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On the Logics with Propositional Quantifiers Extending S5

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On the Logics with Propositional Quantifiers Extending S5 On the Logics with Propositional Quantifiers Extending S5Π Yifeng Ding (voidprove.com) Aug. 27, 2018 @ AiML 2018 UC Berkeley Group of Logic and the Methodology of Science • Kit Fine systematically studied a few modal logic systems with propositional quantifers based on S5. • We provide an analogue of Scroggs's theorem for modal logics with propositional quantifiers using algebraic semantics. • More generally, it is interesting to see how classical results generalize when using algebraic semantics. Introduction • We have expressions that quantifies over propositions: \Everything I believe is true." (Locally) 2 • We provide an analogue of Scroggs's theorem for modal logics with propositional quantifiers using algebraic semantics. • More generally, it is interesting to see how classical results generalize when using algebraic semantics. Introduction • We have expressions that quantifies over propositions: \Everything I believe is true." (Locally) • Kit Fine systematically studied a few modal logic systems with propositional quantifers based on S5. 2 • More generally, it is interesting to see how classical results generalize when using algebraic semantics. Introduction • We have expressions that quantifies over propositions: \Everything I believe is true." (Locally) • Kit Fine systematically studied a few modal logic systems with propositional quantifers based on S5. • We provide an analogue of Scroggs's theorem for modal logics with propositional quantifiers using algebraic semantics. 2 Introduction • We have expressions that quantifies over propositions: \Everything I believe is true." (Locally) • Kit Fine systematically studied a few modal logic systems with propositional quantifers based on S5. • We provide an analogue of Scroggs's theorem for modal logics with propositional quantifiers using algebraic semantics. • More generally, it is interesting to see how classical results generalize when using algebraic semantics. 2 Outline Review of Kripke Semantics Algebraic Semantics Main Theorems Future Research 3 Review of Kripke Semantics Language Definition Let LΠ be the language with the following grammar ' ::= p j > j :' j (' ^ ') j ' j 8p' where p 2 Prop, a countably infinite set of propositional variables. Other Boolean connectives, ?, and ♦ are defined as usual. 4 Under this semantics, it is natural to call this language Second Order Propositional Modal Logic, SOPML for short. M Examples: 8p(p ! p) does not depend on V and is precisely the set of reflexiveJ pointsK in M. M 8p(♦p ! ♦p) is not first-order definable. J K Kripke semantics Every subset is a proposition! • A pointed model hW ; R; V i; w makes 8p' true iff for all X ⊆ W , hW ; R; V [p 7! X ]i; w makes ' true. M T M[p7!X ] • Equivalently, 8p' = X ⊆M ' . J K J K 5 M Examples: 8p(p ! p) does not depend on V and is precisely the set of reflexiveJ pointsK in M. M 8p(♦p ! ♦p) is not first-order definable. J K Kripke semantics Every subset is a proposition! • A pointed model hW ; R; V i; w makes 8p' true iff for all X ⊆ W , hW ; R; V [p 7! X ]i; w makes ' true. M T M[p7!X ] • Equivalently, 8p' = X ⊆M ' . J K J K Under this semantics, it is natural to call this language Second Order Propositional Modal Logic, SOPML for short. 5 M 8p(♦p ! ♦p) is not first-order definable. J K Kripke semantics Every subset is a proposition! • A pointed model hW ; R; V i; w makes 8p' true iff for all X ⊆ W , hW ; R; V [p 7! X ]i; w makes ' true. M T M[p7!X ] • Equivalently, 8p' = X ⊆M ' . J K J K Under this semantics, it is natural to call this language Second Order Propositional Modal Logic, SOPML for short. M Examples: 8p(p ! p) does not depend on V and is precisely the set of reflexiveJ pointsK in M. 5 Kripke semantics Every subset is a proposition! • A pointed model hW ; R; V i; w makes 8p' true iff for all X ⊆ W , hW ; R; V [p 7! X ]i; w makes ' true. M T M[p7!X ] • Equivalently, 8p' = X ⊆M ' . J K J K Under this semantics, it is natural to call this language Second Order Propositional Modal Logic, SOPML for short. M Examples: 8p(p ! p) does not depend on V and is precisely the set of reflexiveJ pointsK in M. M 8p(♦p ! ♦p) is not first-order definable. J K 5 Theorem Full second-order logic can be embedded into SOPML (preserving satisfiability) when R is S4.2 or weaker. Theorem When R = W × W , SOPML is expressively equivalent to MSO. Kripke semantics Another example: M ♦p ^ 8q((p ! q) _ (p !:q)) J K is the set of points that can access to exactly one element in V (p). Call this formula atom(p). 6 Theorem When R = W × W , SOPML is expressively equivalent to MSO. Kripke semantics Another example: M ♦p ^ 8q((p ! q) _ (p !:q)) J K is the set of points that can access to exactly one element in V (p). Call this formula atom(p). Theorem Full second-order logic can be embedded into SOPML (preserving satisfiability) when R is S4.2 or weaker. 6 Kripke semantics Another example: M ♦p ^ 8q((p ! q) _ (p !:q)) J K is the set of points that can access to exactly one element in V (p). Call this formula atom(p). Theorem Full second-order logic can be embedded into SOPML (preserving satisfiability) when R is S4.2 or weaker. Theorem When R = W × W , SOPML is expressively equivalent to MSO. 6 Algebraic Semantics • It is natural. Order-theoretically, 8p' is the weakest proposition that entails all instances of '. • It helps raising intersting questions. What if we drop atomicity? What if we drop complete multiplicativity? How much lattice-completeness do we need for the semantics to be well-defined? • We use it to prove an analogue of Scroggs's theorem. Algebraic semantics: reasons • Kripke frames corresponds to complete, atomic, completely multiplicative modal algebras. We are forced to accept 9p(p ^ atom(p)) when is S5. And we are forced to accept Barcan: 8p' $ 8p'. 7 • It helps raising intersting questions. What if we drop atomicity? What if we drop complete multiplicativity? How much lattice-completeness do we need for the semantics to be well-defined? • We use it to prove an analogue of Scroggs's theorem. Algebraic semantics: reasons • Kripke frames corresponds to complete, atomic, completely multiplicative modal algebras. We are forced to accept 9p(p ^ atom(p)) when is S5. And we are forced to accept Barcan: 8p' $ 8p'. • It is natural. Order-theoretically, 8p' is the weakest proposition that entails all instances of '. 7 • We use it to prove an analogue of Scroggs's theorem. Algebraic semantics: reasons • Kripke frames corresponds to complete, atomic, completely multiplicative modal algebras. We are forced to accept 9p(p ^ atom(p)) when is S5. And we are forced to accept Barcan: 8p' $ 8p'. • It is natural. Order-theoretically, 8p' is the weakest proposition that entails all instances of '. • It helps raising intersting questions. What if we drop atomicity? What if we drop complete multiplicativity? How much lattice-completeness do we need for the semantics to be well-defined? 7 Algebraic semantics: reasons • Kripke frames corresponds to complete, atomic, completely multiplicative modal algebras. We are forced to accept 9p(p ^ atom(p)) when is S5. And we are forced to accept Barcan: 8p' $ 8p'. • It is natural. Order-theoretically, 8p' is the weakest proposition that entails all instances of '. • It helps raising intersting questions. What if we drop atomicity? What if we drop complete multiplicativity? How much lattice-completeness do we need for the semantics to be well-defined? • We use it to prove an analogue of Scroggs's theorem. 7 Algebraic semantics: reasons • Kripke frames corresponds to complete, atomic, completely multiplicative modal algebras. We are forced to accept 9p(p ^ atom(p)) when is S5. And we are forced to accept Barcan: 8p' $ 8p'. • It is natural. Order-theoretically, 8p' is the weakest proposition that entails all instances of '. • It helps raising intersting questions. What if we drop atomicity? What if we drop complete multiplicativity? How much lattice-completeness do we need for the semantics to be well-defined? • We use it to prove an analogue of Scroggs's theorem. 7 General Π logics Definition A (normal) Π-logic is a set Λ of formulas in LΠ such that it is first of all a (normal modal logic) propositional modal logic and that it contains •8 p(' ! ) ! (8p' ! 8p ) •8 p'(p) ! '( ) • ' ! 8p' when p is not free and is closed under universalization: '=8p'. The smallest normal Π-logic containing a normal modal logic L is called LΠ. 8 Of course this is because the atomicity. General algerbaic semantics gives precisely S5Π. S5Π S5Π does not derive 9p(p ^ atom(p)). But on Kripke models where R is an equivalence relation, 9p(p ^ atom(p)) is valid. 9 S5Π S5Π does not derive 9p(p ^ atom(p)). But on Kripke models where R is an equivalence relation, 9p(p ^ atom(p)) is valid. Of course this is because the atomicity. General algerbaic semantics gives precisely S5Π. 9 Algebraic semantics Definition For any modal algebra B, a valuation V on B is a function from Prop to B. It naturally extends to Vb : L! B in the usual way. When B is complete, any such valuation can then be extended to an LΠ-valuation Vb : LΠ ! B by setting • Vb(8p') = VfV [\p 7! b](') j b 2 Bg. A formula φ 2 LΠ is valid on a complete modal algebra B, written as B φ, if for all valuations V on B, Vb(φ) = 1. 10 Questions Which normal Π-logics are complete? Characterize those Λ such that Λ = Log(Alg(Λ)).
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