Forgive me if this sounds ignorant. Although I am an applied person, my experience in using logic in academia is through mathematical proofs, primarily in the real analysis. I have a friend who is a philosophy major who tells me that philosophy and mathematics has a lot of overlap (which I am heavily aware of). I can see the logic aspect of philosophy used widely in proofs about the behavior of functions, I am in no doubt denying that. Without logic, there is no mathematics.
However, I understand there are other subdivisions of philosophy other than logic. These include ethics and metaphysics among others. Do all subdivisions of philosophy use formal logic in mathematics? I am very open to changing my perspective and by my own admission I might come off as abrasive not because I am defensive, but because I am ignorant. How can you use logic in ethics for example?
Isn't the construct of logical argument biased to the beliefs held by the arguer? For example, 2 people could argue on what has a correct truth value. For example, assume you are trying to argue against the right to die.
Someone pro euthanasia could say A⇒B∨C
Where A is "Patient is very sick", B = "Can live with assistance", C = "Allowed to die with dignity"
But, someone who is very pro-life could say
D=∅, where D is the set of justifiable conditions that can warrant humans to take another human's life.
Both have used logic, but both have different views. Who is to say once side of an "if, then" statement has a truth value of T or F? Again, I apologize for my ignorance. It's just that I am used to doing proofs with universally accepted definitions with constraints, like "a is in the rational set iff it can be expressed as the quotient of p and q, with p and q being in the set of integers such that q is not equal to 0".