It is shown that a complete axiomatization of classical non-tautologies can be obtained by taking F (falsehood) as the sole axiom along with the two inference rules: (i) if A is a substitution instance of B, then A |– B; and (ii) if A is obtained from B by replacement of equivalent sentences, then A |– B (counting as equivalent the pairs {T, ~F}, {F, F&F}, {F, F&T}, {F, T&F}, {T, T&T}). Since the set of tautologies is also specifiable by (...) purely syntactic means, the resulting picture gives an improved syntactic account of classical sentential logic. The picture can then be completed by considering related systems that prove adequate to specify the set of contradictions, the set of non-contradictions, and the set of contingencies respectively. (shrink)

The purpose of this paper is to take some of the mystery out of what is known as nonmonotonic logic, by showing that it is not as unfamiliar as may at first sight appear. In fact, it is easily accessible to anybody with a background in classical propositional logic, provided that certain misunderstandings are avoided and a tenacious habit is put aside. In effect, there are logics that act as natural bridges between classical consequence and the principal kinds of nonmonotonic (...) logic to be found in the literature. Like classical logic, they are perfectly monotonic, but they already display some of the distinctive features of the nonmonotonic systems. As well as providing easy conceptual passage to the nonmonotonic case these logics, which we call paraclassical, have an interest of their own. (shrink)

A way is found to add axioms to sequent calculi that maintains the eliminability of cut, through the representation of axioms as rules of inference of a suitable form. By this method, the structural analysis of proofs is extended from pure logic to free-variable theories, covering all classical theories, and a wide class of constructive theories. All results are proved for systems in which also the rules of weakening and contraction can be eliminated. Applications include a system of predicate logic (...) with equality in which also cuts on the equality axioms are eliminated. (shrink)