Giles's Game and the Proof Theory of Lukasiewicz ... - Semantic Scholar

Giles’s Game and the Proof Theory of Lukasiewicz Logic Chris Ferm¨ uller joint work with George Metcalfe

Technische Universität Wien Theory and Logic Group www.logic.at/people/chrisf/ Pontignano, November 2009

Motivation Suppose I’d start by telling you that I want to talk about the logic given by the following Hilbert-style system:

Motivation Suppose I’d start by telling you that I want to talk about the logic given by the following Hilbert-style system: 1 2 3 4 5 6 7 8 9

(A ⊃ B) ⊃ C ) ⊃ ((((B ⊃ A) ⊃ C ) ⊃ C ) (A ⊃ B) ⊃ ((B ⊃ C ) ⊃ (A ⊃ C )) ⊥⊃A ((A ⊃ ⊥) ⊃ ⊥)) ⊃ A (A & B) ⊃ A (A & B) ⊃ (B & A) (A & (A ⊃ B)) ⊃ (B & (B ⊃ A)) ((A & B) ⊃ C ) ⊃ (A ⊃ (B ⊃ C )) (A ⊃ (B ⊃ C )) ⊃ ((A & B) ⊃ C ) Modus Ponens is the only inference rule

Motivation Suppose I’d start by telling you that I want to talk about the logic given by the following Hilbert-style system: 1 2 3 4 5 6 7 8 9

(A ⊃ B) ⊃ C ) ⊃ ((((B ⊃ A) ⊃ C ) ⊃ C ) (A ⊃ B) ⊃ ((B ⊃ C ) ⊃ (A ⊃ C )) ⊥⊃A ((A ⊃ ⊥) ⊃ ⊥)) ⊃ A (A & B) ⊃ A (A & B) ⊃ (B & A) (A & (A ⊃ B)) ⊃ (B & (B ⊃ A)) ((A & B) ⊃ C ) ⊃ (A ⊃ (B ⊃ C )) (A ⊃ (B ⊃ C )) ⊃ ((A & B) ⊃ C ) Modus Ponens is the only inference rule

You were justified to loose interest in my talk because of this obviously inadequate presentation of a logic!

An inconsequential improvement: Suppose I replace the axiom list by 1 2 3 4

A ⊃ (B ⊃ A) (A ⊃ B) ⊃ ((B ⊃ C ) ⊃ (A ⊃ C )) (¬A ⊃ ¬B) ⊃ (B ⊃ A) ((A ⊃ B) ⊃ B) ⊃ ((B ⊃ A) ⊃ A)

An inconsequential improvement: Suppose I replace the axiom list by 1 2 3 4

A ⊃ (B ⊃ A) (A ⊃ B) ⊃ ((B ⊃ C ) ⊃ (A ⊃ C )) (¬A ⊃ ¬B) ⊃ (B ⊃ A) ((A ⊃ B) ⊃ B) ⊃ ((B ⊃ A) ⊃ A)

A much improved start: I want to talk about ◮

one of three fundamental fuzzy logics

◮

the logic based on the t-norm max(0, x + y − 1)

◮

the logic of all MV-algebras

◮

the logic where formulas represent McNaughton functions

◮

the logic of Mundici’s Ulam-Renyi game semantics

◮

the only fuzzy logic where all truth functions are continuous

◮

...

In other words: Lukasiewicz logic L!

Formal reasoning The above remarks seem to suggest: ◮

proof theoretic (syntactic) presentations are uninformative

◮

algebraic (semantic) characterizations are needed



◮


But what if we focus on formal reasoning (within the logic)?!



◮


But what if we focus on formal reasoning (within the logic)?! Hilbert style systems are problematic (also) for this purpose!



◮


But what if we focus on formal reasoning (within the logic)?! Hilbert style systems are problematic (also) for this purpose! But: think of Gentzen’s characterization of classic vs. intuitionistic inference in terms of the sequent calculus! NB. The following systems are closely related: ◮

analytic tableaux

◮

natural deduction

◮

calculus of structures

◮

...

NB. Not all logics have such ‘nice’ analytic sequent systems like Gentzen’s LK for classic logic and LJ for intuitionistic logic.

NB. Not all logics have such ‘nice’ analytic sequent systems like Gentzen’s LK for classic logic and LJ for intuitionistic logic. Some argue (viz., e.g., Jean-Yves Girard!): (Algebraic) semantics is secondary to analytic proofs. If there is no analytic proof system the ‘logic’ in question is inadequate for models of reasoning!


Hypersequent calculi A hypersequent is a multiset of sequent: ‘Sequents live in a (finite) context of other sequents’.


Hypersequent calculi A hypersequent is a multiset of sequent: ‘Sequents live in a (finite) context of other sequents’. In all other respects hypersequent calculi are like sequent calculi: ◮

logical rules: context-free, systematic, for each connectives

◮

structural rules: allow to manipulate contexts without any reference to logical form

◮

analytic: completeness without cut (≈ modus ponens)

◮

subformula property: top down proof search consists in systematic decomposition of formulas into subformulas

HL – A hypersequent system for Lukasiewicz logic: Initial sequents: A ⊢ A (ID)

⊢ (EMPTY )

⊥, Γ ⊢ A (⊥, l)

Logical rules: B, Γ ⊢ ∆, A | Γ ⊢ ∆ | H (⊃, l) A ⊃ B, Γ ⊢ ∆ | H

A, Γ ⊢ ∆, B | H Γ⊢∆|H (⊃, r ) Γ ⊢ ∆, A ⊃ B | H

A, Γ ⊢ ∆ | B, Γ ⊢ ∆ | H (∧, l) A ∧ B, Γ ⊢ ∆ | H

Γ ⊢ ∆, A | H Γ ⊢ ∆, B | H (∧, r ) Γ ⊢ ∆, A ∧ B | H

A, B, Γ ⊢ ∆ | H ⊥, Γ ⊢ ∆ | H ( & , l) A & B, Γ ⊢ ∆ | H

Γ ⊢ ∆, A, B | Γ ⊢ ∆, ⊥ | H ( & , r) Γ ⊢ ∆, A & B | H

Structural rules: H (EW ) Γ⊢∆|H

Γ⊢∆|Γ⊢∆|H (EC ) Γ⊢∆|H

Γ1 , Γ2 ⊢ ∆1 , ∆2 | H (SPLIT ) Γ1 ⊢ ∆2 | Γ2 ⊢ ∆1 | H

Γ⊢∆|H (IW ) A, Γ ⊢ ∆ | H

Γ1 ⊢ ∆1 | H Γ2 ⊢ ∆2 | H (MIX ) Γ1 , Γ2 ⊢ ∆1 , ∆2 | H

HL has nice properties: ◮

sound and complete for L

◮

(potentially much shorter, but hard to find) proofs using H | Γ1 ⊢ ∆ 1 , A A, Γ2 ⊢ ∆2 | H (CUT ) Γ 1 , Γ2 ⊢ ∆ 1 , ∆ 2 | H can be stepwise transformed into cut-free proofs [CM03]

◮

applications of structural rules – trivially: except (EW) – can be limited to atomic hypersequents

◮

the ‘purely logical’ version of HL reduces all complex hypersequents to atomic hypersequents, for which validity can be checked in PTIME

HL has nice properties: ◮

sound and complete for L

◮

(potentially much shorter, but hard to find) proofs using H | Γ1 ⊢ ∆ 1 , A A, Γ2 ⊢ ∆2 | H (CUT ) Γ 1 , Γ2 ⊢ ∆ 1 , ∆ 2 | H can be stepwise transformed into cut-free proofs [CM03]

◮

applications of structural rules – trivially: except (EW) – can be limited to atomic hypersequents

◮

the ‘purely logical’ version of HL reduces all complex hypersequents to atomic hypersequents, for which validity can be checked in PTIME

Nevertheless: is HL a really convincing analysis of actual reasoning?!

Dialogue games and the meaning of connectives Lorenzen’s idea: The meaning of a logical connective is given by the following dialogue game rules:

Dialogue games and the meaning of connectives Lorenzen’s idea: The meaning of a logical connective is given by the following dialogue game rules: Let X/Y stand for me/you or for you/me X asserts A⊃B

‘attack’ by Y A

answer by X B

Dialogue games and the meaning of connectives Lorenzen’s idea: The meaning of a logical connective is given by the following dialogue game rules: Let X/Y stand for me/you or for you/me X asserts A⊃B A∨B

‘attack’ by Y A ‘?’

answer by X B A or B (X chooses)

Dialogue games and the meaning of connectives Lorenzen’s idea: The meaning of a logical connective is given by the following dialogue game rules: Let X/Y stand for me/you or for you/me X asserts A⊃B A∨B A∧B A&B

‘attack’ by Y A ‘?’ ‘l?’ or ‘r?’ (Y chooses) ‘?’

answer by X B A or B (X chooses) A or B (accordingly) A and B

Dialogue games and the meaning of connectives Lorenzen’s idea: The meaning of a logical connective is given by the following dialogue game rules: Let X/Y stand for me/you or for you/me X asserts A⊃B A∨B A∧B A&B ∀x A(x) ∃x A(x)

‘attack’ by Y A ‘?’ ‘l?’ or ‘r?’ (Y chooses) ‘?’ ’t?’ (Y chooses) ‘?’

Note: ¬A abbreviates A ⊃ ⊥. The assertion ‘⊥’ is always false.

answer by X B A or B (X chooses) A or B (accordingly) A and B A(t) A(t) (X chooses)

The rules in sequent-style format State of the game: [A1 , . . . , An B1 , . . . , Bm ] I assert B1 , . . . , Bm , while you assert A1 , . . . , An

The rules in sequent-style format State of the game: [A1 , . . . , An B1 , . . . , Bm ] I assert B1 , . . . , Bm , while you assert A1 , . . . , An The rules from my point of view (for brevity, only ⊃ and & ): [B, Γ ∆, A] [A ⊃ B, Γ ∆] [A, B, Γ ∆] [A & B, Γ ∆]

(⊃, me) ( & , me)

[A, Γ ∆, B] [Γ ∆, A ⊃ B] [Γ ∆, A, B] [Γ ∆, A & B]

(⊃, you) ( & , you)

Note: the labels refer to the attacking player

The rules in sequent-style format State of the game: [A1 , . . . , An B1 , . . . , Bm ] I assert B1 , . . . , Bm , while you assert A1 , . . . , An The rules from my point of view (for brevity, only ⊃ and & ): [B, Γ ∆, A] [A ⊃ B, Γ ∆] [A, B, Γ ∆] [A & B, Γ ∆]

(⊃, me) ( & , me)

[A, Γ ∆, B]

(⊃, you)

[Γ ∆, A ⊃ B] [Γ ∆, A, B] [Γ ∆, A & B]

( & , you)

Note: the labels refer to the attacking player ◮

complex statements are decomposed exactly once

◮

no ‘hedging’ or ‘refuse to attack’ is allowed

◮

arbitrary states are reduced to atomic states

◮

no winning conditions formulated yet!

Dialogues as evaluation games NB: If we add an evaluation function – assigning real numbers to atomic states – to the dialogue rules we obtain an evaluation game

Dialogues as evaluation games NB: If we add an evaluation function – assigning real numbers to atomic states – to the dialogue rules we obtain an evaluation game A simple, but interesting example:

Dialogues as evaluation games NB: If we add an evaluation function – assigning real numbers to atomic states – to the dialogue rules we obtain an evaluation game A simple, but interesting example: 1. assign an arbitrary pay-off value v (p) ∈ R to each atom p

Dialogues as evaluation games NB: If we add an evaluation function – assigning real numbers to atomic states – to the dialogue rules we obtain an evaluation game A simple, but interesting example: 1. assign an arbitrary pay-off value v (p) ∈ R to each atom p P P 2. define v ([p1 , . . . , pn q1 , . . . , qm ]) = i v (pi ) − j v (qj )

Dialogues as evaluation games NB: If we add an evaluation function – assigning real numbers to atomic states – to the dialogue rules we obtain an evaluation game A simple, but interesting example: 1. assign an arbitrary pay-off value v (p) ∈ R to each atom p P P 2. define v ([p1 , . . . , pn q1 , . . . , qm ]) = i v (pi ) − j v (qj ) 3. =⇒ finite 2-person game with perfect information: guaranteed minimal pay-off for me can be calculated using induction following the max-min strategy for finite game trees

Dialogues as evaluation games NB: If we add an evaluation function – assigning real numbers to atomic states – to the dialogue rules we obtain an evaluation game A simple, but interesting example: 1. assign an arbitrary pay-off value v (p) ∈ R to each atom p P P 2. define v ([p1 , . . . , pn q1 , . . . , qm ]) = i v (pi ) − j v (qj ) 3. =⇒ finite 2-person game with perfect information: guaranteed minimal pay-off for me can be calculated using induction following the max-min strategy for finite game trees The resulting logic is Slaney’s Abelian logic (which coincides with one of Casari’s logic of comparison): ◮

‘truth value set’ is R

◮

truth function for conjunction: addition

◮

truth function for implication: subtraction

◮

validity: value ≥ 0 under all assignments

Dialogues as evaluation games (ctd.) To obtain Lukasiewicz logic we have to do three things:

Dialogues as evaluation games (ctd.) To obtain Lukasiewicz logic we have to do three things: (1) restrict to v (p) ∈ [0, 1] for atoms p; v (⊥)=0 (2) allow refusion to attack (no player is forced to attack) (3) allow hegding of maximal loss: instead of defending my(your) assertion I(you) can replace it by asserting ⊥ A simplification: (2) is only relevant for implication (⊃). (3) is only relevant for strong conjunction ( & ).

Dialogues as evaluation games (ctd.) To obtain Lukasiewicz logic we have to do three things: (1) restrict to v (p) ∈ [0, 1] for atoms p; v (⊥)=0 (2) allow refusion to attack (no player is forced to attack) (3) allow hegding of maximal loss: instead of defending my(your) assertion I(you) can replace it by asserting ⊥ A simplification: (2) is only relevant for implication (⊃). (3) is only relevant for strong conjunction ( & ). The resulting rules are: [B, Γ ∆, A] [A ⊃ B, Γ ∆] [A, B, Γ ∆]

(⊃, me) [⊥, Γ ∆]

[A & B, Γ ∆]

[Γ ∆] [A ⊃ B, Γ ∆] ( & , me)

(⊃, me)

[A, Γ ∆, B] [Γ ∆, A & B]

[A, Γ ∆, B]

[Γ ∆]

[Γ ∆, A ⊃ B] ( & , you)

(⊃, you)

[Γ ∆, ⊥] [Γ ∆, A & B]

( & , you)

What is the relation to Giles’s game? Robin Giles [A non-classical logic for physics, 74/79] talks about: ◮

payments to the opponent for each false assertion

◮

dispersive experiments that decide about the truth/falsity of atomic assertions

◮

probabilities associated with experiments

◮

minimizing risk (expected amount of payments)

It seems that we lost the connection to Giles’s approach!?!

What is the relation to Giles’s game? Robin Giles [A non-classical logic for physics, 74/79] talks about: ◮

payments to the opponent for each false assertion

◮

dispersive experiments that decide about the truth/falsity of atomic assertions

◮

probabilities associated with experiments

◮

minimizing risk (expected amount of payments)

It seems that we lost the connection to Giles’s approach!?! However: Giles’s story about dispersive experiments etc. is only a proposal to attach tangible meaning to v (p) and to v ([p1 , . . . , pn q1 , . . . , qm ])

Giles’s original motivation: formalize reasoning in physics. An elementary (yes/no) experiment E may show dispersion; i.e., may not always have the same outcome on every trial. Fixed success probabilities π(E ) are associated with experiments.

Giles’s original motivation: formalize reasoning in physics. An elementary (yes/no) experiment E may show dispersion; i.e., may not always have the same outcome on every trial. Fixed success probabilities π(E ) are associated with experiments. Probabilities obtain tangible meaning ‘á la Finetti’: ‘I assert that q is at least as probable as p’ means: ‘I am willing to pay you 1¿ if experiment deciding q fails, if you pay me 1¿ if the experiment for p fails.’ =⇒ Players (you and me) try to minimize their (perceived) risk: ‘Winning the game’ now means: no expected loss at the end.

Giles’s original motivation: formalize reasoning in physics. An elementary (yes/no) experiment E may show dispersion; i.e., may not always have the same outcome on every trial. Fixed success probabilities π(E ) are associated with experiments. Probabilities obtain tangible meaning ‘á la Finetti’: ‘I assert that q is at least as probable as p’ means: ‘I am willing to pay you 1¿ if experiment deciding q fails, if you pay me 1¿ if the experiment for p fails.’ =⇒ Players (you and me) try to minimize their (perceived) risk: ‘Winning the game’ now means: no expected loss at the end. MyP expected loss P for [p1 , . . . , pn q1 , . . . , qm i] can be calculated to be i hqi i − j hpj i¿, where hpi is short for the risk associated with the corresponding experiment Ep : hpi = 1 − π(Ep ) P P NB: this just inverts i v (pi ) − j v (qj ) minimizing expected payments to you corresponds to maximizing v

From evaluation games to hypersequent systems Giles’s game – and its variants – are evaluation games, i.e., interactive forms of determining truth values (Giles: risk values), given particular assignments. While the rules can be presented in sequent format we still seem to be far from a hypersequent calculus like HL for checking validity.

From evaluation games to hypersequent systems Giles’s game – and its variants – are evaluation games, i.e., interactive forms of determining truth values (Giles: risk values), given particular assignments. While the rules can be presented in sequent format we still seem to be far from a hypersequent calculus like HL for checking validity. However, there is a generic way to turn (finite, finitely-branching) evaluation games into games for checking validity: Keep all choices available: states −→ disjunctive states

From evaluation games to hypersequent systems Giles’s game – and its variants – are evaluation games, i.e., interactive forms of determining truth values (Giles: risk values), given particular assignments. While the rules can be presented in sequent format we still seem to be far from a hypersequent calculus like HL for checking validity. However, there is a generic way to turn (finite, finitely-branching) evaluation games into games for checking validity: Keep all choices available: states −→ disjunctive states Resulting disjunctive strategies can be seen as – either referring to a generalized, parallel version of the game – or simply a bookkeeping device that collects all relevant ordinary strategies into one combined structure (tree)

From evaluation games to hypersequent systems Giles’s game – and its variants – are evaluation games, i.e., interactive forms of determining truth values (Giles: risk values), given particular assignments. While the rules can be presented in sequent format we still seem to be far from a hypersequent calculus like HL for checking validity. However, there is a generic way to turn (finite, finitely-branching) evaluation games into games for checking validity: Keep all choices available: states −→ disjunctive states Resulting disjunctive strategies can be seen as – either referring to a generalized, parallel version of the game – or simply a bookkeeping device that collects all relevant ordinary strategies into one combined structure (tree) Evaluation of atomic disjunctive states: winning means: at least one component state is winning (for me)

From strategies to disjunctive strategies Suppose players me and you have the following choices: or (my choice)

S0I

S0I

S1You S3

S2You

S4

S6 .

S5

corresponding disjunctive strategy: S0I

S0I W

S1You

S0I W I S0

W I S3 S0 W You S3 S2 S3

W

S5

S3

W

W I S4 S0 W You S4 S2 S6

S4

W

S5

S4

W

S6 .

Disjunctive winning strategy for (p ⊃ q) ∨ (q ⊃ p) [ (p ⊃ q) ∨ (q ⊃ p)]You [ (p ⊃ q) ∨ (q ⊃ p)]You [ (p ⊃ q) ∨ (q ⊃ p)]I [ p ⊃ q]You

W

[ p ⊃ q]You

W

W

[ (p ⊃ q) ∨ (q ⊃ p)]You

[ (p ⊃ q) ∨ (q ⊃ p)]You

[ (p ⊃ q) ∨ (q ⊃ p)]You

W

[ (p ⊃ q) ∨ (q ⊃ p)]I

[ p ⊃ q]You

W

[ q ⊃ p]You

[ p ⊃ q]You

W

[ q ⊃ p]You

[p q]

W

[ q ⊃ p]You

[ ]

W

[ q ⊃ p]You

[p q]

W

[ q ⊃ p]You

[ ]

W

[ q ⊃ p]You

[p q]

W

[q p]

[p q]

W

[ ]

[ ]

W

[q p]

[ ]

W

[ ].

Disjunctive game rules are hypersequent rules! Rules of the disjunctive game: W [Γ ∆] H (⊃, l) W [A ⊃ B, Γ ∆] H

[B, Γ ∆, A]

W

W [B, Γ ∆] H (∧, l) W [A ∧ B, Γ ∆] H

[A, Γ ∆]

[A, B, Γ ∆]

W

W

H

[⊥, Γ ∆] W [A & B, Γ ∆] H

W

H

[A, Γ ∆, B]

W

W

H

[Γ ∆] W [Γ ∆, A ⊃ B] H

[Γ ∆, A]

( & , l)

H

[Γ ∆, B] W [Γ ∆, A ∧ B] H

[Γ ∆, A, B]

W

W

H

W

H

[Γ ∆, ⊥] W [Γ ∆, A & B] H

W

(⊃, r )

(∧, r )

H

( & , r)

Disjunctive game rules are hypersequent rules! Logical rules of HL: B, Γ ⊢ ∆, A | Γ ⊢ ∆ | H (⊃, l) A ⊃ B, Γ ⊢ ∆ | H

A, Γ ⊢ ∆, B | H Γ⊢∆|H (⊃, r ) Γ ⊢ ∆, A ⊃ B | H

A, Γ ⊢ ∆ | B, Γ ⊢ ∆ | H (∧, l) A ∧ B, Γ ⊢ ∆ | H

Γ ⊢ ∆, A | H Γ ⊢ ∆, B | H (∧, r ) Γ ⊢ ∆, A ∧ B | H

A, B, Γ ⊢ ∆ | H ⊥, Γ ⊢ ∆ | H ( & , l) A & B, Γ ⊢ ∆ | H

Γ ⊢ ∆, A, B | Γ ⊢ ∆, ⊥ | H ( & , r) Γ ⊢ ∆, A & B | H

What happened to structural rules and initial sequents? Initial sequents:

A ⊢ A (ID)

Structural rules: H (EW ) Γ⊢∆|H Γ 1 , Γ2 ⊢ ∆ 1 , ∆ 2 | H Γ1 ⊢ ∆ 2 | Γ2 ⊢ ∆ 1 | H

⊢ (EMPTY )

Γ⊢∆|Γ⊢∆|H Γ⊢∆|H (SPLIT )

(EC )

Γ1 ⊢ ∆ 1 | H

⊥, Γ ⊢ A (⊥, l) Γ⊢∆|H A, Γ ⊢ ∆ | H Γ2 ⊢ ∆ 2 | H

Γ 1 , Γ2 ⊢ ∆ 1 , ∆ 2 | H

(IW ) (MIX )


A ⊢ A (ID)


⊢ (EMPTY )


(EC )

Γ1 ⊢ ∆ 1 | H

⊥, Γ ⊢ A (⊥, l) Γ⊢∆|H A, Γ ⊢ ∆ | H Γ2 ⊢ ∆ 2 | H

Γ 1 , Γ2 ⊢ ∆ 1 , ∆ 2 | H

(IW ) (MIX )

Remember: the structural rules of HL are only needed at the atomic level. (For proving sequents (EW) is redundant.) If we are satisfied with more complex initial sequents then the structural rules are redundant!


A ⊢ A (ID)


⊢ (EMPTY )


(EC )

Γ1 ⊢ ∆ 1 | H

⊥, Γ ⊢ A (⊥, l) Γ⊢∆|H A, Γ ⊢ ∆ | H Γ2 ⊢ ∆ 2 | H

Γ 1 , Γ2 ⊢ ∆ 1 , ∆ 2 | H

(IW ) (MIX )

Remember: the structural rules of HL are only needed at the atomic level. (For proving sequents (EW) is redundant.) If we are satisfied with more complex initial sequents then the structural rules are redundant! Should we be satisfied with complex initial sequents?


A ⊢ A (ID)


⊢ (EMPTY )


(EC )

Γ1 ⊢ ∆ 1 | H

⊥, Γ ⊢ A (⊥, l) Γ⊢∆|H A, Γ ⊢ ∆ | H Γ2 ⊢ ∆ 2 | H

Γ 1 , Γ2 ⊢ ∆ 1 , ∆ 2 | H

(IW ) (MIX )

Remember: the structural rules of HL are only needed at the atomic level. (For proving sequents (EW) is redundant.) If we are satisfied with more complex initial sequents then the structural rules are redundant! Should we be satisfied with complex initial sequents? In this case: yes! Reason: it can be checked in PTIME whether a given atomic hypersequent is valid or not.

Summary and Conclusion

Summary and Conclusion ◮

Analytic (‘Gentzen style’) proof systems are needed for effective proof search, but also for analyzing reasoning within a logic like Lukasiewicz logic L.



◮

Hypersequents enable useful analytic systems, but are problematic as formal models of reasoning.



◮


◮

Dialogue games, like Giles’s for L, model reasoning from first principles, but seem only to refer to truth evaluation.



◮


◮

Dialogue games, like Giles’s for L, model reasoning from first principles, but seem only to refer to truth evaluation.

◮

We have shown: Constructing disjunctive strategies for Giles-style games corresponds directly to logical hypersequent rules. Structural rules are only needed to reduce valid atomic hypersequents into simple sequents.

References ◮

C.F., G. Metcalfe: Giles’s Game and the Proof Theory of Lukasiewicz Logic. Studia Logica 92(1): 27-61 (2009).

◮

R. Giles: A non-classical logic for physics. In: R. Wojcicki, G. Malinkowski (eds.) Selected Papers on Lukasiewicz Sentential Calculi. 1977, 13-51 Short version in: Studia Logica 4(33): 399-417 (1974).

◮

A. Ciabattoni, C.F., G. Metcalfe: Uniform Rules and Dialogue Games for Fuzzy Logics. LPAR 2004, Springer LNAI 3452 (2005), 496-510.

◮

C.F.: Revisiting Giles’s Game – Reconciling Fuzzy Logic and Supervaluation. In: Games: Unifying logic, Language, and Philosophy, O. Majer et.al. (eds.), LEUS 15, Springer, 209-228 (2009).

◮

C.F.: Dialogue Games for Many-Valued Logics – an Overview. Studia Logica 90(1): 43-68 (2008).