Universidade Tecnica de Lisboa

Instituto Superior Tecnico

Departamento de Matematica

Algebraization of logics and beyond

Ricardo Joao Rodrigues Goncalves

Diploma Thesis

Applied Mathematics and Computation

Supervisor:

Prof. Carlos Caleiro

July 2004

i

Keywords

Logic

Map of Logics

Strong Representation

Algebraizable Logic

Non-truth-functionality

ii

Acknowledgments

This work would not be possible without the help of so many people that, alongthese 11 months of work, shared with me this journey. To all of them I thankfrom the heart. To some of these I would like to give a special thanks.

First of all, I would like to thank my supervisor, Professor Carlos Caleiro, forall that I learned with him throughout this year of work. Most of this work isfruit of his ideas. I would like to thank also his constant support and infinitepatience to read so carefully all that I wrote. For all this, and for a lot more thatremained to write, thank you Professor Carlos!

I would also thank Professor Amilcar Sernadas and Professor Cristina Ser-nadas for their constant support and motivation, and for their contribution inmy development as a scientist.

I also acknowledge the scientific environment provided to me by the Centerof Logic and Computation (CLC), of which I have become a student member.This work falls within the scope of the FibLog project POCTI/MAT/37239/2001financed by FCT and FEDER, as well as of the new QuantLog initiative.

I must thank my friends and workfellows Pedro Adao and Tiago Reis for theirready helps and LATEXadvices.

I would like to thank all my colleagues and friends, specially to Anabela,Angelo, Bela, Bruno, Carlos, Daniel, Francisco, Joao, Jorge, Lucio, Rui andTiago. Their friendship and constant support was (and still is!) very importantto me.

A simple thank you is not enough to express all the gratitude that I have forall that my parents have done for me. Their infinite love and dedication was, andstill is, essential in all aspects of my life.

I also would like to thank my sisters, Rita and Soraia, for all their supportand love, and for all those special moments we shared.

Finally I would like to thank my love Lurdes, for all her love and infinitesupport and for being always at my side through fair and foul, giving me strengthwhen I was low-spirited and distressed.

I dedicate this work to my parents, to my sisters and to Lurdes. Their lovewas essential for this dream to become true.

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iv

Contents

Acknowledgments iii

1 Introduction 3

2 Logics 5

2.1 Logical consequence . . . . . . . . . . . . . . . . . . . . . . . . . . 5

2.2 Maps of logics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9

2.3 Structural propositional logics . . . . . . . . . . . . . . . . . . . . 12

2.4 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13

3 Algebraization 19

3.1 Algebraizable logic . . . . . . . . . . . . . . . . . . . . . . . . . . 19

3.2 One equivalent definition . . . . . . . . . . . . . . . . . . . . . . . 22

4 Beyond algebraization 27

4.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27

4.2 B-able logic . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27

5 Example 33

5.1 The example of C1 . . . . . . . . . . . . . . . . . . . . . . . . . . 33

6 Conclusions 39

1

2 CONTENTS

Chapter 1

Introduction

Algebraic logic in modern sense was born with the work of Tarski, in particularwith his 1935 paper (see [Tar83]) where we can find for the first time somecharacteristic features of the subject we recognize today. In this paper, he gavethe precise connection between Boolean algebra and classical propositional logic,using the idea of looking at the set of formulas as an algebra with operatorsinduced by the logical connectives. He observed that logical equivalence wasa congruence on the formula algebra, and a quotient algebra could be built.This is the so-called Lindenbaum-Tarski method. It turns out that the quotientalgebra is a Boolean algebra, and the theorems coincide exactly with the formulasequivalent to J.

Using this idea, a number of other logics were algebraized, namely the intu-itionistic propositional logic of Heyting, the multiple-valued logics of Post andLukasiewicz and the modal logics S4 and S5 of Lewis. In contrast to Boolean,cylindric, polyadic and Wajsberg algebras which were known before the Linden-baum-Tarski method was first applied to generate them from the appropriate log-ics, Heyting algebras were first identified precisely by applying the Lindenbaum-Tarski method to intuitionistic propositional logic.

Although the focus of algebraic logic was on finding an algebraic counterpartfor particular classes of logics, there was also interest, when this counterpart wasfound, in investigating the relationship between the metalogical properties of thelogic and the algebraic properties of the algebraic counterpart. These results areusually called bridge theorems and allow us to use powerful methods of modernalgebra in the investigation of metalogical properties of algebraizable logics.

The investigation of particular classes of logics gave place to a systematicinvestigation of broad classes of logics in a more abstract context. The focus hasturned to the process of algebraization itself rather than being centered on thealgebraization of particular classes of logics. Only in the 1989 monograph by Blokand Pigozzi [BP89], the concept of algebraizable logic was given a mathematicalprecise sense.

The classical approach considers that the formulas are elements of a free al-

3

4 CHAPTER 1. INTRODUCTION

gebra and the interpretation structures are logical matrices [Woj88] based onalgebras of the same type and where the value of each formula is calculatedhomomorphicaly. But this approach has inherently some limitations of appli-cability, namely to non-truth-functional logics, where there are constructors forwhich the principle of substitution of equivalents is not valid. The paradigmaticexamples of non-truth-functional logics are the paraconsistent logics of da Costa[dC63, dC74].

The main motivation of this work is precisely to extend the existing theoryof algebraization in order to encompass also these non-truth-functional logics.

Our objective is first to give an alternative characterization of the main con-cepts of the theory introduced by Blok and Pigozzi, using maps between thetarget logic and unsorted equational logic. With this characterization, we canthen replace unsorted equational logic by another base logic. We then character-ize the properties that this base logic has to satisfy in order to maintain some“equational flavor”, that is, we try to abstract away the properties of unsortedequational logic that are essential to the process of algebraization. Another ob-jective is the study of the example of paraconsistent logic C1 of da Costa usingthis new approach. C1 was shown to be non-algebraizable first in [Mor80] andrecently using the Blok and Pigozzi theory in [LMS91]. For this particular ex-ample, we replace unsorted equational logic by two-sorted equational logic, withsorts of formulas and truth-values, using ideas in [CCC03], and prove that C1

is algebraizable in this new context. We refer to [BS81] with respect to detailson universal algebra. In Chapter 2 we introduce some basic concepts. We de-fine our working universe of logics and define the notion of a map of logics, anessential ingredient to the generalization we aim to achieve. We also prove someuseful lemmas about maps of logics and, in the last section, we recall some ex-amples of logics. Then, in Chapter 3, we recall the main concepts of the theoryof algebraization of logics introduced by Blok and Pigozzi [BP89]. We then givean alternative characterization of some of the notions previously introduced bymeans of the existence of maps between the logic in study and unsorted equationallogic. This is the (crucial) first step toward a generalization of these concepts,that will be done in Chapter 4. This generalization is based on substituting un-sorted equational logic by another logic. In this chapter we try to characterizethe properties that this replacing logic has to satisfy in order to guarantee thatthe process of algebraization still works. We also try to establish some naturalanalogue results to those introduced in Chapter 3. Finally, in Chapter 5, westudy in detail the application of this new theory to the concrete example of C1

of DaCosta.

Chapter 2

Logics

In this chapter we will introduce some important definitions, that will be essentialfor the consequent chapters. We begin by the general definition of a logic anddefine what we mean by map between logics giving some useful lemmas. Thenwe present a particular case of logics, the structural propositional logics. In thelast section of this chapter we will give some examples of logics.

2.1 Logical consequence

Definition 2.1.1 (Logic)A logic is a pair L xL,$y, where L is a set (of formulas) and $ 2L L is aconsequence relation satisfying the following conditions [Tar83]:

Reflexivity: if ϕ P Γ then Γ $ ϕ;

Cut: if Γ $ ϕ for all ϕ P Φ, and Φ $ ψ then Γ $ ψ;

Weakening: if Γ $ ϕ and Γ Φ then Φ $ ϕ.

We will consider only these three conditions, though more conditions could beimposed. For example, in [BP89], Blok and Pigozzi considered logics that furthersatisfy the finitariness condition:

Finitariness: If Γ $ ϕ then Γ1 $ ϕ for some finite Γ1 Γ.

When considering several logics, to avoid confusion, we shall sometimes write$L instead of $. In the sequel if Γ,Φ L, we shall write Γ $ Φ wheneverΓ $ ϕ for all ϕ P Φ. We say that ϕ and ψ are interderivable, which is denotedby ϕ %$ ψ, if ϕ $ ψ and ψ $ ϕ. In the same way, given Γ,Φ L we say that Γand Φ are interderivable, if Γ $ Φ and Φ $ Γ.

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6 CHAPTER 2. LOGICS

It is well known that Reflexivity and Cut together imply Weakening:Let Γ Φ L and ϕ P L. Suppose also that Γ $ ϕ. We want to prove thatΦ $ ϕ. Let ψ P Γ, then, because Γ Φ, we have that ψ P Φ. By reflexivity weknow that Φ $ ψ for all ψ P Γ. Using the hypothesis that Γ $ ϕ and using Cut,we conclude that Φ $ ϕ.Despite this fact, we have kept Weakening in the definition for methodologicalreasons, thus explicitly excluding non-monotonic logics from this context.

In the literature it is usual to define a logic as a pair L xL,$ y where L isa set and $ is a closure operator on L (in the sense of Kuratowski [MP96]), thatis, $ : 2L Ñ 2L is a function that satisfies the following properties:

Extensiveness: Γ Γ$;

Monotonicity: If Γ Φ then Γ$ Φ$;

Idempotence: pΓ$q$ Γ$.

Let us see that this is an equivalent way to define a logic:

Let L xL,$y be a logic. Given a set Γ L, we can consider the setΓ$ tϕ P L : Γ $ ϕu. This function on the powerset of L is called theconsequence operator of L and satisfies the Kuratowski axioms.

Extensiveness follows directly from the Reflexivity of $. If ϕ P Γ then byReflexivity Γ $ ϕ and hence, by definition of Γ$, ϕ P Γ$.

Monotonicity follows from the Weakening property of $. Suppose thatΓ Φ and ϕ P Γ$, that is, Γ $ ϕ. By Weakening, Φ $ ϕ, and so, ϕ P Φ$.

Idempotence follows from the Cut property of $. Let ϕ P pΓ$q$, that is,Γ$ $ ϕ. We also know that Γ $ ψ for all ψ P Γ$. Hence by Cut we havethat Γ $ ϕ and this means that ϕ P Γ$.

Let L xL,$ y be a logic in the second sense. We now aim to prove thatthe relation $ 2L L, defined by Γ $ ϕ iff ϕ P Γ$ satisfies Reflexivity,Cut and Weakening.

Reflexivity follows from Extensiveness. If ϕ P Γ, then by Extensivenessϕ P Γ$, that is, Γ $ ϕ.

Cut follows from Idempotence and Monotonicity. Suppose that Γ $ ϕ forall ϕ P Φ, and Φ $ ψ. We want to prove that Γ $ ψ. By hypothesis weknow that Φ Γ$, and then by Monotonicity and Idempotence we havethat Φ$ Γ$. Therefore ψ P Γ$, that is, Γ $ ψ.

Weakening follows from the Monotonicity of $. Suppose that Γ $ ϕ andΓ Φ. We want to prove that Φ $ ϕ. We know that ϕ P Γ$ and so, byMonotonicity, we have that ϕ P Φ$, that is, Φ $ ϕ.

2.1. LOGICAL CONSEQUENCE 7

In the sequel we shall use either the consequence relation or the consequenceoperator, interchangeably, whenever it is more convenient.

The theorems of L are the formulas ϕ such that H $ ϕ. A theory of L,or briefly a L-theory, is a set Γ of formulas such that Γ is closed under theconsequence relation $, that is, such that if Γ $ ϕ then ϕ P Γ. Given a set Γ, wecan consider the set Γ$, the smallest theory containing Γ. The set of all theoriesof L is denoted by ThL.

Definition 2.1.2 (Partial order)A partial order is a pair xR,¤y, where R is a set and ¤ R R is a relationsatisfiying the following properties:

Reflexivity: r ¤ r for all r P R;

Transitivity: r1 ¤ r2 and r2 ¤ r3 implies r1 ¤ r3 for all r1, r2, r3 P R;

Antisymmetry: r1 ¤ r2 and r2 ¤ r1 implies r1 r2 for all r1, r2 P R.

If xP,¤y is a partial order, then P is called a partial ordered set or simply aposet. Let A be a subset of a poset P . An element p P P is an upper bound forA if a ¤ p for every a in A. An element p P P is the least upper bound of A, orsupremum of A (

A) if p is an upper bound of A, and a ¤ b for every a P A

implies p ¤ b (i.e., p is the smallest among the upper bounds of A). Similarlywe can define what it means for p to be a lower bound of A, and for p to be thegreatest lower bound of A, also called the infimum of A (

A).

Definition 2.1.3 (Complete partial order)A poset P is complete if for every subset A of P both

A and

A exist (in P ).

Definition 2.1.4 (Maps of partial orders)Given two partial orders xA,¤Ay and xB,¤By, a map from xA,¤Ay to xB,¤Byis a function h : AÑ B.

The map h is monotone if for all a1, a2 P A,

a1 ¤A a2 implies hpa1q ¤B hpa2q.

The map h is sup-preserving if for all A1 A

hpªA1q ª

hrA1s.

8 CHAPTER 2. LOGICS

It is easy to see that a sup-preserving map is monotone. To prove this, supposeh : xA,¤Ay Ñ xB,¤By is a sup-preserving map and consider a1, a2 P A such thata1 ¤A a2. Then

ta1, a2u a2. Because h is sup-preserving we have thathpta1, a2uq hrta1, a2us, which means that hpa2q thpa1q, hpa2qu, andso hpa1q ¤B hpa2q.

We now want to see that xThL,y forms a complete partial order, where thesupremum and the infimum of a set T of theories are respectively,

L T pYΓPT Γq$andL T ΓPT Γ:

By the properties of is trivial to verify that xThL,y forms a partial order.Hence, all we need to prove is that this is a complete partial order, that is, forevery T ThL the supremum and the infimum of T exists in ThL. For this wewill first prove that

L T andL T are well defined, that is, they are still in

ThL. By definition,L T is always a theory, and hence belongs to ThL. We now

prove that the arbitrary intersection of theories is still a theory. Let I ΓPT Γ.If I $ ϕ then by monotonicity Γ $ ϕ for every Γ P T . Because each Γ P T is atheory, ϕ P Γ for every Γ P T , and hence ϕ P I. We conclude that I P ThL. Wewill now prove that in fact

L T andL T are, respectively, the supremum and

the infimum of T .By definition,

L T is clearly a theory that contains all Γ P T . Let Φ P ThLsuch that

ΓPT Γ Φ, then by Monotonicity, pΓPT Γq$ pΦq$ Φ.

By definition,L T it is clear that

L T Γ for all Γ P T . Let Φ P ThLsuch that Φ Γ for all Γ P T . Then trivially Φ ΓPT Γ L T .

When we want to introduce a particular logic, it is useful to introduce it as adeductive system.

Definition 2.1.5 (Deductive system)A deductive system is a pair D xL,Ry where L is a set (of formulas), and R isa subset of p℘finLq L.

Each element r xΓ, ϕy of R is called an inference rule. We say that the(finite) set Γ is the set of premises of r, which we will denote by Premprq, andthat ϕ is the conclusion of r, which we denote by Concprq. If Premprq H, ris said to be an axiom, as well as Concprq.

Given a deductive system it is easy to see what logic is naturally associatedwith it.

Definition 2.1.6 (Derivability)Given a deductive system D xL,Ry, a formula ϕ P L is derivable from a set offormulas Γ L in D, denoted by Γ $D ϕ, if there exists a sequence γ1, . . . , γm P Lsuch that:

- γm is ϕ;

2.2. MAPS OF LOGICS 9

- for each i 1, . . . ,m, the formula γi is either:

– an element of Γ, or

– there exists a rule r P R such that γi Concprq andPremprq tγ1, . . . , γi1u.

The logic associated with D is LD xL,$Dy. Note that, since all rules havefinite sets of premises, the logic LD is always finitary.

2.2 Maps of logics

Let L xL,$y and L1 xL1,$1y be two logics.

Definition 2.2.1 (Map of logics)A map θ from L to L1 is a function θ : LÑ 2L1 such that the following holds:

if Γ $ ϕ then p¤γPΓ

θpγqq $1 θpϕq.

The cases where θpϕq is a singleton set for every ϕ P L or is a finite set forevery ϕ P L, are usual particular cases of the above definition of map. For thesake of notation we will use θrΓs γPΓ θpγq. Using this notation the conditionof a map can be rewritten to:

if Γ $ ϕ then θrΓs $1 θpϕq.Definition 2.2.2 (Conservative Map)A map θ from L to L1 is conservative when

Γ $ ϕ iff θrΓs $1 θpϕq.Definition 2.2.3 (Strong Representation)A strong representation of L in L1 is a pair pθ, τq of conservative maps θ : LÑ L1and τ : L1 Ñ L such that:

i) For all ϕ P L we have that ϕ %$ τ rθpϕqsii) For all ϕ1 P L1 we have that ϕ1 %$1 θrτpϕ1qs;In fact, in the above definition, one of the conditions i) or ii) is enough, since,

given the conservativeness of the maps θ and τ , i) is equivalent to ii).

Suppose, without loss of generality, that we have ii) and consider ϕ P L.Then, because θpϕq L1, by ii), we have that for all ψ P θpϕq, ψ %$1 θrτpψqs.Then θpϕq %$1 θrτ rθpϕqss. By conservativeness of θ we have ϕ %$ τ rθpϕqs.

10 CHAPTER 2. LOGICS

In fact if we assume the conservativeness of θ and consider any functionτ : L1 Ñ 2L that satisfies ii), then we can conclude that τ is in fact a conser-vative map from L1 to L that also satisfies i).

Suppose θ : LÑ L1 is a conservative map. Let τ : L1 Ñ 2L be any functionthat satisfies ii). We aim to prove that τ is a conservative map that satisfiesi). First we will prove that τ is conservative, that is, for all Γ1 Y tϕ1u L1we have that Γ1 $1 ϕ1 iff τ rΓ1s $ τpϕ1q.

Suppose first that Γ1 $1 ϕ1. By ii) we have θrτ rΓ1ss $1 θrτpϕ1qs and bythe conservativeness of θ we get that τ rΓ1s $ τpϕ1q.Suppose now that τ rΓ1s $ τpϕ1q. Because θ is conservative we get thatθrτ rΓ1ss $1 θrτpϕ1qs. Using ii) we get that Γ1 $1 ϕ1.

We proved above that, assuming the conservativeness of θ and τ , condi-tion i) is equivalent to ii). In this case we know that both θ and τ areconservative, and since we are assuming ii) we also have i).

Note also that, because of the symmetry of the definition, we also have astrong representation pτ, θq of L1 in L.

Given a map θ : L Ñ L1 we can consider the function between completepartial orders θTh : ThL Ñ ThL1 , such that θThpΓq θrΓs$1

. We can alsoconsider the function between complete partial orders θ1 : ThL1 Ñ ThL suchthat θ1p∆q tϕ P L : θpϕq ∆u. Let us prove that θ1 is well-defined, thatis, that θ1p∆q is a L-theory for all L1-theory ∆. Suppose that θ1p∆q $ ϕ. Wewant to see that ϕ P θ1p∆q. Since θ is a map we have that θrθ1p∆qs $1 θpϕq.Since θrθ1p∆qs ∆, by monotonicity we get that ∆ $1 θpϕq. Then θpϕq ∆since ∆ is a L1-theory. By definition of θ1p∆q, ϕ P θ1p∆q.Proposition 2.2.4Suppose θ : LÑ L1 is a map. Then θTh : ThL Ñ ThL1 is sup-preserving.

Proof: Let T ThL. Then

θThpL T q = θThppΓPT Γq$q = pθrpΓPT Γq$sq$1

= pΓPT θrΓs$1q$1

= pΓPT θThpΓqq$1

=L1 θThrT s

QED

Proposition 2.2.5Suppose θ : LÑ L1 is a map. Then θ1 : ThL1 Ñ ThL is monotone.

2.2. MAPS OF LOGICS 11

Proof: Let T1, T2 ThL1 such that T1 T2. We want to show thatθ1rT1s θ1rT2s. Let ϕ P θ1rT1s, that is, ϕ P ∆1PT1

θ1r∆1s. Then becauseT1 T2, ϕ P ∆2PT2

θ1r∆2s, that is, ϕ P θ1rT2s.QED

Proposition 2.2.6Suppose θ : L Ñ L1 is a map. Then for all L-theory Γ and all L1-theory ∆ wehave that:

θ1pθThpΓqq Γ

θThpθ1p∆qq ∆

Proof: We will first prove that θ1pθThpΓqq Γ.

θ1pθThpΓqq = tϕ P L : θpϕq θThpΓqu= tϕ P L : θpϕq θrΓs$1u= tϕ P L : θrΓs $1 θpϕqu

Now it is easy to see that θ1pθThpΓqq Γ.

Let us now prove that θThpθ1p∆qq ∆.

θThpθ1p∆qq = θThptϕ P L : θpϕq ∆uq= pϕPθ1p∆q θpϕqq$1

∆$1

= ∆QED

Proposition 2.2.7Suppose θ : LÑ L1 is a conservative map. Then

θ1pθThpΓqq Γ

Proof: Let us prove the two inclusions. We already proved that θ1pθThpΓqq Γ.To prove that θ1pθThpΓqq Γ consider ϕ P θ1pθThpΓqq. As we have alreadysaw θ1pθThpΓqq tϕ P L : θrΓs $1 θpϕqu, so θrΓs $1 θpϕq. Then, by theconservativeness of θ, we get that Γ $ ϕ. Since Γ is a theory, we have that ϕ P Γ.

QED

Proposition 2.2.8Suppose pθ, τq is a strong representation of the logic L in the logic L1. Then wehave the following:

θ1 τTh

12 CHAPTER 2. LOGICS

θTh τ1

Proof: Since pθ, τq is a strong representation, then θ and τ are both conservativemaps and for all ϕ P L we have that ϕ %$ τ rθpϕqs and for all ϕ1 P L1 we have thatϕ1 %$1 θrτpϕ1qs. First let us prove that θ1 τTh. For this we will prove that,for all L1-theory ∆, θ1p∆q τThp∆q and θ1p∆q τThp∆q. Let ϕ P θ1p∆q,then θpϕq ∆. By reflexivity we get that ∆ $1 θpϕq and, since τ is a map, weget that τ r∆s $ τ rθpϕqs. By hypothesis we know that τ rθpϕqs $ ϕ and then bycut we have that τ r∆s $ ϕ, that is, ϕ P τ r∆s$ which means that ϕ P τThp∆q.We then conclude that θ1p∆q τThp∆q.

Let ϕ P τThp∆q, that is, ϕ P τ r∆s$. Then τ r∆s $ ϕ. By hypothesisϕ $ τ rθpϕqs. Then, using cut, we have that τ r∆s $ τ rθpϕqs. Using the con-servativeness of τ , we have ∆ $1 θpϕq. Since ∆ is a theory, we conclude thatθpϕq ∆, that is, ϕ P θ1p∆q.

The proof of the equality θTh τ1 is analogue. QED

We get an immediate corollary of the above proposition that will be usefullater.

Corollary 2.2.9Suppose pθ, τq is a strong representation of the logic L in the logic L1. Then θTh

is a bijection.

2.3 Structural propositional logics

A propositional signature is an indexed set Σ tΣiuiPN0 where each Σi is theset of i-ary constructors. We consider fixed a set Ξ tξiuiPN of propositionalvariables. Given a signature Σ, the language over Σ, which we denote by LΣ, isbuild inductively in the usual way from Σ over the set Ξ:

• ΞY Σ0 LΣ;

• If ϕ1, . . . , ϕn P LΣ and c P Σn then cpϕ1, . . . , ϕnq P LΣ.

We call Σ-formulas to the elements of LΣ , or simply formulas when Σ isclear from the context. Note that LΣ is precisely the carrier of the free Σ-algebra,LΣpΞq, that has as generators the elements of Ξ and is usually called the algebraof formulas. For simplicity we usually write LΣ instead of LΣpΞq.

Given Ξ1 Ξ, we will consider the set LΣpΞ1q LΣ of all Σ-formulas generatedfrom Ξ1. Of course LΣpΞq LΣ.

Given a signature Σ, a substitution is a function σ : Ξ Ñ LΣ. This functionextends to a unique endomorphism of the formula algebra, which we will denotealso by σ, by requiring that:

2.4. EXAMPLES 13

• σpcq c for all c P Σ0;

• σpcpϕ1, . . . , ϕnqq cpσpϕ1q, . . . , σpϕnqq for all c P Σn and all ϕ1, . . . , ϕn P LΣ.

Note that this is precisely the free extension of σ to LΣ. Given a set Γ LΣ

we can also consider the set σrΓs tσpϕq : ϕ P Γu.Given a formula ϕ P LΣptξ1, . . . , ξkuq we will write ϕpξ1zψ1, . . . , ξkzψkq to

denote the formula σpϕpξ1, . . . , ξkqq whenever σpξ1q ψ1, . . . , σpξkq ψk.

Definition 2.3.1 (Structural propositional logic)A structural propositional logic is a pair L xΣ,$y, where Σ is a propositionalsignature and xLΣ,$y is a logic that also satisfies [Tar83]:

Structurality: if Γ $ ϕ then σrΓs $ σpϕq for every substitution σ.

When we are dealing with such structural propositional logics we can partic-ularize the notion of deductive system, but now using schematic rules.

Definition 2.3.2 (Structural deductive system)A structural deductive system is a pair D xΣ, Ry where Σ is a propositionalsignature, and R is a subset of p℘finLΣq LΣ.

Definition 2.3.3 (Derivability)Given a structural deductive system D xΣ, Ry, the consequence relationassociated with D, $D, is the one associated with the deductive systemD xLΣ, tσprq : r P R, σ substitutionuy.

The logic associated with D is LD xL,$Dy. Note that, since all rules havefinite sets of premises, the logic LD is always finitary.

2.4 Examples

In this section we will present some well known examples of logics. The first twoare examples of propositional structural logics, while the other are not proposi-tional. As expected, we will introduce these examples by means of a deductivesystem.

Example 2.4.1 Paraconsistent logic C1 (da Costa, 1963)

- Signature Σ:

C0 tt, fu; C1 t u; C2 t^,_,u;

14 CHAPTER 2. LOGICS

Ci H for all i ¡ 2.

- P txH, ξ1 pξ2 ξ1qy,xH, pξ1 pξ2 ξ3qq ppξ1 ξ2q pξ1 ξ3qqy,xH, pξ1 ^ ξ2q ξ1y,xH, pξ1 ^ ξ2q ξ2y,xH, ξ1 pξ2 pξ1 ^ ξ2qqy,xH, ξ1 pξ1 _ ξ2qy,xH, ξ2 pξ1 _ ξ2qy,xH, pξ1 ξ3q ppξ2 ξ3q ppξ1 _ ξ2q ξ3qqy,xH, ξ1 ξ1y,xH, ξ1 _ ξ1y,xH, ξ1 pξ1 p ξ1 ξ2qqy,xH, pξ1 ^ ξ2q pξ1 ^ ξ2qy,xH, pξ1 ^ ξ2q pξ1 _ ξ2qy,xH, pξ1 ^ ξ2q pξ1 ξ2qy,xH,t pξ1 ξ1qy,xH,f pξ1 ^ pξ1 ^ ξ1qqy,xtξ1, ξ1 ξ2u, ξ2yuwhere ϕ is an abbreviation of pϕ^ p ϕqq and ϕ ψ is an abbreviation of

pϕ ψq ^ pψ ϕq.

Example 2.4.2 Classical propositional logic CPL

- Signature Σ

C0 H C1 t u C2 tñu Cn H, for all n ¡ 2

- P txH, pξ1 ñ pξ2 ñ ξ1qqy,xH, ppξ1 ñ pξ2 ñ ξ3qq ñ ppξ1 ñ ξ2q ñ pξ1 ñ ξ3qqqy,xH, pp ξ1 ñ ξ2q ñ pξ2 ñ ξ1qqy,xtξ1, ξ1 ñ ξ2u, ξ2yu

Before going to the next examples let us introduce some necessary definitions.An equational signature is a pair Σ xS,Oy, where S is a set, called the set of

2.4. EXAMPLES 15

sorts, and O tOwsuwPS,sPS is a family of disjoint sets. We say that an equa-tional signature Σ xS,Oy is n-sorted if n |S|.

Given an equational signature Σ, a Σ-algebra is a pair A=xtAsusPS, Ay, whereeach As is a non-empty set, called the carrier of sort s, and A assigns to eachoperation o P Os1...sns a function oA : As1 . . . Asn Ñ As. The set of allΣ-algebras will be denoted by AlgΣ.

A homomorphism from the Σ-algebra A to the Σ-algebra B, h : AÑ B,is a set ths : As Ñ BsusPS, such that for all o P Os1...sns, we have thathspoApa1, . . . , anqq oBphs1pa1q, . . . , hsnpanqq.

A congruence on a Σ-algebra A is a set =tsusPS, such that s is anequivalence relation on As and for each o P Os1...sns, we have that, if a1 s1 b1, . . . , an sn bn then oApa1, . . . , anq s oApb1, . . . , bnq. We denote by CongA theset of all congruences on the algebra A. Given θ1, θ2 P CongA, we can define theoperation : CongA CongA Ñ CongA, where for all sort s, xa, by P pθ1 θ2qsiff there exists a c P A such that xa, cy P pθ1qs and xc, by P pθ2qs. Inductivelyone defines r1 r2 . . . rn pr1 r2 . . . rn1q rn. It is not difficult tosee that xCongrA,y is a complete partial order, where for tθiuiPI CongrA,tθiuiPI iPI θi and

tθiuiPI tθi1 θi2 . . . θik : i1, i2, . . . , ik P I, k 8u.Given an equational signature Σ xS,Oy, we will consider fixed a family

X tXsusPS of disjoint sets, where Xs is the set of variables of sort s. Wedenote by TpΣ,Xq the free Σ-algebra with generators X. Each set T pΣ, Xqsis called the set of terms over pΣ, Xq of sort s. Given X 1 X we canconsider TpΣ,X1q the free Σ-algebra generated by X 1. We denote by gT pΣq thefamily T pΣ,Hq. Each set gT pΣqs is called the set of ground terms over Σ of sort s.

We also consider EqpΣ, Xq, the set of equations over pΣ, Xq, definedby EqpΣ, Xq tt1 t2 : t1, t2 P T pΣ, Xqs for some s P Su. Each setEqpΣ, Xqs tt1 t2 : t1, t2 P T pΣ, Xqsu is called the set of equations overpΣ, Xq of sort s. In the same way as above we can consider the set of groundequations, gEqpΣq EqpΣ,Hq. The set gEqpΣqs EqpΣ,Hqs denote the set ofground equations of sort s.

By an assignment of X over a Σ-algebra A we mean a family ρ tρsusPS

such that ρs : Xs Ñ As. Given a assignment ρ of X over A, the denotation ofterms is the homomorphism rr ssρA : TpΣ,Xq Ñ A such that, for all s P S andall x P Xs, rrxssρA ρspxq. Note that, since TpΣq is the free Σ-algebra withgenerators X, the value of a term over a homomorphism h depends only on thevalue h assigns to the variables.

16 CHAPTER 2. LOGICS

Given a Σ-algebra A, an assignment ρ and t1 t2 P EqpΣ, Xq, we writeA, ρ , t1 t2 if rrt1ssρA rrt2ssρA. We also write A , t1 t2 if, for everyassignment ρ, we have that rrt1ssρA rrt2ssρA, and in this case we say that A satisfyt1 t2.

Given a class K of Σ-algebras, we write Γ (ΣK t1 t2 if for all A P K, if

A , γ1 γ2 for all γ1 γ2 P Γ then A , t1 t2.It is well known that (Σ

AlgΣcoincide [EM85] with the consequence relation

associated with the deductive system defined by the rules

reflexivity xH, t ty;symmetry xtt1 t2u, t2 t1y;transitivity xtt1 t2, t2 t3u, t1 t3y;congruence xtt11 t21, . . . , t1n t2nu, opt11, . . . , t1nq opt21, . . . , t2nqy;substitution xtt1 t2u, σpt1q σpt2qy.

Example 2.4.3 Unsorted equational logic

Given an 1-sorted equational signature Σ1 (that can trivially be seen as anunsorted signature as it was introduced in Chapter 2.3) and given a class K ofΣ-algebras, we define

EqnΣK xgEqpΣq,(Σ

Ky.Note that, if K is precisely the class of all Σ-algebras that satisfy a given set ∆

of equations (with variables), then (ΣK also coincides with the deductive system

obtained by adding the equations in ∆ as axioms to the set of rules defined abovefor the case of (Σ

AlgΣ.

Example 2.4.4 Two-sorted Equational LogicGiven the 1-sorted equational signatures Σφ,1 and Στ,1 we can consider the

induced 2-sorted equational signature Σφ,τ xS,Oy where

S tφ, τu, where φ is the sort of formulas and τ is the sort of truth-values.

O tOwsuwPS,sPS, such that:

– Oεφ Σφ,10 Y P ;

– Oφkφ Σφ,1k for k ¡ 0;

– Oφτ tvu;– Oετ Στ,1

0 ;

– Oτkτ Στ,1k for k ¡ 0;

2.4. EXAMPLES 17

– Ows H in the other cases.

Let K be a class of Σφ,τ -algebras. Then we can define the 2-sorted equationallogic

EqnΣφ,τ

K xgEqpΣφ,τ q,(Σφ,τ

K y.Note that, if K is precisely the class of all Σφ,τ -algebras that satisfy a given

set ∆ of equations (with variables), then (Σφ,τ

K also coincides with the deductivesystem obtained by adding the equations in ∆ as axioms to the set of rules definedabove for the case of (Σφ,τ

Algφ,τΣ

.

18 CHAPTER 2. LOGICS

Chapter 3

Algebraization

In this chapter we will recall the abstract theory of algebraization of logics, firstintroduced in a mathematical precise definition by Blok and Pigozzi in [BP89].The idea of Blok and Pigozzi was to extend the classical theory of Lindenbaum-Tarski. In [Tar83] Tarski gave the precise connection between classical proposi-tional logic and Boolean algebras. The technique consists on looking at the set offormulas as an algebra with operators induced by the connectives. Logical equiv-alence is a congruence in the formula algebra and the induced quotient algebraturns out to be a Boolean algebra. This is the so called Lindenbaum - Tarskimethod.

The definition proposed by Blok and Pigozzi of algebraizable logic, is actuallywhat is now called a finitely algebraizable logic [FJP03]. Moreover they con-sidered exclusively finitary logics, that is, logics that also satisfy the finitarinessproperty.

In the sequel, as already stated, we will not restrict ourselves to finitary logicsand will consider the wider notion of algebraizable logic proposed in [FJP03].

After introducing the main concepts, we will propose an equivalent definitionby means of maps between the target logic and unsorted equational logic [EM85].

3.1 Algebraizable logic

As in [BP89], we will restrict ourselves to the study of the algebraizability ofstructural propositional logics. Consider fixed an arbitrary propositional signa-ture Σ and a set X of equational variables. In this chapter we will use unsortedequational logics, EqnΣ

K xgEqpΣq,(ΣKy, as they were introduced in Section 2.4.

Before going to the main definitions, let us introduce some notation.

Given a set of equations Θ tδj λj : j P Ju where δj, λj P T pΣ, txuq, it willbe useful to consider the following sets Θpxzϕq tδjpxzϕq λjpxzϕq : j P Ju andΘrΓs tδjpxzψq λjpxzψq : j P J, ψ P Γu.

Given a set of formulas E tεi : i P Iu LΣptξ1, ξ2uq and a set of equa-

19

20 CHAPTER 3. ALGEBRAIZATION

tions Θ tδj λj : j P Ju and an equation δ λ where δj, δ, λj, λ P T pΣ, txuqwe can consider the following sets Epξ1zδ, ξ2zλq tεipξ1zδ, ξ2zλq : i P Iu andErΘs tεipξ1zδj, ξ2zλjq : i P I, j P JuDefinition 3.1.1 (Algebraic Semantics)Let L xΣ,$y be a structural propositional logic and let K be a class of Σ-algebras. The class K is an algebraic semantics for L if there exists a set ofequations Θ tδj λj : j P Ju where δj, λj P T pΣ, txuq, called the set of definingequations, such that for every Γ LΣ and every ϕ P LΣ,

Γ $ ϕ iff ΘrΓs (ΣK Θpxzϕq.

Let us recall the case of classical propositional logic CPL.

Example 3.1.2 The two-element Boolean algebra

B xtJ,Ku,^,_, ,J,Kyconstitutes by itself an algebraic semantics for CPL. We just need to consider thesingleton set with the equation x J. In fact we have that,

Γ $CPL ϕ iff tψ J : ψ P Γu (B ϕ JA congruence in a Σ-algebra A is compatible with a subset F of A if

whenever a P F and xa, by P then b P F . In this case, F is a union ofequivalence classes of .

Definition 3.1.3 (Leibniz operator)Let L xΣ,$y be a structural propositional logic. Then the Leibniz operator onthe formula algebra can be given by:

Ω : ThL Ñ CongrLΣ

Γ ÞÑ largest congruence compatible with Γ.

We still have to see that, for all L-theory, ΩpΓq is well defined, that is, thereexits the largest congruence compatible with Γ.

Proposition 3.1.4The Leibniz operator Ω can be given as

ΩpΓq txϕ, ψy : for all δ P LΣptξuq, δpξzϕq P Γ iff δpξzψq P Γu.

3.1. ALGEBRAIZABLE LOGIC 21

Proof: Let ΘpΓq txϕ, ψy : for all δ P LΣptξuq, δpξzϕq P Γ iff δpξzψq P Γu.We aim to prove that ΘpΓq ΩpΓq, that is, ΘpΓq is the largest congruencecompatible with Γ. ΘpΓq is clearly a congruence and compatible with Γ. We stillhave to prove that it is the largest one. Let Θ1 be a congruence with this propertyand let δ P LΣptξuq. Since Θ1 is a congruence we have, for every xϕ, ψy P Θ1, that

xδpξzϕq, δpξzψqy P Θ1.Thus by the compatibility of Θ1,

δpξzϕq P Γ iff δpξzψq P Γ.

Hence xϕ, ψy P ΘpΓq, and so Θ1 ΘpΓq.QED

In fact, this last characterization is also used as the definition of the Leibnizoperator. Nevertheless, the first characterization is the most useful for us, inparticular because we have in view a generalization of the notion. As we shallsee, the Leibniz operator is one of the most important tools in the theory ofalgebraization. Indeed in [BP89], Blok and Pigozzi use Ω to give intrinsic charac-terizations of algebraizable logics. Let us see how Ω can be used to define anotherimportant class of logics, the protoalgebraic logics.

Definition 3.1.5 (Protoalgebraic logic)A structural propositional logic L xΣ,$y is protoalgebraic if, for every theory Γ,

if xϕ, ψy P ΩpΓq then ΓY tϕu %$ ΓY tψu.Although in general not algebraizable, the protoalgebraic logics constitute

the main class of logics for which the advanced methods of algebraic logic can beapplied. In the next proposition we begin to notice the importance of the Leibnizoperator, and how we can get important results just by looking at the behaviorof Ω on the lattice of L-theories.

Theorem 3.1.6A structural propositional logic L xΣ,$y is protoalgebraic iff Ω is monotoneon the lattice of L-theories.

When we say that a logic L has an algebraic semantics K, we are sayingthat, in some fashion the consequence relation of L can be captured by theequational consequence relation (Σ

K . Of course we would be happy if we couldalso capture, in some interesting way, the equational consequence within theconsequence apparatus of L. This is precisely the ideia behind the notion ofalgebraizable logic.

22 CHAPTER 3. ALGEBRAIZATION

Definition 3.1.7 (Algebraizable Logic)A structural propositional logic L xΣ,$y is algebraizable if there exists aclass K of Σ-algebras such that K is an algebraic semantics for L with definingequations Θ tδj λj : j P Ju where δj, λj P T pΣ, txuq and there also exists aset of formulas E tεi : i P Iu LΣptξ1, ξ2uq, called the set of equivalentialformulas, such that the following holds:

• for every set of equations Θ1 and every equation ϕ ψ we have

Θ1 (K ϕ ψ iff ErΘ1s $ Epξ1zϕ, ξ2zψq;

• ξ %$ ErΘpxzξqs;• x1 x2 )(K ΘrEpξ1zx1, ξ2zx2qs.The definition given by Blok and Pigozzi in [BP89] of algebraizable logic is

what is now called finitely algebraizable logic, since it assumes that the set Eof equivalential formulas and the set Θ of defining equations are finite sets andmoreover they consider exclusively finitary logics.

As we said before, the Leibniz operator plays an important role in this theoryof algebraization of logics. When Blok and Pigozzi introduced in a precise math-ematical sense the notion of algebraizable logic they wanted to give evidence thattheir definition was the proper one. So they gave some intrinsic natural char-acterizations of the notion of algebraizable logic that all characterize the samenotion. We will present one of those intrinsic characterizations, in which, asalready stated, the Leibniz operator is the main tool.

Theorem 3.1.8A structural propositional logic L xΣ,$y is algebraizable iff the Leibniz operatorsatisfies the following conditions:

i) Ω is injective;

ii) Ω is sup-preserving.

3.2 One equivalent definition

In this chapter we will look at the definitions introduced in the previous sectionand reinterpret them in terms of maps between the logic in study and unsortedequational logic which was introduced as an example in Section 2.4. In particularwe will give an equivalent definition of algebraizable logic. Let us begin with thenotion of algebraic semantics, now given from the point of view of maps.

3.2. ONE EQUIVALENT DEFINITION 23

Theorem 3.2.1Let L xΣ,$y be a structural propositional logic and K a class of Σ-algebras.Then K is an algebraic semantics for L iff there exists a conservative mapθ : LÑ EqnΣ

K, satisfying the following uniformity condition

there exists a set of equations Θ tδi λi : i P Iu where δi, λi P T pΣ, txuqsuch that for all ϕ P LΣ

θpϕq Θpxzϕq.Proof: We will prove each of the implications.

Suppose that L has an algebraic semantics. Then there exists a class Kof Σ-algebras and a set of Σ-equations, Θ tδj λj : j P Ju whereδj, λj P T pΣ, txuq such that, for all Γ LΣ and ϕ P LΣ we have thatΓ $ ϕ iff ΘrΓs (Σ

K Θpxzϕq. Consider θ : LÑ EqnΣK such that for ϕ P LΣ,

θpϕq Θpxzϕq. It is obvious that θ is uniform. We now have to provethat θ is a conservative map. Let Γ LΣ and ϕ P LΣ. Because Θ is aset of defining equations, we have that Γ $ ϕ iff ΘrΓs (Σ

K Θpxzϕq, this isprecisely, θrΓs (Σ

K θpϕq. Then we conclude that θ is a conservative map.

Suppose now that there exists a uniform conservative map θ : L Ñ EqnΣK

for some class K of Σ-algebras. By the uniformity of θ, there exists aset of equations Θ tδj λj : j P Ju where δj, λj P T pΣ, txuq, suchthat θpϕq Θpxzϕq. We want to prove that K is an algebraic semanticsfor L with defining equations Θ. Let Γ LΣ, ϕ P LΣ. Because θ is aconservative map, we have that Γ $ ϕ iff θrΓs (Σ

K θpϕq, this is preciselyΘrΓs (Σ

K Θpxzϕq. Then we conclude that K is an algebraic semantics forL with defining equations Θ.

QED

We stress again the importance of looking at the definition of algebraic se-mantic as the existence of a uniform map between L and the unsorted equationallogic EqnΣ

K for some class K of Σ-algebras.We now give an equivalent definition of algebraizable logic in terms of maps

between the target logic and EqnΣK for some class K of Σ-algebras.

Theorem 3.2.2A structural propositional logic L xΣ,$y is algebraizable iff there exists a strongrepresentation pθ, τq of L in EqnΣ

K, for some class K of Σ-algebras, such that θis uniform and τ satisfies the following uniformity condition:

there exists a set E LΣptξ1, ξ2uq such that for all Σ-equation ϕ ψ

τpϕ ψq Epξ1zϕ, ξ2zψq.

24 CHAPTER 3. ALGEBRAIZATION

Proof: We prove each of the implications.

Suppose first that L is algebraizable. Then there exists a class K ofΣ-algebras such that K is an algebraic semantics for L with defining equa-tions Θ tδj λj : j P Ju where δj, λj P T pΣ, txuq, and there exists a setof equivalence formulas E LΣptξ1, ξ2uq.Take θ : L Ñ EqnΣ

K where θpϕq Θpxzϕq and τ : EqnΣK Ñ L where

τpϕ ψq Epξ1zϕ, ξ2zψq. We will show that pθ, τq is a strong represen-tation of L in EqnΣ

K .

It is obvious that both θ and τ satisfy their respective uniformity condition.Let us now prove that θ and τ are both conservative maps.

Let Γ Y tϕu LΣ. Then, because Θ is a set of defining equations, Γ $ ϕiff ΘrΓs (Σ

K Θpxzϕq and this is precisely θrΓs (ΣK θpϕq, proving that θ is a

conservative map.

Let Θ1 be any set of Σ-equations and ϕ ψ any Σ-equation. Then, be-cause E is a set of equivalence formulas, we have that Θ1 (Σ

K ϕ ψ iffErΘ1s $ Epξ1zϕ, ξ2zψq and this is precisely τ rΘ1s $ τpϕ ψq, provingthat τ is a conservative map.

Let ϕ P LΣ. We have to prove that ϕ %$ τ rθpϕqs. We know thatξ %$ ErΘpxzξqs. Using Structurality we have that ϕ %$ ErΘpxzϕqs, butthis is precisely ϕ %$ τ rθpϕqs.Let ϕ ψ be any Σ-equation. We have to prove thatϕ ψ )(Σ

K θrτpϕ ψqs. We know that x1 x2 )(ΣK ΘrEpξ1zx1, ξ2zx2qs.

Using Structurality we have that ϕ ψ )(ΣK ΘrEpξ1zϕ, ξ2zψqs , but this

is precisely ϕ ψ )(ΣK θrτpϕ ψqs.

So, pθ, τq is a strong representation of L in EqnΣK .

Suppose now that there exists a strong representation pθ, τq of L in EqnΣK for

some class K of Σ-algebras, such that both θ and τ are uniform. Then thereexists a set of Σ-equations Θ tδj λj : j P Ju where δj, λj P T pΣ, txuqsuch that, for all ϕ P LΣ, θpϕq Θpxzϕq. There exists also a set offormulas E LΣptξ1, ξ2uq, such that, for all Σ-equation ϕ ψ we havethat τpϕ ψq Epξ1zϕ, ξ2zψq. We aim to prove that K is an algebraicsemantics for L with Θ the set of defining equations and that E is a set ofequivalential formulas.

Let ΓYtϕu LΣ. Because θ is a conservative map, we have that Γ $ ϕ iffθrΓs (Σ

K θpϕq, but this is the same that ΘrΓs (ΣK θpxzϕq.

Let Θ1 be any set of Σ-equations and ϕ ψ any Σ-equation. Because τ isa conservative map, we have that Θ1 (Σ

K ϕ ψ iff τ rΘ1s $ τpϕ ψq, butthis is the same that ErΘ1s $ Epξ1zϕ, ξ2zψq.

3.2. ONE EQUIVALENT DEFINITION 25

Let us now prove that ξ %$ ErΘpxzξqs. We know that, for allϕ P LΣ, ϕ %$ τ rθpϕqs, in particular ξ %$ τ rθpξqs, but this is precisely,ξ %$ ErΘpxzξqs.Finally, we will prove that x1 x2 )(Σ

K ΘrEpξ1zx1, ξ2zx2qs. We knowthat, for Σ-equation ϕ ψ, we have that ϕ ψ )(Σ

K θrτpϕ ψqs. Inparticular we get that x1 x2 )(Σ

K θrτpx1 x2qs, which is the same asx1 x2 )(Σ

K ΘrEpξ1zx1, ξ2zx2qs.QED

26 CHAPTER 3. ALGEBRAIZATION

Chapter 4

Beyond algebraization

4.1 Introduction

In the previous chapter we gave a definition of algebraizable logic in terms ofmaps between the target logic and unsorted equational logic EqnΣ

K for someclass K of Σ-algebras. What happens if we replace EqnΣ

K by another logic?Which properties of unsorted equational logic are really relevant to the processof algebraization? These are the questions we will explore in this chapter.

4.2 B-able logic

We now consider B xLB,$y a logic and L xΣ,$y a structural propositionallogic, that we call the base logic and the object logic, respectively. UnderlyingLB there exists a many-sorted signature Σ1 xS,Oy with distinguished l, b P Sso that

1. LB gT pΣ1qb;2. “ ” : LΣ Ñ TB gT pΣ1ql;3. “ ” : TB TB Ñ 2LB uniform in the sense that there exists a set

Θ T pΣ1, tx1, x2uqb such that “t1 t2” Θpx1zt1, x2zt2q;4. “t1 t2”,∆pxzt1q $B ∆pxzt2q para ∆ T pΣ1, txuqb5. for all ϕ and ψ P LΣ:

• $B “ϕ ϕ”

• “ϕ ψ” $B “ψ ϕ”

• “ϕ ψ”, “ψ δ” $B “ϕ δ”

27

28 CHAPTER 4. BEYOND ALGEBRAIZATION

where, for ϕ, ψ P LΣ, we abbreviate ““ϕ” “ψ”” by “ϕ ψ”.Item 5 says that this “ equality ” induces an equivalence relation between

L-formulas.When we generalize the definition of algebraizable logic by substituting EqnΣ

K

by another base logic B, we are not interested in any kind of logic. So, we imposedthese conditions to the logic B in order to maintain the “ equational flavor ” ofthe base logic. Note that clearly EqnΣ

K satisfies these conditions.

Definition 4.2.1 (Congruent context)A congruent context is a pair xδ, ξy, where δ P LΣ and ξ P Ξ, such that:

“ϕ ψ” $B “δpξzϕq δpξzψq”.Restricted to the congruent contexts, the relation “ϕ ψ” is a semi-

congruence on LΣ. It is not difficult to see that the set of the semi-congruenceson LΣ, denoted by Semi-CongrLΣ

, is a complete partial order, using the sameideas as in the analogue result for congruences over LΣ.

Generalizing the definition of Eqn-semantics we get:

Definition 4.2.2 (B-semantics)A structural propositional logic L xΣ,$y has a B-semantics if there exists aconservative map θ from L to B, satisfying the following uniformity condition:

there exists a set Θ LBptxuq such that θpϕq Θpxz“ϕ”q.Definition 4.2.3 (Compatibility)A semi-congruence on LΣ is compatible with a L-theory Γ provided that if ϕ P Γand xϕ, ψy P imply ψ P Γ.

Definition 4.2.4Let ThL be the lattice of the theories of L. Then we can define a function “Ω”so that:

“Ω” : ThL Ñ Semi-CongrLΣ

Γ ÞÑ largest semi-congruence compatible with Γ.

We need to prove that “Ω” is well defined, that is, for all L-theory Γ, thereexists the largest semi-congruence compatible with Γ.

Proposition 4.2.5The operator “Ω” can be given as

“Ω”pΓq txϕ, ψy : for all congruent context xδ, ξy, δpξzϕq P Γ iff δpξzψq P Γu.

4.2. B-ABLE LOGIC 29

Proof:Let ΘpΓq txϕ, ψy : for all congruent context xδ, ξy, δpξzϕq P Γ iff δpξzψq P Γu.We aim to prove that ΘpΓq “Ω”pΓq, that is, ΘpΓq is the largest semi-congruencecompatible with Γ. ΘpΓq is clearly a semi-congruence and compatible with Γ.We still have to prove that it is the largest one. Let Θ1 be a semi-congruencewith this property and let xδ, ξy be any congruent context. Since Θ1 is asemi-congruence we have, for every xϕ, ψy P Θ1, that

xδpξzϕq, δpξzψqy P Θ1.Thus by the compatibility of Θ1,

δpξzϕq P Γ iff δpξzψq P Γ.

Hence xϕ, ψy P ΘpΓq, and so Θ1 ΘpΓq. QED

Definition 4.2.6 (Proto-B-able logic)A structural propositional logic L xΣ,$y is proto-B-able if

xϕ, ψy P “Ω”pΓq implies ΓY tϕu %$ ΓY tψu.Proposition 4.2.7 L is proto-B-able if an only if “Ω” is monotone.

Proof: We will prove each of the implications.

Suppose that L is proto-B-able.

Consider Γ ∆. It’s enough to prove that “Ω”pΓq is compatible with ∆.Let xϕ, ψy P “Ω”pΓq. Then Γ, ϕ $ ψ and Γ, ψ $ ϕ. We have then that

if ϕ P ∆ then ΓYtϕu ∆ then, since ∆ is a theory and Γ, ϕ $ ψ, wehave that ψ P ∆;

and in the same fashion for the case where ψ P ∆.

Suppose now that Ω is monotone.

Let xϕ, ψy P “Ω”pΓq then, because Ω is monotone, we have that

xϕ, ψy P “Ω”ppΓ Y tϕuq$q. We know that, since “Ω”ppΓ Y tψuq$q iscompatible with pΓY tψuq$, then ϕ P pΓY tϕuq$ iff ψ P pΓY tϕuq$.Since ϕ P pΓ Y tϕuq$ we also have that ψ P pΓ Y tϕuq$, which isprecisely Γ, ϕ $ ψ;

xϕ, ψy P “Ω”ppΓ Y tψuq$q. We know that, since “Ω”ppΓ Y tψuq$q iscompatible with pΓY tψuq$, then ϕ P pΓY tψuq$ iff ψ P pΓY tψuq$.Since ψ P pΓ Y tψuq$ we also have that ϕ P pΓ Y tψuq$, which isprecisely Γ, ψ $ ϕ.

QED

30 CHAPTER 4. BEYOND ALGEBRAIZATION

Definition 4.2.8 (B-able logic)A structural propositional logic L xΣ,$y is B-able if there exists a strongrepresentation pθ, τq of L in B, so that θ is uniform and τ also satisfies thefollowing uniformity condition:

for all ϑ P LBptx1, . . . , xnuq there exists a set Γ LΣpξ1, . . . , ξnq so that

τpϑpx1z“ϕ1”, . . . , xnz“ϕn”qq Γpξ1zϕ1, . . . , ξnzϕnqWe will now prove some useful lemmas.

Theorem 4.2.9 Assuming that L is B-able with strong representation pθ, τq thenθTh is sup-preserving and injective.

Proof: θTh is sup-preserving by lemma 2.2.4 and is injective by lemma 2.2.8.QED

In the following we will assume that xτp“ϕ ξ”q, ξy is a congruent context.This is a natural condition and is verified in all examples that we have in mind,in particular in the case of C1. We believe this result follows immediately froma natural strengthening of the definition of B-able logic, namely the assumptionthat Γ LΣpξ1, . . . , ξnq is a congruent context in each variable.

Lemma 4.2.10Assuming that L is B-able with strong representation pθ, τq and supposing ξ doesnot occur in γ1, we have that τp“γ1 ξ”qpξzγ2q τp“γ1 γ2”q.Theorem 4.2.11If L is B-able with pθ, τq then

“Ω”pΓq txγ1, γ2y : “γ1 γ2” θThpΓqu.Proof: We prove the two inclusions.

“Ω”pΓq txγ1, γ2y : “γ1 γ2” θThpΓqu.Let Γ be a L-theory and xγ1, γ2y P “Ω”pΓq. Then for all congruent contextxδ, ξy we have that δpξzγ1q P Γ iff δpξzγ2q P Γ. Clearly “γ1 γ1” θThpΓqand, since θTh τ1, we conclude that τp“γ1 γ1”q Γ. Using the factthat xτp“γ1 ξ”q, ξy is a congruent context when ξ do not occur in γ1, wehave that τp“γ1 ξ”qpξzγ1q Γ iff τp“γ1 ξ”qpξzγ2q Γ. Since we al-ready know that τp“γ1 γ1”q Γ then we conclude that τp“γ1 γ2”q Γ.Using Reflexivity we get that Γ $ τp“γ1 γ2”q. Since θ is a conservativemap and using the fact that θrτp“γ1 γ2”qs %$B “γ1 γ2”, we concludethat “γ1 γ2” θThpΓq.

4.2. B-ABLE LOGIC 31

txγ1, γ2y : “γ1 γ2” θThpΓqu “Ω”pΓq.Suppose Γ is a L-theory and take xϕ, ψy P txγ1, γ2y : “γ1 γ2” θThpΓqu.Then “ϕ ψ” θThpΓq, that is θrΓs $B “ϕ ψ”. Let xδ, ξy be anycongruent context. Then “ϕ ψ” $B “δpξzϕq δpξzψq”. Using Cut weconclude that θrΓs $B “δpξzϕq δpξzψq”. Now suppose that δpξzϕq P Γ.We aim to prove that δpξzψq P Γ. Since Γ is a L-theory we have thatΓ $ δpξzϕq. We know that there exists a Θ LBptxuq such that, forall γ P LΣ, we have that θpγq Θpxzγq. Then by conservativeness of θwe conclude that θrΓs $B θpδpξzϕqq Θpxzδpξzϕqq and using property4 of B we have that θrΓs $B Θpxzδpξzψqq θpδpξzψqq. Again by theconservativeness of θ we conclude that Γ $ δpξzψq, and since Γ is a L-theorywe get that δpξzψq P Γ.In the same way we can prove that if δpξzψq P Γ then δpξzϕq P Γ. So weconclude that for all congruent contexts xδ, ξy, we have that δpξzϕq P Γ iffδpξzψq P Γ, that is, xϕ, ψy P “Ω”pΓq.

QED

32 CHAPTER 4. BEYOND ALGEBRAIZATION

Chapter 5

Example

One very interesting example where the theory of algebraization of logics does notapply are non-truth-functional logics. In particular we have the paraconsistentlogics of daCosta [dC63, dC74], as it was proved in [Mor80] and in [LMS91]. Themain motivation for this work was precisely the extension of the theory in orderto handle this kind of logic. In this chapter, we will show that the paraconsistentlogic C1 is B-able, where B is a two-sorted equational logic. This idea of usingtwo-sorted algebras of formulas and truth-values and including a valuation mapas an operator between the two sorts was first used in [CCC03].

5.1 The example of C1

Consider the paraconsistent logic C1 given as an example in Chapter 2.4.1. Inthis example we will use the two-sorted equational logic EqnΣφ,τ

K where we take asK the family of algebras such that the truth-values algebra is a boolean algebra.By simplicity of notation we will denote this logic by Eqnφ,τ

Bool. The two-sorted

signature Σφ,τ xS,Oy of Eqnφ,τBool, induced by the propositional signature Σ of

C1 and the usual signature of Boolean algebras, is as follows:

• S tφ, τu• Oεφ Σ0 Y Ξ;

• Oφkφ Σk for k ¡ 0;

• Oφτ tvu;• Oετ tJ,Ku;• Oττ tu;• Oτττ t[,\,ñu;

33

34 CHAPTER 5. EXAMPLE

• Ow H otherwise;

As we remarked in Chapter 2.4, since the family of boolean algebras is avariety, we will work in this example with the sintatical consequence relation$Eqn2 of the usual two-sorted equational consequence relation plus the additionof the following set Ax of axioms that specifies the family of boolean algebras and,since we want the valuation to be non-truth-functional, we also have valuationaxioms.

• Truth values axioms - specification of the class of all Boolean algebras:

x1 [ x2 x2 [ x1

x1 \ x2 x2 \ x1

x1 [ px2 [ x3q px1 [ x2q [ x3

x1 \ px2 \ x3q px1 \ x2q \ x3

x[ x x

x\ x x

px1 [ x2q \ x2 x2

px1 \ x2q [ x1 x1

x1 [ px2 \ x3q px1 [ x2q \ px1 [ x3q x1 \ px2 [ x3q px1 \ x2q [ px1 \ x3q x\J J K [ x K pxq x

x[ pxq K x\ pxq J px1 [ x2q px1q \ px2q px1 \ x2q px1q [ px2q x1 ñ x2 px1q \ x2

The problem with the the logic C1 in terms of the classical theory is that thenegation connective is non-truth-functional. So the valuation v P Oφτ cannotbe defined as an homomorphism. Indeed we have to impose with axioms theproperties we which our valuation should satisfy.

• Valuation axioms:

vptq J

5.1. THE EXAMPLE OF C1 35

vpfq K vpy1 ^ y2q vpy1q [ vpy2q vpy1 _ y2q vpy1q \ vpy2q vpy1 y2q vpy1q ñ vpy2q vpy1q ¤ vp y1q vp y1q ¤ vpy1q pvpy1q [ pvpy1q [ vp y1qqq K pvpy1q [ vpy2qq ¤ vppy1 ^ y2qq pvpy1q [ vpy2qq ¤ vppy1 _ y2qq pvpy1q [ vpy2qq ¤ vppy1 y2qq

Where we define ¤ as: t1 ¤ t2 iff t1 [ t2 t1. It is important to note thatvp yq vpyq does not follow.

First of all, note that, using “ϕ” vpϕq and “ϕ ψ” tvpϕq vpψqu,the two-sorted equational logic that we have introduce in Chapter 2 satisfies therequirements that we have imposed, in Chapter 4, to the base logic. Now we willshow that the logic C1 is Eqnφ,τ

Bool-able. So, all we have to do is define a strong

representation of C1 in Eqnφ,τBool.

Proposition 5.1.1C1 is Eqnφ,τ

Bool-able.

Proof: Consider the pair pθ, τq with

• θ : C1 Ñ Eqnφ,τBool such that ϕ

θÞÝÑ tvpϕq Ju; and

• τ : Eqnφ,τBool Ñ C1 such that t1 t2

τÞÝÑ tt1 t2 , t2 t1u

where t is defined inductively in the following way:

vpϕq ϕ;

J t;

K f;

ptq t f;

pt1 [ t2q t1 ^ t2 ;

36 CHAPTER 5. EXAMPLE

pt1 \ t2q t1 _ t2 ;

pt1 ñ t2q t1 t2

If Θ is a set of equations, Θ denotes the set teq : eq P Θu.It is trivial to see that both θ and τ satisfy their correspondent uniformity

condition. Note also that, like we remarked after the definition of B-able logic,it is easy to see that, in this particular example, xτp“ϕ ξ”q, ξy is a congruentcontext, since τp“ϕ ξ”q τpvpϕq vpξqq pϕ ξq.

We still have to prove that the pair (θ, τ) is a strong representation be-tween C1 and Eqnφ,τ

Bool. Like we remarked after the definition of strong repre-sentation, it is enough to prove that θ is a conservative map, and also thatt1 t2 %$Eqn2 θrτpt1 t2qs. First, let us prove that θ is a conservative map.

θ is a conservative map:

We aim to prove that Γ $C1 ϕ iff θrΓs $Eqn2 θpϕq. Let us first show thatif Γ $C1 ϕ then θrΓs $Eqn2 θpϕq. Suppose Γ $C1 ϕ. We want to see thatθpΓq $Eqn2 θpϕq, that is, tvpγq J : γ P Γu $Eqn2 vpϕq J. We will prove thisby induction in the length of the derivation of ϕ from Γ in C1. For this, we willprove first that for each axiom ϕ of C1 we have that $Eqn2 vpϕq J.

We will not prove this in detail for all the axioms, since some of the proofs areanalogue and straightforward. The first two axioms of C1 are axioms of CPL then,by the correction of CPL with respect to Boolean algebras, and the adequacy ofEqn2, we get the result. We will only give the idea how the rigorous derivationscan be done. We will do this so that the full detailed derivation can be easilyobtained.

Let us prove that the third axiom ppξ1 ^ ξ2q ξ1q is correct.

We want to prove that $Eqn2 vppy1 ^ y2q y1q J.

vppy1 ^ y2q y1q vpy1 ^ y2q ñ vpy1q vpy1q [ vpy2qq ñ vpy1q ppvpy1q [ vpy2qqq \ vpy1q ppvpy1qq \ pvpy2qqq \ vpy1q ppvpy2qq \ pvpy1qqq \ vpy1q pvpy2qq \ ppvpy1qq \ vpy1qq vpy2q \ J JThe axioms 4,5,6,7,8 are handled similarly.

Let us now prove that $Eqn2 vp y1 y1q J

5.1. THE EXAMPLE OF C1 37

vp y1 y1q vp y1q ñ vpy1q pvp y1qq \ vpy1q ppvp y1q [ vpy1qqq \ vpy1q ppvp y1qq \ pvpy1qqq \ vpy1q pvp y1qq \ ppvpy1qq \ vpy1qq pvp y1qq \ J JLet us now prove that $Eqn2 vppy1 ^ y2q py1 ^ y2qq J.We get that

vppy1 ^ y2q py1 ^ y2qq pvpy1q [ vpy2qq ñ vppy1 ^ y2qq ppvpy1q [ vpy2qq [ pvppy1 ^ y2qqqq ñ vppy1 ^ y2qq ppvpy1q [ vpy2qq [ pvppy1 ^ y2qqqq \ vppy1 ^ y2qq ppvpy1q [ vpy2qq \ pvppy1 ^ y2qqqq \ vppy1 ^ y2qq pvpy1q [ vpy2qq \ ppvppy1 ^ y2qqq \ vppy1 ^ y2qqq pvpy1q [ vpy2qq \ ppvppy1 ^ y2qqq \ vppy1 ^ y2qqq pvpy1q [ vpy2qq \ J JThe remaining axioms are treated in a very similar way.

After treating the axioms, we will now prove that the rule xtξ1, ξ1 ξ2u, ξ2ypreserves conservativeness, that is, tvpy1q J, vpy1 y2q Ju $Eqn2 vpy2q J.From vpy1 y2q J, using the valuation axioms, we get vpy1q ñ vpy2q J.Using the last boolean axiom and transitivity we get pvpy1qq\vpy2q J. Againin an informal way consider the sequence:

vpy2q K \ vpy2q pJq \ vpy2q pvpy1qq \ vpy2q JSo we conclude that tvpy1q J, vpy1 y2q Ju $Eqn2 vpy2q J.

Recall that our intention is to prove that if Γ $C1 ϕ then θpΓq $Eqn2 θpϕq.As we said, we will prove this by induction on the length n of the derivation ofϕ from Γ in C1.

Suppose first that n 1. Then we have two possibilities:

– ϕ P Γ.The result follows from the fact that vpϕq J P θpΓq.

– ϕ is an axiom.We have proved that $Eqn2 θpϕq for all axiom ϕ of C1. Then using

38 CHAPTER 5. EXAMPLE

monotonicity we have the result.

Suppose now, by induction hypothesis, that, for all ψ such that thereexists a derivation of ψ from Γ with length less than n, we have thatθpΓq $Eqn2 θpψq. Let´s look again at the derivation of ϕ from Γ. We havethree cases:

– ϕ P Γ.Equal to the base

– ϕ is an axiom.Equal to the base

– ϕ is obtained by the rule xtξ1, ξ1 ξ2u, ξ2y from ψ and ψ ϕ, bothwith derivations from Γ with length less than n. By induction hypoth-esis, we have that θrΓs $Eqn2 vpψq J and θrΓs $Eqn2 vpψ ϕq J.Since we proved that the rule xtξ1, ξ1 ξ2u, ξ2y preserves conservative-ness we have that θrΓs $Eqn2 vpϕq J.

We still have to prove the reverse implication, that is, if θrΓs $Eqn2 θpϕqthen Γ $C1 ϕ. We will prove this by contraposition, proving that if Γ &C1 ϕthen θrΓs &Eqn2 θpϕq. Recall that the logic C1 is complete for the semantic ofbivaluations. Assume that Γ &C1 ϕ. Then there exists a bivaluation v such thatv is a model of Γ but v does not satisfy ϕ. Now just observe that bivaluationsare particular cases of our models, with just the 2-valued Boolean algebra. Thenthe 2-valued Boolean algebra and the valuation v show that θrΓs *Σφ,τ

Bool θpϕq. Bythe soundness of Eqn2 we have that θrΓs &Eqn2 θpϕq. After proving that θ is aconservative map, we will prove that t1 t2 %$Eqn2 θrτpt1 t2qs. For this let uspresent two useful lemmas (see [CCC03] for details).

Lemma 5.1.2 If t P T pΣ2qτ , then $Eqn2 vptq t.

Lemma 5.1.3 If t1, t2 P T pΣ2qτ , A is a Σ2-algebra that satisfies the boolean andthe valuation axioms and ρ is a assignment over A then

rrt1ssρA rrt2ssρA iff vAprrt1 t2ssρAq JA

The last lemma says precisely that t1 t2 (Σφ,τ

Bool pt1 t2q J and also thatpt1 t2q J (Σφ,τ

Bool t1 t2. This is same as t1 t2 )(Σφ,τ

Bool pt1 t2q J. Interms of the maps θ and τ , this is exactly t1 t2 )(Σφ,τ

Bool θrτpt1 t2qs. By theadequacy of Eqn2 we get that t1 t2 %$Eqn2 θrτpt1 t2qs.

QED

Chapter 6

Conclusions

In Chapter 2 we introduced the notions of logic, maps of logics and gave someexamples of logics. In Chapter 3 we gave an alternative characterization of somenotions of the existing theory of algebraization of logics, in terms of maps betweenthe target logic and unsorted equational logic.

Our main contribution was the generalization of the theory, given in Chapter4. We generalized the notions we characterized in Chapter 3, by abstractingaway from unsorted equational logic the relevant properties for the algebraizationprocess, and then replacing it by another base logic with these properties.

In Chapter 5 we gave a concrete example of the application of our theory tothe paraconsistent logic C1 of daCosta.

It remained to prove that xτp“ϕ ξ”q, ξy is a congruent context. This isa natural condition and we believe that, with a natural strengthening of thedefinition of B-able logic, this condition will immediately follow.

Our focus was not on the characterization of all the notions of the algebraiza-tion hierarchy but on the characterization of the main notion of the theory, thatof algebraizable logic. Nevertheless, we hope that most of these notions could becharacterized and therefore generalized, in particular the notion of equivalentiallogic whose generalization we believe to be the natural converse of the notion ofB-semantics.

We also want to know if every proto-B-able logic is B-able, since it is wellknown that every protoalgebraizable logic is algebraizable, being clear that oureffort is also on generalizing the main results of the existing theory.

In the future we intend to study the semantic consequences of this generaliza-tion, namely using logical matrices. In particular in the example of C1 we intendto give an “ algebraic flavor ” to its non-truth-functional semantic of bivaluations,and further recover the so-called “ daCosta algebras ” now using an inequationalbase. We would also like to study what could be obtain by substituting unsortedequational logic by other interesting base logics, namely multisorted equationallogic, inequational logics based on ordered algebras, and logics based on partialalgebras and on non-deterministic algebras.

39

40 CHAPTER 6. CONCLUSIONS

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