The more recent sense is strife or bickering, though not one you’re likely to have come across, digladiation being as archaic as any word that has ever featured in this section.
Dr Johnson included it in his Dictionary, together with many another strange creation, illustrating it with this quotation from Joseph Glanvill's The Vanity of Dogmatizing, or Confidence in Opinions of 1665, “Aristotle seems purposely to intend the cherishing of controversial digladiations, by his own affectation of an intricate obscurity”.
Thomas McCrie wrote disparagingly about “scholastic wrangling and digladiation” in his work The Life of Andrew Melville of 1819. It appeared a few times after that, as a ponderous and obscurely humorous literary term, in reference especially to courtroom advocatory sparring, but it seems to have died out completely by the end of the nineteenth century.
The link with strife may suggest a connection with gladiator, and indeed physical aggression was the first meaning — in particular hand-to-hand combat with swords. The word is from Latin gladius, the short sword wielded by the gladiators of classical times. To digladiate, you might say, is to cross swords.
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