinguished from the whole by its action, which happens in time.
2.2.5.3. Therefore, something cannot be both the cause of the whole and part of the whole, as it would be prior and subsequent to the whole.
2.2.5.4. However, if the cause of the whole is not part of the whole, the whole will not be the whole, as it will not include its cause.
2.2.5.5. Consequently, the whole has no cause.
2.2.5.6. Therefore, there is no infinite regress in causal dependency relationships.
VI.
2.2.6.1. If A depends on B, C depends on B, and A depends on C, assuming that dependency relationships are causal relationships that occur in space and time, then all elements A, B, and C will be prior and subsequent to A, B, and C, which is absurd.
2.2.6.2. If A causes B and B causes A, then A is and is not superior to B, and B is and is not superior to A, which is absurd.
2.2.6.3. Consequently, there is no circular dependency in causality.
Proof of premise 3:
All truths depend on the principle of non-contradiction, while this principle does not depend on any truth. This must be asserted axiomatically, without the need for further proof. However, we can provide the following demonstration:
3.1. Let's assume that the principle of non-contradiction can be contradicted. That would be absurd.
3.2. Let's also assume that the principle of non-contradiction derives from another truth. Such truth must be true, that is, non-contradictory. Therefore, it will depend on the principle of non-contradiction, so it is false that the principle of non-contradiction derives from another truth.
3.3. The conclusion is that the principle of non-contradiction is neither contradicted by anything nor derived from anything. Consequently, it is absolutely without limits.
Proof of premise 4:
4.1. For every entity x, if x exists, then the existence of x is equivalent to x acting to preserve itself or change another entity. And for every entity x, if x acts, then the action of x is equivalent to the existence of x.
4.2. Therefore, for every entity x, its existence is equivalent to its action preserving itself or changing another entity.
Proof of premise 11:
11.1. Everything that is not self-contradictory, if it is absolutely without limits, cannot be confined to the realm of non-existence.
11.2. Therefore, if the principle of non-contradiction is absolutely without limits, we must identify it with an entity that necessarily exists.
11.3. Everything defective is limited.
11.4. Therefore, everything that is absolutely without limits and necessarily exists also exists perfectly.
11.5. Therefore, the principle of non-contradiction exists necessarily and perfectly.
11.6. If the existence of two or more perfect beings were possible, they would limit or augment one another. If they limit each other, they are not absolutely without limits, nor, for this reason, perfect. If they augment each other, they do not exist perfectly, which also goes against the premise. It follows that the existence of two or more perfect beings is not possible.
11.7. Therefore, everything that exists necessarily and perfectly exists singularly.
11.8. By God, we understand the being that exists necessarily and perfectly.
11.9. If the principle of non-contradiction and God existed necessarily and perfectly while being distinct, they would not exist singularly, which we have concluded to be impossible.
11.10. Therefore, the principle of non-contradiction and God are one and the same being.
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