Stefano Bonzio

Università degli studi di Cagliari | UNICA · Department of Mathematics and Information Technology

In this paper we propose a semantic analysis of Lewis’ counterfactuals. By exploiting the structural properties of the recently introduced boolean algebras of conditionals, we show that counterfactuals can be expressed as formal combinations of a conditional object and a normal necessity modal operator. Specifically, we introduce a class of algebra...

Bochvar algebras consist of the quasivariety BCA playing the role of equivalent algebraic semantics for Bochvar (external) logic, a logical formalism introduced by Bochvar in the realm of (weak) Kleene logics. In this paper, we provide an algebraic investigation of the structure of Bochvar algebras. In particular, we prove a representation theorem...

In the logical context, ignorance is traditionally defined recurring to epistemic logic $S_4$ \cite{Hintikka1962}. In particular, an agent ignores a formula $\varphi$ when s/he does not know neither $\varphi$ nor its negation $\neg\varphi$: $\neg\K\varphi\land\neg\K\neg\varphi$ (where $\K$ is the epistemic operator for knowledge). In other words, i...

A Boolean algebra $\A$ equipped with a (finitely-additive) positive probability measure $m$ can be turned into a metric space $(\A , d_{m})$, where $d_{m}(a,b)= m ((a\wedge\neg b)\vee(\neg a\wedge b))$, for any $a,b\in A$, sometimes referred to as \emph{metric Boolean algebra}. In this paper, we study under which conditions the space of atoms of a...

As we have hinted in the previous chapter, the algebraic construction of Płonka sums yields a convenient representation for algebras defined by means of regular identities, namely, those identities where the same variables appear on both sides. Informally, Płonka sums build new algebras out of families of algebras organised into a certain system, c...

The present book aims at providing readers with a primer on logics of variable inclusion, systems that have attracted more and more attention in the last few years. The main focus of the volume is algebraic, given the prominent role played by the application of Płonka sums to logical matrices. We are not claiming that the algebraic approach is the...

Mathematicians often try to shed new light on the properties of abstract or unfamiliar structures by somehow linking them to more concrete or better understood objects. Representation theorems are a convenient illustration of this situation – think of the commonplace examples of Cayley’s representation of groups as groups of permutations or of Ston...

Since the next three chapters will contain some heavy-duty Abstract Algebraic Logic, we include here some basic notions on the subject to keep this book reasonably self-contained. This chapter is not an Abstract Algebraic Logic primer: for a far more extensive treatment and for the occasional bits of unexplained terminology and notation, the reader...

In Chapter 5 we associated to an arbitrary logic L its left variable inclusion companion Ll.

We close this volume with a focus on one of the most notable and best understood logics of variable inclusion – Paraconsistent Weak Kleene Logic, the left variable inclusion companion of classical logic.

Regular varieties are defined by regular identities, where the same set of variables occurs on both sides. Therefore, we are not taking a wild guess if we assume that there must be some kinship between regular varieties and logics of variable inclusion, of which we have given a cursory preview in Section 1.4. The aim of the next two chapters is, on...

According to a dominant tradition in modern and contemporary philosophy, logic is the paradigmatic example of a discipline consisting of analytic sentences, whose truth or otherwise can be known simply by an analysis of their meanings, and of the meaning of their constituents.

Abstract Individuating the logic of scientific discovery appears a hopeless enterprise. Less hopeless is trying to figure out a logical way to model the epistemic attitude distinguishing the practice of scientists. In this paper, we claim that classical logic cannot play such a descriptive role. We propose, instead, one of the three-valued logics...

The containment companion of a logic $\vdash $ consists of the consequence relation $\vdash ^{r}$ which satisfies all the inferences of $\vdash $, where the variables of the conclusion are contained into those of the set of premises, in case this is not inconsistent. Following the algebraic analysis started in Bonzio and Pra Baldi (2021, Studia Log...

The class of involutive bisemilattices plays the role of the algebraic counterpart of paraconsistent weak Kleene logic. Involutive bisemilattices can be represented as Plonka sums of Boolean algebras, that is semilattice direct systems of Boolean algebras. In this paper we exploit the Plonka sum representation with the aim of counting, up to isomor...

According to the so-called Lockean thesis, a rational agent believes a proposition just in case its probability is sufficiently high, i.e., greater than some suitably fixed threshold. The Preface paradox is usually taken to show that the Lockean thesis is untenable, if one also assumes that rational agents should believe the conjunction of their ow...

The paper introduces the notion of state for involutive bisemilattices, a variety which plays the role of algebraic counterpart of weak Kleene logics and whose elements are represented as Płonka sums of Boolean algebras. We investigate the relations between states over an involutive bisemilattice and probability measures over the (Boolean) algebras...

The paper studies the containment companion (or, right variable inclusion companion) of a logic \(\vdash \). This consists of the consequence relation \(\vdash ^{r}\) which satisfies all the inferences of \(\vdash \), where the variables of the conclusion are contained into those of the set of premises, in case this is not inconsistent. In accordan...

The paper aims at studying, in full generality, logics defined by imposing a variable inclusion condition on a given logic \(\vdash \). We prove that the description of the algebraic counterpart of the left variable inclusion companion of a given logic \(\vdash \) is related to the construction of Płonka sums of the matrix models of \(\vdash \). Th...

In this paper we study the right variable inclusion companion of a logic, also called its containment logic. We show that such logics possess a matrix semantics which is obtained by extending the construction of Plonka sums from algebras to logical matrices. In particular, we provide an appropriate completeness theorem for a wide family of containm...

The paper introduces the notion of state for involutive bisemilattices, a variety which plays the role of algebraic counterpart of paraconsistent weak Kleene logic and whose elements are represented as Plonka sum of Boolean algebras. We investigate the relations between states over an involutive bisemilattice and probability measures over the (Bool...

In this study we analyze a recent controversy within the biomedical world, concerning the evaluation of safety of certain vaccines. This specific struggle took place among experts: the Danish epidemiologist Peter Gøtzsche on one side and a respected scientific institution, the Cochrane, on the other. However, given its relevance, the consequences o...

In this contribution we will present a generalization of de Finetti’s betting game in which a gambler is allowed to buy and sell unknown events’ betting odds from more than one bookmaker. In such a framework, the sole coherence of the books the gambler can play with is not sufficient, as in the original de Finetti’s frame, to bar the gambler from a...

We introduce a topological counterpart to the Płonka sums of algebraic structures: the Płonka product of topological spaces. This leads to a duality when considering spaces that are dually equivalent to the algebras used in the construction of the Płonka sum.

Płonka sums consist of an algebraic construction similar, in some sense, to direct limits, which allows to represent classes of algebras defined by means of regular identities (namely those equations where the same set of variables appears on both sides). Recently, Płonka sums have been connected to logic, as they provide algebraic semantics to log...

We state and prove the " first law of Cubology " , i.e. the solvability criterion, for the n × n × n Rubik's Cube.

In this paper we introduce the notion of near semiring with involution. Generalizing the theory of semirings we aim at represent quantum structures, such as basic algebras and orthomodular lattices, in terms of near semirings with involution. In particular, after discussing several properties of near semirings, we introduce the so-called \L ukasiew...

In this paper we discuss the concept of relational system with involution. This system is called orthogonal if, for every pair of non-zero orthogonal elements, there exists a supremal element in their upper cone and the upper cone of orthogonal elements $x,\,x'$ is a singleton (i.e. $x,\,x'$ are complements each other). To every orthogonal relation...

The aim of the present paper is to generalize the concept of residuated poset, by replacing the usual partial ordering by a generic binary relation, giving rise to relational systems which are residuated. In particular, we modify the definition of adjointness in such a way that the ordering relation can be harmlessly replaced by a binary relation....

We establish a natural duality between the category of involutive bisemilattices and the category of semilattice inverse systems of Stone spaces, using Stone duality from one side and the representation of involutive bisemilattices as P{\l}onka sum of Boolean algebras, from the other. Furthermore, we show that the dual space of an involutive bisemi...

The aim of the present paper is to generalize the concept of residuated poset, by replacing the usual partial ordering by a generic binary relation, giving rise to relational systems which are residuated. In particular, we modify the definition of adjointness in such a way that the ordering relation can be harmlessly replaced by a binary relation....

We introduce a new identity equivalent to the orthomodular law in every ortholattice.

We describe in details the nxnxn Rubik's Cube, namely a Rubik's Cube with n rotating slices in each face. Then we state and prove the "first law of Cubology", i.e. the solvability criterion, for it

The aim of the present paper is to generalize the concept of residuated poset, by replacing the usual partial ordering by a generic binary relation, giving rise to relational systems which are residuated. In particular, we modify the definition of adjointness in such a way that the ordering relation can be harmlessly replaced by a binary relation....

Quantum algorithms can be generally represented as the dynamical evolution of an input quantum register, with the action of each logical gate, as well as of any transmission channel, defined by some quantum propagator. From a global viewpoint, this unitary dynamics is ruled by the flow of a continuous time, and the possible splitting into shorter l...

In this paper we study the deductive properties of a family of 3-valued paraconsistent logics. We define a notion of standard sequent calculus and prove that there is no sound and complete standard sequent calculus for these logics. Moreover, we provide non-standard sound, complete and cut free sequent calculus for Paraconsistent Weak Kleene Logic...

We state and prove the first law of cubology of the Rubik's Revenge and provide necessary and sufficient conditions for a randomly assembled Rubik's Revenge to be solvable.

In this paper, we discuss the concept of relational system with involution. This system is called orthogonal if, for every pair of non-zero orthogonal elements, there exists a supremal element in their upper cone and the upper cone of orthogonal elements (Formula presented.) is a singleton (i.e. x; (Formula presented.) are complements of each other...

Paraconsistent Weak Kleene logic (PWK) is the 3-valued logic with two designated values defined through the weak Kleene tables. This paper is a first attempt to investigate PWK within the perspective and methods of abstract algebraic logic (AAL). We give a Hilbert-style system for PWK and prove a normal form theorem. We examine some algebraic struc...

In this article, we introduce the notion of near semiring with involution. Generalizing the theory of semirings we aim at represent quantum structures, such as basic algebras and orthomodular lattices, in terms of near semirings with involution. In particular, after discussing several properties of near semirings, we introduce the so-called Łukasie...

The model of open quantum systems is adopted to describe the non-local dynamical behaviour of qubits processed by entangling gates. The analysis gets to the conclusion that a distinction between evaluation steps and task-oriented computing steps is justified only within classical computation. In fact, the use of entangling gates permits to reduce t...