Does faith lead to the social virtue of cooperativeness? Yes, “it is… consistent with faith to believe, that [God] gave all human faculties that they might be cultivated and unfolded, not rooted out and consumed”, as said by John Stuart Mill. Furthered by Émile Durkheim, faith can lead to cooperativeness, as “… moral life permeates all the relationships that go to make up cooperation.” Finally, according to Cicero, “the co-operation of men… is secured through wisdom and virtue.” Remembering that faith is a virtue, we can safely say that, yes, faith does lead to the social virtue of cooperativeness.
A study by Joseph Henrich diagrams the key relationships, such as religion, that give rise to the linkage between group beneficial acts, such as cooperation. See below a modified version of Henrich’s Figure 2.

Yes, faith does lead to cooperativeness.
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