Consistent disjunctive sequent calculi and Scott domains

Based on the framework of disjunctive propositional logic, we first provide a syntactic representation for Scott domains. Precisely, we establish a category of consistent disjunctive sequent calculi with consequence relations, and show it is equivalent to that of Scott domains with Scott-continuous functions. Furthermore, we illustrate the approach to solving recursive domain equations by introducing some standard domain constructions, such as lifting and sums. The subsystems relation on consistent finitary disjunctive sequent calculi makes these domain constructions continuous. Solutions to recursive domain equations are given by constructing the least fixed point of a continuous function.

Inconsistency Measurement for Paraconsistent Inference

One of the main aims of the methods developed for reasoning under inconsistency, in particular paraconsistent inference, is to derive informative conclusions from inconsistent bases. In this paper, we introduce an approach based on inconsistency measurement for defining non-monotonic paraconsistent consequence relations. The main idea consists in adapting properties of classical reasoning under consistency to inconsistent propositional bases by involving inconsistency measures (IM). We first exhibit interesting properties of our consequence relations. We then study situations where they bring about consequences that are always jointly consistent. In particular, we introduce a property of inconsistency measures that guarantees the consistency of the set of all entailed formulas. We also show that this property leads to several interesting properties of our IM-based consequence relations. Finally, we discuss relationships between our framework and well-known consequence relations that are based on maximal consistent subsets. In this setting, we establish direct connections between the latter and properties of inconsistency measures.

Fallibilism and Consequence

Alex Worsnip argues in favor of what he describes as a particularly robust version of fallibilism: subjects can sometimes know things that are, for them, possibly false (in the epistemic sense of ‘possible’). My aim in this paper is to show that Worsnip’s argument is inconclusive for a surprising reason: the existence of possibly false knowledge turns on how we ought to model entailment or consequence relations among sentences in natural language. Since it is an open question how we ought to think about consequence in natural language, it is an open question whether there is possibly false knowledge. I close with some reflections on the relation between possibly false knowledge and fallibilism. I argue that there is no straightforward way to use linguistic data about natural language epistemic modals to either verify or refute the fallibilist thesis.

The role of non-connective discourse cues and their interaction with connectives

The disambiguation and processing of coherence relations is often investigated with a focus on explicit connectives, such as but or so. Other, non-connective cues from the context also facilitate discourse inferences, although their precise disambiguating role and interaction with connectives have been largely overlooked in the psycholinguistic literature so far. This study reports on two crowdsourcing experiments that test the role of contextual cues (parallelism, antonyms, resultative verbs) in the disambiguation of contrast and consequence relations. We compare the effect of contextual cues in conceptually different relations, and with connectives that differ in their semantic precision. Using offline tasks, our results show that contextual cues significantly help disambiguating contrast and consequence relations in the absence of connectives. However, when connectives are present in the context, the effect of cues only holds if the connective is acceptable in the target relation. Overall, our study suggests that cues are decisive on their own, but only secondary in the presence of connectives. These results call for further investigation of the complex interplay between connective types, contextual cues, relation types and other linguistic and cognitive factors.

Tree-Like Proof Systems for Finitely-Many Valued Non-deterministic Consequence Relations

The main goal of this paper is to provide an abstract framework for constructing proof systems for various many-valued logics. Using the framework it is possible to generate strongly complete proof systems with respect to any finitely valued deterministic and non-deterministic logic. I provide a couple of examples of proof systems for well-known many-valued logics and prove the completeness of proof systems generated by the framework.

Sequent calculi of first-order logics of partial predicates with extended renominations and composition of predicate complement

We study new classes of program-oriented logical formalisms – pure first-order logics of quasiary predicates with extended renominations and a composition of predicate complement. For these logics, various logical consequence relations are specified and corresponding calculi of sequent type are constructed. We define basic sequent forms for the specified calculi and closeness conditions. The soundness, completeness, and counter-model existence theorems are proved for the introduced calculi.