General remarks on AAL Bridge theorems and transfer theorems Two open problems Ordering protoalgebraic logics Syntax meets semantics in abstract algebraic logic Josep Maria Font University of Barcelona SYSMICS 2016 06 September 2016 Barcelona Josep Maria Font (Barcelona) Syntax meets semantics in abstract algebraic logic SYSMICS 2016 06/09/2016 1 / 21
General remarks on AAL Bridge theorems and transfer theorems Two open problems Ordering protoalgebraic logics Outline and references General remarks on Abstract Algebraic Logic 1 Bridge theorems and transfer theorems 2 Two open problems 3 Ordering protoalgebraic logics 4 F ont , J. M. Abstract Algebraic Logic - An Introductory Textbook vol. 60 of Studies in Logic - Mathematical Logic and Foundations . College Publications, London, 2016. http://www.amazon.com F ont , J. M. Ordering protoalgebraic logics Journal of Logic and Computation . To appear. Josep Maria Font (Barcelona) Syntax meets semantics in abstract algebraic logic SYSMICS 2016 06/09/2016 2 / 21
General remarks on AAL Bridge theorems and transfer theorems Two open problems Ordering protoalgebraic logics Algebraic Logic: the study of algebra-based semantics Logics L = � F m , ⊢ L � �− → algebra-based semantics , i.e., any kind of semantics where : 1) models are : algebras A + additional structure 2) interpretations are : h : F m → A additional structure : semantic: 1 ∈ A , F ⊆ A , C ⊆ P ( A ) or syntactic: τ ( x ) ⊆ Fm × Fm The syntax is hidden inside algebra-based semantics How much similar is the semantics to the logic? Which properties of the logic are shared by the semantics? Are there properties of the logic that are always shared by the semantics? Josep Maria Font (Barcelona) Syntax meets semantics in abstract algebraic logic SYSMICS 2016 06/09/2016 3 / 21
General remarks on AAL Bridge theorems and transfer theorems Two open problems Ordering protoalgebraic logics Assume you work in a framework where there is a criterion or procedure: L �− → K . . . . . . . . . . . . . . . the algebraic an arbitrary logic a class of counterpart of L (perhaps only of a certain kind) algebra-based models Bridge Theorem For every logic L (perhaps only of a certain kind), L satisfies P ⇐ ⇒ K satisfies Q P : a syntactic property of a logic Q : a semantic property of a class of models Algebra Logic Josep Maria Font (Barcelona) Syntax meets semantics in abstract algebraic logic SYSMICS 2016 06/09/2016 4 / 21
General remarks on AAL Bridge theorems and transfer theorems Two open problems Ordering protoalgebraic logics A special kind of Bridge Theorem, when Q is essentially the same as P : Transfer Theorem For every logic L (of a certain kind), L satisfies P ⇐ ⇒ K satisfies P P has to be interpreted (perhaps slightly differently) in both sides (e.g., “to be finitely axiomatizable” , “to be decidable” , etc.). Or not: when P is a property of a generalized matrix, directly: � F m , T h L � or � F m , ⊢ L � satisfies P � A , F i L A � or � A , Fg A L � satisfies P , for all A ? ⇒ = We say: “The property P transfers from L to K ” (converse trivial) Josep Maria Font (Barcelona) Syntax meets semantics in abstract algebraic logic SYSMICS 2016 06/09/2016 5 / 21
General remarks on AAL Bridge theorems and transfer theorems Two open problems Ordering protoalgebraic logics Bridge theorems and transfer theorems are the ultimate justification of Abstract Algebraic Logic (and a major driving force in the evolution of the field) Theorem Let L be a logic. The following conditions are equivalent: (i) L is finitary. (ii) The class Mod L is closed under ultraproducts. (iii) For every algebra A , the closure operator Fg A L is finitary. (i) ⇔ (ii): Bridge (i) ⇔ (iii): Transfer Josep Maria Font (Barcelona) Syntax meets semantics in abstract algebraic logic SYSMICS 2016 06/09/2016 6 / 21
General remarks on AAL Bridge theorems and transfer theorems Two open problems Ordering protoalgebraic logics Characterizing classes in the Leibniz hierarchy Theorem (B lok , P igozzi , C zelakowski , 1986,1992) Let L be a logic. The following conditions are equivalent: (i) L is protoalgebraic. (ii) There is a set ∆ ( x , y ) of formulas (in at most two variables) satisfying: ⊢ L ∆ ( x , x ) (R ∆ ) x , ∆ ( x , y ) ⊢ L y (MP ∆ ) (iii) The class Mod ∗ L is closed under subdirect products. (iv) The Leibniz operator Ω on the formula algebra is monotonic over the theories of L . (v) For every algebra A , the Leibniz operator Ω A is monotonic over the L -filters of A . Josep Maria Font (Barcelona) Syntax meets semantics in abstract algebraic logic SYSMICS 2016 06/09/2016 7 / 21
� � � � � � � � � � � � � � � � � � � General remarks on AAL Bridge theorems and transfer theorems Two open problems Ordering protoalgebraic logics the Leibniz fully Fregean hierarchy finitely regularly fully Fregean algebraizable selfextensional finitely regularly selfextensional algebraizable algebraizable the Frege finitely regularly weakly algebraizable equivalential algebraizable hierarchy weakly equivalential assertional algebraizable protoalgebraic truth-equational Josep Maria Font (Barcelona) Syntax meets semantics in abstract algebraic logic SYSMICS 2016 06/09/2016 8 / 21
General remarks on AAL Bridge theorems and transfer theorems Two open problems Ordering protoalgebraic logics A T arski -style condition: the Inconsistency Lemma ( Reductio ad Absurdum for Intuitionistic Propositional Logic I ℓ ) For all Γ ⊆ Fm and all α 1 , . . . , α n ∈ Fm , Γ ∪ { α 1 , . . . , α n } is inconsistent in I ℓ ⇐ ⇒ Γ ⊢ I ℓ ¬ ( α 1 ∧ · · · ∧ α n ) . Definition (extending R aftery ’s terminology) A sequence � Ψ n ( x 1 , . . . , x n ) : n � 1 � of finite sets defines an Inconsistency Lemma for a generalized matrix � A , C � when for all X ∪ { a 1 , . . . , a n } ⊆ A , Ψ A X ∪ { a 1 , . . . , a n } is C -inconsistent ⇐ ⇒ n ( a 1 , . . . , a n ) ⊆ C ( X ) . ( C is the closure operator associated with the closure system C .) � {¬ ( x 1 ∧ · · · ∧ x n ) } : n � 1 � defines an Inconsistency Lemma for I ℓ . Josep Maria Font (Barcelona) Syntax meets semantics in abstract algebraic logic SYSMICS 2016 06/09/2016 9 / 21
General remarks on AAL Bridge theorems and transfer theorems Two open problems Ordering protoalgebraic logics A T arski -style condition: the Inconsistency Lemma Fact (essentially, F ont and J ansana , 1996) The Inconsistency Lemma does not transfer, in general, from a logic to arbitrary algebras; not even for finitary logics. Counterexample: I ℓ ¬ , ∧ Theorem (R aftery , 2013) Let L be a finitary and protoalgebraic logic. The following conditions are equivalent. (i) L satisfies an Inconsistency Lemma. (ii) For every algebra A , the generalized matrix � A , F i L A � satisfies the same Inconsistency Lemma. (iii) For all A , the join-semilattice F i ω L A is dually pseudo-complemented. (iv) The join-semilattice T h ω L is dually pseudo-complemented. Josep Maria Font (Barcelona) Syntax meets semantics in abstract algebraic logic SYSMICS 2016 06/09/2016 10 / 21
General remarks on AAL Bridge theorems and transfer theorems Two open problems Ordering protoalgebraic logics A T arski -style condition: the Inconsistency Lemma Theorem (R aftery , 2013) Let L be a finitary and finitely algebraizable logic, and let the quasivariety K be its equivalent algebraic semantics. The following conditions are equivalent. (i) L satisfies an Inconsistency Lemma. (iii) For all A , the join-semilattice Con ω K A is dually pseudo-complemented. (v) For all A ∈ K , the join-semilattice Con ω K A is dually pseudo-complemented. Josep Maria Font (Barcelona) Syntax meets semantics in abstract algebraic logic SYSMICS 2016 06/09/2016 11 / 21
General remarks on AAL Bridge theorems and transfer theorems Two open problems Ordering protoalgebraic logics Two general transfer results Theorems (C zelakowski and P igozzi , 2001,2004) Let L be a finitary and protoalgebraic logic. Then: 1. Every property of a logic expressible by a first-order formula α of the language of lattices transfers from L to all algebras; i.e., � T h L , ∩ , ∨ � = ⇒ � F i L A , ∩ , ∨ � � α � α for all A 2. Every property of a logic expressible by an accumulative set G of Gentzen-style rules transfers from L to all algebras; i.e., � F m , T h L � satisfies G = ⇒ � A , F i L A � satisfies G for all A A set G of Gentzen-style rules is accumulative when { Γ i ✄ ϕ i : i ∈ I } { ∆ , Γ i ✄ ϕ i : i ∈ I } ∈ G = ⇒ ∈ G Γ ✄ ϕ ∆ , Γ ✄ ϕ Josep Maria Font (Barcelona) Syntax meets semantics in abstract algebraic logic SYSMICS 2016 06/09/2016 12 / 21
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