Dialogue Games for Minimal Logic Alexandra Pavlova National - - PowerPoint PPT Presentation

Gmin 3 a Sequent Calculus for Minimal Logic Dmin Minimal Dialogue Logic The Correspondence between Gmin 3 a and Dmin 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 3 a Sequent Calculus for Minimal Logic Dmin Minimal Dialogue Logic The Correspondence between Gmin 3 a and Dmin References General Idea Gmin 3 a System Comments on the Rules

General Idea

Minimal Propositional Logic We reject:

Alexandra Pavlova Dialogue Games for Minimal Logic

Gmin 3 a Sequent Calculus for Minimal Logic Dmin Minimal Dialogue Logic The Correspondence between Gmin 3 a and Dmin References General Idea Gmin 3 a System Comments on the Rules

General Idea

Minimal Propositional Logic We reject:

• the law of excluded middle A ∨ ¬A

Alexandra Pavlova Dialogue Games for Minimal Logic

Gmin 3 a Sequent Calculus for Minimal Logic Dmin Minimal Dialogue Logic The Correspondence between Gmin 3 a and Dmin References General Idea Gmin 3 a System Comments on the Rules

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 3 a Sequent Calculus for Minimal Logic Dmin Minimal Dialogue Logic The Correspondence between Gmin 3 a and Dmin References General Idea Gmin 3 a System Comments on the Rules

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 3 a Sequent Calculus for Minimal Logic Dmin Minimal Dialogue Logic The Correspondence between Gmin 3 a and Dmin References General Idea Gmin 3 a System Comments on the Rules

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: ⊥ D = ⇒ − → A ∧ ¬A − → D∗

Alexandra Pavlova Dialogue Games for Minimal Logic

Gmin 3 a Sequent Calculus for Minimal Logic Dmin Minimal Dialogue Logic The Correspondence between Gmin 3 a and Dmin References General Idea Gmin 3 a System Comments on the Rules

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: ⊥ D = ⇒ − → A ∧ ¬A − → D∗ According to the rules of LJ this can be transformed as follows:

Alexandra Pavlova Dialogue Games for Minimal Logic

Gmin 3 a Sequent Calculus for Minimal Logic Dmin Minimal Dialogue Logic The Correspondence between Gmin 3 a and Dmin References General Idea Gmin 3 a System Comments on the Rules

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: ⊥ D = ⇒ − → A ∧ ¬A − → D∗ According to the rules of LJ this can be transformed as follows: Γ − → A ∧ ¬A A − → A

(UEA)

A ∧ ¬A − → A

(NEA)

¬A, A ∧ ¬A − →

(UEA)

A ∧ ¬A, A ∧ ¬A − →

(CL)

A ∧ ¬A − →

(cut)

Γ − →

(WR)

Γ − → D∗

Alexandra Pavlova Dialogue Games for Minimal Logic

Gmin 3 a Sequent Calculus for Minimal Logic Dmin Minimal Dialogue Logic The Correspondence between Gmin 3 a and Dmin References General Idea Gmin 3 a System Comments on the Rules

G min

3

a System

Definition The language Lmin is defined in BNF-style as followsa: 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”) − →.

aWe use Gothic letters to indicate meta-language variables. Alexandra Pavlova Dialogue Games for Minimal Logic

Gmin 3 a Sequent Calculus for Minimal Logic Dmin Minimal Dialogue Logic The Correspondence between Gmin 3 a and Dmin References General Idea Gmin 3 a System Comments on the Rules

G min

3

a System

Definition The language Lmin is defined in BNF-style as followsa: 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”) − →.

aWe use Gothic letters to indicate meta-language variables.

Definition The axiom of the system G min

3

aa is A, Γ − → A (Ax.Int)

aThis 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

• ne.

Alexandra Pavlova Dialogue Games for Minimal Logic

Gmin 3 a Sequent Calculus for Minimal Logic Dmin Minimal Dialogue Logic The Correspondence between Gmin 3 a and Dmin References General Idea Gmin 3 a System Comments on the Rules

G min

3

a System

Definition Given that Γ and Θ are multisets of formulae, the system G min

3

a has the following rules of inference A, Γ − → B Γ − → A ⊃ B ⊃S+ A ⊃ B, Γ − → A and B, A ⊃ B, Γ − → Θ A ⊃ B, Γ − → Θ ⊃A+ Γ − → A and Γ − → B Γ − → A ∧ B ∧S+ A, A ∧ B, Γ − → Θ

• r

B, A ∧ B, Γ − → Θ A ∧ B, Γ − → Θ ∧A+ Γ − → A or Γ − → B Γ − → A ∨ B ∨S+ A, A ∨ B, Γ − → Θ and B, A ∨ B, Γ − → Θ A ∨ B, Γ − → Θ ∨A+ A, Γ − → ⊥ Γ − → ¬A ¬S+ ¬A, Γ − → A ¬A, Γ − → ⊥ ¬A+ where, for all rules, the succedent should contain exactly one formula.

Alexandra Pavlova Dialogue Games for Minimal Logic

Gmin 3 a Sequent Calculus for Minimal Logic Dmin Minimal Dialogue Logic The Correspondence between Gmin 3 a and Dmin References General Idea Gmin 3 a System Comments on the Rules

Why do we need ⊥?

Alexandra Pavlova Dialogue Games for Minimal Logic

Gmin 3 a Sequent Calculus for Minimal Logic Dmin Minimal Dialogue Logic The Correspondence between Gmin 3 a and Dmin References General Idea Gmin 3 a System Comments on the Rules

Why do we need ⊥?

We introduce the ⊥ sign because of the constraints on the succedent.

Alexandra Pavlova Dialogue Games for Minimal Logic

Gmin 3 a Sequent Calculus for Minimal Logic Dmin Minimal Dialogue Logic The Correspondence between Gmin 3 a and Dmin References General Idea Gmin 3 a System Comments on the Rules

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 3 a Sequent Calculus for Minimal Logic Dmin Minimal Dialogue Logic The Correspondence between Gmin 3 a and Dmin References General Idea Gmin 3 a System Comments on the Rules

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 A1, ..., An, †(A1, ..., An) and †∗(A1, ..., An) are interdeducible – i.e., †(A1, ..., An) ⊣⊢ †∗(A1, ..., An) – using imperatively at least one

• f 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 3 a Sequent Calculus for Minimal Logic Dmin Minimal Dialogue Logic The Correspondence between Gmin 3 a and Dmin References General Idea Gmin 3 a System Comments on the Rules

Why do we need ⊥?

Alexandra Pavlova Dialogue Games for Minimal Logic

Gmin 3 a Sequent Calculus for Minimal Logic Dmin Minimal Dialogue Logic The Correspondence between Gmin 3 a and Dmin References General Idea Gmin 3 a System Comments on the Rules

Why do we need ⊥?

Let us consider the following rules for negation (¬A):

Alexandra Pavlova Dialogue Games for Minimal Logic

Gmin 3 a Sequent Calculus for Minimal Logic Dmin Minimal Dialogue Logic The Correspondence between Gmin 3 a and Dmin References General Idea Gmin 3 a System Comments on the Rules

Why do we need ⊥?

Let us consider the following rules for negation (¬A): A, Γ − → Γ − → ¬A ¬S+′ ¬A, Γ − → A ¬A, Γ − → Θ ¬A+′ where Θ is empty (Θ = ∅).

Alexandra Pavlova Dialogue Games for Minimal Logic

Gmin 3 a Sequent Calculus for Minimal Logic Dmin Minimal Dialogue Logic The Correspondence between Gmin 3 a and Dmin References General Idea Gmin 3 a System Comments on the Rules

Why do we need ⊥?

Let us consider the following rules for negation (¬A): A, Γ − → Γ − → ¬A ¬S+′ ¬A, Γ − → A ¬A, Γ − → Θ ¬A+′ where Θ is empty (Θ = ∅). What is wrong with these ¬S+′ and ¬A+′ rules?

Alexandra Pavlova Dialogue Games for Minimal Logic

Gmin 3 a Sequent Calculus for Minimal Logic Dmin Minimal Dialogue Logic The Correspondence between Gmin 3 a and Dmin References General Idea Gmin 3 a System Comments on the Rules

Why do we need ⊥?

Let us consider the following rules for negation (¬A): A, Γ − → Γ − → ¬A ¬S+′ ¬A, Γ − → A ¬A, Γ − → Θ ¬A+′ where Θ is empty (Θ = ∅). What is wrong with these ¬S+′ and ¬A+′ rules? We can use them to prove identity of indiscernibles. We would be able to deduce ¬φ ⊃⊂ ¬∗φ (or, equally, ¬φ ⊣⊢ ¬∗φ) as follows:

Alexandra Pavlova Dialogue Games for Minimal Logic

Gmin 3 a Sequent Calculus for Minimal Logic Dmin Minimal Dialogue Logic The Correspondence between Gmin 3 a and Dmin References General Idea Gmin 3 a System Comments on the Rules

Why do we need ⊥?

Let us consider the following rules for negation (¬A): A, Γ − → Γ − → ¬A ¬S+′ ¬A, Γ − → A ¬A, Γ − → Θ ¬A+′ where Θ is empty (Θ = ∅). What is wrong with these ¬S+′ and ¬A+′ rules? We can use them to prove identity of indiscernibles. We would be able to deduce ¬φ ⊃⊂ ¬∗φ (or, equally, ¬φ ⊣⊢ ¬∗φ) as follows: φ − → φ ¬φ, φ − → ¬φ − → ¬∗φ − → ¬φ ⊃ ¬∗φ φ − → φ ¬∗φ, φ − → ¬∗φ − → ¬φ − → ¬∗φ ⊃ ¬φ − → ¬φ ⊃⊂ ¬∗φ

Alexandra Pavlova Dialogue Games for Minimal Logic

Gmin 3 a Sequent Calculus for Minimal Logic Dmin Minimal Dialogue Logic The Correspondence between Gmin 3 a and Dmin References General Idea Gmin 3 a System Comments on the Rules

Clarification

We can define negation as ¬A

def

= A ⊃ ⊥. Thus, we can see that our rules for negation are just a special case of rules for implication:

Alexandra Pavlova Dialogue Games for Minimal Logic

Gmin 3 a Sequent Calculus for Minimal Logic Dmin Minimal Dialogue Logic The Correspondence between Gmin 3 a and Dmin References General Idea Gmin 3 a System Comments on the Rules

Clarification

We can define negation as ¬A

def

= A ⊃ ⊥. Thus, we can see that our rules for negation are just a special case of rules for implication: A, Γ − → ⊥ Γ − → A ⊃ ⊥ ⊃S+ Γ − → ¬A def A ⊃ ⊥, Γ − → A A ⊃ ⊥, Γ, ⊥ − → ⊥Ax. A ⊃ ⊥, Γ − → ⊥ ⊃A+ ¬A, Γ − → ⊥ def

Alexandra Pavlova Dialogue Games for Minimal Logic

Gmin 3 a Sequent Calculus for Minimal Logic Dmin Minimal Dialogue Logic The Correspondence between Gmin 3 a and Dmin References General Idea Gmin 3 a System Comments on the Rules

Clarification

We can define negation as ¬A

def

= A ⊃ ⊥. Thus, we can see that our rules for negation are just a special case of rules for implication: A, Γ − → ⊥ Γ − → A ⊃ ⊥ ⊃S+ Γ − → ¬A def A ⊃ ⊥, Γ − → A A ⊃ ⊥, Γ, ⊥ − → ⊥Ax. A ⊃ ⊥, Γ − → ⊥ ⊃A+ ¬A, Γ − → ⊥ def We provide our proof of correspondence result for the sequent calculus with branching in ¬A+, but as one can see it is applicable to the variant without branching.

Alexandra Pavlova Dialogue Games for Minimal Logic

Gmin 3 a Sequent Calculus for Minimal Logic Dmin Minimal Dialogue Logic The Correspondence between Gmin 3 a and Dmin References What is Dialogue Logic? Logical Rules Structural Rules Example Additional Definitions

What is Dialogue Logic?

Alexandra Pavlova Dialogue Games for Minimal Logic

Gmin 3 a Sequent Calculus for Minimal Logic Dmin Minimal Dialogue Logic The Correspondence between Gmin 3 a and Dmin References What is Dialogue Logic? Logical Rules Structural Rules Example Additional Definitions

What is Dialogue Logic?

Definition A dialogue is a sequence of attacks and defences that begins with a finite (possibly empty) multiset Π of formulae that are initially granted by O and a finite (nonempty) multiset ∆ of formulae that are initially disputed by Oa.

aIt is possible to dispute over a number of formulae, however, in our account we assume that

there is only one initially disputed formula.

Alexandra Pavlova Dialogue Games for Minimal Logic

Gmin 3 a Sequent Calculus for Minimal Logic Dmin Minimal Dialogue Logic The Correspondence between Gmin 3 a and Dmin References What is Dialogue Logic? Logical Rules Structural Rules Example Additional Definitions

What is Dialogue Logic?

Definition A dialogue is a sequence of attacks and defences that begins with a finite (possibly empty) multiset Π of formulae that are initially granted by O and a finite (nonempty) multiset ∆ of formulae that are initially disputed by Oa.

aIt is possible to dispute over a number of formulae, however, in our account we assume that

there is only one initially disputed formula.

Types of Rules There are two following levels of rules in dialogue logic:

Alexandra Pavlova Dialogue Games for Minimal Logic

Gmin 3 a Sequent Calculus for Minimal Logic Dmin Minimal Dialogue Logic The Correspondence between Gmin 3 a and Dmin References What is Dialogue Logic? Logical Rules Structural Rules Example Additional Definitions

What is Dialogue Logic?

Definition A dialogue is a sequence of attacks and defences that begins with a finite (possibly empty) multiset Π of formulae that are initially granted by O and a finite (nonempty) multiset ∆ of formulae that are initially disputed by Oa.

aIt is possible to dispute over a number of formulae, however, in our account we assume that

there is only one initially disputed formula.

Types of Rules There are two following levels of rules in dialogue logic: I Logical rules define the possible types of attacks and defences for each type of formula, i.e. containing specific logical operators. II Structural rules define the general course of game, i.e. when each player can move and what type of move he is allowed to make.

Alexandra Pavlova Dialogue Games for Minimal Logic

Gmin 3 a Sequent Calculus for Minimal Logic Dmin Minimal Dialogue Logic The Correspondence between Gmin 3 a and Dmin References What is Dialogue Logic? Logical Rules Structural Rules Example Additional Definitions

Logical Rules

Definition (Logical Rules) The system Dmina has the following logical rules: Connective Attack Defence X−! − A ∧ B Y −? − ∧L X−! − A Y −? − ∧R X−! − B X−! − A ∨ B Y −? − ∨ X−! − A X−! − B X−! − A ⊃ B Y −! − A X−! − B X−! − ¬A Y −! − A X−! − ⊥ X−! − ∀xA(x) Y −? − ∀x/n X−! − A[n/x] X−! − ∃xA(x) Y −? − ∃x X−! − A[n/x] where A and B are metavariables, X and Y variables for player (with P and O being the precise roles) with X = Y , ! and ? are used to represent assertion and demand respectively. In case of conjunction and universal quantification the choice is made by the attacker, whereas in case of disjunction and existential quantification the choice is made by the defender.

Alexandra Pavlova Dialogue Games for Minimal Logic

Gmin 3 a Sequent Calculus for Minimal Logic Dmin Minimal Dialogue Logic The Correspondence between Gmin 3 a and Dmin References What is Dialogue Logic? Logical Rules Structural Rules Example Additional Definitions

Structural Rules

Definition (Structural rules) The system Dmina has the following structural rules: Start The first move of the dialogue is carried out by O and consists of an attack on (the unique) initially disputed formula Aa. Alternation Moves strictly alternate between players O and P. Atom P can affirm atomic formulae, including ⊥, iff they were previously stated by O. D11 If it is X’s turn and there is more than one attack by Y that X has not yet defended, only the most recent one may be defended. D12 Any attack may be defended at most once. Attack-rule O can attack P’s one and the same formula only once, whereas P can attack O’s formula at most 2 times

Alexandra Pavlova Dialogue Games for Minimal Logic

Gmin 3 a Sequent Calculus for Minimal Logic Dmin Minimal Dialogue Logic The Correspondence between Gmin 3 a and Dmin References What is Dialogue Logic? Logical Rules Structural Rules Example Additional Definitions

Structural Rules

Definition (Structural rules) The system Dmina has the following structural rules: Start The first move of the dialogue is carried out by O and consists of an attack on (the unique) initially disputed formula Aa. Alternation Moves strictly alternate between players O and P. Atom P can affirm atomic formulae, including ⊥, iff they were previously stated by O. D11 If it is X’s turn and there is more than one attack by Y that X has not yet defended, only the most recent one may be defended. D12 Any attack may be defended at most once. Attack-rule O can attack P’s one and the same formula only once, whereas P can attack O’s formula at most 2 times Minimal rule For each attack players must provide a defence.

aWe count as a zero step the one where P proposes a formula for the dispute. Alexandra Pavlova Dialogue Games for Minimal Logic

Gmin 3 a Sequent Calculus for Minimal Logic Dmin Minimal Dialogue Logic The Correspondence between Gmin 3 a and Dmin References What is Dialogue Logic? Logical Rules Structural Rules Example Additional Definitions

Structural Rules

Alexandra Pavlova Dialogue Games for Minimal Logic

Gmin 3 a Sequent Calculus for Minimal Logic Dmin Minimal Dialogue Logic The Correspondence between Gmin 3 a and Dmin References What is Dialogue Logic? Logical Rules Structural Rules Example Additional Definitions

Structural Rules

Definition (Ending) The game ends if and only if the player whose turn it is to move has no legal move to make.

Alexandra Pavlova Dialogue Games for Minimal Logic

Gmin 3 a Sequent Calculus for Minimal Logic Dmin Minimal Dialogue Logic The Correspondence between Gmin 3 a and Dmin References What is Dialogue Logic? Logical Rules Structural Rules Example Additional Definitions

Structural Rules

Definition (Ending) The game ends if and only if the player whose turn it is to move has no legal move to make. Definition (Winning conditions) The game ends with P winning iff it is O’s turn and she has no possible move left to make and P has satisfied the Minimal rule (cf. 6). The game ends with O winning iff it is P’s turn and she has no possible move left to make or if P has not satisfied the Minimal rule (cf. 6).

Alexandra Pavlova Dialogue Games for Minimal Logic

Gmin 3 a Sequent Calculus for Minimal Logic Dmin Minimal Dialogue Logic The Correspondence between Gmin 3 a and Dmin References What is Dialogue Logic? Logical Rules Structural Rules Example Additional Definitions

Example

Consider the following dialogue: Round Opponent Proponent (0) A ⊃ ¬¬A

Table: An example: A ⊃ ¬¬A

Alexandra Pavlova Dialogue Games for Minimal Logic

Gmin 3 a Sequent Calculus for Minimal Logic Dmin Minimal Dialogue Logic The Correspondence between Gmin 3 a and Dmin References What is Dialogue Logic? Logical Rules Structural Rules Example Additional Definitions

Example

Consider the following dialogue: Round Opponent Proponent (0) A ⊃ ¬¬A I (1) A (Att. 0)

Table: An example: A ⊃ ¬¬A

Alexandra Pavlova Dialogue Games for Minimal Logic

Gmin 3 a Sequent Calculus for Minimal Logic Dmin Minimal Dialogue Logic The Correspondence between Gmin 3 a and Dmin References What is Dialogue Logic? Logical Rules Structural Rules Example Additional Definitions

Example

Consider the following dialogue: Round Opponent Proponent (0) A ⊃ ¬¬A I (1) A (Att. 0) (2) ¬¬A (Def. 1)

Table: An example: A ⊃ ¬¬A

Alexandra Pavlova Dialogue Games for Minimal Logic

Gmin 3 a Sequent Calculus for Minimal Logic Dmin Minimal Dialogue Logic The Correspondence between Gmin 3 a and Dmin References What is Dialogue Logic? Logical Rules Structural Rules Example Additional Definitions

Example

Consider the following dialogue: Round Opponent Proponent (0) A ⊃ ¬¬A I (1) A (Att. 0) (2) ¬¬A (Def. 1) II (3) ¬A (Att. 2)

Table: An example: A ⊃ ¬¬A

Alexandra Pavlova Dialogue Games for Minimal Logic

Gmin 3 a Sequent Calculus for Minimal Logic Dmin Minimal Dialogue Logic The Correspondence between Gmin 3 a and Dmin References What is Dialogue Logic? Logical Rules Structural Rules Example Additional Definitions

Example

Consider the following dialogue: Round Opponent Proponent (0) A ⊃ ¬¬A I (1) A (Att. 0) (2) ¬¬A (Def. 1) II (3) ¬A (Att. 2) (4) ⊥ (Def. 3 )

Table: An example: A ⊃ ¬¬A

Alexandra Pavlova Dialogue Games for Minimal Logic

Gmin 3 a Sequent Calculus for Minimal Logic Dmin Minimal Dialogue Logic The Correspondence between Gmin 3 a and Dmin References What is Dialogue Logic? Logical Rules Structural Rules Example Additional Definitions

Example

Consider the following dialogue: Round Opponent Proponent (0) A ⊃ ¬¬A I (1) A (Att. 0) (2) ¬¬A (Def. 1) II (3) ¬A (Att. 2) III (4) A (Att. 3)

Table: An example: A ⊃ ¬¬A

Alexandra Pavlova Dialogue Games for Minimal Logic

Gmin 3 a Sequent Calculus for Minimal Logic Dmin Minimal Dialogue Logic The Correspondence between Gmin 3 a and Dmin References What is Dialogue Logic? Logical Rules Structural Rules Example Additional Definitions

Example

Consider the following dialogue: Round Opponent Proponent (0) A ⊃ ¬¬A I (1) A (Att. 0) (2) ¬¬A (Def. 1) II (3) ¬A (Att. 2) III (5) ⊥ (Def. 4) (4) A (Att. 3)

Table: An example: A ⊃ ¬¬A

Alexandra Pavlova Dialogue Games for Minimal Logic

Gmin 3 a Sequent Calculus for Minimal Logic Dmin Minimal Dialogue Logic The Correspondence between Gmin 3 a and Dmin References What is Dialogue Logic? Logical Rules Structural Rules Example Additional Definitions

Example

Consider the following dialogue: Round Opponent Proponent (0) A ⊃ ¬¬A I (1) A (Att. 0) (2) ¬¬A (Def. 1) II (3) ¬A (Att. 2) (6) ⊥ (Def. 3 ) III (5) ⊥ (Def. 4) (4) A (Att. 3)

Table: An example: A ⊃ ¬¬A

Alexandra Pavlova Dialogue Games for Minimal Logic

Gmin 3 a Sequent Calculus for Minimal Logic Dmin Minimal Dialogue Logic The Correspondence between Gmin 3 a and Dmin References What is Dialogue Logic? Logical Rules Structural Rules Example Additional Definitions

Let us now consider a dialogue game in the extensive form for this formula. It has the following form:

❄ ❄ t t t t ❄ ❄ t ❄ t ❄ t

0: P−! − A ⊃ ¬¬A

Alexandra Pavlova Dialogue Games for Minimal Logic

Gmin 3 a Sequent Calculus for Minimal Logic Dmin Minimal Dialogue Logic The Correspondence between Gmin 3 a and Dmin References What is Dialogue Logic? Logical Rules Structural Rules Example Additional Definitions

Let us now consider a dialogue game in the extensive form for this formula. It has the following form:

❄ ❄ t t t t ❄ ❄ t ❄ t ❄ t

0: P−! − A ⊃ ¬¬A 1: O−! − A (Att.0)

Alexandra Pavlova Dialogue Games for Minimal Logic

Gmin 3 a Sequent Calculus for Minimal Logic Dmin Minimal Dialogue Logic The Correspondence between Gmin 3 a and Dmin References What is Dialogue Logic? Logical Rules Structural Rules Example Additional Definitions

Let us now consider a dialogue game in the extensive form for this formula. It has the following form:

❄ ❄ t t t t ❄ ❄ t ❄ t ❄ t

0: P−! − A ⊃ ¬¬A 1: O−! − A (Att.0) 2: P−! − ¬¬A (Def.1)

Alexandra Pavlova Dialogue Games for Minimal Logic

Gmin 3 a Sequent Calculus for Minimal Logic Dmin Minimal Dialogue Logic The Correspondence between Gmin 3 a and Dmin References What is Dialogue Logic? Logical Rules Structural Rules Example Additional Definitions

Let us now consider a dialogue game in the extensive form for this formula. It has the following form:

❄ ❄ t t t t ❄ ❄ t ❄ t ❄ t

0: P−! − A ⊃ ¬¬A 1: O−! − A (Att.0) 2: P−! − ¬¬A (Def.1) 3: O−! − ¬A (Att.2)

Alexandra Pavlova Dialogue Games for Minimal Logic

Gmin 3 a Sequent Calculus for Minimal Logic Dmin Minimal Dialogue Logic The Correspondence between Gmin 3 a and Dmin References What is Dialogue Logic? Logical Rules Structural Rules Example Additional Definitions

Let us now consider a dialogue game in the extensive form for this formula. It has the following form:

❄ ❄ t t t t ❄ ❄ t ❄ t ❄ t

0: P−! − A ⊃ ¬¬A 1: O−! − A (Att.0) 2: P−! − ¬¬A (Def.1) 3: O−! − ¬A (Att.2) 4: P−! − A (Att.3)

Alexandra Pavlova Dialogue Games for Minimal Logic

Gmin 3 a Sequent Calculus for Minimal Logic Dmin Minimal Dialogue Logic The Correspondence between Gmin 3 a and Dmin References What is Dialogue Logic? Logical Rules Structural Rules Example Additional Definitions

Let us now consider a dialogue game in the extensive form for this formula. It has the following form:

❄ ❄ t t t t ❄ ❄ t ❄ t ❄ t

0: P−! − A ⊃ ¬¬A 1: O−! − A (Att.0) 2: P−! − ¬¬A (Def.1) 3: O−! − ¬A (Att.2) 4: P−! − A (Att.3) 5: O−! − ⊥ (Def.4)

Alexandra Pavlova Dialogue Games for Minimal Logic

Gmin 3 a Sequent Calculus for Minimal Logic Dmin Minimal Dialogue Logic The Correspondence between Gmin 3 a and Dmin References What is Dialogue Logic? Logical Rules Structural Rules Example Additional Definitions

Let us now consider a dialogue game in the extensive form for this formula. It has the following form:

❄ ❄ t t t t ❄ ❄ t ❄ t ❄ t

0: P−! − A ⊃ ¬¬A 1: O−! − A (Att.0) 2: P−! − ¬¬A (Def.1) 3: O−! − ¬A (Att.2) 4: P−! − A (Att.3) 5: O−! − ⊥ (Def.4) 6: P−! − ⊥ (Def.5)

Alexandra Pavlova Dialogue Games for Minimal Logic

Gmin 3 a Sequent Calculus for Minimal Logic Dmin Minimal Dialogue Logic The Correspondence between Gmin 3 a and Dmin References What is Dialogue Logic? Logical Rules Structural Rules Example Additional Definitions

Example

The same example in the G min

3

a:

Alexandra Pavlova Dialogue Games for Minimal Logic

Gmin 3 a Sequent Calculus for Minimal Logic Dmin Minimal Dialogue Logic The Correspondence between Gmin 3 a and Dmin References What is Dialogue Logic? Logical Rules Structural Rules Example Additional Definitions

Example

The same example in the G min

3

a: Ax. ¬A, A − → A

(¬A+)

¬A, A − → ⊥

(¬S+)

A − → ¬¬A

(⊃S+)

− → A ⊃ ¬¬A

Alexandra Pavlova Dialogue Games for Minimal Logic

Gmin 3 a Sequent Calculus for Minimal Logic Dmin Minimal Dialogue Logic The Correspondence between Gmin 3 a and Dmin References What is Dialogue Logic? Logical Rules Structural Rules Example Additional Definitions

Example

The same example in the G min

3

a: Ax. ¬A, A − → A

(¬A+)

¬A, A − → ⊥

(¬S+)

A − → ¬¬A

(⊃S+)

− → A ⊃ ¬¬A Ax. ¬A, A − → A Ax. ¬A, A, ⊥ − → ⊥

(¬(⊃)A+)

¬A, A − → ⊥

(¬(⊃)S+)

A − → ¬¬A

(⊃S+)

− → A ⊃ ¬¬A

Alexandra Pavlova Dialogue Games for Minimal Logic

Gmin 3 a Sequent Calculus for Minimal Logic Dmin Minimal Dialogue Logic The Correspondence between Gmin 3 a and Dmin References What is Dialogue Logic? Logical Rules Structural Rules Example Additional Definitions

Additional Definitions

Definition A dialogue tree T for a dialogue sequent Π − → A is a rooted directed tree whose nodes are rounds in a dialogue game such that every branch of T is a dialogue with initially granted formulas Π and initially disputed formula A.

Alexandra Pavlova Dialogue Games for Minimal Logic

Gmin 3 a Sequent Calculus for Minimal Logic Dmin Minimal Dialogue Logic The Correspondence between Gmin 3 a and Dmin References What is Dialogue Logic? Logical Rules Structural Rules Example Additional Definitions

Additional Definitions

Definition A dialogue tree T for a dialogue sequent Π − → A is a rooted directed tree whose nodes are rounds in a dialogue game such that every branch of T is a dialogue with initially granted formulas Π and initially disputed formula A. Definition (Winning Strategy) A finite dialogue tree T ′ (T ′ ⊂ T, where T is a full dialogue tree for the a dialogue sequent) is a winning strategy τ for X if and only if each branch of the tree T ′ (regardless of any moves of Y ) ends with the move of X, i.e. player Y has no possible move to make, and the condition specified in the minimal rule is satisfied. ”Note that the dialogue tree is just a representation of the game in extensive form, and that a winning strategy is a subtree of the entire dialogue tree that can be understood as a procedure that P can follow to win the game no matter how O responds to P’s moves.” Alama J., Knoks A., Uckelman S.L. (2011): ”Dialogue Games for Classical Logic”

Alexandra Pavlova Dialogue Games for Minimal Logic

Gmin 3 a Sequent Calculus for Minimal Logic Dmin Minimal Dialogue Logic The Correspondence between Gmin 3 a and Dmin References

Minimal Validity

Alexandra Pavlova Dialogue Games for Minimal Logic

Gmin 3 a Sequent Calculus for Minimal Logic Dmin Minimal Dialogue Logic The Correspondence between Gmin 3 a and Dmin References

Minimal Validity

Theorem (Minimal validity) Let A be any formula of propositional logic. The following conditions are equivalent:

Alexandra Pavlova Dialogue Games for Minimal Logic

Gmin 3 a Sequent Calculus for Minimal Logic Dmin Minimal Dialogue Logic The Correspondence between Gmin 3 a and Dmin References

Minimal Validity

Theorem (Minimal validity) Let A be any formula of propositional logic. The following conditions are equivalent:

1

There is a winning strategy for Proponent in dialogue D(A);

Alexandra Pavlova Dialogue Games for Minimal Logic

Gmin 3 a Sequent Calculus for Minimal Logic Dmin Minimal Dialogue Logic The Correspondence between Gmin 3 a and Dmin References

Minimal Validity

Theorem (Minimal validity) Let A be any formula of propositional logic. The following conditions are equivalent:

1

There is a winning strategy for Proponent in dialogue D(A);

2

There exists a G min

3

a derivation of the formula A (i.e., Γ − → A, where Γ is empty).

Alexandra Pavlova Dialogue Games for Minimal Logic

Gmin 3 a Sequent Calculus for Minimal Logic Dmin Minimal Dialogue Logic The Correspondence between Gmin 3 a and Dmin References

Minimal Validity

Theorem (Minimal validity) Let A be any formula of propositional logic. The following conditions are equivalent:

1

There is a winning strategy for Proponent in dialogue D(A);

2

There exists a G min

3

a derivation of the formula A (i.e., Γ − → A, where Γ is empty). Furthermore, there exists an algorithm turning Proponent’s winning strategy into the G min

3

a derivation and visa versa.

Alexandra Pavlova Dialogue Games for Minimal Logic

Gmin 3 a Sequent Calculus for Minimal Logic Dmin Minimal Dialogue Logic The Correspondence between Gmin 3 a and Dmin References

Minimal Validity

Theorem (Minimal validity) Let A be any formula of propositional logic. The following conditions are equivalent:

1

There is a winning strategy for Proponent in dialogue D(A);

2

There exists a G min

3

a derivation of the formula A (i.e., Γ − → A, where Γ is empty). Furthermore, there exists an algorithm turning Proponent’s winning strategy into the G min

3

a derivation and visa versa. Theorem (Correspondence result) Every winning strategy τ for Proponent with respect to D(A, Γ) (i.e., for a dialogue with initially disputed formula A, where the Opponent initially grants the formulae in the multiset Γ) can be transformed into a G min

3

a derivation of Γ − → A and visa versa.

Alexandra Pavlova Dialogue Games for Minimal Logic

Gmin 3 a Sequent Calculus for Minimal Logic Dmin Minimal Dialogue Logic The Correspondence between Gmin 3 a and Dmin References

Proof

We prove theorem 2 in two steps, by establishing two lemmata. Lemma Every P′s winning strategy τ for D(A, Γ) can be transformed into a G min

3

a derivation of Γ − → A. Given an arbitrary w.s. τ for P, we associate each round of Dmin with a sequence.

Alexandra Pavlova Dialogue Games for Minimal Logic

Gmin 3 a Sequent Calculus for Minimal Logic Dmin Minimal Dialogue Logic The Correspondence between Gmin 3 a and Dmin References

Proof

Claim For every round of the dialogue D(A, Γ) there is a G min

3

a deduction of the sequent corresponding to the dialogue sequent at this round Θ − → Ai , which consists of Θ = Γ ∪ {A1, A2, . . . An}, where A1, A2, . . . An are subformulae of A asserted by O and Ai represents a subformula of A asserted by P. We prove this claim by induction on the depth d of τ. The base case: d = 1. In this case the game terminates at the round #1. P moves last. P has asserted an atomic formula A. Hence, O have already granted A. Thus, the dialogue sequent at round #1 is A, Γ − → A (or Γ − → A if A ∈ Γ). The induction hypothesis: we assume that the claim holds for all n d and show that it also holds at d + 1 step. The inductive step: we proceed by analysing the cases according to the form of the formula that is defended or attacked by P.

Alexandra Pavlova Dialogue Games for Minimal Logic

Gmin 3 a Sequent Calculus for Minimal Logic Dmin Minimal Dialogue Logic The Correspondence between Gmin 3 a and Dmin References

Proof

Lemma Every G min

3

a derivation of Γ − → A can be transformed into a winning strategy τi for D(A, Γ). (i.e., for a dialogue with initially disputed formula A, where the Opponent initially grants the formulae in the multiset Γ). To prove lemma 2 we show that each rule of G min

3

a can be transformed into a corresponding dialogue rule. The web-page with details: For complete proof visit: https://sites.google.com/view/the-correspondence-result/

Alexandra Pavlova Dialogue Games for Minimal Logic

Gmin 3 a Sequent Calculus for Minimal Logic Dmin Minimal Dialogue Logic The Correspondence between Gmin 3 a and Dmin References

Example

Example Round O P (0) (A ∨ B) ⊃ (A ∨ B) I (1) A ∨ B (2) A ∨ B II (3) ? − v (6) A/B III (5) A/B (4) ? − v

Alexandra Pavlova Dialogue Games for Minimal Logic

Gmin 3 a Sequent Calculus for Minimal Logic Dmin Minimal Dialogue Logic The Correspondence between Gmin 3 a and Dmin References

Example

Example Round O P (0) (A ∨ B) ⊃ (A ∨ B) I (1) A ∨ B (2) A ∨ B II (3) ? − v (6) A/B III (5) A/B (4) ? − v Round O P (0) (A ∨ B) ⊃ (A ∨ B) I (1) A ∨ B (4) A ∨ B II (3) A/B (2) ? − v III (5) ? − v (6) A/B

Alexandra Pavlova Dialogue Games for Minimal Logic

Gmin 3 a Sequent Calculus for Minimal Logic Dmin Minimal Dialogue Logic The Correspondence between Gmin 3 a and Dmin References

Example

B, A ∨ B − → B B, A ∨ B − → A ∨ B A, A ∨ B − → A A, A ∨ B − → A ∨ B A ∨ B − → A ∨ B − → (A ∨ B) ⊃ (A ∨ B)

Alexandra Pavlova Dialogue Games for Minimal Logic

Gmin 3 a Sequent Calculus for Minimal Logic Dmin Minimal Dialogue Logic The Correspondence between Gmin 3 a and Dmin References

Some References

1

Alama J., Knoks A., Uckelman S.L. (2011): ”Dialogue Games for Classical Logic” (short paper). In M. Giese and R. Kuznets (Eds.), TABLEAUX 2011: Workshops, Tutorials, and Short Papers (pp. 82–86).

2

Felscher W. (1985): ”Dialogues, strategies, and intuitionistic provability”. Annals of Pure and Applied Logic 28, pp. 217–254.

3

Ferm¨ uller C.G. (2003): ”Parallel Dialogue Games and Hypersequents for Intermediate Logics”. In M.C. Mayer, F. Pirri (Eds.), TABLEAUX 2003 Automated Reasoning with Analytic Tableaux and Related Methods (pp. 48–64).

4

Rahman S., Clerbout N., Keiff L. (2009): ”On Dialogues and natural Deduction”. In G. Primeiro and S. Rahman (Eds.), Acts of Knowledge: History, Philosophy and Logic (pp. 301–355). London: College Publications.

5

Sørensen M.H., Urzyczyn P. (2007): ”Sequent calculus, dialogues, and cut elimination”. In: Reflections on Type Theory, λ-Calculus, and the Mind,

• pp. 253–261.

Alexandra Pavlova Dialogue Games for Minimal Logic

Gmin 3 a Sequent Calculus for Minimal Logic Dmin Minimal Dialogue Logic The Correspondence between Gmin 3 a and Dmin References

Thank you! https://sites.google.com/view/the-correspondence-result/

Alexandra Pavlova Dialogue Games for Minimal Logic