Consequence Relations of Modal Logic

Transcription

1 Consequence Relations of Modal Logic Lauren Coe, Trey Worthington Huntingdon College BLAST 2015 January 6, 2015

2 Outline 1. Define six standard consequence relations of modal logic (Syntactic, Algebraic, Kripke Semantic)

3 Outline 1. Define six standard consequence relations of modal logic (Syntactic, Algebraic, Kripke Semantic) 2. Prove that the three local consequence relations are equivalent

4 Outline 1. Define six standard consequence relations of modal logic (Syntactic, Algebraic, Kripke Semantic) 2. Prove that the three local consequence relations are equivalent 3. Prove that the three global consequence relations are equivalent

5 Interior Algebra An interior algebra is a Boolean algebra with and extra operation σ that satisfies the following axioms:

6 Interior Algebra An interior algebra is a Boolean algebra with and extra operation σ that satisfies the following axioms: 1. σ(x) x 2. σ(σ(x)) = σ(x) 3. σ(x y) = σ(x) σ(y) 4. σ(1) = 1

7 Interior Algebra An interior algebra is a Boolean algebra with and extra operation σ that satisfies the following axioms: 1. σ(x) x 2. σ(σ(x)) = σ(x) 3. σ(x y) = σ(x) σ(y) 4. σ(1) = 1 Notice that topological spaces are interior algebras.

8 Interior Algebra An interior algebra is a Boolean algebra with and extra operation σ that satisfies the following axioms: 1. σ(x) x 2. σ(σ(x)) = σ(x) 3. σ(x y) = σ(x) σ(y) 4. σ(1) = 1 Notice that topological spaces are interior algebras. We will denote the variety of interior algebras by IA.

9 Interior Algebra An interior algebra is a Boolean algebra with and extra operation σ that satisfies the following axioms: 1. σ(x) x 2. σ(σ(x)) = σ(x) 3. σ(x y) = σ(x) σ(y) 4. σ(1) = 1 Notice that topological spaces are interior algebras. We will denote the variety of interior algebras by IA. It is known that IA is an algebraic semantics of S4.

10 Kripke Semantics A model is a triple W, R, V such that: W is a non-empty set, R WxW and R is reflexive and transitive, and V is a value assignment, or valuation, that for each formula φ and each world w in a model the value of φ at w is either equal to 1 or to 0 but not both.

11 Kripke Semantics A model is a triple W, R, V such that: W is a non-empty set, R WxW and R is reflexive and transitive, and V is a value assignment, or valuation, that for each formula φ and each world w in a model the value of φ at w is either equal to 1 or to 0 but not both. We denote the class of Kripke models as KS.

12 Kripke Semantics A model is a triple W, R, V such that: W is a non-empty set, R WxW and R is reflexive and transitive, and V is a value assignment, or valuation, that for each formula φ and each world w in a model the value of φ at w is either equal to 1 or to 0 but not both. We denote the class of Kripke models as KS. It is known that KS is a semantics of S4.

13 Algebraic CR Let Γ {φ} be a set of formulas. We say Γ = g IA φ provided that for every interior algebra A, σ and every substitution π if π(γ) = 1, for every γ Γ π(φ) = 1.

14 Algebraic CR Let Γ {φ} be a set of formulas. We say Γ = g IA φ provided that for every interior algebra A, σ and every substitution π if π(γ) = 1, for every γ Γ π(φ) = 1. Let Γ {φ} be a set of formulas. We say Γ = l IA φ provided that for every interior algebra A, σ, every substitution π and every ultrafilter U if π(γ) U for every γ Γ π(φ) U.

15 Kripke Semantic CR Let Γ {φ} be a set of formulas. We say Γ = g KS φ provided that for every model W, R, V, if V (γ, w) = 1 for every γ Γ and w W then V (φ, w) = 1 for every w W.

16 Kripke Semantic CR Let Γ {φ} be a set of formulas. We say Γ = g KS φ provided that for every model W, R, V, if V (γ, w) = 1 for every γ Γ and w W then V (φ, w) = 1 for every w W. Let Γ {φ} be a set of formulas. We say Γ = l KS φ provided that for every model W, R, V and every w W, if V (γ, w) = 1 for every γ Γ then V (φ, w) = 1.

17 Syntactic CR Let Γ {φ} be a set of formulas. We say Γ l φ if there exists a sequence of formulas x 1, x 2,..., x n, φ such that each of these formulas is either a theorem of S4, a premise (an element of Γ), or follows from previous formulas by modus ponens.

18 Syntactic CR Let Γ {φ} be a set of formulas. We say Γ l φ if there exists a sequence of formulas x 1, x 2,..., x n, φ such that each of these formulas is either a theorem of S4, a premise (an element of Γ), or follows from previous formulas by modus ponens. Let Γ {φ} be a set of formulas. We say Γ g φ if there exists a sequence of formulas x 1, x 2,..., x n, φ such that each of these formulas is either a theorem of S4, a premise (an element of Γ), follows from previous formulas by modus ponens or the rule of necessitation.

19 Our Goal To show that the following global consequence relations are equivalent: 1. Γ = g KS φ 2. Γ = g IA φ 3. Γ g φ

20 Our Goal To show that the following global consequence relations are equivalent: 1. Γ = g KS φ 2. Γ = g IA φ 3. Γ g φ To show that the following local consequence relations are equivalent: 1. Γ = l KS φ 2. Γ = l IA φ 3. Γ l φ

21 Proof To show that Γ = g KS φ is equivalent to Γ =g IA φ we are going to use Jonsson-Tarski duality.

22 Proof To show that Γ = g KS φ is equivalent to Γ =g IA φ we are going to use Jonsson-Tarski duality. Assume there exists a model W, R, V, and construct an interior algebra P(W ), that satisfies the same formulas as the model. For X P(W ), we define X = {a W (a, b) R b X }.

23 Proof To show that Γ = g KS φ is equivalent to Γ =g IA φ we are going to use Jonsson-Tarski duality. Assume there exists a model W, R, V, and construct an interior algebra P(W ), that satisfies the same formulas as the model. For X P(W ), we define X = {a W (a, b) R b X }. Given an interior algebra A, σ and a substitution π, we can construct a model F, R, V (where F is a set of ultrafilters, we define R as a reflexive and transitive relation where (F 1, F 2 ) R if whenever σ(x) F 1, x F 2, and V is the valuation defined by V (φ, Γ) = 1 iff the substiution π(φ) Γ) that satisfies the same formulas as the interior algebra.

24 Proof To show that Γ = l KS φ is equivalent to Γ =l IA φ The same construction at the global level also works locally.

25 Proof To show that Γ = l KS φ is equivalent to Γ =l IA φ The same construction at the global level also works locally. To show that if Γ l φ then, Γ = l IA φ in one direction we use a standard induction proof.

26 Proof To show that Γ = l KS φ is equivalent to Γ =l IA φ The same construction at the global level also works locally. To show that if Γ l φ then, Γ = l IA φ in one direction we use a standard induction proof. For the other direction assume Γ = l IA φ, then in the free interior algebra with denumerably many generators, φ is in the filter generated by Γ.

27 Proof To show that Γ = l KS φ is equivalent to Γ =l IA φ The same construction at the global level also works locally. To show that if Γ l φ then, Γ = l IA φ in one direction we use a standard induction proof. For the other direction assume Γ = l IA φ, then in the free interior algebra with denumerably many generators, φ is in the filter generated by Γ. φ γ 1 γ n, for some γ 1 γ n Γ

28 Proof To show that Γ = l KS φ is equivalent to Γ =l IA φ The same construction at the global level also works locally. To show that if Γ l φ then, Γ = l IA φ in one direction we use a standard induction proof. For the other direction assume Γ = l IA φ, then in the free interior algebra with denumerably many generators, φ is in the filter generated by Γ. φ γ 1 γ n, for some γ 1 γ n Γ γ 1, γ 2, γ 1 (γ 2 γ 1 γ 2 ), γ 2 γ 1 γ 2, γ 1 γ 2, γ 3,, γ 1 γ 2 γ 3,, γ 1 γ 2 γ n, γ 1 γ n φ, φ

29 Proof To show that if Γ g φ then, Γ = g IA φ we use a standard induction proof.

30 Proof To show that if Γ g φ then, Γ = g IA φ we use a standard induction proof. For the other direction assume Γ = g IA φ, then in the free interior algebra with denumerably many generators, φ is in the kernel (a filter that is closed under σ) generated by Γ.

31 Proof To show that if Γ g φ then, Γ = g IA φ we use a standard induction proof. For the other direction assume Γ = g IA φ, then in the free interior algebra with denumerably many generators, φ is in the kernel (a filter that is closed under σ) generated by Γ. φ σ(γ 1 ) σ(γ n ), for some γ 1 γ n Γ

32 Proof To show that if Γ g φ then, Γ = g IA φ we use a standard induction proof. For the other direction assume Γ = g IA φ, then in the free interior algebra with denumerably many generators, φ is in the kernel (a filter that is closed under σ) generated by Γ. φ σ(γ 1 ) σ(γ n ), for some γ 1 γ n Γ γ 1, γ 1, γ 2, γ 2, γ 1 ( γ 2 γ 1 γ 2 ), γ 2 γ 1 γ 2, γ 1 γ 2, γ 3, γ 3,, γ 1 γ 2 γ 3,, γ 1 γ 2 γ n, γ 1 γ n φ, φ

33 Thank You! Thank You!