illogicality vs quality

illogicality and quality both are nouns.

Word	Noun	Adjective	Verb	Adverb
illogicality	Yes	No	No	No
quality	Yes	Yes	No	No

As nouns, quality is a hypernym of illogicality; that is, quality is a word with a broader meaning than illogicality:

illogicality (noun)	quality (noun)
invalid or incorrect reasoning	an essential and distinguishing attribute of something or someone
	(music) the distinctive property of a complex sound (a voice or noise or musical sound)
	a degree or grade of excellence or worth
	a characteristic property that defines the apparent individual nature of something
	high social status

illogicality (adjective)	quality (adjective)
	of superior grade
	of high social status