calculi for Lewis conditional logic V Marianna Girlando, Sara Negri, - PowerPoint PPT Presentation

Equivalence between internal and labelled sequent calculi for Lewis’ conditional logic V Marianna Girlando, Sara Negri, Nicola Olivetti Aix Marseille Universit´ e, Laboratoire des Sciences de l’Information et des Syst` emes; University of Helsinki, Department of Philosophy; Aix Marseille Universit´ e, Laboratoire des Sciences de l’Information et des Syst` emes TICAMORE Meeting 15 - 18 November, Marseille 1 / 46

Equivalence results Why? ◦ Investigate the interrelations between different proof systems; ◦ Deeper understanding of the systems; ◦ Transfer proof-theoretic and model theoretic results between the calculi. Partial references ◦ Fitting (2011); ◦ Poggiolesi (2011); ◦ Gor´ e and Ramanayake (2012). 2 / 46

Our case study Conditional logic V I i G3V V Labelled sequent calculs Internal sequent calculus References ◦ Negri and Olivetti (2015); ◦ Olivetti and Pozzato (2015); Girlando, Lellmann, Olivetti and Pozzato (2016). 3 / 46

Outline (1) Backgrounds and proof systems for V (2) Translation: from I i V to G3V (3) Inverse translation: from G3V to I i V (4) Conclusions 4 / 46

Backgrounds 5 / 46

Logic V Lewis (1973) Conditional operators ◦ Counterfactual conditional operator: A > B “If A were the case, then B would have been the case” ◦ Comparative plausibility operator: A � B “ A is at least as plausible as B ” ◦ The two operators are interdefinable: A > B ≡ ( ⊥ � A ) ∨ ¬ (( A ∧ ¬ B ) � ( A ∧ B )) A � B ≡ (( A ∨ B ) > ⊥ ) ∨ ¬ (( A ∨ B ) > ¬ A ) 6 / 46

Logic V Lewis (1973) Language A , B :: = P | ⊥ | A → B | A � B Axioms and inference rules Axioms and inference rules of classical propositional logic; (CPR) if ⊢ A � B then ⊢ B → A (CPA) ( A � ( A ∨ B )) ∨ ( B � ( A ∨ B )) (TR) ( A � B ) ∧ ( B � C ) → ( A � C ) (CO) ( A � B ) ∨ ( B � A ) 7 / 46

Neighbourhood models for V A neighbourhood model M = � W , I , � � � consists of: ◦ W , non-empty set of elements; ◦ I : W → P ( P ( W )) , function assigning to each x a set I ( x ) . Let α, β, ... be elements of I ( x ) ; ◦ � � : Atm → P ( W ) , propositional evaluation. A model for V satisfies: ◦ (Non-emptiness) for each α ∈ I ( x ) , α � ∅ ; ◦ (Nesting) for each α, β ∈ I ( x ) , α ⊆ β or β ⊆ α . 8 / 46

Comparative plausibility ◦ α � ∃ A i ff ∃ y ∈ α ( y � A ) ◦ x � A � B i ff ∀ α ∈ I ( x )( α � ∃ B implies α � ∃ A ) A α β B I ( x ) 9 / 46

Proof systems 10 / 46

Labelled sequent calculus Negri and Olivetti (2015) Two kinds of labels ◦ labels for worlds: x , y , z . . . ◦ labels for neighbourhoods: a , b , c . . . Expressions employed in the calculus ◦ a ∈ I ( x ) ◦ x ∈ a ◦ a ⊆ b ◦ x : A ◦ a � ∃ A ≡ ∃ x ( x ∈ a and x � A ) ◦ x : A � B ≡ ∀ a ∈ I ( x )( a � ∃ B implies a � ∃ A ) 11 / 46

Labelled sequent calculus Rules of G3V (1) Initial sequents x : p , Γ ⇒ ∆ , x : p x : ⊥ , Γ ⇒ ∆ Rules for local forcing x ∈ a , Γ ⇒ ∆ , x : A , a � ∃ A x ∈ a , x : A , Γ ⇒ ∆ L � ∃ ( x fresh ) R � ∃ a � ∃ A , Γ ⇒ ∆ x ∈ a , Γ ⇒ ∆ , a � ∃ A Propositional rules Γ ⇒ ∆ , x : A x : A , Γ ⇒ ∆ L ¬ R ¬ x : ¬ A , Γ ⇒ ∆ Γ ⇒ ∆ , x : ¬ A Γ ⇒ ∆ , x : A x : B , Γ ⇒ ∆ x : A , Γ ⇒ ∆ , x : B L → R → x : A → B , Γ ⇒ ∆ Γ ⇒ ∆ , x : A → B 12 / 46

Labelled sequent calculus Rules of G3V (2) Rules for comparative plausibility a � ∃ B , a ∈ I ( x ) , Γ ⇒ ∆ , a � ∃ A R � ( a new ) Γ ⇒ ∆ , x : A � B a ∈ I ( x ) , x : A � B , Γ ⇒ ∆ , a � ∃ B a � ∃ A , a ∈ I ( x ) , x : A � B , Γ ⇒ ∆ L � a ∈ I ( x ) , x : A � B , Γ ⇒ ∆ Rules for inclusion a ⊆ a , Γ ⇒ ∆ c ⊆ a , c ⊆ b , b ⊆ a , Γ ⇒ ∆ x ∈ a , a ⊆ b , x ∈ b , Γ ⇒ ∆ Ref Tr L ⊆ Γ ⇒ ∆ c ⊆ b , b ⊆ a , Γ ⇒ ∆ x ∈ a , a ⊆ b , Γ ⇒ ∆ Rule for nesting a ⊆ b , a ∈ I ( x ) , b ∈ I ( x ) , Γ ⇒ ∆ b ⊆ a , a ∈ I ( x ) , b ∈ I ( x ) , Γ ⇒ ∆ Nes a ∈ I ( x ) , b ∈ I ( x ) , Γ ⇒ ∆ 13 / 46

Example Derivation of A � B ∨ B � A ( AX ) a ⊆ b , a ∈ I ( x ) , b ∈ I ( x ) , y ∈ a , y ∈ b , y : B , b � ∃ A ⇒ a � ∃ A , b � ∃ B , y : B R � ∃ a ⊆ b , a ∈ I ( x ) , b ∈ I ( x ) , y ∈ a , y ∈ b , y : B , b � ∃ A ⇒ a � ∃ A , b � ∃ B L ⊆ a ⊆ b , a ∈ I ( x ) , b ∈ I ( x ) , y ∈ a , y : B , b � ∃ A ⇒ a � ∃ A , b � ∃ B L � ∃ a ⊆ b , a ∈ I ( x ) , b ∈ I ( x ) , a � ∃ B , b � ∃ A ⇒ a � ∃ A , b � ∃ B ( ∗ ) Nes a ∈ I ( x ) , b ∈ I ( x ) , a � ∃ B , b � ∃ A ⇒ a � ∃ A , b � ∃ B R � a ∈ I ( x ) , a � ∃ B ⇒ x : B � A , a � ∃ A R � ⇒ x : A � B , x : B � A The right premiss of Nes is derivable in a similar way. ( ∗ ) b ⊆ a , a ∈ I ( x ) , b ∈ I ( x ) , a � ∃ B , b � ∃ A ⇒ a � ∃ A , b � ∃ B 14 / 46

Properties of G3V Basic structural properties ◦ Weakening and contraction are height-preserving admissible; ◦ All the rules are height-preserving invertible; ◦ The cut rule is admissible (syntactic cut elimination). Soundness The rules of G3V are sound with respect to neighbourhood models for V . Completeness We obtain completeness by simulating within G3V the internal sequent calculus. 15 / 46

The calculus I i Olivetti and Pozzato (2015) V Blocks ◦ A block is a pair consisting of a multiset Σ of formulas and a single formula B , written [ Σ ⊳ B ] . ◦ Blocks denote disjunctions of � -formulas: [ A 1 , . . . , A m ⊳ B ] ( A 1 � B ) ∨ ( A 2 � B ) ∨ · · · ∨ ( A m � B ) Sequents ◦ Blocks can occur only in the succedent of a sequent. ◦ The formula interpretation of a sequent is given by: ∆ ′ ∨ � � � � Γ ⇒ ∆ ′ , [ Σ 1 ⊳ B 1 ] , . . . , [ Σ n ⊳ B n ] : = Γ → ( A � B i ) 1 ≤ i ≤ n A ∈ Σ i 16 / 46

The calculus I i V Rules of I i V Initial sequents Γ , ⊥ ⇒ ∆ Γ , p ⇒ ∆ , p Propositional rules (standard) Rules for comparative plausibility Γ ⇒ ∆ , [ A ⊳ B ] � i R Γ ⇒ ∆ , A � B Γ , A � B ⇒ ∆ , [ B , Σ ⊳ C ] Γ , A � B ⇒ ∆ , [ Σ ⊳ A ] , [ Σ ⊳ C ] � i L Γ , A � B ⇒ ∆ , [ Σ ⊳ C ] Rules for the blocks Γ ⇒ ∆ , [ Σ 1 , Σ 2 ⊳ A ] , [ Σ 2 ⊳ B ] Γ ⇒ ∆ , [ Σ 1 ⊳ A ] , [ Σ 1 , Σ 2 ⊳ B ] com i Γ ⇒ ∆ , [ Σ 1 ⊳ A ] , [ Σ 2 ⊳ B ] A ⇒ Σ jump Γ ⇒ ∆ , [ Σ ⊳ A ] 17 / 46

Example Derivation of ( A � B ) ∨ ( B � A ) ( AX ) ( AX ) B ⇒ A , B A ⇒ A , B ⇒ [ A , B ⊳ B ] , [ B ⊳ A ] jump ⇒ [ A ⊳ B ] , [ A , B ⊳ A ] jump com i ⇒ [ A ⊳ B ] , [ B ⊳ A ] ⇒ [ A ⊳ B ] , B � A � i R � i ⇒ A � B , B � A R ⇒ ( A � B ) ∨ ( B � A ) ∨ R 18 / 46

Properties of I i Girlando, Lellmann, Olivetti, Pozzato (2016) V Cut elimination Proved for an equivalent version of the calculus, non-invertible and with contraction rules explicitly defined. Soundness If a formula is derivable in the calculus I i V , then it is valid in V . Completeness If formula A is valid in V , then sequent ⇒ A is derivable in I i V . 19 / 46

Translation: from I i V to G3V 20 / 46

Translation Idea ◦ x � A � B ≡ ∀ α ∈ I ( x )( α � ∃ B implies α � ∃ A ) ◦ [ A 1 , . . . , A n ⊳ B ] = ( A 1 � B ) ∨ ( A 2 � B ) ∨ · · · ∨ ( A n � B ) ◦ Each block is interpreted in the language of the labelled sequent calculus as expressing the semantic condition corresponding to the disjunction of � -formulas: x � [ A 1 , . . . , A n ⊳ B ] i ff ∀ α ∈ I ( x )( α � ∃ B implies α � ∃ ( A 1 ∨ · · · ∨ A n )) ◦ We introduce a new neighbourhood label a ∈ I ( x ) for each block. 21 / 46

Definition From I i V to G3V Given a world label x , countably many neighbourhood labels a , b , c ... and a multiset Σ = F 1 , . . . , F k , define: ◦ Σ t x = x : F 1 , . . . , x : F k ◦ ( Γ ⇒ ∆ , [ Σ 1 ⊳ B 1 ] , . . . , [ Σ n ⊳ B n ]) t x = a 1 , . . . , a n ∈ I ( x ) , a 1 � ∃ B 1 , . . . , a n � ∃ B n , Γ t x ⇒ ∆ t x , a 1 � ∃ Σ 1 , . . . , a n � ∃ Σ n where for Σ i = S 1 i , . . . , S k i a i � ∃ Σ i = a i � ∃ S 1 i , . . . , a i � ∃ S k i 22 / 46

Translation Theorem V , its translation ( Γ ⇒ ∆ ) t x is derivable If a sequent Γ ⇒ ∆ is derivable in I i in G3V . Proof Induction on the height of the derivation of a sequent Γ ⇒ ∆ , and distinction of cases. 23 / 46

Translation Lemma Rule Mon ∃ is admissible in G3V : b ⊆ a , Γ ⇒ ∆ , a � ∃ A , b � ∃ A Mon ∃ b ⊆ a , Γ ⇒ ∆ , a � ∃ A ◦ idea: if α � ∃ A is false , for all neighbourhood β ⊆ α formula β � ∃ A is false; ◦ directly implements rule com i in G3V . 24 / 46