Paul's primary concern for the church in Philippi, as he will exhort them in 2:2, is that their love may abound [yet] more and more. Love is such a common word to us that it is easy to miss Paul's concern. Following the lead of the Septuagint, his use of love first of all points to the character of God, and to God's actions toward his people based on that character. God's love is demonstrated especially in his forbearance and kindness (1 Cor 13:4), manifested ultimately in the death of Christ for his enemies (Rom 5:6-8).
Thus the primary connotation of love is not "affection," as in the preceding phrase about Christ (Phil 1:8), but rather a sober kind of love that places high value on a person and actively seeks that person's benefit. This is what Paul now prays will abound ( be present in an abundant way) yet more and more among the Philippian believers. The rest of the prayer, after all, emphasizes love not as affection but as behavior, behavior that is both pure (stemming from right motives) and blameless (lacking offense).
The more and more indicates that Paul is not getting on them for something they lacked. His concern is rather that "selfish ambition and vain conceit" (2:3) not undermine what has long characterized them, to which 2 Corinthians 8:1-6 bears eloquent testimony. The problem is similar to that occasionally experienced by families, where love is sometimes more easily shown toward those on the outside, who are known very little and with whom one does not have constant association. But actively to love on the inside, to love those with whom one is in constant relationship and where one's own place in the sun is constantly being threatened—that can be another matter. Thus he prays that the love that has long characterized them will overflow still more and more toward one and all.
Paul also prays (item 2) for a similar increase in knowledge and depth of insight. Although this phrase grammatically modifies that your love may abound more and more, it nonetheless moves in a slightly new direction, so that Paul is in effect praying a second thing. Along with an ever-increasing love, he wants them to experience an ever-increasing knowledge (of God and his will) and moral insight.
The primary sense of the word translated knowledge is not so much "knowledge about" something as the kind of "full" or "innate" knowing that comes from experience or personal relationship. The second word denotes moral understanding based on experience, hence something close to "moral insight." Very likely this phrase is something of an abbreviated equivalent to the similar phrase in the (roughly contemporary) prayer in Colossians 1:9 ("that by means of all of the Spirit's wisdom and insight you might be filled with the knowledge of God's will"—my translation).
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