L-algebras and three main non-classical logics
Introduction
In a 2005 paper [38], Siegfried Gottwald raised the question to find a class of algebras that unifies the semantics of three main generalizations of classical logic, to grasp the essence of a major part of algebraic logic. The three types of non-classical logic with their semantics are given by the following table (see [38], Section 1): Heyting algebras and MV-algebras can be unified as classes of residuated lattices [32], [31], [33]. The main drawback that orthomodular lattices resisted such a unification came from the special nature of quantum logic. Gottwald [38] speaks of the “isolated position [of quantum logic] in the whole system of non-classical logics”.
In this note, we will show that nevertheless, there exists a perfectly well-suited overarching system for all three types of non-classical logic, and even a very simple one. The relevant concept is that of an L-algebra [62], given by a set X with a single binary operation → such that the following are satisfied. Firstly, there is a logical unit, that is, an element 1 with for all . Note that a logical unit must be unique. Secondly, X satisfies the equation and thirdly, holds for . The operation → stands for logical implication.
Beyond algebraic logic, the basic equation (2) of an L-algebra occurred in the theory of Garside groups [2], [9], [16], [17], and in connection with set-theoretic solutions to the quantum Yang-Baxter equation [25], [50], [61]. In algebraic logic, it can be found much earlier in [8] and [39] (Eq. (29) of Theorem 2.6), but it was never seriously pursued, and remained almost completely neglected.
Information on L-algebras, their structure groups and applications, can be drawn from [62], [63], [67], [68], [70]. For a very brief introduction, see Section 1.
We will show that there are simple conditions which specialize an L-algebra to the three above mentioned systems of non-classical logic as well as to three intermediate, more general systems. More precisely, we establish an inclusion diagram
for classes of L-algebras which faithfully represents intersections (Theorem 1). Thus, for example, orthomodular MV-algebras () are exactly the Boolean algebras (Bool). Instead of Heyting algebras, we also consider the more general (bounded) Hilbert algebras () where the lattice property is not assumed. The diagram contains three further classes of algebras: as the class of Glivenko algebras (see [64]; cf. [12]) where double negation is an idempotent endomorphism onto an MV-algebra, and as the class of lattice effect algebras [29] which characterize unsharp quantum logic. The subalgebra of sharp elements is an orthomodular lattice [59], [69]. Finally, there is a new class in the diagram, ortho-Hilbert algebras as a quantum version of Hilbert algebras. Here the double negation is an endomorphism onto an orthomodular lattice.
The diagram brings to notice that the three basic logical algebras generate three “coordinate directions” of non-classical logic:
The up-right direction signifies quantization: orthomodular lattices as quantized Boolean algebras, and lattice effect algebras as quantized MV-algebras. The up direction points to “unsharp” logic: MV-algebras as unsharp (=many-valued) Boolean algebras, Glivenko algebras as unsharp Hilbert algebras, and lattice effect algebras as unsharp orthomodular lattices. Finally, there is an up-left direction which features non-involutive double negation.
According to the three generalizing directions, their inverse directions correspond to specializing conditions for an L-algebra. So the inverse of the up-left direction imposes the condition that double negation is involutive. The up direction is inverted by the sharpness condition , and the up-right direction (quantization) is inverted by the commuting condition which characterizes the four classical algebras of the diagram. In complete symmetry to the three basic classes of algebras () which arise from generalizing Boolean algebras, the three specializing directions also belong to three classes of algebras, , , and the new class of ortho-Hilbert algebras. As quantized Hilbert algebras, ortho-Hilbert algebras suggest an alternative approach to non-commutative topology [1], [35], [60]. If Heyting algebras are preferred instead of Hilbert algebras, the diagram has to be modified by assuming all algebras to be lattice-ordered.
At this point, a basic fact on L-algebras must be recalled. Any L-algebra X embeds into a bigger L-algebra with a compatible monoid and ∧-semilattice structure which is self-similar (see Section 1). Every self-similar L-algebra X satisfies . Moreover, admits a left group of fractions , the structure group of X, equipped with a natural map . The structure group can be viewed as a universal group of X, and in favourite cases, the map is an injection.
All of the mentioned algebras in the above diagram are obtained as specializations of a new class of L-algebras which we call implicative (Definition 2). For an implicative L-algebra X, the image of is an L-algebra , and the induced surjection coincides with the double negation which is an idempotent endomorphism of X (Theorem 4). Some versions of Glivenko's theorem [37], [30], [12], [31], [64] are special cases. Moreover, the structure group of X can be identified with the structure group of . The implicative L-algebras of the form are characterized as bounded L-algebras (i.e. with a smallest element 0) satisfying the single condition The structure group of these L-algebras X is a two-sided group of fractions, and X is the interval in (Theorem 6). A first theorem of this type was proved by Mundici [56] for abelian lattice-ordered groups. More generally, we characterize the groups arising as two-sided groups of fractions of an L-algebra as groups with an upper directed ∧-semilattice structure, invariant under right multiplication (Theorem 5).
Section snippets
Implicative L-algebras
In this section, we exhibit a class of L-algebras which specializes to the logical systems mentioned above. For the reader's convenience, we briefly recall some basic properties of L-algebras [62], as far as needed here. It will turn out that the single operation (implication) of an L-algebra suffices to generate the other basic logical operations and relations. Besides the implicational operation, there is an entailment relation which endows each L-algebra X with a partial order. By
Heyting algebras
Brouwer's intuitionistic logic was formalized by A. Heyting [42], which led to the concept of Heyting algebra [54]. Every Heyting algebra is a Brouwerian semilattice, that is, a ∧-semilattice with a greatest element 1 and a binary operation → satisfying More generally, a Hilbert algebra [55], [18] is a set X with a binary operation → and a constant 1 satisfying the equations Proposition 3 Every Hilbert algebra is an L-algebra. An
MV-algebras
The semantics of Łukasiewicz' infinite-valued propositional logic was first studied by Chang [10], [11] who introduced the concept of MV-algebra. Measure and integration in its widest scope can be formalized efficiently in terms of MV-algebras [71]. Gispert and Mundici [36] characterized MV-algebras as commutative monoids with an involution (the negation) such that satisfies , and holds for all . Proposition 5 With , every MV-algebra is an implicative
The L-algebras of quantum logic
Quantum logic was introduced by Birkhoff and von Neumann [7]. In their Hilbert space model, the main difference to classical logic is that propositions are represented by closed subspaces of a Hilbert space instead of subsets of a set. The orthogonal complement then stands for the negation.
The collection of closed subspaces of a Hilbert space is an orthomodular lattice [49], a structure which is widely considered as the proper semantics of quantum logic. Recall that a lattice X is said to be
Ortho-Heyting algebras
There are important algebras which generalize two of the three logical systems studied above. They fit into the following specialization diagram for the category of implicative L-algebras where stands for lattice effect algebras [29], [59], and denotes the category of Glivenko algebras [64]. The class of algebras has not been studied before. We will call them ortho-Hilbert algebras. In the lower part of the diagram, , , , and denotes respectively the full
The structure group of an L-algebra
In this section, we recall some structures associated to any L-algebra X, which are very useful and indispensable for many applications. In particular, X has an associated group which can be obtained in two steps. Namely, there is an embedding into a self-similar L-algebra with a compatible monoid structure, which admits a group of left fractions, the structure group of X. More precisely, an L-algebra X is said to be self-similar [62] if for each the map is a
Application to implicative L-algebras
Now we return to the algebras of the diagram (27), using the results of Section 6. For Brouwerian semilattices, the following definition from [64] is due to Köhler [48]. Definition 6 A sequence of morphisms between L-algebras is said to be short exact if v is surjective with kernel u. The sequence is said to be strongly split if in addition, there is a morphism with and for all .
The following result provides a general Glivenko-type theorem (cf. [37], [30], [12], [31], [64]). Theorem 4
Special implicative L-algebras
In Section 7, we applied the results of Section 6 to general implicative L-algebras. Now we turn our attention to the special categories of diagram (27). Let us start with the case of Boolean algebras. Recall that an element of an ℓ-group G is said to be singular [5], [15] if with implies that . The ℓ-group G is said to be archimedean [15] if with for all implies that . Every archimedean ℓ-group is commutative ([15], Theorem 53.3). By [62], Theorem 5, we
Concluding remarks
Following an anonymous referee's suggestion, we add some further remarks on the L-algebras considered here as well as on L-algebras in general. We have seen that L-algebras are defined by a single operation → which can be viewed as a very general kind of implication, inducing the entailment relation (4) and the logical unit 1. When passing to the self-similar closure, new operations arise: a classical (commutative) conjunction ∧ and a non-classical one, satisfying equations (33)-(38), some of
Acknowledgement
We are grateful to an anonymous referee for spotting inaccuracies and suggesting a reorganization of the paper to improve its readability. In a previous version, we immediately went to the self-similar closure and the structure group after defining L-algebras. The idea was to prove the embedding theorems (Theorem 4, Theorem 6) as early as possible and apply them to the logical algebras. Now we show first that the algebras in the diagram (27) are implicative L-algebras and then pass to the
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