About Kripke Semantics Aim Calculi Proof Reflexive Case Proof Further Work Just some space Axiomatizing a Reflexive Real-Valued Modal Logic Laura Janina Schn¨ uriger Axiomatizing a Reflexive Real-valued Modal Logic Just some space TACL, Praha 29 June 2017 Laura Janina Schn¨ uriger Axiomatizing a Reflexive Real-Valued Modal Logic
About Kripke Semantics Aim Calculi That’s the talk about Proof A Real-Valued Modal Logic Reflexive Case Proof Further Work That’s the talk about Motivation Many-valued logics and modal logics are well-established topics Laura Janina Schn¨ uriger Axiomatizing a Reflexive Real-Valued Modal Logic
About Kripke Semantics Aim Calculi That’s the talk about Proof A Real-Valued Modal Logic Reflexive Case Proof Further Work That’s the talk about Motivation Many-valued logics and modal logics are well-established topics But the connection is still not a very well-studied field of research Laura Janina Schn¨ uriger Axiomatizing a Reflexive Real-Valued Modal Logic
About Kripke Semantics Aim Calculi That’s the talk about Proof A Real-Valued Modal Logic Reflexive Case Proof Further Work A Real-Valued Modal Logic universe (=truth values): R Laura Janina Schn¨ uriger Axiomatizing a Reflexive Real-Valued Modal Logic
About Kripke Semantics Aim Calculi That’s the talk about Proof A Real-Valued Modal Logic Reflexive Case Proof Further Work A Real-Valued Modal Logic universe (=truth values): R (=true truth values): R + designated 0 Laura Janina Schn¨ uriger Axiomatizing a Reflexive Real-Valued Modal Logic
About Kripke Semantics Aim Calculi That’s the talk about Proof A Real-Valued Modal Logic Reflexive Case Proof Further Work A Real-Valued Modal Logic universe (=truth values): R (=true truth values): R + designated 0 connectives interpreted as arithmetic group operations + , − , 0 Laura Janina Schn¨ uriger Axiomatizing a Reflexive Real-Valued Modal Logic
About Kripke Semantics Aim Calculi That’s the talk about Proof A Real-Valued Modal Logic Reflexive Case Proof Further Work A Real-Valued Modal Logic universe (=truth values): R (=true truth values): R + designated 0 connectives interpreted as arithmetic group operations + , − , 0 plus modality � Laura Janina Schn¨ uriger Axiomatizing a Reflexive Real-Valued Modal Logic
About Kripke Semantics Aim Calculi That’s the talk about Proof A Real-Valued Modal Logic Reflexive Case Proof Further Work Language L { & , ¬ , 0 , � } = {→ , � } Laura Janina Schn¨ uriger Axiomatizing a Reflexive Real-Valued Modal Logic
About Kripke Semantics Aim Calculi That’s the talk about Proof A Real-Valued Modal Logic Reflexive Case Proof Further Work Language L { & , ¬ , 0 , � } = {→ , � } 0 = p 0 → p 0 p 0 ∈ Var ¬ ϕ = ϕ → 0 ϕ & ψ = ¬ ϕ → ψ ♦ ϕ = ¬ � ¬ ϕ Laura Janina Schn¨ uriger Axiomatizing a Reflexive Real-Valued Modal Logic
About Kripke Semantics Aim Calculi That’s the talk about Proof A Real-Valued Modal Logic Reflexive Case Proof Further Work Language L { & , ¬ , 0 , � } = {→ , � } 0 = p 0 → p 0 p 0 ∈ Var ¬ ϕ = ϕ → 0 ϕ & ψ = ¬ ϕ → ψ ♦ ϕ = ¬ � ¬ ϕ 0 ϕ = 0 ( n + 1) ϕ = ϕ & n ϕ Laura Janina Schn¨ uriger Axiomatizing a Reflexive Real-Valued Modal Logic
About Kripke Semantics Aim Calculi Frames and models Proof Reflexive Case Proof Further Work Kripke frames frame F = � W , R � Laura Janina Schn¨ uriger Axiomatizing a Reflexive Real-Valued Modal Logic
About Kripke Semantics Aim Calculi Frames and models Proof Reflexive Case Proof Further Work Kripke frames frame F = � W , R � W non-empty set of elements called worlds R ⊆ W × W a binary relation on W Laura Janina Schn¨ uriger Axiomatizing a Reflexive Real-Valued Modal Logic
About Kripke Semantics Aim Calculi Frames and models Proof Reflexive Case Proof Further Work Kripke frames frame F = � W , R � W non-empty set of elements called worlds R ⊆ W × W a binary relation on W serial a frame F is called serial if for all x ∈ W there is y ∈ W such that Rxy Laura Janina Schn¨ uriger Axiomatizing a Reflexive Real-Valued Modal Logic
About Kripke Semantics Aim Calculi Frames and models Proof Reflexive Case Proof Further Work Kripke frames frame F = � W , R � W non-empty set of elements called worlds R ⊆ W × W a binary relation on W serial a frame F is called serial if for all x ∈ W there is y ∈ W such that Rxy reflexive if for all x ∈ W : Rxx Laura Janina Schn¨ uriger Axiomatizing a Reflexive Real-Valued Modal Logic
About Kripke Semantics Aim Calculi Frames and models Proof Reflexive Case Proof Further Work Models Intuition Talk about abelian groups at each world of the frames Laura Janina Schn¨ uriger Axiomatizing a Reflexive Real-Valued Modal Logic
About Kripke Semantics Aim Calculi Frames and models Proof Reflexive Case Proof Further Work Models model M = � W , R , V � Laura Janina Schn¨ uriger Axiomatizing a Reflexive Real-Valued Modal Logic
About Kripke Semantics Aim Calculi Frames and models Proof Reflexive Case Proof Further Work Models model M = � W , R , V � � W , R � is a frame Laura Janina Schn¨ uriger Axiomatizing a Reflexive Real-Valued Modal Logic
About Kripke Semantics Aim Calculi Frames and models Proof Reflexive Case Proof Further Work Models model M = � W , R , V � � W , R � is a frame V : Var × W → [ − r , r ] for some r ∈ R + valuation Laura Janina Schn¨ uriger Axiomatizing a Reflexive Real-Valued Modal Logic
About Kripke Semantics Aim Calculi Frames and models Proof Reflexive Case Proof Further Work Models model M = � W , R , V � � W , R � is a frame V : Var × W → [ − r , r ] for some r ∈ R + valuation extends to V : Fm × W → R via V ( ϕ → ψ ; x ) = V ( ψ ; x ) − V ( ϕ ; x ) V ( � ϕ ; x ) = � { V ( ϕ ; y ) | Rxy } Laura Janina Schn¨ uriger Axiomatizing a Reflexive Real-Valued Modal Logic
About Kripke Semantics Aim Calculi Frames and models Proof Reflexive Case Proof Further Work Models V (0; x ) = 0 V ( ¬ ϕ ; x ) = − V ( ϕ ; x ) V ( ϕ & ψ ; x ) = V ( ϕ ; x ) + V ( ψ ; x ) V ( ♦ ϕ ; x ) = � { V ( ϕ ; y ) | Rxy } Laura Janina Schn¨ uriger Axiomatizing a Reflexive Real-Valued Modal Logic
About Kripke Semantics Aim Calculi Frames and models Proof Reflexive Case Proof Further Work Models K ( A )-model if R is serial Laura Janina Schn¨ uriger Axiomatizing a Reflexive Real-Valued Modal Logic
About Kripke Semantics Aim Calculi Frames and models Proof Reflexive Case Proof Further Work Models K ( A )-model if R is serial KT ( A )-model if R is reflexive Laura Janina Schn¨ uriger Axiomatizing a Reflexive Real-Valued Modal Logic
About Kripke Semantics Aim Calculi Frames and models Proof Reflexive Case Proof Further Work Validity valid ϕ ∈ Fm is K ( A ) / KT ( A ) -valid if V ( ϕ ; x ) ≥ 0 for all x ∈ W in all K ( A ) / KT ( A )-models M = � W , R , V � Laura Janina Schn¨ uriger Axiomatizing a Reflexive Real-Valued Modal Logic
About Kripke Semantics Aim Calculi Frames and models Proof Reflexive Case Proof Further Work Validity valid ϕ ∈ Fm is K ( A ) / KT ( A ) -valid if V ( ϕ ; x ) ≥ 0 for all x ∈ W in all K ( A ) / KT ( A )-models M = � W , R , V � notation � K ( A ) ϕ respectively � KT ( A ) ϕ Laura Janina Schn¨ uriger Axiomatizing a Reflexive Real-Valued Modal Logic
About Kripke Semantics Aim Calculi Frames and models Proof Reflexive Case Proof Further Work Remarks seriality is not necessary if � ∅ = � ∅ := 0 K ( A ) Laura Janina Schn¨ uriger Axiomatizing a Reflexive Real-Valued Modal Logic
About Kripke Semantics Aim Calculi Frames and models Proof Reflexive Case Proof Further Work Remarks seriality is not necessary if � ∅ = � ∅ := 0 K ( A ) � K ( A ) ϕ iff ϕ is valid in all models without restricting R to be serial Laura Janina Schn¨ uriger Axiomatizing a Reflexive Real-Valued Modal Logic
About Kripke Semantics Aim Calculi Frames and models Proof Reflexive Case Proof Further Work Remarks seriality is not necessary if � ∅ = � ∅ := 0 K ( A ) � K ( A ) ϕ iff ϕ is valid in all models without restricting R to be serial KT ( A ) reflexive ⇒ serial Laura Janina Schn¨ uriger Axiomatizing a Reflexive Real-Valued Modal Logic
About Kripke Semantics Aim Calculi Axiom systems Proof Main Theorem 1 Reflexive Case Proof Further Work Aim so far semantical definition of K ( A ) and KT ( A ) Laura Janina Schn¨ uriger Axiomatizing a Reflexive Real-Valued Modal Logic
About Kripke Semantics Aim Calculi Axiom systems Proof Main Theorem 1 Reflexive Case Proof Further Work Aim so far semantical definition of K ( A ) and KT ( A ) aim syntactical description Laura Janina Schn¨ uriger Axiomatizing a Reflexive Real-Valued Modal Logic
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