From Axioms to Rules — a Coalition of Fuzzy, Linear and Substructural Logics

Home , Hypersequent, Structural rule, Substructural logic

From Axioms to Rules — A Coalition of Fuzzy, Linear and Substructural Logics Kazushige Terui National Institute of Informatics, Tokyo Laboratoire d’Informatique de Paris Nord (Joint work with Agata Ciabattoni and Nikolaos Galatos)

Genova, 21/02/08 – p.1/?? Parties in Nonclassical Logics

Modal Logics Default Logic Intermediate Logics (Padova) Basic Logic Paraconsistent Logic

Linear Logic Fuzzy Logics

Substructural Logics

Genova, 21/02/08 – p.2/?? Parties in Nonclassical Logics

Modal Logics Default Logic Intermediate Logics (Padova) Basic Logic Paraconsistent Logic

Linear Logic Fuzzy Logics

Substructural Logics

Our aim: Fruitful coalition of the 3 parties

Genova, 21/02/08 – p.2/?? Basic Requirements

Substractural Logics: Algebraization

µ Ä ´ Î

Genova, 21/02/08 – p.3/?? Basic Requirements

Substractural Logics: Algebraization

µ Ä ´ Î

Fuzzy Logics: Standard Completeness

½ ¼

µ Ä ´ Ã

Genova, 21/02/08 – p.3/?? Basic Requirements

Substractural Logics: Algebraization

µ Ä ´ Î

Fuzzy Logics: Standard Completeness

½ ¼

µ Ä ´ Ã

Linear Logic: Cut Elimination

Genova, 21/02/08 – p.3/?? Basic Requirements

Substractural Logics: Algebraization

µ Ä ´ Î

Fuzzy Logics: Standard Completeness

½ ¼

µ Ä ´ Ã

Linear Logic: Cut Elimination A logic without cut elimination is like a car without engine (J.-Y. Girard)

Genova, 21/02/08 – p.3/?? Outcome

We classify axioms in Substructural and Fuzzy Logics according to the Substructural Hierarchy, which is deﬁned based on Polarity (Linear Logic).

Genova, 21/02/08 – p.4/?? Outcome

We classify axioms in Substructural and Fuzzy Logics according to the Substructural Hierarchy, which is deﬁned based on Polarity (Linear Logic). Give an automatic procedure to transform axioms up to level

¿ ( , in the absense of Weakening) into Hyperstructural

¼ Rules in Hypersequent Calculus (Fuzzy Logics).

Genova, 21/02/08 – p.4/?? Outcome

¿ ( , in the absense of Weakening) into Hyperstructural

¼ Rules in Hypersequent Calculus (Fuzzy Logics). Give a uniform, semantic proof of cut-elimination via DM completion (Substructural Logics)

Genova, 21/02/08 – p.4/?? Outcome

¿ ( , in the absense of Weakening) into Hyperstructural

¼ Rules in Hypersequent Calculus (Fuzzy Logics). Give a uniform, semantic proof of cut-elimination via DM completion (Substructural Logics) To sum up: Every system of substructural and fuzzy logics

¿ deﬁned by axioms (acyclic , in the absense of

¼ Weakening) admits a cut-admissible hypersequent calculus.

Genova, 21/02/08 – p.4/?? Kouan 1: Why uniformity?

The Vienna Group once wrote a META-paper (a paper generator) which

Genova, 21/02/08 – p.5/?? Kouan 1: Why uniformity?

The Vienna Group once wrote a META-paper (a paper generator) which

given a sequent calculus Ä as input

Genova, 21/02/08 – p.5/?? Kouan 1: Why uniformity?

The Vienna Group once wrote a META-paper (a paper generator) which

given a sequent calculus Ä as input generates a paper (with introduction and reference) that

proves the cut elimination theorem for Ä .

Genova, 21/02/08 – p.5/?? Kouan 1: Why uniformity?

The Vienna Group once wrote a META-paper (a paper generator) which

given a sequent calculus Ä as input generates a paper (with introduction and reference) that

proves the cut elimination theorem for Ä . The generated paper was submitted, and accepted.

Genova, 21/02/08 – p.5/?? Outline

1. Preliminary: Commutative Full Lambek Calculus Ä and Commutative Residuated Lattices 2. Background: Key concepts in Substructural, Fuzzy and Linear Logics 3. Substructural Hierarchy 4. From Axioms to Rules, Uniform Semantic Cut Elimination 5. Conclusion

Genova, 21/02/08 – p.6/?? SyntaxÁÅÄÄ of Ä

Formulas : Æ , ² , , , ½ , , , ¼ .

Sequents¥ :

( : multiset of formulas, ¥ : stoup with at most one formula) Inference rules:

¥ ¡

ÙØ

ÒØ ØÝ Á

¥ ¡

¥ ¡ Æ Æ

Ð Æ

Ö Æ

Genova, 21/02/08 – p.7/?? ¥ ¥

SyntaxÁÅÄÄ of Ä

¾ ½

¥ ¥ ¼

Ð ¼

¾ ½

¥ ² ²

Ð ² Ö ² Ö

¥ ½ ½

Ö ½ Ð ½

Ð Ö

Notational Correspondence

ÓÒ Å ÁÑÔ Å ÓÒ ×

ÖÄ ÄÓ Ä ÒÖ ½ Æ ² ¼

ÙÞÞÝÄÓ ×

Genova, 21/02/08 – p.8/?? Commutative Residuated Lattices

A (bounded pointed) commutative residuated lattice is

½ ¼ Æ ² È

¼ ² 1. is a lattice with greatest and ¼ least

½ 2. is a commutative monoid.

3.Æ For any ,

4. .

È µ ´ if ½ for any valuation .

: the variety of commutative residuated lattices.

Genova, 21/02/08 – p.9/?? Algebraization

A (commutative) substructural logic Ä is an extension of Ä

È È µ Ä ´

Ä with axioms ¨ .

µ Ä ´ : the subvariety of corresponding to Ä

Ä for ¨ any

Theorem: For every substructural logic Ä ,

µ Ä ´ Î

Genova, 21/02/08 – p.10/?? Algebraization

Is it trivial? Yes, but the consequences are not. Syntax: existential

Ä ´ is a proof ofµ

Semantics: universal

µ Ä ´ Î

µ È ´ µ Ä ´ È

Consequence: Semantics mirrors Syntax

Syntax Semantics

Argument Argument Property Property

Genova, 21/02/08 – p.11/?? Interporation and Amalgamation

µ ´ : formulas over variables .

Ä admits Interpolation:µ ´ Suppose and

Ä µ ´ Æ .If µ , then ´ there is such that

Ä Ä

Æ Æ

A class of algebras admits Amalgamation if for any

Ô Ê

¾ ½

with embeddingsÔ , there are and

¾ ½ embeddings such that

½ ½

Genova, 21/02/08 – p.12/?? Interporation and Amalgamation

Theorem (Maximova): For any intermediate logic Ä ,

Ä admits interpolationµ Ä ´ admits amalgamation

Extended to substructural logics by Wronski,´ Kowalski, Galatos-Ono, etc. Syntax is to split, semantics is to join.

Genova, 21/02/08 – p.13/?? Basic Requirements

Substractural Logics: Algebraization

µ Ä ´ Î

Fuzzy Logics: Standard Completeness

½ ¼

µ Ä ´ Ã

Linear Logic: Cut Elimination

Genova, 21/02/08 – p.14/?? Linearization

Logically, this amounts to adding the axiom of linearity:

µ Æ ´

µ ´

µ Æ ´ µ Æ ´ µ ´

µµ ´ µ ´ ´ µ ´

Example: Gödelµ logic ´ · = ÁÄ

µµ ´ µ ´ ´ µ ² ´

Complete w.r.t.½ the ¼ valuations s.t.

¼ µ ¼ ´

½ µ ´

µ ´ µ if´ otherwise

Genova, 21/02/08 – p.15/?? Linearization

Other Fuzzy Logics:

Uninormµ Logic ´ · = Ä

µµ Æ ´ ´ Æ µ ² ´ µ ´

Monoidal T-normµ Logic ´ · = ÄÛ

½ ² ½ ²

µ Æ ´ µ Æ ´ µ ´

µ Basic ´ · µ Logic ´ · = ÄÛ

Æ Łukasiewicz´ · µ ´ · µ Logic ´ · = ÄÛ

If axioms are added, cut elimination is lost. We need to ﬁnd corresponding rules.

Genova, 21/02/08 – p.16/?? Hypersequent Calculus

Hypersequent calculus (Avron 87)

Ò Ò ½ ½ ¥ Hypersequent: ¥

½ ² ½ ²

Ò Ò ½ ½ µ ¥ Æ ´ µ Intuition:¥ Æ ´

ÀÄ consists of

Rules of Ä Ext-Weakening Ext-Contraction

Æ ¥ ¥

¥ ¥ Communication Rule:

¾ ¾ ¾ ½ ½ ½

¥ ¡ ¥ ¡

¾ ¾ ½ ½ ½ ¾

¥ ¡ ¥ ¡

µ ´

Genova, 21/02/08 – p.17/?? Hypersequent Calculus

µ ´ · ÀÄ provesµ ´ .

¾ ¾ ¾ ½ ½ ½

¥ ¡ ¥ ¡

¾ ¾ ½ ½ ½ ¾

¥ ¡ ¥ ¡

µ ´

Æ Æ

µ Æ ´

µ Æ ´ µ Æ ´ µ Æ ´ µ Æ ´

µ ´

µ Æ ´ µ Æ ´

µ ´

µ ·´ ÀÁÄ = Gödel Logic. Enjoys cut elimination (Avron 92). Similarly for Monoidal T-norm and Uninorm Logics (cf. Metcalfe-Montagna 07) Semantics is to narrow, Syntax is to widen.

Genova, 21/02/08 – p.18/?? Basic Requirements

Substractural Logics: Algebraization

µ Ä ´ Î

Fuzzy Logics: Standard Completeness

½ ¼

µ Ä ´ Ã

Linear Logic: Cut Elimination

Genova, 21/02/08 – p.19/?? Conservativity

Á ¾ Inﬁnitary extension Ä of Ä : ²

for any for some

Á ¾ Á ¾

² ¥ ²

Ä is a conservative extension of Ä if

Ä Ä

for any set of ﬁnite formulas.

Theorem: For any substructural logic Ä , Ä is a conservative

extension of Ä µ Ä iff´ any È can be embedded into a

completeµ Ä algebra´ È .

½ Syntax is to eliminate, Semantics is to enrich.

Genova, 21/02/08 – p.20/?? Dedekind Completion of Rationals

ForÉ any ,

is closed if

µ · É ´ can be embeddedµ · µ intoÉ ´ ´ with

É µ É ´ is closed

Dedekind completion extends to various ordered algebras (MacNeille).

Genova, 21/02/08 – p.21/?? ½ ½

Dedekind-MacNeille Completion

µ ´

Theorem: Every È can be embedded into a complete

µ ½ ¼ Æ È ² µ È ´ , where´ È

½ ½

(Ono 93; cf. Abrusci 90, Sambin 93) Some axioms (eg. distributivity) are not preserved by DM completion (cf. Kowalski-Litak 07). Genova, 21/02/08 – p.22/?? Cut Elimination via Completion

Syntactic argument: elimination procedure

Cut-ful Proofs Cut-free Proofs Semantic argument: Quasi-DM completion

CRL ‘Intransitive’ CRL

Genova, 21/02/08 – p.23/?? Cut Elimination via Completion

Due to (Okada 96). Algebraically reformulated by (Belardinelli-Ono-Jipsen 01).

the set of multisets ¡ of formulas

the set of sequents¥ ¦

For ¥ and ¦ ,

¥ ¦ ¥ ¦ iff is cut-free provable in Ä

For and ,

¥µ ´¦ ¥µ ´¦

Genova, 21/02/08 – p.24/?? Cut Elimination via Completion

induce a complete CRL È , and

µ ´ closed

is a quasi-homomorphismµ È ´ :

µ ´ µ ´ µ ´

µ ´ ¼ ½ for

² Æ for

If the valuation µ ´ validates , then is cut-free

provable in Ä . Again, not all axioms are preserved by quasi-DM completion. Which axioms are preserved by quasi-DM completion?

Genova, 21/02/08 – p.25/?? Kouan 2: What is Completeness?

Completeness: to establish a correspondence between syntax and semantics.

µ Ä ´ Î

Ä Gödel Completeness: is trivial in the algebraic setting. Meta-Completeness: to describe the Syntax-Semantics Mirror correspondence as precisely as possible. Syntax Semantics Interpolation Amalgamation Hypersequentialization Linearization Conservativity Completion Cut-elimination Quasi-DM completion

Genova, 21/02/08 – p.26/?? Kouan 3: Is Semantic Cut-Elimination Weak?

Proof of Cut-Elimination

sound complete

È È

We have just replaced object cuts with a big META-CUT. Another criticism: it does not give a cut-elimination procedure. Conjecture: If we eliminate the meta-cut, a concrete (object) cut-elimination procedure emerges.

Genova, 21/02/08 – p.27/?? When Cut-Elimination Holds?

Cut-Elimination holds when Ä is extended with a natural structural rule:

Contraction: Æ ¥

¥ It fails when extended with an unnatural one:

Broccoli:Æ ¥

¥ What is ‘natural’?

Genova, 21/02/08 – p.28/?? Girard’s test

(Girard 99) proposes a test for naturality of structural rules. - A structural rule passes Girard’s test if, in every ‘phase

structure’µ ´ ,itpropagates from atomic closed

sets to all closed sets .

“If the rule holds for rationals, it also holds for all reals.” Contraction passes it:

µ closed ´

Broccoli fails it:

µ closed ´

Genova, 21/02/08 – p.29/?? Irony

¾ ½

¥ ¡ ¦ ¥ ¡ ¦

Broccoli is equivalent to Æ Mingle:

¾ ½

¥ ¡ ¦ ¦

Mingle passes Girard’s test.

Genova, 21/02/08 – p.30/?? Broccoli and Mingle

Broccoli does not admit cut elimination:

¬ ¨ « µ ¬ ¨ «

« µ «

¬ µ ¬ ¬ ¬ ¨ « ¨ « µ µ « ¬ ¨ « ¬ ¨ «

ÜÔ

¬ ¨ « µ ¬ ¬ ¨ « µ ¬ ¨ « «

ÙØ

¬ ¨ « µ ¬ «

ÙØ

When Broccoli is replaced with Mingle, the above cut can be eliminated:

¬ µ ¬

« µ «

¬ ¨ « µ « ¬ ¨ « µ ¬

¬ ¨ « µ ¬ «

ÅÒ

Genova, 21/02/08 – p.31/?? Characterization of Cut-Elimination

Additive structural rules:

Ò ½

¡ ¡

¼ Ò ½ such that .

Theorem (Terui 07): For any additive structural ruleµ ´ ,

µ ´ · Ä admits cut-elimination iffµ ´ passes Girard’s test.

Key fact:µ ´ passes Girard’sµ test´ is preserved by (quasi-)DM completion. (Ciabattoni-Terui 06) considerably extends this result.

Genova, 21/02/08 – p.32/?? Questions

Which axiom can be transformed into structural rules? Usually checked one by one. Is there a more systematic way? Which structural rules can be transformed into good ones

admitting cut-elimination? (eg. Broccoli Mingle) How does the situation change if we adopt hypersequent

calculus? (eg. Linearity Communication) Does hypersequent calculus admit semantic cut-elimination? All known proofs are syntactic, tailored to each speciﬁc logic (exception: Metcalfe-Montagna 07), and quite complicated. Semantic one would lead to a uniform, conceptually simpler proof.

(Ciabattoni-Galatos-Terui 08)

Genova, 21/02/08 – p.33/?? Outline

Genova, 21/02/08 – p.34/?? Polarity

Key notion in Linear Logic since its inception (Girard 87) Plays a central role in Efﬁcient Proof Search (Andreoli 90), Constructive Classical Logic (Girard 91), Polarized Linear Logic (Laurent), Game Semantics, Ludics.

Genova, 21/02/08 – p.35/?? Polarity

Positive connectives ¼ ½ have invertible left rules:

and propagate closure operator:

µ ´

Negative connectivesÆ ² have invertible right rules:

¾ ½

and distribute closure operator:

µ ´

Genova, 21/02/08 – p.36/?? Polarity

Connectives of the same polarity associate well.

¼ ¼ ¼ Positives: ½

µ ´ µ ´ µ ´

Æ ½ Æ ²

Negatives:

µ Æ ´ ² µ Æ ´ Æ µ ´

µ Æ ´ ² µ Æ ´ µ ² ´ Æ

(polarity reverses on the LHS of an implication)

Genova, 21/02/08 – p.37/?? ¼ ¼

½ ½ Substructural Hierarchy

¾ ¾

Ò Ò The sets of formulas deﬁned by:

¿ ¿

¼ ¼ (0) the set of atomic formulas

Ô Ô

·½ Ò Ò

Ô Ô

Ô Ô (P1)

Ô Ô

·½ Ò ·½ Ò

¼ ½ (P2)

·½ Ò Ò (N1)

·½ Ò ·½ Ò ² (N2)

·½ Ò ·½ Ò ·½ Ò

Æ (N3) Á

Genova, 21/02/08 – p.38/?? Substructural Hierarchy

Ò Due to lack of Weakening, is too strong. It is also

Ò convenient to consider a subclass :

·½ Ò

½ ²

·½ Ò ·½ Ò

¼ ½

¼ ¼ Intuition:

¼ : Formulas

½ : Disj of multisets

½ : Conj of sequents¥

Ò ½ : Conj of hypersequents

¾ : Conj of structural rules

: Conj of hyperstructural rules

Genova, 21/02/08 – p.39/?? Examples

Class Axiom Name

¾ ½ Æ « « Æ , weakening

« « Æ « contraction

« Æ « « expansion

« Æ « knotted axioms¼ Ñ (Ò )

Ñ Ò

µ « ² « ´ weak contraction

¾ « « excluded middle

µ « Æ ¬ ´ µ ¬ Æ « ´ linearity

µ ½ ² µ « Æ ¬ ´´ µ ½ ² µ ¬ Æ « ´´ prelinearity

« «

µ Ô Æ Ô ´ weak excluded middle

Ä Ä

Kripke models with width

Genova, 21/02/08 – p.40/?? Examples

¿ Axioms in :

µ Æ µ Æ Łukasiewicz´´ Æ µ Æ µ axiomÆ ´´

µµ ² ´ µ ² ´´ Æ Distributivityµµ ´ ² ´

µµ Æ ´ ´ DivisibilityÆ µ ² ´

Genova, 21/02/08 – p.41/?? From Axioms to Rules

A structural rule is

Ò Ò ½ ½

¼ ¼

where© § are sets of metavariables½ (© ). A hyperstructural rule is

½ ½

Ò Ò

¼ ¼ ¼ ¼

Ñ Ñ ½ ½

Theorem:

¾ 1. Any -axiom is equivalent to (a set of) structural rules in Ä .

2. Any -axiom is equivalent to hyperstructural rules in ÀÄ .

¿ 3. Any -axiom is equivalent to hyperstructural rules in ÀÄÛ .

Genova, 21/02/08 – p.42/?? Example

Weak nilpotent minimum (Esteva-Godo 01):

µ Æ ² ´ µ ´

½ ²

½ ² µ Æ ² Its ´ version:µ ´

¼ is equivalent to

Genova, 21/02/08 – p.43/?? Example

Weak nilpotent minimum (Esteva-Godo 01):

µ Æ ² ´ µ ´

½ ²

½ ² µ Æ ² Its ´ version:µ ´

¼ is equivalent to

Æ ² Æ µ ´

Genova, 21/02/08 – p.43/?? Example

Weak nilpotent minimum (Esteva-Godo 01):

µ Æ ² ´ µ ´

½ ²

½ ² µ Æ ² Its ´ version:µ ´

¼ is equivalent to

Æ ²

Genova, 21/02/08 – p.43/?? Example

Weak nilpotent minimum (Esteva-Godo 01):

µ Æ ² ´ µ ´

½ ²

½ ² µ Æ ² Its ´ version:µ ´

¼ is equivalent to

Æ ²

Genova, 21/02/08 – p.43/?? Example

Weak nilpotent minimum (Esteva-Godo 01):

µ Æ ² ´ µ ´

½ ²

½ ² µ Æ ² Its ´ version:µ ´

¼ is equivalent to

Æ ²

Genova, 21/02/08 – p.43/?? Example

Weak nilpotent minimum (Esteva-Godo 01):

µ Æ ² ´ µ ´

½ ²

½ ² µ Æ ² Its ´ version:µ ´

¼ is equivalent to

Genova, 21/02/08 – p.43/?? Example

Weak nilpotent minimum (Esteva-Godo 01):

µ Æ ² ´ µ ´

½ ²

½ ² µ Æ ² Its ´ version:µ ´

¼ is equivalent to

Genova, 21/02/08 – p.43/?? Example

Weak nilpotent minimum (Esteva-Godo 01):

µ Æ ² ´ µ ´

½ ²

½ ² µ Æ ² Its ´ version:µ ´

¼ is equivalent to

Genova, 21/02/08 – p.43/?? Example

Weak nilpotent minimum (Esteva-Godo 01):

µ Æ ² ´ µ ´

½ ²

½ ² µ Æ ² Its ´ version:µ ´

¼ is equivalent to

Genova, 21/02/08 – p.43/?? Example

Weak nilpotent minimum (Esteva-Godo 01):

µ Æ ² ´ µ ´

½ ²

½ ² µ Æ ² Its ´ version:µ ´

¼ is equivalent to

The procedure is automatic (in contrast to the usual practice). Similarity with principle of reﬂection (automatic derivation of inference rules for a logical connective from its deﬁning equation; Sambin-Battilotti-Faggian 00).

Genova, 21/02/08 – p.43/?? Towards Cut Elimination

Not all rules admit cut elimination. They have to be completed. In absence of Weakening, cyclic rules are problematic:

Theorem: 1. Any acyclic hyperstructural rule can be transformed into

an equivalent one in ÀÄ that enjoys cut elimination. 2. Any hyperstructural rule can be transformed into an

equivalent one in ÀÄÛ that enjoys cut elimination.

Genova, 21/02/08 – p.44/?? Towards Cut Elimination

Not all rules admit cut elimination. They have to be completed. In absence of Weakening, cyclic rules are problematic:

Theorem: 1. Any acyclic hyperstructural rule can be transformed into

an equivalent one in ÀÄ that enjoys cut elimination. 2. Any hyperstructural rule can be transformed into an

equivalent one in ÀÄÛ that enjoys cut elimination.

Genova, 21/02/08 – p.44/?? Example

µ Weak Æ nilpotent ² ´ minimum: µ ´

We have obtained:

It is equivalent to

Genova, 21/02/08 – p.45/?? Example

µ Weak Æ nilpotent ² ´ minimum: µ ´

We have obtained:

It is equivalent to

Genova, 21/02/08 – p.45/?? Example

µ Weak Æ nilpotent ² ´ minimum: µ ´

We have obtained:

It is equivalent to

Æ ¡

Genova, 21/02/08 – p.45/?? Example

µ Weak Æ nilpotent ² ´ minimum: µ ´

We have obtained:

It is equivalent to

£ ¡

Æ £ ¡

Genova, 21/02/08 – p.45/?? Example

µ Weak Æ nilpotent ² ´ minimum: µ ´

We have obtained:

It is equivalent to

¥ ¦ Æ £ ¡

¥ ¦ £ ¡

Genova, 21/02/08 – p.45/?? Example

µ Weak Æ nilpotent ² ´ minimum: µ ´

We have obtained:

It is equivalent to

£ ¡

¥ ¦

¥ ¦ £ ¡

Genova, 21/02/08 – p.45/?? Example

µ Weak Æ nilpotent ² ´ minimum: µ ´

We have obtained:

It is equivalent to

¥ ¦ £ £

¥ ¦ £ ¡

Genova, 21/02/08 – p.45/?? Example

µ Weak Æ nilpotent ² ´ minimum: µ ´

We have obtained:

It is equivalent to

¡ £

¥ ¦ £ ¥ ¦

¥ ¦ £ ¡

Genova, 21/02/08 – p.45/?? Example

µ Weak Æ nilpotent ² ´ minimum: µ ´

We have obtained:

It is equivalent to

¥ ¦ £ £ ¥ ¦ ¡ £

¥ ¦ £ ¥ ¦ ¡

¥ ¦ £ ¡

Genova, 21/02/08 – p.45/?? Uniform Cut Elimination

The resulting rule

¥ ¦ £ £ ¥ ¦ ¡ £

¥ ¦ £ ¥ ¦ ¡

¥ ¦ £ ¡

satisﬁes Strong subformula property: Any formula occuring on the LHS (resp. RHS) of a premise also occurs on the LHS (resp. RHS) of the conclusion Conclusion-linearity: No metavariable occurs in the conclusion twice. Coupling: ...

Genova, 21/02/08 – p.46/?? Uniform Cut Elimination

Theorem: If a hyperstructural ruleµ ´ satisﬁes the above

conditions,µ ´ · then ÀÄ admits cut-elimination. Proof: The above are sufﬁcient conditions for a rule to be preserved by quasi-DM completion. Semantics allows for a uniform proof.

Genova, 21/02/08 – p.47/?? Main Results

Theorem:

¾ 1. Any acyclic -axiom is equivalent in Ä to (a set of) structural rules enjoying cut-elimination.

¾ 2. Any -axiom is equivalent in ÄÛ to (a set of) structural rules enjoying cut-elimination.

3. Any acyclic -axiom is equivalent in ÀÄ to

¼ hyperstructural rules enjoying cut-elimination.

¿ 4. Any -axiom is equivalent in ÀÄÛ to hyperstructural rules enjoying cut-elimination. Our results automatically yield: Esteva-Godo’s logic

ÄÛ + (linearity) + (weak nilpotent minimum) admits a cut-admissible hypersequent calculus. Genova, 21/02/08 – p.48/?? Examples

Class Axiom Name

¾ ½ Æ « « Æ , weakening

« « Æ « contraction

« Æ « « expansion

« Æ « knotted axioms¼ Ñ (Ò )

Ñ Ò

µ « ² « ´ weak contraction

¾ « « excluded middle

µ « Æ ¬ ´ µ ¬ Æ « ´ linearity

µ ½ ² µ « Æ ¬ ´´ µ ½ ² µ ¬ Æ « ´´ prelinearity

« «

µ Ô Æ Ô ´ weak excluded middle

Ä Ä

Kripke models with width

Genova, 21/02/08 – p.49/?? Conclusion

Our coalition successfully combines various ideas: Polarity, Girard’s test from Linear Logic Hypersequent calculus from Fuzzy Logic DM completion from Substructural Logic

to establish uniform cut-elimination for extensions of Ä with

¿ axioms. Research directions: Computational meaning of axioms in Fuzzy Logic. Eg. Peirce’s law corresponds to call/cc in Functional

Programming. Whatµ aboutÆ ´ Linearity µ Æ ´ ? Better understanding of Syntax-Semantics Mirror

Genova, 21/02/08 – p.50/?? ¼ ¼

½ ½ Conclusion

¾ ¾

¿ ¿

Uniform treatment of axioms in :

Ô Ô

Ô Ô Łukasiewicz axiom, divisibility, cancellativity,

Ô Ô

Ô Ô distributivity, etc. (Known calculi are tailord to

Ô Ô each speciﬁc logic; cf. Metcalfe-Olivietti-Gabbay

Á 04)

How high can we climb up?

Genova, 21/02/08 – p.51/??