Verse 15 contains an implicit criticism of Paul’s opponents, that they have exceeded their rights to boast because they have based their claims on someone else’s (Paul’s) work.
The first half of verse 15 is basically a restatement of 10.13a. If Paul were to boast and take credit for the missionary work of other evangelists, he would be taking credit for work he had not done.
In other men’s labors may be understood to include the work of women as well as men. Good News Translation, New Revised Standard Version, and Revised English Bible all use a more inclusive word, “others.” Revised English Bible translates 10.15a as follows: “And we do not boast of work done where others have laboured, work beyond our proper sphere.”
But: this seems to mark the contrast between what Paul does not do (boast beyond limit) and what he does do (hope). Good News Translation introduces the change with “instead.”
The Greek grammar of 10.15b is rather difficult, but the point seems to be that, as the Corinthians’ faith grows, Paul and his co-workers will be able to have an even greater ministry among the Christians there. Revised English Bible says “Our hope is rather that, as your faith grows, we may attain a position among you greater than ever before, but still within the limits of our sphere.”
Our field among you may be greatly enlarged: is literally only “among you to be magnified.” The subject of the passive verb “magnified,” translated enlarged, is not actually specified, but most scholars seem to feel that the implied subject is the “area of activity” (New International Version) or field of Paul and his associates. While the meaning of the verb is thought to be enlarged, “magnified” can also mean “praised,” as it probably does here. In this case the implied subject is Paul and his colleagues themselves rather than their field of service. Of the eight occurrences of this verb in the New Testament, six of them have to do with praise. For example, it is translated by the verb “extol” (Acts 10.46; 19.17) and “honor” (Phil 1.20), or “hold in high honor” (Acts 5.13). Anchor Bible therefore translates “[our hope is that as your faith grows] we may be abundantly praised among you.” If this interpretation is followed, the passive construction may be avoided by saying something like “that you may give us abundant praise” or “that you may abundantly praise our work.” Revised English Bible seems to follow this same basic interpretation when it says “that our influence may also grow among you and overflow.” This meaning may be more simply worded as in the footnote alternative of CEV: “you will praise us even more because of our work among you.”
In addition to taking “magnified” as “praised,” Anchor Bible also includes the final clause “in accord with our jurisdiction” and notes that, by repeating a part of the phrase in verse 13, Paul is clarifying the idea of being praised on the basis of the principle presented in that verse.
Quoted with permission from Omanson, Roger L. and Ellingworth, Paul. A Handbook on Paul’s Second Letter to the Corinthians. (UBS Handbook Series). New York: UBS, 1993. For this and other handbooks for translators see here .