Post Completeness in Congruential Modal Logics Peter Fritz University of Oslo peter.fritz@ifikk.uio.no AiML September 2, 2016 1 / 10
Post completeness Let L be a set (the formulas); let C ⊆ P ( L ) such that L ∈ C (the logics). 2 / 10
Post completeness Let L be a set (the formulas); let C ⊆ P ( L ) such that L ∈ C (the logics). Λ ∈ C is Post complete in C iff Λ � = L and there is no Λ ′ ∈ C such that Λ ⊂ Λ ′ ⊂ L . 2 / 10
Post completeness Let L be a set (the formulas); let C ⊆ P ( L ) such that L ∈ C (the logics). Λ ∈ C is Post complete in C iff Λ � = L and there is no Λ ′ ∈ C such that Λ ⊂ Λ ′ ⊂ L . In short: being Post complete is being a co-atom. 2 / 10
Post completeness Let L be a set (the formulas); let C ⊆ P ( L ) such that L ∈ C (the logics). Λ ∈ C is Post complete in C iff Λ � = L and there is no Λ ′ ∈ C such that Λ ⊂ Λ ′ ⊂ L . In short: being Post complete is being a co-atom. Theorem (Makinson 1971): There are two logics Post complete in normal modal logics, Triv = K � p ↔ p and Ver = K � p 2 / 10
Post completeness Let L be a set (the formulas); let C ⊆ P ( L ) such that L ∈ C (the logics). Λ ∈ C is Post complete in C iff Λ � = L and there is no Λ ′ ∈ C such that Λ ⊂ Λ ′ ⊂ L . In short: being Post complete is being a co-atom. Theorem (Makinson 1971): There are two logics Post complete in normal modal logics, Triv = K � p ↔ p and Ver = K � p What about other lattices of modal logics? 2 / 10
Congruential modal logics L : propositional language with operators ⊤ , ¬ , ∧ and � . 3 / 10
Congruential modal logics L : propositional language with operators ⊤ , ¬ , ∧ and � . Modal logic : Λ ⊆ L containing all tautologies and closed under MP and US. 3 / 10
Congruential modal logics L : propositional language with operators ⊤ , ¬ , ∧ and � . Modal logic : Λ ⊆ L containing all tautologies and closed under MP and US. Congruential modal logic (CML) : Modal logic Λ closed under ϕ ↔ ψ/ � ϕ ↔ � ψ . 3 / 10
Congruential modal logics L : propositional language with operators ⊤ , ¬ , ∧ and � . Modal logic : Λ ⊆ L containing all tautologies and closed under MP and US. Congruential modal logic (CML) : Modal logic Λ closed under ϕ ↔ ψ/ � ϕ ↔ � ψ . Modal algebra : A = � A, 1 , − , ⊓ , ∗� such that � A, 1 , − , ⊓� is a Boolean algebra and ∗ : A → A . 3 / 10
Congruential modal logics L : propositional language with operators ⊤ , ¬ , ∧ and � . Modal logic : Λ ⊆ L containing all tautologies and closed under MP and US. Congruential modal logic (CML) : Modal logic Λ closed under ϕ ↔ ψ/ � ϕ ↔ � ψ . Modal algebra : A = � A, 1 , − , ⊓ , ∗� such that � A, 1 , − , ⊓� is a Boolean algebra and ∗ : A → A . Λ( A ), the logic of A : { ϕ ∈ L : ϕ mapped to 1 by all interpretations in A } 3 / 10
Congruential modal logics L : propositional language with operators ⊤ , ¬ , ∧ and � . Modal logic : Λ ⊆ L containing all tautologies and closed under MP and US. Congruential modal logic (CML) : Modal logic Λ closed under ϕ ↔ ψ/ � ϕ ↔ � ψ . Modal algebra : A = � A, 1 , − , ⊓ , ∗� such that � A, 1 , − , ⊓� is a Boolean algebra and ∗ : A → A . Λ( A ), the logic of A : { ϕ ∈ L : ϕ mapped to 1 by all interpretations in A } Theorem (Hansson & G¨ ardenfors 1973): Λ ⊆ L is a CML iff Λ is the logic of some modal algebra. 3 / 10
A continuum of Post complete logics C -Post complete : Post complete in (the lattice of) congruential modal logics. 4 / 10
A continuum of Post complete logics C -Post complete : Post complete in (the lattice of) congruential modal logics. Theorem: The number of C -Post complete modal logics is � 1 . 4 / 10
A continuum of Post complete logics C -Post complete : Post complete in (the lattice of) congruential modal logics. Theorem: The number of C -Post complete modal logics is � 1 . Proof: By Lindenbaum’s Lemma, every consistent CML can be extended to a C -Post complete one. 4 / 10
A continuum of Post complete logics C -Post complete : Post complete in (the lattice of) congruential modal logics. Theorem: The number of C -Post complete modal logics is � 1 . Proof: By Lindenbaum’s Lemma, every consistent CML can be extended to a C -Post complete one. So it suffices to construct � 1 CMLs such that any two of them have an inconsistent join. 4 / 10
A continuum of Post complete logics C -Post complete : Post complete in (the lattice of) congruential modal logics. Theorem: The number of C -Post complete modal logics is � 1 . Proof: By Lindenbaum’s Lemma, every consistent CML can be extended to a C -Post complete one. So it suffices to construct � 1 CMLs such that any two of them have an inconsistent join. We construct one for every set of natural numbers S ⊆ ω . 4 / 10
A continuum of Post complete logics ω A S based on algebra of finite/cofinite subsets of ω . ∅
A continuum of Post complete logics ω A S based on algebra of finite/cofinite subsets of ω . b 0 b 1 b 2 . . . b 3 ∅
A continuum of Post complete logics ω A S based on algebra of finite/cofinite subsets of ω . b 0 b 1 b 2 . . . b 3 ∅
A continuum of Post complete logics ω if n ∈ S A S based on algebra of finite/cofinite subsets of ω . − b n b 0 b 1 b 2 if n / ∈ S . . . b 3 ∅
A continuum of Post complete logics ω if n ∈ S A S based on algebra of finite/cofinite subsets of ω . − b n Consider ϕ n = � ¬ � n � ⊤ b 0 b 1 b 2 if n / ∈ S . . . b 3 ∅
A continuum of Post complete logics ω if n ∈ S A S based on algebra of finite/cofinite subsets of ω . − b n Consider ϕ n = � ¬ � n � ⊤ ϕ n ∈ Λ( A S ) iff n ∈ S ¬ ϕ n ∈ Λ( A S ) iff n / ∈ S b 0 b 1 b 2 if n / ∈ S . . . b 3 ∅
A continuum of Post complete logics ω if n ∈ S A S based on algebra of finite/cofinite subsets of ω . − b n Consider ϕ n = � ¬ � n � ⊤ ϕ n ∈ Λ( A S ) iff n ∈ S ¬ ϕ n ∈ Λ( A S ) iff n / ∈ S b 0 b 1 b 2 if n / ∈ S . . . b 3 � ∅ 5 / 10
Neighborhood Semantics Neighborhood frame: Pair � W, N � such that W is a set and N : P ( W ) → P ( W ). 6 / 10
Neighborhood Semantics Neighborhood frame: Pair � W, N � such that W is a set and N : P ( W ) → P ( W ). � W, N, V � , w � � ϕ iff w ∈ N ( { v ∈ W : � W, N, V � , v � ϕ } ) 6 / 10
Neighborhood Semantics Neighborhood frame: Pair � W, N � such that W is a set and N : P ( W ) → P ( W ). � W, N, V � , w � � ϕ iff w ∈ N ( { v ∈ W : � W, N, V � , v � ϕ } ) Neighborhood frames are (effectively) modal algebras based on powerset algebras. 6 / 10
Neighborhood Semantics Neighborhood frame: Pair � W, N � such that W is a set and N : P ( W ) → P ( W ). � W, N, V � , w � � ϕ iff w ∈ N ( { v ∈ W : � W, N, V � , v � ϕ } ) Neighborhood frames are (effectively) modal algebras based on powerset algebras. Theorem: There are at least ℵ 0 C -Post complete modal logics each of which is the logic of a class of neighborhood frames. 6 / 10
Neighborhood Semantics Neighborhood frame: Pair � W, N � such that W is a set and N : P ( W ) → P ( W ). � W, N, V � , w � � ϕ iff w ∈ N ( { v ∈ W : � W, N, V � , v � ϕ } ) Neighborhood frames are (effectively) modal algebras based on powerset algebras. Theorem: There are at least ℵ 0 C -Post complete modal logics each of which is the logic of a class of neighborhood frames. Proof: We construct one as Λ( A n ) for each n < ω . 6 / 10
Neighborhood Semantics A n based on P ( n ); Λ n = Λ( A n ) n = { 0 , . . . , n − 1 } ∅ = b 0
Neighborhood Semantics A n based on P ( n ); Λ n = Λ( A n ) n = { 0 , . . . , n − 1 } b 2 n − 3 ... b 2 n − 2 b 2 b 1 ∅ = b 0
Neighborhood Semantics A n based on P ( n ); Λ n = Λ( A n ) n = { 0 , . . . , n − 1 } b 2 n − 3 ... b 2 n − 2 b 2 b 1 ∅ = b 0
Neighborhood Semantics A n based on P ( n ); Λ n = Λ( A n ) n = { 0 , . . . , n − 1 } b 2 n − 3 ... b 2 n − 2 b 2 b 1 ∅ = b 0
Neighborhood Semantics A n based on P ( n ); Λ n = Λ( A n ) If n < n ′ , then n = { 0 , . . . , n − 1 } ¬ � 2 n − 1 ⊥ ∈ Λ n ¬ � 2 n − 1 ⊥ / ∈ Λ n ′ b 2 n − 3 ... b 2 n − 2 b 2 b 1 ∅ = b 0
Neighborhood Semantics A n based on P ( n ); Λ n = Λ( A n ) If n < n ′ , then n = { 0 , . . . , n − 1 } ¬ � 2 n − 1 ⊥ ∈ Λ n ¬ � 2 n − 1 ⊥ / ∈ Λ n ′ Let Λ ⊃ Λ n and ϕ ∈ Λ \ Λ n . b 2 n − 3 Mapped to non-top element by ... some interpretation; replace proposition letters by “defini- b 2 n − 2 b 2 tions” accordingly: ϕ ′ . b 1 ∅ = b 0
Neighborhood Semantics A n based on P ( n ); Λ n = Λ( A n ) If n < n ′ , then n = { 0 , . . . , n − 1 } ¬ � 2 n − 1 ⊥ ∈ Λ n ¬ � 2 n − 1 ⊥ / ∈ Λ n ′ Let Λ ⊃ Λ n and ϕ ∈ Λ \ Λ n . b 2 n − 3 Mapped to non-top element by ... some interpretation; replace proposition letters by “defini- b 2 n − 2 b 2 tions” accordingly: ϕ ′ . ¬ � k ϕ ′ ∈ Λ n for some k . � k ⊤ ↔ � k ϕ ′ ∈ Λ. But b 1 � k ⊤ ∈ Λ n ⊆ Λ, so � k ϕ ′ ∈ Λ. So Λ = L . ∅ = b 0
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