algebraizing modal logic - algebraizing modal...
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Algebraizing Modal Logic
Algebraizing modal axiomatics
Wang Haoyu
Advanced modal logic
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Table of contents
1. Introduction
2. Algebraizing modal axiomatics
3. Limits and further results
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Intro: an outline of the proof
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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
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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 ≈ >
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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 ≈ >
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Algebraizing modal axiomatics
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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
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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
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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)))≈.
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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(>).
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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.
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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]).
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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.
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Limits and further results
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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.
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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.
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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
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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.
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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.
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