The opening of the letter to the Ephesians and Magnesians: Ἰγνάτιος, ὁ καὶ Θεοφόρος, and five other of the "genuine" letters.
The second term reads, "also [called] Theophoros," like many people in nearly any age he had a second name or titie. For instance, almost every one of Jesus Twelve Disciples has at least two identifiers. E.g. Simon, Barjona, Peter. John, BarZebedee, Boanerges. Etc. The meaning of any name applied to an individual does not call for literalism.
Moreover, "God-inspired" as a translation of theophoros truly needs some rationale offered. Phorew is a verb meaning "to bear," and the name is sometimes rendered in translation, "God-bearer," that is accoutered with God (so, the word in Rom.13:4, "...he bears not in vain the sword;" or in Jas.2:3, "...him that wears fine clothes." Another plausible connection is the noun phoros, meaning "tribute;" so this name might be glossed, "a tribute to God," or somesuch.
According to CBTEL (Cyclopedia, 1895, an old multivolume reference work) Ignatius himself offered this meaning for Theophorus: "a title which he explained to the emperor Trajan as meaning 'one who has Christ in his heart.'” Given this notice (presumably genuine), why did the translator you read offer such a representation as he did? Does he take the term "inspired" not as in 2Tim.3:16, θεόπνευστος (..."all Scripture is God-breathed") but more like "motivated?" Who knows...
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My point in acknowledging what you acknowledged, about being new to this field, is hopefully so you will recognize that a claim like "no one disputes X," or "this means that," is dilettantish. By what authority could you possibly say so? If it is the opinion of someone else, then folks have a right to assess (so far as possible) the competence of that authority, and have the opportunity to propose an alternate (if desired) of equal or even greater cachet.
When you point to the (apparent) absence of contentions over expressions in this significant Father Ignatius (who is so largely because he wrote a few letters and they survived), we have not even begun to compare his writings to other material from that age; nor have we established the true meaning (what HE intended to communicate) by the terms he used.
Many times, later church authorities have "read into" previous writings and terms used their own developed doctrines. This is pure anachronism; such as when all the trappings and accretions of rank of the monoepiscopate "bishop" of the 4th century and later is attributed to a simpler "bishop" like Ignatius.
Then, must you read Ignatius as if he must mean by his strong terminology views that are quite in keeping with the later monoepiscopal developments? Or, is one perfectly accurate to assess his terminology against the background of the NT, and to look for the basis of his statements? The former method is useful for justifying the "progress" of hierarchical church orders. Ignatius is read "backwards," in that case.
On the other hand, if we read Ignatius primarily against previous revelation, we take note of where the emphasis has turned in his day. We are free to assess the "progress" or "regress" of matters by his short day, in light of the apostolic beginnings.
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There's yet another issue that has yet to be raised in this discussion. Convenient to both the Roman view and the anti-authority views of much modern evangelicalism is the impression that the only two choices are 1) church authority, and 2) Bible/personal authority. While Rome professes a triple-authority (Scripture/Tradition/Magisterium), in practice there can be only one final authority, which for them is the Magisterium--because they assert the current accepted meaning of any Scripture, and which tradition is meaningful or what some tradition means.
The radical individualists are Rome's ideal foils. They are not--indeed they are proud not to be--beholden (they think) to any tradition. They are, indeed, autonomous interpreters of Scripture, and think of themselves as pure biblicists who need no church. For them, institutional church is largely a fellowship gathering, and not an exhibit of the eternal Kingdom complete with order and discipline; exercised through officers that bear a measure of Christ's authority; which power is exercised through the combined ministry of Word and Spirit.
The Reformation church is neither of these things, but a third option, offensive to both the Roman and the Biblicist perspective. Rome doesn't want there to be any church that challenges her over "whose history" goes from A.D. 100-1500. They don't mind so much the church that "disappeared" for 1400yrs or so, according to the sectarians. People with no historical roots can be more easily persuaded first of Rome's historical claims; followed by her interpretive claims as to Scripture; and ultimately her raw claims to know what is spiritual good on no other basis than sheer weight of "inevitability."
The magisterial Reformation neither rejects church history, nor accepts Rome's "inevitability" by means of that history. The best and most faithful of that history is our history, and she cannot take it from us. The Reformers did not exclusively go back to Scripture, except as the final authority. But they appealed constantly to the ancient fathers, and the best of medieval Christianity as well.
Calvin's encyclopedic knowledge of the fathers (especially Augustin) was legendary. In fact it has been said, it was the Reformers who basically invented Patristics as a discipline. Because with quotes from the ancients they challenged the ignorant priests who only parroted: "Quod ubique, quod semper, quod ab omnibus creditum est," and knew none of the facts of the father's faith or the text of Scripture.
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That said, we may ask the question: Is Ignatius' ecclesiology "too high?" Does he too-highly magnify his office as a bishop? Any claim that Ignatius' official authority (authority of office) is at variance with Scriptural authority is a contention that must be specified and demonstrated.
Please give the line where Ignatius claims infallibility as (or for) a bishop. I haven't found it. It doesn't appear in the passage you cite. In that passage, Ignatius seems to counsel the Ephesians to honor the derived authority of her bishops, as those who occupy that position bearing authority delegated down from Christ.
Indeed, this is not typical American congregationalism. Most churchgoing people today think of the pastor as someone who is given his authority "from the people." that is, from below. They may think of him, afterward, as wielding unchallenged authority for the collective. They may think their only choice in disagreement is to "leave," thereby reclaiming their little bit of autonomous power. But this view is not the view of the Presbyterians, or the Reformed, or the Lutherans, Anglicans, etc.
To the typical American evangelical, our view of church authority simply makes us "Roman-lite." The irony is, it is more difficult (though not impossible) for a pastor to exercise tyranny in a Presbyterian church than in a Congregationalist. This is because a pastor like myself is always under authority of my peers in the sister churches; and the congregation is ruled jointly by its own non-clergy elders together with its pastor. I have no delegated "authority" UP from the congregation, by which to rule them all through the exercise of a collective "will."
As minister, I exercise the authority of Christ, giving Word-based advice to the elders, and publicly and privately ministering the Word. I have no authority to speak where the Word has no counsel. Where the Word gives clear guidance, I have full authority to tell the people the mind of God as best I can. And if they believe it is so for me, as a function of my office, then they should pay no mind to my youth (1Tim.4:12), not that I have youth anymore...), but honor that Word as it "really is, the Word of God," not the word of men, 1Ths.2:13.
Thus do I read Ignatius' advice. I do not interpret him as going beyond the Word of God, at least not in your excerpt. I do not interpret him as saying to the Ephesians they should act all "spooked" at the sheer holiness of their minister/bishop, as if he were God on earth. Of course, the "bishop must be blameless," 1Tim.3:2; Tit.1:7. He must be an example as 1Tim.4:12 says, in love, spirituality, faith, and purity. But the deference Ignatius speaks of is not to his person, as rather to his office and his preaching. If he is to be "obeyed," surely that must be exclusively in matters wherein he has the authority from God and Christ to give command; and not simply at a whim.
I hope these are thoughts by which you are able to read these writers freshly. The translation should be quality, but do not ignore good teachers who synthesize this material, together with deep understanding of the times in which they were written, the currents of thought, the social realities of the day. Always keep in the fore the final authority of Scripture.