The correct answer is A. Deductive Systems.
A deductive system is a formal system in which logical consequences of a set of axioms can be derived. The axioms are the starting points of the system, and the theorems are the statements that can be derived from the axioms using the rules of inference.
Inductive systems are a type of machine learning algorithm that learns from data. The system starts with a set of examples, and then tries to find a rule that can be used to predict the output for new examples.
Reasoning with knowledge based systems is a type of artificial intelligence that uses knowledge to make decisions. The system starts with a set of facts, and then uses those facts to reason about new situations.
Search based systems are a type of artificial intelligence that uses search to find solutions to problems. The system starts with a set of possible solutions, and then tries to find the best solution.
In the given question, we are asked to choose the option that is used to demonstrate, on a purely syntactic basis, that one formula is a logical consequence of another formula. The only option that meets this criteria is A. Deductive Systems.