28 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
Epistemic Logics of Structured Intensional Groups
Epistemic logics of intensional groups lift the assumption that membership in
a group of agents is common knowledge. Instead of being represented directly as
a set of agents, intensional groups are represented by a property that may
change its extension from world to world. Several authors have considered
versions of the intensional group framework where group-specifying properties
are articulated using structured terms of a language, such as the language of
Boolean algebras or of description logic. In this paper we formulate a general
semantic framework for epistemic logics of structured intensional groups,
develop the basic theory leading to completeness-via-canonicity results, and
show that several frameworks presented in the literature correspond to special
cases of the general framework.Comment: In Proceedings TARK 2023, arXiv:2307.0400
Moss' logic for ordered coalgebras
We present a finitary coalgebraic logic for -coalgebras, where is a
locally monotone endofunctor of the category of posets and monotone maps that
preserves exact squares and finite intersections. The logic uses a single cover
modality whose arity is given by the dual of the coalgebra functor , and the
semantics of the modality is given by relation lifting. For the finitary
setting to work, we need to develop a notion of a base for subobjects of .
This in particular allows us to talk about a finite poset of subformulas for a
given formula, and of a finite poset of successors for a given state in a
coalgebra. The notion of a base is introduced generally for a category equipped
with a suitable factorisation system.
We prove that the resulting logic has the Hennessy-Milner property for the
notion of similarity based on the notion of relation lifting. We define a
sequent proof system for the logic and prove its completeness
Constraint tableaux for two-dimensional fuzzy logics
We introduce two-dimensional logics based on \L{}ukasiewicz and G\"{o}del
logics to formalize paraconsistent fuzzy reasoning. The logics are interpreted
on matrices, where the common underlying structure is the bi-lattice (twisted)
product of the interval. The first (resp.\ second) coordinate encodes
the positive (resp.\ negative) information one has about a statement. We
propose constraint tableaux that provide a modular framework to address their
completeness and complexity
Monotone sequent calculus and resolution
summary:We study relations between propositional Monotone Sequent Calculus (MLK --- also known as Geometric Logic) and Resolution with respect to the complexity of proofs, namely to the concept of the polynomial simulation of proofs. We consider Resolution on sets of monochromatic clauses. We prove that there exists a polynomial simulation of proofs in MLK by intuitionistic proofs. We show a polynomial simulation between proofs from axioms in MLK and corresponding proofs of contradiction (refutations) in MLK. Then we show a relation between a resolution refutation of a set of monochromatic clauses (CNF formula) and a proof of the sequent (representing corresponding DNF formula) in MLK. Because monotone logic is a part of intuitionistic logic, results are relevant for intuitionistic logic too