You are right to say there is an intuitive use of logic. Even little children can understand basic deductions and use logical operators in their speech. Consider a child who says, "If I go to bed, then tomorrow I can have breakfast AND desert?". While not the only example, clear a mastery of the conditional and the conjunction. It is a trope that children quickly acquire the skills of a lawyer around 3 or 4. The basic operations of quantification, qualification, classification, and logic are a function of neural development.
Keeping in mind that logic is grown from neurons, it helps keep perspective on what logic really is and isn't. Logic isn't some program transferred to the mind like software is uploaded to a computer. Logic is more like the skill of playing a violin: some natural talents are present, but then the skill is slowly developed through practice when the brain is ready. As a former teacher, particularly enlightening are the ideas of Jean Piaget's theory of cognitive development. While your question is about logic, which is a field of study generally placed in philosophy departments, in some ways your question is a question of human development.
Ultimately, there is no monolith called "logic", but rather a series of "logics" that vary in terms of complexity and are adopted one after another. For instance, Aristotelian syllogisms are rather simple and intuitive, but modal logics require some studying, particularly as formalisms are used. Mathematical logic is just another formalism in a sense.
Can the "intuitive logics" be studied? Yes, they are, but not by philosophers. Rather the basis of how the brain infers is more a question pursued by cognitive scientists like those with backgrounds in cognitive semantics, artificial intelligence researchers, and psychologists. Generally the intuition-provided logics are part of a philosophers intuition or metaphysics and often aren't scrutinized.