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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)

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

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..

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 ..

G.E. Hughes & M.J. Cresswell, A New Introduction to Modal Logic, London and New York: Routledge, 1996, pp. 58-62. (5) The System S5 ① S5 Mp ⊃ LMP S5 (or E) Mp ⊃ LMP [R1a] ② Translation into quantification theory ③ S5 (1)-(3) ④ S4 Lp ⊃ LLp (as a theorem of S5) ⑤ S5 (4)-(7) ⑥ S5* (MLp ∧ Mq) ⊃ M(p ∧ q) ⑦ S5 is not a theorem of S4 ∴ S5 properly contains S4 (6) Modalities in S5 ① In any pair of adjac..

G.E. Hughes & M.J. Cresswell, A New Introduction to Modal Logic, London and New York: Routledge, 1996, pp. 53-56. (2) The System S4 ① S4 Lp ⊃ LLp S4 Lp ⊃ LLp ② Translation into quantification theory ③ S4 (1) and S4 (2) S4 (1) MMp ⊃ Mp S4 (2) Lp ≡ LLp [R4] ④ S4 (3)-(7) S4 (3) Mp ≡ MMp S4 (4) MLMp ⊃ Mp S4 (5) LMp ⊃ LMLMp S4 (6) LMp ≡ LMLMp S4 (7) MLp ≡ MLMLp ⑤ S4* (Lp ∧ MMq) ⊃ M (p ∧ q) S4* (Lp ∧ ..

G.E. Hughes & M.J. Cresswell, A New Introduction to Modal Logic, London and New York: Routledge, 1996, pp. 51-53. (1) Iterated Modalities ① Constructing a system stronger than T when we ask, informally, whether Lp ⊃ LLp is valid, the issue we are raising is this: is whatever is necessary necessarily necessary? when something is necessarily so, is the fact that it is necessarily so always itself ..

G.E. Hughes & M.J. Cresswell, A New Introduction to Modal Logic, London and New York: Routledge, 1996, pp. 43-45. (6) The System D ① Interpretation of L as expressing obligationess (moral necessity) (a) Lp ⊃ Mp is not valid in this system, since when it will then mean is that whatever ought to be the case is in fact the case. (b) D Lp ⊃ Mp (Denotic Interpretation) Lp: It is obligatory that P Mp:..

G.E. Hughes & M.J. Cresswell, A New Introduction to Modal Logic, London and New York: Routledge, 1996, pp. 41-43. (5) The System T ⓛ As we previously noted, Lp ⊃ p is not K-valid ② The System T K + T (Lp ⊃ p) ※ T is often called Axiom of Necessity ③ Translation into quantification theory ④ T1 p ⊃ Mp ⑤ T2 M(p ⊃ Lp) ⑥ p is not a rule of T If P were a rule of T, then from it and T2 we could derive ..

G.E. Hughes & M.J. Cresswell, A New Introduction to Modal Logic, London and New York: Routledge, 1996, pp. 33-36. (4) L and M ① K5 Lp ≡ ~M~p K5: Lp ≡ ~M~p Proof 1. p ≡ ~~p PC12 (DN) 2. Lp ≡ ~~Lp 1[Lp/p] 3. Lp ≡ ~~L~~p 2 × 1 × Eq 4. Lp ≡ ~M~P Def M Q. E. D. ※ Clearly K5, by Eq, will entitle us to replace L by ~M~ anywhere in a theorem; and by Def M we may replace M anywhere in a theorem by ~L~. ②..