8 research outputs found
The Jacobson Radical of a Propositional Theory
Alongside the analogy between maximal ideals and complete theories, the Jacobson radical carries over from ideals of commutative rings to theories of propositional calculi. This prompts a variant of Lindenbaum's Lemma that relates classical validity and intuitionistic provability, and the syntactical counterpart of which is Glivenko's Theorem. The Jacobson radical in fact turns out to coincide with the classical deductive closure. As a by-product we obtain a possible interpretation in logic of the axioms-as-rules conservation criterion for a multi-conclusion Scott-style entailment relation over a single-conclusion one.Peer reviewe
Double Negation Semantics for Generalisations of Heyting Algebras
This paper presents an algebraic framework for investigating proposed translations of classical logic into intuitionistic logic, such as the four negative translations introduced by Kolmogorov, Gödel, Gentzen and Glivenko. We view these as variant semantics and present a semantic formulation of Troelstraâs syntactic criteria for a satisfactory negative translation. We consider how each of the above-mentioned translation schemes behaves on two generalisations of Heyting algebras: bounded pocrims and bounded hoops. When a translation fails for a particular class of algebras, we demonstrate that failure via specific finite examples. Using these, we prove that the syntactic version of these translations will fail to satisfy Troelstraâs criteria in the corresponding substructural logical setting
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 . We also consider a paraconsistent expansion of
\KbiG with a De Morgan negation 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 connected via
.
For these two logics, we establish that their decidability and validity are
-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 (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 where and are monotone, define the
same classes of frames in and \KG^c
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
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 Kantâs
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.TÀmÀn opinnÀytetyön aiheena on todistusteorian ja algebrallisen logiikan leikkauspiste. Keskustelun pÀÀaiheen, rakentavan pÀÀttelyn, esitteli 1900-luvun alussa Brouwer, joka seurasi Kantin selitystÀ ihmisen intuitiosta tilamuodoista ja aikapisteistÀ: nÀmÀ rakennetaan askel askeleelta ÀÀrellisessÀ prosessissa tiettyjen sÀÀntöjen mukaan, jotka jÀljittelevÀt suoran ja kompassin konstruktioita ja luonnollisten lukujen konstruktiota.
TÀmÀn opinnÀytetyön tavoitteena on osoittaa, kuinka klassista pÀÀttelyÀ, joka mahdollistaa tietyt epÀsuoran pÀÀttelyn muodot, voidaan tehdÀ rakentavammaksi. Keskeinen työkalu, jota kÀytÀmme, ovat induktioperiaatteet, menetelmÀt, jotka kerÀÀvÀt ÀÀrettömiÀ objektikokoelmia ottamalla huomioon niiden luomisprosessin koko luokan sijaan. Aloitamme tutkimalla vuorovaikutusta tiettyjen induktiota tyydyttÀvien rakenteiden ja joidenkin ei-klassisten logiikan laskelmien vÀlillÀ. Todistamme sitten induktiivisten menetelmien avulla muutamia sÀilymislauseita, jotka auttavat vastaamaan kysymykseen siitÀ, mitkÀ klassisen logiikan ja matematiikan osat voidaan tehdÀ rakentaviksi
Glivenko theorems revisited
Glivenko-type theorems for substructural logics (over FL) are comprehensively studied in the paper (GO06b). Arguments used there are fully algebraic, and based on the fact that all substructural logics are algebraizable (see (GO06a) and also (GJKO07) for the details). As a complementary work to the algebraic approach developed in (GO06b), we present here a concise, proof-theoretic approach to Glivenko theorems for substructural logics. This will show different features of these two approaches