The dog returns to his vomit because that's what dogs do. The sow returns to wallowing in the mire because that is what a hog does. Neither the dog nor the pig has had its nature changed. Neither has been "born again" or been changed into another species. The dog experienced a temporary purging and the sow underwent a washing. Neither "work" had any eternal meaning, no matter how it might have spruced up appearances --like the whitewashing of a tomb (Matthew:23:27Woe unto you, scribes and Pharisees, hypocrites! for ye are like unto whited sepulchres, which indeed appear beautiful outward, but are within full of dead men's bones, and of all uncleanness.
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"But is is happened unto them according to the true proverb, The dog is turned to his own vomit again; and the sow that was washed to her wallowing in the mire." —2 Peter:2:22But it is happened unto them according to the true proverb, The dog is turned to his own vomit again; and the sow that was washed to her wallowing in the mire.
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Links:
[1] https://www.thebereancall.org/bible/1/MAT/23/27#v27
[2] https://www.thebereancall.org/bible/1/2PE/2/22#v22