lecture-3.md (1770B)
1 +++ 2 title = 'Lecture 3' 3 template = 'page-math.html' 4 +++ 5 # Lecture 3 6 ## Preservation of truth and validity 7 ### Substitution 8 Substitution for propositional variables 9 - σ : Var → Form 10 - T and ⊥ not substituted 11 12 If (W,R), V ⊨ φ then not necessarily $(W,R) V \models \phi^{\sigma}$ 13 14 But validity in a frame is preserved under substitution: if F ⊨ φ, then $F \models \phi^{\sigma}$ for any substitution σ. 15 16 Validity is closed under substitution: if F ⊨ φ, then $F \models \phi^{\sigma}$ for any substitution σ. 17 18 ### Alternative semantics 19 The interpretation $[\\![ \phi ]\\!] _{M}$ of a formula φ in model M = (W,R,V) is set of worlds in which φ is true. 20 21 M, w ⊨ φ iff $w \in [\\![\phi]\\!]_{M}$ 22 23 M ⊨ φ iff $[\\![\phi]\\!]_{M} = W$ 24 25 ### Preservation of truth and validity 26 Local truth preserved by modus ponens: if M, w ⊨ φ → ψ and M, w ⊨ φ then M, w ⊨ ψ 27 28 Global truth preserved by modus ponens and necessitation: if M ⊨ φ then M ⊨ □ φ 29 30 Frame validity preserved by modus ponens, necessitation, and substitution: if F ⊨ φ then $F \models \phi^{\sigma}$. 31 32 ## Modal tautologies 33 ⊨ □ (p → q) → □ p → □ q 34 35 If ⊨ φ → ψ and ⊨ φ then ⊨ ψ 36 37 If ⊨ φ then ⊨ □ φ 38 39 If ⊨ φ then $\models \phi^{\sigma}$ 40 41 ## Characterizations of frame properties 42 If F reflexive then F ⊨ □ p → p. 43 This holds in the opposite. 44 So the formula □ p → p characterizes the frame property 'reflexivity'. 45 46 In general, formula φ characterizes the frame property P means: F has property P iff F ⊨ φ. 47 If you need to prove that a formula characterizes a property, you need to prove this bi-implication in _both_ directions. 48 49 ## Modal equivalence 50 Two states M, w and M', w' are modally equivalent if they satisfy the same formulas.