I kept getting pushed out of flow by words like "intuitionistically" and constructions like "derive absurdity" (as opposed to 'derive an absurd conclusion'). 'Intuitive' is a perfectly functional word, no need to make new ones up to sound formal.
"Derive absurdity" comes from the fact that in these logics there is only one statement which is "Absurdity"—capitalized perhaps. It's a unique thing to talk about and "an absurd conclusion" is actually shorthand for "a statement from which you could derive Absurdity".
"intuitionistically" refers to the intuitionistic logic.
Which is one of the reasons I prefer the term "constructive logic"; "intuitionistic" doesn't tell me anything.
Intuitionistic logic is a subtype of constructive logic, though, so it's more specific. That said, a lot of people don't make the distinction!
Ah, not the author's fault then. Just unnecessarily opague terminology in the canon. I'll redirect my irritation to his predecessors (Constructive Logic seems a much better term that doesn't do any violence to the rules of English).
Intuitionism is a philosophy of mathematics that is deeply related to but not the same thing as constructivism.