A strange coincidence: the Greek word for ‘plank’, ἡ δοκός (hē dokós), sounds similar to the word for ‘opinion’ or ‘judgement’, ἡ δόξα (hē dóxa) (or in the accusative singular, τὴν δοκόν [tḕn dokóv] vs τὴν δόξᾰν [tḕn dóxăn]).
δοκός and κάρφος do not, so it seems, carry the secondary meaning of ‘sunbeam’ and ‘dustmote’, and it has been said that the King James translation with ‘mote’ and ‘beam’ is ‘probably incorrect’. [1] Yet such a beautiful image, encoding such wisdom about the possibility for compassionate human relations, cannot have come about by accident, nor does it seem likely that the King James translators would knowingly have added a new layer of meaning to the sacred text. They must have perceived something that authorised their translation. Could the ‘pearl’ hidden in the image of the splinter and the plank depend on such wordplay?
Although the radiance of the King James image has been lost, the insight conveyed with ‘judgement’ substituted for ‘plank’ is the same as with ‘dustmote and sunbeam’, and is not banal like the interpretation with ‘splinter and plank’. ‘First cease making judgement, and then you will see clearly to cast out the flaw in your brother’; again we are called upon not to excoriate ourselves and then turn superciliously to helping our brother do the same, but to give up being judgemental altogether. Whatever engagement we have with our brother as he, like us, seeks to change, we do not adopt a stance of judging his traits to be flaws. Again, to give up judgement is to exercise compassion.
I cannot know whether a reader of koinē Greek would have found it natural to pass from ‘dokóv’ to ‘dóxăn’. Yet the concepts of judging and judgement are introduced in the previous two lines, 7:1‒2, so the reader is certainly in the right frame of mind to make the connection; and the ancients enjoyed word games even more than we do and did them more subtly. [2] There is a mean joke about philosophers in Aristophanes’ The Clouds that seems to turn on the similarity between the words for charcoal, ἄνθρακες [ánthrakes], and man, ἄνθρωπος [ánthrōpos], and there is less of a similarity here than between the words for judgement and plank. [3]
In the end I can only say this: there is plainly a double meaning in the King James translation of κάρφος and δοκός as ‘mote’ and ‘beam’, and it transcends the crass image found in standard works of biblical interpretation; and Matthew plainly asserts that the image in 7:3‒5 has a hidden meaning. Whether the King James translators identified that meaning in the Greek and found their own way to bring it out, or made a free creative addition, I do not know.
Notes
[1] International Critical Commentary (T&T Clark, 1988), p. 671.
[2] As studied for example by Paulo Vasconcello, ‘Obsessive Love, Obsessive Sounds’.
[3] How else would a sophist argue except by fatuous wordplay?
ψυχῶν σοφῶν τοῦτ᾽ ἐστὶ φροντιστήριον.
ἐνταῦθ᾽ ἐνοικοῦσ᾽ ἄνδρες, οἳ τὸν οὐρανὸν
λέγοντες ἀναπείθουσιν ὡς ἔστιν πνιγεύς,
κἄστιν περὶ ἡμᾶς οὗτος, ἡμεῖς δ᾽ ἄνθρακες. [The Clouds 94‒95]
That is the Phrontisterion, for subtle minds.
In there dwell men, those who say that the sky,
so by reason they persuade us, is an oven,
and surrounds us on all sides, we ourselves being the charcoal [ánthrakes].
Babel editing 
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