Post on 13-Mar-2022
transcript
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
Excursion to UA
Special Cons. Ops
Equivalence
Modules
Cons. in Modules
Equiv. in Modules
Fundamental Property
Consequence relations C. Tsinakis - slide #2
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.
AbstractReferencesHistorical Note
Logical Consequence
Excursion to UA
Special Cons. Ops
Equivalence
Modules
Cons. in Modules
Equiv. in Modules
Fundamental Property
Consequence relations C. Tsinakis - slide #3
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.
AbstractReferencesHistorical Note
Logical Consequence
Excursion to UA
Special Cons. Ops
Equivalence
Modules
Cons. in Modules
Equiv. in Modules
Fundamental Property
Consequence relations C. Tsinakis - slide #4
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.
Consequence relations C. Tsinakis - slide #5
Logical Consequence
AbstractReferencesHistorical Note
Logical ConsequenceRelationsOperationsCorrespondenceTheoriesLogical FormA-setsInvarianceFinitarity
Excursion to UA
Special Cons. Ops
Equivalence
Modules
Cons. in Modules
Equiv. in Modules
Fundamental Property
Consequence relations C. Tsinakis - slide #6
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).
AbstractReferencesHistorical Note
Logical ConsequenceRelationsOperationsCorrespondenceTheoriesLogical FormA-setsInvarianceFinitarity
Excursion to UA
Special Cons. Ops
Equivalence
Modules
Cons. in Modules
Equiv. in Modules
Fundamental Property
Consequence relations C. Tsinakis - slide #6
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
AbstractReferencesHistorical Note
Logical ConsequenceRelationsOperationsCorrespondenceTheoriesLogical FormA-setsInvarianceFinitarity
Excursion to UA
Special Cons. Ops
Equivalence
Modules
Cons. in Modules
Equiv. in Modules
Fundamental Property
Consequence relations C. Tsinakis - slide #7
Consequence Operations
An equivalent approach to logical consequence,due to Alfred Tarksi, consists in giving an accountof it via a closure operator.
AbstractReferencesHistorical Note
Logical ConsequenceRelationsOperationsCorrespondenceTheoriesLogical FormA-setsInvarianceFinitarity
Excursion to UA
Special Cons. Ops
Equivalence
Modules
Cons. in Modules
Equiv. in Modules
Fundamental Property
Consequence relations C. Tsinakis - slide #7
Consequence Operations
An equivalent approach to logical consequence,due to Alfred Tarksi, consists in giving an accountof it via a closure operator.
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).
AbstractReferencesHistorical Note
Logical ConsequenceRelationsOperationsCorrespondenceTheoriesLogical FormA-setsInvarianceFinitarity
Excursion to UA
Special Cons. Ops
Equivalence
Modules
Cons. in Modules
Equiv. in Modules
Fundamental Property
Consequence relations C. Tsinakis - slide #7
Consequence Operations
An equivalent approach to logical consequence,due to Alfred Tarksi, consists in giving an accountof it via a closure operator.
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.
AbstractReferencesHistorical Note
Logical ConsequenceRelationsOperationsCorrespondenceTheoriesLogical FormA-setsInvarianceFinitarity
Excursion to UA
Special Cons. Ops
Equivalence
Modules
Cons. in Modules
Equiv. in Modules
Fundamental Property
Consequence relations C. Tsinakis - slide #8
Correspondence
Given a set M , there exists a bijectivecorrespondence between all consequenceoperations Ξ on ℘(M) and all consequencerelations ⊢ over M . More specifically:
AbstractReferencesHistorical Note
Logical ConsequenceRelationsOperationsCorrespondenceTheoriesLogical FormA-setsInvarianceFinitarity
Excursion to UA
Special Cons. Ops
Equivalence
Modules
Cons. in Modules
Equiv. in Modules
Fundamental Property
Consequence relations C. Tsinakis - slide #8
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).
AbstractReferencesHistorical Note
Logical ConsequenceRelationsOperationsCorrespondenceTheoriesLogical FormA-setsInvarianceFinitarity
Excursion to UA
Special Cons. Ops
Equivalence
Modules
Cons. in Modules
Equiv. in Modules
Fundamental Property
Consequence relations C. Tsinakis - slide #8
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).■ 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 .
AbstractReferencesHistorical Note
Logical ConsequenceRelationsOperationsCorrespondenceTheoriesLogical FormA-setsInvarianceFinitarity
Excursion to UA
Special Cons. Ops
Equivalence
Modules
Cons. in Modules
Equiv. in Modules
Fundamental Property
Consequence relations C. Tsinakis - slide #8
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).■ 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⊢
= ⊢.
AbstractReferencesHistorical Note
Logical ConsequenceRelationsOperationsCorrespondenceTheoriesLogical FormA-setsInvarianceFinitarity
Excursion to UA
Special Cons. Ops
Equivalence
Modules
Cons. in Modules
Equiv. in Modules
Fundamental Property
Consequence relations C. Tsinakis - slide #9
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}
AbstractReferencesHistorical Note
Logical ConsequenceRelationsOperationsCorrespondenceTheoriesLogical FormA-setsInvarianceFinitarity
Excursion to UA
Special Cons. Ops
Equivalence
Modules
Cons. in Modules
Equiv. in Modules
Fundamental Property
Consequence relations C. Tsinakis - slide #9
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}
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.
AbstractReferencesHistorical Note
Logical ConsequenceRelationsOperationsCorrespondenceTheoriesLogical FormA-setsInvarianceFinitarity
Excursion to UA
Special Cons. Ops
Equivalence
Modules
Cons. in Modules
Equiv. in Modules
Fundamental Property
Consequence relations C. Tsinakis - slide #9
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}
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).
AbstractReferencesHistorical Note
Logical ConsequenceRelationsOperationsCorrespondenceTheoriesLogical FormA-setsInvarianceFinitarity
Excursion to UA
Special Cons. Ops
Equivalence
Modules
Cons. in Modules
Equiv. in Modules
Fundamental Property
Consequence relations C. Tsinakis - slide #10
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?
AbstractReferencesHistorical Note
Logical ConsequenceRelationsOperationsCorrespondenceTheoriesLogical FormA-setsInvarianceFinitarity
Excursion to UA
Special Cons. Ops
Equivalence
Modules
Cons. in Modules
Equiv. in Modules
Fundamental Property
Consequence relations C. Tsinakis - slide #10
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.
AbstractReferencesHistorical Note
Logical ConsequenceRelationsOperationsCorrespondenceTheoriesLogical FormA-setsInvarianceFinitarity
Excursion to UA
Special Cons. Ops
Equivalence
Modules
Cons. in Modules
Equiv. in Modules
Fundamental Property
Consequence relations C. Tsinakis - slide #11
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 “·”.
AbstractReferencesHistorical Note
Logical ConsequenceRelationsOperationsCorrespondenceTheoriesLogical FormA-setsInvarianceFinitarity
Excursion to UA
Special Cons. Ops
Equivalence
Modules
Cons. in Modules
Equiv. in Modules
Fundamental Property
Consequence relations C. Tsinakis - slide #12
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}.]
AbstractReferencesHistorical Note
Logical ConsequenceRelationsOperationsCorrespondenceTheoriesLogical FormA-setsInvarianceFinitarity
Excursion to UA
Special Cons. Ops
Equivalence
Modules
Cons. in Modules
Equiv. in Modules
Fundamental Property
Consequence relations C. Tsinakis - slide #12
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}.]
Note that ⊢ is action invariant iff the associatedconsequence operation Ξ satisfies the condition
a ◦ Ξ(X) ⊆ Ξ(a ◦X).
AbstractReferencesHistorical Note
Logical ConsequenceRelationsOperationsCorrespondenceTheoriesLogical FormA-setsInvarianceFinitarity
Excursion to UA
Special Cons. Ops
Equivalence
Modules
Cons. in Modules
Equiv. in Modules
Fundamental Property
Consequence relations C. Tsinakis - slide #12
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}.]
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.
AbstractReferencesHistorical Note
Logical ConsequenceRelationsOperationsCorrespondenceTheoriesLogical FormA-setsInvarianceFinitarity
Excursion to UA
Special Cons. Ops
Equivalence
Modules
Cons. in Modules
Equiv. in Modules
Fundamental Property
Consequence relations C. Tsinakis - slide #13
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.
AbstractReferencesHistorical Note
Logical ConsequenceRelationsOperationsCorrespondenceTheoriesLogical FormA-setsInvarianceFinitarity
Excursion to UA
Special Cons. Ops
Equivalence
Modules
Cons. in Modules
Equiv. in Modules
Fundamental Property
Consequence relations C. Tsinakis - slide #13
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.
Consequence relations C. Tsinakis - slide #14
Short Excursion to Universal Algebra
AbstractReferencesHistorical Note
Logical Consequence
Excursion to UAFormula AlgebraHomomorphismsVarieties
Special Cons. Ops
Equivalence
Modules
Cons. in Modules
Equiv. in Modules
Fundamental Property
Consequence relations C. Tsinakis - slide #15
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)
AbstractReferencesHistorical Note
Logical Consequence
Excursion to UAFormula AlgebraHomomorphismsVarieties
Special Cons. Ops
Equivalence
Modules
Cons. in Modules
Equiv. in Modules
Fundamental Property
Consequence relations C. Tsinakis - slide #15
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)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.
AbstractReferencesHistorical Note
Logical Consequence
Excursion to UAFormula AlgebraHomomorphismsVarieties
Special Cons. Ops
Equivalence
Modules
Cons. in Modules
Equiv. in Modules
Fundamental Property
Consequence relations C. Tsinakis - slide #15
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)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.
An L-algebra A = 〈A,∧,∨, ·,→, 0, 1〉 is any algebrain the preceding signature.
AbstractReferencesHistorical Note
Logical Consequence
Excursion to UAFormula AlgebraHomomorphismsVarieties
Special Cons. Ops
Equivalence
Modules
Cons. in Modules
Equiv. in Modules
Fundamental Property
Consequence relations C. Tsinakis - slide #15
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)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.
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.
AbstractReferencesHistorical Note
Logical Consequence
Excursion to UAFormula AlgebraHomomorphismsVarieties
Special Cons. Ops
Equivalence
Modules
Cons. in Modules
Equiv. in Modules
Fundamental Property
Consequence relations C. Tsinakis - slide #16
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).
AbstractReferencesHistorical Note
Logical Consequence
Excursion to UAFormula AlgebraHomomorphismsVarieties
Special Cons. Ops
Equivalence
Modules
Cons. in Modules
Equiv. in Modules
Fundamental Property
Consequence relations C. Tsinakis - slide #16
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).
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 ⋆ ∈ {∧,∨, ·,→}.
AbstractReferencesHistorical Note
Logical Consequence
Excursion to UAFormula AlgebraHomomorphismsVarieties
Special Cons. Ops
Equivalence
Modules
Cons. in Modules
Equiv. in Modules
Fundamental Property
Consequence relations C. Tsinakis - slide #16
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).
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 ⋆ ∈ {∧,∨, ·,→}.
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.
AbstractReferencesHistorical Note
Logical Consequence
Excursion to UAFormula AlgebraHomomorphismsVarieties
Special Cons. Ops
Equivalence
Modules
Cons. in Modules
Equiv. in Modules
Fundamental Property
Consequence relations C. Tsinakis - slide #16
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).
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 ⋆ ∈ {∧,∨, ·,→}.
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
AbstractReferencesHistorical Note
Logical Consequence
Excursion to UAFormula AlgebraHomomorphismsVarieties
Special Cons. Ops
Equivalence
Modules
Cons. in Modules
Equiv. in Modules
Fundamental Property
Consequence relations C. Tsinakis - slide #17
Birkhoff’s Theorem
A class of L-algebras is called a variety provided itis closed under direct products, subalgebras andhomomorphic images.
AbstractReferencesHistorical Note
Logical Consequence
Excursion to UAFormula AlgebraHomomorphismsVarieties
Special Cons. Ops
Equivalence
Modules
Cons. in Modules
Equiv. in Modules
Fundamental Property
Consequence relations C. Tsinakis - slide #17
Birkhoff’s Theorem
A class of L-algebras is called a variety provided itis closed under direct products, subalgebras andhomomorphic images.
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).
AbstractReferencesHistorical Note
Logical Consequence
Excursion to UAFormula AlgebraHomomorphismsVarieties
Special Cons. Ops
Equivalence
Modules
Cons. in Modules
Equiv. in Modules
Fundamental Property
Consequence relations C. Tsinakis - slide #17
Birkhoff’s Theorem
A class of L-algebras is called a variety provided itis closed under direct products, subalgebras andhomomorphic images.
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 relations C. Tsinakis - slide #18
Consequence Operations and Relations onFormula Structures
AbstractReferencesHistorical Note
Logical Consequence
Excursion to UA
Special Cons. Ops
|=K
Equivalence
Modules
Cons. in Modules
Equiv. in Modules
Fundamental Property
Consequence relations C. Tsinakis - slide #19
Consequence Relations on Fm(X) and Fm(X)
Consequence relations overFm(X), Fm(X)k (k > 1), and sequents.
AbstractReferencesHistorical Note
Logical Consequence
Excursion to UA
Special Cons. Ops
|=K
Equivalence
Modules
Cons. in Modules
Equiv. in Modules
Fundamental Property
Consequence relations C. Tsinakis - slide #19
Consequence Relations on Fm(X) and Fm(X)
Consequence relations overFm(X), Fm(X)k (k > 1), and sequents.
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)).
AbstractReferencesHistorical Note
Logical Consequence
Excursion to UA
Special Cons. Ops
|=K
Equivalence
Modules
Cons. in Modules
Equiv. in Modules
Fundamental Property
Consequence relations C. Tsinakis - slide #19
Consequence Relations on Fm(X) and Fm(X)
Consequence relations overFm(X), Fm(X)k (k > 1), and sequents.
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)).
In this setting, we use the term substitutioninvariant instead of action invariant.
AbstractReferencesHistorical Note
Logical Consequence
Excursion to UA
Special Cons. Ops
|=K
Equivalence
Modules
Cons. in Modules
Equiv. in Modules
Fundamental Property
Consequence relations C. Tsinakis - slide #19
Consequence Relations on Fm(X) and Fm(X)
Consequence relations overFm(X), Fm(X)k (k > 1), and sequents.
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)).
In this setting, we use the term substitutioninvariant instead of action invariant.
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).
AbstractReferencesHistorical Note
Logical Consequence
Excursion to UA
Special Cons. Ops
|=K
Equivalence
Modules
Cons. in Modules
Equiv. in Modules
Fundamental Property
Consequence relations C. Tsinakis - slide #19
Consequence Relations on Fm(X) and Fm(X)
Consequence relations overFm(X), Fm(X)k (k > 1), and sequents.
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)).
In this setting, we use the term substitutioninvariant instead of action invariant.
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.
Consequence relations C. Tsinakis - slide #20
Equivalence of Two Consequence Relations
AbstractReferencesHistorical Note
Logical Consequence
Excursion to UA
Special Cons. Ops
EquivalenceAction-Invariant MapsEquivalenceDiagramAlgebraizabilityBlok-PigozziMore . . .
Modules
Cons. in Modules
Equiv. in Modules
Fundamental Property
Consequence relations C. Tsinakis - slide #21
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.
AbstractReferencesHistorical Note
Logical Consequence
Excursion to UA
Special Cons. Ops
EquivalenceAction-Invariant MapsEquivalenceDiagramAlgebraizabilityBlok-PigozziMore . . .
Modules
Cons. in Modules
Equiv. in Modules
Fundamental Property
Consequence relations C. Tsinakis - slide #21
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.
AbstractReferencesHistorical Note
Logical Consequence
Excursion to UA
Special Cons. Ops
EquivalenceAction-Invariant MapsEquivalenceDiagramAlgebraizabilityBlok-PigozziMore . . .
Modules
Cons. in Modules
Equiv. in Modules
Fundamental Property
Consequence relations C. Tsinakis - slide #22
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:
AbstractReferencesHistorical Note
Logical Consequence
Excursion to UA
Special Cons. Ops
EquivalenceAction-Invariant MapsEquivalenceDiagramAlgebraizabilityBlok-PigozziMore . . .
Modules
Cons. in Modules
Equiv. in Modules
Fundamental Property
Consequence relations C. Tsinakis - slide #22
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:
(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.
AbstractReferencesHistorical Note
Logical Consequence
Excursion to UA
Special Cons. Ops
EquivalenceAction-Invariant MapsEquivalenceDiagramAlgebraizabilityBlok-PigozziMore . . .
Modules
Cons. in Modules
Equiv. in Modules
Fundamental Property
Consequence relations C. Tsinakis - slide #22
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:
(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.
AbstractReferencesHistorical Note
Logical Consequence
Excursion to UA
Special Cons. Ops
EquivalenceAction-Invariant MapsEquivalenceDiagramAlgebraizabilityBlok-PigozziMore . . .
Modules
Cons. in Modules
Equiv. in Modules
Fundamental Property
Consequence relations C. Tsinakis - slide #23
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
AbstractReferencesHistorical Note
Logical Consequence
Excursion to UA
Special Cons. Ops
EquivalenceAction-Invariant MapsEquivalenceDiagramAlgebraizabilityBlok-PigozziMore . . .
Modules
Cons. in Modules
Equiv. in Modules
Fundamental Property
Consequence relations C. Tsinakis - slide #23
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
℘(M2)ρ
//
Ξ2
����
℘(M1)
Ξ1
����
τ //℘(M2)
Ξ2
����
Th(Ξ2) Th(Ξ1)oo
ρ−1
oooo Th(Ξ2)oo
τ−1
oooo
AbstractReferencesHistorical Note
Logical Consequence
Excursion to UA
Special Cons. Ops
EquivalenceAction-Invariant MapsEquivalenceDiagramAlgebraizabilityBlok-PigozziMore . . .
Modules
Cons. in Modules
Equiv. in Modules
Fundamental Property
Consequence relations C. Tsinakis - slide #23
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
℘(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)
AbstractReferencesHistorical Note
Logical Consequence
Excursion to UA
Special Cons. Ops
EquivalenceAction-Invariant MapsEquivalenceDiagramAlgebraizabilityBlok-PigozziMore . . .
Modules
Cons. in Modules
Equiv. in Modules
Fundamental Property
Consequence relations C. Tsinakis - slide #24
Algebraizability
A consequence relation ⊢ over Fm is calledalgebraizable provided there exists a class K ofL-algebras such that ⊢ and |=
Kare equivalent.
AbstractReferencesHistorical Note
Logical Consequence
Excursion to UA
Special Cons. Ops
EquivalenceAction-Invariant MapsEquivalenceDiagramAlgebraizabilityBlok-PigozziMore . . .
Modules
Cons. in Modules
Equiv. in Modules
Fundamental Property
Consequence relations C. Tsinakis - slide #24
Algebraizability
A consequence relation ⊢ over Fm is calledalgebraizable provided there exists a class K ofL-algebras such that ⊢ and |=
Kare equivalent.
When the preceding holds, we say that the class Kis an equivalent algebraic semantics for theconsequence relation ⊢.
AbstractReferencesHistorical Note
Logical Consequence
Excursion to UA
Special Cons. Ops
EquivalenceAction-Invariant MapsEquivalenceDiagramAlgebraizabilityBlok-PigozziMore . . .
Modules
Cons. in Modules
Equiv. in Modules
Fundamental Property
Consequence relations C. Tsinakis - slide #24
Algebraizability
A consequence relation ⊢ over Fm is calledalgebraizable provided there exists a class K ofL-algebras such that ⊢ and |=
Kare equivalent.
When the preceding holds, we say that the class Kis an equivalent algebraic semantics for theconsequence relation ⊢.
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)}
AbstractReferencesHistorical Note
Logical Consequence
Excursion to UA
Special Cons. Ops
EquivalenceAction-Invariant MapsEquivalenceDiagramAlgebraizabilityBlok-PigozziMore . . .
Modules
Cons. in Modules
Equiv. in Modules
Fundamental Property
Consequence relations C. Tsinakis - slide #24
Algebraizability
A consequence relation ⊢ over Fm is calledalgebraizable provided there exists a class K ofL-algebras such that ⊢ and |=
Kare equivalent.
When the preceding holds, we say that the class Kis an equivalent algebraic semantics for theconsequence relation ⊢.
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.
AbstractReferencesHistorical Note
Logical Consequence
Excursion to UA
Special Cons. Ops
EquivalenceAction-Invariant MapsEquivalenceDiagramAlgebraizabilityBlok-PigozziMore . . .
Modules
Cons. in Modules
Equiv. in Modules
Fundamental Property
Consequence relations C. Tsinakis - slide #25
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?
AbstractReferencesHistorical Note
Logical Consequence
Excursion to UA
Special Cons. Ops
EquivalenceAction-Invariant MapsEquivalenceDiagramAlgebraizabilityBlok-PigozziMore . . .
Modules
Cons. in Modules
Equiv. in Modules
Fundamental Property
Consequence relations C. Tsinakis - slide #25
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 )).]
AbstractReferencesHistorical Note
Logical Consequence
Excursion to UA
Special Cons. Ops
EquivalenceAction-Invariant MapsEquivalenceDiagramAlgebraizabilityBlok-PigozziMore . . .
Modules
Cons. in Modules
Equiv. in Modules
Fundamental Property
Consequence relations C. Tsinakis - slide #26
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)
AbstractReferencesHistorical Note
Logical Consequence
Excursion to UA
Special Cons. Ops
EquivalenceAction-Invariant MapsEquivalenceDiagramAlgebraizabilityBlok-PigozziMore . . .
Modules
Cons. in Modules
Equiv. in Modules
Fundamental Property
Consequence relations C. Tsinakis - slide #26
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).
Consequence relations C. Tsinakis - slide #27
Modules over Quantales
AbstractReferencesHistorical Note
Logical Consequence
Excursion to UA
Special Cons. Ops
Equivalence
ModulesOverviewQuantalesResiduated MapsModules
Cons. in Modules
Equiv. in Modules
Fundamental Property
Consequence relations C. Tsinakis - slide #28
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}.
AbstractReferencesHistorical Note
Logical Consequence
Excursion to UA
Special Cons. Ops
Equivalence
ModulesOverviewQuantalesResiduated MapsModules
Cons. in Modules
Equiv. in Modules
Fundamental Property
Consequence relations C. Tsinakis - slide #28
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}.
℘(A) is a ringlike object – in which set-union playsthe role of addition and complex product serves asmultiplication.
AbstractReferencesHistorical Note
Logical Consequence
Excursion to UA
Special Cons. Ops
Equivalence
ModulesOverviewQuantalesResiduated MapsModules
Cons. in Modules
Equiv. in Modules
Fundamental Property
Consequence relations C. Tsinakis - slide #28
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}.
℘(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.
AbstractReferencesHistorical Note
Logical Consequence
Excursion to UA
Special Cons. Ops
Equivalence
ModulesOverviewQuantalesResiduated MapsModules
Cons. in Modules
Equiv. in Modules
Fundamental Property
Consequence relations C. Tsinakis - slide #28
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}.
℘(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.
AbstractReferencesHistorical Note
Logical Consequence
Excursion to UA
Special Cons. Ops
Equivalence
ModulesOverviewQuantalesResiduated MapsModules
Cons. in Modules
Equiv. in Modules
Fundamental Property
Consequence relations C. Tsinakis - slide #29
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).
AbstractReferencesHistorical Note
Logical Consequence
Excursion to UA
Special Cons. Ops
Equivalence
ModulesOverviewQuantalesResiduated MapsModules
Cons. in Modules
Equiv. in Modules
Fundamental Property
Consequence relations C. Tsinakis - slide #29
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.
AbstractReferencesHistorical Note
Logical Consequence
Excursion to UA
Special Cons. Ops
Equivalence
ModulesOverviewQuantalesResiduated MapsModules
Cons. in Modules
Equiv. in Modules
Fundamental Property
Consequence relations C. Tsinakis - slide #30
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.
AbstractReferencesHistorical Note
Logical Consequence
Excursion to UA
Special Cons. Ops
Equivalence
ModulesOverviewQuantalesResiduated MapsModules
Cons. in Modules
Equiv. in Modules
Fundamental Property
Consequence relations C. Tsinakis - slide #30
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. [If P and Q are complete, then ϕ isresiduated iff it preserving all joins.]
AbstractReferencesHistorical Note
Logical Consequence
Excursion to UA
Special Cons. Ops
Equivalence
ModulesOverviewQuantalesResiduated MapsModules
Cons. in Modules
Equiv. in Modules
Fundamental Property
Consequence relations C. Tsinakis - slide #30
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. [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.
AbstractReferencesHistorical Note
Logical Consequence
Excursion to UA
Special Cons. Ops
Equivalence
ModulesOverviewQuantalesResiduated MapsModules
Cons. in Modules
Equiv. in Modules
Fundamental Property
Consequence relations C. Tsinakis - slide #30
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. [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. [Again, if P, Q and R
are complete, then ◦ is residuated iff it preservesall joins in each coordinate.]
AbstractReferencesHistorical Note
Logical Consequence
Excursion to UA
Special Cons. Ops
Equivalence
ModulesOverviewQuantalesResiduated MapsModules
Cons. in Modules
Equiv. in Modules
Fundamental Property
Consequence relations C. Tsinakis - slide #30
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. [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. [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.
AbstractReferencesHistorical Note
Logical Consequence
Excursion to UA
Special Cons. Ops
Equivalence
ModulesOverviewQuantalesResiduated MapsModules
Cons. in Modules
Equiv. in Modules
Fundamental Property
Consequence relations C. Tsinakis - slide #31
Modules over Quantales
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).
AbstractReferencesHistorical Note
Logical Consequence
Excursion to UA
Special Cons. Ops
Equivalence
ModulesOverviewQuantalesResiduated MapsModules
Cons. in Modules
Equiv. in Modules
Fundamental Property
Consequence relations C. Tsinakis - slide #31
Modules over Quantales
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 relations C. Tsinakis - slide #32
Consequence Operations in Modules
AbstractReferencesHistorical Note
Logical Consequence
Excursion to UA
Special Cons. Ops
Equivalence
Modules
Cons. in ModulesA − Mod
Cons. OpsQuotientsCyclic Modules
Equiv. in Modules
Fundamental Property
Consequence relations C. Tsinakis - slide #33
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 .
AbstractReferencesHistorical Note
Logical Consequence
Excursion to UA
Special Cons. Ops
Equivalence
Modules
Cons. in ModulesA − Mod
Cons. OpsQuotientsCyclic Modules
Equiv. in Modules
Fundamental Property
Consequence relations C. Tsinakis - slide #33
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.
AbstractReferencesHistorical Note
Logical Consequence
Excursion to UA
Special Cons. Ops
Equivalence
Modules
Cons. in ModulesA − Mod
Cons. OpsQuotientsCyclic Modules
Equiv. in Modules
Fundamental Property
Consequence relations C. Tsinakis - slide #34
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.)
AbstractReferencesHistorical Note
Logical Consequence
Excursion to UA
Special Cons. Ops
Equivalence
Modules
Cons. in ModulesA − Mod
Cons. OpsQuotientsCyclic Modules
Equiv. in Modules
Fundamental Property
Consequence relations C. Tsinakis - slide #34
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 .
AbstractReferencesHistorical Note
Logical Consequence
Excursion to UA
Special Cons. Ops
Equivalence
Modules
Cons. in ModulesA − Mod
Cons. OpsQuotientsCyclic Modules
Equiv. in Modules
Fundamental Property
Consequence relations C. Tsinakis - slide #35
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]
AbstractReferencesHistorical Note
Logical Consequence
Excursion to UA
Special Cons. Ops
Equivalence
Modules
Cons. in ModulesA − Mod
Cons. OpsQuotientsCyclic Modules
Equiv. in Modules
Fundamental Property
Consequence relations C. Tsinakis - slide #36
Cyclic Modules
An A-module P is called cyclic, if there exists anelement u ∈ P , such that P = A ◦ u.
AbstractReferencesHistorical Note
Logical Consequence
Excursion to UA
Special Cons. Ops
Equivalence
Modules
Cons. in ModulesA − Mod
Cons. OpsQuotientsCyclic Modules
Equiv. in Modules
Fundamental Property
Consequence relations C. Tsinakis - slide #36
Cyclic Modules
An A-module P is called cyclic, if there exists anelement u ∈ P , such that P = A ◦ u.
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.
AbstractReferencesHistorical Note
Logical Consequence
Excursion to UA
Special Cons. Ops
Equivalence
Modules
Cons. in ModulesA − Mod
Cons. OpsQuotientsCyclic Modules
Equiv. in Modules
Fundamental Property
Consequence relations C. Tsinakis - slide #36
Cyclic Modules
An A-module P is called cyclic, if there exists anelement u ∈ P , such that P = A ◦ u.
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.
Consequence relations C. Tsinakis - slide #37
Equivalence in the Setting of Modules
AbstractReferencesHistorical Note
Logical Consequence
Excursion to UA
Special Cons. Ops
Equivalence
Modules
Cons. in Modules
Equiv. in ModulesEquivalenceInduced Equivalences
Fundamental Property
Consequence relations C. Tsinakis - slide #38
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.
AbstractReferencesHistorical Note
Logical Consequence
Excursion to UA
Special Cons. Ops
Equivalence
Modules
Cons. in Modules
Equiv. in ModulesEquivalenceInduced Equivalences
Fundamental Property
Consequence relations C. Tsinakis - slide #38
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.
AbstractReferencesHistorical Note
Logical Consequence
Excursion to UA
Special Cons. Ops
Equivalence
Modules
Cons. in Modules
Equiv. in ModulesEquivalenceInduced Equivalences
Fundamental Property
Consequence relations C. Tsinakis - slide #39
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
//
���
P
���
τ //Q
���
Qζ//ϕ−1
// //Pξoo
ρ∗oooo
//ϕ
// //Qζoo
τ∗oooo
Consequence relations C. Tsinakis - slide #40
The Fundamental Categorical Property
AbstractReferencesHistorical Note
Logical Consequence
Excursion to UA
Special Cons. Ops
Equivalence
Modules
Cons. in Modules
Equiv. in Modules
Fundamental PropertyTeaserDefinitionCharacterizationCo-productsSequents
Consequence relations C. Tsinakis - slide #41
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 //___
���
Q
���
Pξ
//ϕ
// // Qζ
(S)
AbstractReferencesHistorical Note
Logical Consequence
Excursion to UA
Special Cons. Ops
Equivalence
Modules
Cons. in Modules
Equiv. in Modules
Fundamental PropertyTeaserDefinitionCharacterizationCo-productsSequents
Consequence relations C. Tsinakis - slide #41
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 //___
���
Q
���
Pξ
//ϕ
// // Qζ
(S)
TheoremThe objects P of the category A −Mod for whichevery square of type (S) can be completed areprecisely
AbstractReferencesHistorical Note
Logical Consequence
Excursion to UA
Special Cons. Ops
Equivalence
Modules
Cons. in Modules
Equiv. in Modules
Fundamental PropertyTeaserDefinitionCharacterizationCo-productsSequents
Consequence relations C. Tsinakis - slide #41
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 //___
���
Q
���
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.
AbstractReferencesHistorical Note
Logical Consequence
Excursion to UA
Special Cons. Ops
Equivalence
Modules
Cons. in Modules
Equiv. in Modules
Fundamental PropertyTeaserDefinitionCharacterizationCo-productsSequents
Consequence relations C. Tsinakis - slide #42
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)
AbstractReferencesHistorical Note
Logical Consequence
Excursion to UA
Special Cons. Ops
Equivalence
Modules
Cons. in Modules
Equiv. in Modules
Fundamental PropertyTeaserDefinitionCharacterizationCo-productsSequents
Consequence relations C. Tsinakis - slide #43
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.
AbstractReferencesHistorical Note
Logical Consequence
Excursion to UA
Special Cons. Ops
Equivalence
Modules
Cons. in Modules
Equiv. in Modules
Fundamental PropertyTeaserDefinitionCharacterizationCo-productsSequents
Consequence relations C. Tsinakis - slide #43
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.
AbstractReferencesHistorical Note
Logical Consequence
Excursion to UA
Special Cons. Ops
Equivalence
Modules
Cons. in Modules
Equiv. in Modules
Fundamental PropertyTeaserDefinitionCharacterizationCo-productsSequents
Consequence relations C. Tsinakis - slide #44
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.
AbstractReferencesHistorical Note
Logical Consequence
Excursion to UA
Special Cons. Ops
Equivalence
Modules
Cons. in Modules
Equiv. in Modules
Fundamental PropertyTeaserDefinitionCharacterizationCo-productsSequents
Consequence relations C. Tsinakis - slide #44
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.
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.
AbstractReferencesHistorical Note
Logical Consequence
Excursion to UA
Special Cons. Ops
Equivalence
Modules
Cons. in Modules
Equiv. in Modules
Fundamental PropertyTeaserDefinitionCharacterizationCo-productsSequents
Consequence relations C. Tsinakis - slide #44
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.
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.
AbstractReferencesHistorical Note
Logical Consequence
Excursion to UA
Special Cons. Ops
Equivalence
Modules
Cons. in Modules
Equiv. in Modules
Fundamental PropertyTeaserDefinitionCharacterizationCo-productsSequents
Consequence relations C. Tsinakis - slide #45
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.