TITLE: Recognizable vs Decidable
QUESTION [60 upvotes]: What is difference between "recognizable" and "decidable" in context of Turing machines?

REPLY [6 votes]: My answer mostly agrees with Aryabhata’s:
A language is “Turing-Recognizable” iff there exists a Turing Machine such that

when encountering a string in that language, the machine terminates and accepts that string;
when encountering a string not in that language, the machine either terminates and rejects that string or doesn’t terminate at all.

A language is “Turing-Decidable” iff there exists a Turing Machine such that

when encountering a string in that language, the machine terminates and accepts that string;
when encountering a string not in that language, the machine terminates and rejects that string.

Note that “Turing-Decidable” is a stronger condition than “Turing-Recognizable”, because, if a language is Turing-Decidable then its corresponding Turing Machine never runs forever.