Using Thesaurus Linguae Graecae (UC Irvine), in conjunction with the assistance of good ole LSJ, I have examined the verb horizo in extra-biblical Greek. Here are the results of my study. There are nine main usages, and these nine can be subdivided into two groups – the first six and the last three.
First, to mark out land by boundaries
“For when he learned of Polycrates’ death, first he set up an altar to Zeus the Liberator and marked out around it that sacred enclosure which is still to be seen in the suburb of the city” (Herodotus 3.142). A sacred enclosure (temenos) is a piece of land marked off from common uses and dedicated to a god.
Second, to limit the range/extent of something
“Now that you have seen fit to confiscate the property of Erasiphon, I relinquish two-thirds to the State … So I have limited my share to one-third of their property, making no exact calculation, but leaving much more than two-thirds to the Treasury” (Lysias 17.6).
“Women are not allowed by law to accept agreements involving too large a sum, but a limit has been set defining the amount to which they may do so” (Dio Chrysostom 74.10).
Third, to decide someone’s fate or destiny
“Then the Delphian lords … decided that my mistress should die, by being cast from the rock” (Euripides, Ion 1222).
“Yes, for my father’s fate has marked out this destiny for you” (Aeschylus, Choephoroe 927).
Fourth, to lay down or ordain (legal/judicial contexts)
“And yet, gentlemen, in cases of this sort your laws lay down the most severe penalties if an Athenian transports corn to any place other than your city” (Lycurgus, Oratio in Leocratem §27).
“Against those who call God as a witness in favour of assertions which are not true, the punishment of death is ordained in the law” (Philo, Spec. 2.252).
Fifth, to fix upon, in the sense of selecting one option within a range of options
“‘Well then,’ said Socrates, ‘that there may be no question raised about my obedience, please fix the age limit below which a man is to be accounted young’” (Xenophon, Memorabilia 1.2.35).
“In this case he shall depart into another country and place, and dwell there as an exile for a year; and should he return within the time fixed by the law or set foot at all within his own country, he shall be put in the public jail by the Law-wardens for the space of two years” (Plato, Leges 864e6).
“For all breaches of the law alike, however small, they fixed upon the death penalty, making no special allowances, in their assessment of the magnitude of crimes, for the individual circumstances of each” (Lycurgus, Oratio in Leocratem §65).
Sixth, to appoint
“You, O Lord, according to the sweetness of your grace, promised forgiveness to those who repent of their sins, and in the multitude of your mercies appointed repentance as salvation for sinners” (Prayer of Manasseh 7b).
The six usages of horizo to this point all have one thing in common: they have to do with an action of marking, limiting, ordaining, fixing, or appointing something that could have been marked, limited, ordained, fixed or appointed differently. The sacred enclosure to Zeus could have been made three feet larger in circumference. The share of property retained by Lysias could have been half rather than a third. The law could have fixed the time of exile at two years rather than one. And so on. There is a sense in which the act of setting limits involves an arbitrary decision, or is at least not dictated by the nature of things.
If we use these first six usages of horizo as the measure for the meaning of the word in Rom 1:4, we might very well incline toward the rendering “appointed Son-of-God-in-power.”
However, the next three usages of horizo push us in a different direction. In these usages, the word has more to do with defining, determining, and marking out that which is the case. The act of determining is not an arbitrary decision, a picking of one option among many, but is one that is dictated by the nature of the case.
Seventh, to define
“Shall we therefore rightly define the pious man as one who knows what is lawful concerning the gods?” (Xenophon, Memorabilia 4.6.4; cp. 4.6.6).
“… those who define the good as pleasure” (Plato, Respublica 505c; cp. Sophista 246b; Gorgias 475a; Theaetetus 190e).
“And life is defined, in the case of animals, by the capacity for sensation; in the case of man, by the capacity for sensation and thought” (Aristotle, Ethica Nicomachea 1170a16; cp. 1104b24).
“For from whence could [geometry] define and pronounce that a point is that which has no parts, that a line is length without breadth,” etc. (Philo, Congr. 147; cp. Deus 44).
“A man who lives in an irrational manner is separated from the life of God; for to live according to God is defined by Moses to consist in loving him; for Moses says to the children of Israel, ‘Your life is to love the living God’” (Deut 30:20) (Philo, Post. 69).
Eighth, to determine by measuring, ascertaining, or dating
“Or is it rather because, since they measured time by the phases of the moon, they observed that in each month the moon undergoes three very important changes?” (Plutarch, Aetia Romana et Graeca 269C).
“Manetho says that the Jews departed out of Egypt, in the reign of Tethmosis, three hundred ninety-three years before Danaus fled to Argos; Lysimaehus says it was under king Bocchoris, that is, one thousand seven hundred years ago; Molo and some others determined it as every one pleased: but this Apion of ours, as deserving to be believed before them, hath determined the exodus exactly to have been in the seventh Olympiad, and the first year of that Olympiad” (Josephus, Contra Apionem 2.17).
“I date his hostility from the very day when he wiped out the Phocians” (Demosthenes 9.19).
Ninth, to mark out what is already there
“Medicine is, of course, a science … Then the temperate man will know, indeed, that the doctor has a certain science; but when he has to put its nature to the proof, must he not consider what its subjects are? Is not each science marked out, not merely as a science, but as a particular one, by the particular subjects it has?” (Plato, Charmides 171a).
“He who fixes his eyes considers perverse things, and he marks out with his lips all evil” (Prov 16:30 LXX).
So there are broadly two types of determination indicated by horizo – determinations that involve fixing upon one option among many, and determinations that involve marking out what is already the case. The only way we know the difference in any case is from the context.
“It depends entirely upon the connection whether a declarative or a determinative decision is meant, whether it means … to determine that one is something, or that one is to be something” (Hermann Cremer, Biblico-Theological Lexicon of New Testament Greek [4th ed.], pp. 461-2).
I would argue that the context, specifically the formula frame, of Rom 1:3-4 supports the declarative meaning. The introductory portion of the frame, “concerning his Son,” with the two genitive participles modifying “Son,” makes clear that it is God’s Son who was born of the seed of David and that it is God’s Son who was then declared with power to be what he was all along.
“The position of the words τοῦ υἱοῦ αὐτοῦ – being placed, so to speak, outside the bracket, they are naturally taken to control both participial clauses alike – would seem to imply that the One who was born of the seed of David was already Son of God before, and independently of, the action denoted by the second participle” (Cranfield, 1.58).
It is not that, by his resurrection, Jesus was appointed Son of God in power, but that he was powerfully marked out to be what he had been all along, namely, Son of God.