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What I know from history books is that Semi-Arians held to the neutral theological position of homoiousians against both strict homoousians and anomoeans.

However, I found a very scant knowledge about their theology and these are what I want to know:

  1. Do Semi-Arians teach that the Son is eternal or not?
  2. Do Semi-Arians teach that the Son is a creature?
  3. What do they believe about the Holy Spirit?
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1 Answer 1

Fr. Hardon's Modern Catholic Dictionary gives this definition of Semi-Arianism:

The teaching of certain theologians who, after the Council of Nicaea (A.D. 325), sought a compromise between Arianism and the doctrine of Christ's consubstantiality with the Father. They were led by Basil, Bishop of Ancyra, and their sympathies were toward orthodoxy, although they substituted homoiousios (similar to) the Father. St. Athanasius treated them kindly and their influence was felt in the reaffirmation of the Nicene Creed at the ecumenical Council of Constantinople in the year 381.

From Fr. Réginald Garrigou-Lagrange, O.P.'s, The Trinity & God the Creator, a commentary on St. Thomas Aquinas's articles on the Trinity in his Summa Theologica, the Treatise on the Trinity (qq. 27-43):

According to Arius, God the Father alone is eternal; the Father created the Son, not of His own substance but out of nothing, and then God made use of the Son as an instrument to create the universe and redeem men.

After Arianism was thus condemned by the Church as a heresy, the Arians tried to dissimulate their error and said that the Son was similar in nature to the Father, homoiousion or homoion, but they refused to say that He was consubstantial or homoousion. Such was the teaching of Basil of Ancyra and Auxentius of Milan, who are called Semi-Arians. Arianism lasted into the sixth century, when it completely disappeared.

Following the principles that misled Arius, Eunomius concluded that the Holy Ghost was not God but a creature made by the Son of God, inferior to Him and similar to the angels. At about the same time, the Macedonians like the Semi-Arians denied the divinity and consubstantiality of the Holy Ghost. Eunomius was refuted by St. Gregory of Nyssa, St. Basil of Caesarea, and St. Ambrose. Macedonianism [and thus also Semi-Arianism] was condemned by [Pope] St. Damasus [I] in the fourth Council of Rome (380) and in the following year by the second ecumenical Council of Constantinople. The most important definition of the Council is: "If anyone shall say that the Holy Ghost is not truly and properly of the Father, like the Son, of the divine substance, and true God, let him be anathema." Thus in the fourth century, opposing these heresies, the Church explicitly taught a Trinity of distinct persons, upheld their divinity and consubstantiality, and so preserved the unity of essence together with the distinction of persons. In the earliest centuries, therefore, the Church explicitly condemned that Unitarianism which the liberal Protestants have recently revived.

In 325 the Council of Nicaea defended the true tradition against Arius, who taught that the Father alone was truly God, that the Word was the most excellent of creatures, created in time out of nothing, and that the Holy Ghost was also a creature, inferior to the Son. After long discussion it was defined that the Word was consubstantial with the Father, homousion: "We believe in one God the Father almighty, maker of all things, visible and invisible. And in one Lord, Jesus Christ, the Son of God, the only-begotten Son of the Father, that is, of the substance of the Father, God of God, light of light, true God of true God, begotten, not made, of one substance with the Father, as the Greeks say, homousion, by whom all things were made. And in the Holy Ghost."

After this condemnation the heretics tried to cover up their error by teaching that the Son was not properly homousion or consubstantial with the Father, that is, of the same essence, but that He was similar in nature, or homoiousion. Such was the teaching of the Semi-Arians; the Acacians said the Son was homoion, that is, similar with regard to form and accidents. These teachings were refuted by St. Alexander, the bishop of Alexandria, and by St. Athanasius.

So, to answer your questions:

  1. Semi-Arians, like the Arians, do not believe in the co-equal dignity or co-eternal existence of the Father and Son (cf. beginning of 3rd ¶ of the "Doctrine" § of this).
  2. Yes, since they teach He is not the same nature as God. (Since they professed to be monotheist, they only believed in the existence of one Godly nature. Thus, according to them, the Son can only be of a creaturely nature.)
  3. They deny the Holy Ghost is divine.
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