>>14105710>what are the differences between formal logic that philosophy professors do and mathematical logic mathematicians do?At one end of the spectrum, you have logic as a branch of philosophy that forms a foundation for mathematical reasoning (or deductive reasoning in general).
Professors of the philosophy of logic seek to understand the nature of logical truth (truth in virtue of logic alone) and what distinguishes valid from invalid logical inference.
This usually entails examining and questioning the axioms of particular logical systems, teasing out their implications, and arguing in favor or against, usually using intuition as a basis (e.g., the relevance logicians' critiques of the arguably intuition-defying paradoxes of classical material implication, like p -> (q -> p)--i.e., that a truth is implied by everything).
There are many different "camps" in the philosophy of logic, like classical vs. non-classical or monist vs. pluralist. For example, Quine is probably the most famous and influential logical monist, and Kripke is easily the most important modal logician.
At the other end of the spectrum you have people trying prove results (typically metatheorems) about a given logic, its semantics, and its proof system(s), which may have practical, or even foundational, import on mathematics, especially if it involves set theory, model theory, or algebraic structures.
Logicians vary in what part of the spectrum they do most of their work, but the point is, even those of a more philosophical bent still usually make contributions at the other end. For example, both Kripke and Quine devised interesting alternative axiomatic set theories (Kripke–Platek Set Theory and New Foundations).
