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1

Relational Syllogistics

Nikolay Ivanov

naivanov@gmail.com

Dimiter Vakarelov

dvak@fmi.uni-sofia.bg

Faculty of Mathematics and Informatics, Sofia University5 James Bourchier Blvd., 1164 Sofia, Bulgaria

Abstract

We present a quantifier-free Hilbert-style axiomatization, based onclassical propositional logic, of a system of relational syllogistic formal-izing the following binary relations between classes (of objects):

a ≤ bdef⇔ ∀x(x ∈ a⇒ x ∈ b)

〈Q1Q2〉(a, b)[α]def⇔ (Q1x ∈ a)(Q2y ∈ b)

(

(x, y) ∈ α)

,

where a, b denote arbitrary classes, Q1, Q2 ∈ {∀, ∃}, and α denotes anarbitrary binary relation between objects. The language of the logiccontains only variables denoting classes, determining the set of classterms, and variables denoting binary relations between objects, deter-mining the set of relational terms. Both classes of terms are closedunder the standard Boolean operations. The set of relational terms isalso closed under taking the converse of a relation α−1. The results ofthe paper are the completeness theorem with respect to the intendedsemantics and the computational complexity of the satisfiability prob-lem.

1

Contents

1 Introduction 2

2 Syntax and semantics 7

2.1 Language . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72.2 Semantics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8

3 Axioms and inference rules 8

4 Theories 11

5 Boolean algebras of classes of terms 14

5.1 The Boolean algebra of classes of Boolean terms . . . . . . . 145.2 The Boolean algebra of classes of relational terms . . . . . . . 145.3 The Boolean algebra of symmetric classes of relational terms 23

6 Completeness 25

7 Complexity 36

8 Concluding remarks 37

1 Introduction

It is a well-known fact that the syllogistic was the first formal theory oflogic introduced in Antiquity by Aristotle. It was presented by Lukasiewiczin [10] as a quantifier-free extension of propositional logic, having as atomsthe expressions A(a, b) (All a are b) and I(a, b) (Some a are b) and their

negations – E(a, b)def= ¬I(a, b) and O(a, b)

def= ¬A(a, b), where a, b are class

variables interpreted in the ordinary language by noun phrases like ‘men’,‘Greeks’, ‘mortal’. An example of an Aristotelian syllogism taken from [10]is: “If all men are mortal and all Greeks are men, then all Greeks are mortal”.The specific axioms for A and I from [10] are (in a different logical notation)the following: L1. A(a, a), L2. I(a, a), L3. A(b, c) ∧ A(a, b) → A(a, c), L4.A(b, c)∧I(b, a)→ I(a, c). The only rules are Modus Ponens and substitutionof a class variable with another class variable. The standard semantics ofthis language consists of interpreting class variables by arbitrary non-emptysets, A(a, b) as set-inclusion a ⊆ b, and I(a, b) as the overlap relation betweensets – a ∩ b 6= ∅. Note that the non-emptiness of classes, which is typicalof the Antiquity, is essential for the axiom I(a, a) to be true. Under thisset-theoretical interpretation of syllogistic one can easily see that all of itsspecific axioms are simple properties of inclusion and overlap.

Wedberg introduced in [32] variations of the Aristotelian syllogistic withthe operation of complementation a′ on class variables interpreted as the

2

Boolean complement of the variable in a given universe. Wedberg’s systemwith unrestricted interpretation on class variables is based on the followingaxioms (containing only A and complementation because I(a, b) can be de-fined by ¬A(a, b′)): W1. A(a, a′′), W2. A(a′′, a), W3. A(a, b) ∧ A(b, c) →A(a, c), W4. A(a, b)→ A(b′, a′). W5. A(a, a′)→ A(a, b).

Simple Henkin-type completeness and decidability proofs for Lukasiewicz’s,Wedberg’s and some other classical syllogistic systems were given by Shep-herdson in [28]. Shepherdson’s completeness proofs are based on the notionof partially ordered set S with an operation of complementation ‘′’ satis-fying the following axioms for all a, b ∈ S: a′′ = a, a ≤ b → b′ ≤ a′, anda ≤ a′ → a ≤ b. Note that such structures are now known as orthoposets (see[16]). The completeness proofs are based on a simple embedding theorem forpartial orders with complementation into Boolean algebras of subsets of agiven universe such that A is interpreted as set inclusion and ‘′’ is interpretedas set complement. Shepherdson also mentioned in [28] systems containingnot only complementation on class terms, but also Boolean intersection. In[29, 30] Sotirov showed that the roots of the idea of complementation andsome other algebraic operations on classes goes back to Leibnitz.

We call the variations of Aristotelian syllogistic, mentioned above, clas-sical syllogistics. All such logics are based on propositional logic, butweaker systems, which do not contain the propositional connectives or con-tain only negation, have also been considered in the literature. For in-stance, Moss in [15, 16], motivating mainly with applications of syllogisticsto natural languages, considers various syllogistics of classical type, based onlanguages containing primitives like A, I,E,O with or without complemen-tation on class variables. The corresponding axiomatic systems are basedon a set of inference rules with finite sets of atomic premises.

For a long time classical syllogistic has been considered only as an intro-ductory course on elementary logic. Nowadays, however, syllogistic theories,extended and modified in various ways, find applications in different areas,mainly in natural language theory [12, 15, 16, 17, 18, 22, 23, 24, 25, 26,27, 31], computer science and Artificial Intelligence [3, 13, 8], generalizedquantifiers [33], argumentation theory [20], cognitive psychology [9, 21] andothers (the list of references is fairly incomplete). Most of the extended syl-logistics generalize the standard syllogistic relations A(a, b), I(a, b), E(a, b)and O(a, b) using in their definitions various non-standard quantifiers arisingfrom natural language. Examples: “At least 5 a are b”, “Exactly 5 a are b”,“Most a are b”, “All except 2 a are not b”, “Many a are not b”, “Only a fewa are not b”, “Usually some a are not b”, etc.

Some of the relations between classes a and b are determined by certainrelations between their members, expressible by some verbs or verb phrasesin the natural language. Examples: “All students read some textbooks”,“Some people don’t like any cat”, “Some vegetarians eat some fish”, “Allvegetarians don’t like any meat”, “At least 5 students read all textbooks”,

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etc. Syllogistics studying such expressions are called by Moss and Pratt-Hartmann [27, 24] relational syllogistics.

The existence of so many syllogistics raises the question of what are thecharacteristic features qualifying a given logic as syllogistic. We list twosuch features:

(1) Syllogistics can be considered as special theories of classes, which donot have variables running over elements of the classes. In this sensesyllogistics are ‘point-free’.

(2) Syllogistics are quantifier-free logics.

One observation related to the above remark is the following: if a givensyllogistic logic is axiomatized and has a completeness theorem with respectto the intended set-theoretic models, it is also open for other interpreta-tions based on intuitions not directly related to interpretations with classes.For instance, Sheperdson’s completeness proof for Wedberg’s system withunrestricted interpretations on class variables goes trough two steps: first,completeness with respect to partially ordered sets with complementation(which are abstract structures, not directly related to interpretations withclasses), and second, via the representation theory for posets with com-plementation, he obtained the completeness with respect to the intendedsyllogistic models. Analogously, the universal theory of Boolean algebrascan also be treated as syllogistic via the representation theory of Booleanalgebras (in a sense this is the largest classical syllogistic). Even more, ax-iomatic versions of some logics, not considered syllogistics, may happen tobe equivalent to some already known or yet unknown syllogistic logics. Insuch cases the abstract axiomatic system can also be treated as a syllogistic.This feature makes it possible to transfer methods and constructions fromalready developed logics with non-syllogistic semantics into new logics withintended syllogistic semantics. Here are some examples.

Orlowska in [19], Section 4, showed that a great variety of informationrelations between objects in Pawlak’s information systems can be interpretedin Vedberg’s and Sheperdson’s syllogistics, which makes it possible to usesyllogistics for reasoning about information relations in information systems.

Other examples are some logics related to the region-based theory ofspace studied in [1, 2]. The language of these logics from [1] contains vari-ables for regions considered as physical bodies, Boolean operations on re-gions, and two basic relations between regions: part-of a ≤ b and contactaCb. The relational semantics of this language is based on structures (W,R).A valuation v in (W,R) maps every region term a to a subset v(a) of W , vacting homomorphically with respect to Boolean operations. The semanticsof part-of and contact relations in the model (W,R, v) are the following:

(W,R, v) � a ≤ b iff v(a) ⊆ v(b) ,

4

(W,R, v) � aCb iff(

∃x ∈ v(a))(

∃y ∈ v(b))

(xRy) .

According to the characteristic features of syllogistics stated above, thislogic is obviously syllogistic. This semantics shows that a ≤ b has themeaning of the syllogistic predicate A(a, b) – “all a are b”. Since our lan-guage contains Boolean complementation on region terms, then all standardsyllogistic relations are definable by ≤ as in Wedberg’s and Shepherdson’ssystems. Note that in [1] the logics of the region-based theory of space arecalled modal, because they possess many features from the ordinary modallogic like modal definability, canonical model construction, filtration andp-morphisms (in modified forms).

In this paper we introduce a quite rich relational syllogistic combiningsome technical ideas from [1, 2] and some semantical ideas from the afore-mentioned papers on relational syllogistics. The language of the logic is sim-ilar to the language of Dynamic Logic and contains both class variables andrelational variables from which we construct complex terms. Both classes ofterms are closed with respect to all Boolean operations while on relationalterms we also have the operation ‘−1’ of taking the converse. We have fiveatomic predicates from which we construct the set of formulas using thepropositional connectives:

(1) a ≤ b

(2) 〈∃∃〉(a, b)[α]

(3) 〈∀∃〉(a, b)[α]

(4) 〈∃∀〉(a, b)[α]

(5) 〈∀∀〉(a, b)[α]

Here a, b are class terms and α is a relational term. The semantical structuresare the same as in Dynamic logic (W,R, v), where R is a mapping fromrelational variables into the set of binary relations on W and v is a mappingfrom class variables into the set of all subsets of W . The semantics of〈∃∃〉(a, b)[α] is the same as the semantics of contact relation:

(W,R, v) � 〈∃∃〉(a, b)[α] iff(

∃x ∈ v(a))(

∃y ∈ v(b))(

xR(α)y)

.

The semantics of the remaining atomic formulas is analogous. Linguisticallythese formulas cover the examples like “All students read some textbooks”,taking all combinations of ‘some’ and ‘all’, considering subject wide scopereading. Having the operation α−1, we may also express in our languagethe object wide scope reading (see [17] for more details). By means ofthe Boolean operations on relations we may express ‘compound verbs’. Forinstance, if α denotes the verb ‘to read’ and β denotes ‘to write’, then α∩−β

5

denotes the compound verb ‘to read but not to write’. Also by ‘−1’ we mayexpress the passive voice of the verbs.

We present a Hilbert-style axiomatic system for the logic based on theaxioms of propositional logic, Modus Ponens and several additional finitaryinference rules satisfying some syntactic restrictions. The list of axiomscontains the finite list of axiom schemes for Boolean algebra plus a finite listof axiom schemes for the basic predicates.

Logics with similar rules, which in a sense imitate quantification, andcanonical constructions for corresponding completeness proofs are studied in[1, 2]. We adopt and modify these canonical techniques. There are, however,new difficulties, which have no analogs in [1, 2], related to the axiomatizationof intersection and complementation of relations. As in modal logic, theseoperations are not modally definable (see [4, 5, 6]). That is why we need tocombine canonical constructions from [1, 2] with a modification of a copyingconstruction from [4, 5, 6]. The formulas of our logic have a translationinto Boolean Modal Logic [4, 5] extended with converse on relational terms.We obtain that the complexity of the satisfiability problem for the logic isthe same as the complexity of BML [11], i.e. NExpTime if the languagecontains an infinite number of relational variables, and ExpTime if only afinite number of relational variables is available.

The present paper is an extended version of the first author’s master’sthesis [7] and was inspired by [27], especially by the presentation of [27]by Moss as an invited lecture at the Conference “Advances in Modal Logic2008” [14].

The paper is organized as follows.In section 2, we introduce the language and semantics of our logic. We

do not use the standard notation for modal formulas. A notation, similarto the one used for modal logics for the regional theory of space, will proveto be more appropriate.

In section 3, we list the axioms and inference rules of our logical sys-tem. We use the axioms for the contact relation from [1] and some addi-tional axioms and inference rules which essentially imitate quantifiers in ourquantifier-free language.

In section 4, we review the technique for building maximal theories inthe presence of inference rules which imitate quantifiers.

In section 5, we define some equivalence relations between terms. Weexamine some Boolean algebras of equivalence classes of terms. The pointsin our canonical model are built from ultrafilters in these Boolean algebras.

In section 6, we prove the completeness of our axiomatic system. Theproof uses some ideas from the completeness proofs for modal logics of thecontact relation [1] and BML [4, 5].

In section 7, we discuss the complexity of the satisfiability problem forthe logic under consideration and some of its fragments.

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2 Syntax and semantics

2.1 Language

The language consists of the following sets of symbols:

(1) an infinite set BVAR of Boolean variables;

(2) the Boolean constants 0 and 1;

(3) a set RVAR of relational variables such that RVAR ∩ BVAR = ∅ and|RVAR| ≤ |BVAR|;

(4) relational constants 0R and 1R;

(5) functional symbols ∩, ∪ and − for the Boolean operations meet, joinand complement;

(6) functional symbol −1;

(7) relational symbols ≤, 〈∃∃〉, 〈∀∃〉, 〈∀∀〉, 〈∃∀〉;

(8) propositional connectives ∧,∨,¬,→,↔;

(9) propositional constants ⊥ and ⊤;

(10) the symbols “(”, “)”, “[”, “]”, “,”.

As the language is uniquely determined by the pair (BVAR,RVAR), wewill also call (BVAR,RVAR) a language. In the first two sections we willkeep the language fixed.

Boolean terms are built from the Boolean constants and Boolean vari-ables by means of the Boolean connectives. If B ⊆ BVAR, we will denoteby BTerm(B) the set of all Boolean terms with variables from B.

We define the set of relational terms RTerm(R) with variables in R ⊆RVAR inductively:

• {0R, 1R} ⊆ RTerm(R);

• R ⊆ RTerm(R);

• If α ∈ RTerm(R) then{

−α,α−1}

⊆ RTerm(R);

• If {α, β} ⊆ RTerm(R) then {α ∩ β, α ∪ β} ⊆ RTerm(R).

Atomic formulas have one of the forms a ≤ b, 〈∃∃〉(a, b)[α], 〈∀∃〉(a, b)[α],〈∀∀〉(a, b)[α] and 〈∃∀〉(a, b)[α], where a and b are Boolean terms and α is arelational term.

Formulas are built from atomic formulas by means of the propositionalconnectives. We will use a = b as an abbreviation for the formula (a ≤b) ∧ (b ≤ a). If B ⊆ BVAR and R ⊆ RVAR, we will denote by Form(B,R)the set of all formulas with Boolean variables from the set B and relationalvariables from R.

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2.2 Semantics

Let W be a set and let R : RVAR→ P (W 2) and v : BVAR→ P (W ) be twofunctions. R is a valuation of the relational variables, which maps everyrelational variable to a relation on W . The valuation v of the Booleanvariables maps Boolean variables to subsets of W . We will call the pair(W,R) a frame and the triple (W,R, v) a model. The set W is called thedomain of that frame or model.

The only frame with an empty domain is

F∅ =(

∅,RVAR× {∅})

and the only model built over that frame is

M∅ =(

∅,RVAR× {∅},BVAR × {∅})

.

We will call them the empty frame and the empty model respectively.We extend the function R to the set of all relational terms by defining

v(0R) = ∅ and v(1R) = W 2 and interpreting the symbols ∩, ∪, − and −1

by intersection, union, complement to W 2 and taking the converse of therelations on W . We extend the function v to the set of all Boolean termsanalogously.

We define the truth and falsity of atomic formulas in a model (W,R, v)by the following equivalences:

(W,R, v) � a ≤ b⇔ v(a) ⊆ v(b)

(W,R, v) � 〈∃∃〉(a, b)[α] ⇔(

∃x ∈ v(a))(

∃y ∈ v(b))(

(x, y) ∈ R(α))

(W,R, v) � 〈∀∃〉(a, b)[α] ⇔(

∀x ∈ v(a))(

∃y ∈ v(b))(

(x, y) ∈ R(α))

(W,R, v) � 〈∀∀〉(a, b)[α] ⇔(

∀x ∈ v(a))(

∀y ∈ v(b))(

(x, y) ∈ R(α))

(W,R, v) � 〈∃∀〉(a, b)[α] ⇔(

∃x ∈ v(a))(

∀y ∈ v(b))(

(x, y) ∈ R(α))

.

The definition is extended to the set of all formulas according to the standardmeaning of the propositional connectives.

These formulas have equivalents in the language of Boolean Modal Logic(BML) [4, 5] extended with a symbol −1 for the converse of the accessibilityrelation. If we denote the global box by G and the global diamond by E,then:

a ≤ b is equivalent to G(a→ b)〈∃∃〉(a, b)[α] is equivalent to E

(

a ∧ 〈α〉b)

〈∀∃〉(a, b)[α] is equivalent to G(

a→ 〈α〉b)

〈∀∀〉(a, b)[α] is equivalent to G(

a→ [−α]¬b)

〈∃∀〉(a, b)[α] is equivalent to E(

a ∧ [−α]¬b)

3 Axioms and inference rules

We will use the following notation: If A is a formula or a term, then BVar(A)and RVar(A) denote respectively the sets of Boolean and relational variables

8

which occur in A. Also, BVar(A1, . . . , An) =⋃n

i=1 BVar(Ai).The idea behind the list of axioms is the following. Since 〈∃∃〉 is the

well-known contact relation from the modal logics of region-based theoriesof space [1], we use the same set of axioms for it. The truth of each of theother three relations 〈Q1Q2〉 is linked to the truth of 〈∃∃〉 by the followingequivalences:

(W,R, v) � 〈∀∃〉(a, b)[α]⇔

⇔(

∀p ⊆W)

(

v(a) ∩ p = ∅ ∨(

∃x ∈ p)(

∃y ∈ v(b))(

(x, y) ∈ R(α))

)

(W,R, v) � 〈∀∀〉(a, b)[α]⇔

⇔(

∀p ⊆W)

(

v(b) ∩ p = ∅ ∨(

∀x ∈ v(a))(

∃y ∈ p)(

(x, y) ∈ R(α))

)

(W,R, v) � ¬〈∃∀〉(a, b)[α]⇔

⇔(

∀p ⊆W)

(

v(a) ∩ p = ∅ ∨ ¬(

∀x ∈ p)(

∀y ∈ v(b))(

(x, y) ∈ R(α))

)

These equivalences express the following simple statement. If ϕ(x) isa property of points x in some set W and A ⊆ W , then (∀x ∈ A)ϕ(x) isequivalent to

(∀X ⊆W )(

X ∩A 6= ∅ ⇒ (∃x ∈ X)ϕ(x))

.

Thus, we expressed the universally quantified property (∀x ∈ A)ϕ(x) by theexistentially quantified property (∃x ∈ X)ϕ(x) and a quantification oversets. Substituting the appropriate formulas in the place of ϕ(x), we get theabove equivalences.

The left-to-right direction of each of these equivalences is a universalformula. We add it to the set of axioms. These are the axioms (A6), (A7),(A8) in the list below. We call them axioms for quantifier change, becausethey link relations 〈Q1Q2〉 and 〈Q′

1Q′2〉, which differ in the first or second

quantifier.The right-to-left directions of the equivalences are not universal formulas.

Since we do not have quantifiers in our language, we cannot write theseconditions as axioms. Instead, we imitate them by inference rules with aspecial variable, corresponding to the quantified variable p in the aboveequivalences, using a technique from [1]. These are the rules (R1), (R2),(R3) from the list below.

We will also use a rule whose only purpose is to derive all formulas ofthe form

a = 0 ∨ ¬〈∀∀〉(a, a)[

α1 ∩ (−α1−1) ∪ α2 ∩ (−α2

−1) ∪ · · · ∪ αk ∩ (−αk−1)

]

.

These formulas state that a pair of points (x, x) ∈W 2 may not be an elementof the valuation of any relational term of the form α ∩ (−α−1).

The set of axioms consists of the following groups of formulas:

9

(1) A sound and complete set of axiom schemes for propositional calculus;

(2) A set of axioms for Boolean algebra in terms of the relation ≤;

(3) Axioms for equality:

〈Q1Q2〉(a, b)[α] ∧ a = c→ 〈Q1Q2〉(c, b)[α] (A1)

〈Q1Q2〉(a, b)[α] ∧ b = c→ 〈Q1Q2〉(a, c)[α] (A2)

(4) Axioms for 〈∃∃〉:

a = 0 ∨ b = 0→ ¬〈∃∃〉(a, b)[α] (A3)

〈∃∃〉(a ∪ b, c)[α] ↔ 〈∃∃〉(a, c)[α] ∨ 〈∃∃〉(b, c)[α] (A4)

〈∃∃〉(a, b ∪ c)[α]↔ 〈∃∃〉(a, b)[α] ∨ 〈∃∃〉(a, c)[α] (A5)

(5) Axioms for quantifier change:

〈∀∃〉(a, b)[α]→ a ∩ c = 0 ∨ 〈∃∃〉(c, b)[α] (A6)

〈∀∀〉(a, b)[α]→ b ∩ c = 0 ∨ 〈∀∃〉(a, c)[α] (A7)

¬〈∃∀〉(a, b)[α]→ a ∩ c = 0 ∨ ¬〈∀∀〉(c, b)[α] (A8)

(6) Axioms for 0R and 1R:

¬〈∃∃〉(a, b)[0R] (A9)

〈∀∀〉(a, b)[1R] (A10)

(7) Axioms for ∩, ∪, − and −1 in relational terms:

〈∀∀〉(a, b)[α ∩ β]↔ 〈∀∀〉(a, b)[α] ∧ 〈∀∀〉(a, b)[β] (A∩)

〈∃∃〉(a, b)[α ∪ β]↔ 〈∃∃〉(a, b)[α] ∨ 〈∃∃〉(a, b)[β] (A∪)

〈∀∀〉(a, b)[−α]↔ ¬〈∃∃〉(a, b)[α] (A−)

〈∃∃〉(a, b)[α−1]↔ 〈∃∃〉(b, a)[α] (A−1)

Inference rules:

(1) Modus ponens (MP): ϕ,ϕ→ ψ ⊢ ψ;

(2) Special rules imitating quantifiers: If p ∈ BVAR \ BVar(ϕ, a, b) then

ϕ→ a ∩ p = 0 ∨ 〈∃∃〉(p, b)[α] ⊢ ϕ→ 〈∀∃〉(a, b)[α] (R1)

ϕ→ b ∩ p = 0 ∨ 〈∀∃〉(a, p)[α] ⊢ ϕ→ 〈∀∀〉(a, b)[α] (R2)

ϕ→ a ∩ p = 0 ∨ ¬〈∀∀〉(p, b)[α] ⊢ ϕ→ ¬〈∃∀〉(a, b)[α] (R3)

ϕ→ a ∩ p = 0 ∨ ¬〈∀∀〉(p, p)[α] ⊢

⊢ ϕ→ a = 0 ∨ ¬〈∀∀〉(a, a)[α ∪ β ∩ (−β−1)] (R4)

The variable p is called the special variable of the rule.

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The notions of proof and theorem are defined in the standard way. Wewill denote by Th(BVAR,RVAR) the set of all theorems in the language(BVAR,RVAR).

Proposition 3.1. All theorems are true in all models.

Proof. All axioms are true in all models and the rule of MP preserves truthin each model. Each of the special rules preserves validity in each frame,that is: if the premise is true in all valuations on a given frame, then so isthe conclusion.

4 Theories

In this section, we review the definition of theories and the construction ofmaximal theories from consistent sets of formulas in the presence of specialrules of inference, which imitate quantifiers [1].

Definition 1 (Theory). Let Γ ⊆ Form(BVAR,RVAR) and let V ⊆ BVAR.We say that the pair (V,Γ) is a theory in the language (BVAR,RVAR) whenthe following conditions hold:

(1) Th(BVAR,RVAR) ⊆ Γ;

(2) If ϕ,ϕ→ ψ ∈ Γ then ψ ∈ Γ;

(3) Let L(p) be a premise of a special rule, where p ∈ BVAR is the specialvariable of the rule. Let R be the conclusion of that rule, p ∈ BVAR \(

V ∪ BVar(R))

and L(p) ∈ Γ. Then R ∈ Γ.

We say that the theory (V,Γ) is consistent if ⊥ /∈ Γ.We say that the theory (V,Γ) in the language (BVAR,RVAR) is a good

theory if |V | < |BVAR|.The theory (V,Γ) is called complete if it is consistent and for each formula

ϕ in its language we have either ϕ ∈ Γ or ¬ϕ ∈ Γ.The theory (V,Γ) in the language (BVAR,RVAR) is called rich if for

each special rule with premise L(p) ∈ Form(BVAR,RVAR) and conclusionR the following implication holds:

R /∈ Γ⇒ (∃p ∈ BVAR)(

L(p) /∈ Γ)

.

(The conclusion R uniquely determines L(p) up to a substitution of p withanother Boolean variable.)

Notation. We define a relation ⊆ between theories in the same language:

(V1,Γ1) ⊆ (V2,Γ2)def⇔ V1 ⊆ V2 ∧ Γ1 ⊆ Γ2.

We will write ϕ ∈ (V,Γ) if ϕ ∈ Γ.

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Definition 2. We say that a set of formulas A is consistent if there is aconsistent theory (V,Γ) such that A ⊆ Γ.

We fix a language (BVAR,RVAR) and introduce the following notation:

Notation. If Γ is a set of formulas and ϕ is a formula,

Γ + ϕdef=

{

ψ ∈ Form(BVAR,RVAR)∣

∣ ϕ→ ψ ∈ Γ}

.

If T = (V,Γ) is a theory and ϕ is a formula,

T ⊕ ϕdef=

(

V ∪ BVar(ϕ),Γ + ϕ)

.

Lemma 4.1. If T = (V,Γ) is a good theory and ϕ is a formula then:

(1) T ⊕ ϕ is a good theory, T ⊆ T ⊕ ϕ and ϕ ∈ T ⊕ ϕ;

(2) T ⊕ ϕ is inconsistent ⇔ ¬ϕ ∈ Γ;

(3) If ψ → L(p) and ψ → R are the premise and conclusion of a specialrule and the theory T⊕¬(ψ → R) is consistent, then there is a Booleanvariable p ∈ BVAR \

(

V ∪ BVar(ψ → R))

, such that T ⊕ ¬(ψ →R)⊕ ¬

(

ψ → L(p))

is a good consistent theory.

Proof. Straightforward verification.

Lemma 4.2 (Lindenbaum). Every good consistent theory T0 = (V0,Γ0) iscontained in a complete rich theory T = (V,Γ).

Proof. Let T0 = (V0,Γ0) be a good consistent theory. Let κ = |BVAR| andForm(BVAR,RVAR) = {ϕα|α < κ}. We will build a sequence of theories{Tα}α<κ with the following properties:

(1) Tα is a good consistent theory;

(2) ¬ϕα ∈ Tα or ϕα ∈ Tα+1;

(3) If ϕα ∈ Tα+1, ϕα = ¬R and R is the conclusion of a special rule, thenthere is a Boolean variable p, such that the negated premise of the rule¬L(p) belongs to Tα+1.

Suppose that Tβ have been defined for β < α. We will define Tα. Weconsider the following cases:

(1) α = β+ 1 for some β and Tβ = (Vβ,Γβ) has already been defined. Weneed to consider two possibilities for the theory Tβ ⊕ ϕβ :

(a) Tβ ⊕ ϕβ is consistent. We have two cases depending on ϕβ :

12

i. ϕβ does not have the form of a negated conclusion of a specialrule. In this case we define Tα = Tβ ⊕ ϕβ.

ii. ϕβ = ¬R and R is a conclusion of a special rule. Let L(p)be the premise of that rule. According to Lemma 4.1 thereis a Boolean variable p ∈ BVAR \

(

Vβ ∪BVar(ϕβ))

such thatTβ ⊕ϕβ ⊕¬L(p) is a good consistent theory. We choose sucha variable p and define Tα = Tβ ⊕ ϕβ ⊕¬L(p).

(b) Tβ ⊕ϕβ is inconsistent. Then Lemma 4.1 tells us that ¬ϕβ ∈ Γβ.We define Tα = Tβ.

(2) α =⋃

α. We define Vα =⋃

{Vβ |β < α}, Γα =⋃

{Γβ|β < α} andTα = (Vα,Γα).

It is easy to verify the three properties of Tα stated above by inductionon α. We define V =

⋃

{Vα|α < κ}, Γ =⋃

{Γα|α < κ} and T = (V,Γ).By the properties of Tα for α < κ it easily follows that T is a complete richtheory.

The Lindenbaum lemma is only applicable to good theories. That is whywe will also need the following lemma:

Lemma 4.3. Let T0 = (V,Γ0) be a consistent theory in a language

(BVAR0,RVAR)

and let BVAR ⊇ BVAR0 with |BVAR| > |V | be an extension of BVAR0

with a set BVAR \ BVAR0 of new Boolean variables. Then there is a goodconsistent theory T = (V,Γ) in the language (BVAR,RVAR) such that Γ0 ⊆Γ.

Proof. Define

Γ ={

ϕ∣

∣ (∃ψ ∈ Γ0)(

ψ → ϕ ∈ Th(BVAR,RVAR))}

.

It is straightforward to check that T = (V,Γ) has the desired properties.

Corollary 4.4. (1) Every consistent set of formulas is contained in a goodconsistent theory in an extension of the language with a set of newBoolean variables.

(2) Every consistent set of formulas is contained in a complete rich theoryin an extension of the language with a set of new Boolean variables.

A complete rich theory is also called a maximal theory.

13

5 Boolean algebras of classes of terms

Let S be a maximal theory. We will associate with S some equivalence re-lations in BTerm(BVAR) and RTerm(RVAR) and will show that the equiv-alence classes form Boolean algebras with respect to some naturally definedoperations.

5.1 The Boolean algebra of classes of Boolean terms

We will associate with S a Boolean algebra of classes of Boolean terms. Wedefine the relations 4 and ≈ on BTerm(BVAR):

a 4 bdef⇔ a ≤ b ∈ S

a ≈ bdef⇔ (a 4 b ∧ b 4 a) .

The relation ≈ is an equivalence relation. We denote by

[a]def=

{

b ∈ BTerm(BVAR)∣

∣ a ≈ b}

the equivalence class of a. We denote by ClB the set of equivalence classes

ClBdef= BTerm(BVAR)

/

≈={

[a]∣

∣ a ∈ BTerm(BVAR)}

.

We define a relation ≤ on ClB:

[a] ≤ [b]def⇔ a 4 b .

We define the operations ∩, ∪ and − on ClB:

[a] ∩ [b]def= [a ∩ b]

[a] ∪ [b]def= [a ∪ b]

−[a]def= [−a] .

The relation ≤ and the operations ∩, ∪ and − are well-defined. Thesix-tuple

(

ClB,∩,∪,−, [0], [1])

is a Boolean algebra. We denote by UltB the set of its ultrafilters.

5.2 The Boolean algebra of classes of relational terms

If Q is a quantifier, we will denote by Q the dual quantifier.

Lemma 5.1. Let a and b be Boolean terms and let α be a relational term.Then, for arbitrary quantifiers Q1 and Q2 the formula

〈Q1Q2〉(a, b)[−α]↔ ¬⟨

Q1Q2

⟩

(a, b)[α]

is a theorem.

14

Proof. One of these four formulas is an axiom. It remains to prove 6 im-plications. In the following proofs p and q are different Boolean variables,which do not occur in the terms a and b. We know that such variables exist,since BVAR is infinite.

(1) ⊢〈∀∃〉(a, b)[α]→ a ∩ p = 0 ∨ 〈∃∃〉(p, b)[α] by (A6)

⊢〈∀∃〉(a, b)[α]→ a ∩ p = 0 ∨ ¬〈∀∀〉(p, b)[−α] by (A−)

⊢〈∀∃〉(a, b)[α]→ ¬〈∃∀〉(a, b)[−α] by (R3)

(2) ⊢¬〈∃∀〉(a, b)[−α]→ a ∩ p = 0 ∨ ¬〈∀∀〉(p, b)[−α] by (A8)

⊢¬〈∃∀〉(a, b)[−α]→ a ∩ p = 0 ∨ 〈∃∃〉(p, b)[α] by (A−)

⊢¬〈∃∀〉(a, b)[−α]→ 〈∀∃〉(a, b)[α] by (R1)

(3) ⊢¬〈∃∃〉(a, b)[−α]→ a ∩ p = 0 ∨ ¬〈∀∃〉(p, b)[−α] by (A6)

⊢¬〈∃∃〉(a, b)[−α]→

→ a ∩ p = 0 ∨ b ∩ q = 0 ∨ ¬〈∀∀〉(p, q)[−α] by (A7)

⊢¬〈∃∃〉(a, b)[−α]→ a ∩ p = 0 ∨ b ∩ q = 0 ∨ 〈∃∃〉(p, q)[α] by (A−)

⊢¬〈∃∃〉(a, b)[−α]→ b ∩ q = 0 ∨ 〈∀∃〉(a, q)[α] by (R1)

⊢¬〈∃∃〉(a, b)[−α]→ 〈∀∀〉(a, b)[α] by (R2)

(4) ⊢¬〈∃∀〉(a, b)[α]→ a ∩ p = 0 ∨ ¬〈∀∀〉(p, b)[α] by (A8)

⊢¬〈∃∀〉(a, b)[α]→ a ∩ p = 0 ∨ 〈∃∃〉(p, b)[−α] by item 3

⊢¬〈∃∀〉(a, b)[α]→ 〈∀∃〉(a, b)[−α] by (R1)

(5) ⊢〈∀∀〉(a, b)[α]→ b ∩ p = 0 ∨ 〈∀∃〉(a, p)[α] by (A7)

⊢〈∀∀〉(a, b)[α]→ b ∩ p = 0 ∨ ¬〈∃∀〉(a, p)[−α] by item 1

⊢〈∀∀〉(a, b)[α]→ b ∩ p = 0 ∨ 〈∀∃〉(a, p)[−− α] by item 4

⊢〈∀∀〉(a, b)[α]→ 〈∀∀〉(a, b)[− − α] by (R2)

⊢〈∀∀〉(a, b)[α]→ ¬〈∃∃〉(a, b)[−α] by (A−)

(6) ⊢〈∀∃〉(a, b)[−α]→ a ∩ p = 0 ∨ 〈∃∃〉(p, b)[−α] by (A6)

⊢〈∀∃〉(a, b)[−α]→ a ∩ p = 0 ∨ ¬〈∀∀〉(p, b)[α] by item 5

⊢〈∀∃〉(a, b)[−α]→ ¬〈∃∀〉(a, b)[α] by (R3)

Lemma 5.2. Let α, β ∈ RTerm(RVAR) and let B = BTerm(BVAR). Thenthe following conditions are equivalent:

(1) (∀a, b ∈ B)(

〈∃∃〉(a, b)[α]→ 〈∃∃〉(a, b)[β] ∈ S)

(2) (∀a, b ∈ B)(

〈∀∃〉(a, b)[α]→ 〈∀∃〉(a, b)[β] ∈ S)

(3) (∀a, b ∈ B)(

〈∀∀〉(a, b)[α]→ 〈∀∀〉(a, b)[β] ∈ S)

15

(4) (∀a, b ∈ B)(

〈∃∀〉(a, b)[α]→ 〈∃∀〉(a, b)[β] ∈ S)

Proof. (1)→(2) Assume that item 1 is true and suppose that there areBoolean terms a and b such that

〈∀∃〉(a, b)[α]→ 〈∀∃〉(a, b)[β] /∈ S .

Since S is a rich theory, there is a Boolean variable p such that

〈∀∃〉(a, b)[α]→ a ∩ p = 0 ∨ 〈∃∃〉(p, b)[β] /∈ S .

Hence〈∀∃〉(a, b)[α] → a ∩ p = 0 ∨ 〈∃∃〉(p, b)[α] /∈ S .

This is a contradiction, since the last formula is a theorem.(2)→(3) The proof is analogous to the previous one.(3)→(4) Assume that item 3 is true. By Lemma 5.1

(∀a, b ∈ B)(

¬〈∃∃〉(a, b)[−α]→ ¬〈∃∃〉(a, b)[−β] ∈ S)

.

For the sake of contradiction suppose that there are Boolean terms a and bsuch that

〈∀∃〉(a, b)[−β]→ 〈∀∃〉(a, b)[−α] /∈ S .

Since S is a rich theory, there is a Boolean variable p such that

〈∀∃〉(a, b)[−β]→ a ∩ p = 0 ∨ 〈∃∃〉(p, b)[−α] /∈ S .

Hence〈∀∃〉(a, b)[−β]→ a ∩ p = 0 ∨ 〈∃∃〉(p, b)[−β] /∈ S .

This is a contradiction, since the last formula is a theorem. Hence

(∀a, b ∈ B)(

〈∀∃〉(a, b)[−β]→ 〈∀∃〉(a, b)[−α] ∈ S)

.

Using Lemma 5.1 again, we conclude that

(∀a, b ∈ B)(

¬〈∃∀〉(a, b)[β]→ ¬〈∃∀〉〉(a, b)[α] ∈ S)

,

which implies item 4.(4)→(1) The proof is analogous to the previous one.

We define the relations 4 and ≈ on the set of all relational terms:

α 4 βdef⇔

(

∀a, b ∈ BTerm(BVAR))(

〈∃∃〉(a, b)[α]→ 〈∃∃〉(a, b)[β] ∈ S)

α ≈ βdef⇔ (α 4 β ∧ β 4 α) .

The relation ≈ is an equivalence relation. We denote by

[α]def=

{

β ∈ RTerm(RVAR)∣

∣ α ≈ β}

16

the equivalence class of α. We denote by ClR the set of equivalence classes

ClRdef= RTerm(RVAR)

/

≈={

[α]∣

∣ α ∈ RTerm(RVAR)}

.

We define a relation ≤ on ClR:

[α] ≤ [β]def⇔ α 4 β .

We define the operations ∩, ∪, − and −1 on ClR:

[α] ∩ [β]def= [α ∩ β]

[α] ∪ [β]def= [α ∪ β]

− [α]def= [−α]

[α]−1 def= [α−1] .

Lemma 5.3. The relation ≤ and the operations ∩, ∪, − and −1 on ClR arewell-defined.

Proof. Let B = BTerm(BVAR).

(1) Clearly, ≤ is well-defined.

(2) Let γ ≈ α and δ ≈ β. Then

(∀a, b ∈ B)(

〈∀∀〉(a, b)[γ]↔ 〈∀∀〉(a, b)[α] ∈ S)

and

(∀a, b ∈ B)(

〈∀∀〉(a, b)[δ] ↔ 〈∀∀〉(a, b)[β] ∈ S)

.

Hence

(∀a, b ∈ B)(

〈∀∀〉(a, b)[γ] ∧ 〈∀∀〉(a, b)[δ] ↔ 〈∀∀〉(a, b)[α] ∧ 〈∀∀〉(a, b)[β] ∈ S)

.

By (A∩)

(∀a, b ∈ B)(

〈∀∀〉(a, b)[γ ∩ δ]↔ 〈∀∀〉(a, b)[α ∩ β] ∈ S)

.

By Lemma 5.2

(∀a, b ∈ B)(

〈∃∃〉(a, b)[γ ∩ δ]↔ 〈∃∃〉(a, b)[α ∩ β] ∈ S)

.

Hence γ ∩ δ ≈ α ∩ β.

17

(3) The implication

(γ ≈ α ∧ δ ≈ β)⇒ γ ∪ δ ≈ α ∪ β

follows by similar reasoning.

(4) Let γ ≈ α. Then

(∀a, b ∈ B)(

〈∃∃〉(a, b)[γ]↔ 〈∃∃〉(a, b)[α] ∈ S)

.

By Lemma 5.1

(∀a, b ∈ B)(

¬〈∀∀〉(a, b)[−γ]↔ ¬〈∀∀〉(a, b)[−α] ∈ S)

.

By Lemma 5.2

(∀a, b ∈ B)(

〈∃∃〉(a, b)[−γ]↔ 〈∃∃〉(a, b)[−α] ∈ S)

.

Thus −γ ≈ −α.

(5) The equivalence α ≈ β ⇔ α−1 ≈ β−1 follows directly from (A−1).

Lemma 5.4. If a, b, c, d are Boolean terms and α, β are relational terms,then the following formulas are theorems:

(1) 〈∃∃〉(a, b)[α] ∧ a ≤ c→ 〈∃∃〉(c, b)[α]and〈∃∃〉(a, b)[α] ∧ b ≤ c→ 〈∃∃〉(a, c)[α]

(2) 〈∀∀〉(a, b)[α] ∧ c ≤ a→ 〈∀∀〉(c, b)[α]and〈∀∀〉(a, b)[α] ∧ c ≤ b→ 〈∀∀〉(a, c)[α]

(3) 〈∀∀〉(a, b)[0R]↔ a = 0 ∨ b = 0

(4) 〈∀∀〉(a, b)[α] ∧ 〈∀∀〉(c, d)[β] → 〈∀∀〉(a ∩ c, b ∩ d)[α ∩ β]

(5) 〈∀∀〉(a, b)[α] ∧ ¬〈∃∃〉(c, d)[α ∩ β]→ 〈∀∀〉(a ∩ c, b ∩ d)[−β]

Proof.

(1) We will prove the first one.⊢〈∃∃〉(a, b)[α] ∧ a ≤ c→

(

〈∃∃〉(a, b)[α] ∨ 〈∃∃〉(c, b)[α])

⊢〈∃∃〉(a, b)[α] ∧ a ≤ c→ 〈∃∃〉(a ∪ c, b)[α] ∧ a ∪ c = c by (A4)

⊢〈∃∃〉(a, b)[α] ∧ a ≤ c→ 〈∃∃〉(c, b)[α] by (A1)

18

(2) We will prove the first one. Let p and q be different Boolean variableswhich do not occur in b and c.⊢〈∀∀〉(a, b)[α] ∧ c ≤ a→ b ∩ q = 0 ∨ 〈∀∃〉(a, q)[α] by (A7)

⊢〈∀∀〉(a, b)[α] ∧ c ≤ a→

→ a ∩ p = 0 ∨ b ∩ q = 0 ∨ 〈∃∃〉(p, q)[α] by (A6)

⊢〈∀∀〉(a, b)[α] ∧ c ≤ a→

→ c ∩ p = 0 ∨ b ∩ q = 0 ∨ 〈∃∃〉(p, q)[α]

⊢〈∀∀〉(a, b)[α] ∧ c ≤ a→ b ∩ q = 0 ∨ 〈∀∃〉(c, q)[α] by (R1)

⊢〈∀∀〉(a, b)[α] ∧ c ≤ a→ 〈∀∀〉(c, b)[α] by (R2)

(3) We will prove the two implications separately.

(→) ⊢〈∀∀〉(a, b)[0R]→ b = 0 ∨ 〈∀∃〉(a, b)[0R] by (A7)

⊢〈∀∀〉(a, b)[0R]→ a = 0 ∨ b = 0 ∨ 〈∃∃〉(a, b)[0R] by (A6)

⊢〈∀∀〉(a, b)[0R]→ a = 0 ∨ b = 0 by (A9)

(→) ⊢a = 0 ∨ b = 0→ ¬〈∃∃〉(a, b)[−0R] by (A3)

⊢a = 0 ∨ b = 0→ 〈∀∀〉(a, b)[0R] by Lemma 5.1

(4) ⊢〈∀∀〉(a, b)[α] ∧ 〈∀∀〉(c, d)[β]→

→ 〈∀∀〉(a ∩ c, b ∩ d)[α] ∧ 〈∀∀〉(a ∩ c, b ∩ d)[β] by item 2

⊢〈∀∀〉(a, b)[α] ∧ 〈∀∀〉(c, d)[β]→ 〈∀∀〉(a ∩ c, b ∩ d)[α ∩ β] by (A∩)

(5) Let p and q be different Boolean variables which do not occur in a, b,c and d.⊢〈∀∀〉(a, b)[α] ∧ ¬〈∃∃〉(c, d)[α ∩ β] ∧ 〈∀∀〉(p, q)[β]→

→ 〈∀∀〉(a ∩ p, b ∩ q)[α ∩ β] ∧ ¬〈∃∃〉(c, d)[α ∩ β] by item 4

⊢〈∀∀〉(a, b)[α] ∧ ¬〈∃∃〉(c, d)[α ∩ β] ∧ 〈∀∀〉(p, q)[β]→

→ a ∩ p ∩ c = 0 ∨ b ∩ q ∩ d = 0 by (A6) and (A7)

⊢〈∀∀〉(a, b)[α] ∧ ¬〈∃∃〉(c, d)[α ∩ β]→

→ a ∩ c ∩ p = 0 ∨ b ∩ d ∩ q = 0 ∨ ¬〈∀∀〉(p, q)[β]

⊢〈∀∀〉(a, b)[α] ∧ ¬〈∃∃〉(c, d)[α ∩ β]→

→ a ∩ c ∩ p = 0 ∨ b ∩ d ∩ q = 0 ∨ 〈∃∃〉(p, q)[−β] by Lemma 5.1

⊢〈∀∀〉(a, b)[α] ∧ ¬〈∃∃〉(c, d)[α ∩ β]→

→ b ∩ d ∩ q = 0 ∨ 〈∀∃〉(a ∩ c, q)[−β] by (R1)

⊢〈∀∀〉(a, b)[α] ∧ ¬〈∃∃〉(c, d)[α ∩ β]→

→ 〈∀∀〉(a ∩ c, b ∩ d)[−β] by (R2)

19

Lemma 5.5. The six-tuple

(ClR,∩,∪,−, [0R], [1R])

is a Boolean algebra.

Proof. We need to verify the following properties for arbitrary relationalterms α, β and γ:

(1) α 4 α(2) (α 4 β ∧ β 4 γ)⇒ α 4 γ(3) (α 4 β ∧ β 4 α)⇒ α ≈ β(4) α ∩ β 4 α(5) α ∩ β 4 β(6) (γ 4 α ∧ γ 4 β)⇒ γ 4 α ∩ β(7) α 4 α ∪ β(8) β 4 α ∪ β(9) (α 4 γ ∧ β 4 γ)⇒ α ∪ β 4 γ

(10) 0R 4 α(11) α 4 1R(12) α ∩ (β ∪ γ) 4 (α ∩ β) ∪ (α ∩ γ)(13) α ∩ −α 4 0R(14) 1R 4 α ∪ −α

The conditions (1), (2) and (3) follow directly from the definition of therelation 4. To prove the other claims, it suffices to prove that the followingformulas are theorems for arbitrary Boolean terms a and b and arbitraryrelational terms α, β and γ:

(1) 〈∀∀〉(a, b)[α ∩ β]→ 〈∀∀〉(a, b)[α]

(2) 〈∀∀〉(a, b)[α ∩ β]→ 〈∀∀〉(a, b)[β]

(3)(

〈∀∀〉(a, b)[γ]→ 〈∀∀〉(a, b)[α])

∧(

〈∀∀〉(a, b)[γ]→ 〈∀∀〉(a, b)[β])

→

→(

〈∀∀〉(a, b)[γ]→ 〈∀∀〉(a, b)[α ∩ β])

(4) 〈∃∃〉(a, b)[α]→ 〈∃∃〉(a, b)[α ∪ β]

(5) 〈∃∃〉(a, b)[β]→ 〈∃∃〉(a, b)[α ∪ β]

(6)(

〈∃∃〉(a, b)[α]→ 〈∃∃〉(a, b)[γ])

∧(

〈∃∃〉(a, b)[β]→ 〈∃∃〉(a, b)[γ])

→

→(

〈∃∃〉(a, b)[α ∪ β]→ 〈∃∃〉(a, b)[γ])

(7) 〈∃∃〉(a, b)[0R]→ 〈∃∃〉(a, b)[α]

(8) 〈∀∀〉(a, b)[α]→ 〈∀∀〉(a, b)[1R]

(9) 〈∀∀〉(a, b)[

α ∩ (β ∪ γ)]

→ 〈∀∀〉(a, b)[

(α ∩ β) ∪ (α ∩ γ)]

20

(10) 〈∀∀〉(a, b)[α ∩ −α]→ 〈∀∀〉(a, b)[0R]

(11) 〈∃∃〉(a, b)[1R]→ 〈∃∃〉(a, b)[α ∪ −α]

(1), (2) and (3) follow from (A∩). (4), (5) and (6) follow analogously from(A∪). (7) follows from the axiom for 0R; (8) follows analogously from theaxiom for 1R. We will prove the remaining three theorems. Let a, b ∈BTerm(BVAR).

(9) We will make use of item 5 in Lemma 5.4. Let p, q ∈ BVAR, p 6= qand {p, q} ∩ BVar(a, b) = ∅.

⊢〈∀∀〉(a, b)[

α ∩ (β ∪ γ)]

∧ ¬〈∃∃〉(p, q)[

(α ∩ β) ∪ (α ∩ γ)]

→

→ 〈∀∀〉(a, b)[α] ∧ 〈∀∀〉(a, b)[β ∪ γ] ∧ ¬〈∃∃〉(p, q)[α ∩ β] ∧

∧ ¬〈∃∃〉(p, q)[α ∩ γ] by (A∩) and (A∪)

⊢〈∀∀〉(a, b)[

α ∩ (β ∪ γ)]

∧ ¬〈∃∃〉(p, q)[

(α ∩ β) ∪ (α ∩ γ)]

→

→ 〈∀∀〉(a, b)[β ∪ γ] ∧ 〈∀∀〉(a ∩ p, b ∩ q)[−β] ∧

∧ 〈∀∀〉(a ∩ p, b ∩ q)[−γ] by Lemma 5.4

⊢〈∀∀〉(a, b)[

α ∩ (β ∪ γ)]

∧ ¬〈∃∃〉(p, q)[

(α ∩ β) ∪ (α ∩ γ)]

→

→ 〈∀∀〉(a, b)[β ∪ γ] ∧ ¬〈∃∃〉(a ∩ p, b ∩ q)[β] ∧

∧ ¬〈∃∃〉(a ∩ p, b ∩ q)[γ] by (A−)

⊢〈∀∀〉(a, b)[

α ∩ (β ∪ γ)]

∧ ¬〈∃∃〉(p, q)[

(α ∩ β) ∪ (α ∩ γ)]

→

→ 〈∀∀〉(a, b)[β ∪ γ] ∧ ¬〈∃∃〉(a ∩ p, b ∩ q)[β ∪ γ] by (A∪)

⊢〈∀∀〉(a, b)[

α ∩ (β ∪ γ)]

∧ ¬〈∃∃〉(p, q)[

(α ∩ β) ∪ (α ∩ γ)]

→

→ a ∩ p = 0 ∨ b ∩ q = 0 by (A6) and (A7)

⊢〈∀∀〉(a, b)[

α ∩ (β ∪ γ)]

→

→ a ∩ p = 0 ∨ b ∩ q = 0 ∨ 〈∃∃〉(p, q)[

(α ∩ β) ∪ (α ∩ γ)]

⊢〈∀∀〉(a, b)[

α ∩ (β ∪ γ)]

→

→ b ∩ q = 0 ∨ 〈∀∃〉(a, q)[

(α ∩ β) ∪ (α ∩ γ)]

by (R1)

⊢〈∀∀〉(a, b)[

α ∩ (β ∪ γ)]

→ 〈∀∀〉(a, b)[

(α ∩ β) ∪ (α ∩ γ)]

by (R2)

21

(10) ⊢〈∀∀〉(a, b)[α ∩ −α]→

→ 〈∀∀〉(a, b)[α] ∧ 〈∀∀〉(a, b)[−α] by (A∩)

⊢〈∀∀〉(a, b)[α ∩ −α]→

→ 〈∀∀〉(a, b)[α] ∧ ¬〈∃∃〉(a, b)[α] by (A−)

⊢〈∀∀〉(a, b)[α ∩ −α]→ a = 0 ∨ b = 0 by (A6) and (A7)

⊢〈∀∀〉(a, b)[α ∩ −α]→ ¬〈∃∃〉(a, b)[−0R] by (A3)

⊢〈∀∀〉(a, b)[α ∩ −α]→ 〈∀∀〉(a, b)[0R] by Lemma 5.1

(11) ⊢¬〈∃∃〉(a, b)[α ∪ −α]→

→ ¬〈∃∃〉(a, b)[α] ∧ ¬〈∃∃〉(a, b)[−α] by (A∪)

⊢¬〈∃∃〉(a, b)[α ∪ −α]→

→ ¬〈∃∃〉(a, b)[α] ∧ 〈∀∀〉(a, b)[α] by Lemma 5.1

⊢¬〈∃∃〉(a, b)[α ∪ −α]→ a = 0 ∨ b = 0 by (A6) and (A7)

⊢¬〈∃∃〉(a, b)[α ∪ −α]→ ¬〈∃∃〉(a, b)[1R] by (A3)

⊢〈∃∃〉(a, b)[1R]→ 〈∃∃〉(a, b)[α ∪ −α]

Lemma 5.6. If a, b are Boolean terms and α, β are relational terms, thenthe formula

〈∀∀〉(a, b)[α ∪ β]→ 〈∃∃〉(a, b)[α] ∨ 〈∀∀〉(a, b)[β]

is a theorem.

Proof. By item 5 in Lemma 5.4, the formula

〈∀∀〉(a, b)[−α] ∧ ¬〈∃∃〉(a, b)[

(−α) ∩ (−β)]

→ 〈∀∀〉(a, b)[β]

is a theorem. Hence

〈∀∀〉(a, b)[α ∪ β] ∧ ¬〈∃∃〉(a, b)[α]→ 〈∀∀〉(a, b)[β]

is a theorem.

We denote by UltR the set of ultrafilters of the Boolean algebra

(ClR,∩,∪,−, [0R], [1R]) .

Lemma 5.7. [1] ≤ [0]⇔ [1R] ≤ [0R].

Proof. (→) Let 1 = 0 ∈ S and a, b ∈ BTerm(BVAR). Then a = 0 ∈ S andb = 0 ∈ S. By (A3), ¬〈∃∃〉(a, b)[1R] ∈ S, and hence

〈∃∃〉(a, b)[1R]→ 〈∃∃〉(a, b)[0R] ∈ S .

22

(←) Assume that [1R] ≤ [0R]. Then

〈∃∃〉(1, 1)[1R]→ 〈∃∃〉(1, 1)[0R] ∈ S .

By (A9), ¬〈∃∃〉(1, 1)[0R] ∈ S, and hence

¬〈∃∃〉(1, 1)[1R] ∈ S .

By (A10), 〈∀∀〉(1, 1)[1R] ∈ S, hence

〈∀∀〉(1, 1)[1R] ∧ ¬〈∃∃〉(1, 1)[1R] ∈ S .

Using (A6) and (A7), we conclude that 1 = 0 ∈ S.

5.3 The Boolean algebra of symmetric classes of relational

terms

Lemma 5.8. Let a, b ∈ BTerm(BVAR) and α, β ∈ RTerm(RVAR). Thefollowing formulas are theorems:

(1) 〈∀∀〉(a, b)[

α−1]

↔ 〈∀∀〉(b, a)[α]

(2) 〈∀∀〉(a, b)[

(α ∩ β)−1]

↔ 〈∀∀〉(a, b)[

(α−1) ∩ (β−1)]

(3) 〈∃∃〉(a, b)[

(α ∪ β)−1]

↔ 〈∃∃〉(a, b)[

(α−1) ∪ (β−1)]

(4) 〈∀∀〉(a, b)[

(−α)−1]

↔ 〈∀∀〉(a, b)[

−(α−1)]

Proof. (1) Let p and q be different Boolean variables which do not occurin b and c.⊢〈∀∀〉(a, b)

[

α−1]

→ b ∩ q = 0 ∨ 〈∀∃〉(a, q)[

α−1]

by (A7)

⊢〈∀∀〉(a, b)[

α−1]

→ a ∩ p = 0 ∨ b ∩ q = 0 ∨ 〈∃∃〉(p, q)[

α−1]

by (A6)

⊢〈∀∀〉(a, b)[

α−1]

→ a ∩ p = 0 ∨ b ∩ q = 0 ∨ 〈∃∃〉(q, p)[α] by (A−1)

⊢〈∀∀〉(a, b)[

α−1]

→ a ∩ p = 0 ∨ 〈∀∃〉(b, p)[α] by (R1)

⊢〈∀∀〉(a, b)[

α−1]

→ 〈∀∀〉(b, a)[α] by (R2)

The proof of ⊢ 〈∀∀〉(b, a)[α] → 〈∀∀〉(a, b)[

α−1]

is similar to the above.

(2) Follows from item 1 and (A∩).

(3) Follows from (A−1) and (A∪).

(4) Follows from item 1, (A−) and (A−1).

We define an operation −1 on P (ClR): For each V ⊆ ClR

V −1 def=

{

[α]−1∣

∣ [α] ∈ V}

.

23

Lemma 5.9. (1) If α ∈ RTerm(RVAR) then(

α−1)−1≈ α.

(2) If X ∈ ClR then(

X−1)−1

= X. If V ⊆ ClR then(

V −1)−1

= V .

(3) Let V ⊆ ClR. If V is a filter, then so is V −1. If V is an ultrafilter,then so is V −1.

Proof. (1) Follows easily from (A−1).

(2) Follows from the previous item and the definition of the operation −1

on ClR.

(3) It suffices to prove that 0R−1 ≈ 0R, 1R

−1 ≈ 1R and, for arbitraryrelational terms α and β, we have the following:

(a) (α ∩ β)−1 ≈ (α−1) ∩ (β−1)

(b) (α ∪ β)−1 ≈ (α−1) ∪ (β−1)

(c) (−α)−1 ≈ −(α−1)

This follows from (A9), (A−1), (A10) and Lemma 5.8.

Definition 3. • x ∈ ClR is called symmetric iff x = x−1.

• V ⊆ ClR is called symmetric iff V = V −1.

Lemma 5.10. The set of symmetric classes of relational terms is a Booleansubalgebra of

(ClR,∩,∪,−, [0R], [1R]) .

Proof. Straightforward verification.

Lemma 5.11. If a is a Boolean term and α1, α2, . . . , αk are relational terms,then the formula

a = 0 ∨ ¬〈∀∀〉(a, a)[

α1 ∩ (−α1−1) ∪ α2 ∩ (−α2

−1) ∪ · · · ∪ αk ∩ (−αk−1)

]

is a theorem.

Proof. Let p1, p2, . . . , pk be different Boolean variables, which do not occurin a. By item 3 in Lemma 5.4, we have

⊢ p1 ∩ · · · ∩ pk ∩ a = 0 ∨ ¬〈∀∀〉(p1 ∩ · · · ∩ pk ∩ a, p1 ∩ · · · ∩ pk ∩ a)[0R]

⊢ p1 ∩ · · · ∩ pk ∩ a = 0 ∨ ¬〈∀∀〉(p1, p1)[0R] by item 2 in Lemma 5.4

⊢ p2 ∩ · · · ∩ pk ∩ a = 0 ∨

∨ ¬〈∀∀〉(p2 ∩ · · · ∩ pk ∩ a, p2 ∩ · · · ∩ pk ∩ a)[α1 ∩ (−α1−1)] by (R4)

24

Similarly we obtain

⊢ p2 ∩ · · · ∩ pk ∩ a = 0 ∨ ¬〈∀∀〉(p2, p2)[α1 ∩ (−α1−1)]

⊢ p3 ∩ · · · ∩ pk ∩ a = 0 ∨

∨ ¬〈∀∀〉(p3 ∩ · · · ∩ pk ∩ a, p3 ∩ · · · ∩ pk ∩ a)[α1 ∩ (−α1−1) ∪ α2 ∩ (−α2

−1)]

Continuing in the same way, we arrive at

⊢ a = 0 ∨ ¬〈∀∀〉(a, a)[

α1 ∩ (−α1−1) ∪ α2 ∩ (−α2

−1) ∪ · · · ∪ αk ∩ (−αk−1)

]

.

6 Completeness

Let S be a maximal theory. We will prove that S has a model. First, wewill consider the case where 1 = 0 ∈ S.

Lemma 6.1. If S is a maximal theory and 1 = 0 ∈ S, then M∅ � S.

Proof. As S is a maximal theory, it suffices to prove the equivalence

M∅ � ϕ⇔ ϕ ∈ S

for atomic formulas ϕ.

(1) Clearly, all formulas in the form of a ≤ b belong to S and are true inM∅.

(2) ϕ is 〈∃∃〉(a, b)[α]. Then M∅ 2 ϕ. By (A3) ϕ /∈ S.

(3) ϕ is 〈∀∃〉(a, b)[α]. Then M∅ � ϕ. For the sake of contradiction supposethat ϕ /∈ S. There exists a Boolean variable p for which

a ∩ p = 0 ∨ 〈∃∃〉(p, b)[α] /∈ S .

This is a contradiction, since a ∩ p = 0 ∈ S.

(4) ϕ is 〈∀∀〉(a, b)[α]. Then M∅ � ϕ. By (A3)

¬〈∃∃〉(a, b)[−α] ∈ S ,

hence, by Lemma 5.1, ϕ ∈ S.

(5) ϕ is 〈∃∀〉(a, b)[α]. Then M∅ 2 ϕ. By item 3

〈∀∃〉(a, b)[−α] ∈ S ,

hence, by Lemma 5.1, ϕ /∈ S.

25

We will now consider the case when 1 = 0 /∈ S. In this case UltB 6= ∅and UltR 6= ∅.

If Q is a quantifier and F (a) is a statement about Boolean terms, suchthat a ≈ b implies F (a) ⇔ F (b), we will use

(

Q[a] ∈ ClB)

F (a) as anabbreviation for (Qx ∈ ClB)(∃a ∈ x)F (a) or the equivalent (Qx ∈ ClB)(∀a ∈x)F (a). We will also use a similar notation for statements about relationalterms.

For each V ⊆ ClR we define a relation R0V ⊆ P (ClB)2:

R0V =

{

(U1, U2) ∈ P (ClB)2∣

∣

∣

(

∀[α] ∈ V)(

∀[a1] ∈ U1

)(

∀[a2] ∈ U2

)(

〈∃∃〉(a1, a2)[α] ∈ S)

}

.

Lemma 6.2. If V ⊆ ClR, then R0V −1 =

(

R0V

)−1.

Proof. Follows from (A−1) and the definition of R0V .

Notation. If a ∈ BTerm(BVAR), we denote by [a) ={

x ∈ ClB∣

∣ [a] ≤ x}

the smallest filter containing [a]. Similarly, if α ∈ RTerm(RVAR), we denoteby [α) =

{

x ∈ ClR∣

∣ [α] ≤ x}

the smallest filter containing [α].

Notation. Let F1 and F2 be filters in the Boolean algebra of ClB and let Gbe a filter in the Boolean algebra of ClR. We will use the following notation:

IF1,F2=

{

[α] ∈ ClR

∣

∣

∣

(

∃[a1] ∈ F1

)(

∃[a2] ∈ F2

)(

〈∃∃〉(a1, a2)[α] /∈ S)

}

IG,F2=

{

[a1] ∈ ClB

∣

∣

∣

(

∃[α] ∈ G)(

∃[a2] ∈ F2

)(

〈∃∃〉(a1, a2)[α] /∈ S)

}

IF1,G ={

[a2] ∈ ClB

∣

∣

∣

(

∃[a1] ∈ F1

)(

∃[α] ∈ G)(

〈∃∃〉(a1, a2)[α] /∈ S)

}

It is easy to check that the Is are ideals in the respective Boolean alge-bras.

Lemma 6.3. Let F1 and F2 be filters in the Boolean algebra of ClB and letG be a filter in the Boolean algebra of ClR. If (F1, F2) ∈ R0

G, then thereare U1, U2 ∈ UltB and V ∈ UltR such that F1 ⊆ U1, F2 ⊆ U2, G ⊆ V and(U1, U2) ∈ R0

V .

Proof. We use the equivalences

(F1, F2) ∈ R0G ⇔ F1 ∩ IG,F2

= ∅ ⇔ G ∩ IF1,F2= ∅ ⇔ F2 ∩ IF1,G = ∅

and apply the separation theorem for filter-ideal pairs in Boolean algebrasthree times.

26

Lemma 6.4. If U is a filter in the Boolean algebra of classes of Booleanterms, then

(U,U) ∈ R0V ⇔

(

∀[α] ∈ V)(

∀[a] ∈ U)(

〈∃∃〉(a, a)[α] ∈ S)

.

Proof. (→) is obvious.(←) Let [α] ∈ V and [a1], [a2] ∈ U . Then [a1 ∩ a2] ∈ U and hence

〈∃∃〉(a1 ∩ a2, a1 ∩ a2)[α] ∈ S .

By Lemma 5.4, 〈∃∃〉(a1, a2)[α] ∈ S.

Lemma 6.5. Let (U1, U2) ∈ UltB2.

(1) (U1, U2) ∈ R0[1R).

(2) There is a V ∈ UltR such that (U1, U2) ∈ R0V .

(3) If U1 = U2, then there exists a symmetric V ∈ UltR such that

(U1, U2) ∈ R0V .

Proof. (1) Suppose this is not true. Since S is a complete theory,

(

∃[a1] ∈ U1

)(

∃[a2] ∈ U2

)(

¬〈∃∃〉(a1, a2)[1R] ∈ S)

.

By the axiom for 1R, 〈∀∀〉(a1, a2)[1R] ∈ S. Using the axioms forquantifier change, we derive a1 = 0∨a2 = 0 ∈ S. Since S is a completetheory, a1 = 0 ∈ S or a2 = 0 ∈ S, hence [a1] = [0] or [a2] = [0]. This isa contradiction, as U1 and U2 are ultafilters. Thus, (U1, U2) ∈ R0

[1R).

(2) By the previous item, (U1, U2) ∈ R0[1R). By Lemma 6.3, there is a

V ∈ UltR such that (U1, U2) ∈ R0V .

(3) Let U = U1 = U2 and let

I0 ={

[α] ∈ ClR

∣

∣

∣α ≈ α−1 ∧

(

∃[a] ∈ U)(

〈∃∃〉(a, a)[α] /∈ S)

}

I1 ={

[α] ∈ ClR

∣

∣

∣α ≈ α−1 ∧ There exists a finite set

{α1, α2, . . . , αk} ⊆ RTerm(RVAR), such that

α 4 α1 ∩ (−α1−1) ∪ α2 ∩ (−α2

−1) ∪ · · · ∪ αk ∩ (−αk−1)

}

I2 ={

x ∈ ClR∣

∣ x = x−1 ∧ (∃y ∈ I0)(∃z ∈ I1)(x ≤ y ∪ z)}

The sets I0, I1, and I2 are ideals in the Boolean algebra of symmetricclasses of relational terms, I0 ⊆ I2 and I1 ⊆ I2. We have

(U,U) ∈ R0[1R) ,

27

and hence [1R] /∈ I0.

Suppose that [1R] ∈ I1. Let

1R 4 α1 ∩ (−α1−1) ∪ α2 ∩ (−α2

−1) ∪ · · · ∪ αk ∩ (−αk−1) .

Since ¬(1 = 0) ∈ S, by Lemma 5.11 we have

¬〈∀∀〉(1, 1)[

α1 ∩ (−α1−1) ∪ α2 ∩ (−α2

−1) ∪ · · · ∪ αk ∩ (−αk−1)

]

∈ S .

By definition of 4 and Lemma 5.2 we obtain

¬〈∀∀〉(1, 1)[1R] ∈ S .

This contradicts (A10).

Hence [1R] /∈ I1 .

Suppose that [1R] ∈ I2. Let [α] ∈ I0 and [β] ∈ I1, where

β = α1 ∩ (−α1−1) ∪ α2 ∩ (−α2

−1) ∪ · · · ∪ αk ∩ (−αk−1) ,

are elements of ClR, such that [1R] ≤ [α] ∪ [β]. Let a ∈ U and

¬〈∃∃〉(a, a)[α] ∈ S .

Since ¬(a = 0) ∈ S, by Lemma 5.11 we have

¬〈∀∀〉(a, a)[β] ∈ S .

By Lemma 5.6 we obtain ¬〈∀∀〉(a, a)[α ∪ β] ∈ S. Hence we get

¬〈∀∀〉(a, a)[1R] ∈ S ,

which contradicts (A10).

Hence [1R] /∈ I2.

Note that [1R) ={

[1R]}

is a filter in the Boolean algebra of symmetricclasses of relational terms and we have

I2 ∩ [1R) = ∅ .

By the separation theorem for filter-ideal pairs, there exist a maximalideal I ⊇ I2 and an ultrafilter Vs ⊇ [1R) in the Boolean algebra ofsymmetric classes of relational terms, such that

I ∩ Vs = ∅ .

LetV =

{

x ∈ ClR∣

∣ x ∩ (x−1) ∈ Vs}

.

V has the following properties:

28

• V ∈ UltR.

We will check the defining properties for ultrafilters.

(a) [1R] ∈ V , since [1R) ⊆ Vs.

(b) Let x ∈ V , y ∈ ClR, and x ≤ y. Then x ∩ (x−1) ∈ Vs andx ∩ (x−1) ≤ y ∩ (y−1). Hence y ∩ (y−1) ∈ Vs and y ∈ V .

(c) Let x, y ∈ V . Then x∩ (x−1) ∈ Vs and y∩ (y−1) ∈ Vs. Hence

x ∩ y ∩ (x−1) ∩ (y−1) ∈ Vs

and x ∩ y ∈ V .

(d) Let x ∈ ClR and suppose that x,−x /∈ V . Then x∩(x−1) /∈ Vsand (−x) ∩ (−x−1) /∈ Vs. Hence (−x) ∪ (−x−1) ∈ Vs andx ∪ x−1 ∈ Vs. Since Vs is a filter, the equivalence class(

(−x) ∪ (−x−1))

∩(

x ∪ x−1)

= x ∩ (−x−1) ∪ (−x) ∩ (x−1)

belongs to Vs. On the other hand, x∩(−x−1)∪(−x)∩(x−1) ∈I1 ⊆ I2 ⊆ I. Thus we obtain a contradiction with I ∩Vs = ∅.Hence

{

x,−x}

∩ V 6= ∅.

(e) [0R] /∈ V , since [0R] /∈ Vs.

• V = V −1.

• (U,U) ∈ R0V .

Since I0 ∩ Vs = ∅,(

∀[α] ∈ V)(

∀[a] ∈ U)(

〈∃∃〉(a, a)[α ∩ α−1] ∈ S)

and hence(

∀[α] ∈ V)(

∀[a] ∈ U)(

〈∃∃〉(a, a)[α] ∈ S)

.

We denote(

Ult2B)+

={

(U1, U2) ∈ UltB2∣

∣ U1 = U2

}

and(

Ult2B)−

={

(U1, U2) ∈ UltB2∣

∣ U1 6= U2

}

.

We choose a set(

Ult2B)−

0⊆

(

Ult2B)−

such that for each (U1, U2) ∈(

Ult2B)−

it contains exactly one x ∈{

(U1, U2), (U2, U1)}

. We denote

(

Ult2B)

0=

(

Ult2B)+∪(

Ult2B)−

0.

Let (U1, U2) ∈ UltB2 and let f be a choice function for UltR. We define

VU1,U2= f

(

{

V ∈ UltR∣

∣ (U1, U2) ∈ R0V ∧ (U1 = U2 ⇒ V = V −1)

}

)

.

We will define a model M = (W,R, v) corresponding to S.Let κ =

∣

∣UltR∣

∣. We consider two cases for κ:

29

(1) κ < ω. Let UltR ={

Vi∣

∣ 1 ≤ i ≤ κ}

with (i 6= j ⇒ Vi 6= Vj). Wedenote Rw = {0, 1, . . . , 2κ}. For m,n ∈ Rw we define

m⊕ n = (m + n) mod (2κ + 1)

m⊖ n = min(

(m− n) mod (2κ+ 1), (n −m) mod (2κ+ 1))

.

We have 0 ≤ m⊖n = n⊖m ≤ κ for all m,n ∈ Rw. Also, (m⊕n)⊖m =n for arbitrary m ∈ Rw and 1 ≤ n ≤ κ. We define an irreflexiverelation ⋖ on Rw

m⋖ n⇔ (n−m) mod (2κ + 1) < (m− n) mod (2κ+ 1) ,

such that for all m,n ∈ Rw either m⋖n, or n⋖m. We have m⋖m⊕nfor arbitrary m ∈ Rw and 1 ≤ n ≤ κ.

(2) κ ≥ ω. Let UltR ={

Vα∣

∣ 0 < α < κ}

with (α 6= β ⇒ Vα 6= Vβ). Wedenote Rw = κ. For α, β ∈ Rw we define α⊕ β = α+ β and

α⊖ β =

{

α− β if β < α,

β − α otherwise.

Again, we have µ⊖ν = ν⊖µ for all µ, ν ∈ Rw, and (µ⊕ν)⊖µ = ν forarbitrary µ ∈ Rw and 0 < ν < κ. As in the previous case, we define arelation ⋖ on Rw, which in this case is just the usual strict total order:

µ⋖ ν ⇔ µ < ν .

We have µ⋖ µ⊕ ν for arbitrary µ ∈ Rw and 0 < ν < κ.

The domain of the canonical model is

W = UltB ×Rw .

If x ∈W , we denote by x1 and x2 its first and second component respec-tively. For each p ∈ BVAR we define

v(p) ={

x ∈W∣

∣ [p] ∈ x1}

.

It is easy to check that for all Boolean terms a

v(a) ={

x ∈W∣

∣ [a] ∈ x1}

using induction on the structure of a.

30

For each V ∈ UltR we define a relation RV ⊆W2:

RV =

{

(x, y) ∈W 2

∣

∣

∣

∣

x2 6= y2 ∧ (x1, y1) ∈ R0Vx2⊖y2

∩R0V −1

x2⊖y2

∧

∧(

x2 ⋖ y2 ∧ V = Vx2⊖y2 ∨ y2 ⋖ x2 ∧ V = V −1x2⊖y2

)

∨

∨x2 6= y2 ∧ (x1, y1) ∈ R0Vx2⊖y2

\R0V −1

x2⊖y2

∧ V = Vx2⊖y2∨

∨x2 6= y2 ∧ (x1, y1) ∈ R0V −1

x2⊖y2

\R0Vx2⊖y2

∧ V = V −1x2⊖y2

∨

∨(

x2 = y2 ∨ (x1, y1) /∈ R0Vx2⊖y2

∪R0V −1

x2⊖y2

)

∧

∧(

(x1, y1) ∈(

Ult2B)

0∧ V = Vx1,y1 ∨ (y1, x1) ∈

(

Ult2B)

0∧ V = V −1

y1,x1

)

}

.

Lemma 6.6. (1)⋃{

RV

∣

∣ V ∈ UltR}

= W 2.

(2) V ′ 6= V ′′ implies RV ′ ∩RV ′′ = ∅.

(3) (x, y) ∈ RV implies (x1, y1) ∈ R0V ;

(4) RV −1 =(

RV

)−1;

(5) If (U1, U2) ∈ R0Vν, then for each µ ∈ Rw it holds that

(

(U1, µ), (U2, µ⊕ ν))

∈ RVν .

Proof. For the first two items it is enough to show that each (x, y) ∈W 2 isan element of RV for exactly one V ∈ UltR. For item 4 it suffices to provethe inclusion

(

RV

)−1⊆ RV −1 .

First, note that the four cases in the definition of RV are mutually ex-clusive. Let (x, y) ∈ RV . We will prove the first four items in the claimtogether by considering the four cases in the definition.

(1) x2 6= y2 ∧ (x1, y1) ∈ R0Vx2⊖y2

∩ R0V −1

x2⊖y2

. We have exactly one of the

following:

• x2 ⋖ y2. Then V = Vx2⊖y2 .

• y2 ⋖ x2. Then V = V −1x2⊖y2

.

In both cases (x1, y1) ∈ R0V . As y2⊖x2 = x2⊖y2 and R0

V −1 =(

R0V

)−1,

we have (y1, x1) ∈ R0Vx2⊖y2

∩ R0V −1

x2⊖y2

. Hence (y, x) falls in the same

case and (y, x) ∈ RV −1 .

(2) x2 6= y2 ∧ (x1, y1) ∈ R0Vx2⊖y2

\ R0V −1

x2⊖y2

. Then V = Vx2⊖y2 and hence

(x1, y1) ∈ R0V . As (y, x) falls in the next case, we have (y, x) ∈ RV −1 .

31

(3) x2 6= y2 ∧ (x1, y1) ∈ R0V −1

x2⊖y2

\ R0Vx2⊖y2

. Then V = V −1x2⊖y2

and hence

(x1, y1) ∈ R0V . As (y, x) falls in the previous case, we have (y, x) ∈

RV −1 .

(4) x2 = y2 ∨ (x1, y1) /∈ R0Vx2⊖y2

∪R0V −1

x2⊖y2

. We consider two subcases:

• x1 = y1. By the definition of Vx1,y1 , it is a symmetric ultrafilter.Since (x1, y1) = (y1, x1) is an element of

(

Ult2B)

0, we have V =

Vx1,y1 = V −1. Hence (x1, y1) ∈ R0V . As (y, x) satisfies the same

condition, it is an element of RV = RV −1 .

• x1 6= y1. We have exactly one of the following:

– (x1, y1) ∈(

Ult2B)

0. Then V = Vx1,y1 . Hence (x1, y1) ∈ R

0V .

As (y, x) falls in the next case, we have (y, x) ∈ RV −1 .

– (y1, x1) ∈(

Ult2B)

0. Then V = V −1

y1,x1. Since (y1, x1) ∈ R

0Vy1,x1

and R0V −1 =

(

R0V

)−1, we have (x1, y1) ∈ R

0V . As (y, x) falls

in the previous case, we have (y, x) ∈ RV −1 .

It remains to prove the last item in the claim. Let (U1, U2) ∈ R0Vν

andµ ∈ Rw. Note that µ ⋖ µ ⊕ ν and hence µ 6= µ ⊕ ν. Consider the pair(

(U1, µ), (U2, µ ⊕ ν))

. We have (µ ⊕ ν) ⊖ µ = µ ⊖ (µ ⊕ ν) = ν. There aretwo possibilities:

• (U1, U2) ∈ R0Vν∩R0

V −1ν

. As µ⋖ µ⊕ ν, we have(

(U1, µ), (U2, µ⊕ ν))

∈

RVν .

• (U1, U2) ∈ R0Vν\R0

V −1ν

. Then(

(U1, µ), (U2, µ⊕ ν))

∈ RVν .

For each α ∈ RVAR we define

R(α) =⋃

{

RV

∣

∣ [α] ∈ V}

.

Lemma 6.7. For each term α ∈ RTerm(RVAR)

R(α) =⋃

{

RV

∣

∣ V ∈ UltR ∧ [α] ∈ V}

.

Proof. For each (x, y) ∈W 2 we denote by V (x, y) the unique V ∈ UltR suchthat (x, y) ∈ RV . We need to prove that

R(α) ={

(x, y) ∈W 2∣

∣ [α] ∈ V (x, y)}

.

We will use induction on the structure of α. Let (x, y) ∈W 2.

• (1) α is 0R. Then [0R] /∈ V (x, y). As this holds for arbitrary (x, y) ∈W 2, we have R(0R) = ∅.

32

(2) α is 1R. Then [1R] ∈ V (x, y). As this holds for arbitrary (x, y) ∈W 2, we have R(1R) = W 2.

• Suppose the claim is true for the relational terms β and γ.

(1) α is β ∩ γ.

(x, y) ∈ R(β ∩ γ)⇔ (x, y) ∈ R(β) ∩R(γ)⇔

⇔ [β] ∈ V (x, y) ∧ [γ] ∈ V (x, y)⇔ [β ∩ γ] ∈ V (x, y) .

(2) α is β ∪ γ.

(x, y) ∈ R(β ∪ γ)⇔ (x, y) ∈ R(β) ∪R(γ)⇔

⇔ [β] ∈ V (x, y) ∨ [γ] ∈ V (x, y)⇔ [β ∪ γ] ∈ V (x, y) .

(3) α is −β.

(x, y) ∈ R(−β)⇔ (x, y) /∈ R(β)⇔

⇔ [β] /∈ V (x, y)⇔ [−β] ∈ V (x, y) .

(4) α is β−1. By item 4 in Lemma 6.6, V (x, y) =(

V (y, x))−1

andhence

(x, y) ∈ R(β−1)⇔ (y, x) ∈ R(β)⇔

⇔ [β] ∈ V (y, x)⇔ [β−1] ∈ V (x, y) .

Notation. If a ∈ BTerm(BVAR), h(a) ={

U ∈ UltB∣

∣ [a] ∈ U}

.If α ∈ RTerm(RVAR), h(α) =

{

V ∈ UltR∣

∣ [α] ∈ V}

.

Lemma 6.8. If a, b ∈ BTerm(BVAR) and α ∈ RTerm(RVAR), then:

〈∀∃〉(a, b)[α] ∈ S ⇔(

∀U ∈ h(a))(

∀[c] ∈ U)(

〈∃∃〉(c, b)[α] ∈ S)

.

Proof. (→) Assume that(

∃U ∈ h(a))(

∃[c] ∈ U)(

〈∃∃〉(c, b)[α] /∈ S)

. By themaximality of S, ¬〈∃∃〉(c, b)[α] ∈ S. [a]∩ [c] 6= [0] and hence ¬a∩ c = 0 ∈ S.By (A6) ¬〈∀∃〉(a, b)[α] ∈ S.

(←) Assume that 〈∀∃〉(a, b)[α] /∈ S. Since S is a rich theory, thereis a Boolean variable p such that a ∩ p = 0 ∨ 〈∃∃〉(p, b)[α] /∈ S. Hencea ∩ p = 0 /∈ S and 〈∃∃〉(p, b)[α] /∈ S. Then [a] ∩ [p] 6= [0] and there is anultrafilter U ⊇

{

[a], [p]}

.

Lemma 6.9. For each formula ϕ ∈ Form(BVAR,RVAR) the followingequivalence holds:

ϕ ∈ S ⇔M � ϕ .

33

Proof. The proof is by induction on the structure of ϕ. Since S is a maximaltheory, we need to consider explicitly only the cases where ϕ is an atomicformula.

(1) ϕ is a1 ≤ a2. By the Stone representation theorem for Boolean alge-bras,

a1 ≤ a2 ∈ S ⇔ h(a1) ⊆ h(a2)⇔

⇔ v(a1) = h(a1)× Rw ⊆ h(a2)× Rw = v(a2) .

(2) ϕ is 〈∃∃〉(a1, a2)[α].

(→) Let 〈∃∃〉(a1, a2)[α] ∈ S. Then(

[a1), [a2))

∈ R0[α). By Lemma 6.3

(

∃U1 ∈ h(a1))(

∃U2 ∈ h(a2))(

∃V ∈ h(α))(

(U1, U2) ∈ R0V

)

.

Let V = Vµ. Then (U1, 0) ∈ v(a1), (U2, µ) ∈ v(a2) and RV ⊆ R(α).By item 5 in Lemma 6.6,

(

(U1, 0), (U2, µ))

∈ RV .

This shows that M � 〈∃∃〉(a1, a2)[α].

(←) Let M � 〈∃∃〉(a1, a2)[α]. Then

(

∃x ∈ v(a1))(

∃y ∈ v(a2))(

∃V ∈ h(α))(

(x, y) ∈ RV

)

.

Thus, we have [a1] ∈ x1, [a2] ∈ y1, [α] ∈ V , and item 3 in Lemma 6.6gives us (x1, y1) ∈ R0

V . Hence

〈∃∃〉(a1, a2)[α] ∈ S .

(3) ϕ is 〈∀∀〉(a1, a2)[α].

〈∀∀〉(a1, a2)[α] ∈ S ⇔ ¬〈∃∃〉(a1, a2)[−α] ∈ S ⇔

⇔ 〈∃∃〉(a1, a2)[−α] /∈ S ⇔M 2 〈∃∃〉(a1, a2)[−α]⇔

⇔M � 〈∀∀〉(a1, a2)[α]

(4) ϕ is 〈∀∃〉(a1, a2)[α].

(→) Let 〈∀∃〉(a1, a2)[α] ∈ S. By Lemma 6.8

(

∀U1 ∈ h(a1))(

∀[c] ∈ U1

)(

〈∃∃〉(c, a2)[α] ∈ S)

.

Hence(

∀U1 ∈ h(a1))

(

(

U1, [a2))

∈ R0[α)

)

.

34

By Lemma 6.3

(

∀U1 ∈ h(a1))(

∃U2 ∈ h(a2))(

∃V ∈ h(α))(

(U1, U2) ∈ R0V

)

.

By item 5 in Lemma 6.6,

(

∀U1 ∈ h(a1))(

∃U2 ∈ h(a2))(

∃ν ∈ Rw

)(

∀µ ∈ Rw

)

(

(

(U1, µ), (U2, µ⊕ ν))

∈ RVν ∧ Vν ∈ h(α))

and hence

(

∀U1 ∈ h(a1))(

∀µ ∈ Rw

)(

∃U2 ∈ h(a2))(

∃ν ∈ Rw

)

(

(

(U1, µ), (U2, µ ⊕ ν))

∈ R(α))

Therefore

(

∀x ∈ v(a1))(

∃y ∈ v(a2))(

(x, y) ∈ R(α))

.

That is, M � 〈∀∃〉(a1, a2)[α].

(←) Assume that M � 〈∀∃〉(a1, a2)[α]. Then

(

∀x ∈ v(a1))(

∃y ∈ v(a2))(

∃V ∈ h(α))(

(x, y) ∈ RV

)

.

By item 3 in Lemma 6.6,

(

∀x ∈ v(a1))(

∃y ∈ v(a2))(

∃V ∈ h(α))(

(x1, y1) ∈ R0V

)

.

As Rw 6= ∅,

(

∀U1 ∈ h(a1))(

∃U2 ∈ h(a2))(

∃V ∈ h(α))(

(U1, U2) ∈ R0V

)

.

Hence(

∀U1 ∈ h(a1))(

∀[c] ∈ U1

)(

〈∃∃〉(c, a2)[α] ∈ S)

.

Then, by Lemma 6.8

〈∀∃〉(a1, a2)[α] ∈ S .

(5) ϕ is 〈∃∀〉(a1, a2)[α].

〈∃∀〉(a1, a2)[α] ∈ S ⇔ ¬〈∀∃〉(a1, a2)[−α] ∈ S ⇔

⇔ 〈∀∃〉(a1, a2)[−α] /∈ S ⇔M 2 〈∀∃〉(a1, a2)[−α]⇔

⇔M � 〈∃∀〉(a1, a2)[α]

35

Theorem 6.10 (Completeness). If Γ ⊆ Form(BVAR,RVAR), then

Γ is consistent⇔ Γ has a model .

Proof. (→) Let Γ be a consistent set of formulas. By item 2 in Corollary 4.4there is a maximal theory S which contains Γ. S has a model which is alsoa model of Γ.

(←) Let M � Γ and let

∆ ={

ϕ ∈ Form(BVAR,RVAR)∣

∣M � ϕ}

.

As (MP) preserves the truth in every model, the set ∆ is closed under (MP).Since M 2 ⊥, ⊥ /∈ ∆. Hence (BVAR,∆) is a consistent theory. As Γ ⊆ ∆,the set Γ is consistent.

7 Complexity

In this section, we consider the complexity of the satisfiability problem forour logic. It is decidable in NExpTime, since the formulas of our logic aretranslatable (in polynomial time) into the NExpTime-decidable two-variablefragment of first-order predicate logic. We argue that the complexity is thesame as the complexity of BML, i.e. NExpTime if the language contains aninfinite number of relational variables, and ExpTime if only a finite numberof relational variables is available [11]. Also, the complexity does not dependon whether we allow −1 in the language.

In the case of an infinite number of relational variables, the lower NEx-pTime bound is proved in [11] by a reduction from an NExpTime-completetiling problem. The BML formula, used to encode the tiling, is a conjunc-tion of a formula, which describes the initial condition for the problem, andseveral conjuncts, which ensure that every model satisfying the formula isindeed a tiling. All conjuncts but the one for the initial condition can betranslated into our fragment. The formula for the initial condition can bereplaced by a formula from our fragment, such that the whole conjunctionis satisfiable iff the original one is satisfiable.

In the case of a finite number of relational variables, the lower ExpTimebound of BML follows from the ExpTime-completeness of Ku (the basicmodal logic enriched with the universal modality). However, the intersectionof Ku with our fragment is also ExpTime-hard, hence the ExpTime-hardnessof our logic.

The upper ExpTime bound for BML is proved in [11] by reduction tothe satisfiability problem for the basic multimodal logic enriched with theuniversal modality. The same reduction is applicable in the presence of −1,and multimodal Ku enriched with −1 is also ExpTime-complete.

These high complexities are due to the presence of 〈∀∃〉 and 〈∃∀〉 in thelanguage. If we remove these symbols from the language, the resulting logic

36

has an NP-complete satisfiability problem, as it possesses the polysize modelproperty. This can be proved by selection of points from a model in the wayit is done in [2] for the dynamic logics of the region-based theory of discretespaces.

8 Concluding remarks

The first completeness proof for a non-classical relational syllogistic (i.e.one that contains relational terms) was given by Nishihara, Morita, andIwata in [18]. Their fragment contains variables for proper nouns and n-ary relational terms closed only under complementation and does not allowBoolean operations on class terms.

Later works on relational syllogistics, devoted mainly to the compu-tational complexity problems, are McAllester and Givan [12] and Pratt-Hartmann [22, 23, 24, 26]. The paper [27] is devoted both to completeaxiomatizations and some computational complexity results. A successor of[27] is Moss [17], devoted to axiomatizations and completeness proofs for anumber of relational syllogistics.

Our logic differs in expressiveness from all systems of relational syllogisticmentioned above. One of the reasons is that we have quite rich structurebased on both class terms and relational terms, while the other logics arebased on weaker or incomparable with our system languages, some of themdealing only with atomic formulas. Such is, for instance, the system ofMcAllester and Givan [12] and some systems studied in Moss and Pratt-Hartmann [27] and Moss [17]. The following fragment of our language isconsidered in Moss [17]. All propositional connectives are omitted, onlyclass variables are considered as class terms, and the language contains asingle relational variable (the relational terms are α and α−1). This fragmentis axiomatized in [17] by means of an infinite set of finitary rules of inferenceand it is noted that there is no finite one, while our system with only onerelational variable has a finitary axiomatization. If in the above fragment weallow only two relational terms α and −α and all class terms are variablesor complemented variables, then the fragment coincides with the system R†

studied by Moss and Pratt-Hartmann in [27].In the present paper we have proved the completeness of a syllogistic logic

with a set of binary relations closed under the Boolean operations and undertaking the converse. The completeness proof can be easily generalized to thecase of n-ary relations for some arbitrary n. The difference is that we need toconsider all transformations of a relation by permuting its arguments insteadof just taking the converse. In order to generalize the construction of thecanonical model, it is necessary to consider Boolean algebras of symmetricrelational terms for every possible set of permutations of the n argumentsof a relation.

37

The construction of the canonical model in our logic is similar to thatfor BML. It is also possible to use the construction of the relations RV fromR0

V to prove the completeness of BML extended with a symbol −1 for theconverse of the accessibility relation.

Acknowledgements

The authors would like to thank Ian Pratt-Hartmann for valuable commentson the paper.

This work was supported by the European Social Fund through theHuman Resource Development Operational Programme 2007–2013 undercontract BG051PO001-3.3.04/27/28.08.2009 and by Sofia University undercontract 136/2010.

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