76 research outputs found
Proof Theory for Positive Logic with Weak Negation
Proof-theoretic methods are developed for subsystems of Johansson's logic
obtained by extending the positive fragment of intuitionistic logic with weak
negations. These methods are exploited to establish properties of the logical
systems. In particular, cut-free complete sequent calculi are introduced and
used to provide a proof of the fact that the systems satisfy the Craig
interpolation property. Alternative versions of the calculi are later obtained
by means of an appropriate loop-checking history mechanism. Termination of the
new calculi is proved, and used to conclude that the considered logical systems
are PSPACE-complete
MacNeille Completion and Buchholz\u27 Omega Rule for Parameter-Free Second Order Logics
Buchholz\u27 Omega-rule is a way to give a syntactic, possibly ordinal-free proof of cut elimination for various subsystems of second order arithmetic. Our goal is to understand it from an algebraic point of view. Among many proofs of cut elimination for higher order logics, Maehara and Okada\u27s algebraic proofs are of particular interest, since the essence of their arguments can be algebraically described as the (Dedekind-)MacNeille completion together with Girard\u27s reducibility candidates. Interestingly, it turns out that the Omega-rule, formulated as a rule of logical inference, finds its algebraic foundation in the MacNeille completion.
In this paper, we consider a family of sequent calculi LIP = cup_{n >= -1} LIP_n for the parameter-free fragments of second order intuitionistic logic, that corresponds to the family ID_{<omega} = cup_{n <omega} ID_n of arithmetical theories of inductive definitions up to omega. In this setting, we observe a formal connection between the Omega-rule and the MacNeille completion, that leads to a way of interpreting second order quantifiers in a first order way in Heyting-valued semantics, called the Omega-interpretation. Based on this, we give a (partly) algebraic proof of cut elimination for LIP_n, in which quantification over reducibility candidates, that are genuinely second order, is replaced by the Omega-interpretation, that is essentially first order. As a consequence, our proof is locally formalizable in ID-theories
Hybridization of institutions
Extended version including all proofsModal logics are successfully used as specification logics for reactive systems. However, they are not expressive enough to refer to individual states and reason about the local behaviour of such systems. This limitation is overcome in hybrid logics which introduce special symbols for naming states in models. Actually, hybrid logics have recently regained interest, resulting in a number of new results and techniques as well as applications to software specification.
In this context, the first contribution of this paper is an attempt to ‘universalize’ the hybridization idea. Following the lines of [DS07], where a method to modalize arbitrary institutions is presented, the paper introduces a method to hybridize logics at the same institution-independent level. The method extends arbitrary institutions with Kripke semantics (for multi-modalities with arbitrary arities) and hybrid features. This paves the ground for a general result: any encoding (expressed as comorphism) from an arbitrary institution to first order logic (FOL) deter- mines a comorphism from its hybridization to FOL. This second contribution opens the possibility of effective tool support to specification languages based upon logics with hybrid features.Fundação para a Ciência e a Tecnologia (FCT
Robinson consistency in many-sorted hybrid first-order logics
In this paper we prove a Robinson consistency theorem for a class of many-sorted hybrid logics as a consequence of an Omitting Types Theorem. An important corollary of this result is an interpolation theorem
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