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UNIVERSIDADE TÉCNICA DE LISBOA

INSTITUTO SUPERIOR TÉCNICO

Fusion of General Modal Logics Labelled with Truth Values

Karina Girardi Roggia

Supervisor: Doctor Maria Cristina de Sales Viana Serôdio SernadasCo-Supervisor: Doctor João Filipe Quintas dos Santos Rasga

Thesis approved in public session to obtain the PhD Degree inMathematics

Jury final classification: Pass

Jury

Chairperson: Chairman of the IST Scientific Board

Members of the Committee:Doctor José Manuel Cunha Leal Molarinho CarmoDoctor Maria Cristina de Sales Viana Serôdio SernadasDoctor Luca ViganòDoctor Carlos Manuel Costa Lourenço CaleiroDoctor João Filipe Quintas dos Santos Rasga

2012

UNIVERSIDADE TÉCNICA DE LISBOA

INSTITUTO SUPERIOR TÉCNICO

Fusion of General Modal Logics Labelled with Truth Values

Karina Girardi Roggia

Supervisor: Doctor Maria Cristina de Sales Viana Serôdio SernadasCo-Supervisor: Doctor João Filipe Quintas dos Santos Rasga

Thesis approved in public session to obtain the PhD Degree inMathematics

Jury final classification: Pass

Jury

Chairperson: Chairman of the IST Scientific Board

Members of the Committee:

Doctor José Manuel Cunha Leal Molarinho Carmo – FullProfessor, Universidade da Madeira – PortugalDoctor Maria Cristina de Sales Viana Serôdio Sernadas– Full Professor, Instituto Superior Técnico, Universidade Téc-nica de Lisboa – PortugalDoctor Luca Viganò – Associate Professor, Università diVerona – ItalyDoctor Carlos Manuel Costa Lourenço Caleiro – AssociateProfessor, Instituto Superior Técnico, Universidade Técnica deLisboa – PortugalDoctor João Filipe Quintas dos Santos Rasga – AssistantProfessor, Instituto Superior Técnico, Universidade Técnica deLisboa – Portugal

Funding Institution

FCT – Fundação para a Ciência e TecnologiaResearch grant SFRH/BD/23663/2005

2012

Título Fusão de Lógicas Modais com Semântica Generalizada Etiquetadas com Va-

lores de Verdade

Nome Karina Girardi Roggia

Doutoramento em Matemática

Orientadora Doutora Maria Cristina de Sales Viana Serôdio Sernadas

Co-Orientador Doutor João Filipe Quintas dos Santos Rasga

Resumo

Fusão é uma forma de combinação de lógicas bem conhecida e amplamente estudada

principalmente no contexto de lógicas modais normais sendo que a investigação para

lógicas modais não-normais ainda é incipiente. Um dos objetivos deste trabalho é

apresentar a fusão de lógicas modais não-normais preservando correção e comple-

tude. Para essa finalidade são apresentados sistemas lógicos etiquetados com valores

de verdade. O cálculo para estes sistemas é baseado em um cálculo de sequentes

etiquetados. Para lógicas que podem ser expressas por estruturas do tipo Kripke,

por exemplo, isto significa que as fórmulas podem ser etiquetadas não só por um

único mundo, mas na verdade por um conjunto de mundos. A semântica é dada

por álgebras bisortidas que permitem capturar a semântica generalizada de lógicas

modais. Tanto lógicas modais normais quanto lógicas modais não-normais são então

apresentadas neste ambiente. Morfismos entre estes sistemas de lógicas são definidos

com o intuito de utilizar uma abordagem categórica para definir a fusão como uma

soma amalgamada de dois monomorfismos. A combinação prevista é obtida e provas

da preservação da correção e da completude são dadas. Alguns exemplos de sistemas

combinados são mostrados, incluindo fusões de duas lógicas modais onde pelo menos

uma é não-normal.

Palavras-chave: Lógica modal (normal e não-normal), fusão, semântica genera-lizada, valores de verdade, sequentes etiquetados, dedução etiquetada, abordagemcategorial, correção, completude, resultados de preservação.

iii

Title: Fusion of General Modal Logics Labelled with Truth Values

Abstract:

Fusion is a well know and widely studied form of combining logics. However it is

investigated mainly for normal modal logics while the research for non-normal modal

logics is still incipient. One of the aims of this work is to introduce fusion of non-

normal modal logics preserving soundness and completeness. For this purpose, logic

systems labelled with truth values are adopted. The calculus is based in a labelled

sequent calculus. For instance, in what concerns logics that can be expressed by

Kripke-like structures this means that formulas can be labelled not only by a single

world, but actually by a set of worlds. The semantics is given by bisorted algebras,

that can capture the general semantics of modal logics. Both normal and non-

normal modal logics are then presented in this framework. Morphisms between

these logic systems are defined in order to use a categorical approach to define fusion

as a pushout of two monomorphisms. The envisaged combination is so obtained

and preservation of soundness and completeness is then proved. Some examples of

combined logic systems are given, including fusions of two modal logics where at

least one is non-normal.

Keywords: (Normal and non-normal) modal logic, fusion, general semantics, truthvalues, labelled sequents, labelled deduction, categorical approach, soundness, com-pleteness, preservation results.

v

Acknowledgements

I have always thought that this would be the easy part to do when writing the thesis.I was wrong. There are so many people that I would like to say thanks that if Imentioned each one I should be forgetting someone, but I must name some for allthat they help me.

First I would like to thank my supervisors Professor Cristina Sernadas and Pro-fessor João Rasga for their support, guidance and, most of all, patience throughthese years that made this work possible.

I would like to thank everyone at the Security and Quantum Information Groupat IT. Thanks to Carlos, Paula, Jaime, Miguel, João and Pedro to their preciouscompany in so many lunches and good conversation through them (scientific or not).Special thanks to my fellow and office mate Pedro Baltazar that helped me a lotspecially during the first years.

I cannot forget to mention the support that I had for three years living in Bal-daques Residence. Thanks to all the staff and all good friends from so differentplaces in the world that I have made there.

Finally thank you from the heart to some people that cannot understand aword that is here written, but without their support and love this could not bepossible: my mother Elenir, my father Neris, my sisters Fernanda and Patríciaand my beautiful niece Laura – my first family and the basis of what I am. CecíliaRosário do Nascimento, an angel from São Tomé that gave me so much strength. Mymother-in-law Magda, that came from Brazil to take care of what is more preciousto me to let me writing this thesis. And at last, to the family that I started withthe best man that I couldn’t even think that exists: to Daniel and to my lovely sonLucas. Daniel, I couldn’t make it without your support and love: this is for you.

This work was supported by Fundação para a Ciência e a Tecnologia researchgrant FCT SFRH/BD/23663/2005.

vii

Contents

1 Introduction 1

1.1 Background . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1

1.1.1 Modal logics . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1

1.1.2 Labelled systems . . . . . . . . . . . . . . . . . . . . . . . . . 2

1.1.3 Combination of logics . . . . . . . . . . . . . . . . . . . . . . . 3

1.2 Aims . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5

1.3 Overview . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5

1.4 Claim of Contributions . . . . . . . . . . . . . . . . . . . . . . . . . . 6

2 Logic Systems Labelled with Truth Values 7

2.1 Language . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8

2.2 Deduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11

2.3 Metatheorems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19

2.4 Semantics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23

2.5 Logic Systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27

2.5.1 Soundness . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28

2.5.2 Completeness . . . . . . . . . . . . . . . . . . . . . . . . . . . 29

3 Modal Logics Labelled with Truth Values 35

3.1 (Normal) Modal Logics . . . . . . . . . . . . . . . . . . . . . . . . . . 35

3.2 Non-normal Modal Logics . . . . . . . . . . . . . . . . . . . . . . . . 43

ix

x CONTENTS

4 Fusion of Systems Labelled with Truth Values 61

4.1 Algebraic Presentation of Fusion . . . . . . . . . . . . . . . . . . . . . 62

4.2 Categorical Presentation of Fusion . . . . . . . . . . . . . . . . . . . . 62

4.2.1 Category of logic systems . . . . . . . . . . . . . . . . . . . . 63

4.2.2 Categorical fusion . . . . . . . . . . . . . . . . . . . . . . . . . 71

4.3 Soundness and Completeness Preservation in Fusion . . . . . . . . . . 73

5 Combined Logic Systems 75

5.1 Systems Obtained by Fusion . . . . . . . . . . . . . . . . . . . . . . . 75

5.2 Preservation of Soundness and Completeness . . . . . . . . . . . . . . 82

6 Conclusion 85

6.1 Final Remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 85

6.2 Future Work . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 86

Bibliography 87

Table of Symbols 97

Subject Index 99

Chapter 1

Introduction

Fusion is a well know and widely studied form of combining logics [KW91, FS96,KW97, Wol98, Kur06]. However it is investigated mainly for normal modal logicswhile the research for non-normal modal logics is still incipient [FF05, FF02]. Oneof the aims of this work is to introduce fusion of non-normal modal logics whilepreserving soundness and completeness. For this purpose, it is adopted the logicsystems labelled with truth values introduced in [MSSV04].

1.1 Background

The three main building subjects of this thesis are modal logics, labelled systemsand combination of logics, in particular fusion.

1.1.1 Modal logics

Modal logics started with Lewis defining systems with strict implication instead ofmaterial implication [Lew18]. Based in this work [LL32] defines the systems S1–S5,where S4 and S5 are well known normal modal systems. These systems were con-structed independently of the classical propositional logic. In fact, it was Gödel thathad the idea to build a modal logic system as an extension of the classical proposi-tional calculus, when constructing a system similar to S4 in [Göd33]. Lemmon usedthis approach to construct Lewis’ systems in [Lem57].

Despite that the original motivation of modal systems was to deal with strictimplication, for many years modal logic was seen as “the logic of necessity and

1

2 Chapter 1. Introduction

possibility”. Only after the introduction of Kripke semantics they were seen also as“simple yet expressive languages for talking about relational structures” [BdRV01].A great multitude of modal systems, with various kinds of modal operators, eachof them with different meanings or properties, have then been constructed in orderto provide effective formalisms for talking about time, space, knowledge, beliefs,actions, obligations, among others.

The interpretation of modal logic simply as “the logic of necessity” restricts notonly its applications but also the range of modal systems. In fact, modal logic canalso be seen as “the study of modalities – logical operations that qualify assertionsabout the truth of statements” [Gol93]. In this sense, the necessitation rule (fromϕ infers 2ϕ) can be relaxed. This brings a bunch of modal systems known asnon-normal, whose semantics was defined only in the 1970’s by Montague [Mon70]and Scott [Sco70]. The Scott-Montague semantics, or also known as neighbourhoodsemantics, uses the notion of set of worlds as Kripke semantics does, but insteadof a relation between worlds, frames are composed by a function N from the set ofworlds W to the powerset of the powerset of W (N(w) is called the set of neighboursof the world w ∈ W ). The satisfaction of a modality 2ϕ in a certain world w isthen given iff the denotation of the formula ϕ belongs to the neighbours of w.

The use of these “weaker” modal logics can avoid some undesirable propertiesbrought by the normality. In epistemic logics, for instance, normal systems havethe problem of logical omniscience [Hoc72]: due to the necessity rule, taking themodality as a knowledge operator, all theorems must be known (in practice, this isnot true). In [vBP11] a semantics of evidence is proposed based on neighbourhoodmodels. Similar problems are found either in deontic systems where, for instance,Ross paradox (2ϕ⊃2(ϕ∨ψ)) should be avoided [CJ02], or when using the modalityas a likelihood operator based on some fixed probability (where 2ϕ∧2ψ⊃2(ϕ∧ψ)can be false) [Pac07].

In order to have contact with modal logics the reader can consult the books[Che80, BdRV01, Kra99] dedicated to this subject.

1.1.2 Labelled systems

While the extension of a Hilbert calculus with new axioms is not very complex, theuse of this kind of deduction systems is not simple in what concerns theorem proving.In fact, the calculus of a logic system, if designed to be easy for theorem proving,should be presented by as natural deduction, tableaux, or by a sequent system.However this kind of calculus are particularly suitable for classical propositionallogic (or for first-order calculus) but they are in general neither uniform nor modular

1.1. Background 3

for non-classical logics. In many cases a tailored calculus has to be provided for eachmodal logic, for instance, in the use of simple sequent calculus [Wan02]. That is,the definition cannot be given using extensions as in the case of Hilbert calculi.

A solution to avoid this problem was to incorporate, in some sense, semanticalinformation at the deduction level. Labelled deduction systems [Gab96b, Vig00,BDG+00, RSSV02b] are based in this idea. In these systems the deduction calculusmanipulates pairs label : formula instead of reasoning only about formulas. Herein,labels can be worlds, truth values, resources, names of agents, nodes of a net orany piece of structured information that facilitates the derivation process. This ideatraces back to the work of Prior [Pri67] and, for the case of modal labelled deduction,to the work of Fitting [Fit83]. The latter work covers deductive systems for mostcommon modal logics and also for a number of regular, non-normal modal logicsand some quasi-regular logics as S2 and S3.

Labelled deduction was used to define natural deduction calculus for intuition-istic logic [BMV98b], relevance logics [BMV98b, RSSV02b], substructural logics[BFR99] (this class of logics is defined using labelled tableaux systems in [DG94]),many-valued logics [RSSV02b] (labelled sequent calculus for many-valued logics ispresented in [BFSZ98]) and first-order logics [Ras03]. For modal logics the listhas several guises: tableaux [BR05], sequent systems [GR00, MSSV04], naturaldeduction [Rus96, BMV97, BMV98b, RSSV02b, SVRS03, dQG99]. Note that in[RSSV02b, SVRS03, MSSV04] labels are truth values instead of simple worlds. Ob-serve that section 1.6 of [dQG99] mentions the possibility to use labelled systems todefine a calculus for non-normal modal logics. Quantified modal logic by labellednatural deduction was proposed in [BMV98a] while [ABGR96, GM02] use tableauxsystems. Intuitionistic modal logics was also defined with labelled natural deduc-tion in [Sym94]. With such an extensive list of logics using the labelled deductionapproach, the problem of combining them arose naturally, as is analysed in thefollowing subsection.

In order to have contact with labelled systems the reader can consult the books[Gab96b, Vig00, BDG+00] dedicated to this subject.

1.1.3 Combination of logics

Reuse. This word is a slogan in many areas nowadays. Reuse to preserve the envi-ronment, in ecology. Reuse to save money, say economists. Reuse to don’t rewritecode for the same functionality, from software engineers. So in logics: reuse. This isone of the reasons for combination of logics: get new logics from well known ones,while preserving properties, in order to have more expressive and complex systems


and to simplify problems. This is a subject concerning not only mathematical logicbut also philosophical and applied logics as well. So preservation of properties fromthe components is crucial and in fact is the main goal when investigating combina-tion of logics. In fact it is where most of the effort of the research in this field isconcentrated.

The concept of get new logics from existing ones goes back to Ramón Lull inthe XIII century and to Gottfried W. Leibniz in the XVII century with the idea ofbuilding schemes where different logics could cooperate instead of competing. Butit was only at the end of the last century that rigorous techniques for combinationwere developed and studied.

Combination started in modal logics, with the work of Fitting [Fit69] that pro-vided the first examples of fusion of modal logics. However it was Thomason in1984 that introduced the term and of the technique in [Tho84]. In fusion of normalmodal logics there is no interaction between the modalities. On the other hand,in the product of modal logics, introduced independently by [Seg73] and [She78],modalities have interactions like commutativity (♦ϕ ⊃ ♦ϕ) and the Church-Rosser property (♦ϕ⊃♦ϕ) in these logics.

Dov Gabbay in the 1990’s introduced the mechanism of fibring of (normal) modallogics [Gab96a, Gab96c] from which fusion is a particular case, and which per-mits some bridge principles between the logics to be combined. A more generalpresentation of fibring, the algebraic fibring, was developed in [SSC99] to allowcombinations of a wider class of logics using a categorical approach. Categori-cally, unconstrained fibring (when the signatures of the component logics do notshare connectives at all) is defined as a coproduct in the category of logic sys-tems, and constrained fibring is defined as a cocartesian lifting. Research on thissubject included preservation of properties [ZSS01, SSZ02, CRS08a, CSS11], thecollapsing problem [SRC02] and extending the mechanism to several kinds of logics[CCC+03, RSSV02a, SSCM00, CSS03, CFSS08, SSRC09] just to name a few.

Other forms of combination of logics related in some sense with fusion are tem-poralization [FG92], parametrization [SSC97] and importing [RSSnt]. The main dis-tinction of these forms of combination in respect to fibring is that their language areintrinsically asymmetric and the rules of the temporalized/parametrizated/importedlogic can only be applied to its own formulas.

In order to study combination of logics with a wider scope, one can consult thebook [CCG+08] which is very complete technically and contains several examples ofapplications, or the article [CC11] which is a survey emphasizing the philosophicalaspects of combining logics.

1.2. Aims 5

1.2 Aims

The main aims of this thesis are:

• to extend the Logic Systems Labelled with Truth Values (LSLTV) approachto a wider class of logics besides normal modal logics;

• to define the fusion of LSLTV;

• to study the preservation of properties of the defined fusion, namely soundnessand completeness;

• to combine non-normal modal logics.

1.3 Overview

This thesis is organized in five more chapters in a sequence to introduce definitionsand to show results in one chapter and presenting examples in the chapter thereafter.

Chapter 2 is dedicated to the definition of Logic Systems Labelled with TruthValues (LSLTV) and to prove some of their properties. LSLTV are systems where thededuction part is presented by a sequent calculus including assertions about labelledformulas and terms. As in [MSSV04] assertions about terms include symbols todescribe components of the intended semantics. The semantics consists of bisortedalgebras: one sort for formulas and the other for terms. Soundness and completenessresults are proved at the end of the chapter.

In Chapter 3 the framework of LSLTV is used to present modal logic systems:normal and non-normal. Each section in this chapter follows the order established inthe previous chapter: the language comes first, followed by the deduction systemsand the semantical counterpart, and the section finishes with examples of logicsystems. Most of the results of Section 3.1, about normal modal logics, are from[MSSV04] while Section 3.2 is new and concentrates on non-normal modal logics.

Chapter 4 is about fusion of LSLTV. Since fusion is a kind of combination definedin the literature for (normal) modal logics, the chapter starts by presenting thecombination for this kind of logic systems labelled with truth values, by definingfusion algebraically in Section 4.1. The section afterwards presents a categoricalapproach for the definition of fusion in order to extend the kind of systems allowedto combine. For that purpose first it is presented the notion of LSLTV morphism andshowed some of its properties. The second part of Section 4.2 gives the definition


of categorical fusion for a wider class of systems in comparison with the originalpresentation. This presentation of fusion not only generalizes the classes of systemsthat can be combined but also permits sharing of different sets of connectives (andoperators, since the framework is LSLTV), so, for instance, a modality can be sharedin a fusion. Section 4.3 establishes the conditions to the preservation of soundnessand completeness in this combination.

Chapter 5 concentrates on applications of fusion by presenting different combinedsystems. In Section 5.1 not only fusion of normal modal logic systems is given (withone example of sharing of the modality), but also the fusion of two non-normalmodal logic systems, and the fusion of a normal modal logic system with a non-normal one are given. Section 5.2 analyses soundness and completeness of some ofthese examples and of a particular system obtained by fusion of two non-normalmodal logic systems whose modalities have strong interactions.

The last chapter compiles the main contributions of this thesis with some finalremarks and pointing several directions of research from the results here given.

1.4 Claim of Contributions

The following are some of the contributions of this thesis:

• the modelling of non-normal modal logic systems as LSLTV, defining a mod-ularized sequent calculus to this class of logics;

• the definition of fusion of LSLTV as a categorical operation, amplifying notonly the class of logics allowed to use this kind of combination but also allow-ing different sets of shared connectives and operators between the componentlogics;

• the preservation of soundness and completeness in the defined fusion of LSLTV,where completeness requires only mild conditions to holds;

• the fusion of non-normal modal logics: not only fusion between two non-normalmodal logics but also the fusion between a normal and a non-normal system.

Chapter 2

Logic Systems Labelled with Truth

Values

This chapter presents the framework used to describe logic systems throughout thisthesis. The idea is to label logic systems by truth values as introduced in [MSSV04].The Logic Systems Labelled with Truth Values (LSLTV) introduced in that work areherein reshaped in order to be used to present a wider class of logics. Observe thatlabelling by truth values allows, when in the context of a modal system, that thedenotation of terms is a range of worlds instead of a single one as in [Vig00, Rus96].The results in this chapter are proved in a similar way to the corresponding ones in[MSSV04].

The chapter is divided in five sections, Section 2.1 concentrates on the languageof LSLTV, concerned with assertions about formulas and terms (that represent truthvalues). In Section 2.2 it is introduced a Gentzen calculus of LSLTV, and derivationand consequence are defined. In Section 2.3 general metatheorems are shown tohold in this framework, adapting some results of [CRS08b] to the labelled case. InSection 2.4, about the two-sorted algebraic semantics of LSLTV, the basic conceptsof denotation, satisfaction and entailment are defined. With these three main piecesdefined: language, deduction and semantics, Section 2.5 about logic systems comesnaturally, and results about soundness and completeness are showed.

Some simple examples are given along this chapter. In Chapter 3 a wholeplethora of logic systems is described in this framework.

7

8 Chapter 2. Logic Systems Labelled with Truth Values

2.1 Language

The objective of this section is to introduce the language of the assertions of LogicSystems Labelled with Truth Values. Basically, the language is composed of expres-sions of the form θ ≤ ϕ expressing that the truth value associated with the termθ is less than or equal to the denotation of formula ϕ. As it will be seen, in logicsendowed with a Kripke semantics, truth values can be seen as sets of worlds.

The language is built from a signature where connectives, operators and sets ofvariables are specified.

Definition 2.1.1. A signature is a tuple Σ = 〈C,O,\, X, Y, Z〉 where:

• C = Ck : k ∈ N where each Ck is a countable set;

• O = Ok : k ∈ N such that ⊥,⊤ ∈ O0 where each Ok is a countable set;

• \ = \k : k ∈ N+ where each \k is a countable set;

• X, Y and Z are countable sets.

All these sets are pairwise disjoints.

The elements of each Ck are known as (formula) constructors or connectives ofarity k. Those of each Ok are known as (truth value) operators of arity k. Moreoverthe elements of each \k are relations (between terms) of arity k. There are threekinds of variables: truth value unbound variables (usually represented as x, elementof X), truth value bound variables (usually represented as y, element of Y ) andformula unbound variables (usually represented as z, element of Z). Sets X andZ are needed by the completeness proof while Y is useful when writing rules withfresh variables.

Example 2.1.2. A signature for the classical propositional logic is

ΣP = 〈C,O,\, X, Y, Z〉

where:

• C0 = t, f ∪ pi : i ∈ N, C1 = ¬, C2 = ∧,∨,⊃, Cn = ∅ for all n ≥ 3;

• O0 = ⊤,⊥, On = ∅ for all n ≥ 1;

• \1 = Ω, \2 = ⊑, \n = ∅ for all n ≥ 3; and

2.1. Language 9

• X, Y , Z are disjoint countable sets of variables.

The next three definitions show how to construct formulas, terms and assertions.Formulas and terms are defined as expected. For that, assume given the followingthree sets Ξ = ξi : i ∈ N, T = τi : i ∈ N and G = Γi : i ∈ N, whose elementsare called meta-variables of formulas, terms and bags of assertions, respectively.

Definition 2.1.3. The set F (Σ) of (schema) simple formulas over Σ is inductively

defined as follows:

i. ξi ∈ F (Σ) for every i ∈ N;

ii. z ∈ F (Σ) for every z ∈ Z; and

iii. c(ϕ1, ..., ϕk) ∈ F (Σ) whenever c ∈ Ck and ϕ1, . . . , ϕk ∈ F (Σ).

The set gF (Σ) of ground simple formulas is composed of the elements in F (Σ)without meta-variables and cF (Σ), the set of closed simple formulas, is the set ofelements of gF (Σ) without variables.

Given the signature for classical propositional logic in Example 2.1.2 the followingformulas can be constructed: ξ1 ⊃ p3, ξ2 ⊃ ¬ ξ1, z1 ∨ (¬p1 ⊃ z2) and ¬ f . Only thelast two belong to gF (Σ) and only ¬ f belongs to cF (Σ).

As mentioned before, terms are constructed similarly to formulas. For the la-belled logic systems considered in this work, terms are seen as labels, representingtruth values.

Definition 2.1.4. The set T (Σ) of (schema) terms over Σ is inductively defined as

follows:

i. τi ∈ T (Σ) for every i ∈ N;

ii. x ∈ T (Σ) for every x ∈ X;

iii. y ∈ T (Σ) for every y ∈ Y ;

iv. o(θ1, ..., θk) ∈ T (Σ) whenever o ∈ Ok and θ1, . . . , θk ∈ T (Σ); and

v. #ϕ ∈ T (Σ) whenever ϕ ∈ F (Σ).

The set gT (Σ) of ground terms is composed of the elements in T (Σ) without meta-variables and cT (Σ), the set of closed terms, is the set of elements of gT (Σ) withoutvariables.


For instance, assuming that o ∈ O2 the following belong to T (Σ): τ3, #ξ1 ando(y2,x1). Among them, only o(y2,x1) ∈ gT (Σ) and no one belongs to cT (Σ). Theintended purpose of #ϕ is to represent syntactically the truth value associated tothe formula ϕ.

Reasoning in the logic systems considered herein is done on assertions that canbe a labelled formula or a relation between terms as defined in the sequel.

Definition 2.1.5. The set A(Σ) of (schema) assertions over Σ is composed of

expressions of the following forms:

• θ ≤ ϕ and θ ϕ (positive and negative labelled formula, respectively) foreach θ ∈ T (Σ) and ϕ ∈ F (Σ),

• (θ1, . . . , θn) and 6(θ1, . . . , θn) (positive and negative relation between terms,respectively) with ∈ \n and θ1, . . . , θn ∈ T (Σ).

The set gA(Σ) of ground assertions is composed of the elements in A(Σ) withoutmeta-variables and cA(Σ), the set of closed assertions, is the set of elements of gA(Σ)without variables.

The notion of conjugate δ is defined as follows:

• θ ≤ ϕ is θ ϕ;

• θ ϕ is θ ≤ ϕ;

• (θ1, . . . , n) is 6(θ1, . . . , θn);

• 6(θ1, . . . , θn) is (θ1, . . . , n).

In the sequel the unary relation Ω ∈ \1 is used to express that a truth valueis atomic (there is no truth value strictly smaller than it besides falsum) and thebinary relation ⊑∈ \2 to compare two truth values. The Ω relation is needed toestablish the notion of local derivation, for instance. Both relations are used in ordersequent calculus (Definition 2.2.7).

For instance, in the context of ΣP it can be considered as assertions expressionsas Ωy1, t ≤ y2, and ⊤ ξ1 ∧ ξ2.

In the sequel, Bf (U) denotes the set of all finite bags of elements in the set U .

Substitutions are maps over meta-variables as is described in the definition below.They are used in derivations to assign concrete values to meta-variables.

2.2. Deduction 11

Definition 2.1.6. A (schema) substitution over A(Σ) is a map σ such that, for all

i ∈ N:

i. σ(ξi) ∈ F (Σ);

ii. σ(τi) ∈ T (Σ); and

iii. σ(Γi) ∈ Bf (A(Σ) ∪ G).

The set of (schema) substitutions over A(Σ) is denoted by Sbs(A(Σ)).

A ground substitution over A(Σ) is a schema substitution ρ such that, for alli ∈ N:

i. ρ(ξi) ∈ gF (Σ);

ii. ρ(τi) ∈ gT (Σ); and

iii. ρ(Γi) ∈ Bf (gA(Σ)).

The set of ground substitutions over A(Σ) is denoted by gSbs(A(Σ)).

In what follows, being ω a substitution, Γω denotes ω(Γ) and the same appliesfor formulas, and truth values terms.

2.2 Deduction

This work will consider labelled logic systems adopting a Gentzen calculus forpresenting the notion of deduction. Sequent calculi were introduced by GerhardGentzen in [Gen69] with the aim “to set up a formalism that reflects as accurate aspossible the actual logic reasoning involved in mathematical proofs”. In contrast,for instance, with Hilbert systems, the process of proving a theorem or a deductionis simpler and more natural.

Definition 2.2.1. A sequent over a set of assertions A(Σ) is a pair s = 〈∆1,∆2〉,

written ∆1 → ∆2, where ∆1,∆2 ∈ Bf (A(Σ) ∪ G).1

1Remember that the elements of Bf(U) are finite bags of elements of U .


A sequent, described as

ϑ1, . . . , ϑn → ζ1, . . . , ζk

with ϑ1, . . . , ϑn, ζ1, . . . , ζk being assertions, means intuitively that the union of theassertions on the left side has at least one of the assertions on the right side as aconsequence (that is, the disjunction of the right-side assertions is a consequence ofthe conjunction of the left-side assertions). If n = 0 then the left side is seen as ⊤and if k = 0, the right side is seen as ⊥.

Sometimes application of rules in derivations should be restricted by constraints.Here, constraints are presented in the form of provisos, that is, restrictions on sub-stitutions.

Definition 2.2.2. A (local) proviso over A(Σ) is a map π : gSbs(A(Σ)) → 0, 1.

The unit proviso up is such that up(ρ) = 1 for every ρ ∈ gSbs(A(Σ)).

The zero proviso zp is such that zp(ρ) = 0 for every ρ ∈ gSbs(A(Σ)).

Given two provisos π and π′, their intersection is the proviso (π ∩ π′) such that(π∩π′)(ρ) = π(ρ)×π′(ρ). The expression π ⊆ π′ denotes that π(ρ) ≤ π′(ρ) for eachground substitution ρ.

Given a schema substitution σ and a proviso π, the proviso (πσ) is defined as(πσ)(ρ) = π(σρ) for every ρ ∈ gSbs(A(Σ)). Notice that, for a ground substitutionρ, (πρ) is either up or zp.

In this work the following two provisos are needed:

• (τk : y)(ρ) = 1 iff ρ(τk) ∈ Y , and

• (τk 6∈∆)(ρ) = 1 iff ρ(τk) does not occur in ∆ρ.

The first proviso only allows ground substitutions where τk is replaced by a truthvalue bound variable. The other proviso accepts a ground substitution ρ only if τkis replaced by a ground term not occurring in ∆ρ.

Definition 2.2.3. A rule over A(Σ) is a triple r = 〈s1, ..., sp, s, π〉 written

s1 . . . sps π

where s1, ..., sp, s are sequents over A(Σ) and π is a proviso over A(Σ).

2.2. Deduction 13

The unit proviso up is usually omitted. Given a substitution σ and a rule r bothover A(Σ), the instance of r by σ

s1σ . . . spσsσ πσ

is denoted by rσ where, being s a sequent 〈∆1,∆2〉, sσ is the instance 〈∆1σ,∆2σ〉.

Definition 2.2.4. A rule is said to be endowed with a persistent proviso if its proviso

does not change value when the context is enriched with closed assertions.

In this work, usually the context of a rule are the bags of assertions Γ1 and Γ2.The “fresh bound variable” proviso, with the form

τ2 : y; τ2 6∈ τ1,Γ1,Γ2,

is persistent in every rule. Indeed, for every ground substitution ρ, it holds

(τ2 : y ∩ τ2 6∈ τ1,Γ1,Γ2)(ρ) = (τ2 : y ∩ τ2 6∈ τ1,Γ1,Γ2, δ)(ρ)

as long as δ is a closed expression. Clearly, every rule endowed with the unit provisoup is endowed with a persistent proviso.

Definition 2.2.5. A (sequent) calculus is a pair C = 〈Σ,R〉 where Σ is a signature

and R is a finite set of rules over A(Σ).

Definition 2.2.6. A sequent calculus C = 〈Σ,R〉 is structural if R contains the

axioms, weakening, contraction, conjugation and cut rules presented in Table 2.1.

Structural rules in Table 2.1 have the suffix ‘F’ when its main assertion is alabelled formula and ‘ρ’ when its main assertion is about the relation ρ ∈ \. ‘Ax’rules are the axioms and there also is one cut rule for each type of assertion. Thereare the weakening rules (Lw and Rw), the contraction rules (Lc and Rc) and therules about the conjugation of the main assertion (Lxi, Lxe, Rxi and Rxe).

Definition 2.2.7. An order sequent calculus C = 〈Σ,R〉 is a structural sequent

calculus such that Ω ∈ \1, ⊑∈ \2 and R has the rules presented in Table 2.2.

The symbol 0 is used instead of 6Ω and ⊑ 〈t1, t2〉 is written t1 ⊑ t2. Theintended meaning of Ωt is that t is atomic, that is, there is no term strictly smallerthan it besides falsum. Moreover ⊑ is intended to be the comparison between termsestablishing an order of truth values: t1 ⊑ t2 is read as “(the denotation of) thetruth value t1 is less than or equal to (the denotation of) the truth value t2”.


AxF τ1 ≤ ξ1,Γ1 → Γ2, τ1 ≤ ξ1 Ax (τ1, . . . , τn),Γ1 → Γ2, (τ1, . . . , τn)

LwF

Γ1 → Γ2

τ1 ≤ ξ1,Γ1 → Γ2 RwF

Γ1 → Γ2

Γ1 → Γ2, τ1 ≤ ξ1

Lw

Γ1 → Γ2

(τ1, . . . , τn),Γ1 → Γ2 Rw

Γ1 → Γ2

Γ1 → Γ2, (τ1, . . . , τn)

LcF

τ1 ≤ ξ1, τ1 ≤ ξ1,Γ1 → Γ2

τ1 ≤ ξ1,Γ1 → Γ2 RcF

Γ1 → Γ2, τ1 ≤ ξ1, τ1 ≤ ξ1

Γ1 → Γ2, τ1 ≤ ξ1

Lc

(τ1, . . . , τn), (τ1, . . . , τn),Γ1 → Γ2

(τ1, . . . , τn),Γ1 → Γ2 Rc

Γ1 → Γ2, (τ1, . . . , τn), (τ1, . . . , τn)

Γ1 → Γ2, (τ1, . . . , τn)

LxiF

Γ1 → Γ2, τ1 ≤ ξ1

τ1 ξ1,Γ1 → Γ2 RxiF

τ1 ≤ ξ1,Γ1 → Γ2

Γ1 → Γ2, τ1 ξ1

Lxi

Γ1 → Γ2, (τ1, . . . , τn)

6 (τ1, . . . , τn),Γ1 → Γ2 Rxi

(τ1, . . . , τn),Γ1 → Γ2

Γ1 → Γ2, 6 (τ1, . . . , τn)

LxeF

Γ1 → Γ2, τ1 ξ1

τ1 ≤ ξ1,Γ1 → Γ2 RxeF

τ1 ξ1,Γ1 → Γ2

Γ1 → Γ2, τ1 ≤ ξ1

Lxe

Γ1 → Γ2, 6 (τ1, . . . , τn)

(τ1, . . . , τn),Γ1 → Γ2 Rxe

6 (τ1, . . . , τn),Γ1 → Γ2

Γ1 → Γ2, (τ1, . . . , τn)

cutF

Γ1 → Γ2, τ1 ≤ ξ1 τ1 ≤ ξ1,Γ1 → Γ2

Γ1 → Γ2

cut

Γ1 → Γ2, (τ1, . . . , τn) (τ1, . . . , τn),Γ1 → Γ2

Γ1 → Γ2

Table 2.1: Structural rules

Most of these rules come from the intuition that the truth values are subsetsof worlds in a Kripke semantics [Kri63]. Notice also that the rule Ω⊤ allows thepassage from local reasoning to global reasoning while rules RgenF and LgenF canbe seen as the passage from an atomic truth value τ2 of a formula to a higher truthvalue τ1 such that τ2 belongs to τ1.

Example 2.2.8. The order sequent calculus CP = 〈ΣP ,RP 〉 where ΣP is the sig-

nature in Example 2.1.2 and RP contains the rules presented in Table 2.3 and therule

1 Ωτ1,Γ1 → Γ2,⊤ ⊑ τ1

is a calculus for the classical propositional logic.

It is worthwhile to explain the 1 rule: in classical logic it is sufficient to considerthe truth values ⊤ and ⊥, so it can be assumed that any atomic truth value τ1 isactually ⊤.

2.2. Deduction 15

Ω⊤

Ωτ1,Γ1 → Γ2, τ1 ≤ ξ1

Γ1 → Γ2,⊤ ≤ ξ1τ1 : y1, τ1 6∈Γ1,Γ2

Ω

Γ1 → Γ2, τ1 6⊑⊥ Ωτ2, τ2 ⊑ τ1,Γ1 → Γ2, τ1 ⊑ τ2

Γ1 → Γ2,Ωτ1τ2 : y, τ2 6∈ τ1,Γ1,Γ2

Ω⊥

Γ1 → Γ2,Ωτ1Γ1 → Γ2, τ1 6⊑⊥ ⊤ Γ1 → Γ2, τ1 ⊑ ⊤

L#

τ1 ≤ ξ1,Γ1 → Γ2

τ1 ⊑ #ξ1,Γ1 → Γ2 R#

Γ1 → Γ2, τ1 ≤ ξ1

Γ1 → Γ2, τ1 ⊑ #ξ1

⊥T Γ1 → Γ2,⊥ ⊑ τ1 ⊥F Γ1 → Γ2,⊥ ≤ ξ1

cons Γ1 → Γ2,⊤ 6⊑⊥ ref Γ1 → Γ2, τ1 ⊑ τ1

Lasym

Ωτ1,Ωτ2, τ1 ⊑ τ2,Γ1 → Γ2

Ωτ1,Ωτ2, τ2 ⊑ τ1,Γ1 → Γ2 Rasym

Ωτ1,Ωτ2,Γ1 → Γ2, τ1 ⊑ τ2

Ωτ1,Ωτ2,Γ1 → Γ2, τ2 ⊑ τ1

transT

Γ1 → Γ2, τ1 ⊑ τ2 Γ1 → Γ2, τ2 ⊑ τ3

Γ1 → Γ2, τ1 ⊑ τ3

transF

Γ1 → Γ2, τ1 ⊑ τ2 Γ1 → Γ2, τ2 ≤ ξ1

Γ1 → Γ2, τ1 ≤ ξ1

LgenT

Ωτ2, τ2 ⊑ τ3,Γ1 → Γ2 Ωτ2,Γ1 → Γ2, τ2 ⊑ τ1 τ1 ⊑ τ3,Γ1 → Γ2,Ωτ2τ1 ⊑ τ3,Γ1 → Γ2

RgenT

Ωτ2, τ2 ⊑ τ1,Γ1 → Γ2, τ2 ⊑ τ3

Γ1 → Γ2, τ1 ⊑ τ3τ2 : y, τ2 6∈ τ1, τ3,Γ1,Γ2

LgenF

Ωτ2, τ2 ≤ ξ1,Γ1 → Γ2 Ωτ2,Γ1 → Γ2, τ2 ⊑ τ1 τ1 ≤ ξ1,Γ1 → Γ2,Ωτ2τ1 ≤ ξ1,Γ1 → Γ2

RgenF

Ωτ2, τ2 ⊑ τ1,Γ1 → Γ2, τ2 ≤ ξ1

Γ1 → Γ2, τ1 ≤ ξ1τ2 : y, τ2 6∈ τ1,Γ1,Γ2

Table 2.2: Order rules

The next notion of this section is derivation. It is a sequence of steps where eachstep is either an instance of an axiom or a hypothesis or results from the applicationof an instance of some rule of the calculus.

Definition 2.2.9. Given a sequent calculus C, a sequent s′ is said to be derived

from a set S of sequents with proviso π, written

S ⊢C s′ π,

if there is a derivation sequence (d1, π1)...(dn, πn) such that: d1 is s′ and π ⊆ π1; andfor every i = 1, ..., n:

1. either di ∈ S and πi is up;

2. or there is an assertion that occurs in both sides of di and πi is up;


Lf

τ1 ⊑ ⊥,Γ1 → Γ2

τ1 ≤ f ,Γ1 → Γ2 Rf

Γ1 → Γ2, τ1 ⊑ ⊥

Γ1 → Γ2, τ1 ≤ f

Lt

τ1 ⊑ ⊤,Γ1 → Γ2

τ1 ≤ t,Γ1 → Γ2 Rt

Γ1 → Γ2, τ1 ⊑ ⊤

Γ1 → Γ2, τ1 ≤ t

L∧

τ1 ≤ ξ1, τ1 ≤ ξ2,Γ1 → Γ2

τ1 ≤ (ξ1 ∧ ξ2),Γ1 → Γ2 R∧

Γ1 → Γ2, τ1 ≤ ξ1 Γ1 → Γ2, τ1 ≤ ξ2

Γ1 → Γ2, τ1 ≤ (ξ1 ∧ ξ2)

L¬

Ωτ1,Γ1 → Γ2, τ1 ≤ ξ1

Ωτ1, τ1 ≤ (¬ ξ1),Γ1 → Γ2 R¬

Ωτ1, τ1 ≤ ξ1,Γ1 → Γ2

Ωτ1,Γ1 → Γ2, τ1 ≤ (¬ ξ1)

L⊃

Γ1 → Γ2, τ1 ≤ ξ1 τ1 ≤ ξ2,Γ1 → Γ2

τ1 ≤ (ξ1 ⊃ ξ2),Γ1 → Γ2 R⊃

Ωτ1, τ1 ≤ ξ1,Γ1 → Γ2, τ1 ≤ ξ2

Ωτ1,Γ1 → Γ2, τ1 ≤ (ξ1 ⊃ ξ2)

L∨

Ωτ1, τ1 ≤ ξ1,Γ1 → Γ2 Ωτ1, τ1 ≤ ξ2,Γ1 → Γ2

Ωτ1, τ1 ≤ (ξ1 ∨ ξ2),Γ1 → Γ2 R∨

Γ1 → Γ2, τ1 ≤ ξ1, τ1 ≤ ξ2

Γ1 → Γ2, τ1 ≤ (ξ1 ∨ ξ2)

Table 2.3: Rules for classical connectives

3. or there are r ∈ R, ρ ∈ Sbs(A(Σ)), p ∈ N and i1, ..., ip ∈ i+1, ..., n such that

rρ =

di1 . . . dip

diπ′

and πi = π′ ∩ πi1 ∩ . . . ∩ πip .

The result that follows shows basic properties of ⊢: it is projective, finitary,extensive, monotonic and idempotent. However these properties include provisoswhenever needed. Notice that the projective property is not common in the usualconsequence relations but they seem natural when dealing with provisos.

Proposition 2.2.10. For any sequent calculus C = 〈Σ,R〉, the relation ⊢ is:

projective: if S ⊢C s′ π and π′ ⊆ π then S ⊢C s

′ π′;

finitary: if S ⊢C s′ π then there is a finite S1 ⊆ S such that S1 ⊢C s

′ π;

extensive: S ⊢C s for each sequent s ∈ S;

monotonic: if S ⊆ S1 and S ⊢C s′ π then S1 ⊢C s

′ π;

idempotent: if S1 ⊢C s πs for each s in a finite set S of sequents and S ⊢C s′ π

then S1 ⊢C s′ π ∩ (

⋂

s∈S πs).

Proof. Indeed:

projective: Suppose S ⊢C s′ π with a derivation sequence d = (d1, π1) . . . (dn, πn),

so by Definition 2.2.9 d1 = s′ and π ⊆ π1. Since π′ ⊆ π, by transitivity of ⊆,π′ ⊆ π1, then S ⊢C s

′ π′.

2.2. Deduction 17

finitary: Suppose S ⊢C s′ π with a derivation sequence d = (d1, π1) . . . (dn, πn).

Define inductively the sequence S ′0, . . . , S

′n as follows:

S ′0 = ∅

S ′i =

S ′i−1 ∪ di if di ∈ S

S ′i−1 otherwise

for i = 1, . . . , n. Take S1 = S ′n.

extensive: Directly from Definition 2.2.9 item 1.

monotonic: Suppose S ⊢C s′ π with derivation sequence d = (d1, π1) . . . (dn, πn).

Thus the same derivation can be used to show that S1 ⊢C s′π. This is proved

by induction on the length of the derivation.

Base: d = (d1.π1). Then (d1, π1) is such that d1 = s′ and

1. either d1 ∈ S and π1 = up. So π ⊆ up and, since S ⊆ S1, d1 ∈ S1.Therefore S1 ⊢C s

′ π;

2. or there is an assertion that occurs in both sides of d1 and π1 = up. SoS1 ⊢C s

′ π;

3. or there are r ∈ R and ρ ∈ Sbs(A(Σ)) such that rρ = 〈∅, d1, π1〉, soS1 ⊢C s

′ π.

Step: The induction hypothesis is that (d2, π2) . . . (dn, πn) is a derivation fromthe set S1. Then (d1, π1) is such that d1 = s′ and (the cases 1. and 2. haveproofs similar to the base)

3. there are r ∈ R, ρ ∈ Sbs(A(Σ)), p ∈ N and i1, . . . , ip ∈ 2, . . . , n suchthat

rρ =

di1 . . . dipd1

π′

and π1 = π′∩πi1 ∩ . . .∩πip. Since, for each dij with j = 1, . . . , p, S1 ⊢C dijby induction hypothesis, then S1 ⊢C s

′ π.

idempotent: Since ⊢ is projective, S1 ⊢C s π ∩ (⋂

s∈S πs) for each s ∈ S andS ⊢C s

′ π ∩ (⋂

s∈S πs). Denote by ds = (ds1s, πs1s) . . . (d

sns, πs

ns) a derivation se-

quence for S1 ⊢C sπ∩(⋂

s∈S πs) for each s ∈ S and by d = (d1, π1) . . . (dm, πm)a derivation sequence for S ⊢C s

′ π ∩ (⋂

s∈S πs). Let d∗ be a derivation se-quence obtained from d by replacing each (di, πi) with di ∈ S by the derivationsequence ddi. It is now showed by induction on the length of d that d∗ is aderivation sequence for S1 ⊢C s

′ π ∩ (⋂

s∈S πs).

Base: d = (d1, π1). Then (d1, π1) is such that d1 = s′ and


1. either d1 ∈ S and π1 = up. So d∗ = dd1 is a derivation sequence forS1 ⊢C s

′ π ∩ (⋂

s∈S πs).

2. or there is an assertion that occurs in both sides of d1 and π1 = up. Sod∗ = (d1, π1) is a derivation sequence for S1 ⊢C s

′ π ∩ (⋂

s∈S πs).

3. or there are r ∈ R and ρ ∈ Sbs(A(Σ)) such that rρ = 〈∅, d1, π1〉. Sod∗ = (d1, π1) is a derivation sequence for S1 ⊢C s

′ π ∩ (⋂

s∈S πs).

Step: The induction hypothesis is such that for any derivation sequence (dj, πj). . . (dm, πm) from d with j = 2, . . . , m there is a derivation sequence d•j ofS1 ⊢C dj π ∩ (

⋂

s∈S πs). Then (d1, π1) is such that d1 = s′ and (the cases 1.and 2. have proofs similar to the base)

3. there are r ∈ R, ρ ∈ Sbs(A(Σ)), p ∈ N and i1, . . . , ip ∈ 2, . . . , m suchthat

rρ =

di1 . . . dipd1

π′

and π1 = π′ ∩ πi1 ∩ . . . ∩ πip . So, using the induction hypotheses, d∗ =(d1, π1)d

•i1. . . d•ip is a derivation sequence for S1 ⊢C s

′ π ∩ (⋂

s∈S πs).

The use of truth value as labels allows the distinction between local and globalreasoning that is interesting to logics such modal logics, for instance.

Definition 2.2.11. In an order calculus C = 〈Σ,R〉 a formula ϕ is:

• globally derived from formulas ψ1, . . . , ψk, denoted by

ψ1, . . . , ψk ⊢gC ϕ

iff ∅ ⊢C ⊤ ≤ ψ1, . . . ,⊤ ≤ ψk → ⊤ ≤ ϕ;

• locally derived from formulas ψ1, . . . , ψk, denoted by

ψ1, . . . , ψk ⊢lC ϕ

iff ∅ ⊢C Ωy1,y1 ≤ ψ1, . . . ,y1 ≤ ψk → y1 ≤ ϕ.

Notice that in the proof of a local or a global consequence the derivation doesnot use hypothesis.

2.3. Metatheorems 19

Proposition 2.2.12. Let C be an order calculus, then the relation ⊢d is transitive

both when d = l or d = g.

Proof. Suppose without loss of generality that ψ ⊢lC ϕ and ϕ ⊢l

C γ. Denote by d1 aderivation for ⊢C Ωy,y ≤ ψ → y ≤ ϕ, and by d2 a derivation for ⊢C Ωy,y ≤ ϕ →y ≤ γ. Then,

1. Ωy,y ≤ ψ → y ≤ γ cutF 2,32. y ≤ ϕ,Ωy,y ≤ ψ → y ≤ γ LwF 43. Ωy,y ≤ ψ → y ≤ γ,y ≤ ϕ RwF 54. y ≤ ϕ,Ωy → y ≤ γ d25. Ωy,y ≤ ψ → y ≤ ϕ d1.

The thesis follows immediately for ⊢gC by applying the cut rule.

2.3 Metatheorems

In this section metatheorems are investigated in two different approaches. Firstlyit is established that three metatheorems hold in the context of structural sequentcalculi. These results are originally presented in [MSSV04] and are used afterwardsto obtain completeness results.

After that, general forms of metatheorems of deduction and modus ponens, usingsets of metavariables, are defined. They are presented in the level of the sequentconsequence relation ⊢ with special attention to ⊢l and ⊢g, the local and globalderivation relations, respectively. These definitions adapt similar ones in [CRS08b]for the calculus defined in this work.

The metatheorem of conjugation is the first to be presented. In what follows, ∆denotes the bag of conjugates δ : δ ∈ ∆.

Theorem 2.3.1. Let C be a structural sequent calculus. Then, for every set S of

ground sequents and every ground sequent ∆′ → ∆′′:

S ⊢C ∆′ → ∆′′ iff S ⊢C → ∆′′,∆′.

Proof. Suppose ∆′ = δ′1, . . . , δ′n where each δ′i for i = 1, . . . , n is a ground assertion.

Given a derivation d for S ⊢C ∆′ → ∆′′, a derivation of → ∆′′,∆′ is obtained byprefixing d with n applications of Rx rules, one for each δ′i ∈ ∆′. On the other way,given a derivation for S ⊢C → ∆′′,∆′, a derivation for S ⊢C ∆′ → ∆′′ is obtained byapplying n times the Lx rules.


Next the metatheorem of contradiction is proved to hold in structural sequentcalculus. It states that if a calculus derives a bag of ground assertions and theirconjugations, then it can derive any ground assertion.

Theorem 2.3.2. Let C be a structural sequent calculus. Then, for every set S of

ground sequents and every ground sequent → ∆, if

S ⊢C→ ∆

S ⊢C→ δ for every δ ∈ ∆

then S ⊢C→ v for every ground assertion v.

Proof. Suppose without loss of generality that ∆ is a set with two assertions δ1 andδ2. Denote by dδi a derivation for S ⊢C → δi for each δi ∈ ∆, and by d∆ a derivationfor S ⊢C → ∆. Then,

1. → v cut 2,32. → v, δ1 cut 4,53. δ1 → v Lx 104. → v, δ1, δ2 Rw 65. δ2 → v, δ1 Lx 76. → δ1, δ2 d∆7. → v, δ1, δ2 Rw 8

8. → δ1, δ2 Rw 9

9. → δ2 dδ210. → v, δ1 Rw 11

11. → δ1 dδ1

In the context of sequent calculi, it is expected two different metatheorems ofdeduction. This first one is defined over a set of sequents and the consequencerelation. This metatheorem will be used, for instance, in the proof of completenessfor logic systems based in this type of calculus.

Theorem 2.3.3. Let C be a structural sequent calculus with rules endowed with

persistent provisos. Then, for every set S of ground sequents and closed sequentδ′1, . . . , δ

′m → ∆′′:

S ⊢C δ′1, . . . , δ

′m → ∆′′ iff S,→ δ′1, . . . ,→ δ′m ⊢C→ ∆′′.

Proof. (⇒) Assume S ⊢C δ′1, . . . , δ

′m → ∆′′ and let d be a derivation sequence for it.

Then,

2.3. Metatheorems 21

1. → ∆′′ cut 2,32. → ∆′′, δ′1 Rw (several times) 43. δ′1 → ∆′′ cut 5,64. → δ′1 hyp5. δ′1 → ∆′′, δ′2 Rw (several times) 76. δ′2, δ

′1 → ∆′′ cut 9,10

7. δ′1 → δ′2 Lw 88. → δ′2 hyp

...δ′m, . . . , δ

′1 → ∆′′ d

(⇐) Assume S, → δ′1, . . . , → δ′m ⊢C → ∆′′ and let d = d1, . . . , dn be a derivationsequence for it. Changing each di = Γi

1 → Γi2 to d′i = δ′1, . . . , δ

′m,Γ

i1 → Γi

2 and ineach di =→ δ′j where the justification is “hyp”, in d′i the justification is replacedby “axiom”, a derivation S ⊢C δ

′1, . . . , δ

′m → ∆′′ is then build. The new sequence

d′1, . . . , d′n is actually a derivation since δ′1, . . . , δ

′m are closed assertions and therefore

any proviso is still fulfilled (remembering that in all rules there are only persistentprovisos).

The metatheorem of deduction and the metatheorem of modus ponens are nowdefined in a different context, over a set of assertions and the sequent. In whatfollows, AΞ′,T′

(Σ) is the set of assertions generated from the sets of meta-variablesΞ′ and T′ for formulas and terms, respectively (instead of the sets Ξ and T).

Definition 2.3.4. A calculus C has the metatheorem of deduction (MTD) if there

is a finite set of assertions Λ ⊆ Aξ1,ξ2,τ(Σ) such that:

if ⊢C δ1, . . . , δn, t ≤ ϕ1 → t ≤ ϕ2 π then ⊢C δ1, . . . , δn → Λ(ϕ1, ϕ2, t) π

where δ1, . . . , δn ∈ AΞ,T(Σ), t ∈ T (Σ), ϕ1, ϕ2 ∈ F (Σ) and Λ(ϕ1, ϕ2, t) is obtainedfrom Λ by substituting ξi by ϕi for i = 1, 2 and τ by t.

A calculus has the metatheorem of modus ponens (MTMP) if there is a finite setof assertions Λ ⊆ Aξ1,ξ2,τ(Σ) such that the converse holds. We refer to Λ as thebase set.

Definition 2.3.5. A calculus C has local metatheorem of deduction (l-MTD) if the

Definition 2.3.4 holds for δ1 = Ωy, δj = y ≤ γj−1 for j = 2, . . . , n and t = y.The calculus C has global metatheorem of deduction (g-MTD) if δi = ⊤ ≤ γi fori = 1, . . . , n and t = ⊤.

Similarly for l-MTMP and g-MTMP.


Example 2.3.6. The classical propositional calculus has l-MTD, g-MTD, l-MTMP

and g-MTMP with Λ = τ ≤ ξ1 ⊃ ξ2.

The derivation to show the l-MTD is obtained simply by applying the R⊃ rule.For l-MTMP the derivation is shown below:

1. Ωy1,y1 ≤ γ1, . . . ,y1 ≤ γn,y1 ≤ ϕ1 → y1 ≤ ϕ2 cutF 2,32. Ωy1,y1 ≤ γ1, . . . ,y1 ≤ γn,y1 ≤ ϕ1 → y1 ≤ ϕ2,y1 ≤ ϕ1 ⊃ ϕ2 RwF 43. y1 ≤ ϕ1 ⊃ ϕ2,Ωy1,y1 ≤ γ1,

. . . ,y1 ≤ γn,y1 ≤ ϕ1 → y1 ≤ ϕ2 L⊃ 5,64. Ωy1,y1 ≤ γ1, . . . ,y1 ≤ γn,y1 ≤ ϕ1 → y1 ≤ ϕ1 ⊃ ϕ2 LwF 75. Ωy1,y1 ≤ γ1, . . . ,y1 ≤ γn,y1 ≤ ϕ1 → y1 ≤ ϕ2,y1 ≤ ϕ1 axiom6. Ωy1,y1 ≤ ϕ2,y1 ≤ γ1, . . . ,y1 ≤ γn,

y1 ≤ ϕ1 → y1 ≤ ϕ2 axiom7. Ωy1,y1 ≤ γ1, . . . ,y1 ≤ γn → y1 ≤ ϕ1 ⊃ ϕ2 hyp

The proof of the g-MTD is the derivation:

1. ⊤ ≤ γ1, . . . ,⊤ ≤ γn → ⊤ ≤ ϕ1 ⊃ ϕ2 RgenF 22. Ωy1,y1 ⊑ ⊤,⊤ ≤ γ1, . . . ,⊤ ≤ γn → y1 ≤ ϕ1 ⊃ ϕ2 R⊃ 33. Ωy1,y1 ≤ ϕ1,y1 ⊑ ⊤,⊤ ≤ γ1,

. . . ,⊤ ≤ γn → y1 ≤ ϕ2 transF 4,54. Ωy1,y1 ≤ ϕ1,y1 ⊑ ⊤,⊤ ≤ γ1,

. . . ,⊤ ≤ γn → y1 ⊑ ⊤ axiom5. Ωy1,y1 ≤ ϕ1,y1 ⊑ ⊤,⊤ ≤ γ1,

. . . ,⊤ ≤ γn → ⊤ ≤ ϕ2 LgenF 6,7,86. Ω⊤,⊤ ≤ ϕ1,Ωy1,y1 ⊑ ⊤,⊤ ≤ γ1,

. . . ,⊤ ≤ γn → ⊤ ≤ ϕ2 LwT 97. Ω⊤,Ωy1,y1 ⊑ ⊤,⊤ ≤ γ1, . . . ,⊤ ≤ γn → ⊤ ≤ ϕ2,⊤ ⊑ y1 1

8. y1 ≤ ϕ1,Ωy1,y1 ⊑ ⊤,⊤ ≤ γ1,

. . . ,⊤ ≤ γn → ⊤ ≤ ϕ2,Ω⊤ Ω 11,129. Ω⊤,Ωy1,⊤ ≤ ϕ1,⊤ ≤ γ1, . . . ,⊤ ≤ γn → ⊤ ≤ ϕ2 LwΩ 10

10. ⊤ ≤ ϕ1,⊤ ≤ γ1, . . . ,⊤ ≤ γn → ⊤ ≤ ϕ2 hyp11. y1 ≤ ϕ1,Ωy1,y1 ⊑ ⊤,⊤ ≤ γ1,

. . . ,⊤ ≤ γn → ⊤ ≤ ϕ2,⊤ 6⊑⊥ cons12. Ωy2,y2 ⊑ ⊤,y1 ≤ ϕ1,Ωy1,y1 ⊑ ⊤,

⊤ ≤ γ1, . . . ,⊤ ≤ γn → ⊤ ≤ ϕ2,⊤ ⊑ y2 1

And, finally, the g-MTMP comes from the following derivation:

2.4. Semantics 23

1. ⊤ ≤ γ1, . . . ,⊤ ≤ γn,⊤ ≤ ϕ1 → ⊤ ≤ ϕ2 cutF 2,32. ⊤ ≤ γ1, . . . ,⊤ ≤ γn,⊤ ≤ ϕ1 → ⊤ ≤ ϕ2,⊤ ≤ ϕ1 ⊃ ϕ2 Lw 43. ⊤ ≤ ϕ1 ⊃ ϕ2,⊤ ≤ γ1, . . . ,

⊤ ≤ γn,⊤ ≤ ϕ1 → ⊤ ≤ ϕ2 L⊃ 6,74. ⊤ ≤ γ1, . . . ,⊤ ≤ γn → ⊤ ≤ ϕ2,⊤ ≤ ϕ1 ⊃ ϕ2 Rw 55. ⊤ ≤ γ1, . . . ,⊤ ≤ γn → ⊤ ≤ ϕ1 ⊃ ϕ2 hyp6. ⊤ ≤ γ1, . . . ,⊤ ≤ γn,⊤ ≤ ϕ1 → ⊤ ≤ ϕ2,⊤ ≤ ϕ1 axiom7. ⊤ ≤ ϕ2,⊤ ≤ γ1, . . . ,⊤ ≤ γn,⊤ ≤ ϕ1 → ⊤ ≤ ϕ2 axiom

2.4 Semantics

The semantics of LSLTV is based on a two-sorted algebra: a sort for truth valuesand another for denotations of formulas. This allows to encompass, in the frameworkof LSLTV, not only general Kripke structures as shown in Section 3.1 but also, forinstance, the description of modal algebras. It is worthwhile to observe that thesplitting in two sorts allows the representation of general Kripke structures insteadof the standard ones.

Definition 2.4.1. Let Σ be a signature. A Σ-algebra is a triple A = 〈F, T, ·A〉 where

F and T are sets and ·A is a map such that:

• cA : F k → F for each c ∈ Ck;

• oA : T k → T for each o ∈ Ok,

• #A : F → T ;

• ≤A⊆ T × F ;

• A ⊆ T k for each ∈ \k.

The notion of reduct of an algebra by a signature will be useful along this thesis,specially when defining fusion algebraically in Section 4.1.

Definition 2.4.2. Given a Σ-algebra A = 〈F, T, ·A〉 and a signature Σ′ such that

Σ′ ⊆ Σ, the reduct of A by Σ′ is the Σ′-algebra A|Σ′ = 〈F, T, ·A|Σ′〉 such that:

• cA|Σ′= cA for all c ∈ Σ′,

• oA|Σ′= oA for all o ∈ Σ′,

• #A|Σ′= #A,


• ≤A|Σ′=≤A, and

• A|Σ′= A for each ∈ \.

In order to define the denotation of formulas and terms, the notion of assignmentover a Σ-algebra is needed. Basically it specifies the meaning of the variables in theΣ-algebra.

Definition 2.4.3. An unbound variable assignment over A is a function α that

maps each element of X to an element of T and each element of Z to an element ofF . A bound variable assignment over A is a map β from Y to T .

The definition of denotation is split in two parts: the denotation of formulas(where only unbound variable assignments may be used) and the denotation ofterms (where both kinds of assignments are needed).

Definition 2.4.4. The denotation of ground simple formulas at a Σ-algebra A for

an unbound variable assignment α, is inductively defined in the following way:

• JzKAα = α(z);

• Jc(ϕ1, . . . , ϕk)KAα = cA(Jϕ1KAα, . . . , JϕkKAα).

The denotation at A for assignments α, β over A of ground terms is inductivelydefined in the following way:

• JxKAαβ = α(x);

• JyKAαβ = β(y);

• Jo(θ1, . . . , θk)KAαβ = oA(Jθ1KAαβ , . . . , JθkKAαβ);

• J#ϕKAαβ = #A(JϕKAα).

The definition of satisfaction of assertions and sequents can be given capitalizingon the denotation of terms and formulas presented above.

Definition 2.4.5. The satisfaction by A for α and β of ground assertions and

sequents is defined as follows:

• Aαβ (θ1, . . . , θn) iff 〈Jθ1KAαβ , . . . , JθnKAαβ〉 ∈ A;

2.4. Semantics 25

• Aαβ 6(θ1, . . . , θn) iff 〈Jθ1KAαβ , . . . , JθnKAαβ〉 6∈ A;

• Aαβ θ ≤ ϕ iff 〈JθKAαβ , JϕKAα〉 ∈≤A;

• Aαβ θ ϕ iff 〈JθKAαβ , JϕKAα〉 6∈ ≤A;

• Aαβ ∆′ → ∆′′ iff Aαβ δ for some δ ∈ ∆′′ ∪∆′.

Futhermore, the satisfaction by A for α for ground assertions and sequents issuch that for every bound variable assignment β over A, Aα δ iff Aαβ δ for δbeing an assertion and, for a sequent ∆′ → ∆′′, Aα ∆′ → ∆′′ iff Aαβ ∆′ → ∆′′.

Entailment is now defined over a class of algebras.

Definition 2.4.6. Given a class A of Σ-algebras, a ground sequent s is A-entailed

by the ground sequents s1, . . . , sp, written

s1, . . . , sp A s,

iff, for each A ∈ A and unbound variable assignment α over A, Aα s wheneverAα si for every i = 1, . . . , p.

A sequent s is A-entailed by the sequents s1, . . . , sp with proviso π, writtens1, . . . , sp A s π, iff s1ρ, . . . , spρ A sρ for every ground substitution ρ over A(Σ)such that π(ρ) = 1.

Similarly to Proposition 2.2.10 where properties for ⊢ are proved, the followingstatement shows properties of .

Proposition 2.4.7. Given a class A of Σ-algebras, the relation is:

projective: if S A s′ π and π′ ⊆ π then S A s′ π′;

extensive: S A s for each sequent s ∈ S;

monotonic: if S ⊆ S1 and S A s′ π then S1 A s′ π;

idempotent: if S1 A sπs for each s in a finite set S of sequents and S A s′ π

then S1 A s′ π ∩ (⋂

s∈S πs).


Proof. The first three properties are immediate to prove from the Definition 2.4.6.With respect to the last one, assume S1 A s πs for each sequent s ∈ S andS A s′ π. Since is projective, S1 A s π ∩ (

⋂

s∈S πs) for each s and S A

s′ π ∩ (⋂

s∈S πs). So, for every ρ such that π ∩ (⋂

s∈S πs)(ρ) = 1, S1ρ A sρ foreach s ∈ S and Sρ A s′ρ. Therefore, for every such ρ, every A ∈ A and everyunbound assignment α over A: for every s ∈ S,Aα sρ whenever Aα s1ρ forevery s1 ∈ S1; and Aα s′ρ whenever Aα sρ for every s ∈ S. So, for every suchρ, every A ∈ A and α, if Aα s1ρ for every s1 ∈ S1 then Aα s′ρ.

Like in Definition 2.2.11, it is possible to distinguish between local and globalentailment.

Definition 2.4.8. Let A be a class of Σ-algebras. A formula ϕ is globally entailed

from formulas ψ1, . . . , ψk, denoted by ψ1, . . . , ψk gA ϕ iff A ⊤ ≤ ψ1, . . . ,⊤ ≤

ψk → ⊤ ≤ ϕ. A formula ϕ is locally entailed from formulas ψ1, . . . , ψk, denoted byψ1, . . . , ψk

lA ϕ iff Ω ∈ \1 and A Ωy1,y1 ≤ ψ1, . . . ,y1 ≤ ψk → y1 ≤ ϕ.

Finally, as for the deductive consequence, entailment is also preserved by substi-tutions.

Proposition 2.4.9. Given a class A of Σ-algebras, for every substitution σ, if

S A s′ π then Sσ A s′σ πσ.

Proof. Observe that, if S A s′ π then for every ground substitution ρ′ such thatπ(ρ′) = 1 it happens that Sρ′ A s′ρ′. Take an arbitrary ground substitution ρ suchthat (πσ)(ρ) = 1, then σρ is ground and π(σρ) = (πσ)ρ = 1. Hence, using thehypothesis, S(σρ) A s′(σρ), and so the thesis follows since for any substitutionsδ(σρ) = (δσ)ρ.

The section finishes by introducing a definition needed to establish a connectionbetween deduction and semantics.

Definition 2.4.10. A class of algebras A is called appropriate for the rule r =

〈s1, ..., sp, s, π〉 if s1, . . . , sp A s π. Moreover an algebra A is said to be appro-priate for a rule if so is the class A. The class of algebras is full for a calculus〈Σ,R〉 if it is the class of all Σ-algebras that are appropriate for all rules in R.

Given a calculus C = 〈Σ,R〉, it will be denoted by app(C) the class of algebrasappropriate for C.

2.5. Logic Systems 27

2.5 Logic Systems

The definition of logic system is now presented. A logic system puts together thedifferent components of a logic: the syntax, the deductive system and the semantics.

Definition 2.5.1. A logic system is a triple L = 〈Σ,R,A〉 where 〈Σ,R〉 is a sequent

calculus and A is a class of Σ- algebras.

A logic system is said to be:

• sound if A is appropriate for each rule in R;

• full if A is the class of all Σ-algebras that are appropriate for each rule in R;

• complete if s1, . . . , sp ⊢R s whenever s1, . . . , sp A s for any closed sequentss, s1, . . . , sp.

Corollary 2.5.2. Every full logic is sound.

The next definition presents the class of logic systems mostly used in this work,the relational LSLTV. Intuitively a relational LSLTV is a structural logic systemwhere its semantics can be expressed in terms of “Kripke-like” models.

Definition 2.5.3. A relational LSLTV LR is a tuple 〈Σ,R,A〉 such that:

• Σ contains ΣP together with I ∈ O1 and lb ∈ O2;

• R contains the structural rules, the order rules, rules for constants and foreach propositional connective, and rules I,ΩI, lb1 and lb2 presented in Table2.4;

• A is a class of Σ-algebras.

I Γ1 → Γ2, I(τ1) ⊑ τ1 ΩI

Γ1 → Γ2, τ1 6⊑⊥

Γ1 → Γ2,ΩI(τ1)

lb1 Γ1 → Γ2, lb(τ1, τ2) ⊑ τ1 lb2 Γ1 → Γ2, lb(τ1, τ2) ⊑ τ2

Table 2.4: Specific rules for operators

The operator lb(t1, t2) represents a lower bound of the terms t1 and t2, and I(t)represents an atomic truth value contained in the denotation of the term t. Therefore


rules I and ΩI impose that I(t) represents an atomic truth value contained in thedenotation of t, as long as t is not ⊥.

It is important to notice that for a relational LSLTV 〈Σ,R,A〉 based in classicalpropositional logic, the rule 1 presented in Example 2.2.8 is not necessarily in R. Infact, it is usually not in most of the typical relational logic systems since intuitively,it means that there are only two truth values in the system.

The following two subsections bring results from [MSSV04].

2.5.1 Soundness

This subsection starts by showing that Σ-algebras are suitable models for structuralcalculus.

Theorem 2.5.4. The class of all Σ-algebras is appropriate for every structural rule

over A(Σ).

Proof. The proof consists in verifying s1, . . . , sn A s for every structural rule〈s1, . . . , sn, s, π〉 in Table 2.1. Only the case of the rule RwF is shown sincefor the other rules the proof follows similarly. Taken any Σ-algebra A, assignmentα and ground substitution ρ, assume that Aα Γ1ρ → Γ2ρ. That is, for everyassignment β, Aαβ Γ1ρ → Γ2ρ. Thus, for every β there is δ ∈ Γ2ρ ∪ Γ1ρ suchthat Aαβ δ. So, for every β it happens that there is δ ∈ Γ2ρ ∪ τ1ρ ≤ ξ1ρ ∪ Γ1ρ

such that Aαβ δ. Therefore, Aα Γ1ρ→ Γ2ρ, τ1ρ ≤ ξ1ρ.

Hence a general result for soundness can now be established.

Theorem 2.5.5. A logic system L = 〈Σ,R,A〉 is sound iff s1, . . . , sp A s π

whenever s1, . . . , sp ⊢R s π for every sequents s1, . . . , sp, s.

Proof. Indeed:

(⇐) It is sufficient to show that A is appropriate for 〈Σ,R〉. So, given a ruler = 〈s′1, . . . , s

′p, s

′, π′〉 ∈ R it is straightfoward to build a derivation of s′1, . . . , s′p ⊢R

s′ π′ and, so, by hypothesis, s′1, . . . , s′p A s′ π′. So, A is appropriate for r.

(⇒) Assume that L is sound and s1, . . . , sp ⊢R s π with derivation d = (d1, π1) . . .(dn, πn). The proof of s1, . . . , sp A s π is by induction on the length of d. Toshow that s1, . . . , sp A d1 π1 it have to be considered three cases that justify thederivation:


hypothesis By Proposition 2.4.7 A is extensive and projective. So s1, . . . , sp A

si π1 for any i = 1, . . . , p.

axiom Obviously A δ → δ and by weakening proved in Theorem 2.5.4 A δ,Γ1 →Γ2, δ. Therefore, by monotonicity and projection from Proposition 2.4.7,s1, . . . , sp A δ,Γ1 → Γ2, δ π1.

rule Assume that r ∈ R was used with substitution σ for justifying d1. Let

rσ =

di1 . . . diq

d1π′

with i1, . . . , iq ∈ 2, . . . , n and π1 ⊆ π′ ∩ πi1 ∩ . . .∩ πiq . Since A is projective(Proposition 2.4.7), it is enough to show s1, . . . , sp A d1 π′ ∩ πi1 ∩ . . . ∩πiq . Considering that A is appropriate for R and using Proposition 2.4.9then di1, . . . , diq A d1 π′. On the other hand, by induction hypothesis,s1, . . . , sp A dij πij for j = 1, . . . , q. Therefore, by idempotence shown inProposition 2.4.7 the envisaged result is obtained.

2.5.2 Completeness

In this subsection it is proved that completeness holds under mild conditions inlogic systems with structural calculus. The conditions are that the rules of thecalculus must have persistent provisos. The theorem is proved using a variant of theLindenbaum-Tarski technique. This variance is expected once the logic systems dealherein uses truth values as labels. First, it starts by defining a maximal consistentset and the syntactical algebra.

Definition 2.5.6. A set S of closed sequents over A(Σ) is said to be consistent if for

no closed assertion δ both S ⊢→ δ and S ⊢→ δ hold. And it is said to be maximalconsistent if for every closed assertion δ either → δ ∈ S or → δ ∈ S but not both.

Definition 2.5.7. Given a sequent calculus C = 〈Σ,R〉 and a maximal consistent

set S of closed sequents over A(Σ), the syntactic algebra induced by C and S is thefollowing Σ-algebra:

A(C, S) = 〈cF (Σ), cT (Σ), ·A(C,S)〉

where

• cA(C,S) = λf1 . . . fk.c(f1, . . . , fk);


• oA(C,S) = λt1 . . . tk.o(t1, . . . , tk);

• #A(C,S) = λf.#f ;

• 〈τ, ϕ〉 ∈≤A(C,S) iff S ⊢C→ τ ≤ ϕ;

• 〈τ1, . . . , τn〉 ∈ A(C,S) iff S ⊢C→ (τ1, . . . , τn).

Let ϕ ∈ gF (Σ) and θ ∈ gT (Σ). Given an unbounded variable assignment α anda bounded variable assignment β both over a syntactic algebra A(C, S), ϕα denotesthe closed simple formula obtained from ϕ by replacing each variable z ∈ Z by α(z);and θαβ denotes the closed term obtained from θ by replacing each variable x ∈ X

by α(x) and each variable y ∈ Y by β(y). This notation is extended to groundassertions and bags of ground assertions by identifying ϕαβ with ϕα.

Lemma 2.5.8. Let C be a structural calculus, S a maximal consistent set of closed

sequents, α an unbound variable assignment and β a bound variable assignment(both over A(C, S)), ϕ a ground simple formula, θ a ground term and δ a groundassertion. Then:

i. JϕKA(C,S)αβ = ϕα;

ii. JθKA(C,S)αβ = θαβ;

iii. A(C, S)αβ δ iff S ⊢C→ δαβ;

Proof. (i) By structural induction on ϕ.

- ϕ is z.

JϕKA(C,S)αβ = JzKA(C,S)αβ = α(z) = ϕα.

- ϕ is c(ϕ1, . . . , ϕn).

JϕKA(C,S)αβ = Jc(ϕ1, . . . , ϕn)KA(C,S)αβ = c(Jϕ1KA(C,S)αβ, . . . , JϕnKA(C,S)αβ = (byinduction hypothesis) c(ϕ1α, . . . , ϕnα) = ϕα.

(ii) By structural induction on θ.

- θ is x.

JθKA(C,S)αβ = JxKA(C,S)αβ = α(x) = θαβ.

- θ is y.

JθKA(C,S)αβ = JyKA(C,S)αβ = β(y) = θαβ.


- θ is o(θ1, . . . , θn).

JθKA(C,S)αβ = Jo(θ1, . . . , θn)KA(C,S)αβ = o(Jθ1KA(C,S)αβ , . . . , JθnKA(C,S)αβ) = (byinduction hypothesis) o(θ1αβ, . . . , θnαβ) = θαβ.

- θ is #ϕ with ϕ being a ground formula.

JθKA(C,S)αβ = J#ϕKA(C,S)αβ = #JϕKA(C,S)αβ = by i. #ϕα = θαβ.

(iii) There are four cases to analyze:

- δ is θ ≤ ϕ.

Then A(C, S)αβ θ ≤ ϕ iff 〈JθKA(C,S)αβ , JϕKA(C,S)αβ〉 ∈≤A(C,S) iff (by i. and ii.)〈θαβ, ϕα〉 ∈≤A(C,S) iff S ⊢C → θαβ ≤ ϕα iff S ⊢C → (θ ≤ ϕ)αβ.

- δ is ρ(θ1, . . . , θn).

Then A(C, S)αβ (θ1, . . . , θn) iff 〈Jθ1KA(C,S)αβ, . . . , JθnKA(C,S)αβ〉 ∈ A(C,S) iff(by ii.) 〈θ1αβ, . . . , θnαβ〉 ∈ A(C,S) iff S ⊢C → (θ1αβ, . . . , θnαβ) iff S ⊢C →((θ1, . . . , θn))αβ.

- δ is θ ϕ.

Then A(C, S)αβ θ ϕ iff 〈JθKA(C,S)αβ , JϕKA(C,S)αβ〉 6∈ ≤A(C,S) iff (by i. and ii.)〈θαβ, ϕα〉 6∈ ≤A(C,S) iff S 6⊢C → θαβ ≤ ϕα iff (since S is maximal consistent)S ⊢C → θαβ ϕα iff S ⊢C → (θ ϕ)αβ.

- δ is 6(θ1, . . . , θn).

Then A(C, S)αβ 6(θ1, . . . , θn) iff 〈Jθ1KA(C,S)αβ , . . . , JθnKA(C,S)αβ〉 6∈ A(C,S) iff(by ii.) 〈θ1αβ, . . . , θnαβ〉 6∈ A(C,S) iff S 6⊢C → (θ1αβ, . . . , θnαβ) iff (since S ismaximal consistent) S ⊢C → 6(θ1αβ, . . . , θnαβ) iff S ⊢C→ 6(θ1, . . . , θn)αβ.

Item iii. of Lemma 2.5.8 is the fundamental lemma to establish the completenesssince makes a correspondence between the satisfaction in the syntactical algebra andthe deduction in the calculus for any ground sequent.

Lemma 2.5.9. Let C be a structural calculus, S a maximal consistent set of closed

sequents, α an unbound variable assignment and β a bound variable assignment(both over A(C, S)) and ∆′ → ∆′′ a ground sequent. Then:

A(C, S)αβ ∆′ → ∆′′ iff S ⊢C ∆′αβ → ∆′′αβ.


Proof. Indeed, A(C, S)αβ ∆′ → ∆′′ iff A(C, S)αβ δ for some δ ∈ ∆′′ ∪ ∆′

iff (by Lemma 2.5.8) S ⊢C → δαβ for some δ ∈ ∆′′ ∪ ∆′ iff S ⊢C → δ for someδ ∈ ∆′′αβ ∪ ∆′αβ iff (by the justification below) S ⊢C → ∆′′αβ,∆′αβ iff (byTheorem 2.3.1) S ⊢C ∆′αβ → ∆′′αβ. It remains to explain that:

(1) If S ⊢C → δ for some δ ∈ ∆′′αβ ∪ ∆′αβ then S ⊢C → ∆′′αβ,∆′αβ. Thisfollows by several applications of right weakening.

(2) If S ⊢C→ ∆′′αβ,∆′αβ then S ⊢C→ δ for some δ ∈ ∆′′αβ ∪ ∆′αβ. Indeed,otherwise, since S is maximal consistent, S ⊢C→ δ for every δ ∈ ∆′′αβ ∪ ∆′αβ.Then, using Theorem 2.3.2, every closed assertion is derivable from S, thereforecontradicting that S is consistent.

Taken a sequent calculus and a class of syntactical algebras induced by it, it isnow shown that these algebras are appropriate for this calculus.

Lemma 2.5.10. The class of all syntactic algebras induced by a sequent calculus

is appropriate for it.

Proof. Let r =s1 . . . sp

s be a ground instance of a (proper) rule of a sequent calculusC. Let α be an arbitrary unbound variable assignment over a syntactic algebraA(C, S). Assume that A(C, S)α si for each i = 1, . . . , p. That is, for every boundvariable assignment β, A(C, S)αβ si. So, by Lemma 2.5.9, for every such β,S ⊢C siαβ with derivation sequence dsiαβ, for each such i. Then, it is straightforwardto build, for every pair α, β, a derivation sequence for S ⊢C sαβ using rule r and thosederivation sequences. Thus, again by Lemma 2.5.9, for every such β, A(C, S)αβ s.That is, A(C, S) s.

The next lemma is crucial to prove completeness. It makes possible to considerthe consistent extension of a consistent set of closed sequents.

Lemma 2.5.11. Let C be a structural sequent calculus with rules endowed with

persistent provisos. If S is a consistent set of closed sequents and S 6⊢C → v1, . . . , vmfor closed assertions v1, . . . , vm then the set S ∪→ v1, . . . ,→ vm is still consistent.

Proof. Assume S∪→ v1, . . . ,→ vm is inconsistent. So there is δ, closed assertion,such that S,→ v1, . . . ,→ vm ⊢C→ δ and S,→ v1, . . . ,→ vm ⊢C→ δ. By Theorem2.3.2, S,→ v1, . . . ,→ vm ⊢C→ v1, using Theorem 2.3.3 S ⊢C v1, . . . , vm → v1and applying Theorem 2.3.1 S ⊢C→ v1, v1, . . . , vm. By right contraction S ⊢C→v1, . . . , vm, contradicting the hypothesis that S 6⊢C → v1, . . . , vm. Therefore, S ∪ →v1, . . . ,→ vm is consistent.


Theorem 2.5.12. Every full structural sequent logic system with rules endowed

with persistent provisos is complete.

Proof. Consider the logic L = 〈Σ,R,A〉 and let C = 〈Σ,R〉. Assume that S 6⊢C ∆′ →

∆′′ with S ∪ ∆′ → ∆′′ composed of closed sequents.

Given an enumeration vn with n ∈ N of the set of closed assertions (notice thatthis set is enumerable since the components of a signature are enumerable sets),start by extending S to a maximal consistent set S• as follows:

• S0 = S ∪ → δ : δ ∈ ∆′′ ∪∆′;

• Sn+1 =

S ∪ → vn provided that Sn ⊢C→ vnS ∪ → vn otherwise

;

• S• = ∪n∈NSn.

Observe that S• is still consistent thanks to Lemma 2.5.11. Furthermore, by con-struction, it is maximal consistent. Therefore, S• 6⊢C ∆

′ → ∆′′ because otherwiseS• ⊢C δ for some δ ∈ ∆′′ ∪ ∆′ and, hence, S• would be inconsistent. Thus, byLemma 2.5.9 applied to a closed sequent, A(C, S•) 6 ∆′ → ∆′′.

On the other hand, for every s ∈ S, S ⊢C s holds, and thus, again thanks toLemma 2.5.9, A(C, S•) s.

Since the logic is full and taking into account Lemma 2.5.10, A(C, S•) is in A.Hence, S 6A∆′ → ∆′′.

Chapter 3

Modal Logics Labelled with Truth

Values

Modal logics [BdRV01] are probably one of the most well known non-classical logicsstudied nowadays. They are presented by Kripke semantics, modal algebras, Hilbertsystems, Gentzen systems, natural deduction and in many other ways. They havebeen applied in a wide list of areas like philosophy, mathematics, computer science,economy, etc. However most of the development and research of the field is con-centrated in a fragment of the class of modal systems, namely the normal modalsystems. In fact, usually the term “normal” is dropped and “modal logics” is quiteoften synonym of normal modal logics.

In this chapter a wide class of modal logics are presented as LSLTV: the nor-mal modal logics are in Section 3.1 since they were already defined in [MSSV04,RSSV02b, RRS10]. The development of non-normal modal systems as LSLTV isnew and is presented in Section 3.2. It is interesting to notice that accommodatingnon-normal modal calculi and semantics in this framework did not involve significantmodifications.

3.1 (Normal) Modal Logics

This section presents (normal) modal logics as systems labelled with truth values.First the signature and the rules of the calculus are defined and illustrated with anexample of a derivation. After that some remarks are made about the d-MTD andd-MTMP properties (defined in 2.3.4) for modal calculus. Then it is showed howa ΣM -algebra is induced by a general Kripke structure and the other way around.

35

36 Chapter 3. Modal Logics Labelled with Truth Values

For illustration purposes, several well known (normal) modal logic systems are thenpresented.

Definition 3.1.1. A signature ΣM = 〈C,O,\, X, Y, Z〉 for a modal logic with a

modality is such that:

• C0 = t, f ∪ pi : i ∈ N;

• C1 = ¬,2;

• C2 = ∧,∨,⊃;

• Ck = ∅ for k ≥ 3;

• O0 = ⊤,⊥;

• O1 = I,N;

• O2 = lb;

• Ok = ∅ for k ≥ 3;

• \1 = Ω;

• \2 = ⊑;

• \k = ∅ for k ≥ 3;

• X, Y, Z are disjoint countable sets of variables.

The formula constructors are the usual ones in modal logic. The operatorsare used namely to reason about the accessibility relation and to relate terms. Inthe case of the signature presented, N is the neighbourhood operator representingthe collection of neighbours according to the accessibility relation associated to thedenotation of a term, that is, if terms are seen as set of worlds, N(τ1) is the set ofall worlds accessible from at least one of the worlds in τ1.

For (normal) modal logics several kinds of rules are needed: rules for connectives,rules for operators, structural rules, etc. See Table 3.1 for the specific rules for themodal constructor and the operator N.

Observe the intrinsic relation between the N operator and the modality. Mean-while, the rules for N state that the neighbourhood of a truth value is induced bythe neighbourhoods of the atomic truth values contained in it.

3.1. (Normal) Modal Logics 37

L2

N(τ1) ≤ ξ1,Γ1 → Γ2

τ1 ≤ (2ξ1),Γ1 → Γ2 R2

Γ1 → Γ2,N(τ1) ≤ ξ1

Γ1 → Γ2, τ1 ≤ (2ξ1)

LNΩ

τ3 ⊑ τ1,Ωτ3,Ωτ2,Γ1 → Γ2, τ2 6⊑N(τ3)

Ωτ2,Γ1 → Γ2, τ2 6⊑N(τ1)τ3 : y, τ3 6∈ τ1, τ2,Γ1,Γ2

RNΩ

Ωτ2,Γ1 → Γ2,Ωτ3 Ωτ3,Ωτ2,Γ1 → Γ2, τ3 ⊑ τ1 Ωτ3,Ωτ2,Γ1 → Γ2, τ2 ⊑ N(τ3)

Ωτ2,Γ1 → Γ2, τ2 ⊑ N(τ1)

Table 3.1: Specific rules for the modal constructor and operator

Definition 3.1.2. A sequent calculus for a normal modal logic with a modality is

the pair CM = 〈ΣM ,RM〉 where ΣM is the signature in Definition 3.1.1 and RM

is the set containing the rules in Tables 2.3, 3.1, 2.4, 2.2 and 2.1 (where the rulesshowed with are presented in Table 3.2).

It is now presented an example of a derivation in the context of a sequent calculusfor a normal modal logic.

Example 3.1.3. In the context of the sequent calculus introduced in Definition

3.1.2, the following holds: Ωy1,y1 ≤ ψ1,y1 ≤ ψ2 → y1 ≤ ϕ ⊢R→ ⊤ ≤ (ψ1∧ψ2)⊃ϕ.

1. → ⊤ ≤ (ψ1 ∧ ψ2)⊃ ϕ RgenF 2

2. Ωy1,y1 ⊑ ⊤ → y1 ≤ (ψ1 ∧ ψ2)⊃ ϕ R⊃ 3

3. Ωy1,y1 ⊑ ⊤,y1 ≤ ψ1 ∧ ψ2 → y1 ≤ ϕ L∧ 4

4. Ωy1,y1 ⊑ ⊤,y1 ≤ ψ1,y1 ≤ ψ2 → y1 ≤ ϕ Lw 5

5. Ωy1,y1 ≤ ψ1,y1 ≤ ψ2 → y1 ≤ ϕ hyp

Example 3.1.4. Each sequent calculus for (normal) modal logic has l-MTD, l-

MTMP and g-MTMP, all with Λ = τ1 ≤ ξ1 ⊃ ξ2.

The proofs are the same as in Example 2.3.6 in classical propositional logic. Itis worthwhile to notice that the proof of g-MTD in that example is not sound in thecontext of (normal) modal logic since the rule 1 does not belong to RM .

One of the most common ways to express the semantics of modal logics is byKripke structures. Among Kripke structures, general Kripke structures deserve aparticular importance since they provide a complete semantics to some modal logicsnot complete with respect to the standard Kripke semantics. For instance, thesystem GL, the normal modal logic generated by the Löb axiom

2(2ϕ⊃ ϕ)⊃ 2ϕ


AxT τ1 ⊑ τ2,Γ1 → Γ2, τ1 ⊑ τ2 AxΩ Ωτ1,Γ1 → Γ2,Ωτ1

LwT

Γ1 → Γ2

τ1 ⊑ τ2,Γ1 → Γ2 RwT

Γ1 → Γ2

Γ1 → Γ2, τ1 ⊑ τ2

LwΩΓ1 → Γ2

Ωτ1,Γ1 → Γ2 RwΩΓ1 → Γ2

Γ1 → Γ2,Ωτ1

LcT

τ1 ⊑ τ2, τ1 ⊑ τ2,Γ1 → Γ2

τ1 ⊑ τ2,Γ1 → Γ2 RcT

Γ1 → Γ2, τ1 ⊑ τ2, τ1 ⊑ τ2Γ1 → Γ2, τ1 ⊑ τ2

LcΩ

Ωτ1,Ωτ1,Γ1 → Γ2

Ωτ1,Γ1 → Γ2 RcΩ

Γ1 → Γ2,Ωτ1,Ωτ1Γ1 → Γ2,Ωτ1

LxiT

Γ1 → Γ2, τ1 ⊑ τ2

τ1 6⊑ τ2,Γ1 → Γ2 RxiT

τ1 ⊑ τ2,Γ1 → Γ2

Γ1 → Γ2, τ1 6⊑ τ2

LxiΩ

Γ1 → Γ2,Ωτ10τ1,Γ1 → Γ2 RxiΩ

Ωτ1,Γ1 → Γ2

Γ1 → Γ2,0τ1

LxeT

Γ1 → Γ2, τ1 6⊑ τ2τ1 ⊑ τ2,Γ1 → Γ2 RxeT

τ1 6⊑ τ2,Γ1 → Γ2

Γ1 → Γ2, τ1 ⊑ τ2

LxeΩ

Γ1 → Γ2,0τ1Ωτ1,Γ1 → Γ2 RxeΩ

0τ1,Γ1 → Γ2

Γ1 → Γ2,Ωτ1

cutT

Γ1 → Γ2, τ1 ⊑ τ2 τ1 ⊑ τ2,Γ1 → Γ2

Γ1 → Γ2 cutΩ

Γ1 → Γ2,Ωτ1 Ωτ1Γ1 → Γ2

Γ1 → Γ2

Table 3.2: Structural rules for \

is not strongly complete with respect to standard Kripke structures (see [BdRV01,p. 211]). A general Kripke structure is a tuple K = 〈W,B,;, V 〉 where W is theset of possible worlds, B ⊆ 2W is the set of admissible valuations, ;⊆ W 2 is theaccessible relation for the modal constructor 2 and V : Π → B is the valuation map.These general Kripke structures can be induced as ΣM -algebras as showed in thesequel.

Definition 3.1.5. From any general Kripke structure K = 〈W,B,;, V 〉, it can be

defined an algebra AK = 〈F, T, ·AK〉 where:

• F = B;

• T = 2W ;

• #AK= λb.b;

• a ∈ ΩAKiff a is a singleton;

• 〈a, a′〉 ∈⊑AKiff a ⊆ a′;

• 〈a, b〉 ∈≤AKiff a ⊆ b;


• ⊥AK= ∅;

• ⊤AK= W ;

• IAK= λa.ι(a);

• NAK= λa.w′ ∈ W : exists w ∈ a such that w ; w′;

• lbAK= λaa′.a ∩ a′;

• fAK= ∅;

• tAK= W ;

• piAK= V (pi);

• ¬AK= λb.W \ b;

• 2AK= λb.w ∈ W : NAK

(w) ⊆ b;

• ∧AK= λbb′.b ∩ b′;

• ∨AK= λbb′.b ∪ b′;

• ⊃AK= λbb′.(W \ b) ∪ b′;

with ι being a choice function for W .

Definition 3.1.6. From a ΣM -algebra A = 〈F, T, ·A〉 appropriate for CM , it can be

defined a general Kripke structure KA = 〈W,;,B, V 〉 where:

• W = ΩA;

• t ; t′ iff t′ ⊑A NA(t);

• B = 〈f〉A : f ∈ F with 〈f〉A = t ∈ ΩA : t ≤A f;

• V (pi) = 〈piA〉A.

Entailment is preserved (both local and global, as in Definition 2.4.8) between ageneral Kripke structure K and its induced algebra AK , that is,

ψ1, . . . , ψk dK ϕ iff ψ1, . . . , ψk

dAK

ϕ for d = l, g,

as well between an algebra A and its induced general Kripke structure KA, that is,

ψ1, . . . , ψk dA ϕ iff ψ1, . . . , ψk d

KAϕ for d = l, g,

where K and KAare the usual entailment relation for general Kripke structures.

Both proofs can be found in section 3.3 of [MSSV04] as well as the proof that for


every general Kripke structure K the induced ΣM -algebra AK is appropriate foreach rule in RM .

The next example shows that the axiom for reflexivity is satisfied by the ΣM -algebra induced by a general Kripke structure that is not reflexive. Actually, theaxiom characterizes reflexivity only among standard frames. But it is possible tocharacterize general frames with a reflexive accessibility relation using rule T

Γ1 → Γ2, τ1 ⊑ N(τ1).

Example 3.1.7. Consider the general Kripke structure 〈W,B,;, V 〉 where W =

w1, w2, ;= 〈w1, w2〉, 〈w2, w1〉, B = ∅,W and V such that p1 = ∅ and p2 =W . Let A be the ΣM -algebra induced by this general Kripke structure according toExample 3.1.5. Thus, A ⊤ ≤ (2pi)⊃ pi for i = 1, 2 as showed below. In fact:

• J⊤KA =W ;

• Jp1KA = ∅;

• Jp2KA =W ;

• 2A(∅) = ∅;

• 2A(W ) =W .

Then

A ⊤ ≤ (2p1)⊃ p1 iff 〈J⊤KA, J(2p1)⊃ p1KA〉 ∈≤A

iff 〈W,⊃A(J2p1KA, Jp1KA)〉 ∈≤A

iff 〈W,⊃A(2A(Jp1KA), Jp1KA)〉 ∈≤A

iff 〈W,⊃A(2A(∅),∅)〉 ∈≤A

iff 〈W,⊃A(∅,∅)〉 ∈≤A

iff 〈W,W 〉 ∈≤A .

And, for p2

A ⊤ ≤ (2p2)⊃ p2 iff 〈J⊤KA, J(2p2)⊃ p2KA〉 ∈≤A

iff 〈W,⊃A(J2p2KA, Jp2KA)〉 ∈≤A

iff 〈W,⊃A(2A(Jp2KA), Jp2KA)〉 ∈≤A

iff 〈W,⊃A(2A(W ),W )〉 ∈≤A

iff 〈W,⊃A(W,W )〉 ∈≤A

iff 〈W,W 〉 ∈≤A


It is now presented the definition of modal logic system. When the set of rulesis only the one included in Definition 3.1.2, the system characterized is the smallernormal modal system usually called K.

Definition 3.1.8. A (normal) modal logic system LM is a relational logic system

〈ΣM ,RM ,A〉 where the signature ΣM is as Definition 3.1.1, the rules are at leastthose described in Definition 3.1.2 and A is a class of ΣM -algebras as in Definition2.4.1.

Observe that in a modal logic system the following consequences hold:

• ϕ ⊢gR 2ϕ and

• ⊢gR 2(ϕ⊃ ψ)⊃ (2ϕ⊃ 2ψ).

As practical examples of modal logic systems, a deontic logic system, a Löbprovability logic system and a knowledge logic system are presented below.

Example 3.1.9. Consider the modal logic system SDL = 〈ΣD,RD,AD〉 for the

Standard Deontic Logic, see [Hil02, section 6], where:

• ΣD is a modal signature as in Definition 3.1.1 where 2ϕ is intended to meanthat ϕ is obligatory;

• RD is the set of rules as in Definition 3.1.2 plus the rule D:

Ωτ1,Γ1 → Γ2,N(τ1) 6⊑⊥

expressing that for any atomic truth value there is at least another truth valueacceptable by it, i.e., everything obligatory associated to the atomic truth valueholds in that other truth value;

• AD is the class of all ΣD-algebras appropriate for 〈ΣD,RD〉. It worthwhile tonotice that AD includes the ΣD-algebras induced by general Kripke structureswhere the accessible relation is right-unbounded. This is guaranteed by acharacterization theorem given in [MSSV04, p.257–258].

Deontic logic systems are concerned with what is obligatory at a certain point.It has been widely used in a multitude of fields ranging from computer science tophilosophy or even in law. However the definition of deontic systems as a normalmodal logic has some side-effects, as theorems that belongs to normal modal logicsthat should be relaxed for the deontic reasoning. For instance, if the system has


¬2⊥ as an axiom with the intended meaning that there isn’t impossible obligations,then in a normal system the statement ¬(2ϕ ∧ 2¬ϕ), with the intended meaningthat there is no conflict of obligations, always holds. Nevertheless the second formulashould be relaxed if the existence of moral dilemmas in the system could occur. Thiscan be solved using non-normal modal logics as the ones in Section 3.2.

It worthwhile to remember the concept of conversely well-founded relation forthe following example. A relation is conversely well-founded if and only if there noinfinite ascending sequences.

The logic system GL is concerned with reasoning about provability, more specif-ically about what can be expressed by arithmetical theories about their provabilitypredicates, and was considered for instance, when dealing with the logical omni-science problem, and in dealing with reflection in artificial intelligence, automateddeduction and verification.

Example 3.1.10. Consider the modal logic system GL = 〈ΣGL,RGL,AGL〉 cor-

responding to the modal Löb provability logic for a certain theory T, see [AB04],where:

• ΣGL is a modal signature as in Definition 3.1.1 where 2ϕ is intended to meanthat ϕ is provable in T;

• RGL is the set of rules described in Definition 3.1.2 plus the rule W:

Ωτ1,Ωτ3, τ3 ⊑ N(τ1),Γ1 → Γ2, τ3 ⊑ τ2,N(τ3) 6⊑ τ2

Ωτ1,Γ1 → Γ2,N(τ1) ⊑ τ2τ3 : y, τ3 6∈ τ1, τ2,Γ1,Γ2

• AGL is the class of all ΣGL-algebras appropriate for 〈ΣGL,RGL〉. It is worth-while to notice that this includes the ΣGL-algebras induced by general Kripkestructures where the accessibility relation is transitive and conversely well-founded (by the characterization theorem in [MSSV04, p.257–258]).

A multi-modal logic system, for a knowledge logic system is now presented. Thissystem can actually be viewed as an example of fusion of n modal logic systems, asit will be clear when fusion is introduced later on.

Example 3.1.11. Consider a knowledge logic system K for n agents where each

modal formula Kiϕ is intended to mean that agent i knows ϕ, see [Hal95]. That is,K = 〈ΣK ,RK ,AK〉 is such that:

• ΣK is a modal signature as described in Definition 3.1.1 with C1 = ¬, K1, . . . ,

Kn and O1 = I,N1, . . . ,Nn;

3.2. Non-normal Modal Logics 43

• RK is the set of rules from Definition 3.1.2 with n copies of the rules L2,R2, LNΩ and RNΩ renamed as LKi, RKi, LNiΩ and RNiΩ for i = 1, . . . , n,together with the rules listed in Table 3.3. As it is clear from the inspection ofthe rules, Ti imposes that each associated accessibility relation is reflexive, 4ithat it is transitive, and the rule 5i that each accessibility relation is Euclidean.Together they impose that each relation is an equivalence;

• AK is the class of all ΣK-algebras appropriate for 〈ΣK ,RK〉. Observe thatthis class includes the ΣK-algebras induced by general Kripke structures wherethere are n accessibility relations which are equivalence relations (see [MSSV04,p.257–258]).

Γ1 → Γ2, τ1 ⊑ Ni(τ1)Ti

Γ1 → Γ2,Ni(Ni(τ1)) ⊑ Ni(τ1)4i

Ωτ1,Ωτ2,Ωτ3,Γ1 → Γ2, τ1 ⊑ Ni(τ3) Ωτ1,Ωτ2,Ωτ3,Γ1 → Γ2, τ2 ⊑ Ni(τ3) Ωτ1,Ωτ2,Γ1 → Γ2,Ωτ3

Ωτ1,Ωτ2,Γ1 → Γ2, τ1 ⊑ Ni(τ2)5i

Table 3.3: Additional rules for the Knowledge Logic System K

3.2 Non-normal Modal Logics

Non-normal modal logics are an extension of propositional logics with a modal con-structor 2. The calculus usually does not include the necessitation rule (from ϕ infer2ϕ) and/or the K axiom 2(ϕ1 ⊃ ϕ2)⊃ (2ϕ1 ⊃ 2ϕ2). The semantics is a variant ofKripke semantics usually called in the literature neighbourhood semantics.

Despite that the invention of neighbourhood semantics and the minimal modelssemmantics started in the 1970’s (independent publications [Sco70] and [Mon70]),they seem not be so used in comparison with applications as normal ones. Still someproperties were investigated by Segerberg [Seg71] and Chellas [Che80], for instance.However nowadays it is possible to find non-normal modal systems for a little bunchof areas like law [Jon90], game theory [Par85], coalition logics [Pau02], concurrentpropositional dynamic logic [Gol92], and belief modalities [HL00] for instance.

The framework of LSLTV can be easily used to deal with non-normal modallogics and, as presented in this section, the whole hierarchy presented in [Che80]of non-normal modal systems can be expressed in this context. With this kind oflogics represented as LSLTV and taking into account the definition of combinationin Chapter 4, new results about fusion of non-normal modal logics can be obtained.

For a gentle introduction to non-normal modal systems, see [Che80].


Definition 3.2.1. A structure S for a non-normal modal logic is a triple 〈W, f, V 〉

where:

• W is a set of worlds,

• f : 2W → 2W is a function, and

• V : pi : i ∈ N → 2W is a valuation map.

The denotation of a formula ψ in S is inductively defined as follows:

• JψKS = V (ψ), if ψ is a propositional symbol,

• J¬ψKS =W \ JψKS ,

• Jψ ∧ ψ′KS = JψKS ∩ Jψ′KS ,

• Jψ ⊃ ψ′KS = Jψ′KS ∪W \ JψKS ,

• J2ψKS = f(JψKS).

In order to represent structures for non-normal modal logics in the LSLTV ap-proach, it is convenient to assume that they use a function f : 2W → 2W instead ofeither the addition of a set Q ⊆W of queer worlds in a Kripke structure as proposedby [Kri65], or the operator N : W → 22

W

as the minimal models in [Che80]. In fact,this formulation with the function f is pointwise equivalent with minimal models(see [Che80, p. 211]).

The denotation of the non-modal fragment is as in normal modal logics. Forthe modality 2ϕ the denotation is directly the application of the funtion f to thedenotation of the formula ϕ. It is worthwhile to notice that no restrictions are madeabout f .

A signature for non-normal modal logic in the context of LSLTV is as follows.

Definition 3.2.2. A signature ΣN = 〈C,O,\, X, Y, Z〉 for a non-normal modal logic

with a modality is such that:

• C0 = t, f ∪ pi : i ∈ N;

• C1 = ¬,2;

• C2 = ∧,∨,⊃;


• Ck = ∅ for k ≥ 3;

• O0 = ⊤,⊥;

• O1 = I,F;

• O2 = lb;

• Ok = ∅ for k ≥ 3;

• \1 = Ω;

• \2 = ⊑;

• \k = ∅ for k ≥ 3;

• X, Y, Z are disjoint countable sets of variables.

The only difference between the signature of a non-normal and a normal modallogic is the operator F instead of N. This new operator will play the role of thefunction f in Definition 3.2.1.

The rules for the modal constructor and the F operator are presented in Table3.4 and are different from the corresponding ones for normal modal logics. It can beseen right away the relation between the rules for 2 and the last item in Definition3.2.1. The rule eqF states the only condition about the function f that must bereflected in the F operator: for equal terms, the result of F must be the same.

L2

τ1 ⊑ F(#ξ1),Γ1 → Γ2

τ1 ≤ 2ξ1,Γ1 → Γ2 R2

Γ1 → Γ2, τ1 ⊑ F(#ξ1)

Γ1 → Γ2, τ1 ≤ 2ξ1

eqF

Γ1 → Γ2, τ1 ⊑ τ2 Γ1 → Γ2, τ2 ⊑ τ1

Γ1 → Γ2,F(τ1) ⊑ F(τ2)

I Γ1 → Γ2, I(τ1) ⊑ τ1 ΩI

Γ1 → Γ2, τ1 6⊑⊥

Γ1 → Γ2,ΩI(τ1)

lb1 Γ1 → Γ2, lb(τ1, τ2) ⊑ τ1 lb2 Γ1 → Γ2, lb(τ1, τ2) ⊑ τ2

Table 3.4: Rules for the modal constructor and the operators in non-normal calculi

Definition 3.2.3. A sequent calculus for a non-normal modal logic with a modality

is the pair CN = 〈ΣN ,RN〉 where ΣN is the signature in Definition 3.2.2 and RN isthe set constituted at least by the rules in Tables 2.3, 3.4, 2.2 and 2.1 (where therules showed with are presented in Table 3.2).


In the sequel it is proved that 2 preserves equivalence theoremhood, i.e., it ispresented a derivation for 2ϕ1⇔2ϕ2 assuming as hypothesis ϕ1⇔ϕ2. Actually, thisis the only property with respect to 2 that is imposed in non-normal modal logics.For this purpose the rules for the connective ⇔ presented in Table 3.5, obtained byseeing ⇔ as the abbreviation ξ1 ⇔ ξ2 ≡ (ξ1 ⊃ ξ2) ∧ (ξ2 ⊃ ξ1), are used.

L⇔Γ1 → Γ2, τ1 ≤ ξ1, τ1 ≤ ξ2 τ1 ≤ ξ1, τ1 ≤ ξ2,Γ1 → Γ2

τ1 ≤ ξ1 ⇔ ξ2,Γ1 → Γ2

R⇔Ωτ1, τ1 ≤ ξ1,Γ1 → Γ2, τ1 ≤ ξ2 Ωτ1, τ1 ≤ ξ2,Γ1 → Γ2, τ1 ≤ ξ1

Ωτ1,Γ1 → Γ2, τ1 ≤ ξ1 ⇔ ξ2

Table 3.5: Rules for the equivalence connective

Example 3.2.4. In the context of a sequent calculus for non-normal modal logics

as in Definition 3.2.3, → ⊤ ≤ ϕ1 ⇔ ϕ2 ⊢CN→ ⊤ ≤ 2ϕ1 ⇔ 2ϕ2.

1. → ⊤ ≤ 2ϕ1 ⇔ 2ϕ2 RgenF 2

2. Ωy1,y1 ⊑ ⊤ → y1 ≤ 2ϕ1 ⇔ 2ϕ2 R⇔ 3,3’

3. Ωy1,y1 ⊑ ⊤,y1 ≤ 2ϕ1 → y1 ≤ 2ϕ2 R2 4

3’. Ωy1,y1 ⊑ ⊤,y1 ≤ 2ϕ2 → y1 ≤ 2ϕ1 see note (B)

4. Ωy1,y1 ⊑ ⊤,y1 ≤ 2ϕ1 → y1 ⊑ F(#ϕ2) L2 5

5. Ωy1,y1 ⊑ ⊤,y1 ⊑ F(#ϕ1) → y1 ⊑ F(#ϕ2) transT 6,7

6. Ωy1,y1 ⊑ ⊤,y1 ⊑ F(#ϕ1) → y1 ⊑ F(#ϕ1) axiom

7. Ωy1,y1 ⊑ ⊤,y1 ⊑ F(#ϕ1) → F(#ϕ1) ⊑ F(#ϕ2) Lw’s 8

8. → F(#ϕ1) ⊑ F(#ϕ2) eqF 9,9’

9. → #ϕ1 ⊑ #ϕ2 cutF 10,11

9’. → #ϕ2 ⊑ #ϕ1 see note (A)

10. → #ϕ1 ⊑ #ϕ2,#ϕ1 ≤ ϕ1 ⇔ ϕ2 transF 12,13

11. #ϕ1 ≤ ϕ1 ⇔ ϕ2 → #ϕ1 ⊑ #ϕ2 L⇔ 14,15

12. → #ϕ1 ⊑ #ϕ2,#ϕ1 ⊑ ⊤ ⊤

13. → #ϕ1 ⊑ #ϕ2,⊤ ≤ ϕ1 ⇔ ϕ2 hyp

14. → #ϕ1 ⊑ #ϕ2,#ϕ1 ≤ ϕ1,

#ϕ1 ≤ ϕ2 cutT 16,17

15. #ϕ1 ≤ ϕ1,#ϕ1 ≤ ϕ2 → #ϕ1 ⊑ #ϕ2 R# 19

16. → #ϕ1 ⊑ #ϕ2,#ϕ1 ≤ ϕ1,

#ϕ1 ≤ ϕ2,#ϕ1 ⊑ #ϕ1 ref

17. #ϕ1 ⊑ #ϕ1 → #ϕ1 ⊑ #ϕ2,#ϕ1 ≤ ϕ1,

#ϕ1 ≤ ϕ2 L# 18


18. #ϕ1 ≤ ϕ1 → #ϕ1 ⊑ #ϕ2,#ϕ1 ≤ ϕ1,

#ϕ1 ≤ ϕ2 axiom

19. #ϕ1 ≤ ϕ1,#ϕ1 ≤ ϕ2 → #ϕ1 ≤ ϕ2 axiom

Notes:

(A) The proof of line 9’ is given by lines 10 to 19 changing all the occurrencesof ϕ1 by ϕ2 and vice-versa.

(B) The proof of line 3’ is given by lines 4 to 19 changing all the occurrences ofϕ2 by ϕ1 and vice-versa.

Similarly to the concept of general Kripke structure, there are general structuresfor non-normal modal logic. As in the normal case, there are systems that arecomplete only with these kind of structures [Ger75]. About this subject, see [Han03,KW99].

Definition 3.2.5. A general structure S for a non-normal modal logic is 〈W, f,A, V 〉

where:

• W is a set of worlds;

• f : 2W → 2W is a function;

• A ⊆ 2W is such that

– ∅ ∈ A,

– if X, Y ∈ A then X ∪ Y ∈ A,

– if X ∈ A then W \X ∈ A,

– if X ∈ A then f(X) ∈ A;

• V : pi : i ∈ N → A is a valuation map.

It is possible to induce a ΣN -algebra from a general structure for a non-normallogic presented in Definition 3.2.5. Moreover global and local entailment are equiv-alent in both semantics.

Definition 3.2.6. Given a general structure S = 〈W, f,A, V 〉 of non-normal modal

logic the corresponding ΣN -algebra AS = 〈F, T, ·AS〉 is such that:

• F = A;

• T = 2W ;


• #AS= λa.a;

• x ∈ ΩASiff x is a singleton;

• 〈x, x′〉 ∈⊑ASiff x ⊆ x′;

• 〈x, a〉 ∈≤ASiff x ⊆ a;

• ⊥AS= ∅;

• ⊤AS= W ;

• IAS= λx.ι(x);

• lbAS= λxx′.x ∩ x′;

• FAS= λx.f(x);

• fAS= ∅;

• tAS= W ;

• piAS= V (pi);

• ¬AS= λa.W \ a;

• 2AS= λa.FAS

(a);

• ∧AS= λaa′.a ∩ a′;

• ∨AS= λaa′.a ∪ a′;

• ⊃AS= λaa′.a′ ∪W \ a;

with ι being a choice function for W .

The following facts are straightforward to verify and, because of them, in theproofs of the following semantic results it is enough to prove statements aboutf ,pi,¬,∨ and 2.

• tAS= ¬AS

(fAS),

• ⊃AS(a, a′) = ∨AS

(¬AS(a), a′),

• ∧AS(a, a′) = ¬AS

(∨AS(¬AS

(a),¬AS(a′)))

It is showed now that the proposed entailment for non-normal modal logics isequivalent to the original one given in Definition 3.2.1.

Lemma 3.2.7. Let S be a general structure for a non-normal modal logic. Then,

for every closed simple formula ϕ, JϕKS = JϕKAS.


Proof. The result follows by structural induction on ϕ.

• ϕ is f . Then JfKS = ∅ = fAS= JfKAS

.

• ϕ is pi. Then JpiKS = V (pi) = piAS

= JpiKAS.

• ϕ is ¬ϕ′. Then J¬ϕ′KS =W \ Jϕ′KS =W \ Jϕ′KAS= ¬AS

(Jϕ′KAS) = J¬ϕ′KAS

.

• ϕ is 2ϕ′. Then J2ϕ′KS = f(Jϕ′KS) = FAS(Jϕ′KAS

) = 2AS(Jϕ′KAS

) = J2ϕ′KAS.

• ϕ is ϕ′∨ϕ′′. Then Jϕ′∨ϕ′′KS = Jϕ′KS ∪ Jϕ′′KS = Jϕ′KAS∪ Jϕ′′KAS

= ∨AS(Jϕ′KAS

,

Jϕ′′KAS) = Jϕ′ ∨ ϕ′′KAS

.

With this result it is possible to prove the satisfiability of a formula in a structureand in the correspondent algebra. This is showed in the next two propositions.

Proposition 3.2.8. Given a general structure S for non-normal modal logic and a

closed simple formula ϕ, S ϕ iff AS ⊤ ≤ ϕ.

Proof. S ϕ iff JϕKS = W iff W ⊆ JϕKS iff ⊤AS⊆ JϕKAS

iff J⊤KAS⊆ JϕKAS

iff〈J⊤KAS

, JϕKAS〉 ∈≤AS

iff AS ⊤ ≤ ϕ.

Proposition 3.2.9. Given a general structure S for non-normal modal logic, an

unbound variable assignment β over AS such that β(y1) = w and a closed simpleformula ϕ, S, w ϕ iff ASβ y1 ≤ ϕ.

Proof. S, w ϕ iff w ∈ JϕKS iff w ⊆ JϕKS iff β(y1) ⊆ JϕKASiff Jy1KAS

⊆ JϕKASiff

〈Jy1KAS, JϕKAS

〉 ∈≤ASiff ASβ y1 ≤ ϕ.

So, from a class of structures for non-normal modal logics, global and localentailment are preserved by the algebras of Definition 3.2.6.

Theorem 3.2.10. Let S be a class of general structures for non-normal modal logics,

A(S) = AS : S ∈ S and ψ1, . . . , ψk, ϕ closed simple formulas. Then

i. ψ1, . . . , ψk gSϕ iff ψ1, . . . , ψk

g

A(S) ϕ,

ii. ψ1, . . . , ψk lSϕ iff ψ1, . . . , ψk

lA(S) ϕ.


Proof. (i.) ψ1, . . . , ψk gSϕ iff, for every S ∈ S, S ϕ whenever S ψi for

i = 1, . . . , k iff (by Proposition 3.2.8), for every A ∈ A(S), A ⊤ ≤ ϕ wheneverA ⊤ ≤ ψi for i = 1, . . . , k iff ψ1, . . . , ψk

g

A(S) ϕ.

(ii.) ψ1, . . . , ψk lSϕ iff, for every S ∈ S and w ∈ W , S, w ϕ whenever

S, w ψi for i = 1, . . . , k iff (by Proposition 3.2.9), for every A ∈ A(S) and everyassignment β such that β(y1) = w, Aβ y1 ≤ ϕ whenever Aβ y1 ≤ ψi fori = 1, . . . , k iff, for every A ∈ A(S) and every assignment β, Aβ y1 ≤ ϕ wheneverAβ y1 ≤ ψi for i = 1, . . . , k and Aβ Ωy1 iff ψ1, . . . , ψk

lA(S) ϕ.

Theorem 3.2.11. For every general structure S for non-normal modal logic, its

corresponding ΣN -algebra AS is appropriate for each rule in RN .

Proof. For structural and order rules the proof is the same of the case of inducedalgebras by general Kripke structures in normal modal logics given in [MSSV04].In the following, let ρ be any ground substitution and α be an arbitrary unboundassignment over AS . For the specific rules, it is analysed here first the case of eqF.

Assume that ASα Γ1ρ → Γ2ρ, τ1ρ ⊑ τ2ρ and ASα Γ1ρ → Γ2ρ, τ2ρ ⊑ τ1ρ,that is, for every bound variable assignment β over AS there are:

• δ1 ∈ Γ1ρ ∪ Γ2ρ ∪ τ1ρ ⊑ τ2ρ such that ASαβ δ1; and

• δ2 ∈ Γ1ρ ∪ Γ2ρ ∪ τ2ρ ⊑ τ1ρ such that ASαβ δ2.

For each β there are two cases to be considered:

i. ASαβ δi with δi ∈ Γ1ρ∪Γ2ρ for some i = 1, 2 which immediately establishesthe conclusion of the rule;

ii. otherwise, ASαβ τ1ρ ⊑ τ2ρ and ASαβ τ2ρ ⊑ τ1ρ. Let W ′,W ′′ ⊆ W besuch that Jτ1ρKASαβ =W ′ and Jτ2ρKASαβ = W ′′, thus W ′ ⊆W ′′ and W ′′ ⊆W ′,that is, W ′ = W ′′. So FAS

(Jτ1ρKASαβ) = FAS(W ′) = FAS

(W ′′) ⊆ FAS(W ′′) =

FAS(Jτ2ρKASαβ) which establishes that ASαβ F(τ1ρ) ⊑ F(τ2ρ) and, so, the

conclusion of the rule holds.

Now it is analysed the case of the rule R2. The proof for L2 is similar.

Assume that ASα Γ1ρ → Γ2ρ, τ1ρ ⊑ F(#ξ1ρ). So, for every bound variableassignment β, there is δβ ∈ Γ1ρ∪Γ2ρ∪ τ1ρ ⊑ F(#ξ1ρ) such that ASαβ δβ. Foreach β, two cases have to be considered:


• there is δβ in Γ1ρ ∪ Γ2ρ such that ASαβ δβ which immediately establishesASαβ Γ1ρ→ Γ2ρ and, hence, the conclusion of the rule holds;

• otherwise δβ is τ1ρ ⊑ F(#ξ1ρ), that is, ASαβ τ1ρ ⊑ F(#ξ1ρ). SupposeJτ1ρKASαβ = W ′ for a W ′ ⊆ W . Thus W ′ ⊆ FAS

(#ASJξ1ρKASα). Since

#AS= λa.a, W ′ ⊆ FAS

(Jξ1ρKASα), that is, Jτ1ρKASαβ ⊆ 2AS(Jξ1ρKASα) which

establishes that ASαβ τ1ρ ≤ 2ξ1ρ and, so, the conclusion of the rule holds.

Given a ΣN -algebra A, there also is an induced general structure such thatentailment is preserved as shown in the sequel. In what follows given a ∈ T , 〈a〉A =a′ ∈ ΩA : a′ ⊑A a, and given b ∈ F , 〈b〉A = a ∈ ΩA : a ≤A b.

Definition 3.2.12. Given a ΣN -algebra A = 〈F, T, ·A〉 appropriate for a sequent

calculus for a non-normal modal logic, the general structure induced by A is SA =〈W, f,A, V 〉 where:

• W = ΩA;

• f(X) =

〈FA(a)〉A if there is a ∈ T such that X = 〈a〉A

∅ otherwise, for X ∈ 2ΩA;

• A = 〈b〉A : b ∈ F;

• V (pi) = 〈piA〉A.

In the sequel, 〈t〉A will be denoted by 〈t〉 when the underlying algebra is clearfrom the context.

It remains to prove that f(X) is well defined.

Lemma 3.2.13. If a, a′ ∈ T are such that 〈a〉 = 〈a′〉 then 〈FA(a)〉 = 〈FA(a′)〉.

Proof. Since A is appropriate and

Ωτ0, τ0 ⊑ τ1,Γ1 → Γ2, τ0 ⊑ τ2 Ωτ0, τ0 ⊑ τ2,Γ1 → Γ2, τ0 ⊑ τ1

Γ1 → Γ2,F(τ1) ⊑ F(τ2)τ0 : y, τ0 6∈ τ1, τ2,Γ1,Γ2

is a derived rule (using rules eqF and RgenT), then, by soundness and using theappropriate assignments, FA(a) ⊑A FA(a

′) and FA(a′) ⊑A FA(a) and so, since the

sequents

Ωy1,F(a) ⊑ F(a′),y1 ⊑ F(a) → y1 ⊑ F(a′)


andΩy1,F(a

′) ⊑ F(a),y1 ⊑ F(a′) → y1 ⊑ F(a)

are derivable (both by transT rule), then 〈FA(a)〉 = 〈FA(a′)〉.

To show the preservation of entailment first it is showed that the structure pre-viously defined is, in fact, a general non-normal structure.

Proposition 3.2.14. Given a ΣN -algebra A appropriate for a sequent calculus CN

for a non-normal modal logic, the tuple SA of Definition 3.2.12 is a general structurefor a non-normal modal logic.

Proof. It has to be shown that W is non-empty and A satisfies the conditions givenin 3.2.5.

(A) W is non-empty. In fact, by absurd, assume that W = ΩA = ∅. Since A isappropriate for CN , it is so for rules cons and Ω, then ⊤A ∈ ΩA. Therefore W 6= ∅.

(B) ∅ ∈ A. Indeed, ∅ = 〈JfKA〉 since A is appropriate for rule Rf .

(C) A is closed for unions. It has to be shown that if 〈b〉, 〈b′〉 ∈ A then (〈b〉∪〈b′〉) ∈ A.Take the set

〈∨A(b, b′)〉 = a ∈ W : a ≤A ∨A(b, b

′)

which is in A. The following facts show that

〈∨A(b, b′)〉 = a ∈ W : a ≤A b or a ≤A b

′

where the latter set is precisely 〈b〉 ∪ 〈b′〉.

(i) a ∈ W : a ≤A b or a ≤A b′ ⊆ a ∈ W : a ≤A ∨A(b, b′). Assume that

a ≤A b or a ≤A b′ for a ∈ ΩA. By satisfaction, choosing α such that α(x1) = a,α(z1) = b and α(z2) = b′ it holds Aα Ωx1 → x1 ≤ z1,x1 ≤ z2. Observe that

Ωx1 → x1 ≤ z1,x1 ≤ z2 ⊢CN Ωx1 → x1 ≤ z1 ∨ z2

by an application of rule R∨. Since the calculus is sound (by Theorem 2.5.5),

Ωx1 → x1 ≤ z1,x1 ≤ z2 A(CN ) Ωx1 → x1 ≤ z1 ∨ z2

where A(CN) is the class of the algebras appropriate for CN . Hence, Aα Ωx1 →x1 ≤ z1 ∨ z2. Again by satisfaction and taking into account the choice of α, a ≤A

∨A(b, b′).

(ii) a ∈ W : a ≤A ∨A(b, b′) ⊆ a ∈ W : a ≤A b or a ≤A b′. The proof is

similar taking into account that Ωx1 → x1 ≤ z1 ∨ z2 ⊢CN Ωx1 → x1 ≤ z1,x1 ≤ z2as showed below:


1. Ωx1 → x1 ≤ z1,x1 ≤ z2 cutF 2,32. Ωx1 → x1 ≤ z1,x1 ≤ z2,x1 ≤ z1 ∨ z2 RwF 43. x1 ≤ z1 ∨ z2,Ωx1 → x1 ≤ z1,x1 ≤ z2 L∨ 5,64. Ωx1 → x1 ≤ z1 ∨ z2 hyp5. x1 ≤ z1,Ωx1 → x1 ≤ z1,x1 ≤ z2 axiom6. x1 ≤ z2,Ωx1 → x1 ≤ z1,x1 ≤ z2 axiom

(D) A is closed for complements. It has to be shown that if 〈b〉 ∈ A then (W \〈b〉) ∈A. Take the set

〈¬A(b)〉 = a ∈ W : a ≤A ¬

A(b)

which is in A. The following facts show that

〈¬A(b)〉 = a ∈ W : a A b

where the latter set is precisely W \ 〈b〉.

(i) a ∈ W : a A b ⊆ a ∈ W : a ≤A ¬A(b). The proof is similar to thecase (i) of (C) taking into account that Ωx1 → x1 z1 ⊢CN Ωx1 → x1 ≤ ¬ z1 byapplication of rules R¬ and LxeF.

(ii) a ∈ W : a ≤A ¬A(b) ⊆ a ∈ W : a A b. Again the proof is similar takinginto account Ωx1 → x1 ≤ ¬ z1 ⊢CN Ωx1 → x1 z1 as showed below:

1. Ωx1 → x1 z1 cutF 2,32. Ωx1 → x1 z1,x1 ≤ ¬ z1 RwF 43. x1 ≤ ¬ z1,Ωx1 → x1 z1 L¬ 54. Ωx1 → x1 ≤ ¬ z1 hyp5. Ωx1 → x1 z1,x1 ≤ z1 RxiF 66. x1 ≤ z1,Ωx1 → x1 ≤ z1 axiom

(E) A is closed under f . It has to be shown that if 〈b〉 ∈ A then f(〈b〉) ∈ A. From〈b〉 ∈ A it can be seen that there is X ∈ 2ΩA such that X = 〈b〉, so

f(〈b〉) = 〈FA(b)〉 = a ∈ W : a ⊑A FA(b)

. The following facts show that the latter set is equal to

a ∈ W : a ≤A 2A(b) = 〈2A(b)〉

which is in A.

(i) a ∈ W : a ⊑A FA(b) ⊆ a ∈ W : a ≤A 2A(b). The proof is similar to thecase (i) of (C) taking into account that Ωx1 → x1 ⊑ F(#z1) ⊢CN Ωx1 → x1 ≤ 2z1by applying the rule R2.


(ii) a ∈ W : a ≤A 2A(b) ⊆ a ∈ W : a ⊑A FA(b). Again the proof issimilar taking into account that Ωx1 → x1 ≤ 2z1 ⊢CN Ωx1 → x1 ⊑ F(#z1) by thefollowing derivation:

1. Ωx1 → x1 ⊑ F(#z1) cutF 2,32. Ωx1 → x1 ⊑ F(#z1),x1 ≤ 2z1 RwF 43. x1 ≤ 2z1,Ωx1 → x1 ⊑ F(#z1) L2 54. Ωx1 → x1 ≤ 2z1 hyp5. x1 ⊑ F(#z1),Ωx1 → x1 ⊑ F(#z1) axiom

The following results establish the preservation of the semantics of closed simpleformulas when moving to general non-normal structures from the bi-sorted algebrasappropriate for a sequent calculus for a non-normal modal logic. It is given now alemma about the denotation of this kind of formulas.

Lemma 3.2.15. Let A be a ΣN -algebra appropriate for CN . Then, for every closed

simple formula ϕ, 〈JϕKA〉 = JϕKSA.

Proof. The proof is by structural induction on ϕ.

• ϕ is f . Then 〈JfKA〉 = 〈fA〉 = u ∈ ΩA : u ≤A fA. Since A is appropriate forrule Rf the latter set coincides with u ∈ ΩA : u ≤A ⊥A. Moreover, this set isequal to ∅ since A is also appropriate for rule Ω⊥. Therefore 〈JfKA〉 = JfKSA.

• ϕ is pi. Then 〈JpiKA〉 = 〈piA〉 = V (pi) = JpiKSA.

• ϕ is ¬ϕ′. Then 〈J¬ϕ′KA〉 = 〈¬AJϕ′KA〉 = u ∈ ΩA : u ≤A ¬A(Jϕ′KA) which

coincides with u ∈ ΩA : u A Jϕ′KA as seen in part (D) of the proof ofProposition 3.2.14. Therefore,

〈J¬ϕ′KA〉 = ΩA \ u ∈ ΩA : u ≤A Jϕ′KA = ΩA \ 〈Jϕ′KA〉.

By the induction hypothesis, the latter is equal to ΩA \ Jϕ′KSA = J¬ϕ′KSA.

• ϕ is 2ϕ′. Then 〈J2ϕ′KA〉 = 〈2AJϕ′KA〉 = u ∈ ΩA : u ≤A 2A(Jϕ′KA) which

coincides with u ∈ ΩA : u ≤A f(Jϕ′KA) as seen in part (E) of the proof ofProposition 3.2.14. Therefore,

〈J2ϕ′KA〉 = f(u ∈ ΩA : u ≤A Jϕ′KA).

By the induction hypothesis, the latter is equal to f(Jϕ′KSA) which is J2ϕ′KSA.


• ϕ is ϕ′ ∨ ϕ′′. Then 〈Jϕ′ ∨ ϕ′′KA〉 = 〈∨A(Jϕ′KA, Jϕ

′′KA)〉 which coincides withu ∈ ΩA : u ≤A Jϕ′KA or u ≤A Jϕ′′KA as seen in part (C) of Proposition3.2.14. Therefore,

〈Jϕ′ ∨ ϕ′′KA〉 = u ∈ ΩA : u ≤A Jϕ′KA ∪ u ∈ ΩA : u ≤A Jϕ′′KA.

By the induction hypothesis, the latter is equal to Jϕ′KSA∪Jϕ′′KSA = Jϕ′∨ϕ′′KSA.

Now it is shown the preservation of satisfaction for closed simple formulas inorder to have results for local and global entailment.

Proposition 3.2.16. Given a ΣN -algebra A appropriate for a sequent calculus CN

for a non-normal modal logic and a closed simple formula ϕ, A ⊤ ≤ ϕ iff SA ϕ.

Proof. Assume A ⊤ ≤ ϕ. Then ⊤A ≤A JϕKA. Since A is appropriate for rules ⊤and transF, for every t ∈ T , t ≤A JϕKA. In particular for every u ∈ ΩA, u ≤A JϕKA.Therefore, u ∈ ΩA : u ≤A JϕKA = ΩA. So 〈JϕKA〉 = ΩA and JϕKSA = W by Lemma3.2.15, that is, SA ϕ.

For the other direction, assume SA ϕ, then u ∈ ΩA : u ≤A JϕKA = ΩA. SinceA is appropriate for rule Ω⊤, ⊤A ≤A JϕKA so A ⊤ ≤ ϕ.

Proposition 3.2.17. Given a ΣN -algebra A appropriate for CN , assignments α and

β over A such that β(y1) = u ∈ ΩA and a closed simple formula ϕ, Aαβ y1 ≤ ϕ

iff SA, u ϕ.

Proof. Observe that Aαβ y1 ≤ ϕ iff β(y1) ≤A JϕKA iff u ∈ 〈JϕKA〉 iff u ∈ JϕKSA iffSA, u ϕ.

Finally, the theorem about the equivalence between entailments.

Theorem 3.2.18. Given a class A of ΣN -algebras appropriate for a sequent calculus

CN for a non-normal modal logic and ψ1, . . . , ψk, ϕ closed simple formulas, let SA =SA : A ∈ A, then for d = l, g:

ψ1, . . . , ψk dA ϕ iff ψ1, . . . , ψk d

SAϕ.

Proof. The proof is immediate using Propositions 3.2.16 and 3.2.17.


The smallest non-normal modal system is denoted as E in [Che80] and the el-ements of class of all systems in which the semantics is given by minimal modelsis called classical systems [Seg71]. Two other classes of non-normal modal systemsare referred in the literature: the class of monotonic systems, consisted of those inwhich the rule

ϕ1 ⊃ ϕ2

2ϕ1 ⊃ 2ϕ2

is admissible (that is, every formula that can be derived using this rule is alreadyderivable in the system without it), and the class of regular systems, consisted ofsystems in which the rule

(ϕ1 ∧ ϕ2)⊃ ϕ3

(2ϕ1 ∧ 2ϕ2)⊃ 2ϕ3

is admissible. It is known that the following inclusions hold:

Normal systems ⊆ Regular systems ⊆ Monotonic systems ⊆ Classical systems.

In the following, labelled non-normal modal logic systems are defined. Theclassical systems are characterized by the smallest set of rules of the next definition.

Definition 3.2.19. A labelled non-normal modal logic system is a relational logic

system LN = 〈ΣN , RN ,A〉 where the signature ΣN is as Definition 3.2.2, the rulesare at least those described in Definition 3.2.3 and A is a class of ΣN -algebras as inDefinition 3.2.6.

The examples in the sequel characterize the other classes mentioned above aslogic systems labelled with truth values. First, the monotonic systems.

Example 3.2.20. Consider the non-normal modal logic system MT = 〈ΣMT ,RMT ,

AMT 〉 corresponding to monotonic systems, see [Che80], where:

• ΣMT is a non-normal modal signature as in Definition 3.2.2;

• RMT is the set of rules as in Definition 3.2.3 plus the rule M:

Ωτ1,#ξ1 ⊑ #ξ2, τ1 ⊑ F(#ξ1),Γ1 → Γ2, τ1 ⊑ F(#ξ2)

• AMT is a class of ΣMT -algebras as in Definition 3.2.6 full for 〈ΣMT ,RMT 〉 cor-responding to supplemented structures, i.e., to structures where the conditionif f(X ∩ Y ) then f(X) and f(Y ) holds.


In a system equipped with the rule M presented in the Example above, the rule

ϕ1 ⊃ ϕ2

2ϕ1 ⊃ 2ϕ2

is globally derived as it can be seen by the following derivation.

1. → ⊤ ≤ 2ϕ1 ⊃ 2ϕ2 RgenF 22. Ωy1,y1 ⊑ ⊤ → y1 ≤ 2ϕ1 ⊃ 2ϕ2 R⊃ 33. Ωy1,y1 ⊑ ⊤,y1 ≤ 2ϕ1 → y1 ≤ 2ϕ2 R,L2 44. Ωy1,y1 ⊑ ⊤,y1 ⊑ F(#ϕ1) → y1 ⊑ F(#ϕ2) cutT 5,65. Ωy1,y1 ⊑ ⊤,y1 ⊑ F(#ϕ1) → y1 ⊑ F(#ϕ2),#ϕ1 ⊑ #ϕ2 cutF 7,86. #ϕ1 ⊑ #ϕ2,Ωy1,y1 ⊑ ⊤,

y1 ⊑ F(#ϕ1) → y1 ⊑ F(#ϕ2) M7. Ωy1,y1 ⊑ ⊤,y1 ⊑ F(#ϕ1) → y1 ⊑ F(#ϕ2),#ϕ1 ⊑ #ϕ2,

#ϕ1 ≤ ϕ1 ⊃ ϕ2 transF 9,108. #ϕ1 ≤ ϕ1 ⊃ ϕ2,Ωy1,y1 ⊑ ⊤,

y1 ⊑ F(#ϕ1) → y1 ⊑ F(#ϕ2),#ϕ1 ⊑ #ϕ2 L⊃ 11,129. Ωy1,y1 ⊑ ⊤,y1 ⊑ F(#ϕ1) → y1 ⊑ F(#ϕ2),#ϕ1 ⊑ #ϕ2,

#ϕ1 ⊑ ⊤ ⊤10. Ωy1,y1 ⊑ ⊤,y1 ⊑ F(#ϕ1) → y1 ⊑ F(#ϕ2),#ϕ1 ⊑ #ϕ2,

⊤ ≤ ϕ1 ⊃ ϕ2 hyp11. Ωy1,y1 ⊑ ⊤,y1 ⊑ F(#ϕ1) → y1 ⊑ F(#ϕ2),#ϕ1 ⊑ #ϕ2,

#ϕ1 ≤ ϕ1 cutT 13,1412. #ϕ1 ≤ ϕ2,Ωy1,y1 ⊑ ⊤,

y1 ⊑ F(#ϕ1) → y1 ⊑ F(#ϕ2),#ϕ1 ⊑ #ϕ2 R# 1613. Ωy1,y1 ⊑ ⊤,y1 ⊑ F(#ϕ1) → y1 ⊑ F(#ϕ2),#ϕ1 ⊑ #ϕ2,

#ϕ1 ≤ ϕ1,#ϕ1 ⊑ #ϕ1 ref14. #ϕ1 ⊑ #ϕ1,Ωy1,y1 ⊑ ⊤, → y1 ⊑ F(#ϕ2),#ϕ1 ⊑ #ϕ2,

y1 ⊑ F(#ϕ1) #ϕ1 ≤ ϕ1 L# 1515. #ϕ1 ≤ ϕ1,Ωy1,y1 ⊑ ⊤, → y1 ⊑ F(#ϕ2),#ϕ1 ⊑ #ϕ2,

y1 ⊑ F(#ϕ1) #ϕ1 ≤ ϕ1 axiom16. #ϕ1 ≤ ϕ2,Ωy1,y1 ⊑ ⊤,

y1 ⊑ F(#ϕ1) → y1 ⊑ F(#ϕ2),#ϕ1 ≤ ϕ2 axiom

Example 3.2.21. Consider the non-normal modal logic system Reg = 〈ΣR,RR,AR〉

corresponding to regular systems, see [Che80], where:

• ΣR is a non-normal modal signature as in Definition 3.2.2;

• RR is the set of rules described in Definition 3.2.3 plus the rules M (presentedin Example 3.2.20) and C:

Ωτ1, τ1 ⊑ F(#ξ1), τ1 ⊑ F(#ξ2),Γ1 → Γ2, τ1 ⊑ F(#(ξ1 ∧ ξ2))


• AR is a class of ΣR-algebras as in Definition 3.2.6 full for 〈ΣR,RR〉 correspond-ing to the class of quasi-filters. A structure belongs to the class of quasi-filtersif it is supplemented and if it is closed under intersections.

In system equipped with rules M and C, presented in the Example 3.2.21, therule

(ϕ1 ∧ ϕ2)⊃ ϕ3

(2ϕ1 ∧ 2ϕ2)⊃ 2ϕ3

is globally derived as it can be seen by the following derivation.

1. → ⊤ ≤ (2ϕ1 ∧ 2ϕ2)⊃ 2ϕ3 RgenF 2

2. Ωy1,y1 ⊑ ⊤ → y1 ≤ (2ϕ1 ∧ 2ϕ2)⊃ 2ϕ3 R⊃ 3

3. Ωy1,y1 ⊑ ⊤,y1 ≤ 2ϕ1 ∧ 2ϕ2 → y1 ≤ 2ϕ3 L∧ + R2 4

4. Ωy1,y1 ⊑ ⊤,y1 ≤ 2ϕ1,y1 ≤ 2ϕ2 → y1 ⊑ F(#ϕ3) L2 + L2 5

5. Ωy1,y1 ⊑ ⊤,y1 ⊑ F(#ϕ1),y1 ⊑ F(#ϕ2) → y1 ⊑ F(#ϕ3) cutF 6,7

6. Ωy1,y1 ⊑ ⊤,y1 ⊑ F(#ϕ1), → y1 ⊑ F(#ϕ3),y1 ⊑ F(#ϕ2) #(ϕ1 ∧ ϕ2) ≤ (ϕ1 ∧ ϕ2)⊃ ϕ3 transF 8,9

7. #(ϕ1 ∧ ϕ2) ≤ (ϕ1 ∧ ϕ2)⊃ ϕ3,Ωy1,

y1 ⊑ ⊤,y1 ⊑ F(#ϕ1),y1 ⊑ F(#ϕ2) → y1 ⊑ F(#ϕ3) L⊃ 10,11

8. Ωy1,y1 ⊑ ⊤,y1 ⊑ F(#ϕ1), → y1 ⊑ F(#ϕ3),y1 ⊑ F(#ϕ2) #(ϕ1 ∧ ϕ2) ⊑ ⊤ ⊤

9. Ωy1,y1 ⊑ ⊤,y1 ⊑ F(#ϕ1), → y1 ⊑ F(#ϕ3),y1 ⊑ F(#ϕ2) ⊤ ≤ (ϕ1 ∧ ϕ2)⊃ ϕ3 weak’s + hyp

10. Ωy1,y1 ⊑ ⊤,y1 ⊑ F(#ϕ1), → y1 ⊑ F(#ϕ3),y1 ⊑ F(#ϕ2) #(ϕ1 ∧ ϕ2) ≤ (ϕ1 ∧ ϕ2) cutT 12,13

11. #(ϕ1 ∧ ϕ2) ≤ ϕ3,Ωy1,y1 ⊑ ⊤,

y1 ⊑ F(#ϕ1),y1 ⊑ F(#ϕ2) → y1 ⊑ F(#ϕ3) cutT 15,16

12. Ωy1,y1 ⊑ ⊤, y1 ⊑ F(#ϕ3),y1 ⊑ F(#ϕ1),y1 ⊑ F(#ϕ2) → #(ϕ1 ∧ ϕ2) ≤ (ϕ1 ∧ ϕ2),

#(ϕ1 ∧ ϕ2) ⊑ #(ϕ1 ∧ ϕ2) ref

13. #(ϕ1 ∧ ϕ2) ⊑ #(ϕ1 ∧ ϕ2),Ωy1, → y1 ⊑ F(#ϕ3),y1 ⊑ ⊤,y1 ⊑ F(#ϕ1),y1 ⊑ F(#ϕ2) #(ϕ1 ∧ ϕ2) ≤ (ϕ1 ∧ ϕ2) L# 14

14. #(ϕ1 ∧ ϕ2) ≤ (ϕ1 ∧ ϕ2),Ωy1, → y1 ⊑ F(#ϕ3),y1 ⊑ ⊤,y1 ⊑ F(#ϕ1),y1 ⊑ F(#ϕ2) #(ϕ1 ∧ ϕ2) ≤ (ϕ1 ∧ ϕ2) axiom

15. #(ϕ1 ∧ ϕ2) ≤ ϕ3,Ωy1,y1 ⊑ ⊤, → y1 ⊑ F(#ϕ3),y1 ⊑ F(#ϕ1),y1 ⊑ F(#ϕ2) y1 ⊑ F(#(ϕ1 ∧ ϕ2)) C

16. y1 ⊑ F(#(ϕ1 ∧ ϕ2)),Ωy1,y1 ⊑ ⊤,

#(ϕ1 ∧ ϕ2) ≤ ϕ3,y1 ⊑ F(#ϕ1),y1 ⊑ F(#ϕ2) → y1 ⊑ F(#ϕ3) cutT 17,18


17. y1 ⊑ F(#(ϕ1 ∧ ϕ2)),Ωy1,y1 ⊑ ⊤,

#(ϕ1 ∧ ϕ2) ≤ ϕ3,y1 ⊑ F(#ϕ1), → y1 ⊑ F(#ϕ3),y1 ⊑ F(#ϕ2) #(ϕ1 ∧ ϕ2) ⊑ #ϕ3 R# 19

18. #(ϕ1 ∧ ϕ2) ⊑ #ϕ3,Ωy1,

#(ϕ1 ∧ ϕ2) ≤ ϕ3,y1 ⊑ ⊤,

y1 ⊑ F(#(ϕ1 ∧ ϕ2)),y1 ⊑ F(#ϕ1),y1 ⊑ F(#ϕ2) → y1 ⊑ F(#ϕ3) M

19. y1 ⊑ F(#(ϕ1 ∧ ϕ2)),Ωy1,y1 ⊑ ⊤,

#(ϕ1 ∧ ϕ2) ≤ ϕ3, → y1 ⊑ F(#ϕ3),y1 ⊑ F(#ϕ1),y1 ⊑ F(#ϕ2) #(ϕ1 ∧ ϕ2) ≤ ϕ3 axiom

Finally, in the LSLTV approach an alternative definition of System K is presentedby enriching the systems described in Example 3.2.21 with a rule inferring that theapplication of f to ⊤ contains ⊤ itself, that is, making 2⊤ a theorem of the system.

Example 3.2.22. Consider the system RN = 〈ΣRN ,RRN ,ARN〉 where:

• ΣRN is a non-normal modal signature as in Definition 3.2.2;

• RRN is the set of rules as in Example 3.2.21 plus the rule N:

Γ1 → Γ2,⊤ ⊑ F(⊤)

• ARN is a class of ΣRN -algebras as in Definition 3.2.6 corresponded to the classof filters, full for 〈ΣMT ,RMT 〉. A structure belongs to the class of filters if isa quasi-filter with unit.

The system RN is actually equivalent to the smallest normal modal logic systemK.

Chellas [Che80] also mentions about the well-known axioms for the constructionof more systems like T, 4,D and gives the characterization of them in terms ofminimal models. Table 3.6 presents the corresponding rules.

Γ1 → Γ2,F(τ1) ⊑ τ1 TNN

Γ1 → Γ2,F(τ1) ⊑ F(F(τ1)) 4NN

Ωτ1, τ1 ⊑ F(#ξ1),Γ1 → Γ2, τ1 6⊑F(#¬ ξ1) DNN

Table 3.6: Rules for T, 4 and D axioms in non-normal systems


As mentioned after Example 3.1.9, deontic logics could be defined as non-normalsystems in order to avoid some theorems. In fact, deontic systems would be definedalternatively as a non-normal modal logic system of Definition 3.2.19 equipped withrules C, N and DNN. Notice that this system is not normal and not even monotonicsince the formula 2(ϕ1 ∧ ϕ2)⊃ 2ϕ1 ∧ 2ϕ2 is not derived from the rules.

Chapter 4

Fusion of Systems Labelled with

Truth Values

Introduced by Thomason [Tho84], fusion is a well known way to combine normalmodal logics. Basically it is defined simply as the union of the given modal axiomaticsystems sharing propositional connectives, so the resulting system has no axiom thatuses together the connectives that are not shared. Several results about transferenceof properties by fusion are known and can be found in [KW91, FS96, KW97, Wol98,Kur06].

The main aim of this chapter is to extend the concept of fusion to other kinds ofsystems besides normal modal logics. The chapter starts by describing an algebraicapproach to the definition of fusion of Logic Systems Labelled with Truth Valuesfollowing the original concept of the combination. Then the definition of fusion isgeneralized by defining it over the category of Logic Systems Labelled with TruthValues. To this purpose Section 4.2 starts by presenting logic system morphisms andsome of their properties. Next, in Subsection 4.2.2 the categorical presentation offusion is introduced as a colimit of two inclusions morphisms. This categorical defi-nition in fact, besides having the algebraic fusion as a particular case, encompasses amuch wider class of combinations besides fusions. Even for fusions, it allows the fu-sion of systems not previously considered. Finally in Section 4.3 preservation resultsare established, namely preservation of soundness by fusion as well as preservationof completeness under mild assumptions.

61

62 Chapter 4. Fusion of Systems Labelled with Truth Values

4.1 Algebraic Presentation of Fusion

In this section an algebraic account of fusion of (normal) modal logic systems basedin [CCG+08] is described. In this algebraic account, propositional connectives andoperators and their rules are shared between the modal logic systems but modalitiesare not, following the original concept of fusion. In Subsection 4.2.2, the fusion oflogic systems possibly sharing modalities is analysed.

Definition 4.1.1. Let L1 = 〈Σ1,R1,A1〉 and L2 = 〈Σ2,R2,A2〉 be two (normal)

modal logic systems. The fusion of L1 and L2 is the logic system 〈Σ,R,A〉 where:

• Σ is the signature 〈C,O,\, X, Y, Z〉 where C0 = t, f ∪ pi : i ∈ N; C1 =¬ ∪ 2′ ∪ 2′′; C2 = ∧,∨,⊃; Ck = ∅ for all k ≥ 3; O0 = ⊤,⊥;O1 = I ∪ N′ ∪ N′′; O2 = lb; Ok = ∅ for all k ≥ 3; \1 = Ω;\2 = ⊑; \k = ∅ for all k ≥ 3;

• R keeps all the structural rules, the order rules and the logical rules for ⊃, ¬,∨, ∧, lb and I. Furthermore, each rule in L1 (L2) that involves the connective2 or the operator N is in R by renaming the connective 2 to 2′ (2′′) and theoperator N to N′ (N′′);

• A is the class of Σ-algebras AA1,A2: A1 ∈ A1,A2 ∈ A2,A1|Σ

P+= A2|Σ

P+

where AA1,A2is such that AA1,A2

|ΣP+

= A1|ΣP+

, 2′AA1,A2

= 2A1,2′′

AA1,A2=

2A2,N′

AA1,A2= NA1

and N′′AA1,A2

= NA2, where ΣP+ is the signature with

the shared propositional connectives and operators.

Observe that L1 and L2 share all the connectives and operators except the modal-ities and the operator N. Moreover the rules of these systems are the same exceptthe rules for the modal constructors and their N operators.

4.2 Categorical Presentation of Fusion

In this section a categorical description of fusion is given by showing that it is acolimit in Log, the category of LSLTV presented in Subsection 4.2.1. In a generalway, fusion is obtained as a pushout as illustrated by the diagram in Figure 4.1where the morphisms are inclusions.

The section starts with LSLTV morphisms and some of their properties. Sub-section 4.2.2 presents the definition of the envisaged combination of systems.

4.2. Categorical Presentation of Fusion 63

Common Logic SystemG g

tt

w

**❯❯❯❯❯❯

❯❯❯❯❯❯

❯❯❯❯❯

Logic System Ax

**❱❱❱❱❱❱

❱❱❱❱❱❱

❱❱❱❱❱❱

Logic System BfF

tt

Fusion

Figure 4.1: General diagram of fusion

4.2.1 Category of logic systems

In order to define fusion as a categorical operation, logic system morphisms are firstintroduced. They are a pair of maps, one map between signatures and the other (acontravariant map) between classes of algebras. In this work we assume that theset of relations \ between terms is the same in all logical systems. The notion ofmorphism between signatures is the starting point of this subsection.

Definition 4.2.1. Given signatures Σ and Σ′ with \ = \′, a signature morphism

from Σ to Σ′ is a tuple h = 〈hC , hO, hX , hY , hZ〉 where:

• hC = hCn: Cn → C ′

nn∈N is a family of maps;

• hO = hOn: On → O′

nn∈N, is a family of maps preserving ⊤ and ⊥;

• hX : X → X ′ is a map;

• hY : Y → Y ′ is a map;

• hZ : Z → Z ′ is a map.

It is possible to define h the extension of a signature morphism h to formulas,terms, assertions, substitutions, sequents, provisos and rules as follows:

• For each ϕ ∈ F (Σ), h(ϕ) ∈ F (Σ′) is such that if ϕ is ξi, h(ϕ) = ξi; if ϕ is z,h(ϕ) = hZ(z); if ϕ is c(ϕ1, . . . , ϕk), h(ϕ) = hCk

(c)(h(ϕ1), . . . , h(ϕk)).

• For each θ ∈ T (Σ), h(θ) ∈ T (Σ′) is such that if θ is τi, h(θ) = τi; if θis x, h(θ) = hX(x); if θ is y, h(θ) = hY (y); if θ is o(θ1, . . . , θk), h(θ) =hOk

(o)(h(θ1), . . . , h(θk)); if θ is #ϕ, h(θ) = #h(ϕ).

• For each a ∈ A(Σ), h(a) ∈ A(Σ′) is such that if a is θ ≤ ϕ [θ ϕ], h(a) =

h(θ) ≤ h(ϕ) [h(a) = h(θ) h(ϕ)]; if a is (θ1, . . . , θk) [6(θ1, . . . , θk)], h(a) =

(h(θ1), . . . , h(θk)) [h(a) = 6(h(θ1), . . . , h(θk))].


• For each ρ ∈ Sbs(A(Σ)), h(ρ) ∈ Sbs(A(Σ′)) is the map h ρ.

• For each sequent s = 〈∆1,∆2〉 over A(Σ), h(s) is the sequent 〈h(∆1), h(∆2)〉where h(∆i) ∈ Bf (A(Σ

′) ∪ G) for i = 1, 2 is such that for each δ ∈ ∆i if

δ ∈ A(Σ) then h(δ) ∈ A(Σ′) is as defined above and if δ ∈ G then h(δ) = δ.

• For each proviso π over A(Σ), h(π) is the proviso over A(Σ′) such that h(π) :h(Sbs(A(Σ))) → 0, 1.

• For each rule r = 〈s1, . . . , sp, s, π〉 over A(Σ),

h(r) = 〈h(s1), . . . , h(sp), h(s), h(π)〉

is a rule over A(Σ′).

In the sequel, h will be denoted simply by h.

Signatures and signature morphisms constitute a category, named Sig, that hasall limits and colimits.

For the purpose of the definition of the map between algebras, the concept ofreduct of an algebra by a signature map is now introduced.

Definition 4.2.2. Given a Σ′-algebra A′ = 〈F, T, ·A′〉 and a signature morphism

h : Σ → Σ′, the reduct of A′ by h is a Σ-algebra A′|h = 〈F, T, ·A′|h〉 where:

• for each connective c ∈ Σ, cA′|h = h(c)A′;

• for each operator o ∈ Σ, oA′|h = h(o)A′ ;

• #A′|h = #A′ , ≤A′|h=≤A′ and for each ρ ∈ Σ, A′|h = A′.

Definition 4.2.3. A logic system morphism m : L → L′ where L = 〈Σ,R,A〉 and

L′ = 〈Σ′,R′,A′〉 is a pair m = 〈h, a〉 where h : Σ → Σ′ is a signature morphismsuch that for each r = 〈s1, ..., sp, s, π〉 ∈ R there is a derivation in R′

h(s1), . . . , h(sp) ⊢C′ h(s) h(π),

and a : A′ → A is a map such that a(A′) = A′|h.

Concerning the definition of logic system morphisms, the map between algebrasis contravariant in order to guarantee the preservation of entailment: it is essentialthat each model A′ in L′ have an image by the morphism in L.


Proposition 4.2.4. Logic systems and logic system morphisms constitute a cate-

gory named Log.

Some important preservation properties hold in the context of Log as describedbelow. The first one is preservation of derivation.

Proposition 4.2.5. Given m : L → L′ if S ⊢C s′ π then h(S) ⊢C′ h(s′) h(π).

Proof. The proof follows by induction on the size of a derivation d = (d1, π1) . . .(dn, πn) for S ⊢C s

′ π.

Base: d = (d1, π1). So d1 = s′. Then:

• either d1 ∈ S and π1 = up and so h(d1) ∈ h(S) and h(π1) = up;

• or there is an assertion that occurs in both sides of d1 and π1 = up. Then h(d1)also has an assertion that occurs in both sides of the sequent and h(π1) = up;

• or there are r ∈ R and ρ ∈ Sbs(A(Σ)) such that rρ =

d1π1.

By the definition of m there is a derivation of h(r) in C′. Using the fact thath(rρ) = h(r)h(ρ) then ⊢C′ h(d1) h(π1).

Step: d = (d1, π1)(d2, π2) . . . (dn, πn) is a derivation in L. Then:

• either d1 ∈ S and π1 = up and so h(d1) ∈ h(S) and h(π1) = up;

• or there is an assertion that occurs in both sides of d1 and π1 = up. Then h(d1)also has an assertion that occurs in both sides of the sequent and h(π1) = up;

• or there are r = 〈s1, . . . , sq, s, π′〉 ∈ R, ρ ∈ Sbs(A(Σ)), q ∈ N and i1, ..., iq ∈2, 3, ..., n such that rρ =

di1 . . . diq

d1π

and π1 = π ∩ πi1 ∩ . . . ∩ πiq . By the definition of m there is a derivation dr =(dr1, πr1) . . . (drm , πrm) showing h(s1), . . . , h(sq) ⊢C′ h(s) h(π′). Observe thatfor j = 1, . . . , q, h(ρ)(h(sj)) = h(sjρ) = h(dij ) and h(ρ)(h(s)) = h(sρ) = h(d1).By the induction hypothesis, (d2, π2) . . . (dn, πn) has an “equivalent” derivationthat will be denoted as dStep. So the final derivation in C′ is h(ρ)(dr)dStepwhere each justification ‘hyp’ in dr is replaced by the correspondent referenceof h(dij ) in dStep.


Given a morphism between logic systems and assignments over the destinationalgebra, it is possible to consider corresponding assignments over the source algebrain order to establish interesting preservation properties.

Given a logic system morphism m = 〈h, a〉 and an algebra A′ ∈ A′ it is denotedby α and β the assignments over A = a(A′) induced by the assignments α′ and β ′

over A′ such that the diagrams in Figure 4.2 commute.

Yβ

//

hY

T Xα //

hX

T Zα //

hZ

F

Y ′β′

>>⑦⑦⑦⑦⑦⑦⑦X ′

α′

>>⑥⑥⑥⑥⑥⑥⑥Z ′

α′

>>⑥⑥⑥⑥⑥⑥⑥

Figure 4.2: Commutative diagrams for induced assignments

The following lemma establishes the preservation of the denotation of a formulaby a logic system morphism.

Lemma 4.2.6. Let ϕ be a formula, 〈h, a〉 : L → L′ a logic system morphism,

A′ ∈ A′, α′ an assignment over A′, and α be α′ h. Then JϕKa(A′)α = Jh(ϕ)KA′α′ .

Proof. The proof follows by induction on the structure of ϕ.

• ϕ is z. Then:

JzKa(A′)α = α(z) = α′(hZ(z)) = Jh(z)KA′α′ .

• ϕ is c(ϕ1, . . . , ϕk). Then:

Jc(ϕ1, . . . , ϕk)Ka(A′)α = ca(A′)(Jϕ1Ka(A′)α, . . . , JϕkKa(A′)α) = h(c)A′(Jh(ϕ1)KA′α′ ,

. . . , Jh(ϕk)KA′α′) = Jh(c(ϕ1, . . . , ϕk))KA′α′ .

The next two lemmas show the preservation of the denotations of the relationsused in assertions. They are used in the proof of Proposition 4.2.9.

Lemma 4.2.7. Given A′ ∈ A′ and assignments α′ and β ′ over A′,

〈JθKa(A′)αβ , JϕKa(A′)α〉 ∈≤a(A′) iff 〈Jh(θ)KA′α′β′, Jh(ϕ)KA′α′〉 ∈≤A′ ,

where α and β are as indicated before.


Proof. The proof follows by structural induction on θ and ϕ.

• ϕ is z ∈ Z.

- θ is x ∈ X. Then:

〈JxKa(A′)αβ , JzKa(A′)α〉 ∈ ≤a(A′)

iff 〈α(x), α(z)〉 ∈ ≤a(A′)

iff 〈α′(hX(x)), α′(hZ(z))〉 ∈ ≤A′

iff 〈Jh(x)KA′α′β′, Jh(z)KA′α′〉 ∈ ≤A′ .

- θ is y ∈ Y . Then:

〈JyKa(A′)αβ , JzKa(A′)α〉 ∈ ≤a(A′)

iff 〈β(y), α(z)〉 ∈ ≤a(A′)

iff 〈β ′(hY (y)), α′(hZ(z))〉 ∈ ≤A′

iff 〈Jh(y)KA′α′β′ , Jh(z)KA′α′〉 ∈ ≤A′ .

- θ is o(θ1, . . . , θk) with o ∈ Ok and θ1, . . . , θk ∈ T (Σ). Then:

〈Jo(θ1, . . . , θk)Ka(A′)αβ , JzKa(A′)α〉 ∈ ≤a(A′)

iff 〈oa(A′)(Jθ1Ka(A′)αβ , . . . , JθkKa(A′)αβ), α(z)〉 ∈ ≤a(A′)

iff 〈h(o)A′(Jh(θ1)KA′α′β′, . . . , Jh(θk)KA′α′β′), α′(hz(z))〉 ∈ ≤A′

iff 〈Jh(o)(h(θ1), . . . , h(θk))KA′α′β′, Jh(z)KA′α′〉 ∈ ≤A′

iff 〈Jh(o(θ1, . . . , θk))KA′α′β′, Jh(z)KA′α′〉 ∈ ≤A′ .

- θ is #ψ with ψ being z ∈ Z. Then:

〈J#zKa(A′)αβ , JzKa(A′)α〉 ∈ ≤a(A′)

iff 〈#a(A′)(JzKa(A′)αβ), α(z)〉 ∈ ≤a(A′)

iff 〈#a(A′)(α(z)), α(z)〉 ∈ ≤a(A′)

iff 〈#A′(α′(hZ(z))), α′(hZ(z))〉 ∈ ≤A′

iff 〈#A′(Jh(z)KA′α′), Jh(z)KA′α′〉 ∈ ≤A′

iff 〈J#h(z)KA′α′ , Jh(z)KA′α′〉 ∈ ≤A′

iff 〈Jh(#z)KA′α′ , Jh(z)KA′α′〉 ∈ ≤A′ .

- θ is #ψ with ψ being c(ψ1, . . . , ψk) where c ∈ Ck and ψ1, . . . , ψk ∈ F (Σ).Then:


〈J#c(ψ1, . . . , ψk)Ka(A′)αβ , JzKa(A′)α〉 ∈ ≤a(A′)

iff 〈#a(A′)(Jc(ψ1, . . . , ψk)Ka(A′)α), α(z)〉 ∈ ≤a(A′)

iff 〈#a(A′)(ca(A′)(Jψ1Ka(A′)α, . . . , JψkKa(A′)α)), α(z)〉 ∈ ≤a(A′)

iff 〈#A′(h(c)A′(Jh(ψ1)KA′α′ , . . . , Jh(ψk)KA′α′)), α′(hZ(z))〉 ∈ ≤A′

iff 〈#A′(Jh(c)(h(ψ1), . . . , h(ψk))KA′α′), Jh(z)KA′α′〉 ∈ ≤A′

iff 〈J#h(c(ψ1, . . . , ψk))KA′α′ , Jh(z)KA′α′〉 ∈ ≤A′

iff 〈Jh(#c(ψ1, . . . , ψk))KA′α′ , Jh(z)KA′α′〉 ∈ ≤A′ .

• ϕ is c(ϕ1, . . . , ϕk) with c ∈ Ck and ϕ1, . . . , ϕk ∈ F (Σ).

It is showed only the case where θ is x ∈ X, all other cases are similar to thecases presented above.

〈JxKa(A′)αβ , Jc(ϕ1, . . . , ϕk)Ka(A′)α〉 ∈ ≤a(A′)

iff 〈α(x), ca(A′)(Jϕ1Ka(A′)α, . . . , JϕkKa(A′)α)〉 ∈ ≤a(A′)

iff 〈α′(hX(x)), h(c)A′(Jh(ϕ1)KA′α′, . . . , Jh(ϕk)KA′α′)〉 ∈ ≤A′

iff 〈Jh(x)KA′α′β′, Jh(c)(h(ϕ1), . . . , h(ϕk))KA′α′〉 ∈ ≤A′

iff 〈Jh(x)KA′α′β′, Jh(c(ϕ1, . . . , ϕk))KA′α′〉 ∈ ≤A′ .

The lemma that follows establishes the preservation of satisfaction for the rela-tions ∈ \.

Lemma 4.2.8. Given A′ ∈ A′ and assignments α′ and β ′ over A′,

〈Jθ1Ka(A′)αβ , . . . , JθkKa(A′)αβ〉 ∈ a(A′) iff 〈Jh(θ1)KA′α′β′ , . . . , Jh(θk)KA′α′β′〉 ∈ A′ ,

where α and β are as in Figure 4.2.

Proof. The proof follows by structural induction on each θi for i = 1, . . . , k. Withoutloss of generality, only the case where ∈ \1 is shown. The proof for \n with n ≥ 2is similar.

• θ1 is x ∈ X.

JxKa(A′)αβ ∈ a(A′) iff α(x) ∈ a(A′) iff α′(hX(x)) ∈ A′ iff Jh(x)KA′α′β′ ∈ A′ .

• θ1 is y ∈ Y .

JyKa(A′)αβ ∈ a(A′) iff β(y) ∈ a(A′) iff β ′(hY (y)) ∈ A′ iff Jh(y)KA′α′β′ ∈ A′ .


• θ1 is o(θ′1, . . . , θ′l) with o ∈ Ol and θ′1, . . . , θl ∈ T (Σ).

Jo(θ′1, . . . , θ′l)Ka(A′αβ) ∈ a(A′)

iff oa(A′)(Jθ′1Ka(A′)αβ , . . . , Jθ

′lKa(A′)αβ) ∈ a(A′)

iff h(o)A′(Jh(θ′1)KA′α′β′, . . . , Jh(θ′l)KA′α′β′) ∈ A′

iff Jh(o)(h(θ′1), . . . , h(θ′l))KA′α′β′ ∈ A′

iff Jh(o(θ′1, . . . , θ′l))KA′α′β′ ∈ A′ .

• θ1 is #ϕ where ϕ is z ∈ Z.

J#zKa(A′)αβ ∈ a(A′)

iff #a(A′)JzKa(A′)αβ ∈ a(A′)

iff #a(A′)α(z) ∈ a(A′)

iff #A′α′(hZ(z)) ∈ A′

iff #A′Jh(z)KA′α′ ∈ A′

iff J#h(z)KA′α′β′ ∈ A′

iff Jh(#z)KA′α′β′ ∈ A′ .

• θ1 is #ϕ where ϕ is c(ϕ1, . . . , ϕl), c ∈ Cl and ϕ1, . . . , ϕl ∈ F (Σ).

J#c(ϕ1, . . . , ϕl)Ka(A′)αβ ∈ a(A′)

iff #a(A′)Jc(ϕ1, . . . , ϕl)Ka(A′)α ∈ a(A′)

iff #a(A′)ca(A′)(Jϕ1Ka(A′)α, . . . , JϕlKa(A′)α) ∈ a(A′)

iff #A′h(c)A′(Jh(ϕ1)KA′α′ , . . . , Jh(ϕl)KA′α) ∈ A′

iff #A′Jh(c)(h(ϕ1), . . . , h(ϕl))KA′α′ ∈ A′

iff J#h(c(ϕ1, . . . , ϕl))KA′α′β′ ∈ A′

iff Jh(#c(ϕ1, . . . , ϕl))KA′α′β′ ∈ A′ .

The next proposition shows that logic system morphisms preserve satisfaction ofassertions.

Proposition 4.2.9. Given m : L → L′, A′ ∈ A′, and α′ and β ′ assignments over

A′,a(A′)αβ L γ iff A′α′β ′ L′ h(γ)

where α and β are as in Figure 4.2 and γ is either an assertion or a sequent.

Proof. Suppose γ is an assertion, the proof follows by case analysis.

(i). a(A′)αβ θ ≤ ϕ iff 〈JθKa(A′)αβ , JϕKa(A′)αβ〉 ∈ ≤a(A′) iff (by Lemma 4.2.7)〈Jh(θ)KA′α′β′ , Jh(ϕ)KA′α′β′〉 ∈ ≤A′ iff A′α′β ′ h(θ) ≤ h(ϕ) iff A′α′β ′ h(θ ≤ ϕ).


(ii). a(A′)αβ θ ϕ iff 〈JθKa(A′)αβ , JϕKa(A′)αβ〉 6∈ ≤a(A′) iff (by Lemma 4.2.7)〈Jh(θ)KA′α′β′ , Jh(ϕ)KA′α′β′〉 6∈≤A′ iff A′α′β ′ h(θ)h(ϕ) iff A′α′β ′ h(θ ϕ).

(iii). a(A′)αβ (θ1, . . . , θk) iff 〈Jθ1Ka(A′)αβ , . . . , JθkKa(A′)αβ〉 ∈ a(A′) iff (by Lemma4.2.8) 〈Jh(θ1)KA′α′β′ , . . . , Jh(θk)KA′α′β′〉 ∈ A′ iff A′α′β ′ (h(θ1), . . . , h(θk)) iffA′α′β ′ h((θ1, . . . , θk)).

(iv). a(A′)αβ 6(θ1, . . . , θk) iff 〈Jθ1Ka(A′)αβ , . . . , JθkKa(A′)αβ〉 6∈ a(A′) iff (by Lemma4.2.8) 〈Jh(θ1)KA′α′β′ , . . . , Jh(θk)KA′α′β′〉 6∈ A′ iff A′α′β ′ 6(h(θ1), . . . , h(θk)) iffA′α′β ′ h( 6(θ1, . . . , θk)).

Suppose now that γ is a sequent ∆′ → ∆′′. Then a(A′)αβ ∆′ → ∆′′ iff a(A′)αβ δ

for some δ ∈ ∆′′ ∪∆′ iff by (i)–(iv) above A′α′β ′ h(δ) with h(δ) ∈ h(∆′′ ∪∆′) iffA′α′β ′ h(∆′ → ∆′′).

The following result states that soundness of rules is preserved by logic systemsmorphism.

Proposition 4.2.10. Given a sound logic system L and a logic system morphism

m : L → L′ where m = 〈mh, ma〉 with mh injective, if r = 〈s1, . . . , sn, s, π〉 ∈ Rthen mh(s1), . . . , mh(sn) L′ mh(s) mh(π).

Proof. Let A′ ∈ A′, α′ an unbound assignment over A′ and ρ a ground substitutionover mh(A(Σ)) such that mh(π)(ρ) = 1. Suppose

A′α′ mh(si)ρ for all i = 1, . . . , n.

Since mh is injective, m−1h is a substitution over A(Σ) such that π(m−1

h (ρ)) = 1. So,by Proposition 4.2.9,

a(A′)α sim−1h (ρ) for all i = 1, . . . , n

with α = α′ mh. But r = 〈s1, . . . , sn, s, π〉 ∈ R and since L is sound

s1, . . . , sn L s π.

By definition of entailment, if Aα∗ siσ for i = 1, . . . , n, with σ any groundsubstitution over A(Σ) such that π(σ) = 1 and any unbound assignment α∗ over Athen Aα∗ sσ. So in particular for a(A′), α and m−1

h (ρ),

a(A′)α s m−1h (ρ).

Again by Proposition 4.2.9 A′α′ mh(s)ρ. Thus

mh(s1), . . . , mh(sn) L′ mh(s) mh(π).


4.2.2 Categorical fusion

In this subsection, the concept of fusion is generalized from (normal) modal logicto relational LSLTV as introduced in Definition 2.5.3. A categorical approach isadopted to attain this goal.

The categorical fusion is given by the next definition: it is the pushout in Log ofthe diagram of Figure 4.3 where LC is a subsystem of L1 and L2 in what concernsthe signature and the set of rules.

Definition 4.2.11. Let L1 = 〈Σ1,R1,A1〉 and L2 = 〈Σ2,R2,A2〉 be two relational

LSLTV. The fusion by sharing LC of L1 and L2 is the object of the pushout in Logof the diagram of Figure 4.3, where LC = 〈ΣC ,RC ,AC〉 is a relational LSLTV suchthat:

• ΣC ⊆ Σi for i = 1, 2,

• RC ⊆ Ri for i = 1, 2,

• AC is the class of all ΣC-algebras,

and the morphisms inci = 〈(inci)h, (inci)a〉 are inclusions where (inci)a(Ai) =Ai|(inci)h for i = 1, 2 .

LCN n

inc1

⑤⑤⑤⑤⑤⑤⑤⑤

p

inc2

!!

L1 L2

Figure 4.3: Diagram for fusion

It is worthwhile to observe that the algebraic fusion of a given pair L1 and L2

of (normal) modal logics, mentioned in Section 4.1, coincides with the object of thepushout described in Definition 4.2.11 when LC is such that:

• ΣC is the signature ΣP together with I ∈ O1 and lb ∈ O2;

• RC is the set constituted by the rules I, ΩI, lb1, lb2 and the rules in Tables2.1, 2.2 and 2.3;

• AC is the class of all ΣC-algebras.


It is now provided a more detailed characterization of the colimit of the diagramof Figure 4.3, that is, a description of the components of the object and of themorphisms that constitute the pushout. It should be stressed that the object of thepushout is the fusion of the logic systems.

Proposition 4.2.12. The triple 〈L, m1, m2〉 where L = 〈Σ,R,A〉 and mi = 〈(mi)h,

(mi)a〉 : Li → L for i = 1, 2, are such that

• 〈Σ, (m1)h, (m2)h〉 is the pushout in Sig of the morphisms (inc1)h and (inc2)hintroduced in Definition 4.2.11;

• R = (m1)h (inc1)h(RC) ∪ (m1)h(R1\RC) ∪ (m2)h(R2\RC);

• A = A1_A2 = 〈F, T, ·A1_A2〉 : (inc1)a(A1) = (inc2)a(A2) = 〈F, T, ·AC

〉 and(m1)h(c)A1_A2

= cA1for c ∈ C1, (m2)h(c)A1_A2

= cA2for c ∈ C2, (m1)h(o)A1_A2

= oA1for o ∈ O1, (m2)h(o)A1_A2

= oA2for o ∈ O2,#A1_A2

= #A1,≤A1_A2

=≤A1,

A1_A2= A1

where C i and Oi are, respectively, the set of connectives andthe set of operators of Σi for i = 1, 2;

• (mi)a(A1_A2) = Ai for i = 1, 2

is the pushout of the diagram of Figure 4.3. So, L is the fusion of L1 and L2.

Proof. Suppose there is 〈L′, l1, l2〉 such that l1 : L1 → L′, l2 : L2 → L′ and l1 inc1 = l2 inc2. It has to be shown that there is a unique morphism u : L → L′

such that u m1 = l1 and u m2 = l2. Let ua(A′) = ((l1)a(A′))_((l2)a(A′)).Unicity of u: Suppose there is n : L → L′ such that n m1 = l1 and n m2 = l2.Then nh = uh since 〈Σ, (m1)h, (m2)h〉 is the pushout in Sig. Let A′ ∈ A′ andna(A′) = Ax_Ay. So, (m1)a na(A′) = (m1)a(Ax_Ay) = Ax = (l1)a(A′) and(m2)a na(A′) = (m2)a(Ax_Ay) = Ay = (l2)a(A′). Thus, na = ua. Therefore,n = u.

An important point about the categorical account of fusion is that it is possibleto change the shared connectives and operators by simply changing the “commonlogic system”, LC , of Definition 4.2.11. This contrasts with the usual definition offusion, where the shared connectives are fixed, being the propositional connectives.On the other hand, the definition of categorical fusion assumes at least that LC isthe common core of L1 and L2. So it is possible to consider extensions of that logic.For instance, if LC is system K then the fusion will shared the modality as well.

4.3. Soundness and Completeness Preservation in Fusion 73

4.3 Soundness and Completeness Preservation in Fu-

sion

This section shows that soundness and completeness are preserved by the categor-ical fusion of Definition 4.2.11. While preservation of soundness is immediate, forpreservation of completeness it has to be assumed that both component systemshave rules endowed with persistent provisos.

Theorem 4.3.1. The fusion of sound relational LSLTV is sound.

Proof. Consider the logic systems L1 and L2 and their fusion L using a logic systemLC for specifying the shared connectives as described in Definition 4.2.11.

The proof follows by applying Proposition 4.2.10 to each rule r= 〈s1, . . . , sn,s, π〉 of R. Several cases have to be considered:

• r ∈ (m1)h(R1\RC), that is, it is a rule from L1. As L1 is sound, by Proposition4.2.10 (m1)h(s1), . . . , (m1)h(sn) L (m1)h(s) (m1)h(π). The reasoning isanalogous if r ∈ (m2)h(R2\RC);

• r ∈ RC . Then r = (m1)h(r′) = (m2)h(r

′′) for some r′ ∈ R1 and r′′ ∈ R2, so itis the same case as before.

Therefore, each rule in R is entailed in L. So the class of algebras A is appropriatefor 〈Σ,R〉.

Using the result for completeness given in Theorem 2.5.12, it is possible to showthe preservation of completeness by the fusion under mild conditions.

Theorem 4.3.2. The fusion of two full relational LSLTV with rules endowed with

persistent provisos is complete.

Proof. From Theorem 2.5.12 it can be concluded that if a logic system is full withrules endowed with persistent provisos, then it is complete.

Let Li = 〈Σi,Ri,Ai〉 with i = 1, 2 be the component logic systems and L =〈Σ,R,A〉 their fusion.

As the rules of R come only from R1 or R2 and both have only rules endowedwith persistent provisos, so it will have R.

Taking an appropriate Σ-algebra A for the rules in R it must be shown thatA ∈ A.


Observe that A|Σ1is appropriate for the calculus 〈Σ1,R1〉 since all rules r =

〈s1,. . . ,sp, s, π〉 that belongs to R1 are also in R and since if s1, . . . , sp A s π

then s1, . . . , sp A|Σ1s π (notice that s1, . . . , sp, s and π are over A(Σ1)). So

A|Σ1∈ A1. The same holds for A|Σ2

. So A = A|Σ1_A|Σ2

∈ A.

Chapter 5

Combined Logic Systems

This chapter is dedicated to illustrate the results established in the preceding chap-ter. Several combinations of LSLTV are investigated and preservation of soundnessand completeness are analysed. A special emphasis is put in the analysis of the com-pleteness of the fusion of the non-normal partition logic (introduced in [FF05]) withitself. In [FF05] this specific combination was the counter-example to the strategyof defining fusion of non-normal modal logics as successive modalizations.

5.1 Systems Obtained by Fusion

This section starts by illustrating the algebraic account of fusion. It is described thecombination of the standard deontic logic system introduced in Example 3.1.9 andthe Löb provability logic introduced in Example 3.1.10.

Example 5.1.1. The fusion of the logic system SDL presented in Example 3.1.9

and the logic system GL presented in Example 3.1.10 is such that:

• the signature is Σ as described in Definition 4.1.1, where, for simplicity, 2D

is used instead of 2′ (the modality of SDL) and 2GL for 2

′′ (the modality ofGL). Note that expressions like 2D2GLϕ have the intended meaning that itis obligatory that ϕ is provable in a certain theory T;

• the set of rules is as described in Definition 4.1.1 (actually, it is RD ∪ RGL

with the appropriate substitutions of 2 and N);

• the class A is constituted by the algebras induced by generalized Kripke struc-tures (see Definition 3.1.5) with two relations such that one accessibility rela-

75

76 Chapter 5. Combined Logic Systems

tion is right-unbounded and the other accessibility relation is transitive andright linear.

An example of a deduction in the context of the sequent calculus resulting fromthe fusion of SDL and GL is now presented.

Example 5.1.2. It is now presented a deduction for

2D(2GL(ξ1 ⊃ ξ2)⊃ (2GLξ1 ⊃ 2GLξ2))

in the context of the logic system resulting of the fusion of SDL and GL, as describedin Example 5.1.1:

1. → ⊤ ≤ 2D(2GL(ξ1 ⊃ ξ2)⊃ (2GLξ1 ⊃ 2GLξ2)) R2D 2

2. → ND(⊤) ≤ 2GL(ξ1 ⊃ ξ2)⊃ (2GLξ1 ⊃ 2GLξ2)) transF3,4

3. → ND(⊤) ⊑ ⊤ ⊤

4. → ⊤ ≤ 2GL(ξ1 ⊃ ξ2)⊃ (2GLξ1 ⊃ 2GLξ2)) RgenF 5

5. Ωy1,y1 ⊑ ⊤ → y1 ≤ 2GL(ξ1 ⊃ ξ2)⊃ (2GLξ1 ⊃ 2GL1ξ2) R⊃ 6

6. Ωy1,y1 ⊑ ⊤,

y1 ≤ 2GL(ξ1 ⊃ ξ2) → y1 ≤ 2GLξ1 ⊃ 2GLξ2 R⊃ 7

7. Ωy1,y1 ⊑ ⊤,y1 ≤ 2GLξ1y1 ≤ 2GL(ξ1 ⊃ ξ2) → y1 ≤ 2GLξ2 R2GL 8

8. Ωy1,y1 ⊑ ⊤,y1 ≤ 2GLξ1y1 ≤ 2GL(ξ1 ⊃ ξ2) → NGL(y1) ≤ ξ2 RgenF 9

9. Ωy1,y1 ⊑ ⊤,y1 ≤ 2GLξ1y1 ≤ 2GL(ξ1 ⊃ ξ2) → y2 ≤ ξ2 L2GL 10

Ωy2,y2 ⊑ NGL(y1)

10. Ωy1,y1 ⊑ ⊤,NGL(y1) ≤ ξ1y1 ≤ 2GL(ξ1 ⊃ ξ2) → y2 ≤ ξ2 LgenF

Ωy2,y2 ⊑ NGL(y1) 11,12,13

11. Ωy1,y1 ⊑ ⊤,y2 ≤ ξ1y1 ≤ 2GL(ξ1 ⊃ ξ2) → y2 ≤ ξ2 L2GL14

Ωy2,y2 ⊑ NGL(y1),Ωy2

12. Ωy1,y1 ⊑ ⊤y1 ≤ 2GL(ξ1 ⊃ ξ2) → y2 ≤ ξ2,y2 ⊑ NGL(y1) ax

Ωy2,y2 ⊑ NGL(y1)

13. Ωy1,y1 ⊑ ⊤,NGL(y1) ≤ ξ1y1 ≤ K1(ξ1 ⊃ ξ2) → y2 ≤ ξ2,Ωy2 ax

Ωy2,y2 ⊑ NGL(y1)

14. Ωy1,y1 ⊑ ⊤,y2 ≤ ξ1NGL(y1) ≤ ξ1 ⊃ ξ2 → y2 ≤ ξ2 LgenF

Ωy2,y2 ⊑ NGL(y1),Ωy2 15,16,17

15. Ωy1,y1 ⊑ ⊤,y2 ≤ ξ1NGL(y1) ≤ ξ1 ⊃ ξ2 → y2 ≤ ξ2,Ωy2 ax


5.1. Systems Obtained by Fusion 77

16. Ωy1,y1 ⊑ ⊤,y2 ≤ ξ1Ωy2,y2 ⊑ NGL(y1) → y2 ≤ ξ2,y2 ⊑ NGL(y1) ax

Ωy2,Ωy2

17. Ωy1,y1 ⊑ ⊤,y2 ≤ ξ1Ωy2,y2 ≤ ξ1 ⊃ ξ2 → y2 ≤ ξ2 L⊃18,19


18. Ωy1,y1 ⊑ ⊤,y2 ≤ ξ1Ωy2,Ωy2 → y2 ≤ ξ2,y2 ≤ ξ1 ax

Ωy2,y2 ⊑ NGL(y1)

19. Ωy1,y1 ⊑ ⊤,y2 ≤ ξ1Ωy2,y2 ≤ ξ2 → y2 ≤ ξ2 ax


The first example of categorical fusion, Example 5.1.3 illustrates the combinationof two normal modal systems: one is the standard deontic logic and the other is usedfor knowledge logic. This system reasons about knowledge and obligation and maybe used, for instance, in multi-agent systems in computer science since it allows totalk about obligatory behaviours and knowledge of the agents.

Example 5.1.3. The fusion of the standard deontic modal system SDL of Example

3.1.9 with the knowledge modal system K of Example 3.1.11 sharing LC where:

• ΣC coincides with ΣP where ΣP is the signature of propositional logic;

• RC is the set with the structural rules of Table 2.1, the order rules of Table2.2 and the rules for the classical connectives presented in Table 2.3;

• AC is the class of all ΣC-algebras;

is obtained by taking the morphisms inc1 : LC → SDL and inc2 : LC → K, whereinc1 = 〈(inc1)h, (inc1)a〉 and inc2 = 〈(inc2)h, (inc2)a〉 are such that:

• (inc1)h : ΣC → ΣD and (inc2)h : ΣC → ΣK are the inclusions of ΣC in ΣD andΣK respectively,

• (inc1)a(AD) = AD|(inc1)h and (inc2)a(AK) = AK |(inc2)h for AD ∈ AD andAK ∈ AK (note that both reducts are ΣC-algebras and so, are in AC).

The object of the pushout of 〈inc1, inc2〉 is the fusion of SDL and K and, up toisomorphism, is such that:


• the signature is ΣSDK that is constituted by ΣC where C1 is enriched with theconnectives 2D, K1, . . . , Kn and O1 is enriched with the operators ND,N1, . . . ,

Nn;

• the set of rules is RSDK = RC∪LNDΩ,RNDΩ,L2D,R2D,D∪LNiΩ,RNiΩ,LKi, RKi,Ti, 4i, 5i : i = 1, . . . , n;

• the class A of appropriate algebras for 〈ΣSDK,RSDK〉 such that it includes the

class of algebras induced by general Kripke structures (see Definition 3.1.5)equipped with n accessibility relations that are equivalence relations togetherwith a relation that is right-unbounded.

Note that, in the context of the logic system resulting from the fusion describedin Example 5.1.3, formulas as 2DK1ϕ mean that it is obligatory that agent 1 knowsϕ.

However, as mentioned in the end of Chapter 3, deontic logic should be presentedas non-normal modal system. The next example defines a deontic logic system asa non-normal modal logic. In order to differentiate the deontic modality from theprevious example with the non-normal deontic modality now presented the latterone will be denoted by ©.

Example 5.1.4. Let DN = 〈ΣN ,RDN ,ADN〉 be a non-normal deontic modal logic

system such that:

• ΣN is as in Definition 3.2.2 where 2 is replaced by ©,

• RDN is the union of the set RN of Definition 3.2.3 and the set C,N,DNN ofrules presented in Section 3.2,

• ADN is the class of all ΣN -algebras appropriate for 〈ΣN ,RDN〉 which in-cludes the ΣN -algebras induced by general structures (see Definition 3.2.6)〈W, f,A, V 〉 such that, for every w ∈ W and X, Y ∈ A, all the followingconditions hold:

(c) if w ∈ f(X) and w ∈ f(Y ) then w ∈ f(X ∩ Y )

(d) if w ∈ f(X) then w 6∈ f(W \X)

(n) w ∈ f(W ).

Proposition 5.1.5. The ΣN -algebras induced by general structures described in

Example 5.1.4 are in ADN .


Proof. In Theorem 3.2.11 it is established that the induced algebras from generalstructures for non-normal modal logics are appropriate. It remains to show thateach additional rule (C, N and DNN) has its corresponding property ((c), (n) and(d)) in the general structure.

(rule N) It is needed to show that

for every w ∈ W,w ∈ f(W )

iff

for every Γ1,Γ2, ρ, α and β,ASαβ Γ1ρ→ Γ2ρ,⊤ρ ⊑ F(⊤ρ).

(⇒) Assume that, for every w ∈ W , w ∈ f(W ), that is, f(W ) = W . It is sufficientto show that ASαβ → ⊤ρ ⊑ F(⊤ρ) which is the case since J⊤ρKASαβ = W andFAS

= λx.f(x), using the hypothesis, J⊤ρKASαβ = W ⊆ f(W ) = JF(⊤ρ)KASαβ.

(⇐) Assume that, for every Γ1,Γ2, ρ, α and β, ASαβ Γ1ρ → Γ2ρ,⊤ρ ⊑ F(⊤ρ).Then, for each w ∈ W , choose Γ1,Γ2 = ∅. Therefore ASαβ ⊤ρ ⊑ F(⊤ρ) and soW ⊆ f(W ) and in particular w ∈ f(W ).

(rule DNN). It is needed to show that

for every w ∈ W and X ∈ A if w ∈ f(X) then w 6∈ f(W \X)

iff

for every Γ1,Γ2, ρ, α and β,ASαβ Ωτ1ρ, τ1ρ ⊑ F(#ξ1ρ),Γ1 → Γ2, τ1ρ 6⊑F(#¬ ξ1ρ).

(⇒) Assume that, for every w ∈ W and X ∈ A if w ∈ f(X) then w 6∈ f(W \X). Itis sufficient to show that

ASαβ Ωτ1ρ, τ1ρ ⊑ F(#ξ1ρ) → τ1ρ 6⊑F(#¬ ξ1ρ)

that is, ASαβ → τ1ρ 6⊑F(#¬ ξ1ρ),0τ1ρ, τ1ρ 6⊑F(#ξ1ρ). Consider two cases:

a. ASαβ 0τ1ρ which immediately establishes the result,

b. otherwise ASαβ Ωτ1ρ, that is, Jτ1ρKASαβ = w for some w ∈ W . From thedefinition of AS , Jξ1ρKASα = Y for some Y ∈ A and so J¬ ξ1ρKASα = W \ Yand since #AS

= λa.a, J#ξ1ρKASα = Y and J#¬ ξ1ρKASα = W \ Y . From thehypothesis, if w ⊆ f(X) then w * f(W \X) for any X ∈ A. So eitherJτ1ρKASαβ = w * f(Y ) = FAS

(J#ξ1ρKASαβ) then ASαβ τ1ρ 6⊑F(#ξ1ρ)or, if w ∈ f(Y ) then Jτ1ρKASαβ = w * f(W \ Y ) = FAS

J#¬ ξ1ρKASαβ soASαβ τ1ρ 6⊑F(#¬ ξ1ρ).


(⇐) Assume for every Γ1,Γ2, ρ, α and β that

ASαβ Ωτ1ρ, τ1ρ ⊑ F(#ξ1ρ),Γ1 → Γ2, τ1ρ 6⊑F(#¬ ξ1ρ).

Then, for each w ∈ W choose Γ1,Γ2 = ∅ such that ρ(τ1) = x1, ρ(ξ1) = z1 andα(x1) = w, α(z1) = X. Therefore

ASα τ1ρ 6⊑F(#ξ1ρ)

orASα τ1ρ 6⊑F(#¬ ξ1ρ),

that is, w * f(X) or w * f(W\X). Thus, if w ⊆ f(X) then w * f(W\X).

The proof for the rule C is similar.

In the sequel it is given again a combination between deontic and knowledgelogics, but now with the deontic system being a non-normal modal logic. So, inthe resulting system, agents can reason about obligations that does not necessarilyrespect the formula ©(ϕ ∧ ψ)⊃ (©ϕ ∧©ψ).

Example 5.1.6. The fusion of DN with the system K of Example 3.1.11 sharing

LC as in Example 5.1.3 is obtained by taking the morphisms inc1 : LC → DN andinc2 : LC → K where inci = 〈(inci)h, (inci)a〉 for i = 1, 2 are such that

• (inc1)h : ΣC → ΣN and (inc2)h : ΣC → ΣK are the obvious inclusions,

• (inc1)a(ADN) = ADN |(inc1)h and (inc2)a(AK) = AK |(inc1)h for ADN ∈ ADN andAK ∈ AK .

The object of the pushout of 〈inc1, inc2〉 is the fusion of DN and K and, upto isomorphism, is such that (the modality from DN is herein renamed to © inorder to differentiate this system from the one resulting of the fusion of the previousexample):

• the signature is ΣNDK that is constituted by ΣC where C1 is enriched with themodalities ©, K1, . . . , Kn and O1 is enriched with the operators F,N1, . . . ,Nn;

• the set of rules is RNDK = RC∪L©,R©, eqF,C,N,DNN∪LNiΩ,RNiΩ,LKi,

RKi,Ti, 4i, 5i : i = 1, . . . , n;

• the class of algebras of the logic resulting from the combination is the class ofappropriate algebras for 〈ΣNDK ,RNDK〉 which includes the algebras inducedby general “mixed” structures of the form 〈W, f,R1, . . . , Rn,B, V 〉 where eachRi is an equivalence relation for i = 1, . . . , n and f respects conditions (c),(d) and (n).


The possibility of sharing modalities is now illustrated with an example of thefusion of two well known normal modal logics: a logic where the accessibility relationof the modality is reflexive and other where the accessibility relation is transitive.

Example 5.1.7. Let L4 = 〈ΣM ,RM ∪ 4,A4〉 be the modal logic system where 4

is the rule 4i in Table 3.3 and A4 is the class of all algebras (modulo equivalence)induced by general Kripke structures (see Definition 3.1.5) over transitive accessibil-ity relations, and LT = 〈ΣM ,RM ∪ T,AT 〉 be the modal logic system where T isthe rule Ti in Table 3.3 and AT is the class of all algebras (modulo equivalence) in-duced by general Kripke structures (see Definition 3.1.5) over reflexive accessibilityrelations.

Take LC = 〈ΣM ,RM ,AM〉, the “common logic system” in the fusion accordingto Definition 4.2.11, that is ΣM is the signature described in Definition 3.1.1, RM isthe set of rules described in Definition 3.1.2, and AM is the class of ΣM -algebras fullfor 〈ΣM ,RM〉. That is, LC is the normal modal logic system known in the literatureas K.

The fusion of L4 and LT sharing LC is obtained by taking the morphisms inc1 :LC → L4 and inc2 : LC → LT where inci = 〈(inci)h, (inci)a〉 for i = 1, 2 are suchthat:

• (inci)h : ΣM → ΣM for i = 1, 2 is the obvious bijection;

• (inc1)a(A4) = A4|(inc1)h for A4 ∈ A4 and (inc2)a(AT ) = AT |(inc2)h for AT ∈ AT .

So, the resulting system is LS4 = 〈ΣM ,RM ∪4,T,AS4〉 where AS4 is the classof appropriate algebras for the calculus of LS4 which includes the algebras inducedby general Kripke structures such that the accessibility relation is transitive andreflexive, commonly known as the S4 modal system.

Example 5.1.7 shows the importance of LC . If LC was a system without themodality, as the one in Example 5.1.3, then the resulting combination was the bi-modal logic system where one modality has a reflexive relation and the other has atransitive relation.

It is also possible to share non-normal modalities, as the next example illustrates.

Example 5.1.8. Let M = 〈ΣMT ,RMT ,AMT 〉 be the monotonic non-normal modal

logic of Example 3.2.20 and CNN = 〈ΣN ,RCN ,ACN〉 a non-normal modal logicsystem where ΣN is the non-normal modal signature of Definition 3.2.2, RCN =RNN ∪ C with RNN being the minimum set of rules described in Definition 3.2.3


and C the rule

Ωτ1, τ1 ⊑ F(#ξ1), τ1 ⊑ F(#ξ2),Γ1 → Γ2, τ1 ⊑ F(#ξ1 ∧ ξ2),

and ACN is the class of ΣN -algebras full for 〈ΣN ,RCN〉.

The fusion of M and CNN sharing E (the smallest non-normal system that canbe defined in Definition 3.2.19) where inc1 and inc2 are the obvious inclusions is thesystem Reg of regular systems defined in Example 3.2.21.

5.2 Preservation of Soundness and Completeness

In this section preservation of soundness and completeness by fusion is analysed forsome of the examples of the previous section. More specifically it is analysed thefusion of the SDL and GL logic systems described in Example 5.1.1, the fusion of theK and SDL logic systems presented in Example 5.1.3, and the fusion of the normaland non-normal systems in Example 5.1.6. After that, the logic system described in[FF05] (counter-example of the strategy of define fusion of non-normal modal logicsas successive modalizations) is presented as LSLTV and it is shown that the logicsystem resulting from its fusion with itself is sound and complete.

Example 5.2.1. Consider the logic system resulting from the fusion of the standard

deontic logic system and the provability logic system, presented in Example 5.1.1.Both component logic systems are sound by Corollary 2.5.2, so, by Theorem 4.3.1their fusion is also sound. The logic system resulting from the fusion is complete,by Theorem 4.3.2, since all the rules of the standard deontic and Löb provabilitysystems are endowed with persistent provisos and both classes of algebras are full.

Now consider the logic system resulting from the fusion of knowledge logic anda standard deontic logic as described in Example 5.1.3. The knowledge and thedeontic systems are sound (again by Corolary 2.5.2), so the logic system resultingfrom the fusion is also sound. Moreover it is complete by Theorem 4.3.2 since all therules of knowledge and deontic systems are endowed with persistent provisos andtheir classes of algebras are full.

Finally, with respect to Example 5.1.6 that presents the fusion of the non-normaldeontic system DN with the knowledge logic system K the results are analogous:both systems involved in the fusion are sound, all rules are endowed with persistentprovisos and all classes of algebras are full, so the logic system resulting from thefusion is sound and complete.

5.2. Preservation of Soundness and Completeness 83

The non-normal modal partition logic MP was defined in [FF05] as a non-normalmodal system containing the axiom

P (2ϕ⇔ ϕ) ∨ (2ψ⇔¬ψ).

This system is particularly interesting for fusion since its combination with itselfcontains strong interactions between the modalities as, for instance, commutativ-ity (2122ϕ ⇔ 2221ϕ). It is now shown that the preservation of soundness andcompleteness by the fusion of MP with itself is obtained directly by the resultsestablished in this thesis.

Example 5.2.2. Let MPi= 〈ΣMPi

,RMPi,AMPi

〉 with i = 1, 2 be the non-normal

modal logic systems where:

• ΣMPiis the signature ΣN of Definition 3.2.2,

• RMPiis the union of the set RN of Definition 3.2.3 with the set of rules

presented in Table 5.1,

• AMPiis the class of all ΣMPi

-algebras appropriate for 〈ΣMPi,RMPi

〉 whichincludes the algebras induced by general structures 〈W, f,A, V 〉 (see Definition3.2.6) such that, for every w ∈ W ,

either w ∈ f(X) for all X ⊆W such that w 6∈X,

or w ∈ f(X) for all X ⊆W such that w ∈ X.

RP

Ωτ1,Γ1 → Γ2, τ1 ⊑ τ2 Ωτ1, τ1 6⊑ τ3,Γ1 → Γ2, τ1 6⊑F(τ3)

Ωτ1,Γ1 → Γ2, τ1 ⊑ F(τ2)

LP

Ωτ1,Γ1 → Γ2, τ1 6⊑ τ2 Ωτ1, τ1 ⊑ τ3,Γ1 → Γ2, τ1 ⊑ F(τ3)

Ωτ1,Γ1 → Γ2, τ1 6⊑F(τ2)

Table 5.1: Rules for the non-normal partition logic

Take LC as in Example 5.1.3 and inci = 〈(inci)h, (inci)a〉 be such that (inci)h :ΣC → ΣMPi is the obvious inclusion and (inci)a(Ai) = Ai|(inci)h for Ai ∈ AMPi

.

The resulting system of the fusion by sharing LC of MP1and MP2

is soundby Theorem 4.3.1 since each MPi

is sound by Corollary 2.5.2 and is complete byTheorem 4.3.2 since all rules are endowed with persistent provisos and both classesof algebras are full.

Chapter 6

Conclusion

As the epilogue of this thesis, some remarks are made in Section 6.1, discussing sometopics that can open new directions to the research outlined in Section 6.2.

6.1 Final Remarks

In Chapter 2 the framework used during the thesis was presented. LSLTV wasalready used in [MSSV04] to present normal modal logic systems but herein a subtlemodification was made in the presentation incorporating the set \ in the signatureof the language. This was made to allow the framework to be able to capture ina more efficient way different structures of truth values that, for instance, do nothave the notion of atomic values as can be the case of some many-valued logics.The interesting point about this change in the language is the possibility to capturethe exact language of labelled systems where labels are simple worlds [Vig00]. Inthis case the signature for modal logic would be the tuple 〈C,O,\, X, Y, Z〉 whereC,X, Y and Z are as in Definition 3.1.1 but O would have only ⊤,⊥ ∈ O0 andOk = ∅ for all k ≥ 1 and \ would have only a binary relation R that modelsdirectly the accessibility relation in a Kripke structure.

Chapter 3 presented the general class of modal logic systems: both normal andnon-normal are defined in a modular calculus which constitutes a novelty for the non-normal case, achieving one of the aims of this work. Using the same principles it ispossible to describe several logic systems using LSLTV. Intuitionistic and relevancelogics are some cases that can be studied in this context after doing perhaps someminor adaptations. For intuitionistic logic, the signature could be defined as thesignature for normal modal logic systems without the connective 2 (but maintaining

85

86 Chapter 6. Conclusion

the operator N). In terms of rules, rules for reflexivity and transitivity (T and 4)of the relation must be in the calculus as well as a rule capturing the persistenceproperty. The specific rules for ⊃ and ¬ connectives also need to be modified. Forrelevance logics, that also have a “Kripke-like” semantics, the fact that they aresubstructural logics require special attention: most of the results presented hereshould be adapted, as, for instance, the completeness result.

Chapter 4 is the core of the thesis, where the aims of defining fusion of LSLTVand the study about the preservation of soundness and completeness are attained.Part of these results were already been published in [RRS10]. The use of a categoricalapproach for fusion led to the possibility of using sets of shared symbols different fromthe usual ones in the literature (namely the propositional connectives). Howeverthe definition presented in this work is still restricted in some cases due to themorphisms that are used. This is expected since fusion is in fact a restricted kind ofcombination. For instance, there is no morphism between a normal modal systemand a non-normal modal system (the operators N and F cannot be mapped one tothe other) for the definition used in this work.

Chapter 5 attains the final goal of the thesis. Fusion of two modal logics whereat least one of them is non-normal is defined.

6.2 Future Work

Although the categorical definition of fusion in Definition 4.2.11 can be appliedto logic systems that are not necessarily (normal) modal systems (generalizing thescope of application of the algebraic definition of fusion), it is possible to generalizeeven more the class of logic systems to which fusion can be applied. Some of thegeneralizations that could be proposed includes:

• generalization of the universe of logic systems involved in the combination;

• generalization of the common logic system LC ;

• generalization of the conditions for a pair of logic systems to be eligible forcombination.

An interesting direction would be to investigate whether or not the generaliza-tions preserve the results established in Section 4.3.

To what concerns the universe of logic systems involved in the combination, theidea is to generalize Definition 4.2.11 by allowing structural logic systems. Ob-

6.2. Future Work 87

serve that this generalization brings the combination to the border of what can beconsidered fusion since it is applied to systems not necessarily modal.

The second generalization consists in removing the condition ΣP ⊆ ΣC . Thatis, to allow ΣC to be a smaller set than ΣP while assuming that the morphism toeach component logic is a monomorphism, so an injection of ΣC to each Σi. Morerigorously, ΣC should be non-empty and should exists an injection from ΣC to eachΣi with i = 1, 2.

The third generalization is about the set of rules RC and consists on removingthe condition that it is a subset of each Ri while imposing that there are rules in RC

for the shared connectives and operators. This generalization changes the object ofthe pushout given before.

In what respects the categorical approach, different definitions of the morphismbetween LSLTV can be investigated. One idea is to modify the mapping betweenalgebras, allowing different sets of sorts between the source and target algebra.

Another direction of research is to investigate preservation of properties besidessoundness and completeness. It is possible that the techniques for proving preser-vation of Craig interpolation presented in [CRS08b] can be adapted to the contextof labelled sequents.

88 Chapter 6. Conclusion

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96 BIBLIOGRAPHY

Table of Symbols

A|h62AK , 36A|Σ′23A(C, S), 29AS , 45app(C), 26A(Σ), 10AΞ′,T′

(Σ), 21

Bf (U), 10

δ, 10∆, 19

⊢, 15

E, 53, 24

F (Σ), 9

GL, 35, 40

K, 38K, 40KA, 37

LC , 69⊑, 10Log, 62

MP , 81MT , 53

Ω, 100, 13

Reg, 55\, 8RN , 56

S4, 79 , 24SA, 49SDL, 39→, 11#ϕ, 9Sig, 62Σ, 8

T (Σ), 9

97

98 TABLE OF SYMBOLS

Subject Index

classical propositional logiccalculus, 14signature, 8

completenessof LSLTV, 32preservation by fusion, 71

derivation, 15global, 18local, 18

entailment, 24global, 25local, 25with proviso, 24

fusion, 4, 59algebraic, 60categorical, 69–70

Gentzen systems, see sequent calculus

Kripke semantics, 8, 14general structure, 35

labelled deduction systems, 3logic systems, 26

classical, 53deontic, 39, 56full, 26knowledge, 40monotonic, 53normal modal, 38, 56provability, 40regular, 53, 55

relational, 26

metatheoremof conjugation, 19of contradiction, 19of deduction (for assertions), 21of deduction (for sequents), 20of modus ponens, 21

morphismof LSLTV, 62of signatures, 61

non-normal modal logic, 2, 41–57calculus, 43general structure, 45partition logic, 81signature, 42structure, 41

normal modal logic, 1, 33–41calculus, 35signature, 34

satisfaction, 24sequent calculus, 11, 13

for classical propositional logic, 14for non-normal modal logic, 43

soundnessby morphisms, 68of LSLTV, 27preservation by fusion, 71

truth values, 7–10

99

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