Matthias’s argument from contingency

[Matthias123]

1. 
   *]If and only if a non-contingent contingent entity exists ξ then for every necessitator contingent to ξ (x), x is a member of the set of particulars explaining ξ’s necessity S, and x is not necessary
   *]If and only if a necessary contingent entity exists ξ then for every necessitator contingent to ξ (x), x is a member of the set of particulars explaining ξ’s necessity S, and x is necessary.
   
   *]A necessary contingent entity that was one time not necessary exists ξ
   
   *]It is not possible for x or ξ to necessitate themselves.
   
   *]Therefore we need to postulate the existence of an absolutely necessary entity ₦ to explain the necessity of ξ.
   1. ¬□ξ ⇔ ∀x(x ∈ Ѕ ∧ ¬□ x)
      a. Ѕ: ¬□ …x4 → x3 →x2 → x1→x0] ⇒¬□ξ
   2 □ξ ⇔ ∀x(x ∈ Ѕ ∧ □ x) ⇔ ₦
   a. Ѕ: □ …x4 → x3 →x2 → x1→x0] ⇒□ξ
   b. Ѕ: …□ x4 → □ x3 →□ x2 → □ x1→□ x0] ⇒□ξ
   c. ₦ → Ѕ: …□ x4 → □ x3 →□ x2 → □ x1→□ x0] ⇒□ξ

 

[NowAgnostic]

1. 
   *]If and only if a non-contingent contingent entity exists ξ then for every necessitator contingent to ξ (x), x is a member of the set of particulars explaining ξ’s necessity S, and x is not necessary
   *]If and only if a necessary contingent entity exists ξ then for every necessitator contingent to ξ (x), x is a member of the set of particulars explaining ξ’s necessity S, and x is necessary.
   
   *]A necessary contingent entity that was one time not necessary exists ξ
   
   *]It is not possible for x or ξ to necessitate themselves.
   
   *]Therefore we need to postulate the existence of an absolutely necessary entity ₦ to explain the necessity of ξ.
   1. ¬□ξ ⇔ ∀x(x ∈ Ѕ ∧ ¬□ x)
      a. Ѕ: ¬□ …x4 → x3 →x2 → x1→x0] ⇒¬□ξ
   2 □ξ ⇔ ∀x(x ∈ Ѕ ∧ □ x) ⇔ ₦
   a. Ѕ: □ …x4 → x3 →x2 → x1→x0] ⇒□ξ
   b. Ѕ: …□ x4 → □ x3 →□ x2 → □ x1→□ x0] ⇒□ξ
   c. ₦ → Ѕ: …□ x4 → □ x3 →□ x2 → □ x1→□ x0] ⇒□ξ

1. 
   I suspect there are a few typos here. What is a “non-contingent contingent entity” or a “necessary contingent entity”. And then, 3 onwards don’t correspond with any line in your symbolic notation. And what is a “necessitator”.

 

[Matthias123]

I suspect there are a few typos here. What is a “non-contingent contingent entity” or a “necessary contingent entity”. And then, 3 onwards don’t correspond with any line in your symbolic notation. And what is a “necessitator”.

The lines in my symbolic notations are only for the axioms that are really crucial. There is no reason to symbolize the rest, unless I just wanted to confuse people.

“non-contingent contingent entity”

Yeah this is a typo, this is non-necessary contingent entity.

“necessary contingent entity”

My language is rather vague, they are the only words I could think up to express the content. When I say “necessary contingent entity” I mean a necessity. entity that can’t explain it’s own necessity.

A necessitator is something that gives something necessity, but it too cannot explain it’s own necessity.

Then I go on to show that if ξ was at one time not-necessary, and is necessary now, the only was for this to happen is for ₦ to exist as necessary by it’s own essence.

Also remember that the necessity of every x is contingent on the necessity of ξ so that if ξ was not necessary then no x is necessary.

 

[Matthias123]

My language is rather vague, they are the only words I could think up to express the content. When I say “necessary contingent entity” I mean a necessary entity that can’t explain it’s own necessity.

I am bad for typos today

 

[punkforchrist]

Hi Matthias,

The best word, in my opinion, to express metaphysical (but not necessarily logical) contingency is “dependent.” So, let’s say that A exists in all possible worlds. This would make A logically necessary. However, A may also be dependent on B, so that in every possible world in which B is instantiated, A is reliant upon B and exists wherever B exists.

This is one of the reasons why the TCA isn’t suspect to the “modal collapse objection” that some of the traditional Leibnizian versions of the cosmological argument run into.

Of course, a stress on “dependency” changes the argument a bit. Instead of saying that the set of all contingent entities is itself contingent (this is certainly true), we are now saying that the set of all dependent things is dependent (which I agree with, but its demonstrability requires more argumentation). We could illustrate the soundness of this claim by various analogies. If, say, the set of one type of dependent entity relies upon the set of another, and vice-versa, then the union of both sets is insufficient to explain the existence of this union.

For example, the set of all oak trees is dependent upon the set of acorns. Yet, the reverse is also true. What this shows is that if we suggest that oak trees and acorns are the only members of the set causing the union of oak trees and acorns, then we are engaging in circular reasoning, since we do not have an explanation for the existence of the union as a whole.

I think an easier way of arguing this point is not in terms of explanation, but in terms of causation qua rank or source. If the regress of dependent beings does not have a source in an independent (and arguably self-existent) being, then we have an infinite regress of dependent beings. However, given the finitude of dependent beings, it is much more likely that the regress as a whole is likewise finite (this is an inductive argument). An independent First Cause is then posited to end the regress.

 

[hatsoff]

I have changed your OP according to your follow-up post. Corrections appear in boldface:

A necessitator is something that gives something necessity, but it too cannot explain it’s own necessity.

1. 
   *]If and only if a non-necessary contingent entity exists ξ then for every necessitator contingent to ξ (x), x is a member of the set of particulars explaining ξ’s necessity S, and x is not necessary
   *]If and only if a necessity entity that can’t explain it’s own necessity exists ξ then for every necessitator contingent to ξ (x), x is a member of the set of particulars explaining ξ’s necessity S, and x is necessary.
   
   *]A necessary contingent entity that was one time not necessary exists ξ
   
   *]It is not possible for x or ξ to necessitate themselves.
   
   *]Therefore we need to postulate the existence of an absolutely necessary entity ₦ to explain the necessity of ξ.
   
   **If ξ was at one time not-necessary, and is necessary now, the only was for this to happen is for ₦ to exist as necessary by it’s own essence.
   
   Also remember that the necessity of every x is contingent on the necessity of ξ so that if ξ was not necessary then no x is necessary.**
   1. ¬□ξ ⇔ ∀x(x ∈ Ѕ ∧ ¬□ x)
      a. Ѕ: ¬□ …x4 → x3 →x2 → x1→x0] ⇒¬□ξ
   2 □ξ ⇔ ∀x(x ∈ Ѕ ∧ □ x) ⇔ ₦
   a. Ѕ: □ …x4 → x3 →x2 → x1→x0] ⇒□ξ
   b. Ѕ: …□ x4 → □ x3 →□ x2 → □ x1→□ x0] ⇒□ξ
   c. ₦ → Ѕ: …□ x4 → □ x3 →□ x2 → □ x1→□ x0] ⇒□ξ

1. 
   This still does not make much sense. For example, you say that “ξ” is an “entity” of some sort. Okay, but then what is “ξ (x)” ? And, when you say “necessary” entity, do you mean one for which the formula “∃ξ, where ξ is a particular entity” is necessarily true—that is, true in all possible worlds—? If so, then to say that such a formula is “non-necessary” and “contingent” is redundant, since all contingently true formulas are not necessarily true.
   
   Also, what is “a necessity entity that can’t explain it’s own necessity”?

 

[Matthias123]

I have changed your OP according to your follow-up post. Corrections appear in boldface:

This still does not make much sense. For example, you say that “ξ” is an “entity” of some sort. Okay, but then what is “ξ (x)” ? And, when you say “necessary” entity, do you mean one for which the formula “∃ξ, where ξ is a particular entity” is necessarily true—that is, true in all possible worlds—? If so, then to say that such a formula is “non-necessary” and “contingent” is redundant, since all contingently true formulas are not necessarily true.

Also, what is “a necessity entity that can’t explain it’s own necessity”?

If I have failed my failure is due to my lack of understanding of symbolic logic.

"Okay, but then what is “ξ (x)”.

There is no ξ (x) that would mean that ξ has the property of causing necessity in another entity – this is not so for our reflection as I stop at ξ

“∃ξ, where ξ is a particular entity” – ξ is what the entities in set S’s necessity is contingent on. All I am saying is that if ξ is not necessary all of the entities in set S are not necessary. This is true for every “necessitator” (An entity that causes another entity to be necessary.) Think of ξ as an effect.

The type of scenario this argument is aiming at is when an entity is made necessary in relation to something else. Say we have a house that’s roof is necessarily depended on its walls. Now these particular walls are only necessary for on this particular roof once the house has been actualized. Now if this house has not been actualized, then these particular walls are not necessary to this particular roof, as the effect has not yet been actualized, then even if the necessitator regresses to actual infinity, the fact that these entities were not necessary before and now are necessary requires a supreme necessary.

Another example would be forming a card castle. The cards on the bottom of the card castle are not necessary to the effect of the cards on the top standing, until the cards are in a configuration to make this necessary.