How many fights has this caused in the church over the centuries!! Calvinists vs Arminians, to name just one of them; splits in the English and German churches and many more.
The verb in question is τάσσω (tassó) which occurs 9 times in the NT, sometimes with a specific military meaning (Matt 8:9, Luke 7:8). BDAG provides two basic meanings:
- to bring about an order by arranging, arrange, put in place, eg, Rom 13:1, Matt 8:9, Luke 7:8, Acts 13:48, 1 Cor 16:15
- to give instructions as to what must be done, order, fix, determine, appoint, eg, Acts 15:2, 18:2, 22:10, 28:23, Matt 28:16.
Significantly, BDAG puts the OP's verse, Acts 13:48 in the first category. Just as significantly, the verb in Acts 13:48 is Perfect Participle Middle or Passive - Nominative Masculine Plural. Thus, the verb is either in the middle or passive voice.
Most English versions translate this as though the verb is definitely in the passive voice and thus give something like, "all who were appointed for eternal life believed".
However, if the verb is understood in the middle voice (more likely), then it should be translated something like, "as many as believed had set themselves for eternal life".
Ellicott appears to agree:
As many as were ordained to eternal life believed.--Better, as many as
were disposed for. The words seem to the English reader to support the
Calvinistic dogma of divine decrees as determining the belief or
unbelief of men, and it is not improbable, looking to the general
drift of the theology of the English Church in the early part of the
seventeenth century, that the word "ordained" was chosen as expressing
that dogma. It runs, with hardly any variation, through all the chief
English versions, the Rhemish giving the stronger form "pre-ordinate."
The Greek word, however, does not imply more than that they fell in
with the divine order which the Jews rejected. They were as soldiers
who take the place assigned to them in God's great army. The
quasi-middle force of the passive form of the verb is seen in the
Greek of Acts 20:13, where a compound form of it is rightly rendered
"for so he had appointed," and might have been translated for so he
was disposed. It lies in the nature of the case that belief was
followed by a public profession of faith, but the word "believed" does
not, as some have said, involve such a profession.