Gmin a Sequent Calculus for Minimal Logic 3 Dmin Minimal Dialogue Logic The Correspondence between Gmin a and Dmin 3 References Dialogue Games for Minimal Logic Alexandra Pavlova National Research University Higher School of Economics pavlova.alex22@gmail.com PhDs in Logic X Prague, May the 1st, 2018 Alexandra Pavlova Dialogue Games for Minimal Logic
Gmin a Sequent Calculus for Minimal Logic 3 General Idea Dmin Minimal Dialogue Logic Gmin a System The Correspondence between Gmin a and Dmin 3 3 Comments on the Rules References General Idea Minimal Propositional Logic We reject: Alexandra Pavlova Dialogue Games for Minimal Logic
Gmin a Sequent Calculus for Minimal Logic 3 General Idea Dmin Minimal Dialogue Logic Gmin a System The Correspondence between Gmin a and Dmin 3 3 Comments on the Rules References General Idea Minimal Propositional Logic We reject: • the law of excluded middle A ∨ ¬ A Alexandra Pavlova Dialogue Games for Minimal Logic
Gmin a Sequent Calculus for Minimal Logic 3 General Idea Dmin Minimal Dialogue Logic Gmin a System The Correspondence between Gmin a and Dmin 3 3 Comments on the Rules References General Idea Minimal Propositional Logic We reject: • the law of excluded middle A ∨ ¬ A • the principle of explosion A , ¬ A ⊢ B Alexandra Pavlova Dialogue Games for Minimal Logic
Gmin a Sequent Calculus for Minimal Logic 3 General Idea Dmin Minimal Dialogue Logic Gmin a System The Correspondence between Gmin a and Dmin 3 3 Comments on the Rules References General Idea Minimal Propositional Logic We reject: • the law of excluded middle A ∨ ¬ A • the principle of explosion A , ¬ A ⊢ B So we get the following rule transformation: Alexandra Pavlova Dialogue Games for Minimal Logic
Gmin a Sequent Calculus for Minimal Logic 3 General Idea Dmin Minimal Dialogue Logic Gmin a System The Correspondence between Gmin a and Dmin 3 3 Comments on the Rules References General Idea Minimal Propositional Logic We reject: • the law of excluded middle A ∨ ¬ A • the principle of explosion A , ¬ A ⊢ B So we get the following rule transformation: ⊥ − → A ∧ ¬ A D = ⇒ − → D ∗ Alexandra Pavlova Dialogue Games for Minimal Logic
Gmin a Sequent Calculus for Minimal Logic 3 General Idea Dmin Minimal Dialogue Logic Gmin a System The Correspondence between Gmin a and Dmin 3 3 Comments on the Rules References General Idea Minimal Propositional Logic We reject: • the law of excluded middle A ∨ ¬ A • the principle of explosion A , ¬ A ⊢ B So we get the following rule transformation: ⊥ − → A ∧ ¬ A D = ⇒ − → D ∗ According to the rules of LJ this can be transformed as follows: Alexandra Pavlova Dialogue Games for Minimal Logic
Gmin a Sequent Calculus for Minimal Logic 3 General Idea Dmin Minimal Dialogue Logic Gmin a System The Correspondence between Gmin a and Dmin 3 3 Comments on the Rules References General Idea Minimal Propositional Logic We reject: • the law of excluded middle A ∨ ¬ A • the principle of explosion A , ¬ A ⊢ B So we get the following rule transformation: ⊥ − → A ∧ ¬ A D = ⇒ − → D ∗ According to the rules of LJ this can be transformed as follows: A − → A ( UEA ) A ∧ ¬ A − → A ( NEA ) ¬ A , A ∧ ¬ A − → ( UEA ) A ∧ ¬ A , A ∧ ¬ A − → ( CL ) Γ − → A ∧ ¬ A A ∧ ¬ A − → ( cut ) Γ − → ( WR ) Γ − → D ∗ Alexandra Pavlova Dialogue Games for Minimal Logic
Gmin a Sequent Calculus for Minimal Logic 3 General Idea Dmin Minimal Dialogue Logic Gmin a System The Correspondence between Gmin a and Dmin 3 3 Comments on the Rules References G min a System 3 Definition The language L min is defined in BNF-style as follows a : p | A | ¬ A | A ∧ B | A ∨ B | A ⊃ B | ⊥ We also make use of the meta-language sign of ”syntactic consequence” (or the ”sequent sign”) − → . a We use Gothic letters to indicate meta-language variables. Alexandra Pavlova Dialogue Games for Minimal Logic
Gmin a Sequent Calculus for Minimal Logic 3 General Idea Dmin Minimal Dialogue Logic Gmin a System The Correspondence between Gmin a and Dmin 3 3 Comments on the Rules References G min a System 3 Definition The language L min is defined in BNF-style as follows a : p | A | ¬ A | A ∧ B | A ∨ B | A ⊃ B | ⊥ We also make use of the meta-language sign of ”syntactic consequence” (or the ”sequent sign”) − → . a We use Gothic letters to indicate meta-language variables. Definition a a is The axiom of the system G min 3 ( Ax . Int ) A , Γ − → A a This is a restricted version of a more general axiom A , Γ − → Θ , A where A is atomic and Θ = ∅ . This restriction is used both for minimal and intuitionistic calculus, but not for the classical one. Alexandra Pavlova Dialogue Games for Minimal Logic
Gmin a Sequent Calculus for Minimal Logic 3 General Idea Dmin Minimal Dialogue Logic Gmin a System The Correspondence between Gmin a and Dmin 3 3 Comments on the Rules References G min a System 3 Definition Given that Γ and Θ are multisets of formulae, the system G min a has the 3 following rules of inference A , Γ − → B A ⊃ B , Γ − → A and B , A ⊃ B , Γ − → Θ → A ⊃ B ⊃ S + ⊃ A + Γ − A ⊃ B , Γ − → Θ ∧ S + A , A ∧ B , Γ − → Θ B , A ∧ B , Γ − → Θ or Γ − → A and Γ − → B ∧ A + Γ − → A ∧ B A ∧ B , Γ − → Θ ∨ S + A , A ∨ B , Γ − → Θ and B , A ∨ B , Γ − → Θ Γ − → A or Γ − → B ∨ A + Γ − → A ∨ B A ∨ B , Γ − → Θ A , Γ − → ⊥ ¬ A , Γ − → A ¬ S + → ⊥ ¬ A + Γ − → ¬ A ¬ A , Γ − where, for all rules, the succedent should contain exactly one formula. Alexandra Pavlova Dialogue Games for Minimal Logic
Gmin a Sequent Calculus for Minimal Logic 3 General Idea Dmin Minimal Dialogue Logic Gmin a System The Correspondence between Gmin a and Dmin 3 3 Comments on the Rules References Why do we need ⊥ ? Alexandra Pavlova Dialogue Games for Minimal Logic
Gmin a Sequent Calculus for Minimal Logic 3 General Idea Dmin Minimal Dialogue Logic Gmin a System The Correspondence between Gmin a and Dmin 3 3 Comments on the Rules References Why do we need ⊥ ? We introduce the ⊥ sign because of the constraints on the succedent. Alexandra Pavlova Dialogue Games for Minimal Logic
Gmin a Sequent Calculus for Minimal Logic 3 General Idea Dmin Minimal Dialogue Logic Gmin a System The Correspondence between Gmin a and Dmin 3 3 Comments on the Rules References Why do we need ⊥ ? We introduce the ⊥ sign because of the constraints on the succedent. But why can’t we have intuitionistic constraints on the succedent, i.e. that Θ should not contain more than one element, but it can possibly be empty? Alexandra Pavlova Dialogue Games for Minimal Logic
Gmin a Sequent Calculus for Minimal Logic 3 General Idea Dmin Minimal Dialogue Logic Gmin a System The Correspondence between Gmin a and Dmin 3 3 Comments on the Rules References Why do we need ⊥ ? We introduce the ⊥ sign because of the constraints on the succedent. But why can’t we have intuitionistic constraints on the succedent, i.e. that Θ should not contain more than one element, but it can possibly be empty? It is important because in minimal logic the condition of uniqueness of negation is not satisfied. We understand uniqueness as follows: if two n-ary operators † and † ∗ are governed by the same inference rules, then for all A 1 , ..., A n , † ( A 1 , ..., A n ) and † ∗ ( A 1 , ..., A n ) are interdeducible – i.e., † ( A 1 , ..., A n ) ⊣⊢ † ∗ ( A 1 , ..., A n ) – using imperatively at least one of the rules of † or † ∗ and, when needed, the reflexivity axiom rule in order to close the derivation. No other rules are admitted. Naibo, Alberto and Petrolo, Mattia. (2015): ”Are Uniqueness and Deducibility of Identicals the Same?”. Alexandra Pavlova Dialogue Games for Minimal Logic
Gmin a Sequent Calculus for Minimal Logic 3 General Idea Dmin Minimal Dialogue Logic Gmin a System The Correspondence between Gmin a and Dmin 3 3 Comments on the Rules References Why do we need ⊥ ? Alexandra Pavlova Dialogue Games for Minimal Logic
Gmin a Sequent Calculus for Minimal Logic 3 General Idea Dmin Minimal Dialogue Logic Gmin a System The Correspondence between Gmin a and Dmin 3 3 Comments on the Rules References Why do we need ⊥ ? Let us consider the following rules for negation ( ¬ A ): Alexandra Pavlova Dialogue Games for Minimal Logic
Gmin a Sequent Calculus for Minimal Logic 3 General Idea Dmin Minimal Dialogue Logic Gmin a System The Correspondence between Gmin a and Dmin 3 3 Comments on the Rules References Why do we need ⊥ ? Let us consider the following rules for negation ( ¬ A ): A , Γ − → ¬ A , Γ − → A ¬ S + ′ → Θ ¬ A + ′ Γ − → ¬ A ¬ A , Γ − where Θ is empty (Θ = ∅ ). Alexandra Pavlova Dialogue Games for Minimal Logic
Gmin a Sequent Calculus for Minimal Logic 3 General Idea Dmin Minimal Dialogue Logic Gmin a System The Correspondence between Gmin a and Dmin 3 3 Comments on the Rules References Why do we need ⊥ ? Let us consider the following rules for negation ( ¬ A ): A , Γ − → ¬ A , Γ − → A ¬ S + ′ → Θ ¬ A + ′ Γ − → ¬ A ¬ A , Γ − where Θ is empty (Θ = ∅ ). What is wrong with these ¬ S + ′ and ¬ A + ′ rules? Alexandra Pavlova Dialogue Games for Minimal Logic
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