Thinking about "illation" brought me to "intuition", and I discovered that contrary to what I had assumed, it is not cognate with "tuition". I had always thought "intuition"
was comprehension not acquired from teachers.

tuition [tju¢°'©¥ʃən]
noun
1 instruction, esp. that received in a small group or individually

2 the payment for instruction, esp. in colleges or universities
[ETYMOLOGY: 15th Century: from Old French tuicion, from Latin tuitio a guarding, from tueri to watch over]
tu'itional adjective


intuition [ˌ©¥ntjʊ'©¥ʃən]
noun
1 knowledge or belief obtained neither by reason nor by perception

2 instinctive knowledge or belief

3 a hunch or unjustified belief

4 (Philosophy) immediate knowledge of a proposition or object such as Kant's account of our knowledge of sensible objects

5 the supposed faculty or process by which we obtain any of these
[ETYMOLOGY: 15th Century: from Late Latin intuitio a contemplation, from Latin intueri to gaze upon, from tueri to look at]
"intu'itional adjective
"intu'itionally adverb(ial)