Manţiq, or logic, as studied in the Islamic sciences, is really about two things:
1. How to make a proper definition of a concept.
2. How to construct a sound proof or argument, and detect flaws in a faulty argument.
There is nothing mysterious about it. As such, logic is undeniably needed in all sciences, and most particularly in Kalaam. In Kalaam only the strongest proofs are of significance, so a good understanding of logical principles is needed to assess the strength of a proof.
In general, a solid modern education really teaches what logic was used for in the old days. Back then education relied mostly on memorization, so logic was needed to teach students how to think.
Today, however, most well educated people will be aware of what makes a solid definition, and how to detect flaws in an argument, especially if one has been exposed to Boolean Algebra. If you know how to construct a good search in Google, you know how to use logic. Yet to understand advanced books in kalaam and Uşuulu-l-Fiqh (how to derive Islamic judgments from the Qur’aan and ĥadiitħ), one needs to be familiar with the terminology of Islamic Logic, so it is important to read at least one book in this science.
By “Islamic Logic” I mean books on logic that have been purified from Greek theology and what relates to it. Beware that the criticism of some scholars against the study of logic is meant for logic mixed with Greek philosophy. After all, no one in their right mind will forbid studying how to make a definition or construct a sound proof.
One more thing: some people think that Aristotales invented logic and therefore that using it makes on a follower of his. This is nonsense, because humans have been using logic in all ages, or at least as long as they have been arguing, because they needed to be able to detect flaws in arguments and define concepts properly. All Aristotales did was codify the principles of logic so that it could be studied systematically.