Admissible Rules of (Fragments of) R-Mingle. Admissible Rules of (Fragments of) R-Mingle. Laura Janina Schnüriger

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1 Admissible Rules of (Fragments of) R-Mingle Admissible Rules of (Fragments of) R-Mingle joint work with George Metcalfe Universität Bern Novi Sad 5 June 2015

2 Table of contents 1. What and why? 1.1 What and why? Notations 2.2 Corresponding algebraic semantics 2.3 Sugihara Monoids 2.4 This talk The bases 4.

3 What and why? What and why? Motivation admissible Shorten and fasten proofs by adding short-cuts. But we don t want to have new theorems. Such a rule is called admissible.

4 Notations Corresponding algebraic semantics Sugihara Monoids 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}

5 Notations Corresponding algebraic semantics Sugihara Monoids 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

6 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 min{x, y} if x = y x y := y if x < y x if x > y Z 2n = { n,..., 1, 1,..., n}, min, max,,,, 1 Z 2n+1 = { n,..., 1, 0, 1,..., n}, min, max,,,, 1

7 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 RM t {ψ ψ ψ Γ} SM ϕ ϕ : Γ SM ϕ Γ RM t ϕ for any rule Γ/ϕ.

8 Notations Corresponding algebraic semantics Sugihara Monoids This talk This talk L 1 = {, t} Bases for admissible rules of the fragments of RM t with the following languages L 2 = {,, t} L m = SM L i Remark Raftery, Olson {,, t} = {,,, t} multiplicative fragment. algebraic semantics corresponding to the L i -fragment of RM t, i {1, 2, m} RM t {,, t} has empty basis (= it is structurally complete).

9 The bases Recall SM = V(SM) = V(Z ) Lemma S. V(SM L i ) = V(Z 4 L i ), i {1, 2, m}

10 The bases Recall that for two varieties V 1 and V 2 we have: V 1 = V 2 iff ( V1 ϕ V2 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. Then: Axiomatize the quasivarieties generated by these algebras to get an axiomatization of the admissible rules of our fragments.

11 The bases Theorem Let B be an algebra and F B (ω) the free algebra of V(B) on countably infinite many generators. Then Γ/ϕ is B-admissible Γ FB (ω) ϕ.

12 The bases Lemma The following are equivalent: (i) Γ/ϕ is B-admissible Γ A ϕ (ii) Q(A) = Q(F B (ω)) So we want to find A which is easy to axiomatize - but how? Lemma A F B (ω), B H(A) Q(A) = Q(F B (ω))

13 The bases The algebras in our case Lemma S. Let Z 4 (Z 2 Z 3 ) L 1, Z 4 (Z 2 Z 3 ) L 2, (Z 2 Z 3 ) L m be the algebras pictured. Then (i) Q(F Z4 L 1 (ω)) = Q(Z 4) (ii) Q(F Z4 L 2 (ω)) = Q(Z 4) (iii) Q(F Z4 L m (ω)) = Q((Z 2 Z 3 ) L m ) Figure: Z 4 and Z 4

14 The bases Definition ψ := ψ ψ ϕ ψ := (ϕ ψ ) (ϕ ψ) {p, p q}/q ϕ ψ := (ϕ ψ) (ψ ϕ) (A) { ( p 1... p n )}/q (R n ), n N.

15 The bases Lemma S. The Bases We have the following axiomatizations: (i) RM t L 1 + (A) has equivalent q.v. Q(Z 4) (ii) RM t L 2 + (A) has equivalent q.v. Q(Z 4) (iii) RM t L m + (A) + {(R n )} n N has eq. q.v. Q((Z 2 Z 3 ) L m ) Theorem S. Then as a Corollary of this lemma (i) {(A)} is a basis for the {, t}- and {,, t}-fragment of RM t. (ii) {(A)} {(R n )} n N is a basis for RM t {,, t}

16 G. Metcalfe and C. Röthlisberger. Admissibility in finitely generated quasivarieties. Logical Methods in Computer Science, vol. 9 (2013), no. 2, pp C. Röthlisberger. Admissibility in finitely generated quasivarieties. PhD thesis, 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