25 research outputs found
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
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 Troelstra’s 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 Troelstra’s criteria in the corresponding substructural logical setting
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
Involutive Commutative Residuated Lattice without Unit: Logics and Decidability
We investigate involutive commutative residuated lattices without unit, which
are commutative residuated lattice-ordered semigroups enriched with a unary
involutive negation operator. The logic of this structure is discussed and the
Genzten-style sequent calculus of it is presented. Moreover, we prove the
decidability of this logic.Comment: 16 page
Abstraktnà studium úplnosti pro infinitárnà logiky
V tĂ©to dizertaÄŤnĂ práci se zabĂ˝váme studiem vlastnostĂ Ăşplnosti infinitárnĂch vĂ˝rokovĂ˝ch logik z pohledu abstraktnĂ algebraickĂ© logiky. CĂlem práce je pochopit, jak lze základnĂ nástroj v dĹŻkazech uplnosti, tzv. Lindenbaumovo lemma, zobecnit za hranici finitárnĂch logik. Za tĂmto účelem studujeme vlastnosti Ăşzce souvisejĂcĂ s LindenbaumovĂ˝m lemmatem (a v dĹŻsledku takĂ© s vlastnostmi Ăşplnosti). UvidĂme, Ĺľe na základÄ› tÄ›chto vlastnostĂ lze vystavÄ›t novou hierarchii infinitárnĂch vĂ˝rokovĂ˝ch logik. TakĂ© se zabĂ˝váme studiem tÄ›chto vlastnostĂ v pĹ™ĂpadÄ›, kdy naše logika má nÄ›jakĂ© (pĹ™ĂpadnÄ› hodnÄ› obecnÄ› definovanĂ©) spojky implikace, disjunkce a negace. Mimo jinĂ© uvidĂme, Ĺľe pĹ™Ătomnost danĂ˝ch spojek mĹŻĹľe zajist platnost Lindenbaumova lemmatu. Keywords: abstraktnĂ algebraická logika, infinitárnĂ logiky, Lindenbau- movo lemma, disjunkce, implikace, negaceIn this thesis we study completeness properties of infinitary propositional logics from the perspective of abstract algebraic logic. The goal is to under- stand how the basic tool in proofs of completeness, the so called Linden- baum lemma, generalizes beyond finitary logics. To this end, we study few properties closely related to the Lindenbaum lemma (and hence to com- pleteness properties). We will see that these properties give rise to a new hierarchy of infinitary propositional logic. We also study these properties in scenarios when a given logic has some (possibly very generally defined) connectives of implication, disjunction, and negation. Among others, we will see that presence of these connectives can ensure provability of the Lin- denbaum lemma. Keywords: abstract algebraic logic, infinitary logics, Lindenbaum lemma, disjunction, implication, negationKatedra logikyDepartment of LogicFaculty of ArtsFilozofická fakult