I have this assignment about different types of formal logic systems, like Lewis S5, Fuzzy Logic and some others, but now they ask me to search for any non logicist system, but I've search a lot and didn't find any example or clue about this type of systems, can anyone give me a clue or an example?
My research:
I've search something about this at the Stanford Encyclopedia of Philosophy(and some google search) but got nothing else than the definition and some story. They defined us a formal logicist system as one that has 1) Logic constants. 2)Logic axioms, 3) that the inference rules must be true for any given interpretation of a logic symbol.
Natural Deduction and Sequent Calculus are examples of logical systems without logical axioms: only rules.
You may consider also Post canonical systems or Formal languages as examples of formal systems that do not manage logical constants.