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Definition of arbitrarily : In an arbitrary manner; by will only; despotically; absolutely.

Synonyms:

promiscuously, willy-nilly, helter-skelter, haphazard, at random, indiscriminately, every which way

erratically, changeably, frivolously, fitfully

personally, individually, subjectively

ambivalently, spastically, fleetingly, sporadically, whimsically, adriftly, capriciously, irregularly, randomly, waywardly, indecisively, haphazardly, non-systematically, carelessly, fancifully, impulsively

Usage examples:

• As I have every reason to deem your power great, I desire for it the corresponding great incitement; for nothing here can be arbitrarily substituted or added: genuine strength can only create from necessity. - "Correspondence of Wagner and Liszt, Volume 1", Francis Hueffer (translator).
• Of course it would have been awkward for England's Prime Minister to make Achilles say that " every man must love his concubine, if he has sense and virtue;" so he arbitrarily changes the meaning of the word and then begs us to notice the moral beauty of this sentiment and the " dignity" of the relation between Achilles and Briseis! - "Primitive Love and Love-Stories", Henry Theophilus Finck.
• They forgot- as Godwin, who was opposed to sudden change, should not have forgotten- that laws made for men in general cannot be arbitrarily altered to suit each man in particular. - "Mary Wollstonecraft", Elizabeth Robins Pennell.