In our discussions of "new logic," we should probably recall some other
categorizations of logic before we wander into the desert. There are three that
have been around a very long time. Their present definition is mostly from
Pierce. They are:
-- Deduction, which is reasoning where necessary conclusions are made
from statements - FOL,
which are arguments by
inference, i.e., by likelihood, usually from experience,
-- Abduction, which forms hypotheses from observed
Notice that induction and abduction are particularly phenomenological.
Deductive logic is the most worked out, with first order varieties arising from
reducing the number of the usual base axioms, or restricting the
definition of the usual axioms; and higher order varieties gotten by
expanding the number of axioms, as in adding possibility to get modal logics.