grishin algebras and cover systems for classical bilinear...

Robert Goldblatt Grishin Algebras and CoverSystems for ClassicalBilinear Logic

Dedicated to Ryszard Wojcicki on his 80th birthday.

Abstract. Grishin algebras are a generalisation of Boolean algebras that provide alge-

braic models for classical bilinear logic with two mutually cancelling negation connectives.

We show how to build complete Grishin algebras as algebras of certain subsets (“proposi-

tions”) of cover systems that use an orthogonality relation to interpret the negations.

The variety of Grishin algebras is shown to be closed under MacNeille completion, and

this is applied to embed an arbitrary Grishin algebra into the algebra of all propositions

of some cover system, by a map that preserves all existing joins and meets.

This representation is then used to give a cover system semantics for a version of

classical bilinear logic that has first-order quantifiers and infinitary conjunctions and dis-

junctions.

Keywords: Grishin algebra, bilinear logic, residuated lattice-ordered monoid, quantale,

cover system, orthogonality relation, Kripke-Joyal semantics, MacNeille completion.

1. Introduction

Grishin algebras were defined by Lambek [14, 15] as “a generalisation ofBoolean algebras which do not obey Gentzen’s three structural rules”. Themotivation was to study algebraic models for classical bilinear propositionallogic, described as “a non-commutative version of linear logic which allowstwo negations”. Such models were first considered by V. N. Grishin [13].

A Grishin algebra is a residuated lattice-ordered monoid. Its monoidoperation ⊗ is thought of as a non-commutative conjunction, whose leftand right residuals are viewed as implication operations, which we denoteby ⇒l and ⇒r. It has two negation/complementation operations that wedenote −l and −r, and which are mutually cancelling, i.e. satisfy double-negation elimination, in the sense that the equations −l−r a = a = −r −l aare satisfied. There is a distinguished element 0 such that −l a = a⇒l 0 and−r a = a⇒r 0. Also, there is a definable operation ⊕, a kind of De Morgan

Presented by ; Received 6 December 2010

Studia Logica (2011) 0: 1–25 c©Springer 2011

2 Robert Goldblatt

dual of ⊗, that has 0 as an identity element, and has

a⊕ b = −l(−r b⊗−r a) = −r(−l b⊗−l a).

In this paper we introduce new, fully representative, examples of theabstract and equationally defined notion of a Grishin algebra. These con-crete examples are based on cover systems, motivated by topological ideasunderlying the Kripke-Joyal intuitionistic semantics from topos theory. Acover system assigns to each point certain sets of points called “covers” in away that it formally similar to the neighbourhood semantics of modal log-ics. Covers are used to give a non-classical interpretation of disjunction andexistential quantification (see [16, 2]).

Topologically motivated cover systems have a certain algebra of subsetsthat forms a Heyting algebra, whereas a Grishin algebra need not even bedistributive as a lattice. So to construct Grishin algebras we need to adoptweaker properties than those defining the cover systems of Kripke-Joyal se-mantics (see Remark 3.6). In an earlier paper [10] we used such structuresto provide a semantics for the logic of residuated partially-ordered monoidswithout negation operations. This involved combining cover systems withcertain models, based on ordered groupoids (S,4, ·), that have been de-veloped for various connectives in substructural logic [22, 20, 4, 5]. Thesegroupoid models allow a non-commutative conjunction connective &, corre-sponding to ⊗, to be given the semantics

x |= ϕ&ψ iff ∃y ∃z : y · z 4 x and y |= ϕ and z |= ψ.

Here we extend these ideas to model two negation connectives, denoted ¬land ¬r and corresponding to −l and −r, by introducing a binary relation ⊥on the points of a model, and requiring that

x |= ¬lϕ iff ∀y, y |= ϕ implies x ⊥ y;

x |= ¬rϕ iff ∀y, y |= ϕ implies y ⊥ x.

We can think of ⊥ as being a relation of orthogonality or incompatibilitybetween points of S. This kind of modelling of negation was developedby the author for the logic of ortholattices in [8], a context in which ⊥ issymmetric and the two negations collapse to one. Here we assume that ourmodels have a distinguished subset 0, and define x ⊥ y to mean that x·y ∈ 0,an approach that is used in the phase space semantics of linear logic [7].1

1Cover systems with a symmetric orthogonality relation are used in [12, Chapter 6] to

Grishin Algebras and Cover Systems for Classical Bilinear Logic 3

In what follows, we introduce Grishin algebras, consider various alterna-tive definitions, and discuss their relation to quantales. We then develop anotion of a strong classical residuated cover system, and prove that certainsubsets of such a system, which we call “propositions”, form a complete Gr-ishin algebra (and a unital quantale). It is then shown, conversely, that anycomplete Grishin algebra is isomorphic to the algebra of all propositions ofsome strong classical residuated cover system (Theorem 4.3). This repre-senting cover system is defined by an abstract version of the cover system ofa topological space (see Examples 3.1 and Theorem 3.8).

Our main result is that every Grishin algebra has an isomorphic embed-ding into the algebra of all propositions of some strong classical residuatedcover system, by a map that preserves all existing joins and meets (Theo-rem 6.3). This representation makes use of the celebrated completion of apartially-ordered set due to MacNeille, and involves showing that the varietyof Grishin algebras is closed under MacNeille completion (Theorem 6.2).

Preservation of joins and meets is a powerful tool, since these operationscan be used to interpret the quantifiers ∃ and ∀, as well as the standardlattice-modelled disjunctions

∨and conjunctions

∧. In the final section, we

use our representation of Grishin algebras to obtain a sound and completesemantics over cover systems for a version of classical bilinear predicatelogic that has these first-order quantifiers as well as infinitary disjunctionsand conjunctions.

We will see that the notion of Grishin algebra is equivalent to Ono’snotion of classical FL-algebra [20]. Thus some observations about Grishinalgebras may correspond to, or be implicit in, existing literature on thealgebraic logic of residuated lattices [6].

2. Residuated Posets and Grishin Algebras

We begin with some terminology about partially ordered algebras. Givena poset (L,v), comprising a partial ordering v on a set L, we write

⊔X

for the join (= least upper bound), anddX for the meet (= greatest lower

bound), of a set X ⊆ L, when these bounds exist. The smaller symbols tand u are used for the binary join and meet operations. A poset is order-complete, or just complete, if every subset has a join, or equivalently if every

give a new semantics for relevant logic, a logic that has distribution of conjunction overdisjunction. For further background and discussion of the cover semantics methodology seealso [11], which develops interpretations of intuitionistic modal logic giving the diamondmodality ♦ its standard Kripkean semantics, without validating distribution of ♦ overdisjunction.

4 Robert Goldblatt

subset has a meet.A posemigroup (L,v,⊗) has an associative binary operation ⊗ that is

monotone (i.e. order preserving) in each argument, meaning that b v cimplies a⊗ b v a⊗ c and b⊗ a v c⊗ a. This is a pomonoid if in addition⊗ has a unit element, i.e. an element 1 such that a⊗ 1 = 1⊗ a = a. Analgebra with such a unit is called unital.

A posemigroup is residuated if there are binary operations ⇒l and ⇒r

on L,2 called the left and right residuals of ⊗, satisfying

a v b⇒l c iff a⊗ b v c iff b v a⇒r c. (2.1)

Thus (a ⇒l b)⊗ a v b and a⊗(a ⇒r b) v b. The two residual operationsare monotone in their right arguments and antitone in their left, i.e. if b v c,then a ⇒l b v a ⇒l c and c ⇒l a v b ⇒l a; and similarly for ⇒r. Theseresiduals are identical precisely when ⊗ is commutative.

In any residuated posemigroup, the following equations hold wheneverthe joins they refer to exist:

(⊔X)⊗ a =

⊔x∈X

(x⊗ a) (2.2)

a⊗(⊔X) =

⊔x∈X

(a⊗x). (2.3)

A quantale is a complete poset with an associative ⊗ such that these lastequations hold for every set X ⊆ L. Thus a complete residuated posemi-group is a quantale. The converse is also true: every quantale is residuatedwith a⇒l b =

⊔{x : x⊗ a ≤ b} and a⇒r b =

⊔{x : a⊗x ≤ b}.

There is a standard theory of certain closure operators on quantales thatwe make significant use of. Recall that a closure operator on a poset is aunary function j that is monotone: a v b implies ja v jb; inflationary :a v ja; and idempotent : jja = ja. An element a is called j-closed if ja = a,i.e. if a is a fixed point under j. If L is a complete poset, then the set Lj

of j-closed elements is closed under meetsdX, and so is order-complete

under the same partial ordering. The join operation⊔j in Lj is given by⊔j X = j(

⊔X).

A quantic nucleus is a closure operator on a posemigroup that satisfies

ja⊗ jb v j(a⊗ b). (2.4)

Such an operator has j(a⊗ b) = j(a⊗ jb) = j(ja⊗ b) = j(ja⊗ jb), as wellas j(a ⇒l b) v a ⇒l jb = ja ⇒l jb, and likewise with ⇒r in place of ⇒l.

2Notation: in the literature on residuation, a⇒l b is often written as b/a, and a⇒r bas a\b.


Using such facts, it can be shown that if j is a quantic nucleus on a quantale(L,v,⊗), then the complete poset (Lj ,v) of j-closed elements is a quantaleunder the operation a⊗j b = j(a⊗ b) [19, Theorem 2.1]. Moreover Lj isclosed under the residuals of ⊗, and indeed both a⇒l b and a⇒r b belongto Lj whenever b ∈ Lj [23, Prop. 3.1.2]. From this it can be shown that theresiduals of ⊗j on Lj are just the restrictions of the residuals of ⊗ on L toLj . Thus for a, b, c ∈ Lj ,

a v b⇒l c iff j(a⊗ b) v c iff b v a⇒l c. (2.5)

In addition, if 1 is a unit for ⊗ in L, then j1 is a unit for ⊗j in Lj , sincej(a⊗ j1) = a = j(j1⊗ a) for all a ∈ Lj .

The quantale (Lj ,v,⊗j) is called a quantic quotient of (L,v,⊗) [23,p. 32]. This construction will be applied in the next section (see Theorem3.5).

A Grishin algebra is defined in [14] to be an algebra of the form

L = (L,v,t,u,T,F,⊗, 1,−l,−r, 0),

such that:

• (L,t,u,T,F) is a lattice under the partial ordering v, with greatestelement T and least element F.

• (L,v,⊗, 1) is a pomonoid with identity element 1.

• −l and −r are unary operations,3 and 0 an element of L, satisfying theequations

−l−r a = a = −r −l a, (2.6)

and the conditions

a v b iff a⊗−r b v 0 iff −l b⊗ a v 0. (2.7)

We refer to (2.6) as the law of double-negation elimination (DNE).

By a residuated lattice-ordered monoid we mean a residuated pomonoidthat is a lattice under its partial ordering, with greatest and least elements.It can be shown that the notion of a residuated lattice-ordered monoid isdefinable by equations (see e.g. [6, p. 94]). Lambek gave the following alter-native characterisation of Grishin algebras, which shows that they too areequationally definable:

3Notation: in [14, 15], −r a is written a⊥ and −l a is written aT.

6 Robert Goldblatt

Theorem 2.1. [14, Proposition 2.1]

A Grishin algebra may be described as a residuated lattice-ordered monoidwith operations −l and −r satisfying the equations (2.6) and

−l 1 = −r 1, a⇒l b = −l(a⊗−r b), a⇒r b = −r(−l b⊗ a).

Proof. (Sketch.) Any Grishin algebra has −l 1 = 0 and −r 1 = 0, and isresiduated when a ⇒l b is defined to be −l(a⊗−r b) and a ⇒r b is definedas −r(−l b⊗ a).

Conversely, given an algebra as described in the statement of the theo-rem, one defines 0 to be −l 1 (= −r 1) and derives the equations (2.7).

Here we will make use of another simpler equational characterisation:

Theorem 2.2. A Grishin algebra may be described as a residuated lattice-ordered monoid with a distinguished element 0 satisfying

(a⇒r 0)⇒l 0 = a = (a⇒l 0)⇒r 0. (2.8)

Proof. In any Grishin algebra we have −l 0 = −l−r 1 = 1 by double-negation elimination, and similarly −r 0 = 1. Then by Theorem 2.1 we getthat the algebra is residuated, has

a⇒l 0 = −l(a⊗−r 0) = −l(a⊗ 1) = −l a,

and similarly −r a = a⇒r 0. So (2.8) follows by DNE (2.6).

For the converse, given a residuated lattice-ordered monoid with 0 sat-isfying (2.8), define −l a to be a⇒l 0, and −r a to be a⇒r 0. Then DNE isgiven by (2.8), so we have only to prove (2.7) to show that we have a Grishinalgebra. But now

a⊗−r b v 0 iff a v (−r b)⇒l 0 = −l−r b = b,

and similarly −l b⊗ a v 0 iff a v (−l b) ⇒r 0 = −r −l b = b, as required.

This result shows that the notion of a Grishin algebra is equivalent tothat of a classical FL-algebra as defined in [20, p. 264].


3. Residuated Cover Systems

We work now with structures S = (S,4,C, . . . ) that include the followingcomponents:

• A binary relation 4 on S that is a preorder, i.e. reflexive and transitive.We sometimes write y < x when x 4 y, and say that y refines x.

• A binary relation C from S to its powerset PS. When x C C, wherex ∈ S and C ⊆ S, we say that x is covered by C, and write this also asC B x, saying that C covers x or that C is an x-cover.

An up-set is a subset X of S that is closed upwardly under the preorder:y < x ∈ X implies y ∈ X. For an arbitrary X ⊆ S,

↑X = {y ∈ S : (∃x ∈ X)x 4 y}

is the smallest up-set including X. It consists of those points that refinesome member of X. Thus X is an up-set iff ↑X = X. For x ∈ S,

↑x := ↑{x} = {y : x 4 y}

is the smallest up-set containing x. A subset Y of S refines a subset X ifY ⊆ ↑X, i.e. if every member of Y refines some member of X.

We write Up(S) for the collection of all up-sets of S. It is a completeposet under the partial order ⊆ of set inclusion, with the join

⊔X and meetd

X of any collection X of up-sets being the set union⋃X and intersection⋂

X respectively, while F = ∅ and T = S.For each subset X of S, define

jX = {x ∈ S : ∃C (x C C ⊆ X)}. (3.1)

Now we think of a condition as being locally true of x if there is some C suchthat x C C and each member of C satisfies the condition, i.e. if x is coveredby a set of members that have this condition. But the defining condition“x C C ⊆ X” for jX in (3.1) states that C is an x-cover that consists ofmembers of X. So x belongs to jX just when the property of being a memberof X is locally true of x, i.e. when x is covered by a set of members of X.Thus we may think of jX as the collection of “local members” of X.

X is called localised if jX ⊆ X, i.e. if every local member of X is anactual member of X. Now a property whose satisfaction is implied by itsown local satisfaction is said to be of local character, or more briefly a localproperty. Thus X is a localised set when membership of X is a local propertyin this sense.

8 Robert Goldblatt

Examples 3.1.

To illustrate these notions, let S be the set of open subsets of sometopological space, with x 4 y iff x ⊇ y and x C C iff x ⊆

⋃C. Then C

captures the usual notion of open cover; and “Y refines X”, i.e. Y ⊆ ↑X,has its usual meaning for open covers that every member of Y is includedin a member of X [18, p. 245]. For a different interpretation, think of Sas a collection of states that each contain certain information, with x 4 ywhen the information content of y includes that of x. So 4 is a relation ofrefinement in the sense of increase of information. Interpret covering as arelation expressing commonality of information content, so that if C B xthen the information content of x consists of that which is common to allthe states in C. Further discussion of these and other motivations can befound in [11] and [12, Chapter 6].

We call S a cover system if it satisfies the following axioms, for all x ∈ S:

• Existence: there exists an x-cover C ⊆ ↑x;

• Transitivity : if x C C and for all y ∈ C, y C Cy, then x C⋃y∈C Cy.

• Refinement : if x 4 y, then every x-cover can be refined to a y-cover, i.e.if C B x, then there exists a C ′ B y with C ′ ⊆ ↑C.

Lemma 3.2. In any cover system, the function j defined by (3.1) is a closureoperator on the complete poset (Up(S),⊆) of up-sets.

Proof. This is shown in [10, Theorem 5] and [11, Lemma 3.3]. Briefly: theRefinement axiom ensures that if X is an up-set, then so is jX; and theExistence axiom ensures that j is an inflationary operator on up-sets, i.e.X ⊆ jX for all X ∈ Up(S). Transitivity ensures that j(jX) ⊆ jX holdsfor any X. In particular, j is idempotent on up-sets, i.e. j(jX) = jX for allX ∈ Up(S). That j is monotone, i.e. preserves set inclusion, follows directlyfrom its definition.

An up-set X in a cover system will be called a proposition if it is localised,i.e. if jX ⊆ X. Since j is inflationary on Up(S), an up-set X is a propositioniff jX = X. In general, a set X is a proposition iff X = ↑X = jX. We writeProp(S) for the set of all localised up-sets of a cover system S. j↑X is thesmallest proposition that includes an arbitrary X, and j↑x is the smallestproposition containing the element x. The smallest proposition including anup-set X is just jX, so in fact j maps Up(S) to Prop(S). Indeed, Prop(S)is precisely the set of j-closed members of Up(S).


We now add some addition structure to S in order to make Prop(S) intoa complete residuated pomonoid, i.e. a unital quantale. Let · be a binaryrelation on S, which we will call fusion. We lift this to an operation onsubsets of S by putting, for X,Y ⊆ S,

X · Y = {x · y : x ∈ X and y ∈ Y }.

Then we write x · Y for the set {x} · Y , and X · y for X · {y}. Evidently,fusion is ⊆-monotone in each argument.

Define operations ⇒l and ⇒r on subsets of S by

X ⇒l Y = {z ∈ S : z ·X ⊆ Y }, X ⇒r Y = {z ∈ S : X · z ⊆ Y }. (3.2)

These provide left and right residuals to the fusion operation on the completeposet (PS,⊆), i.e. for all X,Y, Z ⊆ S we have

X ⊆ Y ⇒l Z iff X · Y ⊆ Z iff Y ⊆ X ⇒r Z. (3.3)

Consequently, if the lifted operation X · Y on subsets is associative, then(PS,⊆, ·) is a quantale, with residuals given by (3.2). In particular, thisholds when · is associative as an operation on S.

Now define X ◦ Y to be the up-set ↑(X · Y ) generated by X · Y . Then ifZ is an up-set, we have X · Y ⊆ Z iff X ◦ Y ⊆ Z, and hence (3.3) implies

X ⊆ Y ⇒l Z iff X ◦ Y ⊆ Z iff Y ⊆ X ⇒r Z (3.4)

for any X and Y . If the fusion operation · is 4-monotone in each argument,then Y ⇒l Z and X ⇒r Z are up-sets when Z is an up-set. In particular,this implies that Up(S) is closed under ⇒l and ⇒r, and so by (3.4), theseoperations are left and right residuals to ◦ on Up(S). Moreover, if the oper-ation · is associative on S, then 4-monotonicity implies that ◦ is associativetoo. Altogether this shows:

Lemma 3.3. If the fusion operation · is associative and 4-monotone in eachargument, then (Up(S),⊆, ◦) is a quantale, with residuals given by (3.2).

By a residuated cover system we will mean a structure

S = (S,4,C, ·, ε),

such that:

• (S,4,C) is a cover system.

10 Robert Goldblatt

• (S, ·, ε) is a pomonoid, i.e. · is an associative operation on S that is4-monotone in each argument, and has ε ∈ S as a unit.

• Fusion preserves covering : x C C implies x · y C C · y and y · x C y · C.

• Refinement of ε is local : x C C ⊆ ↑ε implies ε 4 x.

The last condition states that if x locally refines ε, in the sense that it has acover consisting of points refining ε, then x itself refines ε. This means thatthe up-set ↑ε of points refining ε is localised, i.e. j↑ε ⊆ ↑ε, and therefore isa proposition.

Lemma 3.4. In any residuated cover system S:

(1) If x C C and y C D, then x · y C C ·D.

(2) If X is an up-set, then X ◦ ↑ε = X = ↑ε ◦X.

(3) The function j defined by (3.1) is a quantic nucleus on the quantale(Up(S),⊆, ◦).

Proof.

(1) Since fusion preserves covering, from y C D we get x · y C x ·D; and foreach element x · d of x ·D, from x C C we get x · d C C · d. Hence bythe Transitivity axiom of cover systems,

x · y C⋃d∈D

C · d = C ·D.

(2) If z ∈ X ◦ ↑ε, then z < x · y for some x ∈ X and some y < ε. Then4-monotonicity of · gives that z < x · y < x · ε = x ∈ X, hence z ∈ X asX is an up-set. Conversely, if z ∈ X then z = z · ε ∈ X ◦ ↑ε.

The proof that X = ↑ε ◦X is similar.

(3) Since j is a closure operator by Lemma 3.2, we just have to show that itsatisfies 2.4 when ⊗ is ◦, i.e. that jX◦jY ⊆ j(X◦Y ). But if z ∈ jX◦jY ,then z < x · y for some x, y such that there is an x-cover C ⊆ X anda y-cover D ⊆ Y . Then by part (1), x · y C C · D ⊆ X · Y . Hence byRefinement, C ·D can be refined to a z-cover E, giving

z C E ⊆ ↑(C ·D) ⊆ ↑(X · Y ) = X ◦ Y.

This shows that z ∈ j(X ◦ Y ) as required.


Theorem 3.5. The set of propositions of a residuated cover system S formsa quantale (Prop(S),⊆,⊗) under a monoidal operation ⊗ with a unit 1,where

X ⊗Y = j(X ◦ Y ) = j↑(X · Y )

1 = ↑εdX =

⋂X⊔

X = j(⋃X )

X ⇒l Y = {z ∈ S : z ·X ⊆ Y }X ⇒r Y = {z ∈ S : X · z ⊆ Y }

T = S

F = j∅ = {x : x C ∅}.

Proof. Prop(S) is the set of all j-closed members of Up(S), and we havejust seen that this j is a quantic nucleus on the quantale (Up(S),⊆, ◦). Hencewe obtain a quantale (Prop(S),⊆, ◦j) by the theory of quantic quotients thatwas described in the previous section. This tells us that the quantic quotientbased on Prop(S) has the same partial ordering ⊆ as Up(S), with the samemeets

dX , which are given by the set intersection

⋂X , and with joins⊔

X given by j(⋃X ) because the joins in Up(S) are given by set union

⋃.

Also, the semigroup operation ⊗ on Prop(S) is ◦j , i.e. X ⊗Y = j(X ◦ Y ).Moreover, the residuals of ◦j on Prop(S) are just the restrictions to Prop(S)of the residuals of ◦ on Up(S), so these are indeed the operations ⇒l and⇒r on Prop(S) given by (3.2).

Since ↑ε is the unit of ◦ on Up(S) (Lemma 3.4(2)), the theory tells usthat j↑ε is the unit of ◦j on Prop(S). But we already observed that, sincerefinement of ε is local in S, j↑ε = ↑ε. Hence indeed we have 1 = ↑ε in thequotient quantale.

Finally, since S is a localised up-set, it is the greatest member T ofProp(S); and since ∅ is an up-set, j∅ is a proposition, and hence is the leastmember F of Prop(S).

Remark 3.6.The cover systems used in Kripke-Joyal semantics typically have the

property that every x-cover is included in the up-set of x, i.e. x C C impliesC ⊆ ↑x (see, e.g. [2]). This is a much stronger constraint than our Existenceaxiom, which requires only that there be at least one x-cover included in ↑x.The stronger condition is also used in a cover semantics for relevance logic in[12, Chapter 6]. It has the effect of making Prop(S) into a complete Heytingalgebra, hence a distributive lattice and a model of intuitionistic logic.

12 Robert Goldblatt

The topological cover systems of Examples 3.1 do not quite have thisstronger property: if x C C means that x ⊆

⋃C, then it does not imply

C ⊆ ↑x. However in this situation, we can replace the cover C by C ′ ={x∩ c : c ∈ C} and have x C C ′ ⊆ ↑x, with C ′ ⊆ ↑C. So in these topologicalcover systems,

every x-cover can be refined to an x-cover that is included in ↑x.

But even that weaker condition is sufficient to make Prop(S) into a completeHeyting algebra, as is shown in [11], and so it too must be abandoned wheninterpreting non-distributive logics.

A residuated cover system S will be called strong if Prop(S) is closedunder ◦, i.e. if X ◦ Y is a proposition whenever X and Y are propositions.In that situation, j(X ◦ Y ) = (X ◦ Y ) = ↑(X · Y ) for propositions X and Y .Hence when S is strong, the quantale Prop(S) has X ⊗Y = ↑(X · Y ).

Lemma 3.7. A residuated cover system S is strong iff the following conditionholds for any propositions X and Y :

(†) if there exists a z-cover refining X ·Y , then there exist x, y with x ·y 4 z;an x-cover X ′ ⊆ X; and a y-cover Y ′ ⊆ Y .

Proof. Let X and Y be propositions.Now X ◦Y is an up-set by definition, so is a proposition iff it is localised,

i.e. iff j(X ◦ Y ) ⊆ (X ◦ Y ). But if z ∈ j(X ◦ Y ), then there is some C withz C C ⊆ ↑(X · Y ), i.e. C refines X · Y , so if (†) holds we infer that there arex ∈ jX = X and y ∈ jY = Y such that x · y 4 z, hence x ∈ X ◦ Y . Thisshows that if (†) holds then X ◦ Y is a proposition.

Conversely, if X ◦ Y is localised, then (†) follows readily.

We now show that every unital quantale is isomorphic to the algebra ofpropositions of some strong residuated cover system. This is a refinementof a representation of quantales without unit given in [10, Theorem 6]. Theconstruction involves an abstraction of topological cover systems.

Theorem 3.8. Every order-complete residuated pomonoid L is isomorphicto the algebra Prop(SL) of propositions of some strong residuated cover sys-tem SL.

Proof. Let L = (L,v,⊗, 1L). Define SL = (S,4,C, ·, ε) by putting S = L;x 4 y iff y v x; x C C iff x v

⊔C; x · y = x⊗ y; and ε = 1. Then 4 is a

preorder, and · is associative and 4-monotone with ε as a unit. Note that


in SL, the up-set ↑x = {y : x 4 y} is equal to {y : y v x}, which is thedown-set of x in (L,v).

Next we show that SL satisfies the axioms of a residuated cover system.

• Existence: every C with x ∈ C ⊆ ↑x has x =⊔C, and so is an x-cover.

• Transitivity : If x v⊔C, and (∀y ∈ C)(y v

⊔Cy), then x v

⊔y∈C(

⊔Cy)

=⊔

(⋃y∈C Cy).

• Refinement : If x C C and x 4 y, then y v x v⊔C, so C itself is a

y-cover.

• Fusion preserves covering : Let x C C. Then x ·y = x⊗ y v (⊔C)⊗ y =⊔

c∈c c⊗ y (by (2.2)) =⊔

(C · y), so x · y C C · y. Similarly y · x C y · Cby (2.3).

• Refinement of ε is local : If x C C ⊆ ↑ε, then x v⊔C v ε, so ε 4 x as

required.

Thus SL is a residuated cover system. To show that it is strong we show(more strongly!) that the condition (†) of Lemma 3.7 holds for arbitrarysubsets X,Y of S, not just propositions. For, suppose there exists a z-coverC ⊆ ↑(X · Y ). Put X ′ = {a ∈ X : (∃b ∈ Y )(∃c ∈ C) a · b 4 c} ⊆ Xand Y ′ = {b ∈ Y : (∃a ∈ X)(∃c ∈ C) a · b 4 c} ⊆ Y . Let x′ =

⊔X ′ and

y′ =⊔Y ′, so that x′ C X ′ ⊆ X and y′ C Y ′ ⊆ Y . It remains to show

x′ · y′ 4 z. Now if c ∈ C, then a · b 4 c for some a ∈ X and b ∈ Y . Thena ∈ X ′ and b ∈ Y ′, so c v a · b v x′ · y′. Hence

⊔C v x′ · y′. But z v

⊔C,

so z v x′ · y′ as required.Now the propositions of SL are in fact precisely the up-sets ↑x generated

by the elements x of L. First, the fact that ↑x is localised, hence a proposi-tion, follows by the same argument given for ↑ε above: if y C C ⊆ ↑x, theny v

⊔C v x, so y ∈ ↑x. But if X is any localised up-set, let x =

⊔X.

Then X ⊆ ↑x, since if y ∈ X, then y v⊔X, i.e. y < x. Also x C X,

so x ∈ jX = X. Since X is an up-set, this implies ↑x ⊆ X, so altogetherX = ↑x.

The map x 7→ ↑x is order-invariant: x v y iff ↑x ⊆ ↑y. Hence this mapis an isomorphism between the complete posets (L,v) and (Prop(SL),⊆),and preserves all joins and meets. It also satisfies

↑(x⊗y) = (↑x) ◦ (↑y) (3.5)

for all x, y ∈ L. For if z ∈ (↑x) ◦ (↑y), then z < x′ · y′ for some x′ < xand y′ < y, hence x′ · y′ < x · y = x⊗ y and so z ∈ ↑(x⊗y). Conversely, ifz ∈ ↑(x⊗y), then z < x · y ∈ (↑x) · (↑y), so z ∈ (↑x) ◦ (↑y).

14 Robert Goldblatt

Now ◦ is the semigroup operation on Prop(SL), since SL is strong, so(3.5) shows that the map x 7→ ↑x preserves the semigroup operations of ourtwo quantales. Since ↑(1L) = ↑ε, it is a full isomorphism of these unitalquantales, preserving also their residual operations.

4. Classical Systems

This section describes the systems whose propositions form a Grishin Alge-bra. Let

S = (S,4,C, ·, ε, 0S) (4.1)

be a residuated cover system with a designated member 0S of Prop(S). Weuse 0S to define unary operations −l and −r on subsets of S, by putting−lX = X ⇒l 0S and −rX = X ⇒r 0S . Since 0S is a proposition of S, sotoo are −lX and −rX for any X ⊆ S. From (3.2) we get

−lX = {z ∈ S : z ·X ⊆ 0S}, −rX = {z ∈ S : X · z ⊆ 0S}. (4.2)

Lemma 4.1. For any subsets X and Y of S:

(1) X ⊆ Y implies −l Y ⊆ −lX and −r Y ⊆ −rX.

(2) X ⊆ −l Y iff Y ⊆ −rX.

(3) −l−r and −r −l are closure operators on (PS,⊆).

(4) −l ↑X = −lX and −r ↑X = −rX.

(5) −lX ⊆ −l jX and −rX ⊆ −r jX.

(6) If X ∈ Up(S), then −l jX = −lX and −r jX = −rX.

Proof. (1) Let X ⊆ Y . Then in general z · Y ⊆ 0S implies z ·X ⊆ 0S , i.e.z ∈ −l Y implies z ∈ −lX (4.2). Similarly −r Y ⊆ −rX.

(2) By (3.3), X ⊆ Y ⇒l 0S iff Y ⊆ X ⇒r 0S .

(3) Monotonicity : By (1), X ⊆ Y implies−l−rX ⊆ −l−r Y and−r −lX ⊆−r −l Y .

Inflationarity : Since −rX ⊆ −rX, (2) gives X ⊆ −l−rX. SimilarlyY ⊆ −r −l Y .

Idempotence: From X ⊆ −l−rX by (1) we get −r −l−rX ⊆ −rX.Hence −r −l−rX = −rX as −r −l is inflationary. It follows that−l−r −l−rX = −l−rX. Similarly, −r −l−r −lX = −r −lX.


(4) Since X ⊆ ↑X, (1) gives −l ↑X ⊆ −lX. Conversely, let z ∈ −lX. Thenfor all y ∈ ↑X we have y < x for some x ∈ X, hence z · y < z · x, andz · x ∈ z ·X ⊆ 0S (4.2), so z · y ∈ 0S as 0S is an up-set. This shows thatz · ↑X ⊆ 0S , hence z ∈ −l ↑X, as required to prove −l ↑X = −lX.

The proof that −r ↑X = −rX is similar.

(5) Let x ∈ −lX. Then if y ∈ jX, there is some C with x C C ⊆ X, sox · y C x · C ⊆ x · X ⊆ 0S . Hence x · y ∈ j0S = 0S . This shows thatx · jX ⊆ 0S , i.e. that x ∈ −l jX.

The proof that −rX ⊆ −r jX is similar.

(6) j is inflationary on up-sets (Lemma 3.2), so if X ∈ Up(S), then X ⊆ jX,hence −l jX ⊆ −lX by (1), implying −l jX = −lX by (5).

Similarly −r jX = −rX.

We now define a cover system S of the form (4.1) to be classical if it has

j↑X = −l−rX = −r −lX, for all X ⊆ S. (4.3)

Thus in a classical system, the least proposition (localised up-set) containingany given X is equal to both −l−rX and −r −lX.

Theorem 4.2. For any residuated cover system S with distinguished propo-sition 0S , the following are equivalent:

(1) S is classical.

(2) jX = −l−rX = −r −lX, for all up-sets X.

(3) Prop(S) is a Grishin algebra.

Proof. (1) implies (2): if (4.3) holds, then (2) is immediate, as ↑X = Xwhenever X ∈ Up(S).

(2) implies (3): Let X ∈ Prop(S). Then X is an upset, and jX = X,so (2) implies X = −l−rX = −r −lX. Thus the residuated lattice-orderedmonoid Prop(S) satisfies the law of double-negation elimination in the form(2.8), so is a Grishin algebra by Theorem 2.2.

(3) implies (1): for any X ⊆ S we have j↑X ∈ Prop(S), so (3) implies

j↑X = −l−r j↑X = −r −l j↑X.

But as ↑X is an up-set, Lemma 4.1(6) gives −r j↑X = −r ↑X, which isequal to −rX by Lemma 4.1(4). Hence −l−r j↑X = −l−rX. Similarly−r −l j↑X = −r −lX. Thus (4.3) holds, making S classical.

16 Robert Goldblatt

Thus we see that the propositions of a classical residuated cover systemform a complete Grishin algebra. In the converse direction, we have:

Theorem 4.3. Every complete Grishin algebra is isomorphic to the algebraof all propositions of some strong classical residuated cover system.

Proof. Let L be a complete Grishin algebra with distinguished element 0L.Let SL be the associated strong residuated cover system of Theorem 3.8, anddefine 0S

Lto be ↑0L ∈ Prop(SL). The map x 7→ ↑x was shown to be an

isomorphism between L and Prop(SL) as complete residuated pomonoids,and it now preserves the given “0-elements” as well. Since it preservesresiduals, it must also preserve the DNE equations (2.8), and so Prop(SL)is a Grishin algebra. Hence by the previous Theorem, SL is classical.

5. Orthogonality

In a residuated cover system of the form (4.1), an alternative description ofthe operations −l and −r of (4.2) can be developed by introducing a certainbinary relation ⊥ on S. This is defined by

z ⊥ y iff z · y ∈ 0S . (5.1)

We think of ⊥ as a relation of orthogonality or incompatibility between pointsof S. It is lifted to a relation between points z and subsets X of S, by puttingz ⊥ X iff z ⊥ y for all y ∈ X; and X ⊥ z iff y ⊥ z for all y ∈ X. Then weget

z ⊥ X iff z ·X ⊆ 0S ,

and similarly X ⊥ z iff X · z ⊆ 0S , so the operations −l and −r of (4.2)satisfy

−lX = {z ∈ S : z ⊥ X}, −rX = {z ∈ S : X ⊥ z}. (5.2)

Moreover, since ε is a unit for ·, we have z · ε ∈ 0S iff z ∈ 0S iff ε · z ∈ 0S , so

0S = {z : z ⊥ ε} = {z : ε ⊥ z}. (5.3)

This suggests an alternative approach to classical cover systems. Insteadof starting with a system S having a distinguished proposition 0S , we beginwith a residuated cover system

S = (S,4,C, ·, ε,⊥) (5.4)

having a binary relation ⊥ satisfying the following conditions:


• z ⊥ y iff z · y ⊥ ε.

• Orthogonality to ε is monotonic: y < z ⊥ ε implies y ⊥ ε.

• Orthogonality to ε is local : x C C ⊥ ε implies x ⊥ ε.

From the first condition we get that in general ε ⊥ z iff z ⊥ ε, so we canwell-define a subset 0S by (5.3). The other two conditions ensure that 0S

thus defined is an up-set that is localised, i.e. a member of Prop(S). Thefirst condition then ensures that z ⊥ X iff z ·X ⊆ 0S , and likewise X ⊥ ziff X · z ⊆ 0S . So operations −l and−r defined from ⊥ by (5.2) agree withthose defined by (4.2), where 0S itself is defined from ⊥ by (5.3).

We leave it to the reader to formulate the precise sense in which thesetwo approaches – starting with 0S or with ⊥ – are equivalent. In eitherapproach, S can be specified to be classical when (4.3) holds.

It is noteworthy that, since 0S is an up-set, we have z · (↑ε) ⊆ 0S iffz · ε ∈ 0S iff z ∈ 0S ; and so −l 1 = 0S , and likewise −r 1 = 0S , in anyresiduated cover system with 0S , independently of the classicality condition(4.3).

6. MacNeille Completion

Each poset L = (L,v) can be extended to a complete poset L = (L,v) suchthat any subset of L having a join or meet in L has the same join or meetin L. In particular, if L is a lattice, then it is a sublattice of L. Moreover,each element of L is both a join of elements of L and a meet of elements ofL. Hence if a ∈ L,

a =⊔{s ∈ L : s v a} =

d{t ∈ L : a v t} (6.1)

(we use the letters a, b, c for general members of L, and reserve s, t for mem-bers of L). This property characterises L uniquely up to isomorphism [3, 24].

MacNeille [17] proved the existence of the completion L by generalisingthe Dedekind completion of the rationals by cuts. Here we will work withthe abstract description of L based on (6.1). It implies that for any a, b ∈ L,

a v b iff ∀s, t ∈ L (s v a and b v t implies s v t). (6.2)

The MacNeille completion of a residuated pomonoid

L = (L,v,⊗, 1L,⇒l,⇒r)

18 Robert Goldblatt

is a unital quantale. This was shown in [20] by a generalisation of MacNeille’sconstruction.4 Here is the abstract version. The operations on L are liftedto L by putting, for a, b ∈ L:

a⊗ b =⊔{s⊗ t : a w s ∈ L and b w t ∈ L}

a⇒l b =d{s⇒l t : a w s ∈ L and b v t ∈ L}

a⇒r b =d{s⇒r t : a w s ∈ L and b v t ∈ L}

1L = 1L.

These operations agree with the corresponding operations of L (i.e. they areequal to the corresponding operations when restricted to L), and they makeL into a residuated lattice-ordered monoid having L as a subalgebra.5

The variety of classical FL-algebras was observed to be closed underMacNeille completion in [20, p. 275] (see also [23, 6.1]). Here we give a proofof the result for Grishin algebras, using the abstract formalism.

Now suppose L is a Grishin algebra with distinguished element 0. Put0L = 0 and use this element to define −l and −r on L by −l a = a⇒l 0, and−r a = a⇒r 0, thereby extending the operations −l and −r of L.

Lemma 6.1. For any a, b ∈ L,

(1) If a v b, then −l b v −l a and −r b v −r a.

(2) a v −l−r a and a v −r −l a.

(3) −l a =d{−l s : a w s ∈ L}, −r a =

d{−r s : a w s ∈ L}.

Proof. (1) and (2) hold for any residuated pomonoid with 0, given thedefinitions of −l and −r.

(1) Residuation is antitonic in the first argument, so a v b implies b⇒l 0 va⇒l 0, i.e. −l b v −l a, and likewise b⇒r 0 v a⇒r 0.

(2) a⊗(a ⇒r 0) v 0, so a v (a ⇒r 0) ⇒l 0 = −l−r 0. Similarly a v (a ⇒l

0)⇒r 0.

(3) Let X = {s ⇒l t : a w s ∈ L and 0 v t ∈ L}. Then −l a = a ⇒l 0 =dX by definition of ⇒l.

Put Y = {−l s : a w s ∈ L}. Then Y ⊆ X, since if a w s ∈ L,then −l s = s ⇒l 0 ∈ X. Hence

dX v

dY . But if 0 v t, then

4The result for commutative ⊗ is discussed in[25, Chapter 8] and [21]. In that case ⇒l

is equal to ⇒r.5See [24, Prop. 3.17] for a proof that ⇒l is left residual to ⊗ under these definitions.


s⇒l 0 v s⇒l t, so each member s⇒l t of X has the member s⇒l 0 ofY below it under v, and therefore has

dY below it. This implies thatd

Y vdX.

ThusdX =

dY , i.e. −l a =

dY as required. The equation for ⇒r is

proved similarly.

Theorem 6.2. The MacNeille completion of a Grishin algebra is also aGrishin algebra.

Proof. Let L be a Grishin algebra, and ⇒l and ⇒r the operations definedon L as above. Since L is a residuated lattice-ordered monoid, by Theorem2.2 it remains only to show that the double-negation elimination law holdsin L.

Given any a ∈ L, we have a v −l−r a by (2) of the above Lemma, sowe need to show that −l−r a v a. By (6.2), it suffices to take any s, t ∈ Lwith s v −l−r a and a v t, and show that s v t. Now if a v t ∈ L, then−l−r a v −l−r t by (1) of the above Lemma. But −l−r t = t as L is aGrishin algebra, so if s v −l−r a then s v t follows as required.

This proves a = −l−r a, and the proof that a = −r −l a is similar.

We can now combine our results into a representation theorem for Grishinalgebras in general.

Theorem 6.3. Every Grishin algebra has an isomorphic embedding into thealgebra of all propositions of some strong classical residuated cover system,by a map that preserves all existing joins and meets.

Proof. If L is a Grishin algebra, then L is a complete Grishin algebra, sois isomorphic to Prop(SL), where SL is the strong classical residuated coversystem of Theorem 4.3. The inclusion of L into L preserves all existing joinsand meets, hence so does its composition with the isomorphism from L ontoProp(SL).

7. Infinitary First-Order Logic

Our representation of Grishin algebras gives rise to a cover system semanticsfor a version of classical bilinear predicate logic that has quantification ofindividual variables, and (infinitary) disjunctions and conjunctions of setsof formulas. To describe the formal language for this, we fix a denumer-able list v0, . . . , vn, . . . of individual variables and a set of predicate letters,

20 Robert Goldblatt

with typical member P , that are k-ary for various k < ω. These are usedto define atomic formulas P (vn1 , . . . , vnk

). A preformula is any expressiongenerated from atomic formulas and the constants T, F, 1, 0 by using thebinary connectives &,→l,→r and the quantifiers ∃vn, ∀vn, and by allowingthe formation of the disjunction

∨Φ and conjunction

∧Φ of any set Φ of

formulas. A formula is a preformula that has only finitely many free vari-ables. Binary disjunctions and conjunctions are defined by taking ϕ ∨ ψ tobe

∨{ϕ,ψ} and ϕ ∧ ψ to be

∧{ϕ,ψ}. Left and right negation connectives

are introduced by defining ¬lϕ to be ϕ→l 0 and ¬rϕ to be ϕ→r 0.

The semantic analysis builds on the one already given in [10] for thislanguage without the constants T, F, 1, 0, or the negations ¬l,¬r, usingquantales and certain cover systems. We take this earlier work as given(see [10, Sections 3.1–3.4, 3.7]), and confine ourselves to explaining what isinvolved in extending it to the present language.

By a classical model we mean a structure M = (S, D, V ), where S is astrong classical residuated cover system; D is a set of individuals; and foreach k-ary predicate letter P , V (P ) is a function assigning a proposition ofS to each k-tuple of elements of D. In symbols: V (P ) : Dk → Prop(S). Tointerpret variables in the model we use D-valuations, which are sequencesσ = 〈σ0, . . . , σn, . . .〉 of elements of D, the idea being that σ assigns valueσn to variable vn. We write σ(d/n) for the valuation obtained from σ byreplacing σn by d. For each formula ϕ we specify a proposition ‖ϕ‖Mσ ∈Prop(S) for each valuation σ. This is defined inductively on the formationof ϕ, using the structure of Prop(S), as follows (cf. Theorem 3.5):

• ‖P (vn1 , . . . , vnk)‖Mσ = V (P )(σn1 , . . . , σnk

).

• ‖T‖Mσ = S.

• ‖F‖Mσ = j∅.

• ‖1‖Mσ = ↑ε.

• ‖0‖Mσ = 0S .

• ‖ϕ&ψ‖Mσ = ‖ϕ‖Mσ ⊗‖ψ‖Mσ .

• ‖ϕ→l ψ‖Mσ = ‖ϕ‖Mσ ⇒l ‖ψ‖Mσ , ‖ϕ→r ψ‖Mσ = ‖ϕ‖Mσ ⇒r ‖ψ‖Mσ .

• ‖∨

Φ‖Mσ =⊔ϕ∈Φ ‖ϕ‖Mσ .

• ‖∧

Φ‖Mσ =dϕ∈Φ ‖ϕ‖Mσ .

• ‖∃vnϕ‖Mσ =⊔d∈D ‖ϕ‖Mσ(d/n).

• ‖∀vnϕ‖Mσ =dd∈D ‖ϕ‖Mσ(d/n).


Consequently, the negation connectives have

• ‖¬lϕ‖Mσ = ‖ϕ‖Mσ ⇒l 0S = −l ‖ϕ‖Mσ .

• ‖¬rϕ‖Mσ = ‖ϕ‖Mσ ⇒r 0S = −r ‖ϕ‖Mσ .

A satisfaction relation is defined by using the notation

M, x |= ϕ[σ]

to mean that x ∈ ‖ϕ‖Mσ . This can be read “ϕ is true/satisfied in M at xunder σ”. Unravelling the operations in Prop(S), and using the fact thatX ⊗Y = ↑(X · Y ) in a strong system, this satisfaction relation is charac-terised as follows (suppressing the symbol M):

x |= P (vn1 , . . . , vnk)[σ] iff x ∈ V (P )(σn1 , . . . , σnk

).x |= T[σ].x |= F[σ] iff x C ∅.x |= 1[σ] iff ε 4 x.x |= 0[σ] iff x ∈ 0S (iff x ⊥ ε, cf. (5.3)).x |= ϕ&ψ[σ] iff for some y and z such that y · z 4 x,

y |= ϕ[σ] and z |= ψ[σ].x |= ϕ→l ψ[σ] iff y |= ϕ[σ] implies x · y |= ψ[σ].x |= ϕ→r ψ[σ] iff y |= ϕ[σ] implies y · x |= ψ[σ].x |=

∨Φ[σ] iff there exists C B x such that for all z ∈ C,

z |= ϕ[σ] for some ϕ ∈ Φ.x |=

∧Φ[σ] iff x |= ϕ[σ] for all ϕ ∈ Φ.

x |= ∃vnϕ[σ] iff there exists C B x such that for all z ∈ C,z |= ϕ[σ(d/n)] for some d ∈ D.

x |= ∀vnϕ[σ] iff x |= ϕ[σ(d/n)] for all d ∈ D.

Furthermore, the negation connectives satisfy:

x |= ¬lϕ[σ] iff y |= ϕ[σ] implies x ⊥ y,x |= ¬rϕ[σ] iff y |= ϕ[σ] implies y ⊥ x

(cf. (5.2)). This kind of modelling of negation via an orthogonality relationgoes back to [8] in the context of orthologic, where ⊥ is symmetric and thetwo negations collapse to one. The definition of ⊥ from a fixed set 0S as in(5.1) comes from [7].

Each model M gives rise to a semantic implication relation |=M onformulas, defined by writing

ϕ |=M ψ

22 Robert Goldblatt

to mean that ‖ϕ‖Mσ ⊆ ‖ψ‖Mσ for all D-valuations σ, i.e. that for all x in Sand all D-valuations σ, M, x |= ϕ[σ] implies M, x |= ψ[σ]. We say that ϕclassically implies ψ, written ϕ |=c ψ, if ϕ |=M ψ for all classical modelsM.

To axiomatise the semantic relation |=c, we take a sequent to be anexpression ϕ ` ψ with ϕ and ψ being formulas. Alternatively, a sequent maybe thought of as an ordered pair of formulas, with the symbol ` denotinga class of sequents, i.e. a binary relation between formulas. Then we writeϕ a` ψ when both ϕ ` ψ and ψ ` ϕ.

For the logic of residuated posemigroups, we gave in [10, Section 3.3] alist of sequent axiom schemes and a list of rules of inference for generatingnew sequents. We take those axioms and rules as given here, and add tothem the following axiom schemes:

• F ` ϕ, ϕ ` T.

• ϕ&1 a` ϕ, 1&ϕ a` ϕ.

• ¬l¬rϕ a` ϕ, ¬r¬lϕ a` ϕ.

Let `c be the smallest class of sequents that includes all instances of theaxiom schemes and is closed under the rules of inference. To prove soundness,i.e.

ϕ `c ψ implies ϕ |=c ψ, (7.1)

it suffices to observe that for any classical model M, the relation |=M in-cludes all instances of the axioms and is closed under the rules, so it includes`c. Thus ϕ `c ψ implies ϕ |=M ψ for all classical models M.

To prove completeness, i.e. the converse of (7.1), we use a Lindenbaumalgebra construction to build a Grishin algebra, and then embed it into thealgebra of propositions of a model by the representation of Theorem 6.3. Butthe use of infinitary disjunctions and conjunctions allows a proper class offormulas to be generated, and we have to restrict this to a set, in order for theLindenbaum algebra to be based on a set. So we define a fragment to be a setF of formulas that includes all atomic formulas and the constants T, F, 1, 0;is closed under the binary connectives ∧,∨,&, →l,→r, and the quantifiers∃vn, ∀vn; and is closed under subformulas and variable substitution. HenceF is closed under the negation connectives ¬l and ¬r.

Let ` be a relation satisfying all our axioms and rules, and fix a fragmentF . The induced relation a` is an equivalence relation on F . Let |ϕ| = {ψ ∈


F : ϕ a` ψ} be the equivalence class of ϕ ∈ F and LF = {|ϕ| : ϕ ∈ F}. Put

|ϕ| v |ψ| iff ϕ ` ψ,|ϕ| u |ψ| = |ϕ ∧ ψ|,|ϕ| t |ψ| = |ϕ ∨ ψ|,|ϕ| ⊗ |ψ| = |ϕ&ψ|,|ϕ| ⇒l |ψ| = |ϕ→l ψ|,|ϕ| ⇒r |ψ| = |ϕ→r ψ|,−l |ϕ| = |¬lϕ|,−r |ϕ| = |¬rϕ|.

The axioms and rules ensure that this yields a well-defined Grishin algebraLF on LF with greatest element |T| and least element |F|; with |1| beingthe identity element of ⊗; and with the distinguished element |0| having−l |ϕ| = |ϕ| →l |0| and −r |ϕ| = |ϕ| →r |0|. Also we have:6

|∃vnϕ| =⊔p<ω|ϕ(vp/vn)|.

|∀vnϕ| =d

p<ω|ϕ(vp/vn)|.

|∨

Φ| =⊔ϕ∈Φ

|ϕ|, when∨

Φ ∈ F .

|∧

Φ| =d

ϕ∈Φ

|ϕ| when∧

Φ ∈ F .

(7.2)

By Theorem 6.3, there exists a strong classical residuated cover system SFand an isomorphic embedding f : LF → Prop(SF ) that preserves all joinsand meets that exist in LF , including those described in (7.2).

Now define the classical model MF = (SF , D, V ), where D is the set ofall variables vn, and V (P )(vn1 , . . . , vnk

) = f |P (vn1 , . . . , vnk)|. Then if σ is

the D-valuation with σn = vn, we get

‖P (vn1 , . . . , vnk)‖MFσ = f |P (vn1 , . . . , vnk

)|.

We then extend this to show inductively that

‖ϕ‖MFσ = f |ϕ| for all ϕ ∈ F . (7.3)

6The proofs of the equations for |∃vnϕ| and |∀vnϕ| in (7.2) are as for finitary first-orderlogic (e.g. [1, Lemma 3.4.1]), and depend on the fact that a formula ϕ has finitely manyfree variables.

24 Robert Goldblatt

This uses the the definition of ‖ϕ‖Mσ , results(7.2), the fact that f preservesthe Grishin algebra operations and any joins and meets existing in SF ;and the general substitutional result that ‖ϕ(vp/vn)‖Mσ = ‖ϕ‖Mσ(σp/n) (which

holds of any σ in any model M).

To prove completeness of `c for |=c, suppose ϕ |=c ψ. Take a fragment Fcontaining the formulas ϕ and ψ, and construct the classical model MFasabove using the relation `c to define LF and hence SF . Then ϕ |=MF ψ, so

‖ϕ‖MFσ ⊆ ‖ψ‖MFσ in Prop(SF ) where σn = vn. Hence |ϕ| v |ψ| in LF by(7.3) and the fact that f is order-invariant; so ϕ `c ψ. This proves that

ϕ |=c ψ implies ϕ `c ψ

as required. Together with (7.1), we have shown that the proof-theoreticdeducibility relation `c is both sound and complete for the semantic conse-quence relation |=c over models on strong classical residuated covers systems.In other words, these two relations are identical.

References

[1] J. L. Bell and A. B. Slomson. Models and Ultraproducts. North-Holland, Amsterdam,

1969.

[2] John L. Bell. Cover schemes, frame-valued sets and their potential uses in spacetime

physics. In Albert Reimer, editor, Spacetime Physics Research Trends, Horizons in

World Physics, volume 248. Nova Science Publishers, 2005. Manuscript at http:

//publish.uwo.ca/~jbell.

[3] B. A. Davey and H. A. Priestley. Introduction to Lattices and Order. Cambridge

University Press, 1990.

[4] Kosta Dosen. Sequent systems and groupoid models II. Studia Logica, 48(1):41–65,

1989.

[5] Kosta Dosen. A brief survey of frames for the Lambek calculus. Zeitschrift fur

Mathematische Logik und Grundlagen der Mathematik, 38:179–187, 1992.

[6] Nikolaos Galatos, Peter Jipsen, Tomasz Kowalski, and Hiroakira Ono. Residuated

Lattices : An Algebraic Glimpse at Substructural Logics, volume 151 of Studies in

Logic and the Foundations of Mathematics. Elsevier, 2007.

[7] Jean-Yves Girard. Linear logic. Theoretical Computer Science, 50:1–102, 1987.

[8] Robert Goldblatt. Semantic analysis of orthologic. Journal of Philosophical Logic,

3:19–35, 1974. Reprinted in [9].

[9] Robert Goldblatt. Mathematics of Modality. CSLI Lecture Notes No. 43. CSLI

Publications, Stanford University, 1993.

[10] Robert Goldblatt. A Kripke-Joyal semantics for noncommutative logic in quantales.

In Guido Governatori, Ian Hodkinson, and Yde Venema, editors, Advances in Modal

Logic, Volume 6, pages 209–225. College Publications, London, 2006. www.aiml.net/

volumes/volume6/.


[11] Robert Goldblatt. Cover semantics for quantified lax logic. Journal of Logic and

Computation, 2010. doi:10.1093/logcom/exq029.

[12] Robert Goldblatt. Quantifiers, Propositions and Identity: Admissible Semantics for

Quantified Modal and Substructural Logics. Number 38 in Lecture Notes in Logic.

Cambridge University Press and the Association for Symbolic Logic, 2011.

[13] V. N. Grishin. On a generalisation of the Ajdukiewicz-Lambek system. In A. I.

Mikhailov, editor, Studies in Non-Classical Logics and Formal Systems, pages 315–

334. Nauka, Moscow, 1983. English translation in V. M. Abrusci and C. Casadio

(eds.), New Perspectives in Logic and Formal Linguistics, Proceedings 5th Roma

Workshop, Bulzoni Editore, Rome, 2002. Corrected version available as pages 1–17

in http://symcg.pbworks.com/f/essllinotesnew.pdf.

[14] J. Lambek. Some lattice models of bilinear logic. Algebra Universalis, 34:541–550,

1995.

[15] Joachim Lambek. Bilinear logic and Grishin algebras. In Ewa Orlowska, editor,

Logic at Work: Essays Dedicated to the Memory of Helena Rasiowa, pages 604–612.

Physica-Verlag, 1999.

[16] Saunders Mac Lane and Ieke Moerdijk. Sheaves in Geometry and Logic: A First

Introduction to Topos Theory. Springer-Verlag, 1992.

[17] H. M. MacNeille. Partially ordered sets. Transactions of the American Mathematical

Society, 42:416–460, 1937.

[18] James R. Munkres. Topology. Prentice Hall, 2000.

[19] Susan B. Niefield and Kimmo I. Rosenthal. Constructing locales from quantales.

Mathematical Proceedings of the Cambridge Philosophical Society, 104:215–234, 1988.

[20] Hiroakira Ono. Semantics for substructural logics. In Peter Schroeder-Heister and

Kosta Dosen, editors, Substructural Logics, pages 259–291. Oxford University Press,

1993.

[21] Hiroakira Ono. Closure operators and complete embeddings of residuated lattices.

Studia Logica, 74(3):427–440, 2003.

[22] Hiroakira Ono and Yuichi Komori. Logics without the contraction rule. The Journal

of Symbolic Logic, 50(1):169–201, 1985.

[23] Kimmo I. Rosenthal. Quantales and Their Applications, volume 234 of Pitman Re-

search Notes in Mathematics. Longman Scientific & Technical, 1990.

[24] Mark Theunissen and Yde Venema. MacNeille completions of lattice expansions.

Algebra Universalis, 57:143–193, 2007.

[25] A. S. Troelstra. Lectures on Linear Logic. CSLI Lecture Notes No. 29. CSLI Publi-

cations, Stanford, California, 1992.

Robert GoldblattVictoria University of Wellington, New [email protected]

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