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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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2 Answers 2

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I would just like to add some nuance to Geremia's excellent answer.

We have to keep in mind that the intention of the Council Fathers at Nicaea (325) was not so much to defend the term ὁμοούσιος (homoousios, consubstantial; the term is derived from οὐσία, which means "essence" or "substance") as to refute the heresy of Arius.

(For those who are not familiar, Arius, a priest originally from Libya but ministering at a prominent church in Alexandria in Egypt, essentially denied the full divinity of the Son, asserting instead that He is the first "creature" or emanation of the Father. We do not have any certain records of what he believed about the Holy Spirit, but we may deduce that he considered the Holy Spirit to be a creature or emanation inferior to the Son. He also denied the Son's—and we may presume the Spirit's—eternity. He is reported as stating, "There was a time when the Son was not.")

The consensus at the Council of Nicaea was that Arius was in error—in other words, that the Son is fully divine, just like the Father—but there was as yet no consensus as to the correct understanding of the relationship between Father and Son. There are essentially three possibilities:

  1. Father and Son could be "fused" into one Person.
  2. Father and Son could be separate "gods" (i.e., be two equal "natures" or "substances").
  3. Father and Son could be two Persons in one Essence or Substance (which is the orthodox position).

Hardly anyone professing to be an orthodox Christian would have taken the second option (which smacks of polytheism), but there were a number who opted for the first: Marcellus of Ancyra, for example. This first option, the "fusion" of the Persons into one, is not very different from the heresy of Modalism, also known as Sabellianism (which professes that the Persons of the Trinity are merely "aspects" or "modes" of the one God).

And here is the problem: the term ὁμοούσιος was the very same one that Sabellius used to justify his heresy. Moreover, the very important distinction, advanced by St. Basil the Great, between οὐσία (ousia; i.e., essence or substance) and ὑπόστασις (hypostasis; i.e., "Person," in our terminology) had not yet been made: in fact, the Council of Nicaea condemns anyone who professes a distinction in "hypostasis" between Father and Son.

For this reason, many otherwise orthodox doctors and bishops viewed the term ὁμοούσιος with suspicion. Moreover, many who used the term ὁμοούσιος (Marcellus, for instance) used it in a way that, in retrospect, resembles Sabellianism.

This is the origin of the "homoeousian" party (note the extra "e"): they derive their name from their preference for the term ὁμοιούσιος (again, the extra iota is important). They professed that the Father and the Son were not the same substance (ὁμοούσιος) but similar in substance (ὁμοιούσιος).

Now, as St. Athanasius discovered after many discussions, it turns out that the Homoeousians were not necessarily heterodox at all, at least as regards the Father and the Son. When they used the term similar (ὁμοίος, homoeos), they understood a profound similarity: the way a human son is "similar" to his human father. They were, in fact, already on the right track: a human father is identical in nature with his human son, but not the same concrete substance (hypostasis). This idea was made much clearer later, when the Cappadocian Fathers (Basil the Great, Gregory of Nazianzus, and Gregory of Nyssa) introduced the distinction between οὐσία and ὑπόστασις, as I mentioned.

It would be a mistake, in my opinion, to characterize Homoeousianism as a "compromise" between "full" Arianism and orthodoxy. Rather, it was an honest, and, so far as it went, orthodox attempt to avoid both Sabellianism and Arianism. For this reason, I consider it misleading to apply the term "Semi-Arian" (which is a modern, anachronistic term anyhow) to this group.

The Homoeousians should be distinguished sharply from the Homoeans, who arose from other circumstances. Even after the Council of Nicaea, there continued to be a lot discussion and polemics regarding the οὐσία (Essence) of the Father and the Son: as we saw, the opinions and choice of terms were all over the map. It occurred to some of the Christian emperors (who wanted unity of religion in the Roman Empire at all costs) to suppress this discussion once and for all. The reasoning was, if they could simply make the bishops and doctors stop talking about οὐσία, then all could agree to a sort of "lowest common denominator": that the Father is similar (ὁμοίος) to the Son, not necessarily in Essence (οὐσία) but just "according to the Scriptures." In other words, the emperors wanted everyone simply to avoid the problem.

This is the origin of the "homoeian" party: those who agreed, basically, to tow the emperor's line and avoid discussion on the Divine Essence. Since this position was imposed practically by force, naturally there were many adherents; at one point, probably a large majority of bishops. It was Arianism in this form that would last for another three or four centures: as St. Jerome put it, "The world woke up and groaned to find itself Arian." If we are going to use the label "Semi-Arian" at all, then it should apply to this form of Arianism, which was a sort of compromise, in an attempt to please all parties.

Returning to the original question, then, regarding the Homoeousians (who do not, in my opinion, merit the epithet of "Semi-Arian") we can answer as follows:

  1. The Homoeousians would have regarded the Son as eternal.
  2. They would not have regarded Him as a creature.
  3. Regarding divinity and co-eternity of the Holy Spirit, they are all over the map. We have to keep in mind that the speculation on the Holy Spirit was only just beginning in the 50 years or so after the Council of Nicaea. It did not stabilize until 381, at the First Council of Constantinople (and the parallel council at Ariminium in Italy). Most of them would probably have considered Him ὁμοιούσιος with the Father and the Son (consistent with their terminology), but we know of doctors and bishops who taught a subordination of the Spirit to the other Persons. (Such doctors were scornfully called Πνευματομάχοι—Spirit fighters, or in modern colloquial terms, "Spirit bashers.") I should point out that there were Homoousians (those who used what would become the established orthodox term, ὁμοούσιος) who were also Πνευματομάχοι.

(For an interesting insight on the efforts of St. Athanasius to reconcile with members of the Homoeousian party, see his Tomus ad Antiochenos.)

Regarding the Homoeans (who might be called "Semi-Arians"):

  1. There was not a unity of opinion regarding the eternity of the Son, just as there was not unity of opinion regarding the fullness of His divinity. The tendency would have been towards a "moderate" Arianism: the Son was not exactly a creature, nor was there a time when He did not exist, but He was subordinate to the Father (not fully divine).

  2. I think I have answered this question above.

  3. Their position regarding the Holy Spirit was varied, for the same reasons as in the case of the Homoeousians. The tendency, however, was to subordinate the Holy Spirit to the Son. (So, basically, if the Son were not "fully" divine, then the Spirit was even "less" divine than the Son.)

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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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