INTRODUCTION Boolean Algebra with operators Ultrafilters Ultrafilter Extensions The Message

INVARIANCE AND UNDEFINABILITY

Attila Molnár Eötvös Loránd University

December 12, 2013 INTRODUCTION Boolean Algebra with operators Ultrafilters Ultrafilter Extensions The Message

Optional Exercise Solving Seminar

A seminar where we solve exercises, e.g. homeworks, together.

I Thursday 12:00-14:00 INTRODUCTION Boolean Algebra with operators Ultrafilters Ultrafilter Extensions The Message

OPERATIONS

Theorem (Goldblatt-Thomason) A first-order definable class K of frames is modally definable iff it is closed under (i) disjoint unions (ii) subframe generation (iii) zig-zag morphisms and reflects (iv) Ultrafilter extensions INTRODUCTION Boolean Algebra with operators Ultrafilters Ultrafilter Extensions The Message

INSTEADOFTHIS WE CHOOSE THE COOL WAY INTRODUCTION Boolean Algebra with operators Ultrafilters Ultrafilter Extensions The Message

BOOLEAN ALGEBRAWITH OPERATORS INTRODUCTION Boolean Algebra with operators Ultrafilters Ultrafilter Extensions The Message

BOOLEAN ALGEBRA WITH OPERATORS

Definition (BOOLEAN ALGEBRA) A Boolean algebra is an algebra

A = hA, +, −, 0i

satisfying the equations

x + y = y + x x · y = y · x (x + y) + z = x + (y + z)(x · y) · z = y · (x · z) x + 0 = x x · 1 = 1 x + (−x) = 1 x · (−x) = 0 x + (y · z) = (x + y) · (x + z) x · (y + z) = (x · y) + (x · z) INTRODUCTION Boolean Algebra with operators Ultrafilters Ultrafilter Extensions The Message

BOOLEAN ALGEBRA WITH OPERATORS

Definition (BOOLEAN ALGEBRA WITH OPERATORS) A Boolean algebra is an algebra

A = hA, +, −, 0, mi

satisfying the equations

x + y = y + x x · y = y · x (x + y) + z = x + (y + z)(x · y) · z = y · (x · z) x + 0 = x x · 1 = 1 x + (−x) = 1 x · (−x) = 0 x + (y · z) = (x + y) · (x + z) x · (y + z) = (x · y) + (x · z)

m(0) = 0 m(x + y) = m(x) + m(y) INTRODUCTION Boolean Algebra with operators Ultrafilters Ultrafilter Extensions The Message

BOOLEAN ALGEBRA WITH OPERATORS

Claim (MONOTONICITY)

a ≤ b =⇒ m(a) ≤ m(b)

Proof. a ≤ b ⇐⇒ a + b = b ⇐⇒ m(a + b) = m(b) m(a + b) = m(a) + m(b) =⇒ m(a) + m(b) = m(b) ⇐⇒ m(a) ≤ m(b) INTRODUCTION Boolean Algebra with operators Ultrafilters Ultrafilter Extensions The Message

LINDENBAUM-TARSKI ALGEBRAS INTRODUCTION Boolean Algebra with operators Ultrafilters Ultrafilter Extensions The Message

FORMULA ALGEBRA Definition

def Form(Φ) = hFORM(Φ), +, −, 0, mi ϕ + ψ def= (ϕ ∨ ψ) −ϕ def= ¬ϕ mϕ def= ϕ.

Definition (Provable equivalence) Let ` denote the derivability in the Frege–Hilbert calculus.

ϕ ≡ ψ ⇔def ` ϕ ↔ ψ

Theorem Provable equivalence is a congruence relation INTRODUCTION Boolean Algebra with operators Ultrafilters Ultrafilter Extensions The Message

LINDENBAUM-TARSKI ALGEBRA Definition

def FORM/≡ = {Γ ⊆ FORM :(∀ϕ, ψ ∈ Γ) ϕ ≡ ψ} Γ = [ϕ] ⇔def ϕ ∈ Γ

The Lindenbaum–Tarski algebra of FORM(Φ) and ` is the algebra def ∗ ∗ ∗ ∗ Form/≡ = hFORM/≡, + , − , 0 , m i, where the functions defined on the equivalence classes (instead of their elements):

0∗ def= [⊥] −∗[ϕ] def= [−ϕ] [ϕ] +∗ [ψ] def= [ϕ + ψ] m∗[ϕ] def= [ ϕ] INTRODUCTION Boolean Algebra with operators Ultrafilters Ultrafilter Extensions The Message

COMPLEX ALGEBRAS INTRODUCTION Boolean Algebra with operators Ultrafilters Ultrafilter Extensions The Message

COMPLEX ALGEBRAS

Definition (POWER SET ALGEBRA) Definition (FULL COMPLEX ALGEBRA) The power set algebra of W: The full complex algebra of F = hW, Ri:

def + def P(W) = hP(W), ∪, W−, ∅i F = hP(W), ∪, W−, ∅, mi

Definition (SET ALGEBRA) Definition (COMPLEX ALGEBRA) A set algebra is a subalgebra of a A complex algebra is a subalgebra of a power set algebra full complex algebra

I Complex Algebras are Boolean algebras with operators.

[[ϕ]]M = θ˜(ϕ) F |= ϕ ⇐⇒ F+ |= ϕ = 1 K |= ϕ ⇐⇒ Cm(K) |= ϕ = 1 F |= ϕ ↔ ψ ⇐⇒ F+ |= ϕ = ψ K |= ϕ ↔ ψ ⇐⇒ Cm(K) |= ϕ = ψ INTRODUCTION Boolean Algebra with operators Ultrafilters Ultrafilter Extensions The Message

ULTRAFILTERS INTRODUCTION Boolean Algebra with operators Ultrafilters Ultrafilter Extensions The Message

FILTERS –GENERALIZATION Let W be an arbitrary set and A be an arbitrary Boolean algebra. Recall that X ⊆ Y ⇔ X ∩ Y = X Remember that a ≤ b ⇔def a · b = a Definition Definition Let F be a set of subsets of W, Let F be a set of elements of A, i.e., F ∈ PP(W). i.e., F ∈ P(A). 1. F is a filter over W if 1. F is a filter of A if I W ∈ F I 1 ∈ F I X, Y ∈ F ⇒ X ∩ Y ∈ F I a, b ∈ F ⇒ a · b ∈ F I X ∈ F, X ⊆ Y ⇒ Y ∈ F I a ∈ F, a ≤ b ⇒ b ∈ F 2. A filter is proper 2. A filter is proper I ∅ ∈/ F I 0 ∈/ F 3. A filter is ultra if 3. A filter is ultra if I X ∈ F or W − X ∈ F I a ∈ F or − a ∈ F

So the ultrafilters over W are the ultrafilters of P(W). INTRODUCTION Boolean Algebra with operators Ultrafilters Ultrafilter Extensions The Message

EXAMPLES AND ANALOGIES

Filter of a BA Filter over a set W Max. cons. sets Filter of P(W) Filter of FormW/≡

1 ∈ F W ∈ F > ∈ Γ a, b ∈ F X, Y ∈ F ϕ, ψ ∈ Γ a · b ∈ F X ∩ Y ∈ F ϕ ∧ ψ ∈ Γ

a ∈ F X ∈ F ϕ ∈ Γ a ≤ b X ⊆ Y ϕ ` ψ b ∈ F Y ∈ F ψ ∈ Γ

0 ∈/ F ∅ ∈/ F ⊥ ∈/ Γ a ∈ F or X ∈ F or ϕ ∈ Γ or −a ∈ F W−X ∈ F ¬ϕ ∈ Γ

HW: Prove that ϕ ` ψ ⇐⇒ ϕ · ψ ≡ ϕ ⇔def ` ϕ ∧ ψ ↔ ϕ. INTRODUCTION Boolean Algebra with operators Ultrafilters Ultrafilter Extensions The Message

MEMENTO

def 0 0 [[ ϕ]]M = l ([[ϕ]]M) = {w ∈ W :(∀w R w) w ∈ [[ϕ]]M} def 0 0 [[ ϕ]]M = m ([[ϕ]]M) = {w ∈ W :(∃w R w) w ∈ [[ϕ]]M}

X↑ def= {Y : X ⊆ Y ⊆ W} w↑ def= {w}↑ a↑ def= {b : a ≤ b ≤ 1}

−(Γ) def= {ϕ : ϕ ∈ Γ} l−(F) def= {X : l(X) ∈ F} +(Γ) def= { ϕ : ϕ ∈ Γ} m+(F) def= {m(X): X ∈ F}

− − ↑ ↑ wRv ⇒ (thM(w)) ⊆ thM(v) wRv ⇔ l (w ) ⊆ v m m + ↑ + ↑ thM(w) ⊇ (thM(v)) w ⊇ m (v )

URueF ⇔def l−(U) ⊆ F m U ⊇ m+(F) INTRODUCTION Boolean Algebra with operators Ultrafilters Ultrafilter Extensions The Message

ULTRAFILTER EXTENSIONS TOBETRUEINAWORLDIS TOBEANELEMENTOFAWORLD

INTRODUCTION Boolean Algebra with operators Ultrafilters Ultrafilter Extensions The Message

ULTRAFILTER EXTENSION Definition Let F = hW, Ri, and M, = hF, vi. The ultrafilter extension of F and M are

Fue def= hUf(W), Ruei Uf(W) def= {F : F is an ultrafilter over W} FRueF 0 ⇔def l−(F) ⊆ F 0 Mue def= hFue, vuei vue(p) def= {F : v(p) ∈ F}

So p is true in a world (ultrafilter) of the ultrafilter-model m the proposition [[p]]M is in the world (ultrafilter). INTRODUCTION Boolean Algebra with operators Ultrafilters Ultrafilter Extensions The Message

ULTRAFILTER EXTENSION Definition Let F = hW, Ri, and M, = hF, vi. The ultrafilter extension of F and M are

Fue def= hUf(W), Ruei Uf(W) def= {F : F is an ultrafilter over W} FRueF 0 ⇔def l−(F) ⊆ F 0 Mue def= hFue, vuei vue(p) def= {F : v(p) ∈ F} OBETRUEINAWORLDIS TSo p is true in a world (ultrafilter) of the ultrafilter-model TOBEANELEMENTOFm AWORLD the proposition [[p]]M is in the world (ultrafilter). INTRODUCTION Boolean Algebra with operators Ultrafilters Ultrafilter Extensions The Message

INVARIANCE OF TRUTH Theorem Truth is invariant under taking ultrafilter extensions.

ue M, w ϕ ⇐⇒ M , w ↑ ϕ

So ϕ is true in a world of the model m ϕ is true in the ultrafilter generated by that world of the ultrafilter model.

Lemma “To be true is to be in the world.”

ue M , F ϕ ⇐⇒ [[ϕ]]M ∈ F

[[ϕ]]Mue = {F : [[ϕ]]M ∈ F} 2. ¬: 3. ϕ ∧ ψ: 4. ϕ:

INTRODUCTION Boolean Algebra with operators Ultrafilters Ultrafilter Extensions The Message

INVARIANCE OF TRUTH

Proof. By structural induction: 1. p: by the definition of v 3. ϕ ∧ ψ: 4. ϕ:

INTRODUCTION Boolean Algebra with operators Ultrafilters Ultrafilter Extensions The Message

INVARIANCE OF TRUTH

Proof. By structural induction: 1. p: by the definition of v ¬ 2. : ue ue M , F ¬ϕ ⇐⇒ M , F 6 ϕ m [[ϕ]]M ∈/ F m [[¬ϕ]]M ∈ F ⇐⇒ W − [[ϕ]]M ∈ F 4. ϕ:

INTRODUCTION Boolean Algebra with operators Ultrafilters Ultrafilter Extensions The Message

INVARIANCE OF TRUTH

Proof. By structural induction: 1. p: by the definition of v 2. ¬:... 3. ϕ ∧ ψ:

ue ue ue M , F ϕ ∧ ψ ⇐⇒ M , F ϕ and M , F ψ m [[ϕ]]M ∈ F and [[ψ]]M ∈ F m [[ϕ ∧ ψ]]M ∈ F ⇐⇒ [[ϕ]]M ∩ [[ψ]]M ∈ F INTRODUCTION Boolean Algebra with operators Ultrafilters Ultrafilter Extensions The Message

INVARIANCE OF TRUTH

Proof. By structural induction: 1. p: by the definition of v 2. ¬:... 3. ϕ ∧ ψ:... 4. ϕ:

ue 0 ue ue 0 M , F ϕ ⇐⇒ (∃F R F)M , F ϕ m 0 − ue 0 (∃F ⊇ l (F))M , F ϕ m + 0 ue 0 (∃ m (F ) ⊆ F)M , F ϕ m ? + 0 0 [[ ϕ]]M ∈ F ⇐⇒ (∃ m (F ) ⊆ F)[[ϕ]]M ∈ F INTRODUCTION Boolean Algebra with operators Ultrafilters Ultrafilter Extensions The Message

INVARIANCE OF TRUTH

Proof. By structural induction: 1. p: by the definition of v 2. ¬:... 3. ϕ ∧ ψ:... 4. ϕ:

[[ ϕ]]M ∈ F ⇐⇒ m([[ϕ]]M) ∈ F ⇑ 0 0 0 (∃F )(∀X ∈ F )m(X) ∈ F and [[ϕ]]M ∈ F m 0 + 0 0 (∃F , m (F ) ⊆ F)[[ϕ]]M ∈ F INTRODUCTION Boolean Algebra with operators Ultrafilters Ultrafilter Extensions The Message

INVARIANCE OF TRUTH

Proof. By structural induction: 1. p: by the definition of v 2. ¬:... 3. ϕ ∧ ψ:...

4. ϕ: [[ ϕ]]M ∈ F ⇐⇒ m([[ϕ]]M) ∈ F ⇓ 0 − 0 (∃F ⊇ l (F))[[ϕ]]M ∈ F

− I Has l (F) ∪ {[[ϕ]]M} f.i.p? − I l (F) is closed under taking intersections. I l(X) ∈ F ⇒ X ∪ [[ϕ]]M 6= ∅? I l(X), [[ ϕ]]M ∈ F ⇒ l(X) ∩ [[ ϕ]]M ∈ F. (closed under ∩) I ∃w ∈ l(X) ∩ [[ ϕ]]M since the filter is proper 0 I (∃w ∈ X ∩ [[ϕ]]M) INTRODUCTION Boolean Algebra with operators Ultrafilters Ultrafilter Extensions The Message

INVARIANCE OF TRUTH

Theorem Truth is invariant under taking ultrafilter extensions.

ue M, w ϕ ⇐⇒ M , w ↑ ϕ

Proof.

ue M, w ϕ ⇐⇒ w ∈ [[ϕ]]M ⇐⇒ [[ϕ]]M ∈ w ↑ ⇐⇒ M , w ↑ ϕ INTRODUCTION Boolean Algebra with operators Ultrafilters Ultrafilter Extensions The Message

INVARIANCE OF VALIDITY

Theorem Ultrafilter extensions reflects validity

Fue |= ϕ =⇒ F |= ϕ INTRODUCTION Boolean Algebra with operators Ultrafilters Ultrafilter Extensions The Message

INVARIANCE OF VALIDITY

Proof. By contraposition:

F 6|= ϕ ⇐⇒ F, v, w 6 ϕ m ue ue ue F 6|= ϕ ⇐ F , v , w ↑6 ϕ INTRODUCTION Boolean Algebra with operators Ultrafilters Ultrafilter Extensions The Message

THE MESSAGE Definition (Representation) “Meaning of X is the worlds (filters) in which it is true (contained)”

Xue def= {F : X ∈ F}

ue Of course, [[ϕ]]M = [[ϕ]]Mue

INTRODUCTION Boolean Algebra with operators Ultrafilters Ultrafilter Extensions The Message

GENERALIZE

Lemma (Truth-preserving) “To be true is to be in the world.”

ue M , F ϕ ⇐⇒ [[ϕ]]M ∈ F

[[ϕ]]Mue = {F : [[ϕ]]M ∈ F}

Can we generalize the latter to propositions? Definition (Representation) “Meaning of X is the worlds (filters) in which it is true (contained)”

Xue def= {F : X ∈ F}

ue Of course, [[ϕ]]M = [[ϕ]]Mue

INTRODUCTION Boolean Algebra with operators Ultrafilters Ultrafilter Extensions The Message

GENERALIZE

Lemma (Truth-preserving) “To be true is to be in the world.”

ue M , F ϕ ⇐⇒ [[ϕ]]M ∈ F

[[ϕ]]Mue = {F : [[ϕ]]M ∈ F}

Can we generalize the latter to propositions? OFCOURSE INTRODUCTION Boolean Algebra with operators Ultrafilters Ultrafilter Extensions The Message

GENERALIZE

Lemma (Truth-preserving) “To be true is to be in the world.”

ue M , F ϕ ⇐⇒ [[ϕ]]M ∈ F

[[ϕ]]Mue = {F : [[ϕ]]M ∈ F}

Can we generalize the latter to propositions? Definition (Representation) “Meaning of X is the worlds (filters) in which it is true (contained)”

Xue def= {F : X ∈ F}

ue Of course, [[ϕ]]M = [[ϕ]]Mue INTRODUCTION Boolean Algebra with operators Ultrafilters Ultrafilter Extensions The Message

GENERALIZE

Lemma (Truth-preserving) “To be true is to be in the world.”

ue M , F ϕ ⇐⇒ [[ϕ]]M ∈ F

[[ϕ]]Mue = {F : [[ϕ]]M ∈ F}

Can we generalize the latter to propositions? Definition (Representation) “Meaning of X is the worlds (filters) in which it is true (contained)” OBETRUEINAWORLDIS T Xue def= {F : X ∈ F}

TOBEue ANELEMENTOFAWORLD Of course, [[ϕ]]M = [[ϕ]]Mue INTRODUCTION Boolean Algebra with operators Ultrafilters Ultrafilter Extensions The Message

EMBEDDING Xue def= {F : X ∈ F}

Homework (EMBEDDING) + + Prove that ue : F  (F )+: 1. (W − X)ue = Uf(W) − Xue 2. (X ∩ Y)ue = Xue ∩ Yue 3 m(X)ue = mue(Xue) ↑ 4.1 X 6= ∅ ⇒ ∅ ∈/ X 4.2 X 6⊆ Y ⇒ X − Y 6= ∅ 4.3 X 6⊆ Y ⇒ X ∩ (W − Y) 6= ∅ 4.4 X 6⊆ Y ⇒ X↑ ∪ {W − Y} has f.i.p. 4.5 X 6= Y ⇒ Xue 6= Yue INTRODUCTION Boolean Algebra with operators Ultrafilters Ultrafilter Extensions The Message

EMBEDDING What if we generalize ue to Boolean algebras?

Homework (EMBEDDING) + + Prove that ue : F  (F )+: 1. (W − X)ue = Uf(W) − Xue 2. (X ∩ Y)ue = Xue ∩ Yue 3 m(X)ue = mue(Xue) ↑ 4.1 X 6= ∅ ⇒ ∅ ∈/ X 4.2 X 6⊆ Y ⇒ X − Y 6= ∅ 4.3 X 6⊆ Y ⇒ X ∩ (W − Y) 6= ∅ 4.4 X 6⊆ Y ⇒ X↑ ∪ {W − Y} has f.i.p. 4.5 X 6= Y ⇒ Xue 6= Yue INTRODUCTION Boolean Algebra with operators Ultrafilters Ultrafilter Extensions The Message

EMBEDDING aue def= {F : a ∈ F}

Homework (EMBEDDING)

Prove that ue : A  A+: 1. (−a)ue = −aue 2. a · bue = aue ∩ bue 3. (f (b))ue = m(bue) 4.1 a 6= 0 ⇒ 0 ∈/ a↑ 4.2 a 6≤ b ⇒ a ∩ −b 6= 0 4.3 a 6≤ b ⇒ a↑ ∪ {−b} has f.i.p. 4.4 a 6= b ⇒ aue 6= bue INTRODUCTION Boolean Algebra with operators Ultrafilters Ultrafilter Extensions The Message

EMBEDDING aue def= {F : a ∈ F}

Homework (EMBEDDING)

Prove that ue : A  A+: 1. (−a)ue = −aue 2. a · bue = aue ∩ bue 3. (f (b))ue = m(bue) 4.1 a 6= 0 ⇒ 0 ∈/ a↑ 4.2 a 6≤ b ⇒ a ∩ −b 6= 0 4.3 a 6≤ b ⇒ a↑ ∪ {−b} has f.i.p. 4.4 a 6= b ⇒ aue 6= bue

Corollary (Stone) Every BA is isomorphic to a set algebra, i.e., every BA is embeddable to a power set algebra, or, every BA is embeddable to its ultrafilter frame. INTRODUCTION Boolean Algebra with operators Ultrafilters Ultrafilter Extensions The Message

EMBEDDING aue def= {F : a ∈ F}

Homework (EMBEDDING)

Prove that ue : A  A+: 1. (−a)ue = −aue 2. a · bue = aue ∩ bue 3. (f (b))ue = m(bue) 4.1 a 6= 0 ⇒ 0 ∈/ a↑ 4.2 a 6≤ b ⇒ a ∩ −b 6= 0 4.3 a 6≤ b ⇒ a↑ ∪ {−b} has f.i.p. 4.4 a 6= b ⇒ aue 6= bue

Corollary (Jónsson-Tarski) Every BAO is isomorphic to a complex algebra, i.e., every BAO embeddable to a full complex algebra, or, every BAO is embeddable to its ultrafilter frame. INTRODUCTION Boolean Algebra with operators Ultrafilters Ultrafilter Extensions The Message

CANONICALMODELS TOBETRUEINAWORLDIS TOBEANELEMENTOFAWORLD

INTRODUCTION Boolean Algebra with operators Ultrafilters Ultrafilter Extensions The Message

CANONICALMODELS

Question What is the embedding algebra of the Lindenbaum-Tarski algebra? TOBETRUEINAWORLDIS TOBEANELEMENTOFAWORLD

INTRODUCTION Boolean Algebra with operators Ultrafilters Ultrafilter Extensions The Message

CANONICALMODELS

Question What is the embedding algebra of the Lindenbaum-Tarski algebra?

Corollary (Strong Completeness)

Γ  ϕ =⇒ Γ ` ϕ INTRODUCTION Boolean Algebra with operators Ultrafilters Ultrafilter Extensions The Message

CANONICALMODELS

Question What is the embedding algebra of the Lindenbaum-Tarski algebra?

Corollary (Strong Completeness)

Γ  ϕ =⇒ Γ ` ϕ

TOBETRUEINAWORLDIS TOBEANELEMENTOFAWORLD INTRODUCTION Boolean Algebra with operators Ultrafilters Ultrafilter Extensions The Message

CANONICALMODELS

Question How can we prove strong completeness theorems for other modal logics than K?

F |= ϕ =⇒`F ϕ INTRODUCTION Boolean Algebra with operators Ultrafilters Ultrafilter Extensions The Message

INCOMPLETENESS INTRODUCTION Boolean Algebra with operators Ultrafilters Ultrafilter Extensions The Message

PUZZLE

Definition (Thomason’s tense logic) The logic Tho is the smallest set of formulas containing K and the schemata

DF : F ϕ → F ϕ

HF : F ϕ∧ F ψ → F (ϕ∧ F ψ) ∨ F (ϕ∧ψ) ∨ F ( F ϕ∧ψ) GLP : P ( P ϕ → ϕ) → P ϕ

Claim The McKinsey schema

MF : F F ϕ → F F ϕ

is invalid in every frame of Tho, i.e., Tho ⊕ M has no frames.

Claim Tho ⊕ M is consistent. INTRODUCTION Boolean Algebra with operators Ultrafilters Ultrafilter Extensions The Message

PUZZLE

Definition (Cofinality) Let hW, Ri be a strict total order and X ⊆ W. X is cofinal in W if for every w ∈ W there is an x ∈ X such that wRx.

Examples: odd numbers, even numbers, primes in hN,

I Let hW, Ri be a frame of Tho and w ∈ W be arbitrary.

def def I Let U = {x ∈ W : wRx} and RU = R  U

I hU, RUi is a strict total order with no upper bound. I There is a S 6= ∅, such that both S and U − S are cofinal. [BRV 4.48] def I Let v(p) = S.

I hW, Ri, w 6 F F p → F F p INTRODUCTION Boolean Algebra with operators Ultrafilters Ultrafilter Extensions The Message

SOLUTION

Homework M is valid on hN, <, Ai where

A def= {X : X is finite or cofinite} INTRODUCTION Boolean Algebra with operators Ultrafilters Ultrafilter Extensions The Message

SOLUTION

Theorem If Γ is consistent, then there is a general frame in which there is a world in which it is valid.

If it is consistent, then it has a Kripke-model M = hW, R, vi such that in a w ∈ W. M, w Γ Take the general frame

def F = hW, R, {[[ϕ]]M : ϕ ∈ Γ}i

with the world w. INTRODUCTION Boolean Algebra with operators Ultrafilters Ultrafilter Extensions The Message

SOLUTION

Theorem Let Λ be a normal modal logic. Then Λ is sound and strongly complete with respect to the class of general Λ-frames.

See 5.64: Copy the trick from the previous slide but with the canonical model, then continue it in the usual way.