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The logic R-Mingle RMt Finding the bases Further Work References

Admissible Rules of (Fragments of) R-Mingle Admissible Rules of (Fragments of) R-Mingle

Laura Janina Schn¨ uriger Admissible Rules of (Fragments of) R-Mingle joint work with George Metcalfe Universit¨ at Bern Admissible Rules of (Fragments of) R-Mingle Les Diablerets 31 January 2015

Laura Janina Schn¨ uriger Admissible Rules of (Fragments of) R-Mingle

The logic R-Mingle RMt Finding the bases Further Work References

Table of contents

• 1. The logic R-Mingle RMt

1.1 Notations 1.2 Corresponding algebraic semantics 1.3 Sugihara Monoids 1.4 This talk

• 2. Finding the bases

2.1 Idea of how to find the bases 2.2 Finding the bases 2.3 The bases

• 3. Further Work

3.1 Our Conjecture

• 4. References

Laura Janina Schn¨ uriger Admissible Rules of (Fragments of) R-Mingle

The logic R-Mingle RMt Finding the bases Further Work References Notations Corresponding algebraic semantics Sugihara Monoids This talk

R-Mingle

RM

Relevance logic R with Mingle

Mingle

p → (p → p)

Laura Janina Schn¨ uriger Admissible Rules of (Fragments of) R-Mingle

The logic R-Mingle RMt Finding the bases Further Work References Notations Corresponding algebraic semantics Sugihara Monoids This talk

R-Mingle

RM

Relevance logic R with Mingle

Mingle

p → (p → p)

RMt

RM with additional constant t

Language

Lt = {∧, ∨, →, ·, ¬, t}

Laura Janina Schn¨ uriger Admissible Rules of (Fragments of) R-Mingle

The logic R-Mingle RMt Finding the bases Further Work References Notations Corresponding algebraic semantics Sugihara Monoids This talk

Definition

rules are denoted by Γ/ϕ for finite Γ ∪ {ϕ} ⊂ FmL

Laura Janina Schn¨ uriger Admissible Rules of (Fragments of) R-Mingle

The logic R-Mingle RMt Finding the bases Further Work References Notations Corresponding algebraic semantics Sugihara Monoids This talk

Definition

rules are denoted by Γ/ϕ for finite Γ ∪ {ϕ} ⊂ FmL Γ/ϕ is derivable in a logic L if Γ ⊢L ϕ

Laura Janina Schn¨ uriger Admissible Rules of (Fragments of) R-Mingle

The logic R-Mingle RMt Finding the bases Further Work References Notations Corresponding algebraic semantics Sugihara Monoids This talk

Definition

rules are denoted by Γ/ϕ for finite Γ ∪ {ϕ} ⊂ FmL Γ/ϕ is derivable in a logic L if Γ ⊢L ϕ Γ/ϕ is admissible in a logic L if for all substitutions (homomorphisms) σ: FmL → FmL: ⊢L σ(ψ) for all ψ ∈ Γ ⇒ ⊢L σ(ϕ)

Laura Janina Schn¨ uriger Admissible Rules of (Fragments of) R-Mingle

The logic R-Mingle RMt Finding the bases Further Work References Notations Corresponding algebraic semantics Sugihara Monoids This talk

Definition

rules are denoted by Γ/ϕ for finite Γ ∪ {ϕ} ⊂ FmL Γ/ϕ is derivable in a logic L if Γ ⊢L ϕ Γ/ϕ is admissible in a logic L if for all substitutions (homomorphisms) σ: FmL → FmL: ⊢L σ(ψ) for all ψ ∈ Γ ⇒ ⊢L σ(ϕ) {Γ/ϕ | Γ/ϕ is admissible in L} =: | ∼L

Laura Janina Schn¨ uriger Admissible Rules of (Fragments of) R-Mingle

The logic R-Mingle RMt Finding the bases Further Work References Notations Corresponding algebraic semantics Sugihara Monoids This talk

Definition

rules are denoted by Γ/ϕ for finite Γ ∪ {ϕ} ⊂ FmL Γ/ϕ is derivable in a logic L if Γ ⊢L ϕ Γ/ϕ is admissible in a logic L if for all substitutions (homomorphisms) σ: FmL → FmL: ⊢L σ(ψ) for all ψ ∈ Γ ⇒ ⊢L σ(ϕ) {Γ/ϕ | Γ/ϕ is admissible in L} =: | ∼L Let R be a set of rules. L + R = smallest logic containing L ∪ R

Laura Janina Schn¨ uriger Admissible Rules of (Fragments of) R-Mingle

The logic R-Mingle RMt Finding the bases Further Work References Notations Corresponding algebraic semantics Sugihara Monoids This talk

Definition

rules are denoted by Γ/ϕ for finite Γ ∪ {ϕ} ⊂ FmL Γ/ϕ is derivable in a logic L if Γ ⊢L ϕ Γ/ϕ is admissible in a logic L if for all substitutions (homomorphisms) σ: FmL → FmL: ⊢L σ(ψ) for all ψ ∈ Γ ⇒ ⊢L σ(ϕ) {Γ/ϕ | Γ/ϕ is admissible in L} =: | ∼L Let R be a set of rules. L + R = smallest logic containing L ∪ R R is a basis for the admissible rules of L if L + R = | ∼L

Laura Janina Schn¨ uriger Admissible Rules of (Fragments of) R-Mingle

The logic R-Mingle RMt Finding the bases Further Work References Notations Corresponding algebraic semantics Sugihara Monoids This talk

Corresponding algebraic semantics

Z◦ =

Z \ {0}, min, max, →, ·, −, 1

→

x → y :=

• max{−x, y}

if x ≤ y min{−x, y} if x > y

·

x · y :=      min{x, y} if |x| = |y| y if |x| < |y| x if |x| > |y|

Laura Janina Schn¨ uriger Admissible Rules of (Fragments of) R-Mingle

The logic R-Mingle RMt Finding the bases Further Work References Notations Corresponding algebraic semantics Sugihara Monoids This talk

Corresponding algebraic semantics

Z◦ =

Z \ {0}, min, max, →, ·, −, 1

→

x → y :=

• max{−x, y}

if x ≤ y min{−x, y} if x > y

·

x · y :=      min{x, y} if |x| = |y| y if |x| < |y| x if |x| > |y|

Z2n =

{−n, . . . , −1, 1, . . . , n}, min, max, →, ·, −, 1

Z2n+1 =

{−n, . . . , −1, 0, 1, . . . , n}, min, max, →, ·, −, 1

Laura Janina Schn¨ uriger Admissible Rules of (Fragments of) R-Mingle

The logic R-Mingle RMt Finding the bases Further Work References Notations Corresponding algebraic semantics Sugihara Monoids This talk

Sugihara Monoids

SM =

V(Z◦) the variety of Sugihara Monoids generated by Z◦. SM provides an equivalent algebraic semantics for RMt {ψ ≈ |ψ| | ψ ∈ Γ} SM ϕ ≈ |ϕ| ⇔: Γ SM ϕ ⇔ Γ ⊢RMt ϕ for any rule Γ/ϕ.

Laura Janina Schn¨ uriger Admissible Rules of (Fragments of) R-Mingle

The logic R-Mingle RMt Finding the bases Further Work References Notations Corresponding algebraic semantics Sugihara Monoids This talk

This talk

Bases for admissible rules of the fragments of RMt with the following languages

L1 =

{→, t}

L2 =

{→, ·, t}

Lm =

{→, ¬, t} = {→, ·, ¬, t} multiplicative fragment.

Laura Janina Schn¨ uriger Admissible Rules of (Fragments of) R-Mingle

The logic R-Mingle RMt Finding the bases Further Work References Notations Corresponding algebraic semantics Sugihara Monoids This talk

This talk

Bases for admissible rules of the fragments of RMt with the following languages

L1 =

{→, t}

L2 =

{→, ·, t}

Lm =

{→, ¬, t} = {→, ·, ¬, t} multiplicative fragment.

SM ↾ Li

algebraic semantics corresponding to the Li-fragment of RMt , i ∈ {1, 2, m}

Laura Janina Schn¨ uriger Admissible Rules of (Fragments of) R-Mingle

The logic R-Mingle RMt Finding the bases Further Work References Notations Corresponding algebraic semantics Sugihara Monoids This talk

This talk

Bases for admissible rules of the fragments of RMt with the following languages

L1 =

{→, t}

L2 =

{→, ·, t}

Lm =

{→, ¬, t} = {→, ·, ¬, t} multiplicative fragment.

SM ↾ Li

algebraic semantics corresponding to the Li-fragment of RMt , i ∈ {1, 2, m}

Remark Raftery, Olson

RMt ↾ {∧, →, t} has empty basis (= it is structurally complete).

Laura Janina Schn¨ uriger Admissible Rules of (Fragments of) R-Mingle

The logic R-Mingle RMt Finding the bases Further Work References Idea of how to find the bases Finding the bases The bases

Idea of how to find the bases

Recall

SM = V(SM) = V(Z◦)

Laura Janina Schn¨ uriger Admissible Rules of (Fragments of) R-Mingle

The logic R-Mingle RMt Finding the bases Further Work References Idea of how to find the bases Finding the bases The bases

Idea of how to find the bases

Recall

SM = V(SM) = V(Z◦)

Lemma S.

V(SM ↾ Li) = V(Z4 ↾ Li), i ∈ {1, 2, m}

Laura Janina Schn¨ uriger Admissible Rules of (Fragments of) R-Mingle

The logic R-Mingle RMt Finding the bases Further Work References Idea of how to find the bases Finding the bases The bases

Idea of how to find the bases

Recall that if for two varieties V1 and V2 we have: V1 = V2 iff (⊢V1 ϕ ⇔ ⊢V2 ϕ for all formulas ϕ).

Laura Janina Schn¨ uriger Admissible Rules of (Fragments of) R-Mingle

The logic R-Mingle RMt Finding the bases Further Work References Idea of how to find the bases Finding the bases The bases

Idea of how to find the bases

Recall that if for two varieties V1 and V2 we have: V1 = V2 iff (⊢V1 ϕ ⇔ ⊢V2 ϕ for all formulas ϕ). A rule is admissible in RMt ↾ Li ⇔ it is admissible in SM ↾ Li ⇔ it is admissible in Z4 ↾ Li

Laura Janina Schn¨ uriger Admissible Rules of (Fragments of) R-Mingle

The logic R-Mingle RMt Finding the bases Further Work References Idea of how to find the bases Finding the bases The bases

Idea of how to find the bases

Recall that if for two varieties V1 and V2 we have: V1 = V2 iff (⊢V1 ϕ ⇔ ⊢V2 ϕ for all formulas ϕ). A rule is admissible in RMt ↾ Li ⇔ it is admissible in SM ↾ Li ⇔ it is admissible in Z4 ↾ Li Interested in algebras s.t. admissibility in Z4 ↾ Li corresponds to validity in these algebras.

Laura Janina Schn¨ uriger Admissible Rules of (Fragments of) R-Mingle

The logic R-Mingle RMt Finding the bases Further Work References Idea of how to find the bases Finding the bases The bases

Idea of how to find the bases

Recall that if for two varieties V1 and V2 we have: V1 = V2 iff (⊢V1 ϕ ⇔ ⊢V2 ϕ for all formulas ϕ). A rule is admissible in RMt ↾ Li ⇔ it is admissible in SM ↾ Li ⇔ it is admissible in Z4 ↾ Li Interested in algebras s.t. admissibility in Z4 ↾ Li corresponds to validity in these algebras. Then: Axiomatize the quasivarieties generated by these algebras to get an axiomatization of the admissible rules

• f our fragments.

Laura Janina Schn¨ uriger Admissible Rules of (Fragments of) R-Mingle

The logic R-Mingle RMt Finding the bases Further Work References Idea of how to find the bases Finding the bases The bases

Finding the bases

Theorem

Let B be an algebra and FB(ω) its free algebra on countably infinite many generators. Then Γ/ϕ is B-admissible ⇔ Γ FB(ω) ϕ.

Laura Janina Schn¨ uriger Admissible Rules of (Fragments of) R-Mingle

The logic R-Mingle RMt Finding the bases Further Work References Idea of how to find the bases Finding the bases The bases

Finding the bases

Lemma

The following are equivalent: (i) Γ/ϕ is B-admissible ⇔ Γ A ϕ (ii) Q(A) = Q(FB(ω))

Laura Janina Schn¨ uriger Admissible Rules of (Fragments of) R-Mingle

The logic R-Mingle RMt Finding the bases Further Work References Idea of how to find the bases Finding the bases The bases

Finding the bases

Lemma

The following are equivalent: (i) Γ/ϕ is B-admissible ⇔ Γ A ϕ (ii) Q(A) = Q(FB(ω)) So we want to find A which is “easy” to axiomatize - but how?

Laura Janina Schn¨ uriger Admissible Rules of (Fragments of) R-Mingle

The logic R-Mingle RMt Finding the bases Further Work References Idea of how to find the bases Finding the bases The bases

Finding the bases

Lemma

The following are equivalent: (i) Γ/ϕ is B-admissible ⇔ Γ A ϕ (ii) Q(A) = Q(FB(ω)) So we want to find A which is “easy” to axiomatize - but how?

Lemma

A ⊆ FB(ω), B ∈ H(A) ⇒ Q(A) = Q(FB(ω))

Laura Janina Schn¨ uriger Admissible Rules of (Fragments of) R-Mingle

The logic R-Mingle RMt Finding the bases Further Work References Idea of how to find the bases Finding the bases The bases

The algebras in our case

Lemma S.

Let Z′

4 ⊂ (Z2 × Z3) ↾ L1,

Z′′

4 ⊂ (Z2 × Z3) ↾ L2,

(Z2 × Z3) ↾ Lm be the algebras pictured. Then (i) Q(FZ4↾L1(ω)) = Q(Z′

4)

(ii) Q(FZ4↾L2(ω)) = Q(Z′′

4)

(iii) Q(FZ4↾Lm(ω)) = Q((Z2 × Z3) ↾ Lm)

Figure: Z′

4 and Z′′ 4

Laura Janina Schn¨ uriger Admissible Rules of (Fragments of) R-Mingle

The logic R-Mingle RMt Finding the bases Further Work References Idea of how to find the bases Finding the bases The bases

Definition

|ψ| := ψ → ψ

Laura Janina Schn¨ uriger Admissible Rules of (Fragments of) R-Mingle

The logic R-Mingle RMt Finding the bases Further Work References Idea of how to find the bases Finding the bases The bases

Definition

|ψ| := ψ → ψ ϕ ⇒ ψ := (ϕ → |ψ|) → (ϕ → ψ)

Laura Janina Schn¨ uriger Admissible Rules of (Fragments of) R-Mingle

The logic R-Mingle RMt Finding the bases Further Work References Idea of how to find the bases Finding the bases The bases

Definition

|ψ| := ψ → ψ ϕ ⇒ ψ := (ϕ → |ψ|) → (ϕ → ψ) {p, p ⇒ q}/q (A)

Laura Janina Schn¨ uriger Admissible Rules of (Fragments of) R-Mingle

The logic R-Mingle RMt Finding the bases Further Work References Idea of how to find the bases Finding the bases The bases

Definition

|ψ| := ψ → ψ ϕ ⇒ ψ := (ϕ → |ψ|) → (ϕ → ψ) {p, p ⇒ q}/q (A) ϕ ↔ ψ := (ϕ → ψ) · (ψ → ϕ)

Laura Janina Schn¨ uriger Admissible Rules of (Fragments of) R-Mingle

The logic R-Mingle RMt Finding the bases Further Work References Idea of how to find the bases Finding the bases The bases

Definition

|ψ| := ψ → ψ ϕ ⇒ ψ := (ϕ → |ψ|) → (ϕ → ψ) {p, p ⇒ q}/q (A) ϕ ↔ ψ := (ϕ → ψ) · (ψ → ϕ) {¬(|p1| ↔ . . . ↔ |pn|)}/q (Rn), n ∈ N.

Laura Janina Schn¨ uriger Admissible Rules of (Fragments of) R-Mingle

The logic R-Mingle RMt Finding the bases Further Work References Idea of how to find the bases Finding the bases The bases

The Bases

Lemma S.

We have the following axiomatizations: (i) RMt ↾ L1 + (A) has equivalent q.v. Q(Z′

4)

(ii) RMt ↾ L2 + (A) has equivalent q.v. Q(Z′′

4)

(iii) RMt ↾ Lm + (A) + {(Rn)}n∈N has eq. q.v. Q((Z2 × Z3) ↾ Lm)

Laura Janina Schn¨ uriger Admissible Rules of (Fragments of) R-Mingle

The logic R-Mingle RMt Finding the bases Further Work References Idea of how to find the bases Finding the bases The bases

The Bases

Lemma S.

We have the following axiomatizations: (i) RMt ↾ L1 + (A) has equivalent q.v. Q(Z′

4)

(ii) RMt ↾ L2 + (A) has equivalent q.v. Q(Z′′

4)

(iii) RMt ↾ Lm + (A) + {(Rn)}n∈N has eq. q.v. Q((Z2 × Z3) ↾ Lm)

Theorem S.

Then as a Corollary of this lemma (i) {(A)} is a basis for the {→, t}- and {→, ·, t}-fragment of RMt. (ii) {(A)} ∪ {(Rn)}n∈N is a basis for RMt ↾ {→, ¬, t}.

Laura Janina Schn¨ uriger Admissible Rules of (Fragments of) R-Mingle

The logic R-Mingle RMt Finding the bases Further Work References Our Conjecture

Our Conjecture

Look again at RM without constant t.

Laura Janina Schn¨ uriger Admissible Rules of (Fragments of) R-Mingle

The logic R-Mingle RMt Finding the bases Further Work References Our Conjecture

Our Conjecture

Look again at RM without constant t.

(B)

{¬|p| ∨ q}/q

Laura Janina Schn¨ uriger Admissible Rules of (Fragments of) R-Mingle

The logic R-Mingle RMt Finding the bases Further Work References Our Conjecture

Our Conjecture

Look again at RM without constant t.

(B)

{¬|p| ∨ q}/q

Conjecture

We hope to prove the following: (i) RM + (B) is almost structurally complete, i.e., Γ| ∼RM ϕ ⇒ Γ ⊢RM ϕ whenever there is a substitution σ: FmL → FmL s.t. for all ψ ∈ Γ, ⊢RM σ(ψ). (ii) The admissible rules of RM have basis {(B)} ∪ {(Rn)}n∈N.

Laura Janina Schn¨ uriger Admissible Rules of (Fragments of) R-Mingle

The logic R-Mingle RMt Finding the bases Further Work References

References

• G. Metcalfe and C. R¨
• thlisberger. Admissibility in

finitely generated quasivarieties. Logical Methods in Computer Science, vol. 9 (2013), no. 2, pp. 119.

• C. R¨
• thlisberger. Admissibility in finitely generated
• quasivarieties. PhD thesis, 2013.
• G. Metcalfe. An Avron rule for fragments of R-mingle.

Journal of Logic and Computation, to appear.

• J. Olson and J. Raftery. Positive Sugihara monoids.

Algebra Universalis, vol. 57 (2005), pp. 7599.

Laura Janina Schn¨ uriger Admissible Rules of (Fragments of) R-Mingle