Algebraizing Modal Logic

Algebraizing modal axiomatics

Wang Haoyu

Advanced modal logic

Table of contents

1. Introduction

2. Algebraizing modal axiomatics

3. Limits and further results

1

Intro: an outline of the proof

Formulas and their relations with systems

Proof systems

1. `C φ

2. `Σ φ

Alternative semantics

3. �C φ

4. M,w � φ

5. F � φ

6. Neighbourhood semantics

7. Algebraic semantics

2

Chapter 5.1

Algebraizing propositional logic

• Theorem 5.9 Set algebraizes classical validity.

• Theorem 5.11 BA algebraizes classical theoremhood.

• Theorem 5.16 Stone representation theorem Any boolean algebra is

isomorphic to a set algebra.

• Corollary 5.17 Soundness and weak completeness

�C φiff (5.9)←→ Set � φ ≈ >

iff l l iff (5.16)

`C φiff (5.11)←→ BA � φ ≈ >

3

Main results of 5.2

Algebraizing modal logic

• Theorem 5.25 CmK algebraizes frame validity.

• Theorem 5.27 VΣ algebraizes modal theoremhood.

• Chapter 5.43 The Jonsson-Tarski theorem Any BAO is embeddable

in the full complex algebra of its ultrafilter frame.

Let K be a class of frames and Σ a set of formulas.

K � φiff (5.25)←→ CmK � φ ≈ >

? l l (5.43)

`Kτ Σ φiff (5.27)←→ VΣ � φ ≈ >

4

Algebraizing modal axiomatics

Review (c.f. B.12)

BAOs semantics

An assignment for a set of variables X w.r.t. an algebra (A, I ) is a

function θ : X → A. We can extend it to a meaning function

θ : Form(X )→ A satisfying:

θ(p) = θ(p) for all p ∈ X θ(⊥) = 0

θ(φ1 ∨ φ2) = θ(φ1) + θ(φ2) θ(¬φ) = −θ(φ)

θ(∇(φ1, . . . , φn)) = f∇(θ(φ1) . . . , θ(φn))

Another version of Normality and Additivity (c.f. Definition 5.19)

Recall that 1 = −0 and x · y = −(−x +−y).

• Norm’: f∇(a1, . . . , an) = 1 whenever ai = 1 for some i ∈ [1, n]

• Add’: f∇(a1, . . . , ai · a′i , . . . , an)

= f∇(a1, . . . , ai , . . . , an) · f∇(a1, . . . , a′i , . . . , an) 5

Soundness

Theorem 5.27 (Algebraic Completeness)

• Let τ be a modal similarity type and Σ a set a of τ -formulas.

• Let φ≈ = φ ≈ >. Let Σ≈ = {σ≈|σ ∈ Σ}.

• Let VΣ be the class of BAOs such that VΣ � Σ≈.

• KτΣ is the normal modal τ logic axiomatized by Σ.

• KτΣ is sound and weakly complete with respect to VΣ,

• i.e. `Kτ Σ φ ⇐⇒ VΣ � φ≈ for all formulas φ.

Proof. =⇒ : Suppose `Kτ Σ φ. We show that VΣ � φ≈ by induction on

the length n of the proof of φ in KτΣ.

n = 1

If φ is an axiom, i.e. φ ∈ Σ, since φ≈ ∈ Σ≈, VΣ � φ≈ holds by

definition.6

Cont. soundness n = 1

• Propositional axioms:

θ(p → (q → p)) = θ(¬p ∨ (¬q ∨ p)

= (−θ(p)) + ((−θ(q)) + θ(p)

= ((−θ(p)) + θ(p)) + (−θ(q)) (By(B0)(B1))

= 1 + (−θ(q)) (By(B0))

= 1(?c .f .Definition 5.10) = θ(>)

The other two axioms can be proved similarly.

• K i (version 1): By Add’, we have

VΣ � (∇(r1, . . . , ri−1, p ∧ q, . . . , rn)↔(∇(r1, . . . , ri−1, p ∧ q, . . . , rn) ∧∇(r1, . . . , ri−1, p ∧ q, . . . , rn)))≈.

7

Cont. soundness n = 1

• K i (version 2):

θ(∇(r1, . . . , ri−1, p → q, . . . , rn)

→ ∇(r1, . . . , ri−1, p, . . . , rn)→ ∇(r1, . . . , ri−1, q, . . . , rn))

= (−θ(∇(. . . , p → q, . . . ))) + (−θ(∇(. . . , p, . . . )))

+θ(∇(. . . , q, . . . ))

= (−(f∇(. . . , θ(p → q), . . . ) · (f∇(. . . , θ(p), . . . ))))

+f∇(. . . , θ(q), . . . ))

= (−f∇(. . . , ((−θ(p)) + θ(q)) · θ(p), . . . ))) +f∇(. . . , θ(q), . . . ))

= (−f∇(. . . , (−θ(p)) · θ(p) + θ(q) · θ(p), . . . ))) + f∇(. . . , θ(q), . . . ))

= (−f∇(. . . , θ(q) · θ(p), . . . ))) + f∇(. . . , θ(q), . . . ))

(?) = 1 = θ(>).

8

cont. Soundness n > 1 (all KτΣ-rules are valid on VΣ)

• MP: Suppse φ follows by MP from ψ and ψ → φ. By IH, VΣ � ψ≈

and VΣ � (ψ → φ)≈. That is, given any assignment θ,

θ(ψ) = θ(¬ψ ∨ φ) = 1. Therefore θ(¬ψ) + θ(φ) = 1. Since

θ(¬ψ) = −1 = 0, we have θ(φ) = 1 = θ(>), i.e. VΣ � φ≈.

• USUB: Suppse φ = ψ(p\π) follows from ψ by USUB. By IH,

VΣ � ψ≈. Given any assignment θ, we define θ′ such that

θ′(p) = θ(π). Therefore θ(φ) = θ′(ψ) = 1 = θ(>), i.e. VΣ � φ≈.

• NEC: Suppse φ = ∇(⊥, . . . , ψ, . . . ,⊥) follows from ψ by NEC. By

IH, VΣ � ψ≈. Thus for every R such that Rww1 . . .wn, there is a wi

on which θ(ψ) = 1. By Norm’, θ(φ) = θ(>), i.e. VΣ � φ≈.

Corollary: VKτ Σ = VΣ

Proof : For any φ ∈ KτΣ, VΣ � φ≈. Therefore, VΣ ⊆ VKτ Σ. Since

Σ ⊆ KτΣ, VKτ Σ ⊆ VΣ. Thus VKτ Σ = VΣ.

9

Towards completeness

For any φ, suppose 6`Kτ Σ φ, we need to find an algebra A such that

A ∈ VΣ(?) and A 6� φ≈(??). (A: Lindenbaum-Tarski algebra)

Let τ be an algebraic similarity type, Φ a set of propositional variables

and Λ a mormal modal τ -logic.

Definition 5.28 (Formula algebra of τ over Φ)

• Form(τ,Φ) = (Form(τ,Φ),+,−,⊥, f∇)∇∈τ

• −φ := ¬φ, φ+ ψ := φ ∨ ψ, f∇(t1, . . . , tn) := ∇(t1, . . . , tn).

Definition 5.29

φ ≡Λ ψ iff `Λ φ↔ ψ iff φ and ψ are equivalent modulo Λ.

Definition (Congruence)

Let A be an algebra. An equivalence relation R on A is a congruence iff

for all f ∈ τ , if Ra1b1, . . . ,Ranbn, then RfA(a1, . . . , an)fA(b1, . . . , bn).

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A congruence relation

Proposition 5.30

• ≡Λ is a congruence relation on Form(τ,Φ).

Proof. Since ↔ is an equivalence relation, ≡Λ is also an equivalence

relation. For the three operations in Form(τ,Φ),

• if φi ≡Λ ψi for i ∈ {0, 1}, then `Λ φi ↔ ψi for i ∈ {0, 1}. By USUB,

`Λ φ0 ∨ ψ0 ↔ φ1 ∨ ψ1, which implies φ0 ∨ ψ0 ≡Λ φ1 ∨ ψ1;

• if φ ≡Λ ψ, then `Λ ¬φ¬ ↔ ψ, followed by ¬φ ≡Λ ¬ψ;

• if φi ≡Λ ψi for i ∈ [1, n], then `Λ φi ↔ ψi for i ∈ [1, n]. If we can

show that `Λ ∇(φ1, . . . , φn)→ ∇(ψ1, . . . , ψn), by the symmetry

between φi and ψi , we would have the desired result.

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A proof of `Λ ∇(φ1, . . . , φn)→ ∇(ψ1, . . . , ψn)

(???)

φi ↔ ψi Assum. (1)

⊥ → φi ↔ ψi P theorem (2)

φi → ψi (∧ − (1)) (3)

∇(φ1 ↔ ψ1,⊥, . . . ) NEC (1) (4)

∇(φ1 → ψ1,⊥, . . . ) NEC (3) (5)

∇(φ1 → ψ1,⊥, . . . )→ ∇(φ1,⊥, . . . )→ ∇(ψ1,⊥, . . . ) K (6)

∇(φ1,⊥, . . . )→ ∇(ψ1,⊥, . . . ) MP(5)(6) (7)

(8)

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An ideal candidate

Corollary.

Let [φ] = {ψ|φ ≡Λ ψ}. The following functions are well-defined.

• [φ] + [ψ] := [φ ∨ ψ]

• −[ψ] := [¬ψ]

• f∇([φ1], . . . , [φn]) := [∇(φ1, . . . , φn)]

Definition 5.31

The Lindenbaum-Tarski algebra of a normal modal τ -logic Λ over the

set of generators, i.e. a set of propositional variables Φ is

LΛ(Φ) := (Form(τ,Φ)/ ≡Λ,+,−, f∇).

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Property (??)

Theorem 5.32 (v.s. Theorem 5.14)

Let τ be a modal similarity type, and Λ a normal modal τ -logic. Let φ be

some propositional formula, and Φ a set of proposition letters of size not

smaller than the number of proposition letters occurring in φ. Then

`Λ φ ⇐⇒ LΛ(Φ) � φ≈.

Proof. Assume that Φ contains all variables occuring in φ.

⇐=: Suppose 6`Λ φ. Then by MP, 6`Λ > → φ. Then 6`Λ > ↔ φ, i.e.

φ 6≡Λ > or [φ] 6= [>]. Then we define an assignment θ s.t.

θ(p) = [p] for all p ∈ Φ. We can show by induction on φ that

θ(φ) = [φ]. So θ(φ) 6= θ(>). Thus LΛ(Φ) 6� φ≈.

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cont. Property (??) Soundness

=⇒: Let θ be an assignment s.t. θ(p) = [φp] for all p ∈ Φ. Let

ρ(ψ) = ψ(p1\φp1 ) . . . (pn\φpn ) where {pi |i ∈ [1, n]} is the set of all

variables occuring in ψ.

Lemma. θ(ψ) = [ρ(ψ)]

Proof. We show it by induction on ψ.

- If ψ = p, then θ(ψ) = θ(p) = [φp] = [ψ(p\φp)] = [ρ(ψ)].

- If ψ = ¬φ, then θ(ψ) = −θ(φ) = −[ρ(φ)] = [¬ρ(φ)] = [ρ(ψ)].

- If ψ = φ1 ∨ φ2, the proof is similar.

- If ψ = ∇(φ1, . . . , φn), then θ(ψ) = f∇(θ(φ1), . . . , θ(φn)) =

f∇([ρ(φ1)], . . . , [ρ(φn)]) = [∇([ρ(φ1)], . . . , [ρ(φn)]] = [ρ(ψ)].

By USUB, `Λ ρ(ψ). Therefore we have ρ(ψ) ≡Λ >, i.e. [ρ(ψ)] = [>]. By

the lemma, we have θ(ψ) = [>], i.e. LΛ(Φ) � φ≈.

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Property (?)

Theorem 5.33

Let τ be a modal similarity type, and Λ a normal modal τ -logic. Then for

any set Φ of propositional letters, LΛ(Φ) ∈ VΛ.

Proof. With 5.32, we only have to show that LΛ(Φ) is a BAO. Clearly it

is a boolean algebra. We only have to show that f∇ is indeed an operator

by verifying the Add’ and Norm’ properties.

• Add’: Since we have `Λ ∇(φ1, . . . , φi ∧ φ′i , . . . , φn)↔∇(φ1, . . . , φi , . . . , φn) ∧∇(φ1, . . . , φ

′i , . . . , φn).

f∇([φ1], . . . , [φi ] · [φ′i ], . . . , [φn]) = f∇([φ1], . . . , [φi ∧ φ′i ], . . . , [φn])

=[f∇(φ1, . . . , φi ∧ φ′i , . . . , φn)]

=[f∇(φ1, . . . , φi , . . . , φn) ∧ f∇(φ1, . . . , φ′i , . . . , φn)]

=[f∇(φ1, . . . , φi , . . . , φn)] · [f∇(φ1, . . . , φ′i , . . . , φn)]

=f∇([φ1], . . . , [φi ], . . . , [φn]) · f∇([φ1], . . . , [φ′i ], . . . , [φn]).

16

cont. Property (?) Normality

• Norm’: Suppose there is a ai ∈ Form(τ,Φ)/ ≡Λ such that ai = 1,

i.e. ai = [>]. Then f∇([φ1], . . . , [>], . . . , [φn]) =

[f∇(φ1, . . . ,>, . . . , φn)] = [∇(φ1, . . . ,>, . . . , φn)] = [>] = 1.

In contrast to frame semantics

• Immediately we have: (Normal ?) modal logics are always complete

w.r.t. the variety (c.f. Definition B.7) of BAOs where their axioms

are valid.

• Note that modal logics are not necessarily complete w.r.t. the class

of frames that they define.

17

Limits and further results

However...

We want completeness w.r.t. complex algebras rather than abstract

BAOs.

Jonsson-Tarski theorem

Every BAO is isomorphic to a complex algebra.

By taking the complex algebra of the ultrafilter frame of a BAO, we

obtain the canonical embedding algebra of the original BAO.

18

The filter of an algebra

Definition 5.34 (Filter of algebra v.s. filter over set)

A filter of a boolean algebra A = (A,+,−, 0) is a subset F ⊆ A satisfying

(F1) 1 ∈ F ,

(F2) If a, b ∈ F then a · b ∈ F ,

(F3) If a ∈ F and a ≤ b then b ∈ F .

A filter is proper if it does not contain the smallest element 0, or,

equivalently, if F = A. An ultrafilter is a proper filter satisfying

(F4) For every a ∈ A, either a or −a belongs to F .

Proposition 5.38 (Ultrafilter theorem)

Let A be a boolean algebra, a an element of A, and F a proper filter of A

that does not contain a. Then there is an ultrafilter extending F that

does not contain a.

19

Theorem 5.16

The Stone representation theorem

Any boolean algebra is isomorphic to a field of sets, and hence, to a

subalgebra of a power of 2. As a consequence, the variety of boolean

algebras is generated by the algebra 2:

BF = V({2})

Outline of proof:

• Let A be a boolean algebra and the representation function

r : A→ P(Uf A) be

r(a) = {u ∈ Uf A|a ∈ u}

• r is a homomorphism.

• r is injective. (by Proposition 5.38)20

The Jonsson-Tarski theorem

Definition 5.40

• The ultrafilter frame of A: A+ = (Uf A,Qf∇)∇∈τ .

• The (canonical) embedding algebra of A: EmA = (A+)+

Theorem 5.43 (The Jonsson-Tarski theorem)

Let A be a BAO. Then the representation function r : A→ P(Uf A)

given by

r(a) = {u ∈ Uf A|a ∈ u}

is an embedding of A into EmA.

21

Canonicity

K � φiff (5.25)←→ CmK � φ ≈ >

? l l (5.43)

`Kτ Σ φiff (5.27)←→ VΣ � φ ≈ >

Exercise 5.2.6 (The complete variety of BAOs)

A variety V is complete if there is a frame class K that generates it, i.e.

V = HSPCmK . A logic Λ is complete iff VΛ is a complete variety.

22