Fregean Logics ?

J. Czelakowski a, D. Pigozzi b,∗,1aOpole University, Institute of Mathematics, Oleska, 48, 45-951 Opole, PolandbIowa State University, Department of Mathematics, Ames, Iowa 50011, USA

Abstract

According to Frege’s principle the denotation of a sentence coincides with its truth-value. The principle is investigated within the context of abstract algebraic logic,and it is shown that taken together with the deduction theorem it characterizesintuitionistic logic in a certain strong sense.

A 2nd-order matrix is an algebra together with an algebraic closed set systemon its universe. A deductive system is a 2nd-order matrix over the formula algebraof some fixed but arbitrary language. A 2nd-order matrix A is Fregean if, for anysubset X of A, the set of all pairs 〈a, b〉 such that X ∪ a and X ∪ b have thesame closure is a congruence relation on A. Hence a deductive system is Fregeanif interderivability is compositional. The logics intermediate between the classicaland intuitionistic propositional calculi are the paradigms for Fregean logics. Normalmodal logics are non-Fregean while quasi-normal modal logics are generally Fregean.

The main results of the paper: Fregean deductive systems that either have thededuction theorem, or are protoalgebraic and have conjunction, are completely char-acterized. They are essentially the intermediate logics, possibly with additional con-nectives. All the full matrix models of a protoalgebraic Fregean deductive system areFregean, and, conversely, the deductive system determined by any class of Fregean2nd-order matrices is Fregean. The latter result is used to construct an example ofa protoalgebraic Fregean deductive system that is not strongly algebraizable.

Key words: abstract algebraic logic, protoalgebraic logic, equivalential logic,algebraizable logic, self-extensional logic, Leibniz congruence, deduction theorem,quasivariety1991 MSC: Primary: 03G99, Secondary: 03B22, 03C05, 08C15

? Part of the research reported on here was done while both authors were in resi-dence at the Centre de Recerca Matematica of the Institut d’Estudis Catalans.∗ Corresponding author.

Email addresses: jczel@math.uni.opole.pl (J. Czelakowski),dpigozzi@iastate.edu (D. Pigozzi).1 Supported by National Science Foundation grant CCR-9593168 and a grant from

Preprint submitted to Elsevier Science 14 July 2003

Introduction

The origin of Fregean logic is Frege’s principle of compositionality. Frege’sseminal insight, as interpreted by Church [1], was to think of a (declarative)sentence in the same way as one thinks of a proper name. A sentence, like everyproper name, must denote or name something (Church’s rendering of Frege’sbedeuten). Church calls the thing it denotes, i.e., its denotation (Bedeutung),its truth-value. According to Frege a sentence also has a sense (Sinn), whichis also assumed to be compositional. But Frege viewed this concept as extra-linguistic and did not attempt to incorporate it in his formal system.

Frege’s analysis of proper names when applied to the denotation of sentencesleads to the principle of compositionality for truth-values: assume a constituentpart ϕ of a sentence ϑ is replaced by another sentence ϕ′ to give ϑ(ϕ′/ϕ). If ϕand ϕ′ both have the same truth-value, then so do ϑ and ϑ(ϕ′/ϕ). Logical sys-tems that uphold the Frege principle are sometimes called truth-functional orextensional. Those that violate it are called nontruth-functional or intensional.Most modal logics are intensional in this sense.

The first one to formally analyze the Frege principle in a general setting wasR. Suszko. In his view the denotation of a sentence is not its truth-value,but rather something more in keeping with Frege’s notion of the sense of asentence. 2 Moreover, he introduced a new binary connective ∆, called theidentity connective, into the language with the idea that the sentence ϕ∆ψ isto be interpreted as the proposition that ϕ and ψ have the same denotationin this new sense, which for the purposes of this introduction we will viewas the proposition that ϕ and ψ have the same meanings. In Suszko’s formalsystem, which he called logic with identity, the principal axioms governing theidentity-of-meaning connective ∆ express its compositionality. Suszko’s systemalso includes all the classical connectives, in particular the biconditional ↔.As in Frege’s system, ϕ ↔ ψ is to be interpreted as the proposition thatϕ and ψ have the same truth-value. It is easily shown that the two binaryconnectives ↔ and ∆ are both compositional only if the sentences ϕ ↔ ψand ϕ∆ψ are themselves logically equivalent for all sentences ϕ and ψ. ThusFrege’s principle that ↔ is compositional can be formalized in Suszko’s systemas the proposition

(x↔ y) ∆ (x∆ y).

Suszko calls this the Fregean axiom. When adjoined to the other axioms of

the Ministerio de Educacion y Ciencia of the government of Spain.2 Suszko looked to Wittgenstein for support for this view. For him the denotationof a sentence is what the sentence says about a certain “situation”. This term waschosen by Suszko to interpret Wittgenstein’s Sachlage—the state of affairs. See [2]and [3].

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logic with identity it gives Fregean logic, and extensions of logic with identityin which it fails to hold are called non-Fregean.

In this paper we investigate the Fregean axiom within the framework of ab-stract algebraic logic. This is a developing area of algebraic logic in which thefocus is on the process by which a class of algebras is associated with a givenlogical system and on the connection between its metalogical and algebraicproperties. We consider a much wider class of deductive systems than thoseencompassed by Suszko’s logic of identity. In particular, we consider deductivesystems that are not assumed a priori to have special connectives dedicatedto representing identity of meanings or of truth-values. This requires that webegin our investigation with a coherent explanation of how these two key no-tions are to be represented in an arbitrary deductive system. We now discuss,very briefly, some consequences of the theory of referential frames that servethis purpose; for a more detailed account of referential frames see [4].

A deductive system S is characterized by its language type Λ, the set FmΛ offormulas over Λ, and by the consequence relation `S that, for each set Γ offormulas, specifies which formulas ϕ are consequences of Γ . A set T of formulasis a theory of S if it is closed under consequence, i.e., ϕ ∈ T whenever T `S ϕ.The set of all theories of S is denoted by ThS. By the theory axiomatized byan arbitrary set Γ of formulas we mean the set of all formulas ϕ such thatΓ `S ϕ.

S is viewed as an “uninterpreted” logic. Its interpretations take the form ofmatrices. A (logical) matrix is a structure of the form A = 〈A, F 〉 where A isan algebra (of the same language type as S), the underlying algebra of A, andF ⊆ A, the designated set of A. An interpretation of S is a matrix A togetherwith a mapping h: Fm → A from the set of Fm of formulas into the universeA of the underlying algebra of A. h(ϕ) is to be thought of as the “sense” or“meaning” of the formula ϕ under the interpretation, and ϕ is “true” or “false”depending on whether or not h(ϕ) ∈ F . Several natural assumptions are madeabout interpretations. First of all, the meaning function h is assumed to be ahomomorphism from the algebra of formulas Fm into A; this is the principleof compositionality of meaning. Secondly, truth and meaning are assumed tobe connected by another well-known principle, due to Leibniz. According tothe Leibniz principle identity can be characterized in second-order logic bythe formula

x ≈ y iff ∀P (P (x) ↔ P (y)),

where P ranges over all unary predicates. The principle is adapted to the in-terpretations of a deductive system S by restricting attention to predicatesthat are “definable” in S by some formula ϑ(x) with a designated variablex (ϑ(x) may have other variables that are treated as parameters). Thus weassume that, if the formulas ϕ and ψ have different meanings in an interpreta-tion, then they can be distinguished by some predicate, i.e., for some formula

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ϑ(x), ϑ(ϕ/x) and ϑ(ψ/x) have different truth-values. This is also known asthe principle of contextual differentiation. Finally, we assume the class of in-terpretations is sound and complete for the consequence relation in the sensethat Γ `S ϕ iff ϕ is true in every interpretation in which each ψ ∈ Γ is true.

A consequence of these assumptions is that the global identity-of-truth-valueand identity-of-meaning relations can be characterized entirely in terms of theconsequence relation, without direct reference to the interpretations. In fact,the identity-of-truth-value relation of S is given by

ΛS = 〈ϕ, ψ〉 : ∀Γ ⊆ FmΛ

(Γ `S ϕ⇔ Γ `S ψ

),

and the identity-of-meaning relation by

ΩS = 〈ϕ, ψ〉 : ∀Γ ⊆ FmΛ ∀ϑ(x) ∈ FmΛ

(Γ `S ϑ(ϕ/x) ⇔ Γ `S ϑ(ψ/x)

).

ΛS and ΩS are called the Frege relation and Leibniz congruence of S, re-spectively. The Fregean axiom for S takes the form ΛS = ΩS. Arbitrarydeductive systems with this property have been identified and investigated inthe literature under the name self-extensional (see [3]).

The paradigms for self-extensional deductive systems are the classical andintuitionistic propositional calculi. But these systems have a stronger property:every interpreted classical and intuitionistic logic also satisfies the Fregeanaxiom, and it is this that is taken to be the defining property of a Fregeandeductive system. For any theory T of a deductive system S define:

ΛS T = 〈ϕ, ψ〉 : ∀Γ ⊆ FmΛ

(T, Γ `S ϕ⇔ T, Γ `S ψ

),

ΩST =

〈ϕ, ψ〉 : ∀Γ ⊆ FmΛ ∀ϑ(x) ∈ FmΛ

(T, Γ `S ϑ(x/ϕ) ⇔ T, Γ `S ϑ(x/ψ)

).

ΩS T is called the Suszko congruence of T with respect to S; it can be ex-pressed in the following more perspicuous form by means of the consequenceoperator CloS of S. ΩS T = 〈ϕ, ψ〉 : ∀ϑ(x) ∈ FmΛ

(CloS(T, ϑ(x/ϕ)) =

CloS(T, ϑ(x/ψ))). Similarly, ΛS T = 〈ϕ, ψ〉 : CloS(T, ϕ) = CloS(T, ψ) .

A deductive system S is Fregean if ΛS T = ΩS T for every theory T of S.

The main result of the paper is that the Fregean axiom together with thededuction theorem is the characteristic property of the intuitionistic calculus.In Theorems 2.20 and 2.22 it is shown that every Fregean deductive systemwith the uniterm deduction-detachment theorem (Def. 1.38) and every pro-toalgebraic (Def. 1.14) deductive system with conjunction is equivalent in astrong sense to an axiomatic extension of the appropriate fragments of the

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intuitionistic propositional calculus, possibly with arbitrarily many additionalconnectives λ that are compatible with intuitionistic logical equivalence in thesense that, if the rank of λ is n, then

((x0 ↔ y0) ∧ · · · ∧ (xn−1 ↔ yn−1)

)→

(λx0 . . . xn−1 ↔ λy0 . . . yn−1) is a theorem of the deductive system.

Another important theme of the paper has to do with an old problem inabstract algebraic logic: why is it that almost all the algebraizable logics inthe literature have varieties of algebras as their algebraic counterparts, whenthe general theory of algebraizable logics (as recently developed in abstractalgebraic logic) indicates that quasivarieties are the natural algebraic coun-terparts? (Algebraizable deductive systems whose algebraic counterpart is avariety are said to be strongly algebraizable.) The paper contains some new in-sights into the solution of this problem, both in the form of original results andof elaborations of recent important results on this problem due to J. M. Fontand R. Jansana [5]. In particular we show that every Fregean deductive systemwith the uniterm deduction-detachment theorem is strongly algebraizable andthat its algebraic counterpart is termwise definitionally equivalent to a vari-ety of Hilbert algebras with compatible operations (Thm. 2.23 and Cor. 2.24).Similarly, it is shown that every protoalgebraic, Fregean deductive system withconjunction is strongly algebraizable, provided that it has at least one theo-rem, and that its algebraic counterpart is termwise definitionally equivalent toa variety of Brouwerian semilattices with compatible operations (Thm. 2.25and Cor. 2.26.) Partial generalizations of these results to self-extensional sys-tems are given in Thms. 2.29 and 2.32.

The other central topic of the paper is an investigation of the relationshipbetween Fregean deductive systems and their matrix semantics. A semanticversion of the Fregean property is defined (Def. 2.13) and it is proved that, ifa protoalgebraic deductive system is Fregean, then every full 2nd-order model(Def. 3.1) of it is Fregean (Cor. 3.5). Conversely, the deductive system de-termined by any class of Fregean 2nd-order matrices is Fregean (Cor. 3.8).The latter result is used to verify that a particular algebraizable, Fregean de-ductive system is not strongly algebraizable; the example is due to P. Idziak.We also outline a proof that the ↔,¬-fragment of the intuitionistic propo-sitional calculus is Fregean and algebraizable but not strongly algebraizable.We conclude that the behavior of protoalgebraic, Fregean deductive systemsthat fail to have the uniterm deduction-detachment theorem differs strikinglyfrom those that do. A protoalgebraic, Fregean deductive system may be eitherstrongly algebraizable or not, but the uniterm deduction-detachment theoremguarantees strong algebraizability in this context by Thm. 2.23. The fact thatthe deduction-detachment system is uniterm is essential. An example of aFregean deductive system with the multiterm deduction-detachment theoremis given in [6].

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Outline of the paper

The first section contains a survey of the basic elements of abstract algebraiclogic that are needed for a systematic study of Fregean logics. The notionsof protoalgebraic, equivalential, and algebraizable deductive systems are re-viewed. The main novelty of the section is a fairly detailed discussion of thenotion of a regularly algebraizable deductive system (Def. 1.29). The highlightof this part is a proof of the fact that a deductive system is regularly algebraiz-able if and only if it is the assertional logic (Def. 1.32) of a relatively point-regular quasivariety (Def. 1.31). See Thm. 1.34. The deduction-detachmenttheorem in abstract algebraic logic is discussed in the last part of the section.

It turns out that the property of being Fregean is expressible as a Gentzen-style, or what we call a 2nd-order, inference rule. The basic properties ofFregean deductive systems are most conveniently developed within the the-ory of 2nd-order rules and 2nd-order matrices. This is done in the first partof Section 2. The properties of the Frege, Leibniz, and Suszko relations andtheir relationships are also developed here. The latter part of the section con-tains the characterizations of Fregean deductive systems with the unitermdeduction-detachment theorem and, alternatively, conjunction; the results onstrong algebraizability mentioned previously can also be found here.

The discussion of the matrix semantics of Fregean systems is contained in thelast section.

Acknowledgments and connections with other work

As explained above, the study of Fregean logics was initiated by R. Suszkoand his collaborators. His formal system of logic with identity, both with andwithout the Fregean axiom, has been investigated in a number of papers ([7–14].) The first published work on Fregean logic, within the context of abstractalgebraic logic, is the monograph of Font and Jansana [5]; an extended abstractcan be found in [15]. About the same time, and essentially independently ofFont and Jansana, the present authors were obtaining many similar results.However we concentrated almost exclusively on protoalgebraic Fregean logicswhile the scope of Font and Jansana’s work was much broader. The studyof nonprotoalgebraic Fregean logic requires the “2nd-order” methods that aredeveloped in Sections 2 and 3. These were pioneered by the Barcelona algebraiclogic group, in particular by Font, Jansana, A. Torrens, and V. Verdu. Thepresent authors wish to acknowledge the influence this work has had on theirown. A detailed investigation of protoalgebraic Fregean deductive systems withthe multiterm deduction-detachment system can be found in [6].

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The paper [16] on essentially the purely algebraic aspects of Fregean logicappeared earlier and had some influence on the metalogical developments.The algebraic theory has been further developed in [17–19].

1 Elements of Abstract Algebraic Logic

1.1 Closed-set systems

Let A be a nonempty set. A family C of subsets of A that is closed underthe intersection of arbitrary subfamilies is called a closed-set system over A.If C is also closed under the union of subfamilies that are (upward) directed(under inclusion), then it is called an algebraic closed-set system. All closed-setsystems considered here are automatically assumed to be algebraic unless oth-erwise indicated. An algebraic closed set system C forms an algebraic lattice〈C,∩,∨〉 under set-theoretic inclusion. Since empty subfamilies are allowed,every closed-set system over A contains A. Closed-set systems will be rep-resented by the calligraphic letters C,D, . . .. The closed sets of a closed-setsystem C (i.e., the members of C) will be called filters and represented by up-per case Latin letters F,G, . . .. The closure operator associated with a givenclosed-set system C is denoted by CloC. Thus CloC :P(A) → P(A), and, foreach X ⊆ A, CloCX =

⋂F : X ⊆ F ∈ C . The algebraicity of C is reflectedin the fact that CloCX =

⋃CloCX′ : X ′ ⊆ω X . (X ′ ⊆ω X means that X ′

is a finite subset of X.)

Let C be an algebraic closed-set system over a nonempty set A. If B is anonempty subset of A, then F ∩ B : F ∈ C is an algebraic closed-setsystem over B. If h:B → A, then h−1(C) := h−1(F ) : F ∈ C is an algebraicclosed-set system over B.

Given any F ∈ P(A), F,A is the smallest closed-set system containing F ;it is obviously algebraic. If C is a closed-set system over A that contains F ,then we define [F )C := G : F ⊆ G ∈ C . It is called the principal (2nd-order) filter of C generated by F . [F )C is obviously also an algebraic closed-setsystem.

1.2 Deductive systems

By a language type we mean a set Λ of connectives or operation symbols,depending on whether we are viewing them from a logical or algebraic per-spective. Each connective has associated with it a natural number, called its

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rank or arity. Λ is pointed if it contains a distinguished constant (i.e., nullaryoperation) symbol, which is usually denoted by >. A fixed, denumerable setVa of variable symbols is assumed, and the set of Λ-formulas (Λ-terms in analgebraic context) is formed in the usual way. The set of Λ-formulas is denotedby FmΛ, and the corresponding algebra of formulas by FmΛ. (The subscriptmay be omitted when only one language type is under consideration.) For anyset X of variables, FmΛ(X) is the set of formulas in which only variables fromX occur, and FmΛ(X) is the corresponding subalgebra of FmΛ. The opera-tion of simultaneously substituting fixed but arbitrary formulas for variablesis identified with the unique endomorphism of FmΛ it determines. Formulasare represented by lower case Greek letters ϕ, ψ, . . ., and sets of formulas byupper case Greek letters Γ,∆, . . ..

Definition 1.1 Let k be a nonzero natural number. By a k-dimensional de-ductive system, or more simply a k-deductive system, we mean an orderedpair

S = 〈FmΛ,ThS〉,

where Λ is an arbitrary language type and ThS is an (algebraic) closed-setsystem over Fmk

Λ, the k-th Cartesian power of FmΛ, that is substitution-invariant in the following sense. σ−1(T ) ∈ ThS for every T ∈ ThS, or moresuccinctly,

σ−1(ThS) ⊆ ThS, for every substitution σ. 2

The closure operator CloThS :P(FmkΛ) → Fmk

Λ is called the consequence oper-ator of S and will be denoted by the simpler expression CnS . The consequencerelation `S of a k-deductive system S is the binary relation between P(Fmk

Λ)and Fmk

Λ defined by Γ `S ϕ iff ϕ ∈ CloS(Γ ), for all Γ ⊆ FmkΛ and ϕ ∈ Fmk

Λ.A deductive system S is often identified with one of the two ordered pairs〈FmΛ,CloThS〉 or 〈FmΛ,`S〉. The reason for our choosing to identify it withthe pair 〈FmΛ,ThS〉 will become clear below in Section 2.

The basic syntactic unit of a k-deductive system is a k-tuple of Λ-formulas;these are called k-formulas. If ϕ = 〈ϕ0, . . . , ϕk−1〉 is a k-formula and σ : FmΛ →FmΛ is a substitution, then the σ-substitution instance of ϕ, σ(ϕ), is definedto be 〈σ(ϕ0), . . . , σ(ϕn−1)〉.

The filters of S, i.e., the members of ThS, are called theories of S, or S-theories. Theories are represented by the uppercase Latin letters T, S, . . ..

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The defining properties of a k-deductive S system in terms of its consequencerelation are as follows. For all Γ,∆ ⊆ Fmk

Λ and ϕ ∈ FmkΛ,

Γ `S ϕ for all ϕ ∈ Γ (1)

Γ `S ϕ and Γ ⊆ ∆ imply ∆ `S ϕ; (2)

Γ `S ϕ and ∆ `S ψ for every ψ ∈ Γ imply ∆ `S ϕ; (3)

Γ `S ϕ implies Γ ′ `S ϕ for some Γ ′ ⊆ω Γ ; (4)

Γ `S ϕ implies σ(Γ ) `S σ(ϕ) for every substitution σ. (5)

Note that (4), which expresses the property that S is finitary, is equivalent tothe algebraicity of ThS. Similarity, (5) expresses the substitution-invarianceof S. We also note that condition (2) is a consequence of the other conditions.

1-deductive systems, the ones we will be mostly concerned with in this paper,can be identified with deductive systems in the classical sense of Tarski. Theseinclude all the familiar sentential logics together with their various fragmentsand refinements—for example, the classical and intuitionistic propositionalcalculi, the intermediate logics, the various modal logics (including Lewis’sS4 and S5), and the multiple-valued logics of Lukasiewicz and Post. The sub-structural logics such as BCK logic, relevance logic, and linear logic can alsobe formulated as 1-deductive systems, although they are often formulated asGentzen-type systems.

The deductive systems of equational logic however can be most naturallyformulated as 2-deductive systems. k-deductive systems were first consideredin [20] and systematically used in [21] as a vehicle for studying algebraizabilityand the deduction theorem in the context of abstract algebraic logic.

By a k-dimensional sequent, or simply a k-sequent, or more simply a sequentwhen k is clear from context, we mean a pair 〈Γ, ϕ〉 where Γ is a set ofk-formulas and ϕ is a single k-formula; the sequent is finite if Γ is finiteand proper if Γ is nonempty. The k-sequent 〈Γ, ϕ〉 is usually written in the

traditional formΓ

ϕ. A k-formula ϕ is a theorem of a k-deductive system S

if `S ϕ (i.e., ∅ `S ϕ). The set of all theorems is denoted by ThmS. The

k-sequentΓ

ϕis a rule of S if Γ `S ϕ.

A k-formula ψ is directly derivable from a set ∆ of k-formulas by the k-

sequentΓ

ϕif there is a substitution σ : FmΛ → FmΛ such that σ(Γ ) ⊆ ∆

and σ(ϕ) = ψ. Every pair of sets Ax, of k-formulas, and Ru, of finite k-sequents, determines a k-deductive system S in the usual way: for ∆ ⊆ Fmk

Λ

and ϕ ∈ FmkΛ, ∆ `S ϕ iff ϕ is contained in the smallest set of k-formulas that

includes ∆, contains all substitution instances of each k-formula in Ax, and is

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closed under direct derivability with respect to each k-sequent in Ru. The pairAx and Ru is called a system of axioms and inference rules for S, and S is saidto be presented by Ax and Ru. Every k-deductive system can be presented bysome, and in fact, many different systems of axioms and inference rules.

Let S and S ′ be k-deductive systems over the language types Λ and Λ′, re-spectively. S ′ is an expansion of S if Λ ⊆ Λ′ and `S ⊆ `S′ . An expansion iscalled an extension if Λ = Λ′; it is said to be axiomatic if a presentation of S ′can be obtained from a presentation of S by adjoining new axioms but no newinference rules. An expansion is conservative if `S′ ∩

(P(Fmk

Λ)×FmkΛ

)= `S .

In this situation S is called a fragment of S ′.

The most important example of a 2-deductive system is equational (or quasi-equational) logic. In this context a 2-formula 〈ϕ, ψ〉 is to be interpreted as anequation ϕ ≈ ψ.

Free equational logic. Let Λ be any language type. The axioms and inferencerules of the system of free equational logic over Λ are the following.

(A1) 〈x, x〉;

(R1)〈x, y〉〈y, x〉

;

(R2)〈x, y〉, 〈y, z〉〈x, z〉

;

(R3λ)〈x0, y0〉, . . . , 〈xn−1, yn−1〉〈λx0 . . . xn−1, λy0 . . . yn−1〉

, for each λ ∈ Λ, n the rank of λ.

In free equational logic the 2-formula 〈ϕ, ψ〉 is identified with the equationϕ ≈ ψ and the 2-sequent

γ0 ≈ δ0, . . . , γn−1 ≈ δn−1

ϕ ≈ ψ

with the quasi-equation γ0 ≈ δ0 ∧ · · · ∧ γn−1 ≈ δn−1 → ϕ ≈ ψ. The theoriesof free equational logic are exactly the congruences on the formula algebrasFmΛ.

Applied equational logic. Each quasivariety defines an extension of free equa-tional logic in the following way.

Definition 1.2 Let Q be a quasivariety over the language type Λ that is de-fined by the identities Id and quasi-identities Qd. By the equational logic 3

3 Equational logic in this sense differs from the equational logic of identities as itis commonly understood in universal algebra. The latter applies only to varieties sothat the set Qd of proper quasi-identities is empty. Moreover, the free equationallogic in this sense includes the rule of substitution, i.e., 〈ϕ,ψ〉 / 〈σ(ϕ), σ(ψ)〉 for

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of Q, in symbols Seql Q, we mean the 2-deductive system presented by (A1),(R1), (R2), and (R3λ), for λ ∈ Λ, together with the following axioms:

〈ϕ, ψ〉, for every identity ϕ ≈ ψ ∈ Id,

and inference rules:

〈γ0, δ0〉, . . . , 〈γn−1, δn−1〉〈ϕ, ψ〉

,for every quasi-identity(∧

i<n γi ≈ δi)→ ϕ ≈ ψ ∈ Qd. 2

The theories of Seql Q are the Q-congruences on FmΛ, i.e., those congruencesΘ on FmΛ such that FmΛ/Θ ∈ Q. The set of Q-congruences on FmΛ isdenoted by CoQ FmΛ; thus

CoQ FmΛ = Th Seql Q and Seql Q = 〈FmΛ,CoQ FmΛ〉.

The Seql Q-theory generated by E ⊆ Fm2Λ is the K-congruence generated by

E, i.e., CnSeql Q(E) = CgFmΛQ (E).

In the following theorem we show that, conversely, if C is any algebraic closed-set system of congruences of FmΛ that is substitution-invariant, then C =CoQ FmΛ for a unique quasivariety Q. A consequence of this way of lookingat equational logic is a useful intrinsic characterization of those sets of con-gruences on the formula algebra that are the relative congruences of somequasivariety.

Theorem 1.3 Let Λ be an arbitrary language type. Let C ⊆ Co FmΛ. ThenC = CoQ FmΛ for some quasivariety Q iff the following conditions hold.

(i) C is closed under intersection;(ii) C is closed under the union of upper-directed sets;

(iii) C is substitution-invariant.

PROOF. If C = CoQ FmΛ, then it is obvious that conditions (i)–(iii) hold.For the reverse implication, assume (i)–(iii) hold, i.e. that C is an algebraicclosed-set system of congruences on FmΛ that is substitution-invariant. ThenS = 〈FmΛ, C〉 is a 2-deductive system, and since theories of S are congruences,(A1) is a theorem of S and (R1)–(R3λ), for λ ∈ Λ, are rules of S. Thus S isan extension of the free equational logic, i.e., an applied equational logic. LetQ be the quasivariety whose identities coincide with the theorems of S andwhose quasi-identities coincide with the rules of S. Then S = Seql Q. 2

every substitution σ. Consequently only identities are derivable.

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This characterization can also be expressed in purely lattice theoretic terms.A complete lattice L is a complete meet-subsemilattice of a complete latticeM if

∧L S =∧M S for every subset S of L; it is said to be continuous if∨L S =

∨M S for every upward directed subset S of L.

Corollary 1.4 Let C be a sublattice of the complete lattice CoFmΛ of allcongruences on FmΛ. Then C = CoQ FmΛ for some quasivariety Q iff Cis substitution-invariant and a complete, continuous meet-subsemilattice ofCoFmΛ. 2

The mapping Q 7→ Seql Q is a one-one correspondence between quasivarietiesover Λ and extensions of the free equational logic over Λ by additional axiomsand inference rules.

It will be useful to define the equational logic of an arbitrary class K of Λ-algebras. For such a class SK and IK respectively denote the class of allsubalgebras and isomorphic images of members of K. By a K-congruence on aΛ-algebra A (not necessarily a member of K) we mean a congruence Θ on Asuch the A/Θ ∈ I SK. The set of all K-congruences on A is denoted by CoK A.We define Seql K to be 〈FmΛ,CoK FmΛ〉. CoK FmΛ is not in general closedunder intersection or directed union, but it is closed under inverse substitution,and it does have an associated consequence relation: for all E ∪ ϕ ≈ ψ ⊆Fm2

Λ,

E `Seql K ϕ ≈ ψ iff ϕ ≈ ψ ∈⋂

Θ ∈ CoK FmΛ : E ⊆ Θ.

Alternatively, E `Seql K ϕ ≈ ψ iff for every homomorphism h: FmΛ → A,h(ϑ) = h(ξ) for all ϑ ≈ ξ ∈ E implies h(ϑ) = h(ξ). Note that in generalSeql K is not finitary, but it is substitution invariant.

1.3 Algebraizable 1-deductive systems

A general theory of algebraizability of logic was presented in [22] and subse-quently refined and extended in a number of papers by several different authors[23,24,5,21]. We restrict ourselves here to 1-deductive systems and to the no-tion of algebraizability as originally presented in [22]. In current terminologythis is called finite algebraizability. From now on, when we speak of a “de-ductive system” (without reference to its dimension) we mean a 1-deductivesystem.

In the following development it is convenient to use the expression K ≈ L asan abbreviation for a set of equations κi ≈ λi : i ∈ I .

12

Definition 1.5 Let S be a 1-deductive system over Λ. S is finitely algebraiz-able if there is a quasivariety Q over Λ, a finite nonempty system

E(x, y) =ε0(x, y), . . . , εn−1(x, y)

of binary formulas (i.e., formulas in two variables), and a finite nonemptysystem

K(x) ≈ L(x) =κ0(x) ≈ λ0(x), . . . , κm−1(x) ≈ λm−1(x)

of equations (2-formulas) in one variable such that the following equivalenceshold between the rules of S and of Seql Q.

For all Γ ∪ ϕ ⊆ω FmΛ, Γ `S ϕ iffκj(ψ) ≈ λj(ψ) : ψ ∈ Γ, j < m

`Seql Q κj(ϕ) ≈ λj(ϕ), for all j < m.

(6)

For all Γ ≈ ∆ ∪ ϕ ≈ ψ ⊆ω Fm2Λ, Γ ≈ ∆ `Seql Q ϕ ≈ ψ iff

εi(γ, δ) : γ ≈ δ ∈ Γ ≈ ∆, i < n`S εi(ϕ, ψ), for all i < n.

(7)

εi

(κj(x), λj(x)

): i < n, j < m

`S x, and x `S εi

(κj(x), λj(x)

),

for all j < m, i < n.

(8)

κj

(εi(x, y)

)≈ λj

(εi(x, y)

): i < n, j < m

`Seql Q x ≈ y, and

x ≈ y `Seql Q κj

(εi(x, y)

)≈ λj

(εi(x, y)

), for all i < n, j < m. 2

(9)

We note that, since Q is a quasivariety, Seql Q as well as S is a finitarydeductive system. So, if the equivalences (6) and (7) hold for finite sets ofequations and formulas, they also hold for infinite sets.

In the sequel when we say simply “algebraizable” we will mean finitely alge-braizable in the above sense.

These four conditions are abbreviated respectively as follows.

Γ `S ϕ iff K(Γ ) ≈ L(Γ ) `Seql Q K(ϕ) ≈ L(ϕ). (10)

Γ ≈ ∆ `Seql Q ϕ ≈ ψ iff E(Γ,∆) `S E(ϕ, ψ). (11)

x a`S E(K(x), L(x)

). (12)

x ≈ y a`Seql Q K(E(x, y)

)≈ L

(E(x, y)

). (13)

Similar abbreviations, which should be self-explanatory, will be used in thesequel without further elaboration.

It is not difficult to show ([22, Corollary 2.9]) that each of the conditions (11)and (12) is derivable from the two conditions (10) and (13), and vice versa.

13

Thus (10) and (13) together are sufficient for algebraizability. Similarly, (11)and (12) are sufficient.

Definition 1.6 Let S be a 1-deductive system and Q a quasivariety over thelanguage type Λ.

(i) A finite system K(x) ≈ L(x) of equations in one variable is said tobe a faithful interpretation of (the consequence relation of) S in (theconsequence relation of) Seql Q if the equivalence (10) holds for all Γ ∪ϕ ⊆ FmΛ.

(ii) A finite system E(x, y) of binary formulas is said to be a faithful in-terpretation of Seql Q in S if the equivalence (11) holds for all Γ ≈∆ ∪ ϕ, ψ ⊆ Fm2

Λ.(iii) The interpretations K(x) ≈ L(x) and E(x, y) are inverses of one another

if the entailments (12) and (13) both hold. 2

Corollary 1.7 A 1-deductive system S is (finitely) algebraizable iff there isan invertible faithful interpretation E(x, y) of Seql Q in S for some quasiva-riety Q. 2

So a 1-deductive system S is algebraizable if it is definitionally equivalent in anatural way to the equational logic of some quasivariety Q. We will see belowin Cor. 1.26 that Q is uniquely determined by S. It is called the equivalentquasivariety of S.

Definition 1.8 An algebraizable 1-deductive system whose equivalent quasi-variety is a variety is said to be strongly algebraizable. 2

Let A be a Λ-algebra. A subset F of Ak is called a filter of A over a k-deductive system S, an S-filter of A, if F contains all interpretations of eachtheorem of S and is closed under all interpretations of each rule of S. Moreprecisely, for every homomorphism h: FmΛ → A, we have that h(ϕ) ∈ F foreach theorem ϕ of S, and h(ψ0), . . . , h(ψn−1) ∈ F imply h(ϕ) ∈ F for each

ruleψ0, . . . , ψn−1

ϕof S. The set of all S-filters of A is denoted by FiS A; it

is an (algebraic) closed-set system over A. Thus it forms an algebraic latticeunder inclusion, which is denoted by FiS A.

We note that ThS = FiS FmΛ. We also note that, for every quasivarietyQ, the (Seql Q)-filters of A are exactly the Q-congruences on A, i.e., thecongruences Θ such that A/Θ ∈ Q. Thus CoQ A = FiSeql Q A. Since Q isclosed under isomorphism and subdirect products, generated Q-congruencesexist, and, for every R ⊆ A2, CgA

Q (X) = CloFiSeql Q A(X). The algebraic latticeof Q-congruences is denoted by CoQ A.

14

Definition 1.9 Let A be a Λ-algebra and F ⊆ Ak. Let

ΩA F :=〈a, b〉 ∈ A2 : ϕA(a, c) ∈ F iff ϕA(b, c) for all ϕ(x, z) ∈ Fmk

Λ and c ∈ A|k|.

ΩA F is clearly a congruence relation on A. It is called the Leibniz congruenceof F on A. 2

Lemma 1.10 ([25, Lemma 1.5]) Let A be a Λ-algebra and F ⊆ Ak. ΩA Fis the largest congruence Θ on A that is compatible with F in the follow-ing sense. If 〈a0, . . . , ak−1〉 ∈ F and ai ≡ bi (mod Θ) for all i < k, then〈b0, . . . , bk−1〉 ∈ F . 2

It is easy to check that there is always a largest congruence with this property.As a mapping from P(Ak) to Co A, ΩA is called the Leibniz operator on A.When A is the formula algebra FmΛ the superscript on Ω is normally omitted;thus Ω = ΩFmΛ

. In the following lemma we verify a fundamental property ofthe Leibniz congruence, its invariance under surjective homomorphisms.

Lemma 1.11 Let A and B be Λ-algebras and h: A B a surjective homo-morphism. Then for any F ⊆ Bk,

ΩA

(h−1(F )

)= h−1(ΩB F ).

PROOF. h−1(F ) := 〈a0, . . . , ak−1〉 : 〈h(a0), . . . , h(ak−1)〉 ∈ F . It is easyto check that h−1(ΩB F ) is a congruence on A compatible with h−1(F ); so

ΩA

(h−1(F )

)⊇ h−1(ΩB F ).

Let Θ = h−1(IdA), the relation kernel of h. Θ is compatible with h−1(F ),

so that Θ ⊆ ΩA

(h−1(F )

). Thus h

(ΩA

(h−1(F )

))is a congruence of B that

is compatible with hh−1(F ), which equals F because h is surjective. Conse-

quently h(ΩA

(h−1(F )

))⊆ ΩB F , and hence

ΩA

(h−1(F )

)= h−1h

(ΩA

(h−1(F )

))⊆ h−1(ΩB F ). 2

It is known that a 1-deductive system S is algebraizable with equivalentquasivariety Q iff Ω induces an isomorphism between the lattices ThS andCoQ FmΛ; see [22, Theorems 3.7(ii) and 4.1]. It then follows from Cor. 1.4that a 1-deductive system S is algebraizable iff Ω is an isomorphism betweenThS and a substitution-invariant, complete, continuous meet-subsemilatticeof CoFmΛ. (The condition that the meet-subsemilattice be substitution-invariant turns out to be redundant.) This is one of the main results of [22];see Theorem 4.2. It is further proved in [22] that, if a 1-deductive system S

15

is algebraizable, then, for every Λ-algebra A, ΩA is an isomorphism betweenFiS A and a continuous sublattice of CoA ([22, Theorem 5.1]).

This characterization gives rise in turn to a useful representation of the equiv-alent quasivariety of any algebraizable deductive system. The mode of repre-sentation is quite general and can be used to associate a class of Λ-algebraswith every deductive system of any given dimension.

By a k-matrix (over Λ) we mean an ordered pair A = 〈A, FA〉, where A isa Λ-algebra and FA is a subset of Ak. FA is called the designated filter of A

and A the underlying algebra of A. The subscript A on the designated filter isusually omitted. 1-matrices are referred to simply as matrices.

A is reduced if ΩA FA = IdA, the identity congruence. If A is an arbitraryk-matrix, A∗ is defined to be the quotient matrix 〈A/ΩA FA, FA/ΩA FA〉.As a consequence of the commutativity of the Leibniz operator with inversesurjective homomorphisms (Lem. 1.11) we have that A∗ is always reduced; itis called the reduction of A. For any given class K of matrices, Alg K is definedto be the class of underlying algebras of members of K and K∗ is the class ofreduced members of K.

Let S be an arbitrary k-deductive system. A k-matrix A = 〈A, F 〉 is a model,more precisely a matrix model, of S if F ∈ FiS A. The class of models of S isdenoted by ModS, and the class of all reduced models (i.e., models that arereduced as matrices) by Mod∗ S. With each k-deductive system we associatethe class of algebras Alg Mod∗ S, that is the class of underlying algebras of re-duced models of S. We will see below in Cor. 1.26 that, if S is an algebraizable1-deductive system, then Alg Mod∗ S is its equivalent quasivariety.

The following easy lemma exactly characterizes the (Alg Mod∗ S)-congruenceson the formula algebra as the Leibniz congruences.

Lemma 1.12 Let S be a 1-deductive system.

CoAlg Mod∗ S(FmΛ) = ΩT : T ∈ ThS .

PROOF. The inclusion from right to left is obvious. LetΘ ∈ CoAlg Mod∗ S FmΛ,i.e., FmΘ/Θ ∈ Alg Mod∗ S. There exists a 〈A, F 〉 ∈ ModS such that ΩA F =IdA and A ∼= FmΛ/Θ. Let h: FmΛ A be a surjective homomorphism suchthat Θ is its relation kernel. Then Θ = h−1(IdA) = h−1(ΩA F ) = Ωh−1(F ),by Lem. 1.11, and h−1(F ) ∈ ThS. 2

Let S be an algebraizable 1-deductive system. In [22, Theorem 2.17] an al-gorithm is given for constructing a presentation of the equational logic of

16

Alg Mod∗ S by identities and quasi-identities from a given presentation of Sby axioms and inference rules. The following is an improvement of this result. 4

Theorem 1.13 Let S be a 1-deductive system presented by a set of axiomsAx and a set of proper inference rules Ru. Assume S is algebraizable withequivalent quasivariety Q. Let

E(x, y) =ε0(x, y), . . . , εn−1(x, y)

be a faithful interpretation of Seql Q in S, and let K(x) ≈ L(x) =

κ0(x) ≈

λ0(x), . . . , κm−1(x) ≈ λm−1(x)

be a faithful interpretation of S in Seql Q that

is the inverse of E(x, y). Then the equational logic Seql Q of Q is presentedby adjoining to

(i) the axioms and rules of free equational logic ((A1),(R1),(R2), and (R3λ)for each λ ∈ Λ),

the identities

(ii) K(ϕ) ≈ L(ϕ), for each ϕ ∈ Ax,

together with the following quasi-identities.

(iii)K(ψ0) ≈ L(ψ0), . . . , K(ψp−1) ≈ L(ψp−1)

K(ϕ) ≈ L(ϕ), for each sequent

ψ0, . . . , ψp−1

ϕin Ru;

(iv)K

(E(x, y)

)≈ L

(E(x, y)

)x ≈ y

PROOF. It is easy to see that each equation and quasi-equation of (i)–(iv) isan identity and quasi-identity of Q, so we only have to prove that, conversely,every identity and quasi-identity of Q is a consequence of (i)–(iv). We first show

that the equations K(E(x, x)

)≈ L

(E(x, x)

), i.e., κj

(εi(x, x)

)≈ λj

(εi(x, x)

),

for i < n, j < m, are all provable (i.e., derivable from the empty set) withrespect to (w.r.t.) (i)–(iv). εi(x, x) is a theorem of S for each i < n. Letη0, . . . , ηk = εi(x, x) be a proof w.r.t. Ax,Ru. Then the sequence of sets ofequations K(η0) ≈ L(η0), . . . , K(ηk) ≈ L(ηk) includes, in the obvious sense, a

proof of all the equations κj

(εi(x, x)

)≈ λj

(εi(x, x)

)w.r.t. (ii) and (iii).

Suppose now thatγ0 ≈ δ0, . . . , γk−1 ≈ δk−1

ϑ ≈ ψis a quasi-identity of Q. We showed

above that K(E(γi, γi)

)≈ L

(E(γi, γi)

)is provable w.r.t. (ii) and (iii). But

4 This improvement was pointed out to the authors by J. M. Font and R. Jansana.

17

K(E(γi, δi)

)≈ L

(E(γi, δi)

)is derivable from γi ≈ δi and K

(E(γi, γi)

)≈

L(E(γi, γi)

)w.r.t. (A1) and (R3λ) for λ ∈ Λ. So, for each i < k,K

(E(γi, δi)

)≈

L(E(γi, δi)

)is derivable from γi ≈ δi alone w.r.t. (i)–(iii). We next observe

that, from the assumption γ0 ≈ δ0, . . . , γk−1 ≈ δk−1 `Seql Q ϑ ≈ ψ, we have that

E(γ0, δ0), . . . , E(γk−1, δk−1) `S E(ϑ, ψ). So each equation of K(E(ϑ, ψ)

)≈

L(E(ϑ, ψ)

)is derivable from K

(E(γi, δi)

)≈ L

(E(γi, δi)

), for i < k, w.r.t.

(iii), and hence from the γi ≈ δi, for i < k, w.r.t. (i)—(iii). An application of(iv) gives the desired derivation of ϑ ≈ ψ from γ0 ≈ δ0, . . . , γk−1 ≈ δk−1. 2

1.4 Protoalgebraic and equivalential deductive systems

As previously noted, algebraizability can be characterized by conditions onthe Leibniz operator Ω on S-theories. By progressively relaxing these restric-tions we obtain a hierarchy of increasingly wider classes of deductive sys-tems that still retain much of the algebraic character of algebraizable systems[25,20,4,26,24,23,27]. We consider two such classes here.

Definition 1.14 A k-deductive system S is protoalgebraic if Ω : ThS →Co FmΛ is monotonic, i.e,

T ⊆ S implies ΩT ⊆ ΩS, for all S-theories T and S. 2

It is an easy consequence of the commutativity of the Leibniz operator withinverse surjective homomorphisms that, if S is protoalgebraic, then ΩA ismonotonic on S-filters of A for every countably generated Λ-algebra A, andit is not difficult to see that this is in fact the case for all Λ-algebras.

The following lemma is a reformulation of a minor generalization of the corre-spondence theorem for the filters of a protoalgebraic 1-deductive system ([25,Theorem 2.4]).

Lemma 1.15 Let S be a protoalgebraic k-deductive system. Let A,B be Λ-algebras and let h: A → B be a surjective homomorphism. Then for everyF ∈ FiS B,

h−1([F )FiS B

)= [h−1(F ) )FiS A.

PROOF. Obviously h−1([F )FiS B

)⊆ [h−1(F ) )FiS A, whether or not S is

protoalgebraic. For the reverse inclusion, suppose h−1(F ) ⊆ G ∈ FiS A. Notethat F ⊆ h(G) since h is surjective. Let Θ be the relation kernel of h, i.e., Θ =h−1(IdA). Then obviously Θ is compatible with h−1(F ). So Θ ⊆ ΩA h

−1(F )by definition of ΩA, and ΩA h

−1(F ) ⊆ ΩAG by protoalgebraicity. So Θ is

18

compatible with G, which implies h(G) ∈ FiS B and h−1h(G) = G. So G ∈h−1

([F )FiS B

). 2

If S is an arbitrary k-deductive system, then Alg Mod∗ S is closed under iso-morphism, but not in general under subalgebras, products, or subdirect prod-ucts.

Theorem 1.16 ([20, Theorem 9.3]) Let S be a k-deductive system. S isprotoalgebraic iff Mod∗ S is closed under subdirect products. Thus, if S is pro-toalgebraic, then Alg Mod∗ S is closed under subdirect products. 2

Let ∆(x, y) = δi(x, y) : i ∈ I be a (possibly infinite) set of 1-formulas intwo variables. ∆(x, y) is called a protoequivalence system 5 for a 1-deductivesystem S if the following are theorems and a rule of S.

∆(x, x) (14)

x, ∆(x, y)

y. (MP∆)

The latter sequent is called ∆-modus ponens or ∆-detachment, and ∆ is saidto have the modus ponens or detachment property (with respect to S) if (MP∆)is a rule of S.

Every protoequivalence system for S includes a finite subset that is also a pro-toequivalence system for S. This easily follows from the assumption that S isfinitary. Consequently, in the sequel we normally assume that protoequivalencesystems are finite.

The following syntactical characterization is due to [20], but see [4, Theorem1.1.3] for a simpler proof.

Theorem 1.17 A 1-deductive system is protoalgebraic iff it has a protoequiv-alence system. 2

For each language type Λ and each nonzero k there is a unique protoalgebraick-deductive system with no theorems; it is presented by the single inference

rulex

y(and no axioms). The system has exactly two theories, ∅ and Fmk

Λ.

The protoequivalence system for this deductive system is empty. The protoe-quivalence systems for every other protoalgebraic k-deductive system are allnonempty. We will be interested only in such systems in the sequel. Thus inthe sequel we assume that all protoalgebraic deductive systems have at leastone theorem.

5 Elsewhere in the literature a system of this kind is referred to either as an equiv-alence system ([20]) or as an implication system.

19

In the following, u denotes a generally infinite sequence u0, u1, u2, . . ., withoutrepetitions, of variables different from x and y. Let

E(x, y, u) = εi(x, y, u) : i ∈ I

be a system of 1-formulas over Λ in two variables, x and y, and an arbitrarynumber of variables from the list u; the latter variables are called parameters.Of course, each individual formula εi(x, y, u) actually contains only a finitenumber of parameters, but the set of parameters that occurs in at least oneof the members of E(x, y, u) maybe infinite and generally is. For all ϕ, ψ ∈FmΛ, let ∀ϑ E(ϕ, ψ, ϑ) stand for the set of substitution instances σ

(εi(x, y, u)

),

where i ranges over all of I and σ ranges over all substitutions such thatσ(x) = ϕ and σ(y) = ψ; i.e.,

∀ϑ E(ϕ, ψ, ϑ) :=εi(ϕ, ψ, ϑ) : i ∈ I, ϑ ∈ Fm

|u|Λ

.

E(x, y, u) is called an equivalence system with parameters of a 1-deductivesystem S if it is nonempty and each εi(x, x, u) is theorem of S, and hence,because of the substitution-invariance of S, each member of

∀ϑ E(x, x, ϑ) (15)

is a theorem of S. In addition, all of the following are rules of S.

∀ϑ E(x, y, ϑ)

∀ϑ E(y, x, ϑ), i.e.,

∀ϑ E(x, y, ϑ)

εi(y, x, ξ)for each i ∈ I and ξ ∈ Fm

|u|Λ , (16)

∀ϑ E(x, y, ϑ), ∀ϑ E(y, z, ϑ)

∀ϑ E(x, z, ϑ), (17)

∀ϑ E(x0, y0, ϑ), . . . ,∀ϑ E(xn−1, yn−1, ϑ)

∀ϑ E(λx0, . . . , xn−1, λy0, . . . , yn−1, ϑ)for all λ ∈ Λ (n is the rank of λ),

(18)

x, ∀ϑ E(x, y, ϑ)

y. (19)

The infinite system of rules (19) is called E(x, y, u)-modus ponens orE(x, y, u)-detachment.

It is easy to see that if E(x, y, u) is an equivalence system with parameters forS, then E(x, y, x, x, x, . . .), the set of formulas obtained by replacing every pa-rameter by x, is a protoequivalence system for S and hence S is protoalgebraic.The next theorem shows that the converse holds, every protoalgebraic systemwith at least one theorem has an equivalence system with parameters. Thisproperty gives an alternative characterization of protoalgebraic 1-deductivesystems.

20

Theorem 1.18 ([20, Theorem 3.10]) Assume that S is a protoalgebraic 1-deductive system. Let ∆(x, y) = δ0(x, y), . . . , δn−1(x, y) be a protoequivalencesystem for S. Let u be a sequence without repetitions of all variables in Va \x, y. Then

E(x, y, u) :=δi

(ξ(x, u), ξ(y, u)

): i < n, ξ(x, u) ∈ FmΛ(Va \ y

∪

δi

(ξ(y, u), ξ(x, u)

): i < n, ξ(x, u) ∈ FmΛ(Va \ y

is an equivalence system with parameters for S. 2

Let E(x, y, u) = εi(x, y, u) : i ∈ I be a system of binary formulas withparameters. For any algebra A and all a, b ∈ A we denote by ∀d EA(a, b, d)

the set of all elements of A of the form h(εi(x, y, u)

)where i ranges over I and

h over all homomorphisms h: FmΛ → A such that h(x) = a and h(y) = b,i.e.,

EA(a, b, u) = εA(a, b, d) : d ∈ A|u| .

A nonempty system E(x, y, u) = εi(x, y, u) : i ∈ I of binary formulas withparameters is said to define Leibniz congruences in a 1-deductive system S if,for every algebra A and F ⊆ A,

ΩA F =〈a, b〉 ∈ A2 : ∀d EA(a, b, d) ⊆ F

.

Theorem 1.19 E(x, y, u) is an equivalence system with parameters for a 1-deductive system S iff E(x, y, u) defines Leibniz congruences in S.

PROOF. Suppose E(x, y, u) is an equivalence system with parameters for a1-deductive system S. Fix an arbitrary algebra A and F ∈ FiS A and define

Φ :=〈a, b〉 ∈ A2 : ∀d EA(a, b, d) ⊆ F

.

The theorems and rules (15)–(18) of S guarantee that Φ is a congruence rela-tion, and the E(x, y, u)-detachment rule, (19), guarantees that Φ is compatiblewith F . So Φ ⊆ ΩA F . To show the inclusion in the opposite direction assumea ≡ b (mod ΩA F ). Then, for each i ∈ I and d ∈ A|u|, εA

i (a, a, d) ≡ εAi (a, b, d)

(mod ΩA F ). But εAi (a, a, d) ∈ F by (15). Thus εA

i (a, b, d) ∈ F since ΩA Fis compatible with F . This shows Φ = ΩA F . Hence E(x, y, u) defines Leibnizcongruences in S.

Suppose now that E(x, y, u) defines Leibniz congruences in S. Since ϕ ≡ ϕ(mod ΩT ) for every T ∈ ThS and ϕ ∈ FmΛ, we have that ∀ϑ E(ϕ, ϕ, ϑ) ⊆ Tfor every T ∈ ThS, i.e., (15) is a set of theorems of S. Suppose ∀ϑ E(ϕ, ψ, ϑ) ⊆T . Then ϕ ≡ ψ (mod ΩT ). Hence ψ ≡ ϕ (mod ΩT ), and consequently∀ϑ E(ψ, ϕ, ϑ) ⊆ T by assumption. Since this holds for every T ∈ ThS and

21

all ϕ, ψ ∈ FmΛ, (16) is an infinite rule of S. In a similar way we get that(17) and (18) are infinite rules of S. Finally, assume ϕ ∪ ∀ϑ E(ϕ, ψ, ϑ) ⊆ T .Then ϕ ≡ ψ (mod ΩT ) and hence ψ ∈ T since ΩT is compatible with T .So E(x, y, u)-detachment is a rule of S, and hence E(x, y, u) is an equivalencesystem with parameters for S. 2

One useful consequence of this theorem is that, under special circumstances, anequivalence system with parameters gives a faithful interpretation of the equa-tional logic of Seql Alg Mod∗ S in S in a restricted sense; compare Def. 1.6(ii).In order to prove this we need the following technical lemma.

The class of all countable models of a deductive system S is denoted Modω1 S,and, for any class K of Λ-algebras, the class of all countable subalgebras ofmembers of K is denoted by Sω1 K.

Lemma 1.20 Assume S is a protoalgebraic 1-deductive system over a count-able language type Λ. Then Sω1 Alg Mod∗ S = SAlg Mod∗ω1

S.

PROOF. The inclusion from right to left is obvious. Let A ∈ Sω1 Alg Mod∗ S.Let 〈B, F 〉 ∈ Mod∗ S such that A ⊆ B and ΩB F = IdB. If ΩA(F ∩A) = IdA,then 〈A, F∩A〉 ∈ Mod∗ S, and hence A ∈ Alg Mod∗ω1

S, and we are done. So weassume ΩA(F ∩A) ) IdA. Let E(x, y, u) be an equivalence system for S withparameters; E(x, y, u) exists by Thm. 1.18 since S is protoalgebraic. We defineby recursion an increasing sequence A = A0 ⊆ A1 ⊆ A2 ⊆ · · · ⊆ An ⊆ · · ·B,n < ω, of countable subalgebras of B as follows. A0 = A, as indicated.Suppose An is defined. For each pair a, a′ of distinct elements of An, choosea term ε(x, y, u0, . . . , un−1) ∈ E(x, y, u) and elements b0, . . . , bm−1 ∈ B suchthat εB(a, b, b0, . . . , bm−1) /∈ F ; such a choice is possible by Thm. 1.19 since byassumption ΩB F = IdB. Let A′n be the extension of An obtained by adjoiningthe b0, . . . , bm−1 obtained in this way for every 〈a, a′〉 ∈ A2

n\IdAn , and let An+1

be the subalgebra of B generated by A′n. Let C =⋃

n<ω An. Then A ⊆ C ⊆ Band C is countable.

We now show that ΩC(F ∩ C) = IdC . Suppose c, c′ ∈ C with c 6= c′. Con-sider any n ≤ ω such that c, c′ ∈ An. By definition of An+1 there is anε(x, y, u0, . . . , um−1) ∈ E(x, y, u) and b0, . . . , bm−1 ∈ An−1 ⊆ C such thatεC(a, b, b0, . . . , bm−1) = εB(a, b, b0, . . . , bm−1) /∈ F∩C. So c 6≡ c′ (mod ΩC(F∩C)) by Thm. 1.19, and hence ΩC(F ∩C) = IdC . Thus 〈C, F ∩C〉 ∈ Mod∗ω1

S,and A ⊆ C ∈ Alg Mod∗ω1

S. hence A ∈ SAlg Mod∗ω1S. 2

We remark that this lemma also holds without the assumption that S isprotoalgebraic, but we apply it only to protoalgebraic systems here.

22

Theorem 1.21 Let S be a protoalgebraic 1-deductive system over a countablelanguage type, and let E(x, y, u) be a system of formulas in two variables andparameters. E(x, y, u) is an equivalence system with parameters for S iff Shas E(x, y, u)-detachment, and, for all Γ ≈ ∆ ∪ ϕ ≈ ψ ⊆ω Fm2

Λ,

Γ ≈ ∆ `Seql Alg Mod∗ S ϕ ≈ ψ iff ∀ϑ E(Γ,∆, ϑ) `S ∀ϑ E(ϕ, ψ, ϑ), (20)

where ∀ϑ E(Γ,∆, ϑ) =⋃

∀ϑ E(γ, δ, ϑ) : γ ≈ δ ∈ Γ ≈ ∆.

PROOF. Assume first of all the E(x, y, u) is an equivalence system withparameters for S. Then by definition S has E(x, y, u)-detachment. We verifythe equivalence (20).

⇐=: We prove the contrapositive. Suppose Γ ≈ ∆ 6`Seql Alg Mod∗ S ϕ ≈ ψ. Let Xbe the finite set of variables that appear in Γ ≈ ∆∪ϕ ≈ ψ. Then there is aΘ ∈ Co FmΛ(X) such that FmΛ(X)/Θ ∈ Sω1 Alg Mod∗ S such that Γ ≈ ∆ ⊆Θ but ϑ ≈ ψ /∈ Θ. By Lem. 1.20, Sω1 Alg Mod∗ S = SAlg Mod∗ω1

S. So thereis a Θ′ ∈ Co FmΛ such that FmΛ/Θ

′ ∈ Alg Mod∗ω1S and Θ′ ∩ Fm2(X) = Θ.

Since FmΛ/Θ′ ∈ Alg Mod∗ω1

S, by Lem. 1.12 Θ′ = ΩT for some S-theory T . SoΘ′ = 〈ξ, η〉 ∈ Fm2

Λ : ∀ϑ (ξ, η, ϑ) ⊆ T by Thm. 1.19. Thus ∀ϑ E(Γ,∆, ϑ) ⊆T , but ∀ϑ E(ϕ, ψ, ϑ) * T , i.e., ∀ϑ E(Γ,∆, ϑ) 6`S E(ϕ, ψ, ϑ).

=⇒: Again we prove the contrapositive. Suppose ∀ϑ (Γ,∆, ϑ) 6`S ∀ϑ E(ϕ, ψ, ϑ).Let T be the S-theory generated by ∀ϑ E(Γ,∆, ϑ), and let Θ = ΩT . ByLem. 1.12, FmΛ/Θ ∈ Alg Mod∗ S, and by Thm. 1.19 Γ ≈ ∆ ⊆ Θ butϕ ≈ ψ /∈ Θ. So Γ ≈ ∆ 6`Seql Alg Mod∗ S ϕ ≈ ψ.

Assume now that S has E(x, y, u)-detachment and the equivalence (20) holds.The remaining defining conditions for an equivalence system with parameters,(15)–(18), are easily verified. Consider, for example, the transitivity rule (17).For all ϕ, ξ, ψ ∈ FmΛ,

∀ϑ E(ϕ, ξ, ϑ) ∪ ∀ϑE(ξ, ψ, ϑ) `S ∀ϑE(ϕ, ψ, ϑ)

iff ϕ ≈ ξ, ξ ≈ ψ `Seql Alg Mod∗ S ϕ ≈ ψ.

The right-hand entailment obviously holds. Thus the left-hand entailment, thetransitivity rule, holds. The other defining rules of an equivalence system withparameters are verified similarly. 2

We note some features of this proof for future reference. We only used thepremiss that the language type is countable to apply Lem. 1.12. So the con-clusion of the theorem holds, without any condition on the cardinality of thelanguage type, whenever Sω1 Alg Mod∗ S = SAlg Mod∗ω1

S, in particular when-ever Alg Mod∗ S is closed under subalgebras. More generally, it holds without

23

qualification if the consequence relation `Seql Alg Mod∗ S in the equivalence (20)is replaced by `Seql Alg Mod∗ω1

S .

Recall that ThmS is the smallest theory of S, the set of theorems of S.

Corollary 1.22 Let S be a protoalgebraic 1-deductive system over a countablelanguage type Λ. Then the set of all identities of Alg Mod∗ S coincides withΩ ThmS.

PROOF. By the theorem, ϕ ≈ ψ is an identity of Alg Mod∗ S iff the entail-ment `S ∀ϑ E(ϕ, ψ, ϑ) holds, i.e., iff ∀ϑ E(ϕ, ψ, ϑ) ⊆ Thm C. The conclusionof the corollary now follows immediately from Thm. 1.19. 2

The quotient algebra FmΛ/Ω(ThmS) is called the Lindenbaum-Tarski alge-bra of S.

An equivalence system E(x, y) = εi(x, y) : i ∈ I for S with an empty se-quence of parameters is called simply an equivalence system. 6 Thus E(x, y)is an equivalence system for a 1-deductive system S if it is a nonempty pro-toequivalence system (i.e., (14) and (MP∆), with E(x, y) in place of ∆(x, y),are respectively theorems and a rule of S), and in addition the following arerules of S.

E(x, y)

E(y, x), (21)

E(x, y), E(y, z)

E(x, z), (22)

E(x0, y0), . . . , E(xn−1, yn−1)

E(λx0, . . . , xn−1, λy0, . . . , yn−1)for all λ ∈ Λ (n is the rank of λ). (23)

In contrast to protoequivalence systems, an equivalence system may not in-clude a finite subset that is also an equivalence system. We remark that (21)and (22) are provable from the remaining inference rules defining an equiva-lence system, although this is not completely obvious.

A 1-deductive system is said to be (finitely) equivalential if has a (finite)equivalence system ([28,26]). Obviously, every equivalential system is protoal-gebraic. The significance of equivalential systems should be apparent after thenext two theorems and their corollaries.

6 also called a weak congruence system ([20]).

24

Theorem 1.23 ([20, Theorem 13.12]) Let S be a 1-deductive system.

(i) S is equivalential iff Mod∗ S is closed under the formation of submatricesand direct products.

(ii) S is finitely equivalential iff Mod∗ S is closed under the formation ofsubmatrices and reduced products. 2

Corollary 1.24 ([26, Corollary I.13]) Alg Mod∗ S is a quasivariety for ev-ery finitely equivalential 1-deductive system S. 2

Theorem 1.25 Let S be a 1-deductive system. Then a nonempty set E(x, y)of binary formulas is an equivalence system for S iff S has E(x, y)-detachmentand E(x, y) is a faithful interpretation of Seql Alg Mod∗ S in S, i.e., for allΓ ≈ ∆ ∪ ϕ ≈ ψ ⊆ Fm2

Λ,

Γ ≈ ∆ `Seql Alg Mod∗ S ϕ ≈ ψ iff E(Γ,∆) `S E(ϕ, ψ). (24)

PROOF. By Thm. 1.21 the equivalence (24) holds under the hypothesis thatthe language type is countable. But since Alg Mod∗ S is closed under subalge-bras, Sω1 Alg Mod∗ S = SAlg Mod∗ω1

S without any restriction on the cardinal-ity of the language type. See the remarks following the proof of Thm. 1.21 2

Corollary 1.26 If a 1-deductive system S is algebraizable, then its equivalentquasivariety is unique and must be Alg Mod∗ S.

PROOF. Assume S is algebraizable with equivalent quasivariety Q and letE(x, y) be a finite, faithful interpretation of Seql Q in S. It is easy to ver-ify that that (14), (())MP∆, (21)–(23) are all theorems or rules of S andhence that E(x, y) is a finite equivalence system for S. Thus by the theo-rem E(x, y) is also a faithful interpretation of Seql Alg Mod∗ S in S. ThusSeql Q = Seql Alg Mod∗ S, and hence Q = Mod∗ S since they are both quasi-varieties. 2

1.5 Regularly algebraizable deductive systems

The deductive systems of traditional algebraic logic are all algebraizable in thespecial sense that “truth” is represented by a single element in the Lindenbaum-Tarski algebra, in opposition to the case for an arbitrary algebraizable deduc-tive system where it may have a multitude of representations.

25

By the E-G-rule relative to a given equivalence system E(x, y) we mean thesequent

x, y

E(x, y). (25)

It follows immediately from Thm. 1.19 that, if S is equivalential, then

x ≡ y (mod Ω(CloSx, y)) iff E(x, y) ⊆ CloSx, y iff x, y `S E(x, y).

So the E-G-rule is equivalent to the so-called meta G-rule, that is, x ≡ y(mod Ω(CloSx, y)). Thus, if the E-G-rule is a rule of S for some equivalencesystem E(x, y), then the E ′-G-rule is a rule of S for every equivalence systemE ′(x, y) of S. We refer to all these rules collectively simply as the “G-rule”.

Assume the G-rule is a rule of S. Then ϕ, ψ `S E(ϕ, ψ). Hence, for ev-ery S-theory T , ϕ ≡ ψ (mod ΩT ) for all ϕ, ψ ∈ T . Conversely, if ϕ ≡ ψ(mod ΩT ) for every theory T and all ϕ, ψ ∈ T , then in particular x ≡ y(mod Ω(CloSx, y)). So the G-rule holds. This shows that the G-rule is arule of an equivalential deductive system iff T is an equivalence class of ΩTfor every theory T .

The following lemma gives a closely related characterization of the G-rule thatproves useful in the sequel.

Lemma 1.27 Let S be any equivalential 1-deductive system with an equiva-lence system E(x, y). The G-rule is a rule of S iff x a`S E(x,>), where > isany theorem of S.

PROOF. Assume the G-rule is rule of S and let > be any theorem. We haveE(x,>) `S x by E-detachment. Conversely, x,> `S E(x,>) by the G-rule.Hence x `S E(x,>) since > is a theorem.

Assume now that x a`S E(x,>). Then x, y `S E(x,>), E(y,>). But by thesymmetry and transitivity of equivalence ((21) and (22)), E(x,>), E(y,>) `SE(x, y). So the G-rule is a rule of S. 2

Theorem 1.28 ([22], Corollary 4.8) Every finitely equivalential deductivesystem S with the G-rule is algebraizable. Furthermore, the singleton x ≈ >is a faithful interpretation of S in Seql Alg Mod∗ S, where > can be taken tobe any theorem of S.

PROOF. Let E(x, y) be a finite equivalence system for S. E(x, y) is a faith-ful interpretation of Seql Alg Mod∗ S in S by Thm. 1.25. Thus it sufficesto verify the invertibility condition (12), i.e., x a`S E

(K(x), L(x)

), with

26

K(x) ≈ L(x) = x ≈ > This takes the form x a`S E(x,>), which holds byLem. 1.27. 2

For earlier, closely related results that anticipated this theorem see [26, The-orem II.1.2 and Proposition II.1.5].

Definition 1.29 A finitely equivalential deductive system with the G-rule issaid to be regularly algebraizable. 2

Theorem 1.30 Let S be a 1-deductive system presented by a set Ax of axiomsand a set Ru of proper inference rules. Assume S is regularly algebraizable withfinite equivalence system E(x, y) =

ε0(x, y), . . . , εn−1(x, y)

. Let > be a fixed

but arbitrary theorem of S. Then the unique equivalent quasivariety of S isdefined by the identities

(i) ϕ ≈ >, for each ϕ ∈ Ax;

together with the following quasi-identities

(ii)ψ0 ≈ >, . . . , ψp−1 ≈ >

ϕ ≈ >, for each inference rule

ψ0, . . . , ψp−1

ϕin Ru;

(iv)E(x, y) ≈ >

x ≈ y.

PROOF. By Thm. 1.28, x ≈ > is a single defining equation for S. Thetheorem is now an immediate corollary of Thm. 1.13. 2

The equivalent quasivarieties of regularly algebraizable deductive systems turnout to be an important class of quasivarieties from the universal algebraic pointof view. Loosely speaking there are the quasivarieties, and varieties, with a reg-ular ideal structure (or filter structure in the present terminology). A quasivari-ety will be said to be pointed if it has a distinguished constant term, i.e., a termϕ(x0, . . . , xn−1) with the property that ϕ(x0, . . . , xn−1) ≈ ϕ(y0, . . . , yn−1) is anidentity, where the y0, . . . , yn−1 are new variables distinct from x0, . . . , xn−1.Every pointed quasivariety is termwise definitionally equivalent to a quasiva-riety over a pointed language type in which > is the designated constant term.In the sequel we assume every pointed quasivariety is over a pointed languagetype and that > is the distinguished constant term.

Definition 1.31 A pointed quasivariety Q is said to be relatively point-regularif each Q-congruence Θ on FmΛ is uniquely determined by its >-equivalenceclass >/Θ. 2

27

Let A be a Λ-algebra. If Q is relatively point-regular, then every Q-congruenceon A is completely determined by its >A-congruence class (this follows forcountably generated A directly from the one-one correspondence between Q-congruences on FmΛ/Θ and Q-congruences on FmΛ that include Θ).

If Q is a pointed quasivariety, then >/Θ : Θ ∈ CoQ FmΛ is an algebraicclosed-set system over FmΛ. Moreover, given any substitution σ : FmΛ →FmΛ, and any Θ ∈ CoQ FmΛ, σ−1(>/Θ) = >/ σ−1(Θ), as is easily verifiedsince σ(>) = >. So >/Θ : Θ ∈ CoQ FmΛ is substitution-invariant. Theseobservations justify the following:

Definition 1.32 Let Q be pointed quasivariety. The assertional logic of Q isthe 1-deductive system

Sasl Q =⟨FmΛ, >/Θ : Θ ∈ CoQ FmΛ

⟩. 2

Corollary 1.33 Let Q be a pointed quasivariety. Then, for all Γ∪ϕ ⊆ FmΛ,

Γ `Sasl Q ϕ iff Γ ≈ > `Seql Q ϕ ≈ >,

where Γ ≈ > = ψ ≈ > : ψ ∈ Γ .

PROOF. Assume Γ `Sasl Q ϕ. Then, for every Θ ∈ CoQ FmΛ, Γ ⊆ >/Θimplies ϕ ∈ >/Θ, i.e.,

ϕ ≡ > (mod CgQ(Γ ≈ >)). (26)

Thus Γ ≈ > `Seql Q ϕ ≈ >.

Assume conversely that Γ ≈ > `Seql Q ϕ ≈ >. Then (26) holds, and henceΓ `Sasl Q ϕ. 2

Thus x ≈ > is faithful interpretation of Sasl Q in Seql Q, and this conditionclearly characterizes Sasl Q.

Theorem 1.34 A 1-deductive system is regularly algebraizable iff it is theassertional logic of a relatively point-regular quasivariety. More precisely, a1-deductive system S is regularly algebraizable iff Alg Mod∗ S is a relativelypoint-regular quasivariety and S = Sasl Alg Mod∗ S.

PROOF. ⇒: Assume S is regularly algebraizable with finite equivalence sys-tem E(x, y). Let Q = Alg Mod∗ S. Then the G-rule guarantees that ϕ ≈ ψ isan identity of Q for any pair of theorems ϕ and ψ of S. Thus Q is a pointedquasivariety with > representing any theorem. Then by Thm. 1.28 x ≈ > is

28

a faithful interpretation of S in Seql Q. But by by Cor. 1.33 x ≈ > is also afaithful interpretation of Sasl Q in Seql Q. So S = Sasl Q. It remains only toprove that Q is relatively point-regular. Consider any Θ ∈ CoQ FmΛ. Clearly>/Φ ⊆ >/Θ for every Φ ∈ Co FmΛ that is compatible with >/Θ; in particu-lar, >/Ω(>/Θ) ⊆ >/Θ. On the other hand, Θ is compatible with >/Θ, andthus Θ ⊆ Ω /(>/Θ), and hence >/Θ ⊆ >/Ω(>/Θ). So >/Ω(>/Θ) = >/Θ.But Ω is a bijection between ThS = >/Θ : Θ ∈ CoQ FmΛ and CoQ FmΛ.So Ω(>/Θ) = Θ for every Θ ∈ CoQ FmΛ. Hence each Θ ∈ CoQ FmΛ isuniquely determined by its >-equivalence class.

⇐: Assume now that S = Sasl Q for a relatively point-regular quasivarietyQ. Let Θ = CgQ(x ≈ y), the Q-congruence on FmΛ generated by the2-formula 〈x, y〉 where x, y are distinct variables. Since Q is relatively point-regular, Θ = CgQ(>/Θ ≈ >). By the substitution-invariance of Seql Q andthe compactness of Θ there is a finite subset E(x, y) of >/Θ, in two-variables,such that Θ = CgQ(E(x, y) ≈ >). When reformulated in the form of sequentsthis equality says that

E(x, y) ≈ >x ≈ y

andx ≈ y

E(x, y) ≈ >(27)

are rules of Seql Q, i.e., quasi-identities of Q. It follows easily from this thatE(x, x) ≈ > (more precisely, each member of E(x, x) ≈ >) is a theorem ofSeql Q and

E(x, y) ≈ >E(y, x) ≈ >

,E(x, y) ≈ >, E(y, z) ≈ >

E(x, z) ≈ >E(x0, y0) ≈ >, . . . , E(xn−1, yn−1) ≈ >E(λx0, . . . , xn−1, λy0, . . . , yn−1) ≈ >

, λ ∈ Λ, (n the rank of λ

are all rules of Seql Q. It follows from Cor. 1.33 that each formula of E(x, x)is a theorem of Sasl Q and and (21)–(23) are rules of Sasl Q. Moreover, sincex ≈ >, E(x, y) ≈ >

y ≈ >is obviously a consequence of (27) (together with the

properties of equality), it is a rule of Seql Q; hence E-detachment is a rule

of Sasl Q. Finally, thatx ≈ >, y ≈ >E(x, y) ≈ >

is a rule of Seql Q is another obvious

consequence of (27). Thus the G-rule is a rule of Sasl Q. So Sasl Q is regularlyalgebraizable. 2

Corollary 1.35 Let Λ be an arbitrary pointed language type. There is a one-one correspondence between relatively point-regular quasivarieties and regularlyalgebraizable deductive systems. Every relatively point-regular quasivariety de-termines a unique regularly algebraizable 1-deductive system, its assertionallogic. Conversely, every regularly algebraizable deductive system S is the as-sertional logic of a unique relatively point-regular quasivariety, its equivalent

29

quasivariety Alg Mod∗ S. Thus, for each regularly algebraizable deductive sys-tem S we have S = Sasl Alg Mod∗ S and, conversely, for every relatively point-regular quasivariety Q we have Q = Alg Mod∗ Sasl Q. 2

That the assertional logic of a relatively point-regular quasivariety Q is al-gebraizable with Q as its equivalent quasivariety was observed in [29] andindependently in [30].

In the next theorem we get, as a consequence of the correspondence of Cor. 1.35,a Mal’cev-type characterization of relatively point-regular quasivarieties thatgeneralizes the well-known characterization of point-regular varieties obtainedin [31,32]; see also [33]. It was announced in [34]; proofs were given in [29] andindependently in [30].

Theorem 1.36 A pointed quasivariety Q is relatively point-regular iff there isa finite system E(x, y) = ε0(x, y), . . . , εn−1(x, y) of binary terms such that

E(x, x) ≈ > (28)

is a set of identities of Q, and

E(x, y) ≈ >x ≈ y

(29)

is a quasi-identity of Q.

PROOF. We verified in the proof of Thm. 1.34 that, if Q is relatively point-regular, then any finite set E(x, y) of generators of CnSeql Q(x ≈ y) in thetwo variables x and y satisfies these conditions. Alternatively, assume Q isrelatively point-regular, so that Sasl Q is regularly algebraizable. Let E(x, y)be a finite equivalence system for Sasl Q. Then (28) are identities and (29) isa quasi-identity of Q by Thm. 1.30. 7

Assume now that (28) is a set of identities and (29) a quasi-identity of Q. LetΘ be a Q-congruence of FmΛ. Consider any ϕ, ψ ∈ FmΛ. If ϕ ≡ ψ (mod Θ),then εi(ϕ, ψ) ≡ > (mod Θ) by the identities (28). Thus E(ϕ, ψ) ⊆ >/Θ.Conversely, if E(ϕ, ψ) ⊆ >/Θ, then, by (29), ϕ ≡ ψ (mod Θ). So, for everyQ-congruence Θ, Θ is completely determined by >/Θ, i.e., Q is relativelypoint-regular. 2

Another consequence of the correspondence is the following useful character-ization of a relatively point-regular quasivariety in terms of its assertional

7 There is an algebraic proof of this result in which the assertional logic entersonly subliminally. It closely parallels the standard proof of the Mal’cev conditionfor point-regular varieties.

30

logic.

Theorem 1.37 Let Q be a relatively point-regular quasivariety. For every Λ-algebra A we have A ∈ Q iff >A ∈ FiSasl Q A and ΩA>A = IdA.

PROOF. Suppose A ∈ Q. Q = Alg Mod∗ Sasl Q. So there is an F ∈ FiSasl Q Asuch that ΩA F = IdA. By the G-rule, F = >A. Conversely, if >A ∈FiSasl Q A and Ω>A = IdA, then 〈A, >A〉 ∈ Mod∗ Sasl Q. 2

1.6 Deduction-detachment theorem

Generalizations of the deduction theorem of the classical and intuitionisticpropositional calculi have played an important role in abstract algebraic logic;see [21,35–38].

The deduction theorem can be formulated for k-deductive systems; see [22,Definition 4.1]. We here we consider the notion only for the 1-deductive sys-tems.

Definition 1.38 Let S be a 1-deductive system. A nonempty set ∆(x, y) =δi(x, y) : i ∈ I of binary formulas is called a deduction-detachment systemfor S if the following equivalence holds for all Γ ∪ ϕ, ψ ⊆ FmΛ.

Γ, ϕ `S ψ iff Γ `S δi(ϕ, ψ), for all i ∈ I. 2

The implication from right to left in above equivalence is obviously equivalentto S having the ∆-detachment property, i.e., to (MP∆) being a rule of S. Theimplication in the opposite direction is called the ∆-deduction theorem or thededuction property of ∆.

Since S is finitary, there is a finite subset J of I such thatx, δi(x, y) : i ∈ J

yis a rule of S. Thus Γ, ϕ `S ψ iff Γ `S δi(ϕ, ψ) for all i ∈ J . So every deduction-detachment system for S includes a finite subsystem that is also a deduction-detachment system. In the sequel we assume all deduction-detachment systemsare finite.

A deductive system S is said to have the multiterm (or generalized) deduction-detachment theorem if it has a finite deduction-detachment system, and theuniterm deduction-detachment theorem if it has a deduction-detachment sys-tem with a single formula.

31

In the context of the Fregean deductive systems there is an important differ-ence between multiterm and uniterm deduction-detachment systems; we shallsee this in the next section. In the remaining part of this section however weconcentrate on the multiterm case.

Corollary 1.39 Every deductive system S with a deduction-detachment sys-tem is protoalgebraic. Moreover, every deduction-detachment system ∆(x, y)for S is also a protoequivalence system for S.

PROOF. Assume∆(x, y) is a deduction-detachment system for S. Since x `Sx holds trivially, we have `S ∆(x, x) by the deduction property of ∆(x, y).The other part of the definition of a protoequivalence system, namely thedetachment property, is also part of the definition of a deduction-detachmentsystem. 2

For any set X, X(ω) denotes the set of all finite sequences of elements of X.

Let ∆(x, y) = δ0(x, y), . . . , δn−1(x, y). Let ϕ = 〈ϕ0, . . . , ϕm−1〉 ∈ Fm(ω)Λ and

ψ ∈ FmΛ. Then ∆∗(ϕ, ψ) ⊆ FmΛ is defined by recursion on m as follows. Ifm = 1, then

∆∗(ϕ, ψ) = ∆(ϕ0, ψ) (= δi(ϕ0, ψ) : i < n ).

Assume m > 1. Let ϕ = 〈ϕ0〉 a ϕ′, where ϕ′ = 〈ϕ1, . . . , ϕm−1〉, and define

∆∗(ϕ, ψ) =δi(ϕ0, ξ) : i = 0, . . . , n− 1 and ξ ∈ ∆∗(ϕ′, ψ)

=

⋃∆(ϕ0, ξ) : ξ ∈ ∆∗(ϕ′, ψ)

.

The results presented in the following two lemmas can be found in [21], andin a more general context in [35].

Lemma 1.40 Let ∆(x, y) = δi(x, y) : i < n be a finite system of binary

1-formulas. For all ϕ0, . . . , ϕm−1, ψ ∈ FmΛ, the sequentϕ0, . . . , ϕm−1

ψis a

consequence of the set of formulas ∆∗(〈ϕ0, . . . , ϕm−1〉, ψ

)∪ ϕ0, . . . , ϕm−1

using only (MP∆).

PROOF. By induction on m. Let ϕ = 〈ϕ0, . . . , ϕm−1〉.

m = 1: ∆∗(ϕ, ψ) = ∆(ϕ0, ψ). ψ is directly derivable from ϕ0 and ∆(ϕ0, ψ) by(MP∆).

32

m > 1: ∆∗(ϕ, ψ) =⋃∆(ϕ0, ξ) : ξ ∈ ∆∗(ϕ′, ψ)

, where ϕ′ = 〈ϕ1, . . . , ϕm−1〉.

From ϕ0 and ∆∗(ϕ, ψ) we can derive ξ for each ξ ∈ ∆∗(ϕ′, ψ) by (MP∆). Thenby the induction hypothesis we can derive ψ from ϕ1, . . . , ϕm−1 and ∆∗(ϕ′, ψ)by (MP∆). 2

Lemma 1.41 Let S be a deductive system with a deduction-detachment sys-

tem ∆(x, y) = δi(x, y) : i < n . Then for each ruleΓ, ϕ0, . . . , ϕm−1

ψof S we

have

Γ `S ∆∗(〈ϕ0, . . . , ϕm−1〉, ψ

).

PROOF. By induction on m. Let ϕ = 〈ϕ0, . . . , ϕm−1〉.

If m = 1, then ∆∗(ϕ, ψ) = ∆(ϕ0, ψ) and the entailment Γ `S ∆∗(ϕ, ψ) followsfrom Γ and the entailment ϕ0 `S ψ by the deduction-detachment theorem.

Assume m > 1. Let ϕ′ = 〈ϕ1, . . . , ϕm−1〉 so that ϕ = 〈ϕ0〉 a ϕ′.

∆∗(ϕ, ψ) =⋃

∆(ϕ0, ξ) : ξ ∈ ∆∗(ϕ′, ψ).

By the induction hypothesis, it follows from Γ, ϕ0, . . . , ϕm−1 `S ψ that Γ, ϕ0 `Sξ for each ξ ∈ ∆∗(ϕ′, ψ). Thus Γ `S ∆(ϕ0, ξ), for each ξ ∈ ∆∗(ϕ′, ψ), by the

deduction-detachment theorem, i.e., Γ `S ∆∗(〈ϕ0, . . . , ϕm−1〉, ψ

). 2

Theorem 1.42 Let S be a deductive system with a deduction-detachment sys-tem ∆(x, y) = δ0(x, y), . . . , δn−1(x, y). Then S has a presentation by axiomsand inference rules in which the only proper inference rule is (MP∆).

PROOF. Consider a fixed by arbitrary presentation Ax, Ru of S, where Axis a set of axioms and Ru is a set of proper inference rules. We construct anew presentation Ax′, Ru′ with the desired properties. Let Ax′ be the union

of Ax and ∆∗(〈ϕ0, . . . , ϕm−1〉, ψ) for every inference rule

ϕ0, . . . , ϕm−1

ψin Ru.

By Lem. 1.41, each member of Ax′ is a theorem of S. Let Ru′ consist only ofthe inference rule (MP∆). Let S ′ be the deductive system axiomatized by Ax′,Ru′. Then, since Ax′ is a set of theorems of S and (MP∆) is a rule of S, wehave `S′ ⊆ `S . On the other hand, Ax ⊆ Ax′ and each inference rule of Ru isderivable in S ′ by Lem. 1.40. So `S ⊆ `S′ . Hence S ′ = S. 2

Theorem 1.43 Let S be a 1-deductive system that is regularly algebraizableand has the multiterm deduction-detachment theorem. Let

E(x, y) =ε0(x, y), . . . , εn−1(x, y)

and ∆(x, y) =

δ0(x, y), . . . , δm−1(x, y)

33

be respectively a finite equivalence system and a finite deduction-detachmentsystem for S. Then the unique equivalent quasivariety of S is defined by theidentities

E(x, x) ≈ >,the two quasi-identities

x ≈ >, ∆(x, y) ≈ >y ≈ >

andE(x, y) ≈ >

x ≈ y,

and additional identities of the form ϕ ≈ >, where ϕ ranges over any fixedset of the axioms of a presentation of S in which modus ponens is the onlyinference rule.

PROOF. The theorem follows immediately from Thms. 1.30 and 1.42. 2

2 Fregean Deductive Systems

In this section all deductive systems are 1-deductive except where specificallyindicated to the contrary.

In Section 1 we considered only properties of deductive systems that can be

defined by sequents of the formψ0, . . . , ψn−1

ϕ. These can be thought of as

“1st-order” sequents. We consider several important properties of deductivesystems, the Fregean property being one of them, that are defined by “2nd-order” sequents. These are sequents of the form

ψ00, . . . , ψ

0n0−1

ϕ0, . . . ,

ψk−10 , . . . , ψk−1

nk−1−1

ϕk−1

ϑ0, . . . , ϑm−1

ξ

. (30)

The natural models of 2nd-order sequents are 2nd-order matrices, which wenow define.

Definition 2.1 Let Λ be an arbitrary language type. A 2nd-order (or general-ized) matrix (over Λ) is a pair A = 〈A, CA〉, where A is a Λ-algebra and CA isan (algebraic) closed-set system over A. CA is called the designated closed-setsystem of A and A the underlying algebra of A. 2nd-order matrices are calledabstract logics in [8,5] and generalized matrices in [3]. 2

Let A,B be 2nd-order matrices over Λ. B is a submatrix of A, in symbolsB ≤ A, if B is a subalgebra of A and CB = B ∩ F : F ∈ CA . Note that,

34

for every X ⊆ B,CloCB

(X) = CloCA(X) ∩B.

B is a finitely generated submatrix of A if B is a finitely generated subalgebraof A. A homomorphism h: A → B is called an matrix homomorphism betweenA and B if h−1(CB) ⊆ CA; equivalently, if

h(CloCA

(X))⊆ CloCB

(h(X)) for all X ⊆ A.

h is a strict matrix homomorphism, in symbols h: A →s B, if h−1(CB) = CA.If h is also surjective, then

h(CloCA

(X))

= CloCB(h(X)) for all X ⊆ A. (31)

In this case we say that B is a strict homomorphic image of A, in symbolsB 4 A. A strict bijective matrix homomorphism is a matrix isomorphism; inthis case we write A ∼= B.

A matrix A = 〈A, F 〉 is sometimes called a 1st-order matrix to contrast it with2nd-order matrices. For a similar reason, the designated closed-set system CA

of a 2nd-order matrix A is occasionally referred to as the designated 2nd-orderfilter of A, and the closed sets of CA 1st-order filters of A. The subscripts A

on FA and CA are often omitted if the matrix A is clear from context. It isconvenient on occasion to identify the 1st-order matrix 〈A, F 〉 with the special2nd-order matrix 〈A, F,A〉.

Definition 2.2 The 2nd-order sequent (30) is valid in a 2nd-order matrixA = 〈A, C〉, and the matrix is a model of the sequent, if for all assignmentsh: FmΛ → A of elements of A to variables,

h(ϕi) ∈ CloCh(ψij) : j < ni for each i < k

implies h(ξ) ∈ CloCh(ϑj) : j < m . 2

The class of all models of a fixed but arbitrary set of 2nd-order sequents iscalled a 2nd-order class or 2nd-order property of 2nd-order matrices.

Theorem 2.3 Let A and B be 2nd-order matrices.

(i) If B ≤ A, then every 2nd-order sequent that is valid in A is also valid inB.

(ii) If B 4 A, then a 2nd-order sequent is valid in B iff it is valid in A.

PROOF. Let A = 〈A, C〉 and B = 〈B,D〉.

(i). Assume B ≤ A and D = F ∩ B : F ∈ C , and that the sequent (30)is valid in A. Let h: FmΛ → B be any evaluation in B such that h(ϕi) ∈

35

CloDh(ψij) : j < ni for all i < k. Then h(ϕi) ∈ CloCh(ψi

j) : j < ni for alli < k, and hence h(ξ) ∈ CloCh(ϑj) : j < m since (30) is valid in A. Thus,since CloDh(ϑj) : j < m = CloCh(ϑj) : j < m ∩ B and h(ξ) ∈ B, wehave h(ξ) ∈ CloDh(ϑj) : j < m . So (30) is valid in B.

(ii). Let h: A B be a surjective algebra homomorphism such that h−1(D) =C. We prove that the 2nd-order sequent (30) is valid in 〈B,D〉 iff it is validin 〈A, C〉. We note first of all that, for every Γ ⊆ FmΛ and every evaluationg: FmΛ → A,

h−1(CloD(h g)(Γ )

)= CloC g(Γ ). (32)

By (31) CloD(h g)(Γ ) = h(CloC g(Γ )

). Thus

h−1(CloD(h g)(Γ )

)= h−1h

(CloC g(Γ )

)= CloC g(Γ ).

The last equality holds since h is surjective.

Suppose that the sequent (30) is valid in 〈B,D〉 and g: FmΛ → A is anevaluation in A such that g(ϕi) ∈ CloCg(ψi

j) : j < ni for all i < k. Then by

(32), g(ϕi) ∈ h−1(CloD (h g)(ψi

j) : j < ni for every i < k. Thus, since the

sequent (30) is valid in 〈B,D〉, (h g)(ξ) ∈ CloD (h g)(ϑj) : j < m , andhence, by (32) again,

g(ξ) ∈ h−1(CloD (h g)(ϑj) : j < m

)= CloC g(ϑj) : j < m

).

Thus (30) is also valid in 〈A, C〉. The proof of the reverse implication is similarand is omitted. 2

The following rather technical lemma of the same kind will also be useful inthe sequel.

Lemma 2.4 A 2nd-order sequent is valid in a 2nd-order matrix A if, forevery finitely generated B ≤ A, there exists a countably generated B′ suchthat B ≤ B′ ≤ A and the sequent is valid in B′.

PROOF. Let A = 〈A, C〉 be a 2nd-order matrix, and assume that, for everyfinitely generated B ≤ A, there exists a countably generated B′ such thatB ≤ B′ ≤ A and the sequent (30) is valid in B′. Let h: FmΛ → A be anevaluation in A such that h(ϕi) ∈ CloCh(ψi

j) : j < ni for all i < k. Let Bbe the subalgebra of A generated by the h-image of the set of all variablesthat occur in the sequent (30). Let B = 〈B,D〉 where D = B ∩F : F ∈ C .B is a finitely generated submatrix of A, and hence by hypothesis there existsa countably generated B′ = 〈B′,D′〉 such that B ≤ B′ ≤ A and (30) is validin B′. Then h(ϕi) ∈ CloD′h(ψi

j) : j < ni for all i < k. Thus by assumption

36

h(ξ) ∈ CloD′h(ϑj) : j < m ⊆ CloCh(ϑj) : j < m . So the sequent (30) isalso valid in A. 2

A 2nd-order matrix A = 〈A, CA〉 is endomorphism-invariant if h−1(CA) ⊆ CA

for every endomorphism h: A → A. Recall that by definition a 1-deductivesystem is an endomorphism-invariant 2nd-order matrix S = 〈FmΛ, ThS〉over the formula algebra (Def. 1.1). Thus we can speak of a 2nd-order sequentbeing valid in a deductive system, and the deductive system being a model ofthe sequent. We also say in this event that the sequent is a 2nd-order rule ofS. In more detail, the sequent (30) is a 2nd-order rule of a 1-deductive systemS if, for all substitutions σ,

σ(ψi0), . . . , σ(ψi

ni−1) `S σ(ϕi) for all i < k implies σ(ϑ0), . . . , σ(ϑm−1) `S σ(ξ).

A class consisting of all deductive systems that are models of some fixedbut arbitrary set of 2nd-order sequents is called a 2nd-order class or 2nd-order property of deductive systems. 2nd-order sequents can be thought of asGentzen-style rules, and a 2nd-order class of deductive systems as the class ofall deductive systems with a fixed but arbitrary Gentzen-style formalization. 8

In Gentzen-style formulations a 1st-order sequentψ0, . . . , ψn−1

ϕif often called

simply a sequent and denoted fby ψ0, . . . , ψn−1 ⇒ ψ, or by some similar nor-mally linear denotation. Thus in the context of a Gentzen formalization the2nd-order sequent (30) would be written

ψ00, . . . , ψ

0n0−1 ⇒ ϕ0; . . . ;ψk−1

0 , . . . , ψk−1nk−1−1 ⇒ ϕk−1

ϑ0, . . . , ϑm−1 ⇒ ξ.

2nd-order matrices were first explicitly used in abstract algebraic logic by Fontand Jansana in [15,5]. The theory of 2nd-order classes of deductive systems issystematically developed in [39]. In the present paper we consider only foursuch classes. The class of all deductive systems over Λ that have the mul-titerm deduction-detachment theorem with a fixed but arbitrary deduction-detachment system ∆(x, y) = δ0(x, y), . . . , δn−1(x, y) of Λ-formulas is a

8 There is however a subtle but important distinction between these two notions.A Gentzen formalization is normally used to define a single deductive system ratherthan a whole class of systems, namely the largest model of the Gentzen rules in thesense that it has the largest number of theories.

37

2nd-order class. It is defined by the set of 2nd-order sequents

z, x

y

z

δi(x, y)

, for all i < n,

were z ranges over all finite sequence of variables, together of course with the1st-order detachment rule for ∆(x, y). In view of this it makes sense to speak ofan arbitrary 2nd-order matrix A having the multiterm deduction-detachmenttheorem with deduction-detachment system ∆(x, y).

The other 2nd-order classes we consider are the self-extensional and the Fregeandeductive systems (see Def. 2.16 below), and the weakly normal modal logics(see Sec. 2.5 below).

Let A = 〈A, F 〉 be a 1st-order matrix. A binary relation R on A is definableover A if there exists a set Φ(x, y) = ϕi(x, y) : i ∈ I of binary 1-formulassuch that

R = 〈a, b〉 ∈ A2 : ϕAi (a, b) ∈ F for all i ∈ I .

Lemma 2.5 ([8]) Let A be a Λ-algebra, and let F ⊆ A. Let R ⊆ A2 bedefinable over the matrix 〈A, F 〉. If R is reflexive, then ΩA F ⊆ R.

PROOF. Assume R is definable by Φ(x, y) = ϕi(x, y) : i ∈ I . Supposea ≡ b (mod ΩA F ). Then, since ΩA F is a congruence relation, for all i ∈ I,ϕA

i (a, a) ≡ ϕAi (a, b) (mod ΩA F ). But ϕA

i (a, a) ∈ F because R is reflexive.So ϕA

i (a, b) ∈ F since ΩA F is compatible with F . Since this is true for alli ∈ I, 〈a, b〉 ∈ R. 2

The definition of Leibniz congruence can be extended from 1st- to 2nd-orderfilters; at the same time we define the closely related notion of the Fregerelation of a 2nd-order filter.

Definition 2.6 Let A be a nonempty set and let C be an arbitrary closed-setsystem over A.

(i) Λ C :=〈a, b〉 ∈ A2 : CloCa = CloCb

. Λ C is called the Frege

relation of C.(ii) Assume now that A is the universe of a Λ-algebra A. Let ΩA C be the set

of all 〈a, b〉 ∈ A2 such that

CloCϕA(a, c) = CloCϕA(b, c), for all ϕ(x, z) ∈ FmΛ and all c ∈ A|z|

38

ΩA C is called the ( 2nd-order) Leibniz congruence, or the Tarski congruence,of C. 2

The Frege relation and the 2nd-order Leibniz congruence were introduced in[15]. Their theory is developed and applied to abstract algebraic logic in [5](where the term Tarski congruence and the notation Ω C are used).

It is easy to see that Λ C is an equivalence relation on A; in fact it is thelargest equivalence relation on A that is compatible with C in the sense thatis it compatible with each F ∈ C. Although not as obvious, it is also notdifficult to see that ΩA C is indeed a congruence relation on A, and is in factthe largest congruence compatible with C. It follows that ΩA C can also becharacterized as the largest congruence that is included in Λ C.

As mappings from closed-set systems over A into the sets of equivalence andcongruence relations on A, Λ and ΩA are called respectively the Frege op-erator and the 2nd-order Leibniz, or Tarski, operator. The ordinary Leibnizcongruence ΩA F associated with a subset F of A can be obtained by ap-plying the 2nd-order operator ΩA to the discrete closed-set system F,A,i.e., ΩA F = ΩAF,A. For contrast we shall refer to the ordinary Leibnizoperator, which ranges over subsets of A, as the 1st-order Leibniz operator.

We can extend the domain of the Frege operator to include subsets of A bydefining ΛF := ΛF,A for each F ⊆ A. This gives us 1st- and 2nd-ordernotions of the Frege operator in analogy to the 1st- and 2nd-order Leibnizoperator. Note that ΛF is simply the equivalence relation associated withthe two-element partition F,A \ F, so the 1st-order Frege operator is littlemore than a curiosity, but it is useful for the symmetry between 1st- and 2nd-order notions it provides. For example, for any algebra A and any F ⊆ A, wehave that a congruence is compatible with F iff it is included in ΛF . ThusΩA F is the largest congruence included in ΛF .

For any closed-set system C over a set A or over the universe A of an algebraA we have,

Λ C =⋂

F∈C ΛF and ΩA C =⋂

F∈C ΩA F. (33)

The 2nd-order Frege and Leibniz operators on closed-set systems are bothantimonotonic in the sense that, if C and D are two closed-set systems overA such that C ⊆ D (i.e., F ∈ C implies F ∈ D), then Λ C ⊇ ΛD andΩA C ⊇ ΩAD. This is an immediate consequence of the characterizations(33). However, neither the 1st-order Frege nor the 1st-order Leibniz operatorsare either monotonic or antimonotonic on arbitrary subsets of A. As is well-known, the 1st-order Leibniz operator is monotonic in special circumstances,for example, when restricted to the filters of a protoalgebraic deductive system.

In analogy with its 1st-order counterpart (Lem. 1.11) the 2nd-order Leibniz

39

operator commutes with inverse surjective homomorphisms, and the Fregeoperator has the same property.

Lemma 2.7 Let A and B be Λ-algebras and h: A B a surjective homo-morphism. Then for any closed-set system C over B,

(i) ΩA

(h−1(C)

)= h−1(ΩB C), and

(ii) Λ(h−1(C)

)= h−1(Λ C).

PROOF. (i):

ΩA

(h−1(C)

)=

⋂ΩA

(h−1(F )

): F ∈ C

, by (33)

=⋂

h−1(ΩB F ) : F ∈ C, by Lem. 1.11

= h−1(⋂

ΩB F : F ∈ C)

= h−1(ΩB C).

The proof of (ii) is similar. 2

In analogy to 1st-order matrices, a 2nd-order matrix A = 〈A, C〉 is reducedif ΩA C = IdA. For an arbitrary A = 〈A, C〉, A∗ is the quotient matrix〈A/ΩA C, C/ΩA C〉, where C/ΩA C = F/ΩA C : F ∈ C . By Lem. 2.7(i),A∗ is always reduced; it is called the reduction of A.

Definition 2.8 Two 2nd-order matrices A and B are said to be reduction-isomorphic if A∗ ∼= B∗. 2

The natural map from A to A/ΩA C is a strict, surjective matrix homomor-phism between A = 〈A, C〉 and its reduction A∗, and A and B are reduction-isomorphic iff there exists a 2nd-order matrix C such that C 4 A and C 4 B,i.e., there exist strict, surjective matrix homomorphisms from A and B to C.

Corollary 2.9 The validity of each 2nd-order sequent is preserved under re-duction isomorphism. I.e., if a 2nd-order matrix A is a model of a 2nd-ordersequent, then so is any 2nd-order matrix B such that B∗ ∼= A∗.

PROOF. Assume A and B are reduction-isomorphic and let C be a 2nd-order matrix such that C 4 A and C 4 B. Then by Thm. 2.3(ii), a 2nd-ordersequent is valid in A iff it is valid in C iff it is valid in B. 2

40

Protoalgebraicity, a concept that was defined previously for 1-deductive sys-tems, a special kind of 2nd-order matrix, can be applied to any 2nd-ordermatrix.

Definition 2.10 A 2nd-order matrix A = 〈A, C〉 is said to be protoalgebraicif the 1st-order Leibniz operator is monotonic on C, i.e., for all F,G ∈ C,F ⊆ G implies ΩA F ⊆ ΩAG. 2

This notion is consistent with and extends the notion of a protoalgebraicdeductive system; in fact, a 1-deductive system S is protoalgebraic in thesense of Def. 1.14 if as a 2nd-order matrix it is protoalgebraic in the sense ofDef. 2.10.

We localize the 2nd-order Frege and Leibniz operators to individual closed setsF in a closed-set system C by applying them to the principal filter [F )C of Cgenerated by F . These will turn out to be important notions and we introducecorresponding notation and terminology.

Definition 2.11 Let C be a closed-set system over a nonempty set A or overthe universe A of a Λ-algebra A. If F ∈ C we define

ΛC F := Λ[F )C and ΩA,C F := ΩA[F )C.

ΛC F and Ω A,CF are respectively called the local 2nd-order Frege relation and

Leibniz congruence of F over C. ΩA,C F is also called the Suszko congruenceof F over C. 2

Note that

ΛC F =⋂ΛG : F ⊆ G ∈ C and ΩA,C F =

⋂ΩAG : F ⊆ G ∈ C .

Note also that ΛC and ΩA,C, as operators on the closed sets of C, are alwaysmonotonic. It is useful to keep in mind that, while the 1st-order Leibniz con-gruence of F is an absolute notion in the sense that it depends only on F andthe underlying algebra A, the Suszko congruence of F depends also on theclosed-set system of which F is a member.

As special cases of Lem. 2.7 we have, for every surjective homomorphismh: A B, every closed set system C over B, and every F ∈ C,

Λh−1(C) h−1(F ) = h−1

(ΛC F

), ΩA,h−1(C) h

−1(F ) = h−1(ΩB,C F

). (34)

The Suszko operator can also be thought of as relativized to a 2nd-ordermatrix A = 〈A, C〉 by setting ΩA F = ΩA,C F , for every F ∈ C.

41

Lemma 2.12 Let A = 〈A, C〉 be a 2nd-order matrix. The following conditionsare equivalent.

(i) A is protoalgebraic;(ii) ΩA F = ΩA F , for every F ∈ C;(iii) ΩA F ⊆ ΛC F , for every F ∈ C.

PROOF. (i) ⇔ (ii): If A is protoalgebraic, then ΩA F ⊆ ΩAG for everyG ∈ [F )C. So ΩA F =

⋂ΩAG : F ⊆ G ∈ C = ΩA F .

Conversely, if ΩA F = ΩA F , then, for each G such that F ⊆ G ∈ C, ΩA F =ΩA F ⊆ ΩAG.

(ii)⇔(iii): Since ΩA F is the largest congruence included in ΛC F and ΩA F is

a congruence that includes ΩA F , clearly ΩA F = ΩA F iff ΩA F ⊆ ΛC F . 2

In general, for every 2nd-order matrix A = 〈A, C〉 and every F ∈ C, we have

ΩA F ⊆ ΩA F and ΩA F ⊆ ΛC F , while ΩA F and ΛC F are incomparable.

2.1 Self-extensional and Fregean 2nd-order matrices

We define the key notions of the paper.

Definition 2.13 Let A = 〈A, C〉 be a 2nd-order matrix.

(i) A is self-extensional 9 if the Frege relation of C is a congruence, i.e., if

ΩA C = Λ C.

(ii) A is Fregean if every principal-filter matrix 〈A, [F )C〉 of A is self-exten-sional, i.e., if

ΩA F = ΛC F, for every F ∈ C. 2

Note that A is self-extensional iff ΩAF0 = ΛC F0, where F0 is the smallest

C-filter.

Self-extensional and Fregean 2nd-order matrices form 2nd-order classes. Infact, it is easy to see that A is self-extensional iff the following 2nd-order

9 Self-extensional 2nd-order matrices are referred to as abstract logics with thecongruence property in [5].

42

sequent is valid in A for every λ ∈ Λ, where n is the rank of λ,

x0

y0

,y0

x0

, . . . ,xn−1

yn−1

,yn−1

xn−1

λx0 . . . xn−1

λy0 . . . yn−1

.

The class of Fregean matrices is also a 2nd-order class but to show this requiresmore work. It depends on the following technical lemma.

A filter F of an algebraic closed-set system C over a set A is finitely generatedif F = CloCX for some X ⊆ω A.

Lemma 2.14 A 2nd-order matrix A = 〈A, C〉 is Fregean iff ΩA F = ΛC Ffor every finitely generated F ∈ C.

PROOF. Suppose ΩA F ( ΛC F for some F ∈ C. Let 〈a, b〉 ∈ ΛC F suchthat 〈a, b〉 /∈ ΩA F . Then, for every G ∈ [F )C, a ∈ G iff b ∈ G, i.e.,

CloC(F ∪ a) = CloC(F ∪ b).

Let X be a finite subset of F such that CloC(X∪a) = CloC(X∪b) (such an

X exists because C is algebraic), and let F ′ = CloC(X). Then 〈a, b〉 ∈ ΛC F′,

but 〈a, b〉 /∈ ΩA F′ since ΩA F

′ ⊆ ΩA F . 2

It follows easily from this lemma that a 2nd-order matrix A is Fregean iff

z, x0

y0

,z, y0

x0

, . . . ,z, xn−1

yn−1

,z, yn−1

xn−1

z, λx0 . . . xn−1

λy0 . . . yn−1

is valid in A for every λ ∈ Λ and every finite sequence of variables z =z0, . . . , zm−1.

By the accumulative form of the 2nd-order sequent (30) we mean the familyof 2nd-order sequents

z, ψ00, . . . , ψ

0n0−1

ϕ0, . . . ,

z, ψk−10 , . . . , ψk−1

nk−1−1

ϕk−1

z, ϑ0, . . . , ϑm−1

ξ

,

where z ranges over all finite sequences of variables. It is easy to see that a2nd-order matrix A = 〈A, C〉 is a model of the accumulative form of a 2nd-

43

order sequent iff 〈A, [F )C〉 is a model of the sequent itself for every finitelygenerated C-filter F , and hence, by the obvious generalization of Lem. 2.14,iff 〈A, [F )C〉 is a model of the sequent for every F ∈ C, without restriction.

A 2nd-order class of 2nd-order matrices is said to be accumulative if it isdefined by the accumulative forms of a set of 2nd-order sequents. The classof Fregean 2nd-order matrices is accumulative. The 2nd-order matrices thathave the deduction-detachment theorem (with a fixed but arbitrary deduction-detachment system ∆(x, y)) form another accumulative class.

We will be especially interested in those 2nd-order matrices A = 〈A, C〉 thatare both protoalgebraic and Fregean. In this case all three of the relations

ΩA F , ΩA F , and ΛC F coincide for every F ∈ C, as we now show.

Theorem 2.15 Let A = 〈A, C〉 be a 2nd-order matrix.

(i) If A is protoalgebraic, then A is self-extensional iff ΩA F0 = Λ C, whereF0 is the smallest C-filter.

(ii) A is protoalgebraic and Fregean iff ΩA F = ΛC F for every F ∈ C.

PROOF. (i) is immediate.

(ii). If A is protoalgebraic and Fregean, then, for every F ∈ C, ΩA F = ΩA F

(by Lem. 2.12) = ΛC F (by Def. 2.13). Conversely, if ΩA F = ΛC F , then

ΩA F = ΩA F because ΩA F is the largest congruence included in ΛC F .Thus A is both protoalgebraic and Fregean. 2

We now specialize to deductive systems. Let S be a 1-deductive system. Recallthat S is by definition the 2nd-order formula matrix 〈FmΛ, ThS〉. We thushave

ΛThS = 〈ϕ, ψ〉 ∈ Fm2Λ : ϕ a`S ψ .

More generally, for any S-theory T ,

ΛThS T = 〈ϕ, ψ〉 ∈ Fm2Λ : T, ϕ a`S T, ψ .

For obvious reasons the Frege relation ΛThS is also called the symmetric

consequence relation, and ΛThS T the symmetric localized consequence relation(at T ). Recall that ThmS is the set of theorems of S, the smallest S-theory,and that we write Ω for ΩFmΛ

.

Definition 2.16 Let S be a 1-deductive system.

44

(i) S is self-extensional if S = 〈FmΛ,ThS〉 is self-extensional, i.e., if

Ω ThS = ΛThS.

(ii) S is Fregean if 〈FmΛ, T 〉 is self-extensional for every S-theory T , i.e., if

ΩS T = ΛThS T, for every T ∈ ThS. 2

The notion of a self-extensional deductive system was first considered byWojcicki; see [3]. Fregean logics first explicitly occurred in the literature in[15], although protoalgebraic Fregean logics were independently considered bythe present authors about the same time in unpublished notes; see [40]. Aclosely related algebraic notion can be found in [16]. For additional referencessee [5], pages 64 and 66.

A characteristic property of Fregean deductive systems is the coalescence ofthe notions of protoalgebraicity and equivalentialness.

Lemma 2.17 Let S be a protoalgebraic 1-deductive system with at least onetheorem, and let ∆(x, y) be a (necessarily nonempty) protoequivalence systemfor S. If S is Fregean, then ∆(x, y) ∪∆(y, x) is an equivalence system for S.

PROOF. Assume S is protoalgebraic and Fregean. Let

∆(x, y) = δ0(x, y), . . . , δn−1(x, y)

be a protoequivalence system for S, and set E(x, y) = ∆(x, y) ∪ ∆(y, x). Toshow that E(x, y) is an equivalence system for S, we first show that E(x, y)defines a subrelation of the Frege relation. Let T ∈ ThS and let R = 〈ϕ, ψ〉 ∈Fm2

Λ : E(ϕ, ψ) ⊆ T . Suppose 〈ϕ, ψ〉 ∈ R. Then since ϕ,∆(ϕ, ψ) `S ψ, bythe detachment rule for ∆, and ∆(ϕ, ψ) ⊆ E(ϕ, ψ) ⊆ T , we have ϕ, T `Sψ. Similarly, from ψ,∆(ψ, ϕ) `S ϕ and ∆(ψ, ϕ) ⊆ E(ϕ, ψ) ⊆ T , we get

ψ, T `S ϕ. Hence 〈ϕ, ψ〉 ∈ ΛThS T . Thus R ⊆ ΛThS T = ΩT , since Sis protoalgebraic and Fregean. But R is a reflexive relation, because of thereflexivity condition for protoequivalence systems (14), and it is definable byE(x, y) over the matrix 〈FmΛ, T 〉. Thus by Lem. 2.5 ΩT ⊆ R. So R = ΩT .This shows that E(x, y) is an equivalence system for S by Thm. 1.19. 2

The following theorem was proved independently by the present authors (see[40]) and Font [15].

Theorem 2.18 Every protoalgebraic Fregean 1-deductive system with at leastone theorem is regularly algebraizable.

45

PROOF. Let S be a protoalgebraic and Fregean deductive system with atleast one theorem, and let ∆(x, y) be a protoequivalence system. By the pre-ceding lemma, E(x, y) = ∆(x, y)∪∆(y, x) is an equivalence system for S. It re-mains only to show that the E-G-rule is a rule of S. Let T ∈ ThS and suppose

ϕ, ψ ∈ T . Then trivially T, ϕ `S ψ and T, ψ `S ϕ. Thus 〈ϕ, ψ〉 ∈ ΛThS T , andhence 〈ϕ, ψ〉 ∈ ΩT because S is protoalgebraic and Fregean. So, by Thm. 1.19,ϕ, ψ ∈ T implies E(ϕ, ψ) ⊆ T , for every T ∈ ThS, i.e., the E-G-rule is a ruleof S. 2

In Lem. 2.17 we proved that, if a protoalgebraic deductive system S with atleast one theorem is Fregean, then ∆(x, y) ∪∆(y, x) is an equivalence systemfor every protoequivalence system ∆(x, y). Clearly the converse fails. Anyequivalential system that fails to be Fregean, for example almost every normalmodal logic, provides a counterexample.

But in the next lemma we prove the converse under stronger assumptionsabout the system ∆(x, y).

2.2 The deduction theorem and conjunction

Recall the definition of a deduction-detachment system given in Def. 1.38, andof the notions of the uniterm and multiterm deduction-detachment theorem.

A nonempty system ∆(x, y) of binary formulas is a deduction-detachmentsystem for S if it defines the localized consequence relation in the sense that,for every T ∈ ThS, T, ϕ `S ψ iff ∆(ϕ, ψ) ⊆ T . Consequently, ∆(x, y)∪∆(y, x)

defines the symmetric localized consequence relation ΛThS T . This observationleads directly to the following result.

Lemma 2.19 Let S be a deductive system with a deduction-detachment sys-tem ∆(x, y) = δi(x, y) : i ∈ I . Then S is Fregean iff ∆(x, y)∪∆(y, x) is anequivalence system for S.

PROOF. Since ∆(x, y) defines the localized consequence relation, ∆(x, y) ∪∆(y, x) defines the symmetric localized consequence relation ΛThS T , for everyT ∈ ThS. If ∆(x, y) ∪ ∆(y, x) is an equivalence system, then it also defines

ΩT by Thm. 1.19. Thus ΩT = ΛThS T , for all T ∈ ThS, and so S is Fregean.Conversely, if S is Fregean, then ∆(x, y) ∪ ∆(y, x) defines ΩT and hence isan equivalence system. 2

Using this lemma we can obtain a very satisfactory characterization of Fregean

46

deductive systems with the uniterm deduction-detachment theorem. In thenext theorem, and in the sequel, ϕ0 → ϕ1 → · · · → ϕn−1 → ϕn will beshorthand for ϕ0 → (ϕ1 → (· · · → (ϕn−1 → ϕn) · · · )).

Theorem 2.20 Let Λ be an arbitrary language type. Let S be a Fregean de-ductive system over Λ with the uniterm deduction-detachment theorem. Letx → y be a single deduction-detachment formula for S. Then S is an ax-iomatic extension of the deductive system presented by the axioms

x→ y → x, (35)

(x→ y → z) → (x→ y) → (x→ z), (36)

(x0 → y0) → (y0 → x0) → · · · → (xn−1 → yn−1) → (yn−1 → xn−1)

→ λx0 . . . xn−1 → λy0 . . . yn−1,

for each λ ∈ Λ (n is the rank of λ),

(37)

and the single inference rule

x, x→ y

yModus Ponens. (MP→)

Conversely, every axiomatic extension of this deductive system is Fregean andhas the uniterm deduction-detachment theorem with x → y as the deduction-detachment formula.

PROOF. That (35) and (36) are theorems of S and (MP→) is an inferencerule are immediate consequences of the assumption that x→ y is a deduction-detachment formula for S. To see this note that, from the trivial entailmentx, y `S x, we get `S x → y → x by the deduction property. We have thatx, x → y, x → y → z `S z by three applications of detachment, and thenthree applications of the deduction property give `S (x → y → z) → (x →y) → (x→ z).

By Lem. 2.19, E(x, y) = x → y, y → x is an equivalence system for S. Soby the replacement rule for equivalence systems (23) we have

(x0 → y0) → (y0 → x0) → · · · → (xn−1 → yn−1) → (yn−1 → xn−1)

`S λx0 . . . xn−1 → λy0 . . . yn−1.

Hence multiple applications of the deduction property give

`S (x0 → y0) → (y0 → x0) → · · · → (xn−1 → yn−1) → (yn−1 → xn−1)

→ λx0 . . . xn−1 → λy0 . . . yn−1.

So (37) is a theorem of S for every λ ∈ Λ. Finally, by Thm. 1.42 we knowthat a presentation of S can be obtained by adjoining additional axioms to

47

(35)–(37) and (MP→).

For the converse, let S be any axiomatic extension of the deductive systemwith presentation (35)–(37) and (MP→). Since (MP→) is the only inferencerule, the deduction property for x → y can be proved by induction on thelength of the derivation using axioms (35) and (36) in exactly the same way itis for the classical and intuitionistic propositional calculi. We omit the details.

That S is Fregean will then follow by Lem. 2.19 from the fact that E(x, y) :=x → y, y → x is an equivalence system for S. This is easily verified.For example, we have x, x → y, y → z `S z by detachment, and henceE(x, y), E(y, z) `S E(x, z) by the deduction property. So the transitivity rule(22) holds. We get the replacement rule from the axioms (37) together withthe deduction and detachment properties. 2

This theorem can be reformulated in a way that shows more clearly the relationbetween Fregean deductive systems with the uniterm deduction-detachmenttheorem and the intuitionistic propositional calculus IPC. Let IPC→,> be the→,>-fragment of IPC. Let Λ be an arbitrary language type with Λ ∩ →,> = ∅, and let IPC→,>

Λ be the expansion (see Sec. 1) of IPC→,> to thelanguage type →,>∪Λ obtained by adding the axioms (37). The new con-nectives are said to be extensional over IPC→,>, and IPC→,>

Λ and its axiomaticextensions are collectively referred to as the expansions of IPC→,> by exten-sional connectives.

Let S be any deductive system over the language type Λ. By a →,>-definitional expansion of S we mean an conservative expansion of S to thelanguage type Λ ∪ →,> such that x → y ≡ ϕ(x, y) (mod Ω ThS ′) forsome binary formula ϕ(x, y) over Λ and > ≡ ψ (mod Ω ThS ′) for some the-orem of ψ of S.

Theorem 2.20 can now be reformulated in the following way. A deductivesystem over the language type Λ is Fregean with the uniterm deduction-detachment theorem iff it has a definitional expansion by the connectives →and > that is an axiomatic extension of IPC→,>

Λ ; in summary, every Fregeansystem with the uniterm deduction-detachment theorem is (up to definitionalexpansion) an expansion of IPC→,> by extensional connectives.

Since conjunction, disjunction, and negation are all extensional over IPC→,>,the (full) intuitionistic propositional calculus IPC itself and all its axiomaticextensions, in particular CPC, fall within the scope of Theorem 2.20, as wellany expansions of these logics by additional extensional connectives. In thecase of CPC or course there are no such connectives (up to definitional ex-pansion) because of the functional completeness of the two-element Booleanalgebra. We know of no connectives that are extensional over IPC apart from

48

the fundamental ones and those definable in their terms. The nontrivial modaloperators all fail to be extensional, and hence Theorem 2.20 does not apply tomodal logics, at least the so-called strong normal modal logics. We will returnto the subject of modal logics below in Section 2.5.

There are many examples of protoalgebraic Fregean deductive systems with-out even the multiterm deduction-detachment theorem. The paradigm for thisclass of deductive systems is IPC↔,>, the biconditional fragment of intuitionis-tic propositional logic; see [19]. However, every Fregean deductive system withconjunction that has at least one theorem has a uniterm deduction-detachmentsystem, as we next show.

A single formula κ(x, y) is a conjunction formula for S if the following sequentsare rules of S.

x, y

κ(x, y),

κ(x, y)

x, and

κ(x, y)

y.

A deductive system is said to be conjunctive if it has a conjunction formula.In the sequel x ∧ y will represent an arbitrary conjunction formula. Whenwriting iterated conjunctions we omit parenthesis and assume association isto the right.

Note that, if x∧y is a conjunction formula, then, for all ϕ, ψ ∈ FmΛ and everyT ∈ ThS,

T, ϕ `S ψ iff T, ϕ a`S T, ϕ ∧ ψ iff 〈ϕ, ϕ ∧ ψ〉 ∈ ΛThS T.

Theorem 2.21 Assume S is a 1-deductive system with a conjunction formulax ∧ y. If S is protoalgebraic and Fregean with at least one theorem, then ithas the uniterm deduction-detachment theorem. More precisely, if ∆(x, y) =δ0(x, y), . . . , δn−1(x, y) is a protoequivalence system for S, then

x→ y := δ0(x, x ∧ y) ∧ · · · ∧ δn−1(x, x ∧ y)

is a single deduction-detachment formula for S; moreover,

x↔ y := (x→ y) ∧ (y → x)

is a single equivalence formula.

PROOF. For all ϕ, ψ ∈ FmS and every T ∈ ThS,

T, ϕ `S ψiff T, ϕ a`S T, ϕ ∧ ψiff

⟨ϕ, ϕ ∧ ψ

⟩∈ ΛThS T

iff ϕ ≡ ϕ ∧ ψ (mod ΩT ), since S is Fregean and protoalgebraic.

49

Consequently, δi(ϕ, ϕ) ≡ δi(ϕ, ϕ ∧ ψ) (mod ΩT ) for each i < n. Thus, sinceδi(ϕ, ϕ) ∈ T (because ∆(x, y) is a protoequivalence system) and ΩT is com-patible with T , we have δi(ϕ, ϕ∧ ψ) ∈ T for each i < n. Hence T `S δ0(x, x∧y) ∧ · · · ∧ δn−1(x, x ∧ y). Conversely, if T `S δ0(x, x ∧ y) ∧ · · · ∧ δn−1(x, x ∧ y),then T, ϕ `S ψ since δ0(x, y), . . . , δn−1(x, y) is a protoequivalence system.So x→ y is a deduction-detachment formula for S.

x → y is clearly a single protoequivalence formula for S. Thus x ↔ y is asingle equivalence formula by Lem. 2.17. 2

Theorem 2.22 Let Λ be an arbitrary language type. Let S be a protoalge-braic Fregean 1-deductive system over Λ with conjunction, and let x ∧ y be aconjunction formula. If S has at least one theorem, then there exists a binaryformula x → y such that S is an axiomatic extension of the deductive sys-tem presented by the axioms (35)–(37) and the single inference rule (MP→) ofThm. 2.20, along with the following additional axioms.

x ∧ y → x, (38)

x ∧ y → y, (39)

x→ y → x ∧ y. (40)

Conversely, every axiomatic extension of this deductive system is protoalge-braic and Fregean and has conjunction with x ∧ y as conjunction formula.

PROOF. Let S be a protoalgebraic Fregean deductive system over Λ with aconjunction and with at least one theorem. Let x∧y be a conjunction formula.By Thm. 2.21 S has a single deduction-detachment formula x → y. Thus byThm. 2.20 S is an axiomatic extension of the system presented by (35)–(37)and (MP→). Clearly (38)–(40) are theorems of S. The converse follows easilyfrom Thm. 2.20. 2

This theorem can be reformulated as Theorem 2.20 was reformulated in theremarks that followed it: every Fregean system with conjunction is (up todefinitional expansion) an expansion of IPC→,∧,> by extensional connectives.We omit the details.

2.3 Strongly algebraizable Fregean deductive systems

Every protoalgebraic Fregean deductive system is algebraizable and in factregularly algebraizable by Theorem 2.18. One of the more interesting chal-lenges in abstract algebraic logic has been to try to explain why almost all

50

the traditional deductive systems are strongly algebraizable, that is, they arefinitely algebraizable and their equivalent semantics is a variety; in the broadercontext of abstract algebraic logic the equivalent semantics of a finitely alge-braizable system is normally a proper quasivariety.

At least for those logics that have a large enough fragment of the classical orintuitionistic propositional calculi at their core, the key to the puzzle appearsto be the Fregean property or, more significantly, self-extensionality; boththese properties can be shown to induce strong algebraizability under a varietyof different circumstances. The first result of this kind presented here is adirect consequence of the characterization of Fregean deductive systems withthe uniterm deduction-detachment theorem presented in Theorem 2.20.

As is well-known, IPC→,>, the →,>-fragment of the intuitionistic proposi-tional calculus, is strongly, regularly algebraizable with equivalent semanticsthe variety of Hilbert algebras (Diego [41]). Hilbert algebras are defined bythe following four identities ([41], p. 7). (Recall that by convention multipleoccurrences of → in a formula are associated to the right.)

x→ x ≈ >; (41)

> → x ≈ x; (42)

x→ y → z ≈ (x→ y) → x→ z; (43)

(x→ y) → (y → x) → x ≈ (x→ y) → (y → x) → y. (44)

The first part of the following theorem was originally proved by Font andJansana [5, Proposition 4.49].

Theorem 2.23 Every Fregean deductive system with the uniterm deduction-detachment theorem is strongly, regularly algebraizable. More precisely, let Λbe an arbitrary language type and let S be a Fregean deductive system overΛ with a single deduction-detachment formula x → y. Then Alg Mod∗ S isa subvariety of the variety defined by the four identities (41)–(44) of Hilbertalgebras, together with the identity

(x0 → y0) →(y0 → x0) → · · ·· · · → (xn−1 → yn−1) → (yn−1 → xn−1) → λx0 . . . xn−1

(45)

≈ (x0 → y0) →(y0 → x0) → · · ·· · · → (xn−1 → yn−1) → (yn−1 → xn−1) → λy0 . . . yn−1

for each λ ∈ Λ (n is the rank of λ).

PROOF. By Thm. 2.20, S is an axiomatic extension of the deductive system

51

presented by the following axioms and inference rules.

x→ y → x,

(x→ y → z) → (x→ y) → x→ z,

(x0 → y0) → (y0 → x0) → · · · → (xn−1 → yn−1) → (yn−1 → xn−1)

→ λx0 . . . xn−1 → λy0 . . . yn−1,

for each λ ∈ Λ (n is the rank of λ),

and the single inference rulex, x→ y

yApplying Thm. 1.43 directly, with E(x, y) = x → y, y → x, we have thatAlg Mod∗ S is defined by adjoining some set of identities of the form ϕ ≈ > tothe following identities and quasi-identities.

x→ x ≈ >,x ≈ >, x→ y ≈ >

y ≈ >, (46)

x→ y ≈ >, y → x ≈ >x ≈ y

, (47)

x→ y → x ≈ >, (48)

(x→ y → z) → (x→ y) → (x→ z) ≈ >, (49)

(x0 → y0) → (y0 → x0) → · · · → (xn−1 → yn−1) → (yn−1 → xn−1)

→ λx0 . . . xn−1 → λy0 . . . yn−1 ≈ >,for each λ ∈ Λ (n is the rank of λ).

(50)

It is proved in [41], p. 7, that the quasi-identity (47) and the two identities(48) and (49) together define Hilbert algebras, i.e., are logically equivalent tothe four identities (41)–(44). (46) is an immediate consequence of (42), and(45) and (50) are easily shown to be equivalent in the presence of (46), (47),and (49). 2

Let Λ be an arbitrary language type disjoint from →,>. Let HIΛ be thevariety of algebras of type →,>∪Λ defined by the four identities of Hilbertalgebras together with the identity (45) for each λ ∈ Λ (n is the rank of λ).HIΛ is called the the variety of Hilbert algebras with compatible operations overΛ. Theorem 2.23 says that, up to termwise definitional equivalence, the sub-varieties of HIΛ are exactly the equivalent quasivarieties of Fregean deductivesystems with the uniterm deduction-detachment theorem.

52

By the theorem, all the intermediate logics, i.e., all the the axiomatic exten-sions of IPC, are strongly algebraizable, and their equivalent varieties are thesubvarieties of the variety of Heyting algebras.

Theorem 2.23 can be given a more algebraic formulation. Recall the definitionof the assertional logic of a pointed quasivariety (Def. 1.32),

Corollary 2.24 Every relatively point-regular quasivariety Q whose asser-tional logic is Fregean and has the uniterm deduction-detachment theorem is avariety. More precisely, Q is termwise definitionally equivalent to a subvarietyof HIΛ, where Λ is the language type of Q.

PROOF. Let Q be a relatively point-regular quasivariety and assume thatits assertional logic Sasl Q is both Fregean and has the uniterm deduction-detachment theorem. By Thm. 2.23 Sasl Q is strongly, regularly algebraiz-able and its equivalent quasivariety Alg Mod∗ Sasl Q is termwise definitionallyequivalent to a subvariety of HIΛ. But Q = Alg Mod∗ Sasl Q (see Thm. 1.34and the following remarks). 2

Every protoalgebraic Fregean deductive system with conjunction and at leastone theorem has the uniterm deduction-detachment theorem (Theorem 2.21),and hence, as a immediate consequence of Theorem 2.23, is strongly, regularlyalgebraizable. The precise situation is described in the following:

Theorem 2.25 Let Λ be an arbitrary language type and let S be a protoal-gebraic Fregean 1-deductive system over Λ with a conjunction formula x ∧ yand at least one theorem. Let x→ y be any deduction-detachment term for S.Then Alg Mod∗ S is a subvariety of the variety defined by the identities (41)–(44) of Hilbert algebras, the identity (45) for each λ ∈ Λ (n is the rank of λ),together with the three identities

x ∧ y → x ≈ >, (51)

x ∧ y → y ≈ >, (52)

x→ y → x ∧ y ≈ >. (53)

PROOF. Similar to the proof of Thm. 2.23 but with Thm. 2.22 in place ofThm.2.20. 2

This theorem provides an alternative way of showing that all the intermediatelogics are strongly algebraizable.

53

Theorems 2.23 and 2.25 do not comprehend all strongly algebraizable Fregeanlogics however. For example, IPC↔,>, the biconditional fragment of intuition-istic propositional logic, is Fregean and strongly algebraizble (see [19]), but ithas neither the deduction-detachment theorem nor conjunction.

Let Λ be an arbitrary language type disjoint from →,∧,>. Let BSΛ be thevariety of algebras of type →,∧,> ∪ Λ defined by the the four identitiesof Hilbert algebras, the identity (45) for each λ ∈ ∧ ∪ Λ, and the threeidentities (51)–(53). BSΛ is the variety of Brouwerian semilattices with com-patible operations over Λ. (See [42].) Theorem 2.25 says that, up to termwisedefinitional equivalence, the subvarieties of BSΛ are exactly the equivalentquasivarieties of protoalgebraic Fregean deductive systems, with at least onetheorem, that have conjunction. The theorem can also be used to show thatall the intermediate logics are strongly algebraizable.

Like Theorem 2.23, Theorem 2.25 can be given a more algebraic formulation.Note that, if a quasivariety is point-regular, its assertional logic must have atleast one theorem.

Corollary 2.26 ([16, Theorem 4.5]) Every relatively point-regular quasi-variety Q whose assertional logic is protoalgebraic, Fregean, and has conjunc-tion is a variety. More precisely, Q is termwise definitionally equivalent to asubvariety of BSΛ, where Λ is the language type of Q.

PROOF. This is an easy consequence of Thms. 2.22 and 2.23. 2

2.4 Strong algebraizability and self-extensionality

Theorems 2.23 and 2.25 give a good account of the strong algebraizability ofthe classical and intuitionistic calculi and their various fragments, but they donot account for the strong algebraizability of normal modal logic, which is notFregean. However, recent results of Font and Jansana [5] on the relationshipbetween self-extensionality and strong algebraizability do encompass modallogic, and, moreover, they go a long way towards giving us a clear picture ofwhy strong algebraizability is such a common phenomenon in classical alge-braic logic. For some of their results we now present new proofs that generalizethe proofs of Theorems 2.23 and 2.25, which themselves generalize the meth-ods used in classical algebraic logic to establish strong algebraizability. Thisleads to some refinements that may shed further light on this problem.

Let x → y be a binary Λ-formula. For all Γ ∪ ϕ ⊆ω FmΛ, we denote by

54

Γ → ϕ the formula

ψ0 →(ψ1 → (· · · → (ψn−1 → ϕ) · · · )),

where ψ0, . . . , ψn−1 is a fixed but arbitrary ordering of Γ . Furthermore, ifΓ0, . . . , Γm−1 is any finite sequence of finite subsets of FmΛ, then

Γ0 → Γ1 → · · · → Γm−1 → ϕ

will stand for Γ0 →(Γ1 → (· · · → (Γm−1 → ϕ) · · · )).

Recall that every deductive system S with the multiterm, and in particular theuniterm, deduction-detachment theorem is protoalgebraic. Indeed, if x → yis a single deduction-detachment formula for S, then ∆(x, y) = x → y is aprotoequivalence system and hence (by Thm. 1.18)

E(x, y, u) :=ξ(x, u) → ξ(y, u) : ξ(x, u) ∈ FmΛ(Va \ y)

∪

ξ(y, u) → ξ(x, u) : ξ(x, u) ∈ FmΛ(Va \ y)

(54)

is an equivalence system with parameters for S.

Lemma 2.27 Let S be a self-extensional 1-deductive system over a countablelanguage type with the uniterm deduction-detachment theorem. Let x → y bea single deduction-detachment formula. Then for all Γ ∪ ϕ, ψ ⊆ω FmΛ,Γ, ϕ a`S Γ, ψ iff Γ → ϕ ≈ Γ → ψ is an identity of Alg Mod∗ S.

PROOF. Assume Γ, ϕ a`S Γ, ψ. Then Γ, Γ → ϕ `S ψ by the detachmentproperty of x → y. So Γ → ϕ `S Γ → ψ by the deduction property ofx → y. By symmetry, Γ → ψ `S Γ → ϕ. Thus Γ → ϕ a`S Γ → ψ. So〈Γ → ϕ, Γ → ψ〉 ∈ ΛThS. Since S is self-extensional, ΛThS = Ω ThS.So Γ → ϕ ≈ Γ → ψ is an identity of Alg Mod∗ S by Cor. 1.22. (Note thatΩ ThS = Ω(ThmS) since S is protoalgebraic).

Conversely, if Γ → ϕ ≈ Γ → ψ is an identity of Alg Mod∗ S, then, again byThm. 1.22, 〈Γ → ϕ, Γ → ψ〉 ∈ Ω ThS ⊆ ΛThS, and hence Γ → ϕ a`SΓ → ψ. We get Γ, ϕ `S Γ → ϕ by the deduction property of x→ y. Similarly,Γ, ψ `S Γ → ψ. So, by detachment, Γ, ϕ a`S Γ, ψ. (Self-extensionality is notneeded for this direction.) 2

For the next lemma recall that ∀ϑ E(ϕ, ψ, ϑ) :=εi(ϕ, ψ, ϑ) : i ∈ I, ϑ ∈

Fm|u|Λ

.

Lemma 2.28 Let S be a self-extensional 1-deductive system with the unitermdeduction-detachment theorem. Let x → y be a single deduction-detachment

55

formula for S, and let E(x, y, u) be the equivalential system for S with param-eters that is formed from x → y as in (54). Then the following are identitiesof Alg Mod∗ S.

(i) (y → y) → x ≈ x, and(ii) x→ y → y ≈ y → y.

(iii) For every quasi-identityϕ0 ≈ ψ0, ϕ1 ≈ ψ1, . . . , ϕn−1 ≈ ψn−1

ξ ≈ ηof Alg Mod∗ S, there is, for each i < n, a finite subset Fi(ϕi, ψi) of∀ϑ E(ϕi, ψi, ϑ) such that

F0(ϕ0, ψ0) → F1(ϕ1, ψ1) → · · · → Fn−1(ϕn−1, ψn−1) → ξ

≈ F0(ϕ0, ψ0) → F1(ϕ1, ψ1) → · · · → Fn−1(ϕn−1, ψn−1) → η

is an identity of Alg Mod∗ S.

PROOF. (i). `S y → y. Thus (y → y) → x `S x by the detachment propertyof x→ y. On the other hand, x `S (y → y) → x by the deduction property ofx → y. So (y → y) → x a`S x, and hence (y → y) → x ≡ x (mod Ω ThS)by self-extensionality. Thus (y → y) → x ≈ x is an identity of Alg Mod∗ S byCor. 1.22.

(ii). From `S y → y we get x `S y → y and hence `S x → y → y bythe deduction property of x → y. So y → y `S x → y → y Clearly x →y → y `S y → y. So x → y → y ≈ y → y is an identity of Alg Mod∗ S byself-extensionality and Cor. 1.22.

(iii). Assumeϕ0 ≈ ψ0, . . . , ϕn−1 ≈ ψn−1

ξ ≈ η(55)

is a quasi-identity of Alg Mod∗ S. Then, by Thm. 1.21,

∀ϑ E(ϕ0, ψ0, ϑ), . . . ,∀ϑ E(ϕn−1, ψn−1, ϑ) `S ∀ϑ E(ξ, η, ϑ),

and hence by E(x, y, u)-detachment

∀ϑ E(ϕ0, ψ0, ϑ), . . . ,∀ϑ E(ϕn−1, ψn−1, ϑ), ξ `S η.

By symmetry, ∀ϑ E(ϕ0, ψ0, ϑ), . . . ,∀ϑ E(ϕn−1, ψn−1, ϑ), η `S ξ. Thus, since Sis finitary, there is, for each i < n, a finite subset Fi(ϕi, ψi) of ∀ϑ E(ϕi, ψi, ϑ)such that

F0(ϕ0, ψ0), . . . , Fn−1(ϕn−1, ψn−1), ξ a`S F0(ϕ0, ψ0), . . . , Fn−1(ϕn−1, ψn−1), η.

The conclusion of (iii) is now an immediate consequence of Lem. 2.27. 2

56

For any class K of Λ-algebras, Qv K will denote the quasivariety generated byK.

Theorem 2.29 ([5], Theorem 4.45) Let S be a 1-deductive system over acountable language type. If S is self-extensional with the uniterm deduction-detachment theorem, then Qv Alg Mod∗ S is a variety.

PROOF. To prove Qv Alg Mod∗ S is a variety it suffices to show that ev-ery quasi-identity of Alg Mod∗ S is a logical consequence of the identities ofAlg Mod∗ S. Let (55) be a quasi-identity of Alg Mod∗ S. Let x → y be a sin-gle deduction-detachment formula for S, and let E(x, y, u) be the equivalencesystem with parameters for S that is constructed from x→ y in (54).

By Lem. 2.28(iii), there exists, for each i < n, a finite subset Fi(ϕi, ψi) of∀ϑ E(ϕi, ψi, ϑ) such that

F0(ϕ0, ψ0) → F1(ϕ1, ψ1) → · · · → Fn−1(ϕn−1, ψn−1) → ξ (56)

≈ F0(ϕ0, ψ0) → F1(ϕ1, ψ1) → · · · → Fn−1(ϕn−1, ψn−1) → η

is an identity of Alg Mod∗ S. Recall that each formula in ∀ϑE(ϕi, ψi, ϑ) is of theform ζ(ϕi, ϑ) → ζ(ψi, ϑ) or ζ(ψi, ϑ) → ζ(ϕi, ϑ) with ζ(x, u) ∈ FmΛ(Va \ y)and ϑ ∈ Fm

|u|Λ (see (54)). Let A be any member of the variety generated by

Alg Mod∗ S, and let h: FmΛ → A be an evaluation in A such that h(ϕi) =h(ψi) for all i < n. Then, by Lem. 2.28(i). for every a ∈ A,(

ζA(h(ϕi), h(ϑ)) →A ζA(h(ψi), h(ϑ)))→A a = a and(

ζA(h(ψi), h(ϑ)) →A ζA(h(ϕi), h(ϑ)))→A a = a.

Therefore,

h(ξ)

= FA0 (h(ϕ0), h(ψ0)) →A · · · →A FA

n−1(h(ϕn−1), h(ψn−1)) →A h(ξ)

= FA0 (h(ϕ0), h(ψ0)) →A · · · →A FA

n−1(h(ϕn−1), h(ψn−1)) →A h(η), by (56)

= h(η).

So the quasi-identity (55) is a logical consequence of the identities (56) and(y → y) → x ≈ x of Alg Mod∗ S. 2

In [5] a stronger result is obtained, namely that Alg Mod∗ S itself is a variety,and without the restriction on the cardinality of language type.

As a corollary we have that every algebraizable self-extensional deductive sys-tem with the uniterm deduction-detachment theorem is strongly algebraizable.

57

In particular, every Fregean deductive system with the uniterm deduction-detachment theorem is strongly algebraizable, so we get an alternative proofof the first part of Thm. 2.23. In the case of algebraizable logics the proof ofThm. 2.29 gives an algorithm for actually generating a set of defining identi-ties for the equivalent variety of S from any given presentation of S by axiomsand rules of inference:

Theorem 2.30 Every algebraizable self-extensional deductive system S withthe uniterm deduction-detachment theorem is strongly algebraizable.

Let E(x, y) be a finite equivalence system and K(x) ≈ L(y) a finite set ofdefining equations for S. Let x→ y be a single deduction-detachment formulafor S. Then, for each presentation of S by axioms Ax and inference rules Ru,the variety Alg Mod∗ S is defined by the following identities.

(i) (y → y) → x ≈ x;(ii) K(ϕ) ≈ L(ϕ) for each ϕ ∈ Ax;(iii) K(E(x, y)) → L(E(x, y)) → x ≈ K(E(x, y)) → L(E(x, y)) → y;

(iv) E(K(ψ0), L(ψ0)

)→ · · · → E

(K(ψn−1), L(ψn−1)

)→ K(ϕ)

≈ E(K(ψ0), L(ψ0)

)→ · · · → E

(K(ψn−1), L(ψn−1)

)→ L(ϕ),

for each sequentψ0, . . . , ψn−1

ϕin Ru.

PROOF. (i) is an identity of Alg Mod∗ S by Lem. 2.28. By Thm. 1.13, theidentities (iv) together with the quasi-identities

K(E(x, y)

)≈ L

(E(x, y

)x ≈ y

and

K(ψ0) ≈ L(ψ0), . . . , K(ψn−1) ≈ L(ψn−1)

K(ϕ) ≈ L(ϕ), for each

ψ0, . . . , ψn−1

ϕin Ru,

define Alg Mod∗ S. From the proofs of Lem. 2.28 and Thm. 2.29 we see that, inthe presence of the identity (i), these quasi-equations are interderivable fromthe equations (iii) and (iv), respectively. 2

By Thm. 2.25, every protoalgebraic, Fregean deductive system with conjunc-tion, that has at least one theorem, is strongly, regularly algebraizable. Thisis proved using the fact that each such system has the uniterm deduction-detachment theorem (Thm. 2.21). If the assumption that the system is Fregeanis weakened to self-extensionality, then the uniterm deduction-detachment the-orem need no longer hold and hence this argument cannot be used. But, as weshall see, a conjunction seems to be just as effective as a deduction-detachmentformula in forcing strong algebraizability in the presence of self-extensionality.

58

A conjunction formula will always be denoted by x ∧ y. For every nonemptyΓ ⊆ω FmΛ, we denote by

∧Γ the formula

ψ0 ∧ (ψ1 ∧ (· · · ∧ (ψn−2 ∧ ψn−1) · · · )),

where ψ0, . . . , ψn−1 is a fixed but arbitrary ordering of the formulas of Γ . IfΓ0, . . . , Γn−1 is any finite, nonempty sequence of finite, nonempty subsets ofFmΛ, then ∧

Γ0 ∧∧Γ1 ∧ · · · ∧

∧Γn−2 ∧

∧Γn−1

denotes ∧Γ0 ∧ (

∧Γ1 ∧ (· · · ∧ (

∧Γn−2 ∧

∧Γn−1) · · · )).

Let S be a self-extensional and protoalgebraic 1-deductive system with con-junction x ∧ y. Let Γ,∆ ⊆ω FmΛ. If Γ a`S ∆, then

∧Γ a`S

∧∆, and

hence∧Γ ≡ ∧

∆ (mod Ω(ThmS)) by self-extensionality. Thus∧Γ ≈ ∧

∆is an identity of Alg Mod∗ S by Cor. 1.22, provided the language type of S iscountable.

Lemma 2.31 Let S be a self-extensional and protoalgebraic 1-deductive sys-tem with conjunction and with at least one theorem. Assume in addition thatthe language type of S is countable. Let ∆(x, y) be a protoequivalence sys-tem for S and let E(x, y, u) be the the equivalence system with parametersconstructed from ∆(x, y) in Thm. 1.18. Finally, let x ∧ y be a conjunctionformula for S.

(i)∧F (y, y) ∧ x ≈ x is an identity of Alg Mod∗ S for every F (y, y) ⊆ω

∀ϑ E(y, y, ϑ).

(ii) For every quasi-identityϕ0 ≈ ψ0, ϕ1 ≈ ψ1, . . . , ϕn−1 ≈ ψn−1

ξ ≈ ηof Alg Mod∗ S, there is, for each i < n, a finite subset Fi(ϕi, ψi) of∀ϑ E(ϕi, ψi, ϑ) such that

∧F0(ϕ0, ψ0) ∧ · · · ∧

∧Fn−1(ϕn−1, ψn−1) ∧ ξ≈ ∧

F0(ϕ0, ψ0) ∧ · · · ∧∧Fn−1(ϕn−1, ψn−1) ∧ η

is an identity of Alg Mod∗ S.

PROOF. (i). `S F (y, y) and hence F (y, y), x a`S x. Thus∧F (y, y)∧ x ≈ x

is an identity of Alg Mod∗ S by self-extensionality.

(ii). Ifϕ0 ≈ ψ0, . . . , ϕn−1 ≈ ψn−1

ξ ≈ ηis a quasi-identity of Alg Mod∗ S, then by

59

Thm. 1.21 and E(x, y, u)-detachment,

∀ϑ E(ϕ0, ψ0, ϑ), . . . ,∀ϑ E(ϕn−1, ψn−1, ϑ), ξ

a`S ∀ϑ E(ϕ0, ψ0, ϑ), . . . ,∀ϑ E(ϕn−1, ψn−1, ϑ), η.

Since S is finitary, there is, for each i < n, a finite subset Fi(ϕi, ψi) of∀ϑ E(ϕi, ψi, ϑ) such that

∧F0(ϕ0, ψ0) ∧ · · · ∧

∧Fn−1(ϕn−1 ψn−1) ∧ ξa`S

∧F0(ϕ0, ψ0) ∧ · · · ∧

∧Fn−1(ϕn−1, ψn−1) ∧ η.

Thus the equation in (iii) is an identity of Alg Mod∗ S by self-extensionality. 2

Theorem 2.32 ([5], Theorem 4.27) Let S be a protoalgebraic 1-deductivesystem with at least one theorem over a countable language type. If S is self-extensional with conjunction, then Qv Alg Mod∗ S is a variety.

PROOF. Similar to the proof of Thm. 2.29 with Lem. 2.31 in place ofLem. 2.28. 2

As in the case of self-extensional systems with the uniterm deduction-detach-ment theorem, the stronger result that Alg Mod∗ S is a variety is obtainedin [5, Theorem 4.27], without the restriction on the language type, and alsowithout the requirement that S be protoalgebraic But as before, in the caseS is algebraizable, the proof of Thm. 2.32 gives an algorithm for actuallygenerating a set of defining identities for the equivalent variety of S from anygiven presentation of S by axioms and rules of inference. We omit the details.

2.5 Strong algebraizability and modal logics

By a normal modal logic we will mean a 1-deductive system M over thelanguage type Λ = ∧,∨,¬,→,>, whose set of theorems contains (x→y) → x → y, in addition to all classical tautologies, and is closed undermodus ponens and necessitation, i.e., the following “nonaccumulative” formsof 2nd-order modus ponens and necessitation are rules of M.

∅x ,

∅x→ y

∅y

and

∅x

∅x

.

60

The accumulative forms of these sequents are easily seen to be equivalent re-

spectively to the 1st-order rules of modus ponens and necessitation:x, x→ y

y

andx

x. Note that the defining properties of normal modal logics involve

only the set of theorems and give no information about the consequence rela-tion of the system. Each of the standard modal logics M, thought of in thisway simply as a set of formulas, can be presented in two different ways asa deductive system. The “weak” system Mw has (1st-order) modus ponensas its only proper rule of inference, while the “strong” system Ms has bothmodus ponens and the 1st-order rule of necessitation. The weak system,Mw, isself-extensional and has the uniterm deduction-detachment theorem, with theclassical implication x→ y as the single deduction-detachment formula. ThusQv Alg Mod∗Mw is a variety by Thm. 2.29; in fact, as noted, Alg Mod∗Mw

itself is a variety. Mw is however not in general algebraizable. On the otherhand, the strong system Ms is generally not self-extensional and, while inmany cases it has the multiterm and even the uniterm deduction-detachmentsystem, the classical implication is a deduction-detachment formula only invery special cases. Ms is however always algebraizable. Moreover, we haveAlg Mod∗Ms = Alg Mod∗Mw (this is shown in [5]). So Ms is always stronglyalgebraizable. For a discussion of the algebraizability of the weak and strongsystems for the modal logic S5 of Lewis, see [21]. For more detailed discussionof the algebraizability of modal logics see [5] and the additional referencescited there.

There are other algebraizable deductive systems S that, like the normal modallogics, have an associated weak form Sw that is self-extensional with theuniterm deduction-detachment theorem, or that is protoalgebraic and self-extensional with conjunction, and such that the equivalent quasivariety of Scoincides with Alg Mod∗ Sw. The strong algebraizability of S then follows fromeither [5, Theorem 4.27] or [5, Theorem 4.45]. The first-order predicate logichas this property, after first being reformulated as a deductive system in thesense of this paper; see [22]. Each of the finitely valued Lukasiewicz logics alsohas this property ([43]); quantum logic is another example ([44]). There are anumber of strongly algebraizable logics however that do not fit this paradigm.Among the so-called substructural logics, the relevance logic R is strongly al-gebraizable ([45]) but not self-extensional ([46]); it has a weak version WRthat is self-extensional but not protoalgebraic. Another example of this kindis the infinitely valued Lukasiewicz logic ([44]). The attempt to find a com-prehensive theory explaining the phenomenon of strong algebraizability is anongoing project.

61

3 Matrix Semantics for Fregean Deductive Systems

The notion of a full 2nd-order model of a 1-deductive system was introducedby Font and Jansana in [5]. It turns out to be a very useful device for studyingthe 2nd-order properties of deductive systems, in particular Fregean systems.In this section we show that, if a deductive system is protoalgebraic, any ac-cumulative 2nd-order property, that is a property defined by the accumulativeforms of some set of 2nd-order sequents, is inherited by its full 2nd-order mod-els. Something even stronger than the converse holds: any 2nd-order propertycommon to all members of a family of 2nd-order matrices is inherited by thedeductive system that they define. This latter result gives us a convenientmethod for constructing Fregean systems with various properties. As an ap-plication we construct a regularly algebraizable Fregean deductive system thatis not strongly algebraizable.

We will consider only 1-deductive systems in this section.

Definition 3.1 A 2nd-order matrix A = 〈A, C〉 is said to be a (2nd-order)model of a 1-deductive system S if C ⊆ FiS A, i.e., 〈A, F 〉 ∈ ModS for eachF ∈ C. By a full (2nd-order) model of S we mean a 2nd-order model that isreduction-isomorphic to one of the form 〈A,FiS A〉; full models of the latterkind are called basic full models. 2

A reduced full 2nd-order model of S is always basic. Thus A is a full model ofS iff 〈B,FiS B〉 4 A for some Λ-algebra B.

For the purpose of proving the next theorem we need to consider formula al-gebras over uncountable sets of variable symbols. For each infinite cardinalα, let FmΛ,α be the set of formulas over a set Vaα of α variable symbols,and let FmΛ,α be the corresponding formula algebra. We assume Vaω = Vaand thus FmΛ,ω = FmΛ. The only property of FmΛ,α we use is that it isabsolutely freely generated by Vaα, i.e., for every Λ-algebra A and every map-ping h: Vaα → A, h has a unique extension extension to a homomorphismh∗: FmΛ,α → A.

Lemma 3.2 Let S be 1-deductive system and let α be an infinite cardinal. A2nd-order sequent is valid in FΛ,α = 〈FmΛ,α, FiS FmΛ,α〉 iff it is a 2nd-orderrule of S.

PROOF. Consider any denumerable subset X of Vaα and let FmΛ,α(X) bethe subalgebra of FmΛ,α generated by X. We claim that the submatrix

FmΛ,α(X) =⟨FmΛ,α(X), F ∩ FmΛ,α(X) : F ∈ FiS FmΛ,α

⟩

62

of FmΛ,α(X) is isomorphic to S (as the 2nd-order matrix 〈FmΛ,ThS〉. Clearly〈FmΛ,α(X), FiS FmΛ,α(X)〉 ∼= S and

FiS FmΛ,α(X) ⊇ F ∩ FmΛ,α(X) : F ∈ FiS FmΛ,α .

So we only have to show the inclusion in the opposite direction. Let F ∈FiS FmΛ,α(X) and let r: FmΛ,α FmΛ,α(X) be a retraction, i.e., a surjectivehomomorphism such that r r = r; a retraction exists because FmΛ,α isabsolutely freely generated Vaα. Thus F ⊆ r−1(F ) and hence F = r−1(F ) ∩FmΛ,α(X). Since r−1(F ) ∈ FiS FmΛ,α, this establishes the claim. It followsimmediately that a 2nd-order sequent is valid in FmΛ,α(X) iff it is a rule of S.

Consider any finitely generated submatrix B of FmΛ,α. Let X be a denumer-able subset of Vaα that includes all the variables occurring in the generatorsof B. Then B ≤ FmΛ,α(X) ≤ FmΛ,α and FmΛ,α(X) is countably generated.Furthermore, a 2nd-order sequent is valid in FmΛ,α(X) iff it is a rule of S. Thusby Lem. 2.4 the 2nd-order sequents valid in FmΛ,α are exactly the 2nd-orderrules of S. 2

Let P be a 2nd-order property of 2nd-order matrices, i.e., a class of 2nd-ordermatrices that is defined by some set of 2nd-order sequents. For instance, P canbe the property of having the multiterm deduction-detachment theorem with afixed deduction-detachment system, or the properties of being self-extensionalor Fregean. Recall that P is said to be accumulative if it is defined by theaccumulative forms of a set of 2nd-order sequents. The deduction-detachmentand Fregean properties are accumulative; self-extensionality is not. A 2nd-order property P of a 1-deductive system S is said to transfer to full modelsif every full 2nd-order model of S has P.

Theorem 3.3 Let P be any accumulative 2nd-order property, and let S be a1-deductive system with P. If S is protoalgebraic, then P transfers to the fullmodels of S.

PROOF. To prove that every full model of S has P it suffices by Cor. 2.9to prove that every basic full model of S has P. Take 〈A,FiS A〉 to be sucha model. Let α be the least infinite cardinal greater or equal to |A|, and leth: FmΛ,α A be a surjective homomorphism. Since S is protoalgebraic, wehave by Lem. 1.15 that

h−1(FiS A) = [h−1(F0) )FiS FmΛ,α,

were F0 =⋂

FiS A, the smallest S-filter on A. Thus, by Thm. 2.3, 〈A,FiA S〉 ∈P iff

⟨FmΛ,α, [h−1(F0) )FiS FmΛ,α

⟩∈ P. But 〈FmΛ,α, FiS FmΛ,α〉 ∈ P by

Lem. 3.2 and the assumption S = 〈FmΛ,α,ThS〉 ∈ P. Hence we have⟨FmΛ,α,

63

[h−1(F0) )FiS FmΛ,α

⟩∈ P because P is defined by the accumulative forms of a

set of 2nd-order sequents (see the remarks following Lem. 2.14). 2

Corollary 3.4 ([36], Theorem 2.2) The property of having the multiterm(uniterm) deduction-detachment theorem transfers to full models. More pre-cisely, if ∆(x, y) is a deduction-detachment system for a 1-deductive systemS, then it is also a deduction-detachment system for every full model of S.

PROOF. Assume ∆(x, y) is a deduction-detachment system for S. Then byCor. 1.39 S is protoalgebraic and ∆(x, y) is a protoequivalence system for S.Then, since the multiterm deduction-detachment theorem is an accumulative2nd-order property, the conclusion of the corollary follows immediately fromThm. 3.3. 2

The following corollary is an improvement of Proposition 3.19 of [5].

Corollary 3.5 For protoalgebraic 1-deductive systems, the property of beingFregean transfers to full models. 2

For some time it was an open problem if the properties of being Fregeanand self-extensional transfer to full models in general. But this has recentlybeen shown to fail in both cases by Babyonyshev [47] and, independently, byBou [48].

We now turn to what can be viewed as the reverse problem, that is, inferringa 2nd-order property of a deductive system from the assumption that a classof 2nd-order models that defines the system has the property.

Every class of 2nd-order matrices defines a deductive system in the naturalway.

Definition 3.6 Let K be any class of 2nd-order matrices. We denote byS K the largest 1-deductive system (under set-theoretic inclusion of algebraicclosed-set systems on FmΛ) such that each matrix in K is a 2nd-order modelof S K.

Alternatively, S K is defined by the condition that, for all Γ ∪ ϕ ⊆ FmΛ,Γ `S K ϕ iff there are finitely many ψ0, . . . , ψn−1 ∈ Γ such that, for every〈A, C〉 ∈ K and every evaluation h: FmΛ → A,

h(ϕ) ∈ CloCh(ψ0), . . . , h(ψn−1)

.

This is the same finitary deductive system that is determined in the usual wayby the class of 1st-order matrices 〈A, F 〉 : 〈A, C〉 ∈ K, F ∈ C

. S K is said

64

to be finitarily determined by K. 2

Theorem 3.7 Let P be any 2nd-order property, and let K be any class of2nd-order matrices. If K ⊆ P then S K ∈ P.

PROOF. Suppose that the 2nd-order sequent (30) is valid in each matrix ofK. Let σ : FmΛ → FmΛ be a substitution such that σ(ϑ0), . . . , σ(ϑk−1) 0S K

σ ξ. Then there exists a 〈A, C〉 ∈ K and an evaluation h: FmΛ → A such

that h(σ(ξ)) 6∈ CloCh(σ(ϑ0)), . . . , h(σ(ϑk−1))

. Since (30) is valid in 〈A, C〉,

for some i < m, h(σ(ϕi)) 6∈ CloCh(σ(ψi

0)), . . . , h(σ(ψini−1))

. Thus, for some

i < m, σ(ψi)0, . . . , σ(ψini−1) 0S K σ(ϕi). Hence the sequent (30) is valid in

S K. 2

Corollary 3.8 Let K be any class of 2nd-order matrices over Λ. If each mem-ber of K is Fregean, so is S K. 2

Similarly, if each member of a class K of 2nd-order matrices is self-extensional,then so is S K. A closely related result can be found in [3], Theorem 5.6.11.

This corollary can be used to construct examples of Fregean deductive sys-tems with various properties. We will use it now to construct a protoalgebraicFregean deductive system that is not strongly algebraizable.

Some special terminology will prove to be useful. Let C be a closed-set systemover a nonempty set A. Let F ∈ C. Elements a, b ∈ A are said to be C-separableover F if there exists a G such that F ⊆ G ∈ C and either a ∈ G and b 6∈ G or

vice versa. Clearly a and b are C-separable over F iff 〈a, b〉 6∈ ΛC F . Thus a 2nd-order matrix A = 〈A, C〉 is Fregean iff, for every F ∈ C, a 6≡ b (mod ΩA F )implies a and b are C-separable over F .

Let⟨0, 1, a, b, +, ·,−, 0, 1

⟩be the 4-element Boolean algebra. Let Λ = →

,>, a,b, where a,b are constant symbols, and let A be the Λ-algebra

A =⟨1, a, b,→A,>A, aA,bA

⟩,

where x →A y = −x + y for all x, y ∈ 1, a, b, >A = 1, aA = a, and

bA = b. We note that the →,>-reduct of A,⟨1, a, b,→A,>A

⟩, is the

→,>-subreduct of a Boolean algebra and hence is a Hilbert algebra (in facta Tarski algebra). In the sequel we omit the superscript on →A. Note that1 → x = x and x → 1 = x → x = 1, for all x ∈ 1, a, b. These threeequalities completely describe the multiplication table for → with the twoexceptions a→ b and its converse. a→ b = b and b→ a = a.

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Let A be the 2nd-order matrix 〈A, C〉, where

C :=1, 1, b, 1, a, b

.

C is obviously an (algebraic) closed-set system over 1, a, b.

Lemma 3.9 A is Fregean.

PROOF. We first determine the Leibniz congruence for each of the threefilters of C. Since ΩA1 is compatible with 1, if it is not the identity relation,a ≡ b (mod ΩA1). But then 1 = a → a ≡ a → b = b (mod ΩA1),contradicting compatibility with 1. So ΩA1 = IdA.

Let Φ be the equivalence relation whose partition is1, b, a

. We verify

that Φ is a congruence. We first observe that b→ x = x and x→ b = b for allx 6= b. If x 6= b, then

1 → x = b→ x = x and x→ 1 = 1 ≡Φ b = x→ b.

If x = b, then

1 → x = b ≡Φ 1 = b→ x and x→ 1 = x→ b = 1.

So Φ is a congruence and is clearly the largest congruence compatible with1, b. So ΩA1, b = Φ. Finally, it is obvious that ΩA1, a, b = A2, theuniversal relation. Thus

ΩA1 = ΩA1 ∩ΩA1, b ∩ΩA1, a, b = IdA,

ΩA1, a = ΩA1, a ∩ΩA1, a, b = Φ,

ΩA1, a, b = ΩA1, a, b = A2.

The pairs 〈1, a〉, 〈1, b〉, and 〈a, b〉 are all clearly C-separable over 1. 〈1, a〉 isC-separable over 1, b since a 6∈ 1, b. Finally, that each pair 〈x, y〉 such thatx 6≡ y (mod ΩA1, a, b) is C-separable over 1, a, b holds vacuously. Thus A

is Fregean. 2

It follows from Cor. 3.8 that the deductive system S A (i.e., SA) (finitar-ily) determined by A is Fregean. S A is also clearly protoalgebraic withsingle protoequivalence formula x → y. So S A is regularly algebraizable byThm. 2.18. Let Q = Alg Mod∗ S A, the equivalent quasivariety of S A. Q isrelatively point-regular and S A is its assertional logic by Thm. 1.34. (See alsothe remark following Thm. 1.34.)

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It is a trivial matter to verify that the 1st-order sequenta

bis valid in A, i.e.,

in each of the three 1st-order matrices 〈A, 1〉, 〈A, 1, b〉, and 〈A, 1, a, b〉.Thus the quasi-equation

a ≈ >b ≈ >

(57)

is a quasi-identity of Q. 〈A, 1〉 is reduced, so A ∈ Q. Let Θ be the equiva-

lence relation on A whose partition is1, a, b

. Θ is a congruence on A.

(It is the image of the congruence Φ with partition1, b, a

under the

automorphism of the →,>-reduct of A that interchanges a and b.) But thequasi-equation (57) is not valid in A/Θ, so A/Θ 6∈ Q. Hence Q is not a varietyand thus S A is a regularly algebraizable, Fregean deductive system that isnot strongly algebraizable.

The first example of a regularly algebraizable, Fregean deductive system thatis not strongly algebraizable was found by P. Idziak; the example just pre-sented is essentially Idziak’s and is included here with his kind permission.Subsequently it was discovered that the equivalence/negation fragment of theintuitionistic propositional calculus also has these properties; this result canbe extracted without difficulty from Kabzinski, Porebska, and Wronski [49].We briefly outline now how this is done.

Let IPC↔¬ be the ↔,¬-fragment of the intuitionistic propositional calculusIPC. IPC↔¬ clearly inherits the property of being Fregean from IPC, andhence it is algebraizable since it is evidently protoalgebraic. A presentation ofIPC↔¬ is obtained from a presentation of the ↔-fragment IPC↔ of IPC byadjoining two new axioms and the following rule of inference (see [49, Theorem2]).

¬x↔ x

y. (58)

Let IPC−↔¬ be the deductive system obtained by just adjoining the two axioms

but not the rule (58). Then IPC−↔¬ is an axiomatic extension of IPC↔ and

hence is strongly algebraizable because IPC↔ is. IPC↔¬ is not an axiomaticextension of IPC−

↔¬, i.e., the rule (58) cannot be replaced by any set of ax-ioms (this follows easily from [49, Lemma 7]). Hence IPC↔¬ is not stronglyalgebraizable.

We may conclude that the behavior of protoalgebraic, Fregean deductive sys-tems that fail to have the uniterm deduction-detachment theorem differs strik-ingly from those that do. A protoalgebraic, Fregean deductive system may beeither strongly algebraizable or not, but the uniterm deduction-detachmenttheorem guarantees strong algebraizability in this context by Thm. 2.23. Thefact that the deduction-detachment system is uniterm is essential. An exam-ple of a Fregean deductive system with the multiterm deduction-detachmenttheorem that is not strongly algebraizable is given in [6].

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3.1 Fregean quasivarieties

In the last part of this section we investigate some of the properties of theequivalent quasivarieties of protoalgebraic Fregean deductive systems.

Definition 3.10 ([19]) A pointed quasivariety Q is congruence-orderable if,for every A ∈ Q and all a, b ∈ A,

CgAQ (a,>A) = CgA

Q (b,>A) implies a = b. (59)

Q is Fregean if it is both relatively point-regular and relatively congruence-orderable. 2

Let Q be any pointed quasivariety. Then the condition

a ≤AQ b if CgA

Q (a,>A) ⊇ CgAQ (b,>A)

obviously defines a quasi-ordering on any Λ-algebra A. It is called the Q-congruence order on A. Q is congruence-orderable iff its congruence order isa partial order.

Theorem 3.11 A pointed quasivariety is Fregean iff its assertional logic isprotoalgebraic and Fregean, with at least one theorem.

PROOF. Let S = Sasl Q. Assume S is protoalgebraic and Fregean, with atleast one theorem. Then S is regularly algebraizable by Thm. 2.18, and henceQ is relatively point-regular by Thm. 1.34. Let A ∈ Q. Then by Thm. 1.37,>A ∈ FiS A and ΩA>A = IdA. Let a, b ∈ A and suppose

CgAQ (a,>A) = CgA

Q (b,>A). (60)

By the correspondence between S-filters and >A-congruence classes, (60) isequivalent to the condition that 〈a, b〉 ∈ Λ(FiS A). Since S is protoalgebraicand Fregean, the 2nd-order matrix 〈A, FiS A〉 is also Fregean by Cor. 3.5.Thus Λ(FiS A) = ΩA(FiS A) = ΩA>A = IdA. So a = b, and hence Q iscongruence-orderable.

Conversely, assume that Q is relatively point-regular and congruence-orderable,i.e., that Λ(FiS A) = IdA for every A ∈ Q. Consider any A ∈ Q. Then

Λ(FiS A/ΩA F ) = IdA/ΩA F for each F ∈ FiS A. So by Lem. 2.7, ΛFiS A F =ΩA F for every F ∈ FiS A. This shows that 〈A, FiS A〉 is a Fregean 2nd-ordermatrix for every A ∈ Q. Since S is clearly the deductive system (finitarily)determined by this class of 2nd-order matrices, we conclude by Cor. 3.8 thatS is Fregean. 2

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A pointed variety is defined in [16] to be Fregean if it is congruence-orderable.In light of this theorem it seems more appropriate to reserve the term Fregeanfor congruence-orderable quasivarieties that are relatively point-regular.

A nontrivial member of a quasivariety Q is said to be relatively subdirectlyirreducible if it cannot be isomorphically represented as a subdirect productof any system of members of Q unless it is isomorphic to at least one of thefactors. It can be shown that every algebra in Q is isomorphic to a subdirectproduct of a system of relatively subdirectly irreducible members of Q, andthat A ∈ Q is relatively subdirectly irreducible iff the set of all nonidentityQ-congruences of A has a smallest member. The following characterization ofthe relatively subdirectly irreducible members of a Fregean quasivariety wasobtained independently in [29] and [19].

Theorem 3.12 Let Q be a Fregean quasivariety, and let A ∈ Q. Then A isrelatively subdirectly irreducible iff there exists an element ? ∈ A \ >A suchthat a ≤A

Q ? for all a ∈ A \ >A.

PROOF. Suppose A ∈ Q is relatively subdirectly irreducible. There ex-ists a smallest nontrivial Q-congruence Θ on A. Since Q is relatively point-regular, we have ? ≡ >A (mod Θ) for some element ? of A \ >A. ClearlyΘ = CgA

Q (?,>A). Then, for every a ∈ A \ >A, we have CgAQ (a,>A) ⊇

CgAQ (?,>A). So a ≤A

Q ?.

Conversely, suppose ? is the largest element ofA\>A under the Q-congruenceorder, i.e., CgA

Q (a,>A) ⊇ CgAQ (?,>A) for every a ∈ A\>A. Then CgA

Q (?,>A)is the smallest nontrivial congruence of A. 2

The algebras of Q which satisfy the condition of Thm. 3.12 are customarilycalled strongly compact in the sense of Q.

Corollary 3.13 Let Q be a relatively point-regular and congruence-orderablequasivariety, and let A ∈ Q be strongly compact (in the sense of Q). Then the(Sasl Q)-filter on A generated by ? is ?,>A. 2

Thm 3.12 shows that the familiar and useful characterization of subdirectlyirreducible Heyting algebras, in terms of their natural order, derives fromthe fact that the intuitionistic propositional calculus, the assertional logic ofHeyting algebras, is Fregean.

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