Large portions of mathematics such as algebra and geometry can be formalized using first-order axiomatizations. In many cases it is even possible to use a very well-behaved class of first-order axioms, namely, what are called coherent or geometric implications. Such class of axioms can be translated to inference rules that can be added to a sequent calculus while preserving its structural properties. In this work, this fundamental result is extended to their infinitary generalizations as extensions of sequent calculi for both (...) classical and intuitionistic infinitary logic. As an application, a simple proof of the infinitary Barr’s theorem without the axioms of choice is shown. (shrink)

Using labelled formulae, a cut-free sequent calculus for intuitionistic propositional logic is presented, together with an easy cut-admissibility proof; both extend to cover, in a uniform fashion, all intermediate logics characterised by frames satisfying conditions expressible by one or more geometric implications. Each of these logics is embedded by the Gödel–McKinsey–Tarski translation into an extension of S4. Faithfulness of the embedding is proved in a simple and general way by constructive proof-theoretic methods, without appeal to semantics other than in the (...) explanation of the rules. (shrink)

A general method for generating contraction- and cut-free sequent calculi for a large family of normal modal logics is presented. The method covers all modal logics characterized by Kripke frames determined by universal or geometric properties and it can be extended to treat also Gödel-Löb provability logic. The calculi provide direct decision methods through terminating proof search. Syntactic proofs of modal undefinability results are obtained in the form of conservativity theorems.

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)

In 1968, Orevkov presented proofs of conservativity of classical over intuitionistic and minimal predicate logic with equality for seven classes of sequents, what are known as Glivenko classes. The proofs of these results, important in the literature on the constructive content of classical theories, have remained somehow cryptic. In this paper, direct proofs for more general extensions are given for each class by exploiting the structural properties of G3 sequent calculi; for five of the seven classes the results are strengthened (...) to height-preserving statements, and it is further shown that the constructive and minimal proofs are identical in structure to the classical proof from which they are obtained. (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)

Various sources in the literature claim that the deduction theorem does not hold for normal modal or epistemic logic, whereas others present versions of the deduction theorem for several normal modal systems. It is shown here that the apparent problem arises from an objectionable notion of derivability from assumptions in an axiomatic system. When a traditional Hilbert-type system of axiomatic logic is generalized into a system for derivations from assumptions, the necessitation rule has to be modified in a way that (...) restricts its use to cases in which the premiss does not depend on assumptions. This restriction is entirely analogous to the restriction of the rule of universal generalization of first-order logic. A necessitation rule with this restriction permits a proof of the deduction theorem in its usual formulation. Other suggestions presented in the literature to deal with the problem are reviewed, and the present solution is argued to be preferable to the other alternatives. A contraction-and cut-free sequent calculus equivalent to the Hubert system for basic modal logic shows the standard failure argument untenable by proving the underivability of DA from A. (shrink)

A constructivisation of the cut-elimination proof for sequent calculi for classical, intuitionistic and minimal infinitary logics with geometric rules—given in earlier work by the second author—is presented. This is achieved through a procedure where the non-constructive transfinite induction on the commutative sum of ordinals is replaced by two instances of Brouwer’s Bar Induction. The proof of admissibility of the structural rules is made ordinal-free by introducing a new well-founded relation based on a notion of embeddability of derivations. Additionally, conservativity for (...) classical over intuitionistic/minimal logic for the seven (finitary) Glivenko sequent classes is here shown to hold also for the corresponding infinitary classes. (shrink)