Question: Can baptized non-Catholics be virtuous?
Does the answer differ for cardinal and theological virtues? Can they be only if they have an at least implicit wish to become catholic, if so, what does this implicit wish consists in?
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The supernatural, theological virtue of faith is not necessary to have moral virtues. Moral virtues* can be acquired by purely natural means. Theological virtues must be infused by God; moral virtues are either acquired or infused.
*The principal moral virtues are the cardinal virtues: prudence, justice, temperance, and fortitude
Discussing the relation of faith with the other virtues, St. Thomas Aquinas writes (Summa Theologica II-II q. 4 a. 1 co.):
Faith, by its very nature (per se), precedes all other virtues.* […] On the other hand, some virtues can precede faith accidentally (per accidens). […] certain virtues may […] remove obstacles to belief. Thus fortitude removes the inordinate fear that hinders faith; humility removes pride, whereby a man refuses to submit himself to the truth of faith. The same may be said of some other virtues, although there are no real virtues, unless faith be presupposed, as Augustine† states (Contra Julian. iv, 3).
* ∵ "the end is the principle in matters of action" and "the last end [God] must of necessity be present to the intellect before it is present to the will" (faith resides in the intellect)
† cf. II-II q. 23 a. 7 ("Whether any true virtue is possible without [the supernatural, theological virtue of] charity?"), which quotes from St. Augustine's Against Julian, a Pelagian. E.g., St. Augustine says: "virtues must be distinguished from vices, not by their functions, but by their ends". He argues that since God is our last end, only the virtues which help us attain God are truly virtues; "Christ died in vain if men without the faith of Christ through other means or power of reasoning may arrive at true faith, at true virtue, at true justice, at true wisdom."
St. Thomas's question "Of the Connection of Virtues" (I-II q. 65), esp. a. 2 ("Whether moral virtues can be without charity?"), explains the acquired/infused virtue distinction and shows how the theological virtues (esp. the greatest, charity) elevate the other virtues:
it is possible by means of human works to acquire moral virtues, in so far as they produce good works that are directed to an end not surpassing the natural power of man: and when they are acquired thus, they can be without charity [or the other theological virtues], even as they were in many of the Gentiles. But in so far as they produce good works in proportion to a supernatural last end, thus they have the character of virtue, truly and perfectly; and cannot be acquired by human acts, but are infused by God.
St. Thomas (Summa Theologica I-II q. 63 a. 4) gives the example of acquired temperance vs. infused temperance:
For instance, in the consumption of food, the mean fixed by human reason, is that food should not harm the health of the body, nor hinder the use of reason: whereas, according to the Divine rule, it behooves man to "chastise his body, and bring it into subjection" (1 Cor. 9:27), by abstinence in food, drink and the like. It is therefore evident that infused and acquired temperance differ in species (specie*); and the same applies to the other virtues.
*Virtues are habits, and habits are distinguished by their objects. In this example, the object of acquired temperance is bodily health; that of infused temperance is subjugating the body to the soul for God's sake.
quoted on pp. 33-34 of Fr. Royo Marín, O.P.'s Theology of Christian Perfection
Explaining the interrelation and growth of the virtues in a single man, St. Thomas gives the analogy in Summa Theologica I-II q. 66 a. 2 ("Whether all the virtues that are together in one man, are equal?") co. that one's virtues grow together like the fingers in a hand:
all the virtues in one man are equal with an equality of proportion, in so far as their growth in man is equal: thus the fingers are unequal in size, but equal in proportion, since they grow in proportion to one another.
Summa Theologica III q. 69 a. 4 ("Whether grace and virtues are bestowed on man by Baptism?") co.:
As Augustine says in the book on Infant Baptism (De Pecc. Merit. et Remiss. i) "the effect of Baptism is that the baptized are incorporated in Christ as His members." Now the fulness of grace and virtues flows from Christ the Head to all His members, according to Jn. 1:16: "Of His fulness we all have received." Hence it is clear that man receives grace and virtues in Baptism.