Mathematical logic simply attempts to identify recursions (functions defined for their own outputs) that automate the making of large numbers of inferences, with the qualification that inference-classes that mathematical logicians aspire to automate are usually not randomly chosen and are of independent interest.
Philosophical logic studies the concepts in terms of which such formalizations are to be understood, such as proposition, negation, truth, falsehood, counterfactual truth, entailment, and probabilification. Mathematical logic uses these concepts but does not scrutinize them. Philosophical logic scrutinizes them. Philosophical logic also identifies the limits of formalization and it attempts to assess the significance of mathematical proofs of the existence of such limits.