This project introduces inductive logic and uncertain reasoning. Inductive logic was introduced by Rudolf Carnap as a logical system that extends the deductive logic to uncertain inferences, and so can formalise reasoning in non-ideal situations. We will in particular emphasise the probabilistic treatment of uncertainty and the study of probabilistic logic. The language of this logic is the same as that of propositional logic, but the truth values range over [0,1] and respect the axioms of probability. As such, it allows us to model uncertainty by moving from the categorical truth values to the the more fine graded setting with probabilities. We will study a probabilistic consequence relation and its complete proof system and will next look into axiomatisations of probabilistic inference. Later on, we will investigate the connections between probabilistic inference and formal epistemology and, specially, the problem of rational belief formation. Along this line, we will study an axiomatisation of the probabilistic inference that uniquely characterises rational belief formation as is advocated by Objective Bayesians, as well as well some other recent developments in formal epistemology.