Information completeness in Nelson algebras of rough sets induced by quasiorders

In this paper, we give an algebraic completeness theorem for constructive logic with strong negation in terms of finite rough set-based Nelson algebras determined by quasiorders. We show how for a quasiorder $R$, its rough set-based Nelson algebra can be obtained by applying the well-known construction by Sendlewski. We prove that if the set of all $R$-closed elements, which may be viewed as the set of completely defined objects, is cofinal, then the rough set-based Nelson algebra determined by a quasiorder forms an effective lattice, that is, an algebraic model of the logic $E_0$, which is characterised by a modal operator grasping the notion of "to be classically valid". We present a necessary and sufficient condition under which a Nelson algebra is isomorphic to a rough set-based effective lattice determined by a quasiorder.Comment: 15 page

Crisp bi-G\"{o}del modal logic and its paraconsistent expansion

In this paper, we provide a Hilbert-style axiomatisation for the crisp bi-G\"{o}del modal logic \KbiG. We prove its completeness w.r.t.\ crisp Kripke models where formulas at each state are evaluated over the standard bi-G\"{o}del algebra on $[0,1]$. We also consider a paraconsistent expansion of \KbiG with a De Morgan negation $\neg$ which we dub \KGsquare. We devise a Hilbert-style calculus for this logic and, as a~con\-se\-quence of a~conservative translation from \KbiG to \KGsquare, prove its completeness w.r.t.\ crisp Kripke models with two valuations over $[0,1]$ connected via $\neg$. For these two logics, we establish that their decidability and validity are $\mathsf{PSPACE}$-complete. We also study the semantical properties of \KbiG and \KGsquare. In particular, we show that Glivenko theorem holds only in finitely branching frames. We also explore the classes of formulas that define the same classes of frames both in $\mathbf{K}$ (the classical modal logic) and the crisp G\"{o}del modal logic \KG^c. We show that, among others, all Sahlqvist formulas and all formulas $\phi\rightarrow\chi$ where $\phi$ and $\chi$ are monotone, define the same classes of frames in $\mathbf{K}$ and \KG^c

Lewis meets Brouwer: constructive strict implication

C. I. Lewis invented modern modal logic as a theory of "strict implication". Over the classical propositional calculus one can as well work with the unary box connective. Intuitionistically, however, the strict implication has greater expressive power than the box and allows to make distinctions invisible in the ordinary syntax. In particular, the logic determined by the most popular semantics of intuitionistic K becomes a proper extension of the minimal normal logic of the binary connective. Even an extension of this minimal logic with the "strength" axiom, classically near-trivial, preserves the distinction between the binary and the unary setting. In fact, this distinction and the strong constructive strict implication itself has been also discovered by the functional programming community in their study of "arrows" as contrasted with "idioms". Our particular focus is on arithmetical interpretations of the intuitionistic strict implication in terms of preservativity in extensions of Heyting's Arithmetic.Comment: Our invited contribution to the collection "L.E.J. Brouwer, 50 years later

Constructivisation through Induction and Conservation

The topic of this thesis lies in the intersection between proof theory and alge- braic logic. The main object of discussion, constructive reasoning, was intro- duced at the beginning of the 20th century by Brouwer, who followed Kant’s explanation of human intuition of spacial forms and time points: these are constructed step by step in a finite process by certain rules, mimicking con- structions with straightedge and compass and the construction of natural numbers, respectively. The aim of the present thesis is to show how classical reasoning, which admits some forms of indirect reasoning, can be made more constructive. The central tool that we are using are induction principles, methods that cap- ture infinite collections of objects by considering their process of generation instead of the whole class. We start by studying the interplay between cer- tain structures that satisfy induction and the calculi for some non-classical logics. We then use inductive methods to prove a few conservation theorems, which contribute to answering the question of which parts of classical logic and mathematics can be made constructive.Tämän opinnäytetyön aiheena on todistusteorian ja algebrallisen logiikan leikkauspiste. Keskustelun pääaiheen, rakentavan päättelyn, esitteli 1900-luvun alussa Brouwer, joka seurasi Kantin selitystä ihmisen intuitiosta tilamuodoista ja aikapisteistä: nämä rakennetaan askel askeleelta äärellisessä prosessissa tiettyjen sääntöjen mukaan, jotka jäljittelevät suoran ja kompassin konstruktioita ja luonnollisten lukujen konstruktiota. Tämän opinnäytetyön tavoitteena on osoittaa, kuinka klassista päättelyä, joka mahdollistaa tietyt epäsuoran päättelyn muodot, voidaan tehdä rakentavammaksi. Keskeinen työkalu, jota käytämme, ovat induktioperiaatteet, menetelmät, jotka keräävät äärettömiä objektikokoelmia ottamalla huomioon niiden luomisprosessin koko luokan sijaan. Aloitamme tutkimalla vuorovaikutusta tiettyjen induktiota tyydyttävien rakenteiden ja joidenkin ei-klassisten logiikan laskelmien välillä. Todistamme sitten induktiivisten menetelmien avulla muutamia säilymislauseita, jotka auttavat vastaamaan kysymykseen siitä, mitkä klassisen logiikan ja matematiikan osat voidaan tehdä rakentaviksi

Structural and universal completeness in algebra and logic

In this work we study the notions of structural and universal completeness both from the algebraic and logical point of view. In particular, we provide new algebraic characterizations of quasivarieties that are actively and passively universally complete, and passively structurally complete. We apply these general results to varieties of bounded lattices and to quasivarieties related to substructural logics. In particular we show that a substructural logic satisfying weakening is passively structurally complete if and only if every classical contradiction is explosive in it. Moreover, we fully characterize the passively structurally complete varieties of MTL-algebras, i.e., bounded commutative integral residuated lattices generated by chains.Comment: This is a preprin

Monadic Fragments of Intuitionistic Control Logic

We investigate monadic fragments of Intuitionistic Control Logic (ICL), which is obtained from Intuitionistic Propositional Logic (IPL) by extending language of IPL by a constant distinct from intuitionistic constants. In particular we present the complete description of purely negational fragment and show that most of monadic fragments are finite