Philosophy of logic, the study, from a philosophical perspective, of the nature and types of logic, including problems in the field and the relation of logic to mathematics and other disciplines.
But in thinking about the relation of logic to grammar, one should not assume a naive conception of the latter.
Whatever the merits of Abû Bishr's view of the relation of logic to language, it weathered Sîrâfî's storm of criticism badly.
The consequence relations of classical logics are monotonic.
In other words, Israel accused defenders of nonmonotonic logic of confusing a theory of defeasible inference (a branch of epistemology) with a theory of genuine consequence relations (a branch of logic).
The consequence relation of a logic system can be given in several ways, some proof-theoretic, others semantic.
If ⊢ is such a relation there is a unique language in the above sense which is the language of ⊢ and we can think of this 'logically loaded' sense of connective as applying to the ordered pair ⟨#, ⊢⟩; thus in the case of intuitionistic implication this would be ⟨→, ⊢I L⟩, where ⊢I L is the consequence relation of intuitionistic logic.
Despite Russell's misgivings about modal notions, it is clear enough from Russell's conception of logic that logical relations between propositions would always obtain in virtue of their form (IMP, 197 98; PLA, 237 39).
The logic of liberal thinking on equality and justice always points towards equality of condition and, since it is difficult to see how such radical equality is compatible with capitalist relations of power, the logic of liberal thinking points beyond itself, towards socialism.
