To determine how much faith Matthew Arnold has in his proposed solution in the poem "Dover Beach," we must first understand the problem and his solution.
The problem, as the poet sees it, is the waning of faith. The speaker and his companion are listening to the ebb and flow of the waves against the pebbles on the beach, and the speaker hears an "eternal note of sadness" in the sound. He is reminded of Sophocles, who once heard the sea and thought about "human misery." But the speaker thinks about faith and how it was once "at the full" all over the earth. Now, though, he thinks that he hears only the "melancholy, long, withdrawing roar" of faith retreating. Here is the problem.
The speaker's solution is found in the last stanza. "Ah, love," he exclaims, "let us be true / To one another!" The world is dark, without joy or love or certitude or peace. War surrounds them, perhaps literally but certainly metaphorically. Only their love offers hope. Yet the poet doesn't seem too certain about his solution. He asks his beloved for faith, yet he ends the poem on a bleak note with a "darkling plain" where "ignorant armies clash by night." The love between the speaker and his beloved cannot stop the darkness or the battles. It cannot seem to add any light or peace or joy to the world. It will not return faith to the world. Indeed, it doesn't seem to be much of a solution at all, and perhaps Arnold knows that.