Jonathan Edwards (1703-1758 AD) was the grandson of Solomon Stoddard (1643 - 1729 AD). Stoddard was called the 'congressional pope of New England' and it was he who established the Half-Way Covenant that was accepted everywhere in New England as the standard for Church Membership. Under the Half-Way Covenant, evidence of a conversion experience was not required to acquire full membership into the church; and because only members of the Church were allowed to have their children baptized, the requirements for obtaining full communion membership was limited to acknowledging what the Church believed without necessarily giving proof that the membership candidate had actually been born again (ie regenerated). So consequently, anyone who desired Church Membership would be granted it, and therefore anyone could have their children baptized, creating a cycle that allowed for individuals to be fully members of a Church without truly believing the Church's dogma. This is why Jonathan Edwards opposed the Half-Way Covenant, and this is also the reasons Edwards was eventually removed from his Church by his congregation.
Edwards wrote a series of short books that he hoped to overturn the Half-Way Covenant's universal acceptance among New England Congregational Churches. Edwards acknowledged that he at first acquiesced to the Half-Way Covenant but later came to the conviction that it was not biblical sound. His congregation used this confession against him when he was removed from his North Hampton Church. There are three particular books on the Communion Controversy by Edwards that are contained in Yale's Works of Jonathan Edwards, Ecclesiastical Writings (WJE Online Vol. 12) , Ed. David D. Hall, that are most relevant to the Half-Way Covenant and this controversy: An Humble Inquiry into the Rules of the Word of God, Concerning the Qualifications Requisite to a Complete Standing and Full Communion in the Visible Christian Church, Misrepresentations Corrected, and Truth Vindicated, and Narrative of Communion Controversy. (The titles may be abbreviated referred to An Humble Inquiry, Misrepresentations Corrected and Narrative respectively.)
The core point of rejection the Half-Way Covenant was that it did not require a person to be converted, born again, regenerated, but only acknowledge the Church's Dogma. Here is an extended quotation from An Humble Inquiry Part II Section 11, where Edwards explains what it means that a man should examine himself before he eats the Lord's Supper. This discussion is particular important today due to the popularity of Federal Vision and padeocommunion which are remarkably similar to the Half-Way Covenant.
11. When the Apostle says, 1 Corinthians 11:28, "Let a man examine himself, and so let him eat…," it seems to be much the most reasonable to understand it of trying himself with regard to the truth of his Christianity, or reality of his grace; the same which the same Apostle directs the same Corinthians to, in his other epistle 2 Corinthians 13:5, where the same word is used in the original. The Greek word (δοϰιμαζέτω) will not allow of what some have supposed to be the Apostle's meaning, viz. that a man should consider and inquire into his circumstances, and the necessities of his case, that he may know what are the wants he should go to the Lord's Table for a supply of. The word properly signifies "proving" or "trying" a thing with respect to its quality and goodness, or in order to determine whether it be true and of the right sort. And so the word is always used in the New Testament; unless that sometimes 'tis used as it were metonymically, and in such places is variously [-- 259 --] translated, either "discerning," or "allowing," "approving," "liking," etc. these being the effects of trial. Nor is the word used more frequently in the New Testament for any sort of trial whatever, than for the trial of professors with regard to their grace or piety. The word (as Dr. Ames, in his Catecheséòs Sciagraphia,5 and Mr. Willard in his Body of Divinity,6 observe) is borrowed from goldsmiths, properly signifying the trial they make of their silver and gold, whether it be genuine or counterfeit: and with a manifest allusion to this original application of the word, is often used in the New Testament for a trying the piety of professors. 'Tis used with this view in all the following texts. 1 Peter 1:7, "That the trial of your faith, being much more precious than of gold that perisheth, though it be tried by fire, might be found unto praise," etc. 1 Corinthians 3:13, "The fire shall try every man's work of what sort it is." James 1:3, "The trying of your faith worketh patience." 1 Thessalonians 2:4, "God who trieth our hearts." The same word is used in 2 Corinthians 8:8, "To prove the sincerity of your love." So, Galatians 6:3-4, "If any man thinketh himself to be something, when he is nothing, he deceiveth himself: but let every man prove his own work." In all these places there's the same word in the Greek with that in the text now under consideration.
When the Apostle directs professing Christians to "try" themselves, using this word indefinitely, as properly signifying the examining or proving a thing whether it be genuine, or counterfeit, the most natural construction of his advice is, that they should try themselves with respect to their spiritual state and religious profession, whether they are disciples indeed, real and genuine Christians, or whether they are not false and hypocritical professors. As if a man should bring a piece of metal that had the color of gold, with the impress of the king's coin, to a goldsmith, and desire him to try that money, without adding any words to limit his meaning, would not the goldsmith naturally understand, that he was to try whether it was true gold, or true money, yea or no?
But here it is said by some, that the context of the passage under [-- 260 --] debate (1 Corinthians 11:28) does plainly limit the meaning of the word in that place; the Apostle there speaking of those things that had appeared among the communicants at Corinth, which were of a scandalous nature, so doubtless unfitting 'em for the Lord's Supper; and therefore when the Apostle directs 'em to examine or prove themselves, 'tis but just, to suppose his meaning to be, that they should try whether they ben't disqualified by scandal. To this I answer, though the Apostle's putting the Corinthians upon trying themselves, was on occasion of the mentioning some scandalous practices found among them, yet this is by no means any argument of its being only his meaning, that they should try themselves whether they were scandalous persons; and not, that they should try whether they were true genuine Christians. The very nature of scandal (as was observed before) is, that which tends to obscure the visibility of the piety of professors, and wound others' charity towards them, by bringing the reality of their grace into doubt; and therefore what could be more natural, than for the Apostle, when mentioning such scandals among the Corinthians, to put them upon trying the state of their souls, and proving their sincerity? This is certainly the case in this Apostle's directing the same persons to prove themselves (2 Corinthians 13:5), using the same word there, which he uses here, and giving his direction on the like occasion. For in the second epistle (as well as in the first) his putting them on examining and proving themselves, was on occasion of his mentioning some scandals found among them; as is plain from the foregoing context. And yet there it is expressly said, that the thing concerning which he directs them to prove themselves, is, whether they be "in the faith," and whether Christ is in them. Nor is there anything more in the preceding context of one place, than in that of the other, obliging or leading us to understand the Apostle to intend only a trying whether they were scandalous, and not whether they were sincere Christians.
And as to the words following in the next verse, "For he that eateth and drinketh unworthily, eateth and drinketh judgment to himself, not discerning the Lord's body": these words, by no means, make it evident (as some hold) that what the Apostle would have them examine themselves about, is, whether they have doctrinal knowledge, sufficient to understand, that the bread and wine in the sacrament signify the body and blood of Christ: but on the contrary, to interpret the Apostle in this sense only, is unreasonable, upon several accounts. (1) None can so much as go about such an examination, without first knowing, that the Lord's body and blood is signified by these elements. [-- 261 --] For merely a man's putting this question to himself, Do I understand that this bread and this wine signify the body and blood of Christ? supposes him already to know it from a previous information; and therefore to exhort persons to go about such an examination, would be absurd. And then (2) 'tis incredible that there should be any such gross ignorance appearing in a number of the communicants in the Corinthian church, if we consider what the Scripture informs us concerning that church: as particularly, if we consider what an able and thorough instructor and spiritual father they had had, even the apostle Paul, who founded that church, brought them out of their heathenish darkness, and initiated them in the Christian religion, and had instructed them in the nature and ends of gospel ordinances, and continued at Corinth, constantly laboring in the word and doctrine for a long while together, no less than a year and six months; and, as we may well suppose, administering the Lord's Supper among them every Lord's day; for the Apostle speaks of it as the manner of that church to communicate at the Lord's Table with such frequency (1 Corinthians 16:2). And the Corinthian church at that day when the Apostle wrote this epistle, was a church noted for excelling in doctrinal knowledge; as is evident by ch. 1 Corinthians 1:5-7 and several other passages in the epistle. Besides, the communicants were expressly told at every communion, every week, when the bread and wine were delivered to them in the administration, that that bread signified the body, and that wine signified the blood of Christ. And then besides, (3) the Apostle by his argument in ch. 1 Corinthians 10:16 supposes the Corinthians doctrinally acquainted with this subject already. It therefore appears to me much more reasonable, to apprehend the case to be thus: the offensive behavior of the communicants at Corinth gave the Apostle reason to suspect, that some of them came to the Lord's Table without a proper impression and true sense of the great and glorious things there signified; having no habitual hunger or relish for the spiritual food there represented, no inward vital and experimental taste of that flesh of the Son of Man, which is meat indeed. The word translated "discerning," signifies to "discriminate" or "distinguish." The taste is the proper sense whereby to discern or distinguish food (Job 34:3). And 'tis a spiritual sense or taste which is that whereby we discern or distinguish spiritual food. Hebrews 5:14, "Those who by reason of use, have their senses exercised to discern both good and evil," Πϱος διάϱισιν, etc. a word of the same root with that rendered "discerning," in 1 Corinthians 11:29. He that has no habitual appetite to and relish of that spiritual food, which is represented [-- 262 --] and offered at the Lord's Table; he that has no spiritual taste wherewith to perceive anything more at the Lord's Supper, than common food; or that has no higher view, than with a little seeming devotion to eat bread, as it were in the way of an ordinance, but without regarding in his heart the spiritual meaning and end of it and without being at all suitably affected with the dying love of Christ therein commemorated; such a one may most truly and properly be said not to discern the Lord's body. When therefore the Apostle exhorts to self-examination as a preparative for the sacramental supper he may well be understood to put professors upon inquiring whether they have such a principle of faith, by means whereof they are habitually in a capacity and disposition of mind to discern the Lord's body practically and spiritually (as well as speculatively and notionally) in their communicating at the Lord's Table: which is what none can do who have but common grace, or a faith short of that which is justifying and saving. It's only a living faith that capacitates men to discern the Lord's body in the sacrament with that spiritual sensation or spiritual gust, which is suitable to the nature and design of the ordinance, and which the Apostle seems principally to intend.
-- Jonathan Edwards , An Humble Inquiry into the Rules of the Word of God, Concerning the Qualifications Requisite to a Complete Standing and Full Communion in the Visible Christian Church, Ecclesiastical Writings (WJE Online Vol. 12) , Ed. David D. Hall, pages 258-262
Although Edwards lost the battle in his day, due to numerous objects, his writings lead to the future demise of the Half-Way Covenant. Unfortunately, Edwards did not live to see his writings vindicated. The value of these writings in Ecclesiastical Writings (WJE Online Vol. 12) is that they do not collapse into a Independentist, Separatist or Baptist position that ushers the Church out of public life into private quarters, especially in regards to Edward's approval of Infant Baptism (padeobaptism.) Edwards remains faithful to Covenantal Reformed Theology, but also understands the necessity of conversion, and this is most clear in his explanation of the underlying Greek word for "examining oneself".