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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..

G.E. Hughes & M.J. Cresswell, A New Introduction to Modal Logic, London and New York: Routledge, 1996, pp. 24-33. (2) The System K ① The system K is one which will turn out to have as its theorems precisely those modal formulae which are K-valid. ② The axioms of the system K consist of (a) all valid wff of PC (b) the single distinctively modal wff, K PC: If α is a valid wff of PC, then α is an a..

G.E. Hughes & M.J. Cresswell, A New Introduction to Modal Logic, London and New York: Routledge, 1996, pp. 23-24. (1) Systems of Modal Logic ① The axiomatic method: it allows us to define a class of wff without any reference to their meanings. ② An axiomatic basis for a logical system (a) a specification of the language in which the formulae of the system will be expressed - i.e. a list of primi..

G.E. Hughes & M.J. Cresswell, A New Introduction to Modal Logic, London and New York: Routledge, 1996, pp. 13-21. (9) Basic modal notions ① In modal logic, there are a number of non-truth-functional concepts. ② Modal Operator L (□) L: necessity operator "necessarily," "must be," "is bound to" Instead of writing "A is bound to be B," we can write "It is bound to be the case that A is B." Lp= Nece..

G.E. Hughes & M.J. Cresswell, A New Introduction to Modal Logic, London and New York: Routledge, 1996, pp. 3-13. (1) The Language of PC (Propositional Carculus) ① Primitive (or undefined) symbols of PC 1. A set of letters: p, q, r, . . . (with or without numerical subscripts). We suppose ourselves to have an unlimited number of these. 2. The following four symbols: ~, V, (, ). ※ Any symbol in th..