1 Corinthians 10:18: A Brief Syntactical Discussion

βλέπετε τὸν Ἰσραὴλ κατὰ σάρκα· οὐχ οἱ ἐσθίοντες τὰς θυσίας κοινωνοὶ τοῦ θυσιαστηρίου εἰσίν (1 Corinthians 10:18-Nestle 1904) It seems that κοινωνοὶ τοῦ θυσιαστηρίου is a genitive of association (Compare Rom. 8:17). Furthermore, the substantival phrase οἱ ἐσθίοντες is the subject here rather than θυσιαστήριον. Additionally, the context suggests that the eater/one approaching the altar consumes the meat, not the altar per se (1 Cor. 10:16-17). For a similar use of altar terminology, see Heb. 13:10. Roy E. Ciampa and Brian S. Rosner (The First Letter to the Corinthians, PNTC Series):

By “participating in the altar” Paul evidently means that those eating the meal from the food taken from the altar are counted as those who offered the worship through the sacrifice that was offered there (and expect to benefit from the efficacy of that offering). The implication for eating food offered to idols is clear. Paul implies that to knowingly eat food that has been clearly identified as such makes one a willing participant of the offering from which it was taken. Such is understood to be the case in Christian participation in the Lord’s Supper and in the offerings made at the temple in Jerusalem as well, and it would be only reasonable to assume that it applies to food offered to idols also. That very implication brings Paul back to the issue of the significance of idols and idol food, an issue that he touched on in 8:4, 7 and that he addresses again in the following verses.