Classical Modal Display Logic in the Calculus of Structures and Minimal Cut-free Deep Inference Calculi for S5

We begin by showing how to faithfully encode the Classical ModalDisplay Logic (CMDL) of Wansing into the Calculus of Structures(CoS) of Guglielmi. Since every CMDL calculus enjoys cut-elimination,we obtain a cut-elimination theorem for all corresponding CoScalculi. We then show how our result leads to a minimal cut-freeCoS calculus for modal logic S5. No other existing CoS calculifor S5 enjoy both these properties simultaneously.     

Operational Concepts of Nonmonotonic Logics Part 2: Autoepistemic Logic

The subject of nonmonotonic reasoning is reasoning with incompleteinformation. One of the main approaches is autoepistemic logic inwhich reasoning is based on introspection. This paper aims at providing a smooth introduction to this logic,stressing its motivation and basic concepts. The meaning (semantics)of autoepistemic logic is given in terms of so-called expansionswhich are usually defined as solutions of a fixed-point equation. Thepresent paper shows a more understandable, operational method fordetermining expansions. By improving applicability of the basicconcepts to concrete examples, we hope to make a contribution to awider usage of autoepistemic logic in practical applications.     

Kripke Semantics for Modal Substructural Logics

We introduce Kripke semantics for modal substructural logics, and provethe completeness theorems with respect to the semantics. Thecompleteness theorems are proved using an extended Ishihara's method ofcanonical model construction (Ishihara, 2000). The framework presentedcan deal with a broad range of modal substructural logics, including afragment of modal intuitionistic linear logic, and modal versions ofCorsi's logics, Visser's logic, Méndez's logics and relevant logics.     

A Logic with Relative Knowledge Operators

We study a knowledge logic that assumes that to each set of agents, an indiscernibility relation is associated and the agents decide the membership of objects or states up to this indiscernibility relation. Its language contains a family of relative knowledge operators. We prove the decidability of the satisfiability problem, we show its EXPTIME-completeness and as a side-effect, we define a complete Hilbert-style axiomatization.     

Substructural Logics with Mingle

We introduce structural rules mingle, and investigatetheorem-equivalence, cut- eliminability, decidability, interpolabilityand variable sharing property for sequent calculi having the mingle.These results include new cut-elimination results for the extendedlogics: FL_m (full Lambek logic with the mingle), GL_m(Girard's linear logic with the mingle) and L_m (Lambek calculuswith restricted mingle).     

Adaptive Fuzzy Logics for Contextual Hedge Interpretation

The article presents several adaptive fuzzy hedge logics. These logics are designed to perform a specific kind of hedge detection. Given a premise set Γ that represents a series of communicated statements, the logics can check whether some predicate occurring in Γ may be interpreted as being (implicitly) hedged by technically, strictly speaking or loosely speaking, or simply non-hedged. The logics take into account both the logical constraints of the premise set as well as conceptual information concerning the meaning of potentially hedged predicates (stored in the memory of the interpreter in question). The proof theory of the logics is non-monotonic in order to enable the logics to deal with possible non-monotonic interpretation dynamics (this is illustrated by means of several concrete proofs). All the adaptive fuzzy hedge logics are also sound and strongly complete with respect to their [0,1]-semantics.     

Analytic Calculi for Logics of Ordinal Multiples of Standard t-Norms

For two propositional fuzzy logics, we present analytic proofcalculi, based on relational hypersequents. The logic consideredfirst, called M, is based on the finite ordinal sums of ukasiewiczt-norms. In addition to the usual connectives—the conjunction, the implication and the constant 0—we use a furtherunary connective interpreted by the function associating witheach truth value a the greatest -idempotent below a. M is aconservative extension of Basic Logic. The second logic, called M, is based on the finite ordinal sumsof the product t-norm on (0, 1]. Our connectives are in thiscase just the conjunction and the implication.     

Knowledge-Driven versus Data-Driven Logics

The starting point of this work is the gap between two distinct traditions in information engineering: knowledge representation and data-driven modelling. The first tradition emphasizes logic as a tool for representing beliefs held by an agent. The second tradition claims that the main source of knowledge is made of observed data, and generally does not use logic as a modelling tool. However, the emergence of fuzzy logic has blurred the boundaries between these two traditions by putting forward fuzzy rules as a Janus-faced tool that may represent knowledge, as well as approximate non-linear functions representing data. This paper lays bare logical foundations of data-driven reasoning whereby a set of formulas is understood as a set of observed facts rather than a set of beliefs. Several representation frameworks are considered from this point of view: classical logic, possibility theory, belief functions, epistemic logic, fuzzy rule-based systems. Mamdani's fuzzy rules are recovered as belonging to the data-driven view. In possibility theory a third set-function, different from possibility and necessity plays a key role in the data-driven view, and corresponds to a particular modality in epistemic logic. A bi-modal logic system is presented which handles both beliefs and observations, and for which a completeness theorem is given. Lastly, our results may shed new light in deontic logic and allow for a distinction between explicit and implicit permission that standard deontic modal logics do not often emphasize.