equivalence of logical consequence relations: an order

Consequence relations C. Tsinakis - slide #1

Equivalence of logical consequence relations:an order-theoretic and categorical approach

Costas TsinakisVanderbilt University

http://www.math.vanderbilt.edu/people/tsinakis

Logical fOundations of Rational Interaction

Centro di Ricerca Matematica E. De Giorgi, PisaNovember 4, 2009

AbstractReferencesHistorical Note

Logical Consequence

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Fundamental Property


Abstract

Equivalences and translations between logicalconsequence relations abound in logic. The aim ofthis talk is to propose a uniform treatment ofvarious constructions and concepts connected withthe study of logical consequence relations. Theapproach is of order-theoretic and categoricalnature, and provides a roadmap for consideringrelated questions in the future.


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References⋆ S. Abramsky and S. Vickers, Quantales, observational logic and process

semantics, Math. Struct. in Computer Science 3 (1993), 161-227.

W. J. Blok and D. Pigozzi, Algebraizable logics, Memoirs of the AMS 77, no. 396,1989.

⋆W. J. Blok and B. Jónsson, Equivalence of consequence operations, Studia Logica83 (2006), no. 1-3, 91-110.

⋆ Augustus De Morgan and Charles S. Peirce [See e.g., C. Brink, Boolean modules,J. Algebra 71 (1981), no. 2, 291-313.]

⋆ N. Galatos and C. Tsinakis, Equivalence of consequence relations: anorder-theoretic and categorical perspective, J. Symbolic Logic 74, no. 3 (2009),780-810.

J. Gil-Férez, Categories of modules over complete residuated lattices, preprint, 2008.

⋆ A. Joyal and M. Tierney, An extension of the Galois theory of Grothendieck,Memoirs of the Am. Math. Soc. 51(309), 1984.

A. Pynko, Definitional equivalence and algebraizability of generalized logical systems,Annals of Pure and Applied Logic 98 (1999), 1-68.

J. Rebagliato and V. Verdú, On the algebraization of some Gentzen systems. Fund.Inform. 18 (1993), no. 2-4, 319–338.

C. Russo, Quantale Modules, with Applications to Logic and Image Processing,Doctoral dissertation, 2007.


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Historical Note

The history of these algebras can be traced backto the work of Augustus De Morgan and Charles S.Peirce. It was De Morgan (1847-1864) who startedformalising the logic of binary relations as ageneralisation of Aristotle’s syllogistic logic.Moreover, he invented relational composition andrelational converse. Peirce (1866-1883) gave thefirst algebraic treatment of the algebra of relationsinteracting with sets.


Logical Consequence


Logical ConsequenceRelationsOperationsCorrespondenceTheoriesLogical FormA-setsInvarianceFinitarity

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Consequence Relations

A consequence relation over the set M is arelation ⊢⊆ ℘ (M) ×M obeying these conditionsfor all u ∈M and for all X,Y ⊆M :

(1) X ⊢ u, whenever u ∈ X (reflexivity);

(2) If X ⊢ u and X ⊆ Y , then Y ⊢ u (monotonicity);

(3) If Y ⊢ u and X ⊢ v for every v ∈ Y , then X ⊢ u(transitivity; cut).



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Consequence Relations

A consequence relation over the set M is arelation ⊢⊆ ℘ (M) ×M obeying these conditionsfor all u ∈M and for all X,Y ⊆M :

(1) X ⊢ u, whenever u ∈ X (reflexivity);

(2) If X ⊢ u and X ⊆ Y , then Y ⊢ u (monotonicity);

(3) If Y ⊢ u and X ⊢ v for every v ∈ Y , then X ⊢ u(transitivity; cut).

Notation:(i) u ⊢ v means: {u} ⊢ v(ii) ⊢ v means: ∅ ⊢ v(iii) X ⊢ Y means: X ⊢ v, for all v ∈ Y



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Consequence Operations

An equivalent approach to logical consequence,due to Alfred Tarksi, consists in giving an accountof it via a closure operator.



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We use the term consequence operation on thepower set ℘(M) of a set M for a closure operatoron ℘(M). That is, a map Ξ : ℘(M) → ℘(M)satisfying the following conditions for all X,Y ⊆M :

(1) if X ⊆ Y, then Ξ(X) ⊆ Ξ(Y );

(2) X ⊆ Ξ(X); and

(3) Ξ(Ξ(X)) = Ξ(X).



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We use the term consequence operation on thepower set ℘(M) of a set M for a closure operatoron ℘(M). That is, a map Ξ : ℘(M) → ℘(M)satisfying the following conditions for all X,Y ⊆M :

(1) if X ⊆ Y, then Ξ(X) ⊆ Ξ(Y );

(2) X ⊆ Ξ(X); and

(3) Ξ(Ξ(X)) = Ξ(X).

(1) whatever follows from X also follows from anysuperset of X;

(2) all members of X are consequences of X; and(3) whatever follows from consequences of X

also follows from X.



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Correspondence

Given a set M , there exists a bijectivecorrespondence between all consequenceoperations Ξ on ℘(M) and all consequencerelations ⊢ over M . More specifically:



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Correspondence

Given a set M , there exists a bijectivecorrespondence between all consequenceoperations Ξ on ℘(M) and all consequencerelations ⊢ over M . More specifically:■ If ⊢ is a consequence relation over M , then the

map Ξ⊢:℘ (M) → ℘ (M) defined by

Ξ⊢ (X) = {u ∈M : X ⊢ u}

is a consequence operation on ℘(M).



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Correspondence



Ξ⊢ (X) = {u ∈M : X ⊢ u}

is a consequence operation on ℘(M).■ Conversely, if Ξ is a consequence operation on℘(M), then the relation ⊢Ξ⊆ ℘ (M) ×M definedby

X ⊢Ξ u iff u ∈ Ξ (X)

is a consequence relation over M .



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Correspondence



Ξ⊢ (X) = {u ∈M : X ⊢ u}

is a consequence operation on ℘(M).■ Conversely, if Ξ is a consequence operation on℘(M), then the relation ⊢Ξ⊆ ℘ (M) ×M definedby

X ⊢Ξ u iff u ∈ Ξ (X)

is a consequence relation over M .

Furthermore, Ξ⊢Ξ= Ξ and ⊢Ξ⊢

= ⊢.



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Theories

Let ⊢ be a consequence relation over M , and let Ξbe the associated consequence operation on℘(M). X ⊆M is said to be a ⊢-theory if it is closedsubset of M under Ξ:

X = Ξ(X) = {u : u ∈M and X ⊢ u}



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Theories


X = Ξ(X) = {u : u ∈M and X ⊢ u}

Note that the poset of ⊢-theories, denoted byTh (⊢) or Th (Ξ), is a closure system over M , thatis, a subset of ℘(M) that is closed under arbitraryintersections.



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Theories


X = Ξ(X) = {u : u ∈M and X ⊢ u}

Note that the poset of ⊢-theories, denoted byTh (⊢) or Th (Ξ), is a closure system over M , thatis, a subset of ℘(M) that is closed under arbitraryintersections.

Th (⊢) completely determines Ξ and ⊢.Furthermore, there exists a bijectivecorrespondence between closure systems andconsequence relations over M , and consequenceoperations on ℘(M).



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Logical Form

One of the most distinctive properties of logicalconsequence is its formal character: what followslogically from a set of premisses can be inferredfrom the premisses themselves purely by virtue ofits logical form.

How can we give a proper account of this feature?



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Logical Form

One of the most distinctive properties of logicalconsequence is its formal character: what followslogically from a set of premisses can be inferredfrom the premisses themselves purely by virtue ofits logical form.

How can we give a proper account of this feature?

We consider operations σ that act on assertions byuniformly substituting their non-logicalcomponents, in such a way as to leave their logicalform unchanged. We assume that such actionscan be concatenated – τ ◦ σ – and allow for anidentity action 1 that changes nothing at all.



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Monoids Acting on Sets

Formally, let M be a nonempty set. A monoidA = (A, ·, 1) is said to act on M (and M is said tobe an A-set) in case an operation ◦ : A×M →Mis defined such that, for all a, b ∈ A and all u ∈M ,

(a · b) ◦ u = a ◦ (b ◦ a) ; and 1 ◦ u = u.

The operation ◦ is called scalar product, and theelements in A are called actions. If no confusionarises, we’ll use plain juxtaposition in place of “·”.



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Action Invariance

Let now M be an A-set. A consequence relation ⊢over M is said to be action-invariant if, for anya ∈ A and any X ∪ {u} ⊆M ,

wheneverX ⊢ u, then a ◦X ⊢ a ◦ u.

[Here, a ◦X = {a ◦ v : v ∈ X}.]



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Action Invariance



[Here, a ◦X = {a ◦ v : v ∈ X}.]

Note that ⊢ is action invariant iff the associatedconsequence operation Ξ satisfies the condition

a ◦ Ξ(X) ⊆ Ξ(a ◦X).



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Action Invariance



[Here, a ◦X = {a ◦ v : v ∈ X}.]

Note that ⊢ is action invariant iff the associatedconsequence operation Ξ satisfies the condition

a ◦ Ξ(X) ⊆ Ξ(a ◦X).

By extension, we call action-invariant anyconsequence operation Ξ that satisfies thepreceding condition.



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Finitary Consequence Relations

A consequence relation ⊢ over M is called finitary,provided for all X ∪ {u} ⊆M , if X ⊢ u, then thereis a finite subset Y of X such that Y ⊢ u.



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Finitary Consequence Relations

A consequence relation ⊢ over M is called finitary,provided for all X ∪ {u} ⊆M , if X ⊢ u, then thereis a finite subset Y of X such that Y ⊢ u.

Note that ⊢ is finitary iff the associatedconsequence operation Ξ satisfies a relatedcondition for all X ⊆M and all u ∈M : if u ∈ Ξ(X),then u ∈ Ξ(Y ), for some finite subset Y of X.

We use the term finitary for any consequenceoperation that satisfies the preceding condition.


Short Excursion to Universal Algebra


Logical Consequence

Excursion to UAFormula AlgebraHomomorphismsVarieties

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The Formula Algebra of Signature L

L: a language of algebras in logicFor example, L = {∧,∨, ·,→, 0, 1}

X: an infinite countable set (whose members arereferred to as variables)


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X: an infinite countable set (whose members arereferred to as variables)The set Fm(X) of L-formulas over X is defined as follows:

(a) Inductive beginning: The constants 0, 1 and every member of X

is a formula.

(b) Inductive steps: If α and β are formulas, then so are α · β,

α ∧ β, α ∨ β and α → β.

(c) All formulas are generated by (a) and (b) above.


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is a formula.


α ∧ β, α ∨ β and α → β.


An L-algebra A = 〈A,∧,∨, ·,→, 0, 1〉 is any algebrain the preceding signature.


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is a formula.


α ∧ β, α ∨ β and α → β.


An L-algebra A = 〈A,∧,∨, ·,→, 0, 1〉 is any algebrain the preceding signature.Boolean algebras and Heyting algebras areexamples of L-algebras. Another example is theformula algebra Fm(X) of signature L over X.


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Homomorphisms and Congruences

A homomorphism ϕ : A → B between twoL-algebras is a map that preserves all operations:that is, for all a, b ∈ A and all ⋆ ∈ {∧,∨, ·,→},ϕ(0) = 0, ϕ(1) = 1, and ϕ(a ⋆ b) = ϕ(a) ⋆ ϕ(b).


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A congruence relation of an L-algebraA = 〈A,∧,∨, ·,→, 0, 1〉 is an equivalence relation Θon A satisfying the following substitution property:whenever a Θ b and c Θ d, then a ⋆ c Θ b ⋆ d, for allthe operations ⋆ ∈ {∧,∨, ·,→}.


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Note that if ϕ : A → B is a homomorphism, thenKer(ϕ) = {(a, b) ∈ A2 : ϕ(a) = ϕ(b)} is acongruence relation of A. Moreover, everycongruence relation is of this form.


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Note that if ϕ : A → B is a homomorphism, thenKer(ϕ) = {(a, b) ∈ A2 : ϕ(a) = ϕ(b)} is acongruence relation of A. Moreover, everycongruence relation is of this form.

Subalgebras and Direct Products


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Birkhoff’s Theorem

A class of L-algebras is called a variety provided itis closed under direct products, subalgebras andhomomorphic images.


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An L-equation is an an ordered pair (α, β) ofL-formulas over X, often written more suggestivelyas α ≈ β. The set Fm(X) × Fm(X) of allL-equations over X will be denoted by Eq(X).


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An L-equation is an an ordered pair (α, β) ofL-formulas over X, often written more suggestivelyas α ≈ β. The set Fm(X) × Fm(X) of allL-equations over X will be denoted by Eq(X).

Theorem [G. Birkhoff, 1935]A class of L-algebras is an equational class iff it isa variety.


Consequence Operations and Relations onFormula Structures


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|=K

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Consequence Relations on Fm(X) and Fm(X)

Consequence relations overFm(X), Fm(X)k (k > 1), and sequents.


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The endomorphism monoid End of Fm(X) acts oneach of the sets above. For example, if(α, β) ∈ Fm(X)2 and σ ∈ End, thenσ ◦ (α, β) = (σ(α), σ(β)).


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In this setting, we use the term substitutioninvariant instead of action invariant.


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LetK bea class ofL-algebras and Σ∪{ε} ⊆ Eq(X).We say that ε is a K-consequence of Σ – Σ |=

Kε –

if for every A ∈ K and every homomorphismϕ : Fm(X) → A, if Σ ⊆ Ker(ϕ), then ε ∈ Ker(ϕ).


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|=K

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LetK bea class ofL-algebras and Σ∪{ε} ⊆ Eq(X).We say that ε is a K-consequence of Σ – Σ |=

Kε –

if for every A ∈ K and every homomorphismϕ : Fm(X) → A, if Σ ⊆ Ker(ϕ), then ε ∈ Ker(ϕ).

|=K

is a substitution invariant consequence relationover Eq(X). It is finitary whenever K is a variety.


Equivalence of Two Consequence Relations


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EquivalenceAction-Invariant MapsEquivalenceDiagramAlgebraizabilityBlok-PigozziMore . . .

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Action-Invariant Maps

The abstract notion of consequence relation wediscussed in the previous section is, for manypurposes, too fine-grained. Two consequencerelations, even on different sets, could count asdistinct not because they validate differententailments, but merely in virtue of the fact thatthey present the same entailments under differentguises. There are circumstances under which itmay be appropriate to identify such consequencerelations with each other. We’ll only consider thecase of action-invariant consequence relations.


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Action-Invariant Maps

The abstract notion of consequence relation wediscussed in the previous section is, for manypurposes, too fine-grained. Two consequencerelations, even on different sets, could count asdistinct not because they validate differententailments, but merely in virtue of the fact thatthey present the same entailments under differentguises. There are circumstances under which itmay be appropriate to identify such consequencerelations with each other. We’ll only consider thecase of action-invariant consequence relations.

Let M1 and M2 be A-sets. A mappingϕ : M1 → ℘(M2) is said to be action-invariant ifϕ(a ◦ u) = a ◦ ϕ(u), for all a ∈ A and all u ∈M1.


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Equivalence of Consequence Relations

Let ⊢1,⊢2 be action-invariant consequencerelations over the sets M1,M2, respectively. Wesay that ⊢1 and ⊢2 are equivalent provided thereexist action-invariant maps maps τ : M1 → ℘ (M2)and ρ : M2 → ℘ (M1) such that the followingconditions hold for every X ∪ {u} ⊆M1 and forevery Y ∪ {v} ⊆M2:


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(S1) X ⊢1 u iff τ (X) ⊢2 τ (u);

(S2) Y ⊢2 v iff ρ (Y ) ⊢1 ρ (v);

(S3) v ⊣⊢2 τ (ρ (v));

(S4) u ⊣⊢1 ρ (τ (u)).

The maps τ and ρ are called transformers.


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(S1) X ⊢1 u iff τ (X) ⊢2 τ (u);

(S2) Y ⊢2 v iff ρ (Y ) ⊢1 ρ (v);

(S3) v ⊣⊢2 τ (ρ (v));

(S4) u ⊣⊢1 ρ (τ (u)).

The maps τ and ρ are called transformers.

The relations ⊢1and ⊢2 are equivalent iff either (S1)and (S3) hold, or else (S2) and (S4) hold.


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The Diagram of Consequence OperatorsThere are equivalent characterizations in terms ofthe corresponding consequent operations (Wewrite Ξ1 for Ξ⊢1

and Ξ2 for Ξ⊢2):

(S1) X ⊢1 u iff τ (X) ⊢2 τ (u)

(S3) v ⊣⊢2 τ (ρ (v))

(S1′) Ξ1 = τ−1Ξ2τ

(S3′) Ξ2τρ = Ξ2


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(S1) X ⊢1 u iff τ (X) ⊢2 τ (u)

(S3) v ⊣⊢2 τ (ρ (v))

(S1′) Ξ1 = τ−1Ξ2τ

(S3′) Ξ2τρ = Ξ2

℘(M2)ρ

//

Ξ2

��

℘(M1)

Ξ1

��

τ //℘(M2)

Ξ2

��

Th(Ξ2) Th(Ξ1)oo

ρ−1

oooo Th(Ξ2)oo

τ−1

oooo


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(S1) X ⊢1 u iff τ (X) ⊢2 τ (u)

(S3) v ⊣⊢2 τ (ρ (v))

(S1′) Ξ1 = τ−1Ξ2τ

(S3′) Ξ2τρ = Ξ2

℘(M2)ρ

//

Ξ2

��

℘(M1)

Ξ1

��

τ //℘(M2)

Ξ2

��

Th(Ξ2) Th(Ξ1)oo

ρ−1

oooo Th(Ξ2)oo

τ−1

oooo

τ−1 ↾Th(Ξ2) is the inverse of ρ−1 ↾Th(Ξ1)

(i.e, ρ−1τ−1Ξ2 = Ξ2 and τ−1ρ−1Ξ1 = Ξ1)


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Algebraizability

A consequence relation ⊢ over Fm is calledalgebraizable provided there exists a class K ofL-algebras such that ⊢ and |=

Kare equivalent.


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Algebraizability


Kare equivalent.

When the preceding holds, we say that the class Kis an equivalent algebraic semantics for theconsequence relation ⊢.


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Algebraizability


Kare equivalent.


BA: the variety of Boolean algebras

CL: classical propositional calculus

τ : Fm→ ℘(Eq), defined by τ(α) = {α ≈ 1}

ρ : Eq → ℘(Fm), defined byρ(γ ≈ δ) = {(γ → δ) ∧ (δ → γ)}


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Algebraizability


Kare equivalent.


BA: the variety of Boolean algebras

CL: classical propositional calculus

τ : Fm→ ℘(Eq), defined by τ(α) = {α ≈ 1}

ρ : Eq → ℘(Fm), defined byρ(γ ≈ δ) = {(γ → δ) ∧ (δ → γ)}

One can check that ⊢CL

and |=BA

are equivalent viaτ and ρ. Thus, ⊢

CLis algebraizable.


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The Blok-Pigozzi Theorem

We have seen that if two action-invariantconsequence relations are equivalent, then thelattices of their theories are isomorphic. Is theconverse true?


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The Blok-Pigozzi Theorem

We have seen that if two action-invariantconsequence relations are equivalent, then thelattices of their theories are isomorphic. Is theconverse true?

Theorem (Wim Blok and Don Pigozzi; 1989)A substitution-invariant consequence relation ⊢over Fm(X) is algebraizable – with equivalentalgebraic semantics a class K of L- algebras– ifand only if there exists a lattice isomorphismbetween Th(⊢) and Th(|=K) that commutes withinverse substitutions.[The latter condition means that if ϕ : Th(⊢) → Th(|=K) is the

lattice isomorphism in question, then for all σ ∈ End(Fm(X)) and all

Y ∈ Th(⊢), ϕ(σ−1(Y )) = σ−1(ϕ(Y )).]


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More . . .

The isomorphism ϕ above is induced by theequivalence, in the sense that the diagram belowcommutes:

℘(Eq(X))ρ

//

Ξ|=K��

℘(Fm(X))

Ξ⊢��

τ //℘(Eq(X))

Ξ|=K��

Th(Ξ|=K) //

ϕ// //Th(Ξ⊢)

//

ϕ−1

// // Th(Ξ|=K)


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Equiv. in Modules



More . . .

The isomorphism ϕ above is induced by theequivalence, in the sense that the diagram belowcommutes:

℘(Eq(X))ρ

//

Ξ|=K��

℘(Fm(X))

Ξ⊢��

τ //℘(Eq(X))

Ξ|=K��

Th(Ξ|=K) //

ϕ// //Th(Ξ⊢)

//

ϕ−1

// // Th(Ξ|=K)

A Short List of Generalizations: W. Blok and D.Pigozzi, 1989 (finite dimensional systems); J.Rebagliato and V. Verdú, 1993 (associativesequents); A. Pynko, 1999 (finite-demensionalsequents for finitary substitution invariantconsequence relations); J. Raftery, 2006(associative sequents).


Modules over Quantales


Logical Consequence

Excursion to UA

Special Cons. Ops

Equivalence

ModulesOverviewQuantalesResiduated MapsModules

Cons. in Modules

Equiv. in Modules



An Overview

Let M be an A-set. The natural action of A on Mextends to an action of the corresponding powersets. More specifically, for B ⊆ A and X ⊆M , wedefine

B ◦X = {b ◦ x : b ∈ B, x ∈ X}.


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An Overview


B ◦X = {b ◦ x : b ∈ B, x ∈ X}.

℘(A) is a ringlike object – in which set-union playsthe role of addition and complex product serves asmultiplication.


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Cons. in Modules

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An Overview


B ◦X = {b ◦ x : b ∈ B, x ∈ X}.

℘(A) is a ringlike object – in which set-union playsthe role of addition and complex product serves asmultiplication.On the other hand, ℘(M) is astructure corresponding to an abelian group, withset-union playing again the role of addition. Theaforementioned action of power sets possessesthe critical property of being residuated, which, inthis instance, means that it preserves arbitraryunions in each coordinate.


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Cons. in Modules

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An Overview


B ◦X = {b ◦ x : b ∈ B, x ∈ X}.

℘(A) is a ringlike object – in which set-union playsthe role of addition and complex product serves asmultiplication.On the other hand, ℘(M) is astructure corresponding to an abelian group, withset-union playing again the role of addition. Theaforementioned action of power sets possessesthe critical property of being residuated, which, inthis instance, means that it preserves arbitraryunions in each coordinate.The preceding considerations lead to the generalconcept of a (left) module, to be defined shortly.


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Quantales

A quantale is an algebraic structure A = 〈A,∨, ·, 1〉

such that:

(Q1) 〈A,∨〉 is a complete join semilattice (and,

hence, a complete lattice);

(Q2) 〈A, ·, 1〉 is a monoid;

(Q3) For all x ∈ A, {yi}i∈I ⊆ A,

x ·∨

i∈I

yi =∨

i∈I

(x · yi)

and(∨

i∈I

yi) · x =∨

i∈I

(yi · x).


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Quantales

A quantale is an algebraic structure A = 〈A,∨, ·, 1〉

such that:

(Q1) 〈A,∨〉 is a complete join semilattice (and,

hence, a complete lattice);

(Q2) 〈A, ·, 1〉 is a monoid;

(Q3) For all x ∈ A, {yi}i∈I ⊆ A,

x ·∨

i∈I

yi =∨

i∈I

(x · yi)

and(∨

i∈I

yi) · x =∨

i∈I

(yi · x).

The multiplication of a quantale is an example of abinary residuated map, which will be definedshortly.


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Residuated MapsLet P and Q be posets. A map ϕ : P → Q is saidto be residuated provided there exists a mapϕ⋆ : Q → P such that

ϕ(x) ≤ y ⇐⇒ x ≤ ϕ⋆(y),

for all x ∈ P and y ∈ Q. We refer to ϕ⋆ as theresidual of ϕ.


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ϕ(x) ≤ y ⇐⇒ x ≤ ϕ⋆(y),

for all x ∈ P and y ∈ Q. We refer to ϕ⋆ as theresidual of ϕ. [If P and Q are complete, then ϕ isresiduated iff it preserving all joins.]


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ϕ(x) ≤ y ⇐⇒ x ≤ ϕ⋆(y),

for all x ∈ P and y ∈ Q. We refer to ϕ⋆ as theresidual of ϕ. [If P and Q are complete, then ϕ isresiduated iff it preserving all joins.]Let P, Q and R be posets. A map ◦ : P × Q → R

is said to be residuated provided there exist maps\◦ : P × R → Q and /◦ : R × Q → P such that

x ◦ y ≤ z ⇐⇒ x ≤ z/◦y ⇐⇒ y ≤ x\◦z,

for all x ∈ P, y ∈ Q, z ∈ R.


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ϕ(x) ≤ y ⇐⇒ x ≤ ϕ⋆(y),



x ◦ y ≤ z ⇐⇒ x ≤ z/◦y ⇐⇒ y ≤ x\◦z,

for all x ∈ P, y ∈ Q, z ∈ R. [Again, if P, Q and R

are complete, then ◦ is residuated iff it preservesall joins in each coordinate.]


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ϕ(x) ≤ y ⇐⇒ x ≤ ϕ⋆(y),



x ◦ y ≤ z ⇐⇒ x ≤ z/◦y ⇐⇒ y ≤ x\◦z,

for all x ∈ P, y ∈ Q, z ∈ R. [Again, if P, Q and R

are complete, then ◦ is residuated iff it preservesall joins in each coordinate.]We refer to the operations \◦ and /◦ as the leftresidual and right residual of ◦, respectively.


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Let A be a quantale and let P a completejoin-semilattice. A (left) module action of A on P isa map ◦ : A× P → P satisfying the followingconditions, for all x ∈ P and for all a, b ∈ A:

(1) 1 ◦ x = x

(2) a ◦ (b ◦ x) = ab ◦ x

(3) ◦ is residuated; equivalently, it satisfies thefollowing distributive law, for all a ∈ A,{ui}i∈I ⊆ P :

a ◦∨

i∈I

ui =∨

i∈I

(a ◦ ui).


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Let A be a quantale and let P a completejoin-semilattice. A (left) module action of A on P isa map ◦ : A× P → P satisfying the followingconditions, for all x ∈ P and for all a, b ∈ A:

(1) 1 ◦ x = x

(2) a ◦ (b ◦ x) = ab ◦ x

(3) ◦ is residuated; equivalently, it satisfies thefollowing distributive law, for all a ∈ A,{ui}i∈I ⊆ P :

a ◦∨

i∈I

ui =∨

i∈I

(a ◦ ui).

We will refer to P = 〈P, ◦〉 is a (left) A-module anddenote the residuals of ◦ by \◦ and /◦. Note that A

is an A-module with respect to its multiplication.


Consequence Operations in Modules


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Modules

Cons. in ModulesA − Mod

Cons. OpsQuotientsCyclic Modules

Equiv. in Modules



The Category A −Mod

Let P and Q be A modules. A map τ : P → Q iscalled a module morphism provided it is residuatedand action-invariant. Of course, the latter conditionmeans that a ◦ τ(x) = τ(a ◦ x), for all a ∈ A and allx ∈ P .


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The Category A −Mod

Let P and Q be A modules. A map τ : P → Q iscalled a module morphism provided it is residuatedand action-invariant. Of course, the latter conditionmeans that a ◦ τ(x) = τ(a ◦ x), for all a ∈ A and allx ∈ P .

For a given quantale A, A −Mod will denote thecategory of A-modules and A-module morphisms.


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Consequence Operations in A −Mod

A closure operator on a complete join-semilatticeP is a map ξ : P → P with the usual properties ofbeing isotone, enlarging (x ≤ ξ(x)), andidempotent. It is completely determined by itsimage Pξ, which is a closure system (that is, asubset of P that is closed under arbitrary meets inP.)


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Consequence Operations in A −Mod

A closure operator on a complete join-semilatticeP is a map ξ : P → P with the usual properties ofbeing isotone, enlarging (x ≤ ξ(x)), andidempotent. It is completely determined by itsimage Pξ, which is a closure system (that is, asubset of P that is closed under arbitrary meets inP.)

If P is an A-module, we use the termaction-invariant consequence operation for aclosure operator ξ on P that satisfiesa ◦ ξ(u) ≤ ξ(a ◦ u), for all a ∈ A and u ∈ P .


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Quotients in A −Mod

LemmaLet P be an A-module and let ξ be an actioninvariant consequence operation on A.

(1) Pξ is an A-module with respect to the scalarmultiplication ◦ξ : A× Pξ → Pξ – defined bya ◦ξ ξ(u) = ξ(a ◦ u), for all a ∈ A and all u ∈ P.

(2) The map ξ : P → Pξ, with the module structureof Pξ defined by (1) above, is a modulemorphism.

(3) Every epimorphic image of P in A −Mod isisomorphic to an A-module of the form Pξ, forsome action invariant consequence operation ξon P.Note: The epimorphisms in A −Mod are thesurjective maps [José Gil-Férez, 2008]


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Cyclic Modules

An A-module P is called cyclic, if there exists anelement u ∈ P , such that P = A ◦ u.


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Cyclic Modules


Example: Let E denote the endomorphism monoidof Fm(X). Then ℘(Fm(X)) is a cyclic℘(E)-module. Indeed, just let u = X or u = p,where p is any fixed variable.


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Cyclic Modules


Example: Let E denote the endomorphism monoidof Fm(X). Then ℘(Fm(X)) is a cyclic℘(E)-module. Indeed, just let u = X or u = p,where p is any fixed variable.

PropositionEvery cyclic A-module is isomorphic to a moduleof the form Aξ, for some action-invariantconsequence operation ξ on A.


Equivalence in the Setting of Modules


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Modules

Cons. in Modules

Equiv. in ModulesEquivalenceInduced Equivalences



Equivalence, Revisited

The preceding discussion shows that theaction-invariant consequence operations on anA-module P correspond bijectively to theepimorphic images of P. Thus, these operationsmay be identified with objects of the categoryA −Mod. Not surprisingly then, we stipulate thatthey are equivalent if the A-modules associatedwith them are isomorphic.


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Cons. in Modules




Equivalence, Revisited

The preceding discussion shows that theaction-invariant consequence operations on anA-module P correspond bijectively to theepimorphic images of P. Thus, these operationsmay be identified with objects of the categoryA −Mod. Not surprisingly then, we stipulate thatthey are equivalent if the A-modules associatedwith them are isomorphic.

DefinitionLet ξ and ζ be consequence operations on theA-modules P and Q, respectively.■ We say that ξ and ζ are equivalent, if there exists

a module isomorphism ϕ : Pξ → Qζ . We refer tothe isomorphism ϕ as an equivalence between ξand ζ.


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Induced Equivalences

Definition

Let ϕ : Pξ → Qη be an equivalence between ξ andζ. We say that the equivalence ϕ is induced by themodule morphisms τ : P → Q and ρ : Q → P, ifϕξ = ζτ and ϕ−1ζ = ξρ. In this case we will saythat ξ and ζ are equivalent via τ and ρ.

Qρ

//

ζ��

P

ξ��

τ //Q

ζ��

Qζ//ϕ−1

// //Pξoo

ρ∗oooo

//ϕ

// //Qζoo

τ∗oooo


The Fundamental Categorical Property


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Cons. in Modules

Equiv. in Modules

Fundamental PropertyTeaserDefinitionCharacterizationCo-productsSequents


Teaser

Assume that P and Q are A-modules, ξ and ζ areaction-invariant consequence operations on P andQ respectively, and ϕ : P

ξ→ Q

ζis an isomorphism

of modules. We wish to find a module morphismτ : P → Q that induces ϕ, that is, it completes thesquare below.

Pτ //___

ξ��

Q

ζ��

Pξ

//ϕ

// // Qζ

(S)


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Modules

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Teaser


ξ→ Q

ζis an isomorphism


Pτ //___

ξ��

Q

ζ��

Pξ

//ϕ

// // Qζ

(S)

TheoremThe objects P of the category A −Mod for whichevery square of type (S) can be completed areprecisely


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Teaser


ξ→ Q

ζis an isomorphism


Pτ //___

ξ��

Q

ζ��

Pξ

//ϕ

// // Qζ

(S)

TheoremThe objects P of the category A −Mod for whichevery square of type (S) can be completed areprecisely the projective objects of A −Mod.


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Definition of a Projective Object

An object P of A −Mod is called projective, ifwhenever there are modules Q and R and modulemorphisms ψ : Q → R and χ : P → R, with ψ anepimorphism, then there exists a morphismϕ : P → Q, such that χ = ψϕ.

Pϕ

//___

χ��

?

?

?

?

?

?

?

?

Q

ψ��

R

(T)


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Characterization of Cyclic Projective Modules

TheoremFor an A-module P = 〈P, ◦〉, the followingstatements are equivalent.

(1) P is cyclic and a projective object in A −Mod.

(2) There exist elements b ∈ A and u ∈ P such thatb ◦ u = u and [(a ◦ u)/◦u]b = ab, for all a ∈ A.


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Cons. in Modules

Equiv. in Modules



Characterization of Cyclic Projective Modules

TheoremFor an A-module P = 〈P, ◦〉, the followingstatements are equivalent.

(1) P is cyclic and a projective object in A −Mod.

(2) There exist elements b ∈ A and u ∈ P such thatb ◦ u = u and [(a ◦ u)/◦u]b = ab, for all a ∈ A.

Example: ℘(Fm(X)) is a projective cyclic℘(E)-module, where E denote the endomorphismmonoid of Fm(X).To verify cyclicity and Condition 2, choose u = {p},where p is any fixed variable, and b = {κp}, whereκp is the substitution that sends all variables to p.


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Co-productsExample: ℘(Eq) is a projective cyclic℘(Σ)-module.To verify cyclicity and Condition 2 of the precedingtheorem, choose a partition {X1,X2} of X withX1,X2 infinite, and choose p ∈ X1, q ∈ X2. Then setu = {p ≈ q}, b = {p ≈ q}/◦X1 × X2.


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Co-products in A −Mod exist. The underlyingalgebra of the co-product

∐Pi, of (Pi|i ∈ I) in

A −Mod, is the direct product∏

Pi. For eachi ∈ I, the associated embedding ϕi : Pi →

∏Pi

sends an element x ∈ Pi to the element whose ithcoordinate is x and whose all other coordinates arethe least elements of the corresponding factors.


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Cons. in Modules

Equiv. in Modules




Co-products in A −Mod exist. The underlyingalgebra of the co-product

∐Pi, of (Pi|i ∈ I) in

A −Mod, is the direct product∏

Pi. For eachi ∈ I, the associated embedding ϕi : Pi →

∏Pi

sends an element x ∈ Pi to the element whose ithcoordinate is x and whose all other coordinates arethe least elements of the corresponding factors.

The co-product of a family of projective objects inA −Mod is projective.


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Consequence Operations on Sequents

Example

Let Seq denote a set of sequents closed undertype. Then ℘(Seq) is in general a non-cyclic℘(E)-module. However, it is a co-product of cyclicprojective modules. Thus, it is a projective℘(E)-module.

equivalence of logical consequence relations: an order

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