Admissible rules of propositional dependence logic Fan Yang Utrecht - PowerPoint PPT Presentation

Propositional dependence logic and its variants Well-formed formulas of propositional dependence logic ( PD ) are given by the following grammar φ ::= p i | ¬ p i | =( p i 1 , . . . , p i n − 1 , p i n ) | φ ∧ φ | φ ⊗ φ Well-formed formulas of propositional intuitionistic dependence logic ( PID ) are given by the following grammar: φ ::= p i | ⊥ | =( p i 1 , . . . , p i n − 1 , p i n ) | φ ∧ φ | φ ∨ φ | φ → φ ( ¬ φ := φ → ⊥ ) PD ∨ is the logic extended from PD by adding the connective ∨ . Team semantics: A valuation is a function v : N → { 0 , 1 } . A team is a set of valuations. p 0 p 1 p 2 . . . v 1 1 0 0 . . . v 2 1 1 0 . . . v 4 0 1 0 . . . . . . . . . . . . . . . . . . 5/19

Team Semantics Let X be a team. X | = p i iff for all v ∈ X , v ( i ) = 1; X | = ¬ p i iff for all v ∈ X , v ( i ) = 0; X | = ⊥ iff X = ∅ ; = =( p i 1 , . . . , p i n ) iff for all v , v ′ ∈ X X | � v ( i 1 ) = v ′ ( i 1 ) , . . . , v ( i n − 1 ) = v ′ ( i n − 1 ) � ⇒ v ( i n ) = v ′ ( i n ); = X | = φ ∧ ψ iff X | = φ and X | = ψ ; X | = φ ⊗ ψ iff there exist teams Y , Z ⊆ X with X = Y ∪ Z such that Y | = φ and Z | = ψ ; p 0 p 1 p 2 X { 1 0 0 v 1 X | = p 0 v 2 1 0 1 v 3 0 1 0 v 4 0 1 1 X | = φ ∨ ψ iff X | = φ or X | = ψ ; 6/19 X | = φ → ψ iff for any team Y ⊆ X ,

Team Semantics Let X be a team. X | = p i iff for all v ∈ X , v ( i ) = 1; X | = ¬ p i iff for all v ∈ X , v ( i ) = 0; X | = ⊥ iff X = ∅ ; = =( p i 1 , . . . , p i n ) iff for all v , v ′ ∈ X X | � v ( i 1 ) = v ′ ( i 1 ) , . . . , v ( i n − 1 ) = v ′ ( i n − 1 ) � ⇒ v ( i n ) = v ′ ( i n ); = X | = φ ∧ ψ iff X | = φ and X | = ψ ; X | = φ ⊗ ψ iff there exist teams Y , Z ⊆ X with X = Y ∪ Z such that Y | = φ and Z | = ψ ; p 0 p 1 p 2 X { 1 0 0 v 1 X | = p 0 v 2 1 0 1 Y { Y | = ¬ p 0 v 3 0 1 0 v 4 0 1 1 X | = φ ∨ ψ iff X | = φ or X | = ψ ; 6/19 X | = φ → ψ iff for any team Y ⊆ X ,

Team Semantics Let X be a team. X | = p i iff for all v ∈ X , v ( i ) = 1; X | = ¬ p i iff for all v ∈ X , v ( i ) = 0; X | = ⊥ iff X = ∅ ; = =( p i 1 , . . . , p i n ) iff for all v , v ′ ∈ X X | � v ( i 1 ) = v ′ ( i 1 ) , . . . , v ( i n − 1 ) = v ′ ( i n − 1 ) � ⇒ v ( i n ) = v ′ ( i n ); = X | = φ ∧ ψ iff X | = φ and X | = ψ ; X | = φ ⊗ ψ iff there exist teams Y , Z ⊆ X with X = Y ∪ Z such that Y | = φ and Z | = ψ ; p 0 p 1 p 2 X { 1 0 0 v 1 X | = p 0 X ∪ Y �| = p 0 v 2 1 0 1 Y { Y | = ¬ p 0 X ∪ Y �| = ¬ p 0 v 3 0 1 0 v 4 0 1 1 X | = φ ∨ ψ iff X | = φ or X | = ψ ; 6/19 X | = φ → ψ iff for any team Y ⊆ X ,

Dependence atoms are definable in the fragment of PID and PD ∨ without dependence atoms: � �� =( p 0 , p 1 ) ≡ p 0 ∧ p 1 ∨ ¬ p 1 ⊗ ¬ p 0 ∧ p 1 ∨ ¬ p 1 ≡ ( p 0 ∨ ¬ p 0 ) → ( p 1 ∨ ¬ p 1 ) p 0 p 1 p 2 v 1 1 0 0 v 2 1 0 1 v 3 0 1 0 v 4 0 1 1 8/19

Dependence atoms are definable in the fragment of PID and PD ∨ without dependence atoms: � �� =( p 0 , p 1 ) ≡ p 0 ∧ p 1 ∨ ¬ p 1 ⊗ ¬ p 0 ∧ p 1 ∨ ¬ p 1 ≡ ( p 0 ∨ ¬ p 0 ) → ( p 1 ∨ ¬ p 1 ) p 0 p 1 p 2 v 1 1 0 0 v 2 1 0 1 v 3 0 1 0 v 4 0 1 1 Observation (de Jongh, Litak) PID − is equivalent to inquisitive logic (Ciardelli, Roelofsen 2011). 8/19

Fix N = { i 1 , . . . , i n } ⊆ N . An n-valuation on N is a function s : N → { 0 , 1 } . An n-team on N is a set of n -valuations on N . 11 10 01 00 There are in total 2 n distinct n -valuations, and 2 2 n n -teams, among which there exists a biggest team (denoted by 2 n ) consisting of all n -valuations on N . 9/19

℘ ( 2 n ) 00 01 10 11 00 01 00 10 00 11 01 10 01 11 10 11 00 01 10 00 01 11 00 10 11 01 10 11 00 01 10 11 10/19

( ℘ ( 2 n ) , ⊇ ) 00 01 10 11 00 01 00 10 00 11 01 10 01 11 10 11 00 01 10 00 01 11 00 10 11 01 10 11 00 01 10 11 10/19

( ℘ ( 2 n ) \ {∅} , ⊇ ) 00 01 10 11 00 01 00 10 00 11 01 10 01 11 10 11 00 01 10 00 01 11 00 10 11 01 10 11 00 01 10 11 10/19

A Medvedev frame: ( ℘ ( 2 n ) \ {∅} , ⊇ ) 00 01 10 11 00 01 00 10 00 11 01 10 01 11 10 11 00 01 10 00 01 11 00 10 11 01 10 11 00 01 10 11 10/19

A Medvedev frame: ( ℘ ( 2 n ) \ {∅} , ⊇ ) 00 01 10 11 p , � q 00 01 00 10 00 11 01 10 01 11 10 11 00 01 10 00 01 11 00 10 11 01 10 11 00 01 10 11 10/19

A Medvedev frame: ( ℘ ( 2 n ) \ {∅} , ⊇ ) 00 01 10 11 p , � q 00 01 00 10 00 11 01 10 01 11 10 11 00 01 10 00 01 11 00 10 11 01 10 11 p → q 00 01 10 11 10/19

A Medvedev frame: ( ℘ ( 2 n ) \ {∅} , ⊇ ) 00 01 10 11 p , � q 00 01 00 10 00 11 01 10 01 11 10 11 ¬¬ p → p 00 01 10 00 01 11 00 10 11 01 10 11 p → q 00 01 10 11 10/19

A Medvedev frame: ( ℘ ( 2 n ) \ {∅} , ⊇ ) 00 01 10 11 p , � q 00 01 00 10 00 11 01 10 01 11 10 11 ¬¬ p → p 00 01 10 00 01 11 00 10 11 01 10 11 p → q 00 01 10 11 [Ciardelli, Roelofsen 2011]: PID − = ML ¬ = { φ | τ ( φ ) ∈ ML , where τ ( p ) = ¬ p } 10/19

A Medvedev frame: ( ℘ ( 2 n ) \ {∅} , ⊇ ) 00 01 10 11 p , � q 00 01 00 10 00 11 01 10 01 11 10 11 ¬¬ p → p 00 01 10 00 01 11 00 10 11 01 10 11 p → q 00 01 10 11 [Ciardelli, Roelofsen 2011]: PID − = ML ¬ = { φ | τ ( φ ) ∈ ML , where τ ( p ) = ¬ p } = KP ¬ = KP ⊕ ¬¬ p → p 10/19

Fix N = { i 1 , . . . , i n } . Put � φ ( p i 1 , . . . , p i n ) � := { X ⊆ 2 n | X | = φ } , ∇ N := {K ⊆ 2 2 n | ∅ ∈ K , � � X ∈ K , Y ⊆ X = ⇒ Y ∈ K } . For each K ∈ ∇ N , consider � X ∈K Θ X . For any n -team Y on N , � Y | = Θ X ⇐ ⇒ ∃ X ∈ K ( Y ⊆ X ) ⇐ ⇒ Y ∈ K . 11/19

Fix N = { i 1 , . . . , i n } . Put � φ ( p i 1 , . . . , p i n ) � := { X ⊆ 2 n | X | = φ } , ∇ N := {K ⊆ 2 2 n | ∅ ∈ K , � � X ∈ K , Y ⊆ X = ⇒ Y ∈ K } . Theorem ( Ciardelli, Huuskonen, Y.) PID , PD ∨ , PD are maximal downwards closed logics, i.e., for L ∈ { PID , PD ∨ , PD } , ∇ N = { � φ � | φ ( p i 1 , . . . , p i n ) is an n-formula of L } . In particular, PID ≡ PD ∨ ≡ PD . For each K ∈ ∇ N , consider � X ∈K Θ X . For any n -team Y on N , � Y | = Θ X ⇐ ⇒ ∃ X ∈ K ( Y ⊆ X ) ⇐ ⇒ Y ∈ K . 11/19

Fix N = { i 1 , . . . , i n } . Put � φ ( p i 1 , . . . , p i n ) � := { X ⊆ 2 n | X | = φ } , ∇ N := {K ⊆ 2 2 n | ∅ ∈ K , � � X ∈ K , Y ⊆ X = ⇒ Y ∈ K } . Theorem ( Ciardelli, Huuskonen, Y.) PID , PD ∨ , PD are maximal downwards closed logics, i.e., for L ∈ { PID , PD ∨ , PD } , ∇ N = { � φ � | φ ( p i 1 , . . . , p i n ) is an n-formula of L } . In particular, PID ≡ PD ∨ ≡ PD . Theorem (Y.) Every instance of ∨ and → is definable in PD , but ∨ and → are not uniformly definable in PD . 11/19

Fix N = { i 1 , . . . , i n } . Put � φ ( p i 1 , . . . , p i n ) � := { X ⊆ 2 n | X | = φ } , ∇ N := {K ⊆ 2 2 n | ∅ ∈ K , � � X ∈ K , Y ⊆ X = ⇒ Y ∈ K } . Theorem ( Ciardelli, Huuskonen, Y.) PID , PD ∨ , PD are maximal downwards closed logics, i.e., for L ∈ { PID , PD ∨ , PD } , ∇ N = { � φ � | φ ( p i 1 , . . . , p i n ) is an n-formula of L } . In particular, PID ≡ PD ∨ ≡ PD . Proof. We only treat PID and PD ∨ . For each K ∈ ∇ N , consider � X ∈K Θ X . For any n -team Y on N , � Y | = Θ X ⇐ ⇒ ∃ X ∈ K ( Y ⊆ X ) ⇐ ⇒ Y ∈ K . 11/19

Fix N = { i 1 , . . . , i n } . Put � φ ( p i 1 , . . . , p i n ) � := { X ⊆ 2 n | X | = φ } , ∇ N := {K ⊆ 2 2 n | ∅ ∈ K , � � X ∈ K , Y ⊆ X = ⇒ Y ∈ K } . Theorem ( Ciardelli, Huuskonen, Y.) PID , PD ∨ , PD are maximal downwards closed logics, i.e., for L ∈ { PID , PD ∨ , PD } , ∇ N = { � φ � | φ ( p i 1 , . . . , p i n ) is an n-formula of L } . In particular, PID ≡ PD ∨ ≡ PD . Proof. We only treat PID and PD ∨ . First, consider an n -team: p 1 p 2 1 1 v 1 X { v 2 1 0 v 3 0 1 For each K ∈ ∇ N , consider � X ∈K Θ X . For any n -team Y on N , � Y | = Θ X ⇐ ⇒ ∃ X ∈ K ( Y ⊆ X ) ⇐ ⇒ Y ∈ K . 11/19

Fix N = { i 1 , . . . , i n } . Put � φ ( p i 1 , . . . , p i n ) � := { X ⊆ 2 n | X | = φ } , ∇ N := {K ⊆ 2 2 n | ∅ ∈ K , � � X ∈ K , Y ⊆ X = ⇒ Y ∈ K } . Theorem ( Ciardelli, Huuskonen, Y.) PID , PD ∨ , PD are maximal downwards closed logics, i.e., for L ∈ { PID , PD ∨ , PD } , ∇ N = { � φ � | φ ( p i 1 , . . . , p i n ) is an n-formula of L } . In particular, PID ≡ PD ∨ ≡ PD . Proof. We only treat PID and PD ∨ . First, consider an n -team: p 1 p 2 Let  for PD ; 1 1 v 1  X {   v 2 1 0 Θ X := for PID . v 3 0 1    Then Y | = Θ X ⇐ ⇒ Y ⊆ X , for any n -team Y . For each K ∈ ∇ N , consider � X ∈K Θ X . For any n -team Y on N , � Y | = Θ X ⇐ ⇒ ∃ X ∈ K ( Y ⊆ X ) ⇐ ⇒ Y ∈ K . 11/19

Fix N = { i 1 , . . . , i n } . Put � φ ( p i 1 , . . . , p i n ) � := { X ⊆ 2 n | X | = φ } , ∇ N := {K ⊆ 2 2 n | ∅ ∈ K , � � X ∈ K , Y ⊆ X = ⇒ Y ∈ K } . Theorem ( Ciardelli, Huuskonen, Y.) PID , PD ∨ , PD are maximal downwards closed logics, i.e., for L ∈ { PID , PD ∨ , PD } , ∇ N = { � φ � | φ ( p i 1 , . . . , p i n ) is an n-formula of L } . In particular, PID ≡ PD ∨ ≡ PD . Proof. We only treat PID and PD ∨ . First, consider an n -team: p 1 p 2 Let  ( p v ( i 1 ) ∧ · · · ∧ p v ( i n ) � ) , for PD ; 1 1 v 1  i 1 i n X {   v 2 1 0 v ∈ X Θ X := for PID . v 3 0 1    Then Y | = Θ X ⇐ ⇒ Y ⊆ X , for any n -team Y . For each K ∈ ∇ N , consider � X ∈K Θ X . For any n -team Y on N , � Y | = Θ X ⇐ ⇒ ∃ X ∈ K ( Y ⊆ X ) ⇐ ⇒ Y ∈ K . 11/19

Fix N = { i 1 , . . . , i n } . Put � φ ( p i 1 , . . . , p i n ) � := { X ⊆ 2 n | X | = φ } , ∇ N := {K ⊆ 2 2 n | ∅ ∈ K , � � X ∈ K , Y ⊆ X = ⇒ Y ∈ K } . Theorem ( Ciardelli, Huuskonen, Y.) PID , PD ∨ , PD are maximal downwards closed logics, i.e., for L ∈ { PID , PD ∨ , PD } , ∇ N = { � φ � | φ ( p i 1 , . . . , p i n ) is an n-formula of L } . In particular, PID ≡ PD ∨ ≡ PD . Proof. We only treat PID and PD ∨ . First, consider an n -team: p 1 p 2 Let  ( p v ( i 1 ) ∧ · · · ∧ p v ( i n ) � ) , for PD ; 1 1 v 1  i 1 i n X {   v 2 1 0 v ∈ X Θ X := � ( p v ( i 1 ) ∧ · · · ∧ p v ( i n ) ¬¬ ) , for PID . v 3 0 1 i 1 i n    v ∈ X Then Y | = Θ X ⇐ ⇒ Y ⊆ X , for any n -team Y . For each K ∈ ∇ N , consider � X ∈K Θ X . For any n -team Y on N , � Y | = Θ X ⇐ ⇒ ∃ X ∈ K ( Y ⊆ X ) ⇐ ⇒ Y ∈ K . 11/19

Theorem ( Ciardelli, Huuskonen, Y.) PID , PD ∨ , PD are maximal downwards closed logics, i.e., for L ∈ { PID , PD ∨ , PD } , ∇ N = { � φ � | φ ( p i 1 , . . . , p i n ) is an n-formula of L } . In particular, PID ≡ PD ∨ ≡ PD . Proof. We only treat PID and PD ∨ . First, consider an n -team: p 1 p 2 Let  � ( p v ( i 1 ) ∧ · · · ∧ p v ( i n ) ) , for PD ; v 1 1 1 i 1 i n  X {   v 2 1 0 v ∈ X Θ X := � ( p v ( i 1 ) ∧ · · · ∧ p v ( i n ) ¬¬ ) , for PID . v 3 0 1  i 1 i n   v ∈ X Then Y | = Θ X ⇐ ⇒ Y ⊆ X , for any n -team Y . For each K ∈ ∇ N , consider � X ∈K Θ X . For any n -team Y on N , � Y | Θ X ⇐ ⇒ ∃ X ∈ K ( Y ⊆ X ) ⇐ ⇒ Y ∈ K . = X ∈K Hence � � X ∈K Θ X � = K . 11/19

Theorem. PID , PD ∨ , PD are sound and complete w.r.t. their deductive systems. Corrolary. For every formula φ of PID and PD ∨ , φ ⊣⊢ � i ∈ I Θ X i . A Hilbert Style deductive system for PID − (Ciardelli, Roelofsen 2011) Axioms: all substitution instances of IPC axioms all substitution instances of ( KP ) ( ¬ p i → ( p j ∨ p k )) → (( ¬ p i → p j ) ∨ ( ¬ p i → p k )) . ¬¬ p i → p i for all propositional variables p i Rules: Modus Ponens Natural deduction systems for PD and PD ∨ (Väänänen, Y.) For L ∈ { PD , PD ∨ } , if φ does not contain any ∨ or dependence atoms, then ⊢ CPC φ ⇐ ⇒ ⊢ L φ . 12/19

Definition A formula φ is said to be flat iff for all teams X , X | = φ ⇐ ⇒ ∀ v ∈ X , { v } | = φ. 13/19

Definition A formula φ is said to be flat iff for all teams X , X | = φ ⇐ ⇒ ∀ v ∈ X , { v } | = φ. 00 01 11 00 01 00 11 01 11 00 01 11 13/19

Definition A formula φ is said to be flat iff for all teams X , X | = φ ⇐ ⇒ ∀ v ∈ X , { v } | = φ. 00 01 11 00 01 00 11 01 11 φ 00 01 11 13/19

Definition A formula φ is said to be flat iff for all teams X , X | = φ ⇐ ⇒ ∀ v ∈ X , { v } | = φ. φ φ φ 00 01 11 00 01 00 11 01 11 φ 00 01 11 13/19

Definition A formula φ is said to be flat iff for all teams X , X | = φ ⇐ ⇒ ∀ v ∈ X , { v } | = φ. φ φ φ 00 01 11 00 01 00 11 01 11 φ 00 01 11 Fact: Formulas with no occurrences of dependence atoms or intuitionistic disjunction ∨ are flat. 13/19

Definition A formula φ is said to be flat iff for all teams X , X | = φ ⇐ ⇒ ∀ v ∈ X , { v } | = φ. φ φ φ 00 01 11 00 01 00 11 01 11 φ 00 01 11 Fact: Formulas with no occurrences of dependence atoms or intuitionistic disjunction ∨ are flat. ( p v ( i 1 ) ∧ · · · ∧ p v ( i n ) �  ) , for PD ; i 1 i n    v ∈ X E.g. Θ X = is flat ( p v ( i 1 ) ∧ · · · ∧ p v ( i n ) � ¬¬ ) , for PID .  i 1 i n   v ∈ X 13/19

Definition A formula φ is said to be flat iff for all teams X , X | = φ ⇐ ⇒ ∀ v ∈ X , { v } | = φ. φ φ φ 00 01 11 00 01 00 11 01 11 φ 00 01 11 Fact: Formulas with no occurrences of dependence atoms or intuitionistic disjunction ∨ are flat. Lemma. In PID , a formula φ is flat if ⊢ φ ↔ ¬¬ φ . ( p v ( i 1 ) ∧ · · · ∧ p v ( i n ) �  ) , for PD ; i 1 i n    v ∈ X E.g. Θ X = is flat ( p v ( i 1 ) ∧ · · · ∧ p v ( i n ) � ¬¬ ) , for PID .  i 1 i n   v ∈ X 13/19

Definition A formula φ is said to be flat iff for all teams X , X | = φ ⇐ ⇒ ∀ v ∈ X , { v } | = φ. Lemma A formula φ is flat iff φ ⊣⊢ Θ X for some X. 14/19

Definition A formula φ is said to be flat iff for all teams X , X | = φ ⇐ ⇒ ∀ v ∈ X , { v } | = φ. Lemma A formula φ is flat iff φ ⊣⊢ Θ X for some X. Proof. “ = ⇒ ”: Suppose φ is flat. Note that φ ⊣⊢ Θ X 1 ∨ · · · ∨ Θ X k , 14/19

Definition A formula φ is said to be flat iff for all teams X , X | = φ ⇐ ⇒ ∀ v ∈ X , { v } | = φ. Fact: Y | = Θ X ⇐ ⇒ Y ⊆ X , for any n -team Y . Lemma A formula φ is flat iff φ ⊣⊢ Θ X for some X. Proof. “ = ⇒ ”: Suppose φ is flat. Note that φ ⊣⊢ Θ X 1 ∨ · · · ∨ Θ X k , where w.l.o.g. we assume that X i ’s are maximal. X 2 X 3 X 1 · · · · · · X k 14/19

Definition A formula φ is said to be flat iff for all teams X , X | = φ ⇐ ⇒ ∀ v ∈ X , { v } | = φ. Fact: Y | = Θ X ⇐ ⇒ Y ⊆ X , for any n -team Y . Lemma A formula φ is flat iff φ ⊣⊢ Θ X for some X. Proof. “ = ⇒ ”: Suppose φ is flat. Note that φ ⊣⊢ Θ X 1 ∨ · · · ∨ Θ X k , where w.l.o.g. we assume that X i ’s are maximal. If k > 1, X 2 X 3 X 1 · · · · · · X k 14/19

Definition A formula φ is said to be flat iff for all teams X , X | = φ ⇐ ⇒ ∀ v ∈ X , { v } | = φ. Fact: Y | = Θ X ⇐ ⇒ Y ⊆ X , for any n -team Y . Lemma A formula φ is flat iff φ ⊣⊢ Θ X for some X. Proof. “ = ⇒ ”: Suppose φ is flat. Note that φ ⊣⊢ Θ X 1 ∨ · · · ∨ Θ X k , where w.l.o.g. we assume that X i ’s are maximal. If k > 1, then pick X 2 X 3 X 1 v 1 · · · · · · X k 14/19

Definition A formula φ is said to be flat iff for all teams X , X | = φ ⇐ ⇒ ∀ v ∈ X , { v } | = φ. Fact: Y | = Θ X ⇐ ⇒ Y ⊆ X , for any n -team Y . Lemma A formula φ is flat iff φ ⊣⊢ Θ X for some X. Proof. “ = ⇒ ”: Suppose φ is flat. Note that φ ⊣⊢ Θ X 1 ∨ · · · ∨ Θ X k , where w.l.o.g. we assume that X i ’s are maximal. If k > 1, then pick X 2 X 3 v 2 X 1 v 1 · · · · · · X k 14/19

Definition A formula φ is said to be flat iff for all teams X , X | = φ ⇐ ⇒ ∀ v ∈ X , { v } | = φ. Fact: Y | = Θ X ⇐ ⇒ Y ⊆ X , for any n -team Y . Lemma A formula φ is flat iff φ ⊣⊢ Θ X for some X. Proof. “ = ⇒ ”: Suppose φ is flat. Note that φ ⊣⊢ Θ X 1 ∨ · · · ∨ Θ X k , where w.l.o.g. we assume that X i ’s are maximal. If k > 1, then pick X 2 X 3 v 2 X 1 v 1 · · · · · · v k X k 14/19

Definition A formula φ is said to be flat iff for all teams X , X | = φ ⇐ ⇒ ∀ v ∈ X , { v } | = φ. Fact: Y | = Θ X ⇐ ⇒ Y ⊆ X , for any n -team Y . Lemma A formula φ is flat iff φ ⊣⊢ Θ X for some X. Proof. “ = ⇒ ”: Suppose φ is flat. Note that φ ⊣⊢ Θ X 1 ∨ · · · ∨ Θ X k , where w.l.o.g. we assume that X i ’s are maximal. If k > 1, then pick X 2 X 3 v 2 X 1 v 1 · · · · · · v k X k { v i } ⊆ X i and { v 1 , . . . , v k } � X i for all 1 ≤ i ≤ k = ⇒ { v i } | = Θ X i and { v 1 , . . . , v k } �| = Θ X i for all 1 ≤ i ≤ k ⇒ { v i } | = φ for all 1 ≤ i ≤ k whereas { v 1 , . . . , v k } �| = = φ 14/19

Definition A formula φ is said to be flat iff for all teams X , X | = φ ⇐ ⇒ ∀ v ∈ X , { v } | = φ. Fact: Y | = Θ X ⇐ ⇒ Y ⊆ X , for any n -team Y . Lemma A formula φ is flat iff φ ⊣⊢ Θ X for some X. Proof. “ = ⇒ ”: Suppose φ is flat. Note that φ ⊣⊢ Θ X 1 ∨ · · · ∨ Θ X k , where w.l.o.g. we assume that X i ’s are maximal. If k > 1, then pick X 2 X 3 v 2 X 1 v 1 · · · · · · v k X k { v i } ⊆ X i and { v 1 , . . . , v k } � X i for all 1 ≤ i ≤ k = ⇒ { v i } | = Θ X i and { v 1 , . . . , v k } �| = Θ X i for all 1 ≤ i ≤ k ⇒ { v i } | = φ for all 1 ≤ i ≤ k whereas { v 1 , . . . , v k } �| = = φ = ⇒ k = 1 and φ ⊣⊢ Θ X 1 . 14/19

We say that a substitution σ of a logic L is well-behaved, if σ is well-defined: σ ( φ ) is a well-formed formula of L ⊢ L is closed under σ : φ ⊢ L ψ implies σ ( φ ) ⊢ L σ ( ψ ) 15/19

We say that a substitution σ of a logic L is well-behaved, if σ is well-defined: σ ( φ ) is a well-formed formula of L ⊢ L is closed under σ : φ ⊢ L ψ implies σ ( φ ) ⊢ L σ ( ψ ) Recall: Well-formed formulas of PD are built from the following grammar: φ ::= p i | ¬ p i | =( p i 1 , . . . , p i k ) | φ ∧ φ | φ ⊗ φ where p i , p i 1 , . . . , p i k are propositional variables. 15/19

We say that a substitution σ of a logic L is well-behaved, if σ is well-defined: σ ( φ ) is a well-formed formula of L ⊢ L is closed under σ : φ ⊢ L ψ implies σ ( φ ) ⊢ L σ ( ψ ) ⊢ ¬¬ p → p , but �⊢ ¬¬ ( p ∨ ¬ p ) → ( p ∨ ¬ p ) For PID : 15/19

We say that a substitution σ of a logic L is well-behaved, if σ is well-defined: σ ( φ ) is a well-formed formula of L ⊢ L is closed under σ : φ ⊢ L ψ implies σ ( φ ) ⊢ L σ ( ψ ) ⊢ ¬¬ p → p , but �⊢ ¬¬ ( p ∨ ¬ p ) → ( p ∨ ¬ p ) For PID : Definition A substitution σ is called a flat substitution if σ ( p ) is flat for all p . Lemma Flat substitutions are well-behaved in PID and PD . Proof. For PID , it follows from (Ciardelli). For PD , nontrivial. 15/19

Definition (Projective formula) Let S be a set of well-behaved substitutions of a logic L. An L-formula φ is said to be S- projective if there exists σ ∈ S such that (1) ⊢ L σ ( φ ) (2) φ, σ ( ψ ) ⊢ L ψ and φ, ψ ⊢ L σ ( ψ ) for all L-formulas ψ . 16/19

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