When we look at verse 12, we run into a MAJOR exegetical uncertainty:
The verb translated as "exercise authority over" (authenteo) is only used here in the NT, and its meaning is HOTLY contested.
TWO things that ARE sure about its meaning--it is NOT the normal word for "authority" (exousia), "exercising authority" (exousiazo), or "power" (kyrieuo); and it is NOT a 'good' thing (suitable for ANYONE to do--even males!)...so Scholer in WS:WAB:204-205:
Another factor basic to the interpretation of 2.11-12 concerns Paul's use of the unusual word authentein (translated "to have authority over" RSV) in the second injunction (2:12). This is the only occurrence of this word in Paul's writings and, indeed, in the entire New Testament. The word is not frequently used in ancient Greek literature. The precise meaning of authentein and its use in 2:12 cannot be completely resolved at this time; scholars are currently in an extended debate on the issue.
Traditionally, authentein has been understood to connote a sense of "domineer" or "to usurp authority" and the term is even associated with murder. Although not all of the evidence and arguments have been fully assessed, two points seem relatively certain. First, the term is unusual. If Paul were referring to the normal exercise of authority, his otherwise constant exousia/exousiazo ("authority/to exercise authority") vocabulary would most likely have been used. The choice of such an unusual term itself indicates that Paul intended a different nuance or meaning. Second,...many uses of the term seem rather clearly to carry the negative sense of "domineer" or "usurp authority." Thus I see the injunctions of 2:11-12 as directed against women involved in false teaching who have abused proper exercise of authority in the church (not denied by Paul elsewhere to women) by usurpation and domination of the male leaders and teachers in the church at Ephesus.
It is VERY important to point out here that it is PURE FOLLY to base an entire doctrine affecting half the human race (!)--"women should not have authority over men"--on the basis of ONE SINGLE VERSE, and even worse--a single verse where the most important verb is (1) unusual; (2) negative; and (3) not even understood clearly!
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The word 'teach' in the verse ("neither teach nor authentein a man") has a major issue associated with it, as well...
The verb is TOO 'big" to NOT be radically restricted in scope by whatever authentein means.
The situation is this. "Teach" takes an object in the accusative case and authentein takes an object in the genitive case. "Man" is in the genitive case, and is therefore the object of authentein. That means that 'teach' (unless it is 'connected' tightly to authentein) is UNRESTRICTED in scope. Paul would be prohibiting women from teaching anybody at any time--in direct contradiction to his plentiful commands for believers to teach/instruct/train each other (e.g. Tit 2.4; 1 Cor 14.26; Col 3.16). So the scope of the application must be limited somewhat in the context.
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The "conjunction" data concerns the fact that there is a 'but' between verse 11 and 12. So, we have Paul saying something like "Let the women study/learn as proper students...BUT I am not (currently) letting them (the students, having been under the influence of the false teachers--cf. 1.4-7; 5.13; 2 Tim 3.6) teach nor letting them 'overthrow' their teachers (until they are ready--cf. 2 Tim 2.2)". [The fact that 'teach' is present, active, indicative is indecisive as to whether it is a short-term or long-term command--the data is very divided in the extant literature.]
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The word "man" in the authentein clause seems suggestive in context.
This appears to be a very gender-specific word (andros), suggesting that authentein was ONLY DIRECTED (whatever it was) at MEN--not at WOMEN. And, since the passage is apparently ABOUT women, we have women authentein-ing MEN only. Since there had been or were godly women teachers already in Ephesus (e.g. Priscilla and the deaconesses of 3.11), this would make a case that the immodest (3.9), gaudy (3.9), self-righteous (3.10b), unlearned (3.11), and disruptive (3.11) women under discussion in 2.9-15 (no doubt a subset of the women in Ephesus, 3.11; 5.2-9) were SPECIFICALLY teaching something about MEN that led them to seek to authentein them.