Frames monotone logics ordered algebras logic-based dualities and completions Definition Let L be a labeled ordered language. An L -frame F = � W , J , R , { T f : f ∈ L }� is an L -preframe such that for all connectives f ( x 1 , . . . , x m ; y 1 , . . . , y n ) s.t. β ( f ) = ✸ : w 2 ∈ W m , � j 1 ,� j 2 ∈ J n , and u 1 , u 2 ∈ W such that (a) For every � w 1 , � w 1 , � j 2 � J � w 2 � W � � j 1 and u 1 � W u 2 , w 1 ,� w 2 ,� if � � j 1 , u 1 � ∈ T f , then � � j 2 , u 2 � ∈ T f .
Frames monotone logics ordered algebras logic-based dualities and completions Definition Let L be a labeled ordered language. An L -frame F = � W , J , R , { T f : f ∈ L }� is an L -preframe such that for all connectives f ( x 1 , . . . , x m ; y 1 , . . . , y n ) s.t. β ( f ) = ✸ : w 2 ∈ W m , � j 1 ,� j 2 ∈ J n , and u 1 , u 2 ∈ W such that (a) For every � w 1 , � w 1 , � j 2 � J � w 2 � W � � j 1 and u 1 � W u 2 , w 1 ,� w 2 ,� if � � j 1 , u 1 � ∈ T f , then � � j 2 , u 2 � ∈ T f . j ) is a closed set of ( · ) ✄✁ for all � w ∈ W m and � w ,� j ∈ J n . (b) T f ( �
Frames monotone logics ordered algebras logic-based dualities and completions Definition Let L be a labeled ordered language. An L -frame F = � W , J , R , { T f : f ∈ L }� is an L -preframe such that for all connectives f ( x 1 , . . . , x m ; y 1 , . . . , y n ) s.t. β ( f ) = ✸ : w 2 ∈ W m , � j 1 ,� j 2 ∈ J n , and u 1 , u 2 ∈ W such that (a) For every � w 1 , � w 1 , � j 2 � J � w 2 � W � � j 1 and u 1 � W u 2 , w 1 ,� w 2 ,� if � � j 1 , u 1 � ∈ T f , then � � j 2 , u 2 � ∈ T f . j ) is a closed set of ( · ) ✄✁ for all � w ∈ W m and � w ,� j ∈ J n . (b) T f ( � Connectives f such that β ( f ) = ✷ need to satisfy a dual requirement.
Frames monotone logics ordered algebras logic-based dualities and completions Definition Let L be a labeled ordered language. An L -frame F = � W , J , R , { T f : f ∈ L }� is an L -preframe such that for all connectives f ( x 1 , . . . , x m ; y 1 , . . . , y n ) s.t. β ( f ) = ✸ : w 2 ∈ W m , � j 1 ,� j 2 ∈ J n , and u 1 , u 2 ∈ W such that (a) For every � w 1 , � w 1 , � j 2 � J � w 2 � W � � j 1 and u 1 � W u 2 , w 1 ,� w 2 ,� if � � j 1 , u 1 � ∈ T f , then � � j 2 , u 2 � ∈ T f . j ) is a closed set of ( · ) ✄✁ for all � w ∈ W m and � w ,� j ∈ J n . (b) T f ( � Connectives f such that β ( f ) = ✷ need to satisfy a dual requirement. We refer to W and J as to the sets of worlds and co-worlds of F respectively.
Frames monotone logics ordered algebras logic-based dualities and completions ◮ A valuation in a L -frame F is a map v : Var → G ( W , J , R ) .
Frames monotone logics ordered algebras logic-based dualities and completions ◮ A valuation in a L -frame F is a map v : Var → G ( W , J , R ) . ◮ We want to define two relations of satisfaction and co-satisfaction of formulas under v , respectively at worlds w ∈ W and co-worlds j ∈ J , in symbols w , v � ϕ and j , v ≻ ϕ.
Frames monotone logics ordered algebras logic-based dualities and completions ◮ A valuation in a L -frame F is a map v : Var → G ( W , J , R ) . ◮ We want to define two relations of satisfaction and co-satisfaction of formulas under v , respectively at worlds w ∈ W and co-worlds j ∈ J , in symbols w , v � ϕ and j , v ≻ ϕ. ◮ For every variable x ∈ Var , we set w , v � x ⇐ ⇒ w ∈ v ( x ) ⇒ j ∈ v ( x ) ✄ . j , v ≻ x ⇐
Frames monotone logics ordered algebras logic-based dualities and completions ◮ A valuation in a L -frame F is a map v : Var → G ( W , J , R ) . ◮ We want to define two relations of satisfaction and co-satisfaction of formulas under v , respectively at worlds w ∈ W and co-worlds j ∈ J , in symbols w , v � ϕ and j , v ≻ ϕ. ◮ For every variable x ∈ Var , we set w , v � x ⇐ ⇒ w ∈ v ( x ) ⇒ j ∈ v ( x ) ✄ . j , v ≻ x ⇐ ◮ Moreover, for every connective f ( � x ; � y ) s.t. β ( f ) = ✸ we set:
Frames monotone logics ordered algebras logic-based dualities and completions ◮ A valuation in a L -frame F is a map v : Var → G ( W , J , R ) . ◮ We want to define two relations of satisfaction and co-satisfaction of formulas under v , respectively at worlds w ∈ W and co-worlds j ∈ J , in symbols w , v � ϕ and j , v ≻ ϕ. ◮ For every variable x ∈ Var , we set w , v � x ⇐ ⇒ w ∈ v ( x ) ⇒ j ∈ v ( x ) ✄ . j , v ≻ x ⇐ ◮ Moreover, for every connective f ( � x ; � y ) s.t. β ( f ) = ✸ we set: u ∈ W m and � ϕ, � i ∈ J n w , v � f ( � ψ ) ⇐ ⇒ w ∈ { r ∈ W : there are � u ,� s.t. � � i , r � ∈ T f and for all k � m , t � n u k , v � ϕ k and i t , v ≻ ψ t } ✄✁ ϕ, � ϕ, � ψ ) } ✄ . j , v ≻ f ( � ψ ) ⇐ ⇒ j ∈ { w ∈ W : w , v � f ( �
Frames monotone logics ordered algebras logic-based dualities and completions ◮ A valuation in a L -frame F is a map v : Var → G ( W , J , R ) . ◮ We want to define two relations of satisfaction and co-satisfaction of formulas under v , respectively at worlds w ∈ W and co-worlds j ∈ J , in symbols w , v � ϕ and j , v ≻ ϕ. ◮ For every variable x ∈ Var , we set w , v � x ⇐ ⇒ w ∈ v ( x ) ⇒ j ∈ v ( x ) ✄ . j , v ≻ x ⇐ ◮ Moreover, for every connective f ( � x ; � y ) s.t. β ( f ) = ✸ we set: u ∈ W m and � ϕ, � i ∈ J n w , v � f ( � ψ ) ⇐ ⇒ w ∈ { r ∈ W : there are � u ,� s.t. � � i , r � ∈ T f and for all k � m , t � n u k , v � ϕ k and i t , v ≻ ψ t } ✄✁ ϕ, � ϕ, � ψ ) } ✄ . j , v ≻ f ( � ψ ) ⇐ ⇒ j ∈ { w ∈ W : w , v � f ( � ◮ A dual definition applied to connectives f ( � x ; � y ) s.t. β ( f ) = ✷ .
Frames monotone logics ordered algebras logic-based dualities and completions Frames F can be transformed into algebras F + as follows:
Frames monotone logics ordered algebras logic-based dualities and completions Frames F can be transformed into algebras F + as follows: ◮ The universe of F + is G ( W , J , R ) .
Frames monotone logics ordered algebras logic-based dualities and completions Frames F can be transformed into algebras F + as follows: ◮ The universe of F + is G ( W , J , R ) . ◮ For every connective f ( z 1 , . . . , z n ) and a 1 , . . . , a n ∈ F + , f F + ( a 1 , . . . , a n ) := { w ∈ W : w , v � f ( z 1 , . . . , z n ) } where v is any valuation in F s.t. v ( z i ) = a i .
Frames monotone logics ordered algebras logic-based dualities and completions Frames F can be transformed into algebras F + as follows: ◮ The universe of F + is G ( W , J , R ) . ◮ For every connective f ( z 1 , . . . , z n ) and a 1 , . . . , a n ∈ F + , f F + ( a 1 , . . . , a n ) := { w ∈ W : w , v � f ( z 1 , . . . , z n ) } where v is any valuation in F s.t. v ( z i ) = a i . Definition Let L be a labeled ordered language.
Frames monotone logics ordered algebras logic-based dualities and completions Frames F can be transformed into algebras F + as follows: ◮ The universe of F + is G ( W , J , R ) . ◮ For every connective f ( z 1 , . . . , z n ) and a 1 , . . . , a n ∈ F + , f F + ( a 1 , . . . , a n ) := { w ∈ W : w , v � f ( z 1 , . . . , z n ) } where v is any valuation in F s.t. v ( z i ) = a i . Definition Let L be a labeled ordered language. 1. An L -general frame is a pair � F , A � where F is an L -frame and A is the universe of a subalgebra of F + .
Frames monotone logics ordered algebras logic-based dualities and completions Frames F can be transformed into algebras F + as follows: ◮ The universe of F + is G ( W , J , R ) . ◮ For every connective f ( z 1 , . . . , z n ) and a 1 , . . . , a n ∈ F + , f F + ( a 1 , . . . , a n ) := { w ∈ W : w , v � f ( z 1 , . . . , z n ) } where v is any valuation in F s.t. v ( z i ) = a i . Definition Let L be a labeled ordered language. 1. An L -general frame is a pair � F , A � where F is an L -frame and A is the universe of a subalgebra of F + . 2. The complex algebra of a general frame � F , A � is � F , A � + := � A , ⊆� where A � F + .
Frames monotone logics ordered algebras logic-based dualities and completions Frames F can be transformed into algebras F + as follows: ◮ The universe of F + is G ( W , J , R ) . ◮ For every connective f ( z 1 , . . . , z n ) and a 1 , . . . , a n ∈ F + , f F + ( a 1 , . . . , a n ) := { w ∈ W : w , v � f ( z 1 , . . . , z n ) } where v is any valuation in F s.t. v ( z i ) = a i . Definition Let L be a labeled ordered language. 1. An L -general frame is a pair � F , A � where F is an L -frame and A is the universe of a subalgebra of F + . 2. The complex algebra of a general frame � F , A � is � F , A � + := � A , ⊆� where A � F + . Remark If � F , A � is an L -general frame, then � F , A � + is an L -algebra.
Frames monotone logics ordered algebras logic-based dualities and completions Contents 1. What is a frame? (for an arbitrary algebraic language) 2. What does it mean that a logic has a local relational semantics? 3. Why do most logics have a semantics of ordered algebras? 4. Are there logic-based dualities/completions for ordered algebras?
Frames monotone logics ordered algebras logic-based dualities and completions Let Fr be a class of L -general frames.
Frames monotone logics ordered algebras logic-based dualities and completions Let Fr be a class of L -general frames. 1. The local consequence relation of Fr is: Γ ⊢ l Fr ϕ ⇐ ⇒ for every valuation v in � F , A � ∈ Fr and w ∈ W if w , v � Γ , then w , v � ϕ.
Frames monotone logics ordered algebras logic-based dualities and completions Let Fr be a class of L -general frames. 1. The local consequence relation of Fr is: Γ ⊢ l Fr ϕ ⇐ ⇒ for every valuation v in � F , A � ∈ Fr and w ∈ W if w , v � Γ , then w , v � ϕ. 2. The colocal consequence relation of Fr is: Γ ⊢ cl Fr ϕ ⇐ ⇒ for every valuation v in � F , A � ∈ Fr and j ∈ J if j , v ≻ Γ , then j , v ≻ ϕ.
Frames monotone logics ordered algebras logic-based dualities and completions Let Fr be a class of L -general frames. 1. The local consequence relation of Fr is: Γ ⊢ l Fr ϕ ⇐ ⇒ for every valuation v in � F , A � ∈ Fr and w ∈ W if w , v � Γ , then w , v � ϕ. 2. The colocal consequence relation of Fr is: Γ ⊢ cl Fr ϕ ⇐ ⇒ for every valuation v in � F , A � ∈ Fr and j ∈ J if j , v ≻ Γ , then j , v ≻ ϕ. Definition Let L be a labeled ordered language. A logic ⊢ is a L -local (resp. colocal) consequence if it is the local (resp. colocal) consequence of a class of L -general frames.
Frames monotone logics ordered algebras logic-based dualities and completions Let Fr be a class of L -general frames. 1. The local consequence relation of Fr is: Γ ⊢ l Fr ϕ ⇐ ⇒ for every valuation v in � F , A � ∈ Fr and w ∈ W if w , v � Γ , then w , v � ϕ. 2. The colocal consequence relation of Fr is: Γ ⊢ cl Fr ϕ ⇐ ⇒ for every valuation v in � F , A � ∈ Fr and j ∈ J if j , v ≻ Γ , then j , v ≻ ϕ. Definition Let L be a labeled ordered language. A logic ⊢ is a L -local (resp. colocal) consequence if it is the local (resp. colocal) consequence of a class of L -general frames. Remark A logic is local consequence iff it is a colocal consequence.
Frames monotone logics ordered algebras logic-based dualities and completions Definition A logic ⊢ is monotone if there is an ordered language L over L ⊢ s.t. every connective f ( x 1 , . . . , x m ; y 1 , . . . , y n ) is increasing in � x and decreasing in � y on Fm w.r.t. ⊢ ,
Frames monotone logics ordered algebras logic-based dualities and completions Definition A logic ⊢ is monotone if there is an ordered language L over L ⊢ s.t. every connective f ( x 1 , . . . , x m ; y 1 , . . . , y n ) is increasing in � x and decreasing in � y on Fm w.r.t. ⊢ , i.e. if for every ϕ and ψ such that ϕ ⊢ ψ we have f ( δ 1 , . . . , δ i − 1 , ϕ, δ i + 1 , . . . , δ m ,� ǫ ) ⊢ f ( δ 1 , . . . , δ i − 1 , ψ, δ i + 1 , . . . , δ m ,� ǫ ) f ( � δ, ǫ 1 , . . . , ǫ j − 1 , ψ, ǫ j + 1 , . . . , ǫ n ) ⊢ f ( � δ, ǫ 1 , . . . , ǫ j − 1 , ϕ, ǫ j + 1 , . . . , ǫ n ) for every � δ and � ǫ .
Frames monotone logics ordered algebras logic-based dualities and completions Definition A logic ⊢ is monotone if there is an ordered language L over L ⊢ s.t. every connective f ( x 1 , . . . , x m ; y 1 , . . . , y n ) is increasing in � x and decreasing in � y on Fm w.r.t. ⊢ , i.e. if for every ϕ and ψ such that ϕ ⊢ ψ we have f ( δ 1 , . . . , δ i − 1 , ϕ, δ i + 1 , . . . , δ m ,� ǫ ) ⊢ f ( δ 1 , . . . , δ i − 1 , ψ, δ i + 1 , . . . , δ m ,� ǫ ) f ( � δ, ǫ 1 , . . . , ǫ j − 1 , ψ, ǫ j + 1 , . . . , ǫ n ) ⊢ f ( � δ, ǫ 1 , . . . , ǫ j − 1 , ϕ, ǫ j + 1 , . . . , ǫ n ) for every � δ and � ǫ . In this case, ⊢ is L -monotone.
Frames monotone logics ordered algebras logic-based dualities and completions Definition A logic ⊢ is monotone if there is an ordered language L over L ⊢ s.t. every connective f ( x 1 , . . . , x m ; y 1 , . . . , y n ) is increasing in � x and decreasing in � y on Fm w.r.t. ⊢ , i.e. if for every ϕ and ψ such that ϕ ⊢ ψ we have f ( δ 1 , . . . , δ i − 1 , ϕ, δ i + 1 , . . . , δ m ,� ǫ ) ⊢ f ( δ 1 , . . . , δ i − 1 , ψ, δ i + 1 , . . . , δ m ,� ǫ ) f ( � δ, ǫ 1 , . . . , ǫ j − 1 , ψ, ǫ j + 1 , . . . , ǫ n ) ⊢ f ( � δ, ǫ 1 , . . . , ǫ j − 1 , ϕ, ǫ j + 1 , . . . , ǫ n ) for every � δ and � ǫ . In this case, ⊢ is L -monotone. Theorem (Syntactic characterization of local consequences) Let L be an ordered language, and β a labeling map. The following conditions are equivalent:
Frames monotone logics ordered algebras logic-based dualities and completions Definition A logic ⊢ is monotone if there is an ordered language L over L ⊢ s.t. every connective f ( x 1 , . . . , x m ; y 1 , . . . , y n ) is increasing in � x and decreasing in � y on Fm w.r.t. ⊢ , i.e. if for every ϕ and ψ such that ϕ ⊢ ψ we have f ( δ 1 , . . . , δ i − 1 , ϕ, δ i + 1 , . . . , δ m ,� ǫ ) ⊢ f ( δ 1 , . . . , δ i − 1 , ψ, δ i + 1 , . . . , δ m ,� ǫ ) f ( � δ, ǫ 1 , . . . , ǫ j − 1 , ψ, ǫ j + 1 , . . . , ǫ n ) ⊢ f ( � δ, ǫ 1 , . . . , ǫ j − 1 , ϕ, ǫ j + 1 , . . . , ǫ n ) for every � δ and � ǫ . In this case, ⊢ is L -monotone. Theorem (Syntactic characterization of local consequences) Let L be an ordered language, and β a labeling map. The following conditions are equivalent: 1. ⊢ is an L -monotone logic.
Frames monotone logics ordered algebras logic-based dualities and completions Definition A logic ⊢ is monotone if there is an ordered language L over L ⊢ s.t. every connective f ( x 1 , . . . , x m ; y 1 , . . . , y n ) is increasing in � x and decreasing in � y on Fm w.r.t. ⊢ , i.e. if for every ϕ and ψ such that ϕ ⊢ ψ we have f ( δ 1 , . . . , δ i − 1 , ϕ, δ i + 1 , . . . , δ m ,� ǫ ) ⊢ f ( δ 1 , . . . , δ i − 1 , ψ, δ i + 1 , . . . , δ m ,� ǫ ) f ( � δ, ǫ 1 , . . . , ǫ j − 1 , ψ, ǫ j + 1 , . . . , ǫ n ) ⊢ f ( � δ, ǫ 1 , . . . , ǫ j − 1 , ϕ, ǫ j + 1 , . . . , ǫ n ) for every � δ and � ǫ . In this case, ⊢ is L -monotone. Theorem (Syntactic characterization of local consequences) Let L be an ordered language, and β a labeling map. The following conditions are equivalent: 1. ⊢ is an L -monotone logic. 2. ⊢ is an L β -local consequence.
Frames monotone logics ordered algebras logic-based dualities and completions Contents 1. What is a frame? (for an arbitrary algebraic language) 2. What does it mean that a logic has a local relational semantics? 3. Why do most logics have a semantics of ordered algebras? 4. Are there logic-based dualities/completions for ordered algebras?
Frames monotone logics ordered algebras logic-based dualities and completions Definition Let ⊢ be a logic and L be an ordered language over L ⊢ .
Frames monotone logics ordered algebras logic-based dualities and completions Definition Let ⊢ be a logic and L be an ordered language over L ⊢ . 1. An L -algebra � A , � � is an L -ordered model of ⊢ if for every a ∈ A the upset ↑ a is a deductive filter of ⊢ .
Frames monotone logics ordered algebras logic-based dualities and completions Definition Let ⊢ be a logic and L be an ordered language over L ⊢ . 1. An L -algebra � A , � � is an L -ordered model of ⊢ if for every a ∈ A the upset ↑ a is a deductive filter of ⊢ . 2. Accordingly, we set Alg � L ( ⊢ ) := {� A , � � : � A , � � is an L -ordered model of ⊢} .
Frames monotone logics ordered algebras logic-based dualities and completions Definition Let ⊢ be a logic and L be an ordered language over L ⊢ . 1. An L -algebra � A , � � is an L -ordered model of ⊢ if for every a ∈ A the upset ↑ a is a deductive filter of ⊢ . 2. Accordingly, we set Alg � L ( ⊢ ) := {� A , � � : � A , � � is an L -ordered model of ⊢} . Remark Alg � L ( ⊢ ) is closed under S and P (and P u if ⊢ is finitary).
Frames monotone logics ordered algebras logic-based dualities and completions Definition Let ⊢ be a logic and L be an ordered language over L ⊢ . 1. An L -algebra � A , � � is an L -ordered model of ⊢ if for every a ∈ A the upset ↑ a is a deductive filter of ⊢ . 2. Accordingly, we set Alg � L ( ⊢ ) := {� A , � � : � A , � � is an L -ordered model of ⊢} . Remark Alg � L ( ⊢ ) is closed under S and P (and P u if ⊢ is finitary). ◮ Non-mathematical thesis: Alg � L ( ⊢ ) should be understood as the class of distinguished ordered models of ⊢ (from the point of view of the ordered language L ).
Frames monotone logics ordered algebras logic-based dualities and completions Theoretic justification of Alg � L ( ⊢ ) Definition Let ⊢ be a logic and � F , A � be an L -general frame.
Frames monotone logics ordered algebras logic-based dualities and completions Theoretic justification of Alg � L ( ⊢ ) Definition Let ⊢ be a logic and � F , A � be an L -general frame. 1. � F , A � is a model of ⊢ if its local consequence extends ⊢ .
Frames monotone logics ordered algebras logic-based dualities and completions Theoretic justification of Alg � L ( ⊢ ) Definition Let ⊢ be a logic and � F , A � be an L -general frame. 1. � F , A � is a model of ⊢ if its local consequence extends ⊢ . 2. � F , A � is a co-model of ⊢ if its co-local consequence extends ⊢ .
Frames monotone logics ordered algebras logic-based dualities and completions Theoretic justification of Alg � L ( ⊢ ) Definition Let ⊢ be a logic and � F , A � be an L -general frame. 1. � F , A � is a model of ⊢ if its local consequence extends ⊢ . 2. � F , A � is a co-model of ⊢ if its co-local consequence extends ⊢ . Theorem Let ⊢ be a logic, L an ordered lang. over L ⊢ , β a labeling map.
Frames monotone logics ordered algebras logic-based dualities and completions Theoretic justification of Alg � L ( ⊢ ) Definition Let ⊢ be a logic and � F , A � be an L -general frame. 1. � F , A � is a model of ⊢ if its local consequence extends ⊢ . 2. � F , A � is a co-model of ⊢ if its co-local consequence extends ⊢ . Theorem Let ⊢ be a logic, L an ordered lang. over L ⊢ , β a labeling map. L ( ⊢ ) = {� F , A � + : � F , A � is an L β -general frame Alg � and a model of ⊢} .
Frames monotone logics ordered algebras logic-based dualities and completions Theoretic justification of Alg � L ( ⊢ ) Definition Let ⊢ be a logic and � F , A � be an L -general frame. 1. � F , A � is a model of ⊢ if its local consequence extends ⊢ . 2. � F , A � is a co-model of ⊢ if its co-local consequence extends ⊢ . Theorem Let ⊢ be a logic, L an ordered lang. over L ⊢ , β a labeling map. L ( ⊢ ) = {� F , A � + : � F , A � is an L β -general frame Alg � and a model of ⊢} . In other words, Alg � L ( ⊢ ) is the class of complex algebras of relational models of ⊢ (from the point of view of L and β ).
Frames monotone logics ordered algebras logic-based dualities and completions Theoretic justification of Alg � L ( ⊢ ) Definition Let ⊢ be a logic and � F , A � be an L -general frame. 1. � F , A � is a model of ⊢ if its local consequence extends ⊢ . 2. � F , A � is a co-model of ⊢ if its co-local consequence extends ⊢ . Theorem Let ⊢ be a logic, L an ordered lang. over L ⊢ , β a labeling map. L ( ⊢ ) = {� F , A � + : � F , A � is an L β -general frame Alg � and a model of ⊢} . In other words, Alg � L ( ⊢ ) is the class of complex algebras of relational models of ⊢ (from the point of view of L and β ). ◮ Rephrasing: Logics may have a semantics of ordered algebras, because they have a local relational semantics.
Frames monotone logics ordered algebras logic-based dualities and completions Empiric justification of Alg � L ( ⊢ ) : semilattice-based logics Theorem Let K be a variety with a semilattice reduct s.t. when ordered under the meet-order is a class of L -algebras.
Frames monotone logics ordered algebras logic-based dualities and completions Empiric justification of Alg � L ( ⊢ ) : semilattice-based logics Theorem Let K be a variety with a semilattice reduct s.t. when ordered under the meet-order is a class of L -algebras. Then Alg � L ( ⊢ � K ) = {� A , � � : A ∈ K and � is the meet-order of A } .
Frames monotone logics ordered algebras logic-based dualities and completions Empiric justification of Alg � L ( ⊢ ) : semilattice-based logics Theorem Let K be a variety with a semilattice reduct s.t. when ordered under the meet-order is a class of L -algebras. Then Alg � L ( ⊢ � K ) = {� A , � � : A ∈ K and � is the meet-order of A } . Examples:
Frames monotone logics ordered algebras logic-based dualities and completions Empiric justification of Alg � L ( ⊢ ) : semilattice-based logics Theorem Let K be a variety with a semilattice reduct s.t. when ordered under the meet-order is a class of L -algebras. Then Alg � L ( ⊢ � K ) = {� A , � � : A ∈ K and � is the meet-order of A } . Examples: ◮ Let K be a variety of modal algebras, and ⊢ the local consequence of the normal modal logic associated with K.
Frames monotone logics ordered algebras logic-based dualities and completions Empiric justification of Alg � L ( ⊢ ) : semilattice-based logics Theorem Let K be a variety with a semilattice reduct s.t. when ordered under the meet-order is a class of L -algebras. Then Alg � L ( ⊢ � K ) = {� A , � � : A ∈ K and � is the meet-order of A } . Examples: ◮ Let K be a variety of modal algebras, and ⊢ the local consequence of the normal modal logic associated with K. Then Alg � L ( ⊢ ) is K with the lattice order (for the natural L ).
Frames monotone logics ordered algebras logic-based dualities and completions Empiric justification of Alg � L ( ⊢ ) : semilattice-based logics Theorem Let K be a variety with a semilattice reduct s.t. when ordered under the meet-order is a class of L -algebras. Then Alg � L ( ⊢ � K ) = {� A , � � : A ∈ K and � is the meet-order of A } . Examples: ◮ Let K be a variety of modal algebras, and ⊢ the local consequence of the normal modal logic associated with K. Then Alg � L ( ⊢ ) is K with the lattice order (for the natural L ). ◮ Let K be a variety of Heyting algebras, and ⊢ the superintuitionistic logic associated with K.
Frames monotone logics ordered algebras logic-based dualities and completions Empiric justification of Alg � L ( ⊢ ) : semilattice-based logics Theorem Let K be a variety with a semilattice reduct s.t. when ordered under the meet-order is a class of L -algebras. Then Alg � L ( ⊢ � K ) = {� A , � � : A ∈ K and � is the meet-order of A } . Examples: ◮ Let K be a variety of modal algebras, and ⊢ the local consequence of the normal modal logic associated with K. Then Alg � L ( ⊢ ) is K with the lattice order (for the natural L ). ◮ Let K be a variety of Heyting algebras, and ⊢ the superintuitionistic logic associated with K. Then Alg � L ( ⊢ ) is K with the lattice order (for the natural L ).
Frames monotone logics ordered algebras logic-based dualities and completions Empiric justification of Alg � L ( ⊢ ) : intensional fragments For the natural ordered languages L :
Frames monotone logics ordered algebras logic-based dualities and completions Empiric justification of Alg � L ( ⊢ ) : intensional fragments For the natural ordered languages L : ◮ Let IPC → be the �→� -fragment of Intuitionistic Logic.
Frames monotone logics ordered algebras logic-based dualities and completions Empiric justification of Alg � L ( ⊢ ) : intensional fragments For the natural ordered languages L : ◮ Let IPC → be the �→� -fragment of Intuitionistic Logic. Then Alg � L ( IPC → ) = Hilbert algebras + Hilbert-order .
Frames monotone logics ordered algebras logic-based dualities and completions Empiric justification of Alg � L ( ⊢ ) : intensional fragments For the natural ordered languages L : ◮ Let IPC → be the �→� -fragment of Intuitionistic Logic. Then Alg � L ( IPC → ) = Hilbert algebras + Hilbert-order . ◮ Let InFL � e be the �· , →� -fragment of the logic preserving degrees of truth of commutative FL-algebras.
Frames monotone logics ordered algebras logic-based dualities and completions Empiric justification of Alg � L ( ⊢ ) : intensional fragments For the natural ordered languages L : ◮ Let IPC → be the �→� -fragment of Intuitionistic Logic. Then Alg � L ( IPC → ) = Hilbert algebras + Hilbert-order . ◮ Let InFL � e be the �· , →� -fragment of the logic preserving degrees of truth of commutative FL-algebras. Then Alg � L ( InFL � e ) = �· , → , � � -subreducts of commutative FL-algebras .
Frames monotone logics ordered algebras logic-based dualities and completions Empiric justification of Alg � L ( ⊢ ) : intensional fragments For the natural ordered languages L : ◮ Let IPC → be the �→� -fragment of Intuitionistic Logic. Then Alg � L ( IPC → ) = Hilbert algebras + Hilbert-order . ◮ Let InFL � e be the �· , →� -fragment of the logic preserving degrees of truth of commutative FL-algebras. Then Alg � L ( InFL � e ) = �· , → , � � -subreducts of commutative FL-algebras . ◮ Let InR � be the �· , → , ¬� -fragment of the logic preserving degrees of truth of De Morgan monoids.
Frames monotone logics ordered algebras logic-based dualities and completions Empiric justification of Alg � L ( ⊢ ) : intensional fragments For the natural ordered languages L : ◮ Let IPC → be the �→� -fragment of Intuitionistic Logic. Then Alg � L ( IPC → ) = Hilbert algebras + Hilbert-order . ◮ Let InFL � e be the �· , →� -fragment of the logic preserving degrees of truth of commutative FL-algebras. Then Alg � L ( InFL � e ) = �· , → , � � -subreducts of commutative FL-algebras . ◮ Let InR � be the �· , → , ¬� -fragment of the logic preserving degrees of truth of De Morgan monoids. Then Alg � L ( InR � ) = �· , → , ¬ , � � -subreducts of De Morgan monoids .
Frames monotone logics ordered algebras logic-based dualities and completions Contents 1. What is a frame? (for an arbitrary algebraic language) 2. What does it mean that a logic has a local relational semantics? 3. Why do most logics have a semantics of ordered algebras? 4. Are there logic-based dualities/completions for ordered algebras?
Frames monotone logics ordered algebras logic-based dualities and completions Logics preserving degrees of truth of Lattice Expansions Let K be a variety with a bounded lattice reduct s.t. when ordered under the lattice-order is a class of L -algebras.
Frames monotone logics ordered algebras logic-based dualities and completions Logics preserving degrees of truth of Lattice Expansions Let K be a variety with a bounded lattice reduct s.t. when ordered under the lattice-order is a class of L -algebras. Then for all � A , � � ∈ Alg � L ( ⊢ � K ) we have:
Frames monotone logics ordered algebras logic-based dualities and completions Logics preserving degrees of truth of Lattice Expansions Let K be a variety with a bounded lattice reduct s.t. when ordered under the lattice-order is a class of L -algebras. Then for all � A , � � ∈ Alg � L ( ⊢ � K ) we have: 1. A ∈ K and � is the lattice order of A .
Frames monotone logics ordered algebras logic-based dualities and completions Logics preserving degrees of truth of Lattice Expansions Let K be a variety with a bounded lattice reduct s.t. when ordered under the lattice-order is a class of L -algebras. Then for all � A , � � ∈ Alg � L ( ⊢ � K ) we have: 1. A ∈ K and � is the lattice order of A . 2. Pol L � A , � � = � W , J , R � is s.t. W = lattice filters and J = lattice ideals.
Frames monotone logics ordered algebras logic-based dualities and completions Logics preserving degrees of truth of Lattice Expansions Let K be a variety with a bounded lattice reduct s.t. when ordered under the lattice-order is a class of L -algebras. Then for all � A , � � ∈ Alg � L ( ⊢ � K ) we have: 1. A ∈ K and � is the lattice order of A . 2. Pol L � A , � � = � W , J , R � is s.t. W = lattice filters and J = lattice ideals. Moreover, ( � A , � � + ) + is the canonical extension of � A , � � .
Frames monotone logics ordered algebras logic-based dualities and completions Logics preserving degrees of truth of Lattice Expansions Let K be a variety with a bounded lattice reduct s.t. when ordered under the lattice-order is a class of L -algebras. Then for all � A , � � ∈ Alg � L ( ⊢ � K ) we have: 1. A ∈ K and � is the lattice order of A . 2. Pol L � A , � � = � W , J , R � is s.t. W = lattice filters and J = lattice ideals. Moreover, ( � A , � � + ) + is the canonical extension of � A , � � . Implicative fragment of IPC For all � A , � � ∈ Alg � L ( IPC → ) we have:
Frames monotone logics ordered algebras logic-based dualities and completions Logics preserving degrees of truth of Lattice Expansions Let K be a variety with a bounded lattice reduct s.t. when ordered under the lattice-order is a class of L -algebras. Then for all � A , � � ∈ Alg � L ( ⊢ � K ) we have: 1. A ∈ K and � is the lattice order of A . 2. Pol L � A , � � = � W , J , R � is s.t. W = lattice filters and J = lattice ideals. Moreover, ( � A , � � + ) + is the canonical extension of � A , � � . Implicative fragment of IPC For all � A , � � ∈ Alg � L ( IPC → ) we have: 1. � A , � � is a Hilbert algebra equipped with the Hilbert-order.
Frames monotone logics ordered algebras logic-based dualities and completions Logics preserving degrees of truth of Lattice Expansions Let K be a variety with a bounded lattice reduct s.t. when ordered under the lattice-order is a class of L -algebras. Then for all � A , � � ∈ Alg � L ( ⊢ � K ) we have: 1. A ∈ K and � is the lattice order of A . 2. Pol L � A , � � = � W , J , R � is s.t. W = lattice filters and J = lattice ideals. Moreover, ( � A , � � + ) + is the canonical extension of � A , � � . Implicative fragment of IPC For all � A , � � ∈ Alg � L ( IPC → ) we have: 1. � A , � � is a Hilbert algebra equipped with the Hilbert-order. 2. Pol L � A , � � = � W , J , R � is s.t. W = implicative filters and J = downsets.
Frames monotone logics ordered algebras logic-based dualities and completions Logics preserving degrees of truth of Lattice Expansions Let K be a variety with a bounded lattice reduct s.t. when ordered under the lattice-order is a class of L -algebras. Then for all � A , � � ∈ Alg � L ( ⊢ � K ) we have: 1. A ∈ K and � is the lattice order of A . 2. Pol L � A , � � = � W , J , R � is s.t. W = lattice filters and J = lattice ideals. Moreover, ( � A , � � + ) + is the canonical extension of � A , � � . Implicative fragment of IPC For all � A , � � ∈ Alg � L ( IPC → ) we have: 1. � A , � � is a Hilbert algebra equipped with the Hilbert-order. 2. Pol L � A , � � = � W , J , R � is s.t. W = implicative filters and J = downsets. Moreover, ( � A , � � + ) + is intrinsically a Heyting algebra.
Frames monotone logics ordered algebras logic-based dualities and completions Intensional fragment of FL � e For all � A , � � ∈ Alg � L ( InFL � e ) we have:
Frames monotone logics ordered algebras logic-based dualities and completions Intensional fragment of FL � e For all � A , � � ∈ Alg � L ( InFL � e ) we have: 1. � A , � � is a �· , → , � � -subreduct of a commutative FL-algebra.
Frames monotone logics ordered algebras logic-based dualities and completions Intensional fragment of FL � e For all � A , � � ∈ Alg � L ( InFL � e ) we have: 1. � A , � � is a �· , → , � � -subreduct of a commutative FL-algebra. 2. Pol L � A , � � = � W , J , R � is s.t. W = upsets and J = downsets.
Frames monotone logics ordered algebras logic-based dualities and completions Intensional fragment of FL � e For all � A , � � ∈ Alg � L ( InFL � e ) we have: 1. � A , � � is a �· , → , � � -subreduct of a commutative FL-algebra. 2. Pol L � A , � � = � W , J , R � is s.t. W = upsets and J = downsets. Moreover, ( � A , � � + ) + is intrinsically a commutative FL-algebra.
Frames monotone logics ordered algebras logic-based dualities and completions Intensional fragment of FL � e For all � A , � � ∈ Alg � L ( InFL � e ) we have: 1. � A , � � is a �· , → , � � -subreduct of a commutative FL-algebra. 2. Pol L � A , � � = � W , J , R � is s.t. W = upsets and J = downsets. Moreover, ( � A , � � + ) + is intrinsically a commutative FL-algebra. Intensional fragment of R � For all � A , � � ∈ Alg � L ( InR � ) we have:
Frames monotone logics ordered algebras logic-based dualities and completions Intensional fragment of FL � e For all � A , � � ∈ Alg � L ( InFL � e ) we have: 1. � A , � � is a �· , → , � � -subreduct of a commutative FL-algebra. 2. Pol L � A , � � = � W , J , R � is s.t. W = upsets and J = downsets. Moreover, ( � A , � � + ) + is intrinsically a commutative FL-algebra. Intensional fragment of R � For all � A , � � ∈ Alg � L ( InR � ) we have: 1. � A , � � is a �· , → , ¬ , � � -subreduct of a De Morgan monoid.
Frames monotone logics ordered algebras logic-based dualities and completions Intensional fragment of FL � e For all � A , � � ∈ Alg � L ( InFL � e ) we have: 1. � A , � � is a �· , → , � � -subreduct of a commutative FL-algebra. 2. Pol L � A , � � = � W , J , R � is s.t. W = upsets and J = downsets. Moreover, ( � A , � � + ) + is intrinsically a commutative FL-algebra. Intensional fragment of R � For all � A , � � ∈ Alg � L ( InR � ) we have: 1. � A , � � is a �· , → , ¬ , � � -subreduct of a De Morgan monoid. 2. Pol L � A , � � = � W , J , R � is s.t. W = intensional filters and J = intensional ideals.
Frames monotone logics ordered algebras logic-based dualities and completions Intensional fragment of FL � e For all � A , � � ∈ Alg � L ( InFL � e ) we have: 1. � A , � � is a �· , → , � � -subreduct of a commutative FL-algebra. 2. Pol L � A , � � = � W , J , R � is s.t. W = upsets and J = downsets. Moreover, ( � A , � � + ) + is intrinsically a commutative FL-algebra. Intensional fragment of R � For all � A , � � ∈ Alg � L ( InR � ) we have: 1. � A , � � is a �· , → , ¬ , � � -subreduct of a De Morgan monoid. 2. Pol L � A , � � = � W , J , R � is s.t. W = intensional filters and J = intensional ideals. Moreover, ( � A , � � + ) + is intrinsically a De Morgan monoid.
Frames monotone logics ordered algebras logic-based dualities and completions A sample of what comes next... ◮ Substructural logics with weakening can be viewed as global consequences in this spirit,
Frames monotone logics ordered algebras logic-based dualities and completions A sample of what comes next... ◮ Substructural logics with weakening can be viewed as global consequences in this spirit, e.g. Łukasiewicz is the global version of the logic preserving degrees of truth of MV-algebras.
Frames monotone logics ordered algebras logic-based dualities and completions A sample of what comes next... ◮ Substructural logics with weakening can be viewed as global consequences in this spirit, e.g. Łukasiewicz is the global version of the logic preserving degrees of truth of MV-algebras. ◮ One can give a relational semantics for every logic, inspired by the Routley-Meyer semantics for Relevance Logic.
Frames monotone logics ordered algebras logic-based dualities and completions A sample of what comes next... ◮ Substructural logics with weakening can be viewed as global consequences in this spirit, e.g. Łukasiewicz is the global version of the logic preserving degrees of truth of MV-algebras. ◮ One can give a relational semantics for every logic, inspired by the Routley-Meyer semantics for Relevance Logic. ◮ We can delete co-worlds from frames in nice cases, e.g. distributive substructural and modal logics.
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