Just found the response on Quora about a similar question, how philosophical and mathematical logic differ. I have pasted one answer below as it tickled me.

“When I was getting my PhD, we had a joint logic seminar with both philosophical and mathematical logicians. I would say the most striking difference is what part of the talk they are interested in.

When a mathematical logician gives a talk in front of an audience that contains philosophical logicians, it often goes something like this. There is a brief introduction, including a couple of definitions. For the mathematical logician, this is just boring routine stuff, something you need to go through before you write down the theorem and gets to the interesting part, the neat techniques he or she invented to prove it.

However, as soon as the definitions are shown, the philosophers raise their hands and want to discuss whether this is the “right” definition. For them, the definition is supposed to clarify what you are studying; the definition itself should captures some underlying basic truth. The mathematical logician just doesn’t care about that. He or she will rather be thinking something along the lines of “Clearly it is the right definition, because that is the definition that lets us prove this extremely cool theorem that I haven’t even gotten to write down yet! Shut up and let me get on with it!””

What is the difference between philosophical logic & mathematical logic? - Quora

Link: https://www.quora.com/What-is-th

What is the difference between philosophical logic & mathematical logic?