In Catholicism, what are the arguments for the reasonableness (see body) of an eternal hell?

Faith & Reason

Not all divinely revealed truths can be known without Divine Revelation. For example, the fact that God is triune (Three Persons of one Divine Substance) cannot be known by natural reason,¹ even though it is not against reason. The existence of God, however, can be known by natural reason,² and so can His divinity.³
^{¹St. Thomas Aquinas's Summa Theologica I q. 32 a. 1 article "Whether the trinity of the divine persons can be known by natural reason?"
²ibid. q. 2 a. 2 "Whether it can be demonstrated that God exists?"
³Rom. 1:20: "For his invisible things, from the creation of the world are seen, being understood by those things that are made: his eternal power also and Divinity: so that they are inexcusable."}

This is an ambiguous statement: "the Church teaches that the truths of the faith are intelligible in the light of natural reason"

What it should say is: "the Church teaches that some of the truths of the faith are intelligible (i.e., "discoverable" or understandable) in from the light of natural reason, and none of them are against reason," although most are above human reason.

St. Thomas Aquinas gives two uses of reason in his Summa Theologica I q. 32 a. 1 article "Whether the trinity of the divine persons can be known by natural reason?":

Reason may be employed in two ways to establish a point:

1. …for the purpose of furnishing sufficient proof of some principle…

and

2. not as furnishing a sufficient proof of a principle, but as confirming an already established principle, by showing the congruity of its results

Way #1 is not possible for all truths of the faith. Way #2 is how reason is employed for understanding the Trinity and other truths of the Faith not "discoverable"/understandable/intelligible by human reason alone.

See the First Vatican Council's document on faith and reason, Dei Filius.

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Determining Whether a Dogma is Reasonable using Human Reason Only

Proving the Reasonableness of a Dogma

Apart from faith, there are at least the following ways to show that a dogma is in accordance with reason:

1. Come up with a proof of way #1 reasoning. This is the strongest reasoning on a merely human level.
   ^{Such a proof is not known to exist for the dogma of hell's eternity, but that does not necessarily imply someone will not discover such a proof in the future. It is also not defined dogma that such a proof cannot be found.}

2. Show that reason is in accordance with the dogma in every possible way.* This isn't humanly possible because there might be an infinite number of ways to show.
   ^{*This would be "to prove a negative, which isn't possible," as per earlier DJClayworth's comments.}

3. Offer individual examples of way #2 reasoning showing where this dogma is in accordance with reason.
   ^{This is the weakest way of reasoning because it only offers probable reason to believe the dogma does not contradict reason.}

Proving the Unreasonableness of a Dogma

Apart from faith, there is least the following way of showing that a dogma contradicts reason:

1. Come up with a counterexample showing that it does contradict reason (e.g., Carol Meyer's reasoning that the eternity of hell contradicts God's love*).
   ^{*This is easily disproved by noting that God is infinitely just. Plus, is her reasoning even "apart from faith"?}

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Showing the Eternity of Hell is in Accordance with Reason

^{It is a de fide ("of faith") dogma that hell is eternal.}

A few such "individual examples of way #2 reasoning" are the following:

1) One can prove from natural reason the eternity of the soul* and conclude from that that it is reasonable that a soul could exist eternally in a state deprivation of the Beatific Vision in hell.
^{*St. Thomas Aquinas argues that the human soul is "subsistent" (eternal) in Summa Theologica I q. 75 a. 2.}

2) As just kings have prisons with life sentences or impose capital punishment [there is no coming back from that]; so must God, a King, Who is infinitely just, have hell with eternal sentences (cf. this).

3) Since the will of the damned is set in its ways toward evil, we could ask: "Could someone's will never change toward the good?" Analogously, we could also ask: "Could someone be blind forever?" In answer to this, St. Thomas Aquinas would say "Yes." He compares God's withdrawal of His grace from the damned to the lack of sight of a blind man, in Summa Theologica I-II q. 85 a. 2 ("Whether the entire good of human nature can be destroyed by sin?") ad 3:

Even in the lost [i.e., the damned] the natural inclination to virtue remains, else they would have no remorse of conscience. That it is not actualized is owing to their being deprived of grace by Divine justice. Thus even in a blind man the aptitude to see remains in the very root of his nature, inasmuch as he is an animal naturally endowed with sight: yet this aptitude is not actualized, for the lack of a cause capable of actualizing it, by forming the organ requisite for sight.

Thus, just as it is possible for a blind man to lack sight, so is it reasonable that the damned lack grace.

4) The preservation of moral and social order would not be sufficiently provided for, if men knew that the time of trial were to be continued after death. (cf. this). Or that no matter how long, the punishment would eventually end.

The more arguments "of way #2 reasoning," the closer we approach to the more perfect reasoning of the second list point under the heading "Proving the Reasonableness of a Dogma" above.