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A gentle introduction to Abstract Algebraic Logic I

Logics, logical matrices, and completeness theorems Petr Cintula

Institute of Computer Science, Academy of Sciences of the Czech Republic Prague, Czech Republic

www.cs.cas.cz/cintula/AAL

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Logic throughout the history

Logic is the science that studies correct reasoning

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Logic throughout the history

Logic is the science that studies correct reasoning Logic started as a part of philosophy All men are mortal and Socrates in a man, therefore Socrates is mortal

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Logic throughout the history

Logic is the science that studies correct reasoning Since 19th century a part of logic has evolved into mathematical logic PA ⊢ ¬∃wProof(w, ‘¯ 0 = ¯ 1’)

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Logic throughout the history

Logic is the science that studies correct reasoning Nowadays logic is applied mainly in computer science ⊢ [α](x = 4) → [α; (x := 2x)](x = 8)

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Logic throughout the history

Logic is the science that studies correct reasoning However its role in the study of (human) reasoning has been diminished

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Logic throughout the history

Logic is the science that studies correct reasoning However its role in the study of (human) reasoning has been diminished Stenning (psychologist) and van Lambalgen (logician) advocate the indispensability of logic for the study of (human) reasoning

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Logic throughout the history

Logic is the science that studies correct reasoning However its role in the study of (human) reasoning has been diminished Stenning (psychologist) and van Lambalgen (logician) advocate the indispensability of logic for the study of (human) reasoning The key component: the transformation of natural reasoning scenarios into formalized ones where various logics are directly applicable

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Various logic(s)

Formal logical systems are many and vary in expressive powers: propositional logics and its modal extensions first- and higher- order logics various type theories dynamic and non-monotonic logics . . .

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Various logic(s)

Formal logical systems are many and vary in expressive powers: propositional logics and its modal extensions first- and higher- order logics various type theories dynamic and non-monotonic logics . . . This tutorial focuses on truth-functional context-independent transitive monotonic propositional cores of these systems: Classical logic Intuitionistic logic Linear logic Superintuitionistic logics Fuzzy logics Modal logics Substructural logics Paraconsistent logics . . .

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Algebraic logic

Algebraic logic: a field of mathematical logic studying logics using Universal Algebra

(a field of mathematics studying classes

• f algebraic structures)

Logic Algebraic counterpart Classical logic Boolean algebras Modal logics Modal algebras Intuitionistic logic Heyting algebras Linear logics Commutative residuated lattices Fuzzy logics Semilinear residuated lattices Relevance logics Commutative contractive residuated lattices . . . . . .

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Abstract Algebraic Logic

AAL is the evolution of algebraic logic that wants to: understand the several ways by which a logic can be given an algebraic semantics build a general and abstract theory of non-classical logics based

• n their relation to algebras

understand the role of connectives in (non-)classical logics classify non-classical logics find general results connecting logical and algebraic properties (bridge theorems) generalize various properties from syntax to semantics (transfer theorems) advance the study of particular (families of) non-classical logics by using the abstract notions and results It works best, by far, when restricted to propositional logics

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Abstract Algebraic Logic

AAL is the evolution of algebraic logic that wants to: understand the several ways by which a logic can be given an algebraic semantics build a general and abstract theory of non-classical logics based

• n their relation to algebras

understand the role of connectives in (non-)classical logics classify non-classical logics find general results connecting logical and algebraic properties (bridge theorems) generalize various properties from syntax to semantics (transfer theorems) advance the study of particular (families of) non-classical logics by using the abstract notions and results

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Basic syntactical notions – formulas and consecutions

Propositional language: a countable set of connective with arities we write c, n ∈ L if L has an n-ary connective c Formulas: build from infinite countable set Var of variables it the usual way . . . by FmL we denote the set of formulas in L

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Basic syntactical notions – formulas and consecutions

Propositional language: a countable set of connective with arities we write c, n ∈ L if L has an n-ary connective c Formulas: build from infinite countable set Var of variables it the usual way . . . by FmL we denote the set of formulas in L Consecution: a pair Γ ⊲ ϕ, where Γ ∪ {ϕ} ⊆ FmL Note: A set L of consecutions can be seen as a relation between sets

• f formulas and formulas. We write ‘Γ ⊢L ϕ’ instead of ‘Γ ⊲ ϕ ∈ L’

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Basic syntactical notions – formulas and consecutions

Propositional language: a countable set of connective with arities we write c, n ∈ L if L has an n-ary connective c Formulas: build from infinite countable set Var of variables it the usual way . . . by FmL we denote the set of formulas in L Consecution: a pair Γ ⊲ ϕ, where Γ ∪ {ϕ} ⊆ FmL Note: A set L of consecutions can be seen as a relation between sets

• f formulas and formulas. We write ‘Γ ⊢L ϕ’ instead of ‘Γ ⊲ ϕ ∈ L’

Substitution: a mapping σ: FmL → FmL, such that for each c, n ∈ L and every ϕ1, . . . , ϕn ∈ FmL: σ(c(ϕ1, . . . , ϕn)) = c(σ(ϕ1), . . . , σ(ϕn))

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Basic syntactical notions – logics

Definition

A set L of consecutions in L is called a logic in L whenever If ϕ ∈ Γ, then Γ ⊢L ϕ. (Reflexivity) If ∆ ⊢L ψ for each ψ ∈ Γ and Γ ⊢L ϕ, then ∆ ⊢L ϕ. (Cut) If Γ ⊢L ϕ, then σ[Γ] ⊢L σ(ϕ) for each substitution σ. (Structurality) A logic L is finitary if furthermore If Γ ⊢L ϕ, then there is finite Γ′ ⊆ Γ and Γ′ ⊢L ϕ. (Finitarity) Observe that reflexivity and cut entail: If Γ ⊢L ϕ, then Γ ∪ ∆ ⊢L ϕ. (Monotonicity) Theorem of L: a formula ϕ such that ∅ ⊢L ϕ

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Basic syntactical notions – some (trivial) examples

The least logic Dumb is described as (note that it has no theorems): Γ ⊢Dumb ϕ iff ϕ ∈ Γ Inconsistent logic Inc: the set all consecutions (equivalently: a logic where all formulas are theorems) Almost Inconsistent logic AInc: the maximum logic without theorems (note that Γ, ϕ ⊢AInc ψ)

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Basic syntactical notions – theories

Theory: a set of formulas T such that if T ⊢L ϕ then ϕ ∈ T Th(L) is the set of all theories of L

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Basic syntactical notions – theories

Theory: a set of formulas T such that if T ⊢L ϕ then ϕ ∈ T Th(L) is the set of all theories of L Th(Dumb) = P(FmL) Th(Inc) = {FmL} Th(AInc) = {∅, FmL}

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Basic syntactical notions – theories

Theory: a set of formulas T such that if T ⊢L ϕ then ϕ ∈ T Th(L) is the set of all theories of L Th(Dumb) = P(FmL) Th(Inc) = {FmL} Th(AInc) = {∅, FmL} Note that the set of all theorems is the least theory the set ThL(Γ) = {ϕ ∈ FmL | Γ ⊢L ϕ} is a theory

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Basic syntactical notions – theories

Theory: a set of formulas T such that if T ⊢L ϕ then ϕ ∈ T Th(L) is the set of all theories of L Th(Dumb) = P(FmL) Th(Inc) = {FmL} Th(AInc) = {∅, FmL} Note that the set of all theorems is the least theory the set ThL(Γ) = {ϕ ∈ FmL | Γ ⊢L ϕ} is a theory actually Th(L) is a closure system (i.e., it contains FmL and is closed under arbitrary intersections) ThL(Γ) is the closure of Γ (i.e., it is the intersection of all theories containing Γ)

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Basic syntactical notions – axiomatization

Axiomatic system: a set AS of consecutions closed under substitutions Terminology: elements of AS of the form ∅ ⊲ ϕ are called axioms,

• thers are deduction rules

Proof of a formula ϕ from a set of formulas Γ in AS: a well-founded tree labeled by formulas such that its root is labeled by ϕ, its leaves are labelled by axioms of AS or elements of Γ, and if a node is labeled by ψ and ∆ = ∅ is the set of labels of its preceding nodes, then ∆ ⊲ ψ ∈ AS. We write Γ ⊢

AS ϕ if there is a proof of ϕ from Γ in AS.

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Basic syntactical notions – presentation

Lemma

⊢

AS is the least logic containing an axiomatic system AS.

Theorem (Ło´ s–Suszko)

Each logic has a presentation, i.e. an axiomatic system AS st L = ⊢

AS.

Lemma

A logic is finitary iff it has a presentation whose all rules have finitely many premises. Note that Inc, AInc, Dumb are finitary because: Inc is axiomatized by axioms {ϕ | ϕ ∈ FmL} AInc is axiomatized by unary rules {ϕ ⊲ ψ | ϕ, ψ ∈ FmL} Dumb is axiomatized by by the empty set

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Examples: classical logic CL and logic BCI

Finitary axiomatic system for CL in LCL = {→, ¬}

A1 ϕ → (ψ → ϕ) A2 (χ → (ϕ → ψ)) → ((χ → ϕ) → (χ → ψ)) A3 (¬ψ → ¬ϕ) → (ϕ → ψ) MP ϕ, ϕ → ψ ⊲ ψ

Finitary axiomatic system for BCI in LBCI = {→}

B (ϕ → ψ) → ((ψ → χ) → (ϕ → χ)) C (ϕ → (ψ → χ)) → (ψ → (ϕ → χ)) I ϕ → ϕ MP ϕ, ϕ → ψ ⊲ ψ

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Basic semantical notions – algebras

L-algebra: A = A, cA | c ∈ L, where A = ∅ and cA : An → A for each c, n ∈ L Algebra of formulas: the algebra FmL with domain FmL and operations cFmL for each c, n ∈ L defined as: cFmL(ϕ1, . . . , ϕn) = c(ϕ1, . . . , ϕn) FmL is the absolutely free algebra in language L with generators Var Homomorphism of algebras: a mapping f : A → B such that for every c, n ∈ L and every a1, . . . , an ∈ A, f(cA(a1, . . . , an)) = cB(f(a1), . . . , f(an)) Note: substitutions are exactly homomorphisms from FmL to FmL

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Basic semantical notions – L-matrices

L-matrix: a pair A = A, F where A is an L-algebra called the algebraic reduct of A F ⊆ A called the filter of A or the set designated elements of A An L-matrix A = A, F is trivial if F = A finite if A is finite Lindenbaum if A = FmL A-evaluation: a homomorphism from FmL to A, i.e. a mapping e: FmL → A, such that for each c, n ∈ L and ϕ1, . . . , ϕn ∈ FmL: e(c(ϕ1, . . . , ϕn)) = cA(e(ϕ1), . . . , e(ϕn))

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Basic semantical notions – semantical consequence

Definition (Semantical consequence)

A formula ϕ is a semantical consequence of a set Γ of formulas w.r.t. a class K of L-matrices, Γ | =K ϕ in symbols, if for each A, F ∈ K and each A-evaluation e, we have: e(ϕ) ∈ F whenever e[Γ] ⊆ F.

Exercise

Let K be a class of L-matrices. Then | =K is a logic in L.

Lemma

If K is a finite class of finite matrices, then the logic | =K is finitary.

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Basic semantical notions – L-matrices and filters

Definition

Let L be a logic in L and A = A, F an L-matrix such that L ⊆ | =A, i.e., for each Γ ∪ {ϕ} ⊆ FmL holds: Γ ⊢L ϕ implies each A-evaluation e, we have: e(ϕ) ∈ F whenever e[Γ] ⊆ F Then we say that A is an L-matrix (or a model of L), A ∈ MOD(L) in symbols F is an L-filter over A, F ∈ FiL(A) in symbols

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Basic semantical notions – L-matrices and filters

Definition

Let L be a logic in L and A = A, F an L-matrix such that L ⊆ | =A, i.e., for each Γ ∪ {ϕ} ⊆ FmL holds: Γ ⊢L ϕ implies each A-evaluation e, we have: e(ϕ) ∈ F whenever e[Γ] ⊆ F Then we say that A is an L-matrix (or a model of L), A ∈ MOD(L) in symbols F is an L-filter over A, F ∈ FiL(A) in symbols FiDumb(A) = P(A) FiInc(A) = {A} FiAInc(A) = {∅, A}

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Examples: classical logic CL and logic BCI

Exercise

• 1. Classical logic: Let A be a Boolean algebra. Then FiCL(A) is the

class of lattice filters on A, in particular for the two-valued Boolean algebra 2: FiCL(2) = {{1}, {0, 1}}.

• 2. The logic BCI: By M we denote the LBCI-algebra with domain

{⊥, ⊤, t, f} and:

→M ⊤ t f ⊥ ⊤ ⊤ ⊥ ⊥ ⊥ t ⊤ t f ⊥ f ⊤ ⊥ t ⊥ ⊥ ⊤ ⊤ ⊤ ⊤

Check that FiBCI(M) = {{t, ⊤}, {t, f, ⊤}, M}.

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Weird algebras and filters

Note: A ∈ MOD(L) whenever AS ⊆ | =A for some presentation AS of L

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Weird algebras and filters

Note: A ∈ MOD(L) whenever AS ⊆ | =A for some presentation AS of L Consider algebra A = [0, 1], ¬, →, where: a → b = 1 if a ≤ b b if a > b ¬a = 1 if a = 0 if a > 0 We can show that A = [0, 1]G, (0, 1] ∈ MOD(CL): ϕ, ϕ → ψ ⊲ ψ if e(ψ) = 0 and e(ϕ) > 0, then e(ϕ → ψ) = 0 ϕ → (ψ → ϕ) check two cases: 1) e(ψ) ≤ e(ϕ) and 2) e(ψ) > e(ϕ) (¬ψ → ¬ϕ) → (ϕ → ψ) check three cases: 1) e(ϕ) ≤ e(ψ); 2) e(ϕ) > e(ψ) = 0; and 3) e(ϕ) > e(ψ) > 0 (χ → (ϕ → ψ)) → ((χ → ϕ) → (χ → ψ)) check many cases

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More on filters

Note: FiL(A) is a closure system

Proposition

For any logic L in a language L, FiL(FmL) = Th(L).

Proof.

Γ ∈ FiL(FmL) and Γ ⊢L ϕ: take FmL-evaluation i(ϕ) = ϕ Then obviously i(Γ) = Γ ⊆ Γ and so ϕ = i(ϕ)∈ Γ T ∈ Th(L), Γ ⊢L ϕ, and e is an FmL-evaluation st e[Γ] ⊆ T: Then T ⊢L χ for each χ ∈ e[Γ] (reflexivity) and e[Γ] ⊢L e(ϕ) (structurality) Thus T ⊢L e(ϕ) (cut) and so e(ϕ) ∈ T (T is a theory).

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The first completeness theorem

Theorem

Let L be a logic. Then for each Γ ∪ {ϕ} ⊆ FmL: Γ ⊢L ϕ iff Γ | =MOD(L) ϕ.

Proof.

Soundness: just observe that | =MOD(L) =

A∈MOD(L) |

=A Completeness: assume that Γ | =MOD(L) ϕ We know: FmL, ThL(Γ) ∈ MOD(L) and i(χ) = χ is an FmL-evaluation Then i[Γ] = Γ ⊆ ThL(Γ) and so ϕ = i(ϕ) ∈ ThL(Γ), i.e., Γ ⊢L ϕ

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Completeness theorem for classical logic

Suppose that Γ ⊢CL ϕ. We showed that Γ | = ϕ in some meaningful semantics, i.e., Γ | =MOD(L) ϕ. 1st completeness theorem

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Completeness theorem for classical logic

Suppose that Γ ⊢CL ϕ. We showed that Γ | = ϕ in some meaningful semantics, i.e., Γ | =MOD(L) ϕ. 1st completeness theorem We define α, β ∈ Ω(Γ) iff Γ ⊢ α ↔ β and show that it is a congruence relation on FmL, FmL/Ω is a Boolean algebra, and Γ | ={A,{1} | A is BA with top 1} ϕ 2nd completeness theorem

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Completeness theorem for classical logic

Suppose that Γ ⊢CL ϕ. We showed that Γ | = ϕ in some meaningful semantics, i.e., Γ | =MOD(L) ϕ. 1st completeness theorem We define α, β ∈ Ω(Γ) iff Γ ⊢ α ↔ β and show that it is a congruence relation on FmL, FmL/Ω is a Boolean algebra, and Γ | ={A,{1} | A is BA with top 1} ϕ 2nd completeness theorem We extend Γ into a maximal consistent theory T ∈ Th(CL) such that ϕ / ∈ T (Lindenbaum Lemma) and show that FmL/Ω(T) is two-valued Boolean algebra and Γ | =2,{1} ϕ. 3rd completeness theorem

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Congruence Property

Proposition

Let ϕ, ψ, χ be formulas. Then: ⊢CL ϕ ↔ ϕ ϕ ↔ ψ ⊢CL ψ ↔ ϕ ϕ ↔ δ, δ ↔ ψ ⊢CL ϕ ↔ ψ ϕ, ϕ ↔ ψ ⊢CL ψ ϕ1 ↔ ψ1, ϕ2 ↔ ψ2 ⊢CL (ϕ1 → ϕ2) ↔ (ψ1 → ψ2) ϕ1 ↔ ψ1 ⊢CL ¬ϕ1 ↔ ¬ψ1

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Congruence Property

Conventions: for L be either CL or BCI, we write: ϕ ↔ ψ instead of {ϕ → ψ, ψ → ϕ} Γ ⊢ ∆ whenever Γ ⊢ χ for each χ ∈ ∆

Proposition

Let ϕ, ψ, χ be formulas. Then: ⊢L ϕ ↔ ϕ ϕ ↔ ψ ⊢L ψ ↔ ϕ ϕ ↔ δ, δ ↔ ψ ⊢L ϕ ↔ ψ ϕ, ϕ ↔ ψ ⊢L ψ {ϕi ↔ ψi | i ≤ n} ⊢L c(ϕ1, . . . , ϕn) ↔ c(ψ1, . . . , ψn), for each c, n ∈ L

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(Logical) Congruences

An equivalence relation θ on A is a congruence on L-algebra A if for each c, n ∈ L and a1, . . . , an, b1, . . . , bn ∈ A we have: if ai, bi ∈ θ for each i ≤ n, then cA(a1, . . . , an), cA(b1, . . . , bn) ∈ θ

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(Logical) Congruences

An equivalence relation θ on A is a congruence on L-algebra A if for each c, n ∈ L and a1, . . . , an, b1, . . . , bn ∈ A we have: if ai, bi ∈ θ for each i ≤ n, then cA(a1, . . . , an), cA(b1, . . . , bn) ∈ θ A congruence θ of A is logical in a matrix A, F if for each a, b ∈ A: if a ∈ F and a, b ∈ θ, then b ∈ F.

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Leibniz congruence – for CL and BCI

Let L be either CL or BCI (for this slide) and A = A, F be an L-matrix. We define the Leibniz congruence ΩA(F) of A as a, b ∈ ΩA(F) iff {a →A b, b →A a} ⊆ F.

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Leibniz congruence – for CL and BCI

Let L be either CL or BCI (for this slide) and A = A, F be an L-matrix. We define the Leibniz congruence ΩA(F) of A as a, b ∈ ΩA(F) iff {a →A b, b →A a} ⊆ F.

Proposition

Let A = A, F be an L-matrix. Then:

1

ΩA(F) is the largest logical congruence of A.

2

a, b ∈ ΩA(F) iff for each χ ∈ FmL and each A-evaluation e: e[p:a](χ) ∈ F iff e[p:b](χ) ∈ F.

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Leibniz congruence – in general

Theorem

For each L-matrix A, F there is largest logical congruence θ on A and a, b ∈ θ iff for each χ ∈ FmL and each A-evaluation e: e[p:a](χ) ∈ F iff e[p:b](χ) ∈ F.

Definition

Let A = A, F be an L-matrix. By ΩA(F) we denote the largest logical congruence on A and we call it Leibniz congruence of A.

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Leibniz congruence – in general

Theorem

For each L-matrix A, F there is largest logical congruence θ on A and a, b ∈ θ iff for each χ ∈ FmL and each A-evaluation e: e[p:a](χ) ∈ F iff e[p:b](χ) ∈ F.

Definition

Let A = A, F be an L-matrix. By ΩA(F) we denote the largest logical congruence on A and we call it Leibniz congruence of A.

Definition

A matrix A, F is reduced, if ΩA(F) = IdA. For a logic L, by MOD∗(L) we denote the class of reduced L-matrices.

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Algebraic counterpart

Definition

An algebra A is L-algebra, A ∈ ALG∗(L) in symbols, if there is a set F ⊆ A such that A, F ∈ MOD∗(L). Note that ΩA(A) = A2. Thus from FiInc(A) = {A} we obtain: A ∈ ALG∗(Inc) iff A is a singleton Note that ΩA(∅) = A2 and FiAInc(A) = {∅, A} thus also: A ∈ ALG∗(AInc) iff A is a singleton i.e., ALG∗(AInc) = ALG∗(Inc).

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Examples: classical logic CL and logic BCI

Exercise

Classical logic: prove that for any Boolean algebra A: ΩA({1}) = IdA i.e., A, {1} ∈ MOD∗(CL) and A ∈ ALG∗(CL). On the other hand, show that: Ω4({a, 1}) = IdA ∪ {1, a, 0, ¬a} i.e., 4, {a, 1} / ∈ MOD∗(CL). BCI: recall the algebra M defined via:

→M ⊤ t f ⊥ ⊤ ⊤ ⊥ ⊥ ⊥ t ⊤ t f ⊥ f ⊤ ⊥ t ⊥ ⊥ ⊤ ⊤ ⊤ ⊤

Show that: ΩM({t, ⊤}) = ΩM({t, f, ⊤}) = IdM i.e., M ∈ ALG∗(BCI).

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More examples

Logic ALG∗(L) Classical logic Boolean algebras Intuitionistic logic Heyting algebras Modal logics varieties of BA with operators Fuzzy logics varieties of semilinear residuated lattices Relevance logic commutative contractive residuated lattices Linear logic commutative residuated lattices . . . . . .

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Factorizing algebras

Let A be a L-agebra and θ a congruence on A. First we define [a]θ = {b | a, b ∈ θ} Then by A/θ we denote the L-algebra with the domain {[a]θ | a ∈ A} and operations: cA/θ([a1]θ, . . . , [an]θ) = [cA(a1, . . . , an)]θ Fact: the mapping a → [a]θ is an homomorphism of A onto A/θ

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Factorizing matrices

Let us take A = A, F ∈ MOD(L). We write: A∗ for A/ΩA(F) [a] for [a]ΩA(F) F∗ = {[a] | a ∈ F} A∗ for A∗, F∗.

Lemma

Let A = A, F ∈ MOD(L) and a, b ∈ A. Then:

1

a ∈ F iff [a] ∈ F∗.

2

A∗ ∈ MOD∗(L). If A is Lindenbaum matrix, then A∗ is called Lindenbaum–Tarski matrix

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The second completeness theorem

Theorem

Let L be a logic. Then for each Γ ∪ {ϕ} ⊆ FmL: Γ ⊢L ϕ iff Γ | =MOD∗(L) ϕ.

Proof.

Soundness: just observe that | =MOD(L) ⊆ | =MOD∗(L) Completeness: assume that Γ | =MOD∗(L) ϕ We know: FmL, ThL(Γ)∗ ∈ MOD∗(L) and e(χ) = [χ] is an Fm∗

L-eval.

We also know: Γ ⊢L χ iff e(χ) ∈ ThL(Γ)∗ Then e[Γ] ⊆ ThL(Γ)∗ and so e(ϕ) ∈ ThL(Γ)∗, i.e., Γ ⊢L ϕ.

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Completeness theorem for classical logic

Suppose that Γ ⊢CL ϕ. We showed that Γ | = ϕ in some meaningful semantics, i.e., Γ | =MOD(L) ϕ. 1st completeness theorem We define α, β ∈ Ω(Γ) iff Γ ⊢ α ↔ β and show that it is a congruence relation on FmL, FmL/Ω is a Boolean algebra, and Γ | ={A,{1} | A is BA with top 1} ϕ 2nd completeness theorem

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Completeness theorem for classical logic

Suppose that Γ ⊢CL ϕ. We showed that Γ | = ϕ in some meaningful semantics, i.e., Γ | =MOD(L) ϕ. 1st completeness theorem We define α, β ∈ Ω(Γ) iff Γ ⊢ α ↔ β and show that it is a congruence relation on FmL, FmL/Ω is a Boolean algebra, and Γ | ={A,{1} | A is BA with top 1} ϕ 2nd completeness theorem We extend Γ into a maximal consistent theory T ∈ Th(CL) such that ϕ / ∈ T (Lindenbaum Lemma) and show that FmL/Ω(T) is two-valued Boolean algebra and Γ | =2,{1} ϕ. 3rd completeness theorem

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Abstract Lindenbaum Lemma

An element of a closure system C is maximal w.r.t. an element a if it is a maximal (w.r.t. inclusion) element in the set {Y ∈ C | a / ∈ Y}

Lemma (Abstract Lindenbaum Lemma)

Let C be a finitary closure system. If T ∈ C and a / ∈ T, then there is T′ ∈ C such that T′ ⊇ T and T′ is maximal with respect to a.

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Special models

Recall that FiL(A) is a closure system.

Definition

Let L be a logic and A = A, F ∈ MOD∗(L). A is relative subdirectly irreducible, A ∈ MOD∗(L)RSI in symbols, if F is maximal element of FiL(A) w.r.t. some a ∈ A.

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Special models

Recall that FiL(A) is a closure system.

Definition

Let L be a logic and A = A, F ∈ MOD∗(L). A is relative subdirectly irreducible, A ∈ MOD∗(L)RSI in symbols, if F is maximal element of FiL(A) w.r.t. some a ∈ A.

Proposition

Let L be a logic and A = A, F ∈ MOD(L). A∗ ∈ MOD∗(L)RSI iff F is maximal of element of FiL(A) w.r.t. some a ∈ A.

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Special models

Recall that FiL(A) is a closure system.

Definition

Let L be a logic and A = A, F ∈ MOD∗(L). A is relative subdirectly irreducible, A ∈ MOD∗(L)RSI in symbols, if F is maximal element of FiL(A) w.r.t. some a ∈ A.

Proposition

Let L be a logic and A = A, F ∈ MOD(L). A∗ ∈ MOD∗(L)RSI iff F is maximal of element of FiL(A) w.r.t. some a ∈ A.

Theorem (Subdirect representation)

If L is a finitary logic, then every matrix in MOD∗(L) is representable as a subdirect product of matrices in MOD∗(L)RSI.

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The third completeness theorem

Theorem

Let L be a logic. Then for each Γ ∪ {ϕ} ⊆ FmL: Γ ⊢L ϕ iff Γ | =MOD∗(L)RSI ϕ.

Proof.

Soundness: just observe that | =MOD(L) ⊆ | =MOD∗(L)RSI Completeness: assume that Γ ⊢L ϕ We know that there is theory T ⊇ ThL(Γ) maximal w.r.t. ϕ (i.e., ϕ / ∈ T) Thus: FmL, T∗ ∈ MOD∗(L)RSI and e(χ) = [χ] is an Fm∗

L-eval.

We also know: χ ∈ T iff e(χ) ∈ T∗ Then e[Γ] ⊆ T∗ and e(ϕ) / ∈ T∗, because Γ L ϕ.

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