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Algebra and Modal Logic 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 INSTEAD OF THIS WE CHOOSE THE COOL WAY INTRODUCTION Boolean Algebra with operators Ultrafilters Ultrafilter Extensions The Message BOOLEAN ALGEBRA WITH 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 ' + def= (' _ ) −' def= :' m' def= ': Definition (Provable equivalence) Let ` denote the derivability in the Frege–Hilbert calculus. ' ≡ ,def ` ' $ Theorem Provable equivalence is a congruence relation INTRODUCTION Boolean Algebra with operators Ultrafilters Ultrafilter Extensions The Message LINDENBAUM-TARSKI ALGEBRA Definition def FORM=≡ = fΓ ⊆ FORM :(8'; 2 Γ) ' ≡ g Γ = ['] ,def ' 2 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= [?] −∗['] def= [−'] ['] +∗ [ ] def= [' + ] m∗['] def= [ '] 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. [[']]M = θ~(') F j= ' () F+ j= ' = 1 K j= ' () Cm(K) j= ' = 1 F j= ' $ () F+ j= ' = K j= ' $ () Cm(K) j= ' = 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 2 PP(W). i.e., F 2 P(A). 1. F is a filter over W if 1. F is a filter of A if I W 2 F I 1 2 F I X; Y 2 F ) X \ Y 2 F I a; b 2 F ) a · b 2 F I X 2 F; X ⊆ Y ) Y 2 F I a 2 F; a ≤ b ) b 2 F 2. A filter is proper 2. A filter is proper I ? 2= F I 0 2= F 3. A filter is ultra if 3. A filter is ultra if I X 2 F or W − X 2 F I a 2 F or − a 2 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 2 F W 2 F > 2 Γ a; b 2 F X; Y 2 F '; 2 Γ a · b 2 F X \ Y 2 F ' ^ 2 Γ a 2 F X 2 F ' 2 Γ a ≤ b X ⊆ Y ' ` b 2 F Y 2 F 2 Γ 0 2= F ? 2= F ? 2= Γ a 2 F or X 2 F or ' 2 Γ or −a 2 F W−X 2 F :' 2 Γ HW: Prove that ' ` () ' · ≡ ' ,def ` ' ^ $ '. INTRODUCTION Boolean Algebra with operators Ultrafilters Ultrafilter Extensions The Message MEMENTO def 0 0 [[ ']]M = l ([[']]M) = fw 2 W :(8w R w) w 2 [[']]Mg def 0 0 [[ ']]M = m ([[']]M) = fw 2 W :(9w R w) w 2 [[']]Mg X" def= fY : X ⊆ Y ⊆ Wg w" def= fwg" a" def= fb : a ≤ b ≤ 1g −(Γ) def= f' : ' 2 Γg l−(F) def= fX : l(X) 2 Fg +(Γ) def= f ' : ' 2 Γg m+(F) def= fm(X): X 2 Fg − − " " 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 TO BE TRUE IN A WORLD IS TO BE AN ELEMENT OF A WORLD 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= fF : F is an ultrafilter over Wg FRueF 0 ,def l−(F) ⊆ F 0 Mue def= hFue; vuei vue(p) def= fF : v(p) 2 Fg 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= fF : F is an ultrafilter over Wg FRueF 0 ,def l−(F) ⊆ F 0 Mue def= hFue; vuei vue(p) def= fF : v(p) 2 Fg O BE TRUE IN A WORLD IS TSo p is true in a world (ultrafilter) of the ultrafilter-model TO BE AN ELEMENTm OF A WORLD 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 ' () M ; w " ' So ' is true in a world of the model m ' 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 ' () [[']]M 2 F [[']]Mue = fF : [[']]M 2 Fg 2. :: 3. ' ^ : 4. ': 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. ' ^ : 4. ': 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 :' () M ; F 6 ' m [[']]M 2= F m [[:']]M 2 F () W − [[']]M 2 F 4. ': 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. ' ^ : ue ue ue M ; F ' ^ () M ; F ' and M ; F m [[']]M 2 F and [[ ]]M 2 F m [[' ^ ]]M 2 F () [[']]M \ [[ ]]M 2 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. ' ^ :... 4. ': ue 0 ue ue 0 M ; F ' () (9F R F)M ; F ' m 0 − ue 0 (9F ⊇ l (F))M ; F ' m + 0 ue 0 (9 m (F ) ⊆ F)M ; F ' m ? + 0 0 [[ ']]M 2 F () (9 m (F ) ⊆ F)[[']]M 2 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. ' ^ :... 4. ': [[ ']]M 2 F () m([[']]M) 2 F * 0 0 0 (9F )(8X 2 F )m(X) 2 F and [[']]M 2 F m 0 + 0 0 (9F ; m (F ) ⊆ F)[[']]M 2 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. ' ^ :... 4. ': [[ ']]M 2 F () m([[']]M) 2 F + 0 − 0 (9F ⊇ l (F))[[']]M 2 F − I Has l (F) [ f[[']]Mg f.i.p? − I l (F) is closed under taking intersections. I l(X) 2 F ) X [ [[']]M 6= ?? I l(X); [[ ']]M 2 F ) l(X) \ [[ ']]M 2 F.
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