Modal logics over finite residuated lattices Amanda Vidal Institute of Computer Science, Czech Academy of Sciences Topology, Algebra and Categories in Logic 2017, Prague, Czech Republic , June 29, 2017 1 / 15
In particular... ◮ Modal expansions of lattice-based logics are in phase of development and understanding. 2 / 15
In particular... ◮ Modal expansions of lattice-based logics are in phase of development and understanding. ◮ (Bou et. al., 2011) does a general study of axiomatizations of these logics over finite residuated lattices. 2 / 15
In particular... ◮ Modal expansions of lattice-based logics are in phase of development and understanding. ◮ (Bou et. al., 2011) does a general study of axiomatizations of these logics over finite residuated lattices. Propose several open problems. We will address some of them 2 / 15
In particular... ◮ Modal expansions of lattice-based logics are in phase of development and understanding. ◮ (Bou et. al., 2011) does a general study of axiomatizations of these logics over finite residuated lattices. Propose several open problems. We will address some of them ◮ only ✷ operator -with the usual lattice-valued interpretation Q1. Both ✷ and ✸ (! ✸ x � = ¬ ✷ ¬ x ) 2 / 15
In particular... ◮ Modal expansions of lattice-based logics are in phase of development and understanding. ◮ (Bou et. al., 2011) does a general study of axiomatizations of these logics over finite residuated lattices. Propose several open problems. We will address some of them ◮ only ✷ operator -with the usual lattice-valued interpretation Q1. Both ✷ and ✸ (! ✸ x � = ¬ ✷ ¬ x ) ◮ local deduction, global over crisp frames Q2. (general) Global deduction 2 / 15
In particular... ◮ Modal expansions of lattice-based logics are in phase of development and understanding. ◮ (Bou et. al., 2011) does a general study of axiomatizations of these logics over finite residuated lattices. Propose several open problems. We will address some of them ◮ only ✷ operator -with the usual lattice-valued interpretation Q1. Both ✷ and ✸ (! ✸ x � = ¬ ✷ ¬ x ) ◮ local deduction, global over crisp frames Q2. (general) Global deduction ◮ Q3. Is an axiomatization for the Global modal logic an x → y axiomatization for the local one + ✷ x → ✷ y ? (Q3’). Similar question restricting to crisp accessibility and x adding ✷ x 2 / 15
Preliminaries ◮ A = � A , · , → , ∧ , ∨ , 0 , 1 � is a (bounded, commutative, integral) residuated lattice when ◮ � A , ∧ , ∨ , 1 , 0 � is a bounded lattice (with order denoted ≤ ), ◮ � A , · , 1 � is a commutative monoid and ◮ for all a , b , c ∈ A it holds a · b ≤ c ⇐ ⇒ a ≤ b → c . 3 / 15
Preliminaries ◮ A = � A , · , → , ∧ , ∨ , 0 , 1 � is a (bounded, commutative, integral) residuated lattice when ◮ � A , ∧ , ∨ , 1 , 0 � is a bounded lattice (with order denoted ≤ ), ◮ � A , · , 1 � is a commutative monoid and ◮ for all a , b , c ∈ A it holds a · b ≤ c ⇐ ⇒ a ≤ b → c . ◮ A c = expansion of A with constants { a : a ∈ A \ { 1 , 0 }} . 3 / 15
Preliminaries ◮ A = � A , · , → , ∧ , ∨ , 0 , 1 � is a (bounded, commutative, integral) residuated lattice when ◮ � A , ∧ , ∨ , 1 , 0 � is a bounded lattice (with order denoted ≤ ), ◮ � A , · , 1 � is a commutative monoid and ◮ for all a , b , c ∈ A it holds a · b ≤ c ⇐ ⇒ a ≤ b → c . ◮ A c = expansion of A with constants { a : a ∈ A \ { 1 , 0 }} . ◮ Fm = formula algebra built in the language of residuated lattices [+ constants]. 3 / 15
Preliminaries ◮ A = � A , · , → , ∧ , ∨ , 0 , 1 � is a (bounded, commutative, integral) residuated lattice when ◮ � A , ∧ , ∨ , 1 , 0 � is a bounded lattice (with order denoted ≤ ), ◮ � A , · , 1 � is a commutative monoid and ◮ for all a , b , c ∈ A it holds a · b ≤ c ⇐ ⇒ a ≤ b → c . ◮ A c = expansion of A with constants { a : a ∈ A \ { 1 , 0 }} . ◮ Fm = formula algebra built in the language of residuated lattices [+ constants]. ◮ Γ | = A ϕ iff for any h ∈ Hom ( Fm , A ) , h ([ Γ ]) ⊆ { 1 } implies h ( ϕ ) = 1 . 3 / 15
Preliminaries ◮ A = � A , · , → , ∧ , ∨ , 0 , 1 � is a (bounded, commutative, integral) residuated lattice when ◮ � A , ∧ , ∨ , 1 , 0 � is a bounded lattice (with order denoted ≤ ), ◮ � A , · , 1 � is a commutative monoid and ◮ for all a , b , c ∈ A it holds a · b ≤ c ⇐ ⇒ a ≤ b → c . ◮ A c = expansion of A with constants { a : a ∈ A \ { 1 , 0 }} . ◮ Fm = formula algebra built in the language of residuated lattices [+ constants]. ◮ Γ | = A ϕ iff for any h ∈ Hom ( Fm , A ) , h ([ Γ ]) ⊆ { 1 } implies h ( ϕ ) = 1 . In the following A will be finite 3 / 15
Preliminaries ◮ M = � W , R , e � is a A-Kripke model when W is a non-empty set, R : W × W → A and e : W × V → A , extended uniquely in order to be in Hom ( Fm , A ) and e ( v , ✷ ϕ ) = � { Rvw → e ( w , ϕ ) } e ( v , ✸ ϕ ) = � { Rvw · e ( w , ϕ ) } w ∈ W w ∈ W It is said crisp if R ⊆ W × W . 4 / 15
Preliminaries ◮ M = � W , R , e � is a A-Kripke model when W is a non-empty set, R : W × W → A and e : W × V → A , extended uniquely in order to be in Hom ( Fm , A ) and e ( v , ✷ ϕ ) = � { Rvw → e ( w , ϕ ) } e ( v , ✸ ϕ ) = � { Rvw · e ( w , ϕ ) } w ∈ W w ∈ W It is said crisp if R ⊆ W × W . ◮ Γ � l M A ϕ iff for any A -Kripke model M , and any v ∈ W , if e ( v , [ Γ ]) ⊆ { 1 } then e ( v , ϕ ) = 1. 4 / 15
Preliminaries ◮ M = � W , R , e � is a A-Kripke model when W is a non-empty set, R : W × W → A and e : W × V → A , extended uniquely in order to be in Hom ( Fm , A ) and e ( v , ✷ ϕ ) = � { Rvw → e ( w , ϕ ) } e ( v , ✸ ϕ ) = � { Rvw · e ( w , ϕ ) } w ∈ W w ∈ W It is said crisp if R ⊆ W × W . ◮ Γ � l M A ϕ iff for any A -Kripke model M , and any v ∈ W , if e ( v , [ Γ ]) ⊆ { 1 } then e ( v , ϕ ) = 1. ◮ Γ � g M A ϕ iff for any A -Kripke model M , if for all v ∈ W , it holds e ( v , [ Γ ]) ⊆ { 1 } then for all v ∈ W it also holds e ( v , ϕ ) = 1. 4 / 15
Preliminaries ◮ M = � W , R , e � is a A-Kripke model when W is a non-empty set, R : W × W → A and e : W × V → A , extended uniquely in order to be in Hom ( Fm , A ) and e ( v , ✷ ϕ ) = � { Rvw → e ( w , ϕ ) } e ( v , ✸ ϕ ) = � { Rvw · e ( w , ϕ ) } w ∈ W w ∈ W It is said crisp if R ⊆ W × W . ◮ Γ � l M A ϕ iff for any A -Kripke model M , and any v ∈ W , if e ( v , [ Γ ]) ⊆ { 1 } then e ( v , ϕ ) = 1. ◮ Γ � g M A ϕ iff for any A -Kripke model M , if for all v ∈ W , it holds e ( v , [ Γ ]) ⊆ { 1 } then for all v ∈ W it also holds e ( v , ϕ ) = 1. ◮ Same valid formulas. 4 / 15
(Some comparisons with classical K ) ◮ No K . (Bou et. al) [ K is valid only if Rvw is idempotent.] 5 / 15
(Some comparisons with classical K ) ◮ No K . (Bou et. al) [ K is valid only if Rvw is idempotent.] ◮ No ✷ = ¬ ✸ ¬ . [Only if ¬ is involutive (eg., MV algebras)]. 5 / 15
(Some comparisons with classical K ) ◮ No K . (Bou et. al) [ K is valid only if Rvw is idempotent.] ◮ No ✷ = ¬ ✸ ¬ . [Only if ¬ is involutive (eg., MV algebras)]. Not known general interdefinability of modalities... 5 / 15
(Some comparisons with classical K ) ◮ No K . (Bou et. al) [ K is valid only if Rvw is idempotent.] ◮ No ✷ = ¬ ✸ ¬ . [Only if ¬ is involutive (eg., MV algebras)]. Not known general interdefinability of modalities.... ◮ Local classical modal logic enjoys DT = ⇒ usually we say "modal logic" for the set of valid formulas or the global consequence. No longer (necessarily) true -nor even LDT. 5 / 15
Existing axiomatization For A c finite RL with canonical constants, Bou et. al propose an axiomatic system complete wrt. the no- ✸ fragment of � l M A ( c ) (with constants). 6 / 15
Existing axiomatization For A c finite RL with canonical constants, Bou et. al propose an axiomatic system complete wrt. the no- ✸ fragment of � l M A ( c ) (with constants). L A ( c ) = Axiomatization for | = A ( c ) + ✷ ◮ ✷ 1, 6 / 15
Existing axiomatization For A c finite RL with canonical constants, Bou et. al propose an axiomatic system complete wrt. the no- ✸ fragment of � l M A ( c ) (with constants). L A ( c ) = Axiomatization for | = A ( c ) + ✷ ◮ ✷ 1, ◮ ✷ ( ϕ ∧ ψ ) ↔ ( ✷ ϕ ∧ ✷ ψ ) , 6 / 15
Existing axiomatization For A c finite RL with canonical constants, Bou et. al propose an axiomatic system complete wrt. the no- ✸ fragment of � l M A ( c ) (with constants). L A ( c ) = Axiomatization for | = A ( c ) + ✷ ◮ ✷ 1, ◮ ✷ ( ϕ ∧ ψ ) ↔ ( ✷ ϕ ∧ ✷ ψ ) , ◮ ✷ ( c → ϕ ) ↔ ( c → ✷ ϕ ) , 6 / 15
Existing axiomatization For A c finite RL with canonical constants, Bou et. al propose an axiomatic system complete wrt. the no- ✸ fragment of � l M A ( c ) (with constants). L A ( c ) = Axiomatization for | = A ( c ) + ✷ ◮ ✷ 1, ◮ ✷ ( ϕ ∧ ψ ) ↔ ( ✷ ϕ ∧ ✷ ψ ) , ◮ ✷ ( c → ϕ ) ↔ ( c → ✷ ϕ ) , ◮ ⊢ ϕ → ψ implies ⊢ ✷ ϕ → ✷ ψ . 6 / 15
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