Extensions in graph normal form

Graph normal form, introduced earlier for propositional logic, is shown to be a normal form also for first-order logic. It allows to view syntax of theories as digraphs, while their semantics as kernels of these digraphs. Graphs are particularly well suited for studying circularity, and we provide some general means for verifying that circular or apparently circular extensions are conservative. Traditional syntactic means of ensuring conservativity, like definitional extensions or positive occurrences guaranteeing exsitence of fixed points, emerge as special cases.publishedVersio

There are only two paradoxes

Using a graph representation of classical logic, the paper shows that the liar or Yablo pattern occurs in every semantic paradox. The core graph theoretic result generalizes theorem of Richardson, showing solvability of finite graphs without odd cycles, to arbitrary graphs which are proven solvable when no odd cycles nor patterns generalizing Yablo's occur. This follows from an earlier result by a new compactness-like theorem, holding for infinitary logic and utilizing the graph representation.Comment: 7 pages, submitted to a journa

Conformal mappings on the Grushin plane

We study conformal mappings in the Grushin plane and provide a number of their characterizations in terms of the Sobolev mappings and their geometry. Furthermore, we connect conformality on the Grushin plane with conformality on the complex plane by using the Meyerson map. Among applications we discuss admissible curves and length-distortion estimates in the Grushin plane, as well as the Carath\'eodory extension theorem.Comment: 16 pages, 4 figure

Modalities as interactions between the classical and the intuitionistic logics

We give an equivalent formulation of topological algebras, interpreting S4, as boolean algebras equipped with intuitionistic negation. The intuitionistic substructure—Heyting algebra—of such an algebra can be then seen as an “epistemic subuniverse”, and modalities arise from the interaction between the intuitionistic and classical negations or, we might perhaps say, between the epistemic and the ontological aspects: they are not relations between arbitrary alternatives but between intuitionistic substructures and one common world governed by the classical (propositional) logic. As an example of the generality of the obtained view, we apply it also to S5. We give a sound, complete and decidable sequent calculus, extending a classical system with the rules for handling the intuitionistic negation, in which one can prove all classical, intuitionistic and S4 valid sequents

Paraconsistent resolution

Digraphs provide an alternative syntax for propositional logic, with digraph kernels corresponding to classical models. Semikernels generalize kernels and we identify a subset of well-behaved semikernels that provides nontrivial models for inconsistent theories, specializing to the classical semantics for the consistent ones. Direct (instead of refutational) reasoning with classical resolution is sound and complete for this semantics, when augmented with a specific weakening which, in particular, excludes Ex Falso. Dropping all forms of weakening yields reasoning which also avoids typical fallacies of relevance