Proofs (1) 2023-2 Modal Logic (Segment 3) Q) Routley-Meyer Semantics for Relevant Logic has this: Prove (using the semantics) the following: 1. A & B .–▶. A v B 2. ¬¬A ◀– –▶ A 3. ¬(A & B) ◀– –▶. ¬A v ¬B 4. ¬A –▶ B .–▶. ¬B –▶ A Logic/Relevance Logic 2024.01.14
Relevant Logic as Paraconsistent Logic 2023-2 Modal Logic (Segment 3) Q) Explain why that (∀x)Roox yields (∀x)Roxo and collapses Relevent logic into classical logic. If is sufficient (for the present purposes) to just prove something that doesn’t hold in Relevant Logic, e.g., ¬A ∧ (A ∨ B) .–Ȝ. B. Why does it seem not to be viable to take relevant logic to be a universal logic (a logic that can capture all others?) #2) Generally speak.. Logic/Relevance Logic 2024.01.14
Derived Rules 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 Logic/Modal Logic 2024.01.04
Modal Conjunctive Normal Form (1) 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 ∨ Lq LMLp ⊃ Lp Logic/Modal Logic 2024.01.04
Proofs (2) 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 Logic/Modal Logic 2024.01.04
Proofs (1) 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 ⊃ Mp Lp ⊃ LLMLp LLMLp ⊃ Lp ② 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 ) Logic/Modal Logic 2024.01.04
The Tableau Method (1) 2023-2 Modal Logic (Segment 1) ① Determine whether the following valid in the systems indicated by the semantic tableau L(Lp ⊃ Lq) ∨ L(Lq ⊃ Lp) ML(p ⊃ LMp) ② Determine which of the following are valid in K, D, T, S4, S5, B by tableaux. L(p ∨ Lq) .⊃. Lq ∨ Lp M(p⊃p) Mp ⊃ LMMp Lp ∧ M(q ⊃ r) :⊃: L(p⊃q) ⊃ M(p ∧ r) Logic/Modal Logic 2024.01.04
Translation into quantification theory (2) 2023-2 Modal Logic (Segment 1) Q) Prove the following by Translation into quantification theory (you may use conditional proof). Trans(R) & Symm(R) ├ Mp ∧ Mq .⊃. M(p ∧ Mq) Trans(R)├ M(Lp ⊃ Mq) ⊃ M(p ⊃ q) Symm(R) ├ p ⊃ LMp Logic/Modal Logic 2024.01.04
Translation into quantification theory (1) 양상 연산자(Modal operator) L(□)과 M(◇)의 중요한 특징 중 하나는 그것들이 '모든(∀)'과 '어떤(∃)'의 개념을 포함하고 있다는 점이다. 즉, (세계 w에서 성립하는) 명제 Lp는 w가 볼 수 있는 모든 세계에서 p가 성립함을 뜻한다는 점에서 '모든'의 개념을 포함한다. 마찬가지로, Mp는 w가 볼 수 있는 어떤(최소한 하나 이상의) 세계에서 p가 성립함을 의미한다는 점에서 '어떤'의 개념을 포함한다. 따라서 우리는 양상 연산자가 쓰인 모든 문장을 양화 이론(quantification theory)으로 번역할 수 있다. ① (Lp)ʷ와 (Mp)ʷ 우선 Lp와 Mp에 대한 정의는 다음과 같다: 여기에서 (Lp)ʷ가 의미하는 바는 w가 볼 수 있는 모든 세계 α에서 p가 성립한다는 것이.. Logic/Modal Logic 2024.01.04
Chapter 3. The Brouwerian System G.E. Hughes & M.J. Cresswell, A New Introduction to Modal Logic, London and New York: Routledge, 1996, pp. 62-64. (8) The Brouwerian System ① B p ⊃ LMp B (or S5 (8)) p ⊃ LMp ② Translation into quantification theory ③ S5 (9) MLp ⊃ p S5 (9) MLp ⊃ p ④ Neither of these theorems is in S4. Indeed, if we were to add either as an extra axiom to S4 we should obtain a system at least as strong as S5. (In .. Logic/Modal Logic 2024.01.04