Algebraic Logic Research Papers - Academia.edu (original) (raw)

- by
- •
- Mathematical Modelling, Discretization, Algebraic Logic, Boundary Value Problems
- by Joseph Wilson
- •
- Algebra, Computer Science, Parallel Algorithms, Computer Architecture

This paper covers an aspect of Naïve or Intuitive set theory. This content is taught in West African Senior High Schools.

- by Tomasz Kowalski
- •
- Universal Algebra, Algebraic Logic, Residuated Lattices

Abstract. SCα, CAα, QAα and QEAα stand for the classes of Pinter's substitution algebras, Tarski's cylindric algebras, Halmos' quasipolyadic algebras, and quasipolyadic equality algebras of dimension α, respectively.... more

- by Tarek Ahmed
- •
- Philosophy and Religious Studies, Mathematical Sciences, Algebraic Logic

The main result gives a sufficient condition for a classK of finite dimensional cylindric algebras to have the property that not every epimorphism in K is surjective. In particular, not all epimorphisms are surjective in the classes CAn of n-dimensional cylindric algebras and the class of representable algebras in CAn for finite n > 1, solving Problem 10 of (28)

- by Tarek Ahmed
- •
- Pure Mathematics, Finite Model Theory, Algebraic Logic, First Order Logic

Connections between Algebraic Logic and (ordinary) Logic. Algebraic co- unterpart of model theoretic semantics, algebraic counterpart of proof theory, and their connections. The class Alg(L) of algebras associated to any logic L. Equivalence theorems stating that L has a certain logical property iff Alg(L) has a certain algebraic property. (E.g. L admits a strongly complete Hilbert- style inference system iff

- by István Németi
- •
- Proof Theory, Algebraic Logic

Epistemic modals have peculiar logical features that are challenging to account for in a broadly classical framework. For instance, while a sentence of the form p ∧ ♦¬p ('p, but it might be that not p') appears to be a contradiction, ♦¬p does not entail ¬p, which would follow in classical logic. Likewise, the classical laws of distributivity and disjunctive syllogism fail for epistemic modals. Existing attempts to account for these facts generally either under-or over-correct. Some theories predict that p∧♦¬p, a so-called epistemic contradiction, is a contradiction only in an etiolated sense, under a notion of entailment that does not allow substitution of logical equivalents; these theories underpredict the infelicity of embedded epistemic contradictions. Other theories savage classical logic, eliminating not just rules that intuitively fail, like distributivity and disjunctive syllogism, but also rules like non-contradiction, excluded middle, De Morgan's laws, and disjunction introduction, which intuitively remain valid for epistemic modals. In this paper, we aim for a middle ground, developing a semantics and logic for epistemic modals that makes epistemic contradictions genuine contradictions and that invalidates distributivity and disjunctive syllogism but that otherwise preserves classical laws that intuitively remain valid. We start with an algebraic semantics, based on ortholattices instead of Boolean algebras, and then propose a more concrete possibility semantics, based on partial possibilities related by compatibility. Both semantics yield the same consequence relation, which we axiomatize. Then we show how to extend our semantics to explain parallel phenomena involving probabilities and conditionals. The goal throughout is to retain what is desirable about classical logic while accounting for the non-classicality of epistemic vocabulary.

- by Wesley Holliday
- •
- Modal Logic, Philosophy, Philosophy Of Language, Logic
- by István Németi
- •
- Philosophy, Pure Mathematics, Symbolic Logic, Algebraic Logic

In this paper, we study three representations of lattices by means of a set with a binary
relation of compatibility in the tradition of Ploscica. The standard representations of
complete ortholattices and complete perfect Heyting algebras drop out as special cases
of the first representation, while the second covers arbitrary complete lattices, as well
as complete lattices equipped with a negation we call a protocomplementation. The
third topological representation is a variant of that of Craig, Haviar, and Priestley. We
then extend each of the three representations to lattices with a multiplicative unary
modality; the representing structures, like so-called graph-based frames, add a second
relation of accessibility interacting with compatibility. The three representations
generalize possibility semantics for classical modal logics to non-classical modal logics,
motivated by a recent application of modal orthologic to natural language semantics.

- by Wesley Holliday
- •
- Mathematics, Modal Logic, Logic, Lattice Theory

- by István Németi
- •
- Philosophy, Logic, Computational Linguistics, Philosophy and Religious Studies

In traditional semantics for classical logic and its extensions, such as modal logic, propositions are interpreted as subsets of a set, as in discrete duality, or as clopen sets of a Stone space, as in topological duality. A point in such a set can be viewed as a "possible world," with the key property of a world being primeness—a world makes a disjunction true only if it makes one of the disjuncts true—which classically implies totality—for each proposition, a world either makes the proposition true or makes its negation true. This chapter surveys a more general approach to logical semantics, known as possibility semantics, which replaces possible worlds with possibly partial "possibilities." In classical possibility semantics, propositions are interpreted as regular open sets of a poset, as in set-theoretic forcing, or as compact regular open sets of an upper Vietoris space, as in the recent theory of "choice-free Stone duality." The elements of these sets, viewed as possibilities, may be partial in the sense of making a disjunction true without settling which disjunct is true. We explain how possibilities may be used in semantics for classical logic and modal logics and generalized to semantics for intuitionistic logics. The goals are to overcome or deepen incompleteness results for traditional semantics, to avoid the nonconstructivity of traditional semantics, and to provide richer structures for the interpretation of new languages.

- by Wesley Holliday
- •
- Mathematics, Modal Logic, Intuitionistic Logic, Set Theory

In this paper, we tell a story about incompleteness in modal logic. The story weaves together a paper of van Benthem [1979], " Syntactic aspects of modal incompleteness theorems, " and a longstanding open question: whether every normal modal logic can be characterized by a class of completely additive modal algebras, or as we call them, V-BAOs. Using a first-order reformulation of the property of complete additivity, we prove that the modal logic that starred in van Benthem's paper resolves the open question in the negative. In addition, for the case of bimodal logic, we show that there is a naturally occurring logic that is incomplete with respect to V-BAOs, namely the provability logic GLB [Japaridze, 1988, Boolos, 1993]. We also show that even logics that are unsound with respect to such algebras do not have to be more complex than the classical propositional calculus. On the other hand, we observe that it is undecidable whether a syntactically defined logic is V-complete. After these results, we generalize the famed Blok Dichotomy [Blok, 1978] to degrees of V-incompleteness. In the end, we return to van Benthem's theme of syntactic aspects of modal incompleteness.

- by Wesley Holliday and +1
- •
- Modal Logic, Logic, Philosophical Logic, Mathematical Logic
- by Ketty Peeva and +1
- •
- Optimization Problem, Algebraic Logic
- by J. Gispert
- •
- Fuzzy Logic, Pure Mathematics, Mathematical Fuzzy Logic, Algebraic Logic
- by Piero Pagliani
- •
- Cognitive Science, Mathematics, Modal Logic, Intuitionistic Logic
- by Tarek Ahmed
- •
- Philosophy, Logic, Computational Linguistics, Philosophy and Religious Studies

We overview the logic of Bunched Implications (BI) and Separation Logic (SL) from a perspective inspired by Hiroakira Ono's algebraic approach to substructural logics. We propose generalized BI algebras (GBI-algebras) as a common framework for algebras arising via " declarative resource reading " , intuitionistic generalizations of relation algebras and arrow logics and the distributive Lambek calculus with intuitionistic implication. Apart from existing models of BI (in particular, heap models and effect algebras), we also cover models arising from weakening relations, formal languages or more fine-grained treatment of labelled trees and semistructured data. After briefly discussing the lattice of subvarieties of GBI, we present a suitable duality for GBI along the lines of Esakia and Priestley and an algebraic proof of cut elimination in the setting of residuated frames of Galatos and Jipsen. We also show how the algebraic approach allows generic results on decidability, both positive and negative ones. In the final part of the paper, we gently introduce the substructural audience to some theory behind state-of-art tools, culminating with an algebraic and proof-theoretic presentation of (bi-) abduction.

We study various generalizations and weakenings of the Rasiowa-Sikorski Lemma for Boolean algebras. Building on previous work from Goldblatt, we extend the Rasiowa-Sikorski Lemma to co-Heyting algebras and modal algebras, and show how this yields completeness results for the corresponding non-classical first-order logics. Moreover, working without the full power of the Axiom of Choice, we generalize the framework of possibility semantics from Humberstone, and more recently Holliday, in order to provide choice-free representation theorems for distributive lattice, Heyting algebras and co-Heyting algebras. We also generalize a weaker version of the Rasiowa-Sikorski Lemma for Boolean algebras, known as Tarski's Lemma, to distributive lattices, HA's and co-HA's, and use these results to define a new semantics for first-order intuitionisitic logic.

- by Guillaume Massas
- •
- Intuitionistic Logic, Boolean Algebra, Stone Duality, Co-Heyting Algebras

A definition and some inaccurate cross-references in the paper A Survey of Abstract Algebraic Logic, which might confuse some readers, are clarified and corrected; a short discussion of the main one is included. We also update a dozen of... more

- by Ramon Jansana
- •
- Philosophy, Logic, Computational Linguistics, Philosophy and Religious Studies
- by Nathan Houser
- •
- Algebraic Logic

Here the emphasis is on the main pillars of Tarskian structuralist approach to logic: relation algebras, cylindric algebras, polyadic algebras, and Boolean algebras with operators. We also tried to highlight the recent renaissance of these areas and their fusion with new trends related to logic, like the guarded fragment or dynamic logic. Tarskian algebraic logic is far too broad and

- by Hajnal Andréka
- •
- Universal Algebra, Boolean Algebra, Space Time, Fuzzy Metric Space
- by Tarek Ahmed
- •
- Pure Mathematics, Finite Model Theory, Algebraic Logic, First Order Logic

Let α ≥ 2 be any ordinal. We consider the class Drsα of relativized diagonal free set algebras of dimension α. With same technique, we prove several important results concerning this class. Among these results, we prove that almost all free algebras of Drsα are atomless, and none of these free algebras contains zero-dimensional elements other than zero and top element. The class Drsα corresponds to first order logic, without equality symbol, with α-many variables and on relativized semantics. Hence, in this variation of first order logic, there is no finitely axiomatizable, complete and consistent theory.

- by Amitayu Banerjee
- •
- Algebra, Model Theory, Mathematical Logic, Algebraic Logic
- by Frederic Sart and +1
- •
- Modal Logic, Possible World Semantics, Tractatus Logico-Philosophicus, Deontic Logic

Multialgebras (or hyperalgebras, or non-deterministic algebras) have been very much studied in Mathematics and in Computer Science. In 2016 Carnielli and Coniglio introduced a class of multialgebras called swap structures, as a semantic framework for dealing with several logics of formal inconsistency (or LFIs) which cannot be semantically characterized by a single finite matrix. In particular, these LFIs are not algebraizable by the standard tools of abstract algebraic logic. In this paper, the first steps towards a theory of non-deterministic algebraization of logics by swap structures are given. Specifically, a formal study of swap structures for LFIs is developed, by adapting concepts of universal algebra to multi-algebras in a suitable way. A decomposition theorem similar to Birkhoff's representation theorem is obtained for each class of swap structures. Moreover , when applied to the 3-valued algebraizable logic J3 the usual class of algebraic models is recovered, and the swap structures semantics became twist-structures semantics (as introduced by Fidel-Vakarelov). This fact, together with the existence of a functor from the category of Boolean algebras to the category of swap structures for each LFI, which is closely connected with Kalman's functor, suggests that swap structures can be considered as non-deterministic twist structures, opening so interesting possibilities for dealing with non-algebraizable logics by means of multi-algebraic semantics.

- by Marcelo Esteban Coniglio and +1
- •
- Paraconsistent logic, Algebraic Logic, Non-deterministic semantics, Multialgebras

- by D. Pigozzi and +1
- •
- Philosophy, Logic, Computational Linguistics, Philosophy and Religious Studies
- by Uwe Wolter and +1
- •
- Category Theory, Model Driven Engineering, Mathematical Sciences, Life Cycle

On the ordered algebraic structure of Hilbert algebras (spanish). M.Sc. Thesis

A b s t r a c t. In this paper we will introduce N-Vietoris families and prove that homomorphic images of distributive nearlattices are dually characterized by N-Vietoris families. We also show a topological approach of the existence of... more

- by Sergio Celani and +1
- •
- Lattice Theory, Algebraic Logic
- by Hans Schoutens
- •
- Pure Mathematics, Algebraic Logic, Characteristic Function
- by Dorel Lucanu
- •
- Ontology, Semantics, Metadata, Software Development

A general construction of the free algebra over a poset in varieties finitely generated is given in \cite{FZ1}. In this paper, we apply this to the varieties of \L ukasiewicz--Moisil algebras, giving a detailed description of the free algebra over a finite poset (X,leq)(X,\leq)(X,leq), mathbfFreeLn((X,leq))\mathbf{Free}_{\L _n}((X,\leq))mathbfFreeLn((X,leq)). As a consequence of this description, the cardinality of mathbfFreeLn((X,leq))\mathbf{Free}_{\L _n}((X,\leq))mathbfFreeLn((X,leq)) is computed for special posets.

- by Carlos Gallardo and +1
- •
- Algebraic Logic
- by J. Gispert
- •
- Fuzzy Logic, Pure Mathematics, Boolean Algebra, Algebraic Logic

In "A Problem in Possible-World Semantics," David Kaplan presented a consistent and intelligible modal principle that cannot be validated by any possible world frame (in the terminology of modal logic, any neighborhood frame). However, Kaplan's problem is tempered by the fact that his principle is stated in a language with propositional quantification, so possible world semantics for the basic modal language without propositional quantifiers is not directly affected, and the fact that on careful inspection his principle does not target the world part of possible world semantics—the atomicity of the algebra of propositions—but rather the idea of propositional quantification over a complete Boolean algebra of propositions. By contrast, in this paper we present a simple and intelligible modal principle, without propositional quantifiers, that cannot be validated by any possible world frame precisely because of their assumption of atomicity (i.e., the principle also cannot be validated by any atomic Boolean algebra expansion). It follows from a theorem of David Lewis that our logic is as simple as possible in terms of modal nesting depth (two). We prove the consistency of the logic using a generalization of possible world semantics known as possibility semantics. We also prove the completeness of the logic (and two other relevant logics) with respect to possibility semantics. Finally, we observe that the logic we identify naturally arises in the study of Peano Arithmetic.

- by Wesley Holliday
- •
- Modal Logic, Philosophy, Philosophy Of Language, Philosophical Logic
- by Wil van der Aalst
- •
- Mathematical Sciences, Process Algebra, Life Cycle, Petri Net

Multialgebras (or hyperalgebras) have been very much studied in the literature. In the realm of Logic, they were considered by Avron and his collaborators under the name of non-deterministic matrices (or Nmatrices) as a useful semantics tool for characterizing some logics (in particular, several logics of formal inconsistency or LFIs) which cannot be characterized by a single finite matrix. In particular, these LFIs are not algebraizable by any method, including Blok and Pigozzi general theory. Carnielli and Coniglio introduced a semantics of swap structures for LFIs, which are Nmatrices defined over triples in a Boolean algebra, generalizing Avron's semantics. In this paper we develop the first steps towards the possibility of defining an algebraic theory of swap structures for LFIs, by adapting concepts of universal algebra to multialgebras in a suitable way.

- by Marcelo Esteban Coniglio and +1
- •
- Algebraic Logic, Non-deterministic semantics, Multialgebras, Nmatrices

In this paper we investigate the class of MV-algebras equipped with two quantifiers which commute as a natural generalization of diagonal--free two--dimensional cylindric algebras . In the 40's, Tasrki first introduced cylindric algebras in order to provide an algebraic apparatus for the study of classical predicate calculus. The diagonal--free two--dimensional cylindric algebras is a special cylindric algebra. The treatment here of MV-algebras is done in terms of implication and negation. This allows us to simplify some results due to Di Nola and Grigolia related to the characterization of a quantifier in terms of some special sub-algebra associated to it. On the other hand, we present a topological duality to this class of algebras and we apply it to characterize the congruences of one algebra via certain closed sets. Finally, we study the subvariety of this class generated by chains of length n+1 (n<omega). We prove that the subvariety is semisimple and we characterize their simple algebras. Using a special functional algebra, we determine all the simple finite algebras of this subvariety.

One of the active fields of research in epistemic logic is modeling interactive multi-agent systems where agents communicate and as a result their knowledge gets updated. This research line has led to the development of dynamic and... more

- by Mehrnoosh Sadrzadeh
- •
- Modal Logic, Epistemic Logic, Multi Agent System, Artificial Intelligent
- by V. Hahanov
- •
- Intellectual Property, Fault Detection, System on Chip, Fault diagnosis
- by Roberto Giuntini
- •
- Theoretical Physics, Quantum Logic, Mathematical Sciences, Physical sciences
- by Peter Jipsen
- •
- Pure Mathematics, Algebraic Logic, Representation Theorem
- by Francesco Paoli and +1
- •
- Theoretical Physics, Quantum Logic, Mathematical Sciences, Physical sciences