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Logic programming is a type of computer programming that is based on formal logic rules. In a sense, logic programming corresponds to the use of mathematical equations and other logical constructs to drive programming outcomes.
One basic way to explain logic programming is that it typically tends to be lists of logical rules that are used to provide programming outcomes. Another great way to characterize logic programming is to contrast it with other paradigms that mainly have developed more recently since the old days of linear BASIC and Fortran.
One of these is function programming, where instead of using logical lists, programming languages utilize modular functions to drive programming outcomes. Another new innovation is object-oriented programming, in which the computer programming language orders its rules and processes by recognizing a series of virtual objects.
In object-oriented programming, utilizing advanced classes with various attributes, the programming languages basically bake a lot of logic into the object-oriented architecture. This has led some people to theorize on how logic programming itself is somewhat obsolete as a main vehicle for compiled code.
However, as pointed out by others, the logic programming is still in place — it's just housed under a different architectural structure.