Relational semantics, ordered algebras, and quantifiers for - - PowerPoint PPT Presentation

Frames monotone logics

• rdered algebras

logic-based dualities and completions

Relational semantics, ordered algebras, and quantifiers for deductive systems

Tommaso Moraschini joint work with Ramon Jansana

Institute of Computer Science of the Czech Academy of Sciences

July 23, 2018

Frames monotone logics

• rdered 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

• rdered algebras

logic-based dualities and completions

Definition

• 1. An order type for an algebraic language L is an assignment to

every symbol f ∈ L of a choice of which arguments of f will be treated as increasing and which ones as decreasing.

Frames monotone logics

• rdered algebras

logic-based dualities and completions

Definition

• 1. An order type for an algebraic language L is an assignment to

every symbol f ∈ L of a choice of which arguments of f will be treated as increasing and which ones as decreasing.

• 2. An ordered language is an algebraic language equipped with an
• rder type.

Frames monotone logics

• rdered algebras

logic-based dualities and completions

Definition

• 1. An order type for an algebraic language L is an assignment to

every symbol f ∈ L of a choice of which arguments of f will be treated as increasing and which ones as decreasing.

• 2. An ordered language is an algebraic language equipped with an
• rder type.
• 3. Let L be an ordered language. An L -algebra is a pair A,

where A is an algebra, a partial order on A, and if f = f ( x, y), then f A is incr. on x and decr. on y w.r.t. .

Frames monotone logics

• rdered algebras

logic-based dualities and completions

Definition

• 1. An order type for an algebraic language L is an assignment to

every symbol f ∈ L of a choice of which arguments of f will be treated as increasing and which ones as decreasing.

• 2. An ordered language is an algebraic language equipped with an
• rder type.
• 3. Let L be an ordered language. An L -algebra is a pair A,

where A is an algebra, a partial order on A, and if f = f ( x, y), then f A is incr. on x and decr. on y w.r.t. . Examples:

Frames monotone logics

• rdered algebras

logic-based dualities and completions

Definition

• 1. An order type for an algebraic language L is an assignment to

every symbol f ∈ L of a choice of which arguments of f will be treated as increasing and which ones as decreasing.

• 2. An ordered language is an algebraic language equipped with an
• rder type.
• 3. Let L be an ordered language. An L -algebra is a pair A,

where A is an algebra, a partial order on A, and if f = f ( x, y), then f A is incr. on x and decr. on y w.r.t. . Examples:

◮ Heyting algebras, modal algebras and residuated lattices

equipped with the lattice order can be seen as L -algebras.

Frames monotone logics

• rdered algebras

logic-based dualities and completions

Definition

A polarity is a triple W , J, R such that W and J are non-empty sets and R ⊆ W × J.

Frames monotone logics

• rdered algebras

logic-based dualities and completions

Definition

A polarity is a triple W , J, R such that W and J are non-empty sets and R ⊆ W × J.

◮ Every polarity W , J, R induces a Galois connection

(·)✄ : P(W ) ← → P(J): (·)✁

Frames monotone logics

• rdered algebras

logic-based dualities and completions

Definition

A polarity is a triple W , J, R such that W and J are non-empty sets and R ⊆ W × J.

◮ Every polarity W , J, R induces a Galois connection

(·)✄ : P(W ) ← → P(J): (·)✁ by setting for A ⊆ W and B ⊆ J A✄ := {j ∈ J : w, j ∈ R for all w ∈ A} B✁ := {w ∈ W : w, j ∈ R for all j ∈ B}.

Frames monotone logics

• rdered algebras

logic-based dualities and completions

Definition

A polarity is a triple W , J, R such that W and J are non-empty sets and R ⊆ W × J.

◮ Every polarity W , J, R induces a Galois connection

(·)✄ : P(W ) ← → P(J): (·)✁ by setting for A ⊆ W and B ⊆ J A✄ := {j ∈ J : w, j ∈ R for all w ∈ A} B✁ := {w ∈ W : w, j ∈ R for all j ∈ B}.

◮ Indeed, we have that B ⊆ A✄ ⇐

⇒ A ⊆ B✁.

Frames monotone logics

• rdered algebras

logic-based dualities and completions

Definition

A polarity is a triple W , J, R such that W and J are non-empty sets and R ⊆ W × J.

◮ Every polarity W , J, R induces a Galois connection

(·)✄ : P(W ) ← → P(J): (·)✁ by setting for A ⊆ W and B ⊆ J A✄ := {j ∈ J : w, j ∈ R for all w ∈ A} B✁ := {w ∈ W : w, j ∈ R for all j ∈ B}.

◮ Indeed, we have that B ⊆ A✄ ⇐

⇒ A ⊆ B✁.

◮ Then (·)✄✁ : P(W ) → P(W ) is a closure operator on W .

Frames monotone logics

• rdered algebras

logic-based dualities and completions

Definition

A polarity is a triple W , J, R such that W and J are non-empty sets and R ⊆ W × J.

◮ Every polarity W , J, R induces a Galois connection

(·)✄ : P(W ) ← → P(J): (·)✁ by setting for A ⊆ W and B ⊆ J A✄ := {j ∈ J : w, j ∈ R for all w ∈ A} B✁ := {w ∈ W : w, j ∈ R for all j ∈ B}.

◮ Indeed, we have that B ⊆ A✄ ⇐

⇒ A ⊆ B✁.

◮ Then (·)✄✁ : P(W ) → P(W ) is a closure operator on W . We

denote its lattice of closed sets by G(W , J, R).

Frames monotone logics

• rdered algebras

logic-based dualities and completions

• 1. A labeling map for an algebraic language L is a function

β : L → {✷, ✸}.

Frames monotone logics

• rdered algebras

logic-based dualities and completions

• 1. A labeling map for an algebraic language L is a function

β : L → {✷, ✸}.

• 2. A labeled language is an algebraic language L equipped with

a labeling map β. Sometimes we write L β.

Frames monotone logics

• rdered algebras

logic-based dualities and completions

• 1. A labeling map for an algebraic language L is a function

β : L → {✷, ✸}.

• 2. A labeled language is an algebraic language L equipped with

a labeling map β. Sometimes we write L β.

Definition

Let L be a labeled ordered language.

Frames monotone logics

• rdered algebras

logic-based dualities and completions

• 1. A labeling map for an algebraic language L is a function

β : L → {✷, ✸}.

• 2. A labeled language is an algebraic language L equipped with

a labeling map β. Sometimes we write L β.

Definition

Let L be a labeled ordered language. An L -preframe is a structure F = W , J, R, {Tf : f ∈ L }

Frames monotone logics

• rdered algebras

logic-based dualities and completions

• 1. A labeling map for an algebraic language L is a function

β : L → {✷, ✸}.

• 2. A labeled language is an algebraic language L equipped with

a labeling map β. Sometimes we write L β.

Definition

Let L be a labeled ordered language. An L -preframe is a structure F = W , J, R, {Tf : f ∈ L } where W , J, R is a polarity such that W and J are partial

• rders,

Frames monotone logics

• rdered algebras

logic-based dualities and completions

• 1. A labeling map for an algebraic language L is a function

β : L → {✷, ✸}.

• 2. A labeled language is an algebraic language L equipped with

a labeling map β. Sometimes we write L β.

Definition

Let L be a labeled ordered language. An L -preframe is a structure F = W , J, R, {Tf : f ∈ L } where W , J, R is a polarity such that W and J are partial

• rders, and for every operation symbol f ∈ L such that

f = f (x1, . . . , xm; y1, . . . , yn) we have:

Frames monotone logics

• rdered algebras

logic-based dualities and completions

• 1. A labeling map for an algebraic language L is a function

β : L → {✷, ✸}.

• 2. A labeled language is an algebraic language L equipped with

a labeling map β. Sometimes we write L β.

Definition

Let L be a labeled ordered language. An L -preframe is a structure F = W , J, R, {Tf : f ∈ L } where W , J, R is a polarity such that W and J are partial

• rders, and for every operation symbol f ∈ L such that

f = f (x1, . . . , xm; y1, . . . , yn) we have: if β(f ) = ✸, then Tf ⊆ W m × Jn × W if β(f ) = ✷, then Tf ⊆ Jm × W n × J.

Frames monotone logics

• rdered algebras

logic-based dualities and completions

Definition

Let L be a labeled ordered language.

Frames monotone logics

• rdered algebras

logic-based dualities and completions

Definition

Let L be a labeled ordered language. An L -frame F = W , J, R, {Tf : f ∈ L } is an L -preframe such that for all connectives f (x1, . . . , xm; y1, . . . , yn) s.t. β(f ) = ✸:

Frames monotone logics

• rdered algebras

logic-based dualities and completions

Definition

Let L be a labeled ordered language. An L -frame F = W , J, R, {Tf : f ∈ L } is an L -preframe such that for all connectives f (x1, . . . , xm; y1, . . . , yn) s.t. β(f ) = ✸: (a) For every w1, w2 ∈ W m, j1, j2 ∈ Jn, and u1, u2 ∈ W such that

• w2W

w1, j2J j1 and u1W u2, if w1, j1, u1 ∈ Tf , then w2, j2, u2 ∈ Tf .

Frames monotone logics

• rdered algebras

logic-based dualities and completions

Definition

Let L be a labeled ordered language. An L -frame F = W , J, R, {Tf : f ∈ L } is an L -preframe such that for all connectives f (x1, . . . , xm; y1, . . . , yn) s.t. β(f ) = ✸: (a) For every w1, w2 ∈ W m, j1, j2 ∈ Jn, and u1, u2 ∈ W such that

• w2W

w1, j2J j1 and u1W u2, if w1, j1, u1 ∈ Tf , then w2, j2, u2 ∈ Tf . (b) Tf ( w, j) is a closed set of (·)✄✁ for all w ∈ W m and j ∈ Jn.

Frames monotone logics

• rdered algebras

logic-based dualities and completions

Definition

Let L be a labeled ordered language. An L -frame F = W , J, R, {Tf : f ∈ L } is an L -preframe such that for all connectives f (x1, . . . , xm; y1, . . . , yn) s.t. β(f ) = ✸: (a) For every w1, w2 ∈ W m, j1, j2 ∈ Jn, and u1, u2 ∈ W such that

• w2W

w1, j2J j1 and u1W u2, if w1, j1, u1 ∈ Tf , then w2, j2, u2 ∈ Tf . (b) Tf ( w, j) is a closed set of (·)✄✁ for all w ∈ W m and j ∈ Jn. Connectives f such that β(f ) = ✷ need to satisfy a dual requirement.

Frames monotone logics

• rdered algebras

logic-based dualities and completions

Definition

Let L be a labeled ordered language. An L -frame F = W , J, R, {Tf : f ∈ L } is an L -preframe such that for all connectives f (x1, . . . , xm; y1, . . . , yn) s.t. β(f ) = ✸: (a) For every w1, w2 ∈ W m, j1, j2 ∈ Jn, and u1, u2 ∈ W such that

• w2W

w1, j2J j1 and u1W u2, if w1, j1, u1 ∈ Tf , then w2, j2, u2 ∈ Tf . (b) Tf ( w, j) is a closed set of (·)✄✁ for all w ∈ W m and j ∈ Jn. 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

• rdered 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

• rdered 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

• rdered 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

• rdered 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

• rdered 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: w, v f ( ϕ, ψ) ⇐ ⇒ w ∈ {r ∈ W : there are u ∈ W m and i ∈ Jn s.t. u, i, r ∈ Tf and for all k m, t n uk, v ϕk and it, v ≻ ψt}✄✁ j, v ≻ f ( ϕ, ψ) ⇐ ⇒ j ∈ {w ∈ W : w, v f ( ϕ, ψ)}✄.

Frames monotone logics

• rdered 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: w, v f ( ϕ, ψ) ⇐ ⇒ w ∈ {r ∈ W : there are u ∈ W m and i ∈ Jn s.t. u, i, r ∈ Tf and for all k m, t n uk, v ϕk and it, v ≻ ψt}✄✁ j, v ≻ f ( ϕ, ψ) ⇐ ⇒ j ∈ {w ∈ W : w, v f ( ϕ, ψ)}✄.

◮ A dual definition applied to connectives f (

x; y) s.t. β(f ) = ✷.

Frames monotone logics

• rdered algebras

logic-based dualities and completions

Frames F can be transformed into algebras F + as follows:

Frames monotone logics

• rdered 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

• rdered 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 (z1, . . . , zn) and a1, . . . , an ∈ F +,

f F +(a1, . . . , an) := {w ∈ W : w, v f (z1, . . . , zn)} where v is any valuation in F s.t. v(zi) = ai.

Frames monotone logics

• rdered 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 (z1, . . . , zn) and a1, . . . , an ∈ F +,

f F +(a1, . . . , an) := {w ∈ W : w, v f (z1, . . . , zn)} where v is any valuation in F s.t. v(zi) = ai.

Definition

Let L be a labeled ordered language.

Frames monotone logics

• rdered 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 (z1, . . . , zn) and a1, . . . , an ∈ F +,

f F +(a1, . . . , an) := {w ∈ W : w, v f (z1, . . . , zn)} where v is any valuation in F s.t. v(zi) = ai.

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

• rdered 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 (z1, . . . , zn) and a1, . . . , an ∈ F +,

f F +(a1, . . . , an) := {w ∈ W : w, v f (z1, . . . , zn)} where v is any valuation in F s.t. v(zi) = ai.

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

• rdered 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 (z1, . . . , zn) and a1, . . . , an ∈ F +,

f F +(a1, . . . , an) := {w ∈ W : w, v f (z1, . . . , zn)} where v is any valuation in F s.t. v(zi) = ai.

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

• rdered 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

• rdered algebras

logic-based dualities and completions

Let Fr be a class of L -general frames.

Frames monotone logics

• rdered 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

• rdered 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

• rdered 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

• rdered 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

• rdered 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 (x1, . . . , xm; y1, . . . , yn) is increasing in x and decreasing in y on Fm w.r.t. ⊢,

Frames monotone logics

• rdered 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 (x1, . . . , xm; y1, . . . , yn) 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 (δ1, . . . , δi−1, ϕ, δi+1, . . . , δm, ǫ) ⊢ f (δ1, . . . , δi−1, ψ, δi+1, . . . , δm, ǫ) f ( δ, ǫ1, . . . , ǫj−1, ψ, ǫj+1, . . . , ǫn) ⊢ f ( δ, ǫ1, . . . , ǫj−1, ϕ, ǫj+1, . . . , ǫn) for every δ and ǫ.

Frames monotone logics

• rdered 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 (x1, . . . , xm; y1, . . . , yn) 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 (δ1, . . . , δi−1, ϕ, δi+1, . . . , δm, ǫ) ⊢ f (δ1, . . . , δi−1, ψ, δi+1, . . . , δm, ǫ) f ( δ, ǫ1, . . . , ǫj−1, ψ, ǫj+1, . . . , ǫn) ⊢ f ( δ, ǫ1, . . . , ǫj−1, ϕ, ǫj+1, . . . , ǫn) for every δ and ǫ. In this case, ⊢ is L -monotone.

Frames monotone logics

• rdered 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 (x1, . . . , xm; y1, . . . , yn) 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 (δ1, . . . , δi−1, ϕ, δi+1, . . . , δm, ǫ) ⊢ f (δ1, . . . , δi−1, ψ, δi+1, . . . , δm, ǫ) f ( δ, ǫ1, . . . , ǫj−1, ψ, ǫj+1, . . . , ǫn) ⊢ f ( δ, ǫ1, . . . , ǫj−1, ϕ, ǫj+1, . . . , ǫn) 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

• rdered 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 (x1, . . . , xm; y1, . . . , yn) 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 (δ1, . . . , δi−1, ϕ, δi+1, . . . , δm, ǫ) ⊢ f (δ1, . . . , δi−1, ψ, δi+1, . . . , δm, ǫ) f ( δ, ǫ1, . . . , ǫj−1, ψ, ǫj+1, . . . , ǫn) ⊢ f ( δ, ǫ1, . . . , ǫj−1, ϕ, ǫj+1, . . . , ǫn) 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

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Definition

A logic ⊢ is monotone if there is an ordered language L over L⊢ s.t. every connective f (x1, . . . , xm; y1, . . . , yn) 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 (δ1, . . . , δi−1, ϕ, δi+1, . . . , δm, ǫ) ⊢ f (δ1, . . . , δi−1, ψ, δi+1, . . . , δm, ǫ) f ( δ, ǫ1, . . . , ǫj−1, ψ, ǫj+1, . . . , ǫn) ⊢ f ( δ, ǫ1, . . . , ǫj−1, ϕ, ǫj+1, . . . , ǫn) 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

• rdered 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

• rdered algebras

logic-based dualities and completions

Definition

Let ⊢ be a logic and L be an ordered language over L⊢.

Frames monotone logics

• rdered 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

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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

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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

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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

• f view of the ordered language L ).

Frames monotone logics

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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

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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

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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

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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

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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. Alg

L (⊢) ={F, A+ : F, A is an L β-general frame

and a model of ⊢}.

Frames monotone logics

• rdered 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. Alg

L (⊢) ={F, A+ : F, A is an L β-general frame

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

• rdered 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. Alg

L (⊢) ={F, A+ : F, A is an L β-general frame

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

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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

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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

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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

• rdered 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

• rdered 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

• rdered 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

• rdered 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

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logic-based dualities and completions

Empiric justification of Alg

L (⊢): intensional fragments

For the natural ordered languages L :

Frames monotone logics

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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

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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

• rdered 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

• rdered 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

• rdered 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

• rdered 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

• rdered 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

• rdered 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

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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

• rdered 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

• rdered 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. PolL A, = W , J, R is s.t.

W = lattice filters and J = lattice ideals.

Frames monotone logics

• rdered 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. PolL 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

• rdered 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. PolL 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

• rdered 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. PolL 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

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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. PolL 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. PolL A, = W , J, R is s.t.

W = implicative filters and J = downsets.

Frames monotone logics

• rdered 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. PolL 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. PolL A, = W , J, R is s.t.

W = implicative filters and J = downsets. Moreover, (A, +)+ is intrinsically a Heyting algebra.

Frames monotone logics

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logic-based dualities and completions

Intensional fragment of FL

e

For all A, ∈ Alg

L (InFL e ) we have:

Frames monotone logics

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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

• rdered 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. PolL A, = W , J, R is s.t.

W = upsets and J = downsets.

Frames monotone logics

• rdered 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. PolL A, = W , J, R is s.t.

W = upsets and J = downsets. Moreover, (A, +)+ is intrinsically a commutative FL-algebra.

Frames monotone logics

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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. PolL 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

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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. PolL 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

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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. PolL 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. PolL A, = W , J, R is s.t.

W = intensional filters and J = intensional ideals.

Frames monotone logics

• rdered 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. PolL 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. PolL 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

• rdered 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

• rdered 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

• rdered 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

• rdered 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.

Frames monotone logics

• rdered 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.

◮ This approach suggests a semantic-based of expanding every

local consequence to the first-order lever with quantifiers and identity, which is axiomatized very transparently by means of meta-rules.

Frames monotone logics

• rdered 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.

◮ This approach suggests a semantic-based of expanding every

local consequence to the first-order lever with quantifiers and identity, which is axiomatized very transparently by means of meta-rules.

◮ This yields a complete alternative relational semantics for all

first-order modal and superintuitionistic logics.

Frames monotone logics

• rdered 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.

◮ This approach suggests a semantic-based of expanding every

local consequence to the first-order lever with quantifiers and identity, which is axiomatized very transparently by means of meta-rules.

◮ This yields a complete alternative relational semantics for all

first-order modal and superintuitionistic logics. ...thank you for coming!