From paradoxical +‎ -ly.

• (UK) IPA^(key): /ˌpaɹəˈdɒksɪkli/
• (US) IPA^(key): /ˌpɛɹəˈdɑksɪkli/

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paradoxically (comparative more paradoxically, superlative most paradoxically)

1. In a paradoxical manner; so as to create a paradox.

   The proliferation of labor-saving devices has not given us more leisure time, paradoxically.

   If there exists a non-empty set of uninteresting natural numbers, there must be a smallest uninteresting number, but this number is paradoxically interesting because it is the smallest uninteresting number.

   • 1949, F. A. Hayek, “The Intellectuals and Socialism”, in University of Chicago Law Review, volume 16, number 3, Chicago: University of Chicago, →DOI, page 418:

     Paradoxically enough, however, in general only the parties of the Left have done most to spread the belief that it was the numerical strength of the opposing material interests which decided political issues, whereas in practice these same parties have regularly and successfully acted as if they understood the key position of the intellectuals.

   • 1997, Gary A. Olson, Todd W. Taylor, Publishing in Rhetoric and Composition, page 218:

     Paradoxically, the same faculty who complain about busyness are most likely to end up waiting passively. While they acquiescently wait to write, they busy themselves with other, often trivial, activities.

   • 2008 May 14, Brent Staples, “Loving v. Virginia and the Secret History of Race”, in New York Times‎:

     But Central Point, which had been a visibly mixed-race community since the 19th century, was home to a secret but paradoxically open interracialism.

   • 2013 September 6, Colin Robinson, “Put down the pen and give us all a break”, in The Guardian Weekly, volume 189, number 13, page 39:

     Paradoxically, the deluge of writing itself contributes to declining readership. It's not just that if you're writing then you can't be reading. It's also that the sheer volume of what is now available acts as a disincentive to settle down with a single text. The literary equivalent of channel surfing replaces the prolonged concentration required to tackle a book.

   • 2020 March 10, Isabel Slone, “Escape Into Cottagecore, Calming Ethos for Our Febrile Moment”, in The New York Times‎:

     Cottagecore is related to grandmacore, faeriecore, farmcore and goblincore; other nostalgia-ridden aesthetic communities that, paradoxically, thrive on many of the most popular internet platforms of the day.

   • 2024 December 18, Ignatiy Vishnevetsky, “Disney's taxidermized IP is brighter and livelier in Mufasa: The Lion King”, in AV Club‎:

     Lacking the anthropomorphic elasticity of their hand-drawn counterparts in the original film, the animal characters of Favreau’s Lion King appeared, paradoxically, both lifelike and lifeless, their voices never matching their limited facial expressions