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2023-2 Modal Logic (Segment 2) Q)UG, MP. E! is primitive, ∼ , ∨ are primitive and we have the usual definitions. Sketch proofs of:① ├FL (∀x)( E!x ≡ (∃y)(x = y)).② ├FL (∀y) ((∀x) ϕx ⊃ ϕy).Now consider the system FL* which is just like the above except that (∀y) ((∀x) ϕx ⊃ ϕy) replaces axiom schemas 2 and 3, and E! is dropped as a primitive sign.Sketch proofs of:③ ├FL* (∀x) (∃y)(x = y)④ ├FL* (∀x)B..

2023-2 Modal Logic (Segment 3) Q) Explain the system S1. In what sense is it the case that the “unrestricted” rules of Necessitation does not hold in S1. Does pc tautology necessitation hold in S1 ? What is the Cresswell argument (See my SS1 paper) that S1 has no theorems of the form LL(p⊃ p)? Explain the semantic system for S2 and S3 and S4 where one introduces odd vs normal worlds. Note that b..

2023-2 Modal Logic (Segment 2) Q) Use tableaux to show the difference between what we have called the A semantics and the B semantics (+ inclusion requirement) and the C-semantics for quantified modal logic. Be sure to explain by reference to the question of the validity of the Carnap-Barcan Formula (BF) and its converse. Be sure to explain the situation withAx1 (∀x) (ϕx) ⊃ ϕy, where y is free f..

2023-2 Modal Logic (Segment 1) Q) Use the technique of Modal Conjunctive Normal form to test the follow for S5 Validity and if Valid construct a proof using the technique.ML(p ⊃ LMp)M(Lp ⊃Mq)⊃ M(p⊃ q)L(p ∨ Lq) .⊃. Lp ∨ LqLMLp ⊃ Lp

2023-2 Modal Logic (Segment 1) ① Give a recipe justifying the following derived rule: ├K p ∨ q ⇒ ├K Lp ∨ Mq ② Some Drived Rules

2023-2 Modal Logic (Segment 2) Q) Quantification theory may be formalized this way: Prove the following in the systems indicated.1. ├ L(∀x)ϕx . ⊃. (∀x)Lϕx2. ├ β (∀x)Lϕx . ⊃. L(∀x)ϕx (Recall that system β is T + p ⊃ LMp. )3. ├ M(∃x)ϕx . ⊃. (∃x)Mϕx (The system is QML + (∀x)Lϕx . ⊃. L(∀x)ϕx)4. ├ (∀x)Lα ⊃ L(∀x)α (The system is QML+ (∀x)L(α ⊃ β) .⊃. L((∀x)α ⊃ (∀x)β)

2023-2 Modal Logic (Segment 1)Q) Give a proof of the following in the systems indicated:├S5 Mp ∧ Mq .⊃. M(p ∧ Mq)├B MLp ⊃ LMp├S5 Lp ⊃ LLp├S5 p ⊃ LMp├S5 L(p ∨ Lq) .⊃. Lp ∨ Lq├S4 M(p ∧ Mq) .⊃. Mp ∧ Mq

2023-2 Modal Logic (Segment 1)① Check to see which of the following are S4 valid or S5 only valid and give a proof in S4 if it is S4 valid and give a proof in S5 if it is S5 only valid:MMLMp ⊃ MpLp ⊃ LLMLpLLMLp ⊃ Lp② Give a proof of the following in the systems indicated:├K L(p ∨ q) .⊃. Lp ∨ Mq├S4 Lp ∨ Lq .⊃. L(p ∨ Lq)├T M(p ⊃Mp )

2023-2 Modal Logic (Segment 2) Q) Evaluate the following wffs in the B-semantics + inclusion requirement and determine in what systems, K, T, S4, β, S5 (if any) they are valid. 1. (∃y)LM(ϕy ⊃ (∀x)ϕx) 2. L(∃x)(Lϕx ∨ ψy) ⊃ (∃x)L(Lϕx ∨ ψy) 3. L(M(∃x)ϕx ⊃ (∃x)Mϕx) 4. L(∃x)ϕx ⊃ (∃x)Lϕx

2023-2 Modal Logic (Segment 3) Q) Use the tableau method (which involves non-normal world idea) to determine whether the following are valid in Lewis’ systems S2, S3:1. Lp ⊃ LLp2. L( M(p ∧ q) ⊃ Mp)3. L(L(p ⊃ q ) ⊃ L(Lp ⊃ Lq))