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Algebra and Logic C. Tsinakis - slide #1

The Interplay of Algebra and Logic

Costas TsinakisVanderbilt University

Logical fOundations of Rational Interaction

Pontignano, November 2, 2009


Introduction

IntroductionHistory (1)History (2)History (3)History (4)History (5)History (6)History (7)History (8)

Intro: LK

The Role of StructuralRules

The Calculus of RLs

Residuated Lattices

Algebra and Logic


Historical Remarks

The study of (predicate) logic as an independentdiscipline began with Aristotle (384-322 BCE). Heespoused two cornerstones of classical logic: theLaw of Excluded Middle and the Law of Contradiction.


Intro: LK


The Calculus of RLs

Residuated Lattices

Algebra and Logic


Historical Remarks

The study of (predicate) logic as an independentdiscipline began with Aristotle (384-322 BCE). Heespoused two cornerstones of classical logic: theLaw of Excluded Middle and the Law of Contradiction.Concerted efforts to study the role of logicalconnectives such as “and", “or" and “if. . . then. . . "were conducted by the Stoic philosophers in thelate 3rd century BCE. The Stoic philosopherChrysippus (280-205 BCE) suggested, amongothers, the following inference rules:■ If the first, then the second; but the first; therefore

the second. (modus ponens)■ If the first, then the second; but not the second;

therefore, not the first. (modus tollens)


Intro: LK


The Calculus of RLs

Residuated Lattices

Algebra and Logic


Historical Remarks

The fathers of modern mathematical logic areAugustus De Morgan (1806-1871) and, especialy,George Boole (1815-1864). Boole’s The MathematicalAnalysis of Logic, published in 1847, marks the officialbirth of modern mathematical logic. An expendedversion of the treatise appeared in 1854 under thetitle An Investigation of the Laws of Thought.


Intro: LK


The Calculus of RLs

Residuated Lattices

Algebra and Logic


Historical Remarks

The fathers of modern mathematical logic areAugustus De Morgan (1806-1871) and, especialy,George Boole (1815-1864). Boole’s The MathematicalAnalysis of Logic, published in 1847, marks the officialbirth of modern mathematical logic. An expendedversion of the treatise appeared in 1854 under thetitle An Investigation of the Laws of Thought.

Here is how Boole summarizes his objectives inthe preface of the 1854 volume:

The design of the following treatise is to investigate the funda-mental laws of the operations of the mind by which reasoningis performed; to give expression to them in the symbolicallanguage of a calculus, and upon this foundation to establishthe science of Logic and construct its method ...


Intro: LK


The Calculus of RLs

Residuated Lattices

Algebra and Logic


Historical Remarks

Boole wanted to provide an algebraic treatment ofAristotelian syllogistic logic. Boole started with theassumption that ordinary logic is concerned withassertions that can be considered as assertionsabout classes of objects. He then translated thelatter into equations in the language of classes.

His approach contained numerous errors, partlydue to his insistence to make the algebra of logicbehave as ordinary algebra, but it offeredsignificant new perspectives.


Intro: LK


The Calculus of RLs

Residuated Lattices

Algebra and Logic


Historical Remarks

As a means of example, let us consider thefollowing syllogism.

All elders are cautious peopleNo cautious people are fast drivers∴ No elders are fast drivers


Intro: LK


The Calculus of RLs

Residuated Lattices

Algebra and Logic


Historical Remarks

As a means of example, let us consider thefollowing syllogism.

All elders are cautious peopleNo cautious people are fast drivers∴ No elders are fast drivers

If we let E = elders, C = cautious people, and F =fast drivers, then the argument becomes:

All E’s are C ’s E ∩ C ′ = ∅No C ’s are F ’s C ∩ F = ∅

∴ No E’s are F ’s ∴ E ∩ F = ∅


Intro: LK


The Calculus of RLs

Residuated Lattices

Algebra and Logic


Historical Remarks

Symbolic logic, being the codification of exactreasoning, is interested in the nature of validity(syntax) as well as the notion of truth (semantics).


Intro: LK


The Calculus of RLs

Residuated Lattices

Algebra and Logic


Historical Remarks


■ George Boole studied the mathematics of logic.


Intro: LK


The Calculus of RLs

Residuated Lattices

Algebra and Logic


Historical Remarks



■ Gottlob Frege, A. N. Whitehead and BertrandRussell studied the logic of mathematics.


Intro: LK


The Calculus of RLs

Residuated Lattices

Algebra and Logic


Historical Remarks



■ Gottlob Frege, A. N. Whitehead and BertrandRussell studied the logic of mathematics.

■ Alfred Tarski and Adolf Lindenbaum recognizedthe complementary nature of these twoapproaches.


Intro: LK


The Calculus of RLs

Residuated Lattices

Algebra and Logic


Historical RemarksA calculus CL for classical propositional logic

The language L of CL: L = {→,¬}.


Intro: LK


The Calculus of RLs

Residuated Lattices

Algebra and Logic



The language L of CL: L = {→,¬}. The set Fm(X) of

L-formulas over a countably infinite set X is defined as follows:

(a) Inductive beginning: Every member of X, referred to as avariable, is a formula.

(b) Inductive steps: If α and β are formulas, then so are α → β and¬α.

(c) All formulas are generated by (a) and (b).


Intro: LK


The Calculus of RLs

Residuated Lattices

Algebra and Logic



The language L of CL: L = {→,¬}. The set Fm(X) of

L-formulas over a countably infinite set X is defined as follows:

(a) Inductive beginning: Every member of X, referred to as avariable, is a formula.

(b) Inductive steps: If α and β are formulas, then so are α → β and¬α.


(4) The set Thm of all theorems of CL is defined inductively asfollows:

(i) All formulas of the form(A1) α → (β → α),

(A2) (α → (β → γ)) → ((α → β) → (α → γ)),(A3) (¬β → ¬α) → (α → β),

where α, β, γ are arbitrary formulas, are called axioms and areconsidered theorems of CL.

(ii) Modus Ponens) If α and α → β are theorems, then so is β.


Intro: LK


The Calculus of RLs

Residuated Lattices

Algebra and Logic


Historical Remarks

Note that α is a theorem of CL iff there exists asequence of formulas α1, . . . , αn = α such that foreach i ∈ {1, . . . , n}, either αi is an axiom, or αi

follows by modus ponens from two previousmembers of the sequence. We refer to such assequence as a proof of the formula α.


Intro: LK


The Calculus of RLs

Residuated Lattices

Algebra and Logic


Historical Remarks


follows by modus ponens from two previousmembers of the sequence. We refer to such assequence as a proof of the formula α.Consider the Boolean algebra 2 = 〈{0, 1},→,¬〉.A truth assignment is a map τ : X → {0, 1}. Weextend τ for all formulas α and β, by definingτ(α → β) = τ(α) → τ(β) and τ(¬α) = ¬τ(α).

[Such an extension is possible becauseFm(X) = 〈Fm(X),→,¬〉 is the absolutely freealgebra of signature (2, 1) over X.]


Intro: LK


The Calculus of RLs

Residuated Lattices

Algebra and Logic


Historical Remarks


follows by modus ponens from two previousmembers of the sequence. We refer to such assequence as a proof of the formula α.Consider the Boolean algebra 2 = 〈{0, 1},→,¬〉.A truth assignment is a map τ : X → {0, 1}. Weextend τ for all formulas α and β, by definingτ(α → β) = τ(α) → τ(β) and τ(¬α) = ¬τ(α).

[Such an extension is possible becauseFm(X) = 〈Fm(X),→,¬〉 is the absolutely freealgebra of signature (2, 1) over X.]A formula α ∈ Fm(X) is called a tautology ifτ(α) = 1, for every truth assignment τ : X → {0, 1}.The set of tautologies of L will be denoted by Tau.


Intro: LK


The Calculus of RLs

Residuated Lattices

Algebra and Logic


Historical Remarks

Completeness Theorem of Propositional LogicThm = Tau


Intro: LK


The Calculus of RLs

Residuated Lattices

Algebra and Logic


Historical Remarks


Proof Sketch■ The relation ≡ of logical equivalence on Fm(X), defined by α ≡ β

iff α → β and β → α are theorems of CL, is an equivalencerelation on Fm(X) (in fact, it is a congruence relation on Fm(X)).


Intro: LK


The Calculus of RLs

Residuated Lattices

Algebra and Logic


Historical Remarks




■ B = F/ ≡ becomes a Boolean algebra B by defining theoperations as follows: [α] → [β] = [α → β] and ¬[α] = [¬α]. We

refer to B as the Tarski-Lindembaum algebra of CL.


Intro: LK


The Calculus of RLs

Residuated Lattices

Algebra and Logic


Historical Remarks





refer to B as the Tarski-Lindembaum algebra of CL.■ It is easy to see that every theorem of CL is a tautology. One

simply needs to check that every axiom is a theorem and notethat modus ponens preserves tautologies.


Intro: LK


The Calculus of RLs

Residuated Lattices

Algebra and Logic


Historical Remarks





refer to B as the Tarski-Lindembaum algebra of CL.■ It is easy to see that every theorem of CL is a tautology. One

simply needs to check that every axiom is a theorem and notethat modus ponens preserves tautologies.

■ Conversely let α be a tautology. Then for every truth assignmentτ, τ(α) = τ(α → α). Hence the identity α ≈ α → α holds for all

Boolean algebras, since 2 “generates" this class. Thus[α] = [α → α] in B and so (α → α) → α is a theorem of CL.

Since α → α is easily seen to be a theorem of CL, modusponens implies that α is a theorem of CL.


The Sequent System LK for ClassicalPropositional Logic

Introduction

Intro: LKLK (1)LK (2)LK (3)LK (4)LK (5)LK (6)


The Calculus of RLs

Residuated Lattices

Algebra and Logic


LK (1)

The Sequent System LK for classical propositionallogic

The language L of LK is the set of logicalconnectives ∧,∨,→, and ¬: L = {∧,∨,→,¬}.

The set Fm(X) of L-formulas over a countablyinfinite set X is defined as follows:

(a) Inductive beginning: Every member of X,referred to as a variable, is a formula.

(b) Inductive steps: If α and β are formulas, thenso are α ∧ β, α ∨ β, α → β and ¬α.


Introduction



The Calculus of RLs

Residuated Lattices

Algebra and Logic


LK (2)

A sequent is an expression of the form Γ ⇒ ∆,read ∆ “follows" from Γ, where Γ and ∆ are finite,possibly empty, sequences of formulas, separatedby commas: α1, . . . , αm ⇒ β1, . . . , βn.

Introduction



The Calculus of RLs

Residuated Lattices

Algebra and Logic


LK (2)

A sequent is an expression of the form Γ ⇒ ∆,read ∆ “follows" from Γ, where Γ and ∆ are finite,possibly empty, sequences of formulas, separatedby commas: α1, . . . , αm ⇒ β1, . . . , βn.

Intended Interpretation: According to GerhardGentzen, the preceding sequent has the sameinformal meaning as α1 ∧ . . . ∧ αm → β1 ∨ . . . ∨ βn.

Put it in another way, the sequent holds in LK, iffor every substitution of the variables occurring inthe formulas αi and βj by elements of 2, we musthave that α1 ∧ α2 . . . ∧ αn ≤ β1 ∨ β2 . . . ∨ βn.

Introduction



The Calculus of RLs

Residuated Lattices

Algebra and Logic


LK (3)

Initial Sequentsα ⇒ α.

Structural RulesWeakening Rules

Γ, ∆ ⇒ Π

Γ, α, ∆ ⇒ Π(w ⇒)

Γ ⇒ Λ, Θ

Γ ⇒ Λ, α, Θ( ⇒ w)

Contraction Rules

Γ, α, α, ∆ ⇒ Π

Γ, α, ∆ ⇒ Π(c ⇒)

Γ ⇒ Λ, α, α, Θ

Γ ⇒ Λ, α, Θ( ⇒ c)

Exchange RulesΓ, α, β, ∆ ⇒ Π

Γ, β, α, ∆ ⇒ Π(e ⇒)

Γ ⇒ Λ, α, β, Θ

Γ ⇒ Λ, β, α, Θ( ⇒ e)

Cut RuleΓ ⇒ α, Θ Σ, α, ∆ ⇒ Π

Σ, Γ, ∆ ⇒ Π, Θ

Introduction



The Calculus of RLs

Residuated Lattices

Algebra and Logic


LK (4)

Rules for Logical Connectives

Γ, α, ∆ ⇒ Π Γ, β, ∆ ⇒ Π

Γ, α ∨ β, ∆ ⇒ Π(∨ ⇒ )

Γ ⇒ Λ, α, Θ

Γ ⇒ Λ, α ∨ β, Θ( ⇒ ∨ 1)

Γ ⇒ Λ, β, Θ

Γ ⇒ Λ, α ∨ β, Θ( ⇒ ∨ 2)

Γ, α, ∆ ⇒ Π

Γ, α ∧ β, ∆ ⇒ Π(∧1 ⇒)

Γ, β, ∆ ⇒ Π

Γ, α ∧ β, ∆ ⇒ Π(∧2 ⇒)

Γ ⇒ Λ, α, Θ Γ ⇒ Λ, β, Θ

Γ ⇒ Λ, α ∧ β, Θ( ⇒ ∧)

Γ ⇒ α, Θ Σ, β, ∆ ⇒ Π

Σ, Γ, α → β, ∆ ⇒ Π, Θ(→ ⇒ )

α, Γ ⇒ β, Θ

Γ ⇒ α → β, Θ( ⇒ →)

Γ ⇒ α, Θ

¬α, Γ ⇒ Θ(¬ ⇒)

Γ, α ⇒ Θ

Γ ⇒ Θ,¬α( ⇒ ¬)

Introduction



The Calculus of RLs

Residuated Lattices

Algebra and Logic


The CUT Rule

A proof in LK is a finite labeled tree, each node ofwhich is labeled by a sequent. The leaves of thetree are labeled by initial sequents and eachsequent at a node is obtained from sequents frompredecessor nodes according to the rules of LK.

A sequent is provable if it labels the root of a proofin LK. A formula α is said to be a theorem of LK

provided the sequent ⇒ α is provable in LK .

Introduction



The Calculus of RLs

Residuated Lattices

Algebra and Logic


The CUT Rule




Completeness Theorem: A formula is a theorem ofLK if and only if it is a tautology.

Introduction



The Calculus of RLs

Residuated Lattices

Algebra and Logic


The CUT Rule




Completeness Theorem: A formula is a theorem ofLK if and only if it is a tautology.

The cut ruleΓ ⇒ α Σ, α,∆ ⇒ β

Σ,Γ,∆ ⇒ βis accepted by most logicians, since it reflects theidea that derivability is transitive.

Introduction



The Calculus of RLs

Residuated Lattices

Algebra and Logic


LK (6)

However, it is desirable to prove its redundancy,whenever possible, since proofs using this rulemay include formulas that disappear in theconclusion. The remaining rules satisfy the socalled subformula property, that is, theirconclusions contain all the formulas in theirpremises as subformulas. In a sequent calculuswith the subformula property, it is possible to set upa "bottom-up" search for the proof of a theorem.

Introduction



The Calculus of RLs

Residuated Lattices

Algebra and Logic


LK (6)

However, it is desirable to prove its redundancy,whenever possible, since proofs using this rulemay include formulas that disappear in theconclusion. The remaining rules satisfy the socalled subformula property, that is, theirconclusions contain all the formulas in theirpremises as subformulas. In a sequent calculuswith the subformula property, it is possible to set upa "bottom-up" search for the proof of a theorem.

A sequent calculus is said to admit cut eliminationif every provable sequent can be proved withoutthe use of the cut rule.

The Hauptsatz [Gerhard Gentzen, 1935]LK admits cut elimination.


The Role of Structural Rules

Introduction

Intro: LK

The Role of StructuralRulesRules (1)Rules (2)Rules (3)

The Calculus of RLs

Residuated Lattices

Algebra and Logic


Dropping Rules

WEAKENING RULES■ The relevant objection Consider the following

proof in LK:α ⇒ α

β, α ⇒ α(w ⇒ )

⇒ α → (β → α)( ⇒ →) twice

The principle α → (β → α), known as the law ofa fortiori, is viewed by relevant logicians as adeficiency of classical implication. “If α holds,then a fortiori (all the more) holds under thehypothesis β."

Introduction

Intro: LK


The Calculus of RLs

Residuated Lattices

Algebra and Logic


Dropping Rules

WEAKENING RULES■ The relevant objection Consider the following

proof in LK:α ⇒ α

β, α ⇒ α(w ⇒ )

⇒ α → (β → α)( ⇒ →) twice

The principle α → (β → α), known as the law ofa fortiori, is viewed by relevant logicians as adeficiency of classical implication. “If α holds,then a fortiori (all the more) holds under thehypothesis β."

■ The non-monotonic objection The left weakeningrule reflects monotonicity: “If β follows from α,then it also follows from an assumptionconsisting of α and some extra information."

Introduction

Intro: LK


The Calculus of RLs

Residuated Lattices

Algebra and Logic


Dropping Rules

CONTRACTION RULES■ The intuitionistic and many-valued objection

Consider the following proof in LK:α ⇒ α⇒ α,¬α ( ⇒ ¬)

⇒ α ∨ ¬α, α ∨ ¬α⇒ α ∨ ¬α ( ⇒ c)

( ⇒ ∨ 1 & ⇒ ∨ 2)

Thus, contraction on the right proves the law ofexcluded middle, α ∨ ¬α, which is rejected inintuitionistic logic and many-valued logic.

■ The linear logic objectionFormulas in linear logic represent types ofresources that cannot be used ad libitum. Asequent Γ ⇒ α is interpreted to mean that eachdatum in Γ is used exactly once to obtain datumα. Thus, the contraction rule (as well as theweakening rule) is not admissible in linear logic.

Introduction

Intro: LK


The Calculus of RLs

Residuated Lattices

Algebra and Logic


Dropping Rules

EXCHANGE RULE

Concerns about this rule can be traced back to thework Changing Arguments (Mǫταπ ıπτoντǫs Λoγoι)by Sextus Empiricus (Σǫ ξτos ‘o ’Eµπǫιρικos).

Introduction

Intro: LK


The Calculus of RLs

Residuated Lattices

Algebra and Logic


Dropping Rules

EXCHANGE RULE

Concerns about this rule can be traced back to thework Changing Arguments (Mǫταπ ıπτoντǫs Λoγoι)by Sextus Empiricus (Σǫ ξτos ‘o ’Eµπǫιρικos).■ The linear logic objection The exchange rule

Γ, α, β,∆ ⇒ γ

Γ, β, α,∆ ⇒ γ

is sometimes inadmissible, since the order ofdata may be essential in the context of actualinformation processing.

Introduction

Intro: LK


The Calculus of RLs

Residuated Lattices

Algebra and Logic


Dropping Rules

EXCHANGE RULE

Concerns about this rule can be traced back to thework Changing Arguments (Mǫταπ ıπτoντǫs Λoγoι)by Sextus Empiricus (Σǫ ξτos ‘o ’Eµπǫιρικos).■ The linear logic objection The exchange rule

Γ, α, β,∆ ⇒ γ

Γ, β, α,∆ ⇒ γ

is sometimes inadmissible, since the order ofdata may be essential in the context of actualinformation processing.

■ The Lambek calculus objectionAnother motivation for the rejection of exchangerules comes from the field of linguistics, andespecially from Lambek calculus.


The Calculus of RLs

Introduction

Intro: LK


The Calculus of RLsRL Calculus (1)RL Calculus (2)RL Calculus (3)

Residuated Lattices

Algebra and Logic


RL Calculus

■ Signature: L = {∧,∨, ·, \, /, 1}■ The succedent of each sequent in RL is a single

formula.

Initial Sequentsα ⇒ α and ⇒ 1

Structural Rules

1-Weakening Rule

Γ,∆ ⇒ β

Γ, 1,∆ ⇒ β(1w ⇒ )

Cut Rule:

Γ ⇒ α Σ, α,∆ ⇒ β

Σ,Γ,∆ ⇒ β

Introduction

Intro: LK



Residuated Lattices

Algebra and Logic


RL Calculus

Rules for Logical Connectives

Γ, α,∆ ⇒ γ Γ, β,∆ ⇒ γ

Γ, α ∨ β,∆ ⇒ γ(∨ ⇒ )

Γ ⇒ α

Γ ⇒ α ∨ γ( ⇒ ∨ 1)

Γ ⇒ γ

Γ ⇒ α ∨ γ( ⇒ ∨ 2)

Γ, α,∆ ⇒ γ

Γ, α ∧ β,∆ ⇒ γ(∧1 ⇒ )

Γ, β,∆ ⇒ γ

Γ, α ∧ β,∆ ⇒ γ(∧2 ⇒ )

Γ ⇒ α Γ ⇒ β

Γ ⇒ α ∧ β( ⇒ ∧)

Introduction

Intro: LK



Residuated Lattices

Algebra and Logic


RL Calculus

Rules for Logical Connectives (continued)Monoid Operation and Implications (Residuals)

Γ, α, β,∆ ⇒ γ

Γ, α · β,∆ ⇒ γ(· ⇒ )

Γ ⇒ α,∆ ⇒ β

Γ,∆ ⇒ α · β ( ⇒ ·)

Γ ⇒ α Σ, β,∆ ⇒ γ

Σ,Γ, α\β,∆ ⇒ γ(\ ⇒ )

α,Γ ⇒ β

Γ ⇒ α\β ( ⇒ \)

Γ ⇒ α Σ, β,∆ ⇒ γ

Σ, β/α,Γ,∆ ⇒ γ(/ ⇒ )

Γ, α ⇒ β

Γ ⇒ β/α( ⇒ /)


Residuated Lattices

Introduction

Intro: LK


The Calculus of RLs

Residuated LatticesRLsBAs & HAsRings & RLs

ℓ-GroupsCompleteness

Algebra and Logic


Residuated Lattices

A residuated lattice (or a residuated lattice-orderedmonoid) is an algebra A = 〈A,∧,∨, ·, \, /, 1〉 suchthat:

(i) 〈A,∧,∨〉 is a lattice;

(ii) 〈A, ·, 1〉 is a monoid; and

(iii) the operation · is residuated with residuals \and /. This means that that, for all x, y, z ∈ A,

x · y ≤ z ⇐⇒ x ≤ z/y ⇐⇒ y ≤ x\z.

Introduction

Intro: LK


The Calculus of RLs



Algebra and Logic


Residuated Lattices

A residuated lattice (or a residuated lattice-orderedmonoid) is an algebra A = 〈A,∧,∨, ·, \, /, 1〉 suchthat:

(i) 〈A,∧,∨〉 is a lattice;

(ii) 〈A, ·, 1〉 is a monoid; and

(iii) the operation · is residuated with residuals \and /. This means that that, for all x, y, z ∈ A,

x · y ≤ z ⇐⇒ x ≤ z/y ⇐⇒ y ≤ x\z.

An algebra A = 〈A,∧,∨, ·, \, /, 1, 0〉 is said to be anFL algebra provided: (i) A = 〈A,∧,∨, ·, \, /, 1〉 is anRL; and (ii) 0 is a distinguished element of A.

Introduction

Intro: LK


The Calculus of RLs



Algebra and Logic


Boolean Algebras and Heyting Algebras

Notation: If x\y = y/x, we write x → y for thecommon value.

Introduction

Intro: LK


The Calculus of RLs



Algebra and Logic




The class of Boolean algebras “is” the class ofFL-algebras that satisfy the identities x · y ≈ x ∧ y,(x → y) → y ≈ x ∨ y and x ∧ 0 = 0.

Introduction

Intro: LK


The Calculus of RLs



Algebra and Logic




The class of Boolean algebras “is” the class ofFL-algebras that satisfy the identities x · y ≈ x ∧ y,(x → y) → y ≈ x ∨ y and x ∧ 0 = 0.

The class of Heyting algebras “is” the class ofFL-algebras that satisfy the identities x · y ≈ x ∧ yand x ∧ 0 = 0.

Introduction

Intro: LK


The Calculus of RLs



Algebra and Logic


Rings and Residuated Lattices

Let R be a ring with identity and let I(R) denotethe lattice of two-sided ideals of R. ThenI(R) = 〈I(R),∩,∨, ·, \, /, R, {0}〉 is an FL-algebra,where, for I, J ∈ I(R),

I · J = {n

∑

k=1

akbk|ak ∈ I; bk ∈ J ;n ∈ N∗}.

Introduction

Intro: LK


The Calculus of RLs



Algebra and Logic


Rings and Residuated Lattices

Let R be a ring with identity and let I(R) denotethe lattice of two-sided ideals of R. ThenI(R) = 〈I(R),∩,∨, ·, \, /, R, {0}〉 is an FL-algebra,where, for I, J ∈ I(R),

I · J = {n

∑

k=1

akbk|ak ∈ I; bk ∈ J ;n ∈ N∗}.

Number theory and ring theory constitute historicallyremarkable sources for residuated structures. Ernst Kummerobserved in the middle of the XIX century that uniquefactorization into primes fails in Z

[√−n]

for several values ofn, although it holds in special cases, such as Gaussianintegers (n = 1). Richard Dedekind introduced in 1871 theconcepts of ring and ring ideal and proved that every ideal insuch a ring is uniquely representable (up to permutation offactors) as a product of prime ideals. Unique factorization isrecovered at the level of ideals!

Introduction

Intro: LK


The Calculus of RLs



Algebra and Logic


Lattice-Ordered GroupsA lattice-ordered group is an algebraG = 〈G,∧,∨, ·,−1 , 1〉 such that (i) 〈G,∧,∨〉 is alattice; (ii) 〈G, ·,−1 , 1〉 is a group; and (iii)multiplication is order-preserving.[Dedekind (1897); Frigyes Riesz, Hans Freudenthal and Leonid

Kantorovich (late 1920s and early 1930s); Garrett Birkhoff (1940)]

Introduction

Intro: LK


The Calculus of RLs



Algebra and Logic




The class of lattice-ordered groups “is” the class ofresiduated lattices that satisfy the identityx · (x\1) ≈ 1.

Introduction

Intro: LK


The Calculus of RLs



Algebra and Logic




The class of lattice-ordered groups “is” the class ofresiduated lattices that satisfy the identityx · (x\1) ≈ 1.

(1) The preceding examples show that the studyof residuated lattices has a long history.

(2) Our interest in lattice-ordered groups is due tothe fact that these objects are intimatelyconnected with the Łukasiewicz infinite-valuedpropositional logic [Daniele Mundici (1986)].

Introduction

Intro: LK


The Calculus of RLs



Algebra and Logic


Completeness Theorem

The following are equivalent:1. α is a theorem of RL.2. α is valid in all residuated lattices; this means

that for all residuated lattices L and for allhomomorphisms ϕ : Fm(X) → L, 1 ≤ ϕ(α).

Introduction

Intro: LK


The Calculus of RLs



Algebra and Logic


Completeness Theorem

The following are equivalent:1. α is a theorem of RL.2. α is valid in all residuated lattices; this means

that for all residuated lattices L and for allhomomorphisms ϕ : Fm(X) → L, 1 ≤ ϕ(α).

What constitutes good semantics?


Algebra and Logic

Introduction

Intro: LK


The Calculus of RLs

Residuated Lattices

Algebra and LogicCut EliminationMore . . .


Cut Elimination


RLw admits cut elimination. [S. Tamura, 1974]

The following results have been proved byalgebraic means:

RLe admits cut elimination. [M. Okada andK. Terui; 1999]

RL admits cut elimination. [P. Jipsen andC. Tsinakis; 2002]

Introduction

Intro: LK


The Calculus of RLs

Residuated Lattices



Cut Elimination


RLw admits cut elimination. [S. Tamura, 1974]

The following results have been proved byalgebraic means:

RLe admits cut elimination. [M. Okada andK. Terui; 1999]

RL admits cut elimination. [P. Jipsen andC. Tsinakis; 2002]

RL, RLw and RLe are decidable.[Hence, the equational theories of RL, IRL, andCRL and are decidable.

Introduction

Intro: LK


The Calculus of RLs

Residuated Lattices



More . . .

The deduction theorem does not hold in RLe, butwe have the following local deduction theorem:{α1, . . . , αn} ⊢RLe

β iff ⊢RLe(∏n

i=1(αi ∧ 1)ki) → β,

for some non-negative integers k1, . . . , kn.The simplest (and original) proof is based on thedescription of congruence relations in acommutative residuated lattice.

Introduction

Intro: LK


The Calculus of RLs

Residuated Lattices



More . . .


β iff ⊢RLe(∏n

i=1(αi ∧ 1)ki) → β,


There is also a parametrized local deductiontheorem for RL, based, again, on the descriptionof congruence relations in a residuated lattice.

Introduction

Intro: LK


The Calculus of RLs

Residuated Lattices



More . . .


β iff ⊢RLe(∏n

i=1(αi ∧ 1)ki) → β,


There is also a parametrized local deductiontheorem for RL, based, again, on the descriptionof congruence relations in a residuated lattice.

An Important Application: Commutative residuatedlattices satisfy the amalgamation property (andother properties concerning co-products). Thecrucial step in the proof is the fact, established byproof-theoretic means, that RLe satisfies Craig’sinterpolation property.

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