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The Collaborative International Dictionary
Illogically

Illogical \Il*log"ic*al\, a. Ignorant or negligent of the rules of logic or correct reasoning; as, an illogical disputant; contrary of the rules of logic or sound reasoning; as, an illogical inference. -- Il*log"ic*al*ly, adv. -- Il*log"ic*al*ness, n.

Wiktionary
illogically

adv. In an illogical manner, without following methods of sane and correct reasoning, in a manner lacking thought or logic.

WordNet
illogically

adv. in an illogical manner; "she acted illogically under the pressure" [ant: logically]

Usage examples of "illogically".

He turned on the TV, hoping illogically that the news would show Northumber being attacked by peaceniks.

If this class of illuminations really exists, we have to afford them some other explanation than that they are emanations of self-inflamed phosphoretted hydrogen, a method of accounting for them which illogically finds a place in many treatises on atmospheric phenomena.

Yet that music which I gave off Sunday mornings, a music of well-being and of well-nourished desperation, was born of an illogically well-aspected Uranus firmly anchored in the 7th House.

Since the Japs ap-peared illogically unwilling to surrender, the naval bombarders set about annihilating them with an oddly good-humored, ribald ferocity.

A roar of encouragement from those who, illogically, hastened to bet on a forlorn cause.