index.md (2519B)
1 +++ 2 title = 'Lecture 1' 3 +++ 4 5 # Intro 6 7 Basic model logic operators: 8 - □: necessary, known, provable 9 - ◇: possible, considered possible 10 11 In a diagram, one of these symbols is exactly one transition step (use multiple for multiple steps). 12 13 - ◇ φ ⇔ ¬□ ¬φ 14 - □ φ ⇔ ¬◇ ¬φ 15 16 # First-order propositional logic 17 Includes variables, T, ⊥, not, and, or, implication. 18 Proofs are given by structural induction. 19 Precedence is ¬, then ∧∨, then →. 20 21 a valuation v : Var → {0,1} maps propositional variables to truth values. 22 23 the semantics of a formula under a valuation is defined with ⟦p⟧ᵥ = v(p), with p ∈ Var 24 25 if ⟦φ⟧ᵥ = 1, we write v ⊨ φ (read "v models φ") 26 - that is, if φ is true in the model v 27 - then, φ has a model, so φ is satisfiable 28 29 If every model of all φᵢ is a model of ψ, we write φ₁,...,φn ⊨ ψ 30 - then ψ is a semantic consequence of φ₁,...,φn 31 32 If v ⊨ φ for all valuations of v, then ⊨ φ (φ is a tautology) 33 34 Soundness: ⊢ implies ⊨ -- what we can derive is true. proved by induction on length of proof 35 36 Completeness: ⊨ implies ⊢ -- we can derive what is true. can be proven using consistency 37 38 # Basic modal logic 39 E.g. "whatever is necessary is possible" == □φ → ◇φ 40 41 □ can also mean "I know", e.g. "I know that someone appreciates me" == ∃x.□A(x, M) 42 43 Loeb's formula: □ (□ p → p) → □ p 44 45 Veridicality: □ φ → φ 46 47 Truth s relative to current situation/world/environment: 48 - formulas evaluated in given structure 49 - necessity (□): truth in all accessible worlds. if there are no accessible worlds, it's true. 50 - possibility (◇): truth in some accessible world (at least one). if there are no accessible worlds, it's false. 51 52 ## Frames 53 A situation is set by a frame F = (W,R) 54 - W ≠ ∅ set of possible worlds/states 55 - R ⊆ W × W an accessibility/transition relation 56 57 A frame is just the states and transitions between them, without a valuation (i.e. without saying what's true in each state). 58 59 A frame could be (ℕ, <), or ({1,2,3,4}, {(1,2), (2,4), (1,3), (3,4), (2,2)}) 60 61 ## Models 62 model: pair M = (F, V) 63 - a frame F = (W,R) 64 - a valuation V : Var → W → {0,1}, or V : Var → P(W) 65 - the valuation says which letters/formulas are true in which states 66 67 pointed model: pair (M,w) of model M and w a world in M 68 69  70 71 ◇T holds if at least one successor. 72 □⊥ holds if there is no successor. 73 74 Dualities: 75 - ◇ φ := ¬ □ ¬ φ 76 - □ φ := ¬ ◇ ¬ φ