Admissible Rules of (Fragments of) R-Mingle Admissible Rules of - PowerPoint PPT Presentation

The logic R-Mingle RM t 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 RM t Finding the bases Further Work References Table of contents 1. The logic R-Mingle RM t 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 RM t Notations Finding the bases Corresponding algebraic semantics Further Work Sugihara Monoids References 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 RM t Notations Finding the bases Corresponding algebraic semantics Further Work Sugihara Monoids References This talk R-Mingle RM Relevance logic R with Mingle Mingle p → ( p → p ) RM t RM with additional constant t Language L t = {∧ , ∨ , → , · , ¬ , t } Laura Janina Schn¨ uriger Admissible Rules of (Fragments of) R-Mingle

The logic R-Mingle RM t Notations Finding the bases Corresponding algebraic semantics Further Work Sugihara Monoids References This talk Definition rules are denoted by Γ /ϕ for finite Γ ∪ { ϕ } ⊂ Fm L Laura Janina Schn¨ uriger Admissible Rules of (Fragments of) R-Mingle

The logic R-Mingle RM t Notations Finding the bases Corresponding algebraic semantics Further Work Sugihara Monoids References This talk Definition rules are denoted by Γ /ϕ for finite Γ ∪ { ϕ } ⊂ Fm L 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 RM t Notations Finding the bases Corresponding algebraic semantics Further Work Sugihara Monoids References This talk Definition rules are denoted by Γ /ϕ for finite Γ ∪ { ϕ } ⊂ Fm L 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 : Fm L → Fm L : ⊢ 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 RM t Notations Finding the bases Corresponding algebraic semantics Further Work Sugihara Monoids References This talk Definition rules are denoted by Γ /ϕ for finite Γ ∪ { ϕ } ⊂ Fm L 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 : Fm L → Fm L : ⊢ 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 RM t Notations Finding the bases Corresponding algebraic semantics Further Work Sugihara Monoids References This talk Definition rules are denoted by Γ /ϕ for finite Γ ∪ { ϕ } ⊂ Fm L 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 : Fm L → Fm L : ⊢ 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 RM t Notations Finding the bases Corresponding algebraic semantics Further Work Sugihara Monoids References This talk Definition rules are denoted by Γ /ϕ for finite Γ ∪ { ϕ } ⊂ Fm L 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 : Fm L → Fm L : ⊢ 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 RM t Notations Finding the bases Corresponding algebraic semantics Further Work Sugihara Monoids References This talk Corresponding algebraic semantics Z ◦ = � Z \ { 0 } , min , max , → , · , − , 1 � � max {− x , y } if x ≤ y → x → y := min {− x , y } if x > y  min { x , y } if | x | = | y |   · 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 RM t Notations Finding the bases Corresponding algebraic semantics Further Work Sugihara Monoids References This talk Corresponding algebraic semantics Z ◦ = � Z \ { 0 } , min , max , → , · , − , 1 � � max {− x , y } if x ≤ y → x → y := min {− x , y } if x > y  min { x , y } if | x | = | y |   · x · y := y if | x | < | y |  x if | x | > | y |  Z 2 n = �{− n , . . . , − 1 , 1 , . . . , n } , min , max , → , · , − , 1 � Z 2 n +1 = �{− n , . . . , − 1 , 0 , 1 , . . . , n } , min , max , → , · , − , 1 � Laura Janina Schn¨ uriger Admissible Rules of (Fragments of) R-Mingle

The logic R-Mingle RM t Notations Finding the bases Corresponding algebraic semantics Further Work Sugihara Monoids References This talk Sugihara Monoids V ( Z ◦ ) the variety of Sugihara Monoids generated by Z ◦ . SM = SM provides an equivalent algebraic semantics for RM t { ψ ≈ | ψ | | ψ ∈ Γ } � SM ϕ ≈ | ϕ | ⇔ : Γ � SM ϕ ⇔ Γ ⊢ RM t ϕ for any rule Γ /ϕ . Laura Janina Schn¨ uriger Admissible Rules of (Fragments of) R-Mingle

The logic R-Mingle RM t Notations Finding the bases Corresponding algebraic semantics Further Work Sugihara Monoids References This talk This talk Bases for admissible rules of the fragments of RM t with the following languages L 1 = {→ , t } L 2 = {→ , · , t } L m = {→ , ¬ , t } = {→ , · , ¬ , t } multiplicative fragment. Laura Janina Schn¨ uriger Admissible Rules of (Fragments of) R-Mingle

The logic R-Mingle RM t Notations Finding the bases Corresponding algebraic semantics Further Work Sugihara Monoids References This talk This talk Bases for admissible rules of the fragments of RM t with the following languages L 1 = {→ , t } L 2 = {→ , · , t } L m = {→ , ¬ , t } = {→ , · , ¬ , t } multiplicative fragment. SM ↾ L i algebraic semantics corresponding to the L i -fragment of RM t , i ∈ { 1 , 2 , m } Laura Janina Schn¨ uriger Admissible Rules of (Fragments of) R-Mingle

The logic R-Mingle RM t Notations Finding the bases Corresponding algebraic semantics Further Work Sugihara Monoids References This talk This talk Bases for admissible rules of the fragments of RM t with the following languages L 1 = {→ , t } L 2 = {→ , · , t } L m = {→ , ¬ , t } = {→ , · , ¬ , t } multiplicative fragment. SM ↾ L i algebraic semantics corresponding to the L i -fragment of RM t , i ∈ { 1 , 2 , m } RM t ↾ {∧ , → , t } has empty basis (= it is structurally Remark Raftery, Olson complete). Laura Janina Schn¨ uriger Admissible Rules of (Fragments of) R-Mingle

The logic R-Mingle RM t Idea of how to find the bases Finding the bases Finding the bases Further Work The bases References Idea of how to find the bases SM = V ( SM ) = V ( Z ◦ ) Recall Laura Janina Schn¨ uriger Admissible Rules of (Fragments of) R-Mingle

The logic R-Mingle RM t Idea of how to find the bases Finding the bases Finding the bases Further Work The bases References Idea of how to find the bases SM = V ( SM ) = V ( Z ◦ ) Recall Lemma S. V ( SM ↾ L i ) = V ( Z 4 ↾ L i ) , i ∈ { 1 , 2 , m } Laura Janina Schn¨ uriger Admissible Rules of (Fragments of) R-Mingle

The logic R-Mingle RM t Idea of how to find the bases Finding the bases Finding the bases Further Work The bases References Idea of how to find the bases Recall that if for two varieties V 1 and V 2 we have: V 1 = V 2 iff ( ⊢ V 1 ϕ ⇔ ⊢ V 2 ϕ for all formulas ϕ ). Laura Janina Schn¨ uriger Admissible Rules of (Fragments of) R-Mingle

The logic R-Mingle RM t Idea of how to find the bases Finding the bases Finding the bases Further Work The bases References Idea of how to find the bases Recall that if for two varieties V 1 and V 2 we have: V 1 = V 2 iff ( ⊢ V 1 ϕ ⇔ ⊢ V 2 ϕ for all formulas ϕ ). A rule is admissible in RM t ↾ L i ⇔ it is admissible in SM ↾ L i ⇔ it is admissible in Z 4 ↾ L i Laura Janina Schn¨ uriger Admissible Rules of (Fragments of) R-Mingle

The logic R-Mingle RM t Idea of how to find the bases Finding the bases Finding the bases Further Work The bases References Idea of how to find the bases Recall that if for two varieties V 1 and V 2 we have: V 1 = V 2 iff ( ⊢ V 1 ϕ ⇔ ⊢ V 2 ϕ for all formulas ϕ ). A rule is admissible in RM t ↾ L i ⇔ it is admissible in SM ↾ L i ⇔ it is admissible in Z 4 ↾ L i Interested in algebras s.t. admissibility in Z 4 ↾ L i corresponds to validity in these algebras. Laura Janina Schn¨ uriger Admissible Rules of (Fragments of) R-Mingle