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No, logic and formal proof have been integral parts of mathematics for thousands of years. The ancient Greeks, such as Euclid and Pythagoras, were known for their use of logical reasoning and formal proofs. However, the development of formal logic as a field of study did occur more recently in the 19th and 20th centuries.
Mathematical logic is a branch of mathematics which brings together formal logic and mathematics. Mathematical logic entails formal systems for defining the basics and then using the deductive power of logic to develop a system of formal proofs.
formal is just study of logic with purely formal content and material is the branch of logic that focuses the content of reasoning.
Formal logic is logic used to examine the form that an argument is presented in. Formal logic looks at the grammar and sentence structure of an argument through a logical approach.
Boolean Algebra is the study of the algebra of logic whilst Mathematical logic is a way of applying Boolean algebra. Other applications include set theory, digital logic and probability.
Mathematical logic.