Traduzco el Argumento Hercúleo

1. ~(∞R)

Proof 1:

1.1.1. ∀N ≠ U, RED(N, U)

1.1.2. ¬(∃N, RED(U, N))

1.1.3. ∴ ~(∞RN)

Proof 2:

1.2.1.1. EPR → EPD

1.2.1.2. ∀(A, B, C) [(CA(A, B) ∧ CA(B, C) ∧ EQ(A, B)) ∧ ¬EQ(B, C) → EPR → ¬EPD → Absurd

1.2.1.3. ∴ ¬EQ(EF, CC)

1.2.2.1. ∀x ¬CA(x, x)

1.2.2.2. ∀x ¬CA(N, x)

1.2.2.3. ∴ ∀(A, B) [CA(A, B) → PRO(B, A)]

1.2.2.4. ∴ ¬SU(EF, CC)

1.2.3.1. ∴ SU(CC, EF)

1.2.4.1. ∀EE, ∃CC ∧ CA(CC, EE)

1.2.4.2. ∃CC ∧ CA(CC, W)

1.2.4.3. ∀(CC, EE) CA(CC, EE ) → SU(CC, EE)

1.2.4.4. ∀(PA, W), PA ∈ W → INF(PA, W)

1.2.4.5. CA(CC, W) ∧ (CC ∈ W) → SU(CC, W) ∧ INF(CC, W) → Absurd

1.2.4.6. ∴ ∀x (CA(x, W) → (C ∉ W) ∧ (W ≠ W) → Absurd

1.2.4.7. ~(∞RC)

1.2.5.1. (D(A, B) ∧ D(C, B) ∧ D(A, C) → (PRI(A, B) ∧ PRI(B, C) ∧ PRI(C,A) ∧ (POST(A, B) ∧ POST(B, C) ∧ POST(C,A)) → Absurd

1.2.5.2. (CA(A, B) ∧ CA(B, A) → SU(A, B) ∧ ¬SU(A, B) → Absurd

1.2.5.3. ∴ ¬CIRC

2. ∀x(T(x) → (∃y(T(y) ∧ D(y, x)) ∧ ~(∞R) → (AT(z) ∧ (z = PNC))))

Proof:

2.1. ∀x∀y (¬(T(x) ∧ ¬T(y) ∧ D(x, y)))

2.2. ∀x (∃y ((T(x) ∧ ¬AT(x)) → (T(y) ∧ D(x, y))))

2.3. ~(∞R)

2.4. ∀x (CT(x) → (∃y (ST(y) ∧ D(x, y))))

2.5. ∀x (T(x) ∧ x ≠ PNC → ¬(∞S(x)))

2.6. ∴ ∃x (AT(x) ∧ x ≡ PNC)

3. ∀x (RR(x) → T(x))

Proof:

3.1. ∀x(CD(x) ↔ (∃p∃q(P(x, p) ∧ P(x, q) ∧ ¬(p ↔ q))))

3.2. (T(CD) → (T(CD) ∧ ¬T(CD))) → Absurd → ¬T(CD)

3.3. ∀x(RR(x) ↔ (E(x) ∧ STI(x)))

3.4. (¬T(RR) → (CD(RR) ∧ STI(RR) ∧ ¬STI(RR))) → Absurd → T(RR)

4. ∀x (RR(x) → ∃y(AT(y) ∧ D(x, y)))

5. ~(∃p, q (AT(p) ∧ T(q) ∧ CC(q, p)))

Proof:

5.1. ∀x∀y(CC(x, y) → D(y, x))

6. ∀x ((RR(x) → ∃y(AT(y) ∧ D(x, y))) → ¬BR(AT))

7. ∀x(E(x) → (E(x) ↔ O(x))) ∧ ∀x(O(x) → (O(x) ↔ E(x))) ∴ ∀x(E(x) ↔ O(x))

8. ∀x∀y(D(x, y) → C(y, x))

Proof:

8.1. ∀x (DE(x) ≠ IE(x))

8.2. ∀x ∀y ((D(x, y) ∧ ¬∃(y)) → IE(x))

8.3. ∀x ∀y (D(x, y) ∧ ∃(y)) → ¬IE(x) ∧ LB(x, y) ∧ CB(x, y))

9. ∀x(T(x) → (∃y(T(y) ∧ D(x, y))) ∧ AT(z)) → ∀x(RR(x) → CB(z, x))

10. ~(¬BR(AT) ∧ BR(x) ∧ CB(x, AT))

11. ¬BR(AT) ∧ ~(∞R) ↔ (¬L(AT) → ∃x(L(x) ∧ D(AT, x)))

Proof:

11.1. ∀x∀y((D(y, x) → L(x, y)) ∧ (¬D(y, x) → ¬L(x, y)))

12. ∀x((L(x) ∧ T(x)) → ¬L(AT)) ∴ ∀x(T(x) → (∃y(AT(y) ∧ ¬L(y))))

13. ∀x [(¬LB(x) → ¬DD(x) → P(x))] ∴ G(x)

* * *

Where:

R is regress in dependency relations.

RED is is reduced to.

U is unity.

RN is regress in numeric dependency relations.

EPR is equal producer.

EPD is equal product.

CA is causality relation.

EQ is equality relation.

EF is effect.

CC is cause.

N is nothing.

PRO is all its elements proceed from.

INF is inferiority relation.

SU is superiority relation.

EE is any entity.

W is the whole.

PA is a part.

RC is regress in causal dependency relations.

D is dependency relation.

PRI is prior to.

POST is posterior to.

CIRC is circular dependency in causality.

T is true.

AT is absolute (or independent) truth.

CT is complex truth.

ST is simpler truth.

S is simplicity.

PNC is the principle of non-contradiction.

RR is real.

CD is a subject carrying two opposite predicates.

STI is space and time.

BR is belong to the set of entities that are real.

E is existence.

O is acting by preserving itself or changing an entity.

C is entities that change.

DE is dependent entity.

IE is independent entity.

LB is is limited by.

CB is is changed by.

L is limit.

DD is is defective.

P is has all perfections.

G is is God.

* * *

Definitions:

- True or truth is a non-contradictory proposition.

- A defective being is a limited being.

* * *

Translation:

1. It is not the case that there is an infinite regress in dependency relations.

Proof 1:

1.1.1. For all numbers N not equal to unity, N can be reduced to unity.

1.1.2. There does not exist any number N such that unity can be reduced to N.

1.1.3. Therefore, there is no infinite regress in numeric dependency relations.

Proof 2:

1.2.1.1. Equal producers create equal products.

1.2.1.2. So that if a cause B, being different 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.

1.2.1.3. Therefore, the effect is not equal to the cause.

1.2.2.1. For all entities x, there is no causality relation between x and itself.

1.2.2.2. For all entities x, there is no causality relation between nothing and x.

1.2.2.3. For all entities A and B, if there is a causality relation between A and B then all the elements of B proceed from A.

1.2.2.4. Therefore, there is no superiority relation between the effect and the cause.

1.2.3.1. Therefore, there is a superiority relation between the cause and the effect.

1.2.4.1. Everything has a cause (= there is an infinite regress in the chain of dependency in causes).

1.2.4.2. The whole (the sum of all entities) has a cause.

1.2.4.3. The cause is superior to the effect.

1.2.4.4. Any part is inferior to the whole (i.e. less inclusive, less representative and less general).

1.2.4.5. 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 posterior to it, which implies that it would be unlimited (and hence not a part of the whole) before it produced its effect.

1.2.4.6. Therefore, the cause of the whole does not belong to the whole as its part, and the whole is not everything, which is absurd.

1.2.4.7. Therefore, the first premise according to which everything has a cause is false. Thus, there is no infinite regress in the chain of dependency in causes.

1.2.5.1. If A depends on B, C depends on B and A depends on C, assuming that the dependency relations are causal relations that happen in space and time, then all elements A, B and C will be prior and posterior to A, B and C, which is absurd.

1.2.5.2. 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.

1.2.5.3. Consequently, there is no circular dependency in causality.

2. For all entities x, if x is true, then there exists an entity y such that y is true and y depends on x, and it is not the case that there is an infinite regress in dependency relations, then z is an absolute truth and z is equivalent to the principle of non-contradiction.

Proof:

2.1. There does not exist any pair of entities x and y such that x is true, y is not true, and x depends on y.

2.2. For all entities x, there exists an entity y such that if x is true and it is not an absolute (or independent) truth, then y is true and x depends on y.

2.3. It is not the case that there is an infinite regress in dependency relations.

2.4. Every complex truth depends on a simpler truth.

2.5. For all entities x, if x is a truth and x is not the principle of non-contradiction, then x cannot have infinite simplicity.

2.6. Therefore, there is an absolute truth and it is equivalent to the principle of non-contradiction.

3. For all entities x, if x is real, then x is true.

Proof:

3.1. For all entities 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 not equivalent (meaning they are opposite).

3.2. If a subject carrying two opposite predicates is true, then it would be both true and not true, which leads to an absurd conclusion. Therefore, a subject carrying two opposite predicates is not true.

3.3. For all entities x, x is real if and only if x exists and is in space and time.

3.4. If it is not true that the real is true, then the real is a subject carrying two opposite predicates and both exists and does not exist in space and time, which leads to an absurd conclusion. Therefore, the real is true.

4. For all entities x, if x is real, then there exists an entity y that is an absolute truth and x depends on y.

5. There do not exist entities p and q such that p is an absolute truth, q is true, and q causes p.

Proof:

5.1. For all entities x and y, if x causes y, then y depends on x.

6. For all entities x, if x is real and stands in a dependency relation with an absolute truth, then that absolute truth does not belong to the set of entities that are real.

7. For all entities x, if x exists, then the existence of x is equivalent to x acting by preserving itself or changing another entity. And for all entities x, if x acts, then the acting of x is equivalent to the existence of x. Therefore, for every entity x, its existence is equivalent to its acting preserving itself or changing another entity.

8. For all entities x and y, if x stands in a dependency relation with y, then y causes a change in x.

Proof:

8.1. For all entities x, being a dependent entity is not equal to being an independent entity.

8.2. For all entities x and y, if x is dependent on y and y does not exist, then x is an independent entity.

8.3. For all entities x and y, if x is dependent on y and y exists, then x is not an independent entity, x is limited by y, and x is changed by y.

9. For all entities x, if x is true and there exists a y such that y is also true and y is in a dependency relation with x, and if z is an absolute truth, then for all entities x, if x is real, then x is changed by z.

10. It is not the case that there exists an entity x that belongs to the set of entities that are real, that changes an absolute truth entity that does not belong to the set of entities that are real.

11. An absolute truth does not belong to the set of entities that are real, and there is no infinite regress in dependency relations, if and only if, if an absolute truth does not have a limit, then there exists an entity that has a limit and stands in a dependency relation with the absolute truth.

Proof:

11.1. For all entities x and y, if y depends on x, then x limits y; and if y does not depend on x, then x does not limit y.

12. For all entities x, if x has a limit and it is true, then an absolute truth does not have a limit. Therefore, for all entities x, if x is true, then there exists an entity y such that y is an absolute truth and y is unlimited.

13. For all entities x, if x is not limited, then x is not defective, then x has all perfections. Therefore, x is God.