The New English Translation translates it appropriately,
10 so that at the name of Jesus
every knee will bow
—in heaven and on earth and under the earth— NET, ©1996
Many Greek learners fall into the trap of translating every occurrence of a subjunctive verb into English with a preceding “may,” “might,” or “should,” all expressing some semblance of uncertainty.
Contextually, there is no doubt that this is incorrect. Isaiah 45:23, which the apostle Paul quotes, states,
23 I have sworn by Myself. The word has gone out of My mouth in righteousness and shall not return: “Every knee shall bow to Me, every tongue shall swear.”
The verbs in Hebrew are both conjugated in the imperfect tense and appropriately translated into English by the future tense. With Yahveh Himself speaking, and in fact, swearing by Himself, there is no uncertainty that every knee will bow and every tongue will swear to Yahveh. The apostle Paul understood this when he quoted Isa. 45:23.
Daniel Wallace wrote,1
Paul here is not declaring only God’s intention in exalting Christ. Much more than that. The apostle is indicating that what God intends he will carry out. The evidence for this is that he is quoting Isa 45:23 here, though weaving it into his text in such a way that he alters it by turning it into a purpose clause (in the LXX it is a declarative statement using future indicatives after ὅτι)...
The point is simply that since Paul is not directly or formally quoting the OT here, but has worked that quotation into his text by making it the purpose of the exaltation, the subjunctive is required after ἵνα.
Moreover, he wrote,2
Not only is ἵνα used for result in the NT, but also for purpose-result. That is, it indicates both the intetion and its sure accomplishment... In other words, the NT writers employ the language to reflect their theology: what God purposes is what happens and, consequently, ἵνα is used to express both the divine purpose and the result.
The Greek verbs in Phil. 2:10 are conjugated in the subjunctive mood because of the preceding conjunction ἵνα. As others have stated, this is a classic ἵνα clause expressing purpose, also known as a telic subjunctive.
As Heinrich Meyer commented,3
Observe, moreover, how he emphasizes the universality of the divine purpose (ἵνα) with regard to the bowing the knees and confession with the tongue so strongly by πᾶν γόνυ and πᾶσα γλῶσσα, that the arbitrary limitation which makes him mean only those who desire to give God the glory (Hofmann) is out of the question.
So, how would I translate the clause? Replace “should” with “will.”
1 Wallace, p. 474
2 id., p. 473
3 Meyer, p. 84
Meyer, Heinrich August Wilhelm. Critical and Exegetical Handbook to the Epistles to the Philippians and Colossians, and to Philemon. Trans. Moore, John C. Ed. Dickson, William P. New York: Funk, 1889.
Wallace, Daniel B. Greek Grammar, Beyond the Basics: An Exegetical Syntax of the New Testament. Grand Rapids: Zondervan, 1996.
THE NET BIBLE®, New English Translation (NET). Richardson: Biblical Studies Press, 1996.