34 research outputs found
Sequent calculi and interpolation for non-normal modal and deonticlogics
G3-style sequent calculi for the logics in the cube of non-normal modal
logics and for their deontic extensions are studied. For each calculus we prove
that weakening and contraction are height-preserving admissible, and we give a
syntactic proof of the admissibility of cut. This implies that the subformula
property holds and that derivability can be decided by a terminating proof
search whose complexity is in PSPACE. These calculi are shown to be equivalent
to the axiomatic ones and, therefore, they are sound and complete with respect
to neighbourhood semantics. Finally, it is given a Maehara-style proof of
Craig's interpolation theorem for most of the logics considered
Labelled calculi for quantified modal logics with definite descriptions
We introduce labelled sequent calculi for quantified modal logics with
definite descriptions. We prove that these calculi have the good structural
properties of G3-style calculi. In particular, all rules are height-preserving
invertible, weakening and contraction are height-preserving admissible and cut
is admissible. Finally, we show that each calculus gives a proof-theoretic
characterization of validity in the corresponding class of models
Proof-theoretic pluralism
Starting from a proof-theoretic perspective, wheremeaning is determined by the inference rules governing logical operators, in this paper we primarily aim at developing a proof-theoretic alternative to the model-theoretic meaning-invariant logical pluralism discussed in Beall and Restall (Logical pluralism, Oxford University Press, Oxford, 2006). We will also outline how this framework can be easily extended to include a form of meaning-variant logical pluralism. In this respect, the framework developed in this paper\u2014which we label two-level proof-theoretic pluralism\u2014is much broader
in scope than the one discussed in Beall and Restall\u2019s book
Labelled sequent calculi for logics of strict implication
n this paper we study the proof theory of C.I. Lewis’ logics of strict conditional S1-
S5 and we propose the first modular and uniform presentation of C.I. Lewis’ systems.
In particular, for each logic Sn we present a labelled sequent calculus G3Sn and we
discuss its structural properties: every rule is height-preserving invertible and the
structural rules of weakening, contraction and cut are admissible. Completeness of
G3Sn is established both indirectly via the embedding in the axiomatic system Sn
and directly via the extraction of a countermodel out of a failed proof search. Finally,
the sequent calculus G3S1 is employed to obtain a syntactic proof of decidability of
S1
Nested Sequents for Quantified Modal Logics
This paper studies nested sequents for quantified modal logics. In
particular, it considers extensions of the propositional modal logics definable
by the axioms D, T, B, 4, and 5 with varying, increasing, decreasing, and
constant domains. Each calculus is proved to have good structural properties:
weakening and contraction are height-preserving admissible and cut is
(syntactically) admissible. Each calculus is shown to be equivalent to the
corresponding axiomatic system and, thus, to be sound and complete. Finally, it
is argued that the calculi are internal -- i.e., each sequent has a formula
interpretation -- whenever the existence predicate is expressible in the
language.Comment: accepted to TABLEAUX 202
Nested Sequents for Quantified Modal Logics
This paper studies nested sequents for quantified modal logics. In particular, it considers extensions of the propositional modal logics definable by the axioms D, T, B, 4, and 5 with varying, increasing, decreasing, and constant domains. Each calculus is proved to have good structural properties: weakening and contraction are height-preserving admissible and cut is (syntactically) admissible. Each calculus is shown to be equivalent to the corresponding axiomatic system and, thus, to be sound and complete. Finally, it is argued that the calculi are internal—i.e., each sequent has a formula interpretation—whenever the existence predicate is expressible in the language
Constructive Cut Elimination in Geometric Logic
Funding Information: Partially funded by the Academy of Finland, research project no. Publisher Copyright: © 2022 Schloss Dagstuhl- Leibniz-Zentrum fur Informatik GmbH, Dagstuhl Publishing. All rights reserved.A constructivisation of the cut-elimination proof for sequent calculi for classical and intuitionistic infinitary logic with geometric rules - given in earlier work by the second author - is presented. This is achieved through a procedure in which the non-constructive transfinite induction on the commutative sum of ordinals is replaced by two instances of Brouwer's Bar Induction. Additionally, a proof of Barr's Theorem for geometric theories that uses only constructively acceptable proof-theoretical tools is obtained.Peer reviewe
Interpolation in extensions of first-order logic
We prove a generalization of Maehara's lemma to show that the extensions of
classical and intuitionistic first-order logic with a special type of geometric
axioms, called singular geometric axioms, have Craig's interpolation property.
As a corollary, we obtain a direct proof of interpolation for (classical and
intuitionistic) first-order logic with identity, as well as interpolation for
several mathematical theories, including the theory of equivalence relations,
(strict) partial and linear orders, and various intuitionistic order theories
such as apartness and positive partial and linear orders.Comment: In this up-dated version of the paper a more general notion of
singular geometric theory is provided allowing the extension of our
interpolation results to further fundamental mathematical theorie
Interpolation in Extensions of First-Order Logic
We prove a generalization of Maehara\u2019s lemma to show that the extensions
of classical and intuitionistic first-order logic with a special type of geometric axioms,
called singular geometric axioms, have Craig\u2019s interpolation property. As a corollary, we
obtain a direct proof of interpolation for (classical and intuitionistic) first-order logic with
identity, as well as interpolation for several mathematical theories, including the theory
of equivalence relations, (strict) partial and linear orders, and various intuitionistic order
theories such as apartness and positive partial and linear orders
Proof theory of quantified modal logics
We introduce labelled sequent calculi for indexed modal logics. We prove that the structural rules of weakening and contraction are height-preserving admissible, that all rules are invertible, and that cut is admissible. Then we prove that each calculus introduced is sound and complete with respect to the appropriate class of transition frames