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Was John 19:12-13 intended to be a prophecy on the Day of Judgment? If so, then it would portend Jesus Christ to be a humiliated, mocked figure on that awful Day; a pretender to the role of supreme judge who is to have authority regarding life and death.
For that reason alone, the text cannot be prophetic of the Day of Judgment. Yet there is more reason to show that point to be correct. The need here is to go to the Greek text of scripture. Never mind what philosophers (either ancient or modern) say about words. What does God's word actually say?
Although I have a couple of William Barclay's books, his "New Testament Words" does not list bema or kathizien. However, all the Greek words he does list have copious quotations from modern critical scholars, and from Homer, Epicurus, Herodotus, Sophocles, Euripides, Xenophon, Aristotle etc. Classical Greek and the Classics form a considerable part of this book, so much so, that if all that were to be removed, the book would be reduced from 288 pages to about half that. The reason why this is troubling, is that Classical Greek should not be mixed up with koine Greek. The Holy Spirit inspired the writing of the Greek Scriptures, using koine Greek, so that the in-depth meaning of the Bible must come from that, and not language or philosophers of different centuries.
So, what does the koine Greek actually say? Here is the pertinent reading from a very old, respected text, that of the Greek text of Stephens, 1550:
"Pilate therefore having heard this word led out Jesus and sat down upon the judgment-seat at a place called Pavement but in Hebrew Gabattha." The Englishmans Greek New Testament by Robert Estienne, page 303
I have two other Greek interlinears which support that. Indeed, one is the literal word-for-word translation of the Greek text by Westcott and Hort (approved of by William Barclay) and it is even clearer:
"The therefore Pilate having heard of the words these led outside the Jesus, and he sat down upon step into place being said Stone pavement, in Hebrew but Gabattha." John 19:13
This clearly shows that Pilate sat down on that judgment seat. Barclay could only arrive at his view by calling upon an apocryphal writing, and (as is his want) citing Classical language and philosophers. By sticking to what the Bible actually says, and comparing John 19 with Revelation 20:11, it's obvious there is no comparison with the disgraceful, sinful human attempt at judging the Son of God. If you want to find a prophetic parallel, it is in the second Psalm.