index.md (4295B)
1 +++ 2 title = 'Lecture 8' 3 +++ 4 # Lecture 8 5 ## Modal tautologies 6 7  8 9 ## Proof systems 10 Hilbert systems: 11 - proof is sequence of numbered formulas 12 - every formula is: an axiom, or result of applying a derivation rule 13 14 Sufficient to have 2 axioms and a rule: 15 - K: φ → ψ → φ 16 - S: (φ → ψ → θ) → (φ → ψ) → (φ → θ) 17 - modus ponens: if φ and (φ → ψ), then ψ 18 19 Rules will be given, don't need to be memorized. 20 21 Admissible rule: 22 23  24 25 Proof system K is sound and complete with respect to all frames, ⊢K φ iff ⊨ φ. 26 27 <!-- TODO: add this to anki --> 28 Need to memorize soundness and completeness results 29 - K sound and complete for all frames 30 - T sound and complete for all _reflexive_ frames 31 - T: K with □ p → p 32 - S4 sound and complete for all _reflexive-transitive_ frames 33 - S4: T with □ p → □ □ p 34 - S5 sound and complete for all frames with R an equivalence relation 35 - S5: S4 with ¬ □ p → □ ¬ □ p 36 37 ## Example of derivation 38 Give derivation in K of ◇ φ ∧ □ (φ → ψ) → ◇ ψ. 39 Same as example in book <abbr title='Modal Logic for Open Minds (Benthem)'>MLOM</abbr> page 52. 40 41 First, work backwards from the goal towards an axiom or tautology: 42 43 ``` 44 ◇ φ ∧ □ (φ → ψ) → ◇ ψ ≡ □ (φ → ψ) → (◇ φ → ◇ ψ) [you can rewrite a conjunction as an implication] 45 ≡ □ (φ → ψ) → (¬ □ ¬ φ → ¬ □ ¬ ψ) [rewrite diamond to ¬ □ ¬] 46 ≡ □ (φ → ψ) → (□ ¬ ψ → □ ¬ φ) [rewrite contrapositive (¬ a → ¬ b) to (b → a)] 47 ≡ □ (φ → ψ) → □ (¬ ψ → ¬ φ) [box distribution over implication] 48 ≡ □ (φ → ψ) → □ (φ → ψ) [again contrapositive] 49 ≡ (φ → ψ) → (φ → ψ) [because if derivable (a → b), then derivable (□ a → □ b)] 50 ``` 51 52 We arrive at a tautology. 53 54 Then you write it out in a Hilbert-style proof, starting with the tautology/axiom. 55 PROP means rewriting in propositional logic. 56 57 1. (φ → ψ) → (¬ ψ → ¬ φ). PROP. 58 2. □ (φ → ψ) → □ (¬ ψ → ¬ φ). DISTR, 1. 59 3. □ (p → q) → □ p → □ q. modal distribution (i.e., this is an axiom in K that we use) 60 4. □ (¬ ψ → ¬ φ) → □ ¬ ψ → □ ¬ φ. substitution, 3 (i.e., substitute stuff in the axiom). 61 5. □ (φ → ψ) → □ ¬ ψ → □ ¬ φ. PROP, 2, 4. 62 6. □ (φ → ψ) → ¬ ◇ ψ → ¬ ◇ φ. definition of ◇, □. 63 7. □ (φ → ψ) → ◇ φ → ◇ ψ. PROP, 6. 64 8. ◇ φ ∧ □ (φ → ψ) → ◇ ψ. PROP, 7. 65 66 <details> 67 <summary>Why you can rewrite a conjunction as an implication</summary> 68 69 You can safely rewrite a conjunction to an implication: (a ∧ b → c) ≡ a → (b → c). 70 Remember that implication is right-associative! 71 72 If you don't trust me, I didn't trust myself either so I made a truth table: 73 74 <table> 75 <thead> 76 <tr> 77 <th>a</th> 78 <th>b</th> 79 <th>c</th> 80 <th>b → c</th> 81 <th>a ∧ b</th> 82 <th>a → c</th> 83 <th>a ∧ b → c</th> 84 <th>b → (a → c)</th> 85 <th>a → (b → c)</th> 86 </tr> 87 </thead> 88 <tbody> 89 <tr> 90 <td>0</td> 91 <td>0</td> 92 <td>0</td> 93 <td>1</td> 94 <td>0</td> 95 <td>1</td> 96 <td>1</td> 97 <td>1</td> 98 <td>1</td> 99 </tr> 100 101 <tr> 102 <td>0</td> 103 <td>0</td> 104 <td>1</td> 105 <td>1</td> 106 <td>0</td> 107 <td>1</td> 108 <td>1</td> 109 <td>1</td> 110 <td>1</td> 111 </tr> 112 113 <tr> 114 <td>0</td> 115 <td>1</td> 116 <td>0</td> 117 <td>0</td> 118 <td>0</td> 119 <td>1</td> 120 <td>1</td> 121 <td>1</td> 122 <td>1</td> 123 </tr> 124 125 <tr> 126 <td>0</td> 127 <td>1</td> 128 <td>1</td> 129 <td>1</td> 130 <td>0</td> 131 <td>1</td> 132 <td>1</td> 133 <td>1</td> 134 <td>1</td> 135 </tr> 136 137 <tr> 138 <td>1</td> 139 <td>0</td> 140 <td>0</td> 141 <td>1</td> 142 <td>0</td> 143 <td>0</td> 144 <td>1</td> 145 <td>1</td> 146 <td>1</td> 147 </tr> 148 149 <tr> 150 <td>1</td> 151 <td>0</td> 152 <td>1</td> 153 <td>1</td> 154 <td>0</td> 155 <td>1</td> 156 <td>1</td> 157 <td>1</td> 158 <td>1</td> 159 </tr> 160 161 <tr> 162 <td>1</td> 163 <td>1</td> 164 <td>0</td> 165 <td>0</td> 166 <td>1</td> 167 <td>0</td> 168 <td>0</td> 169 <td>0</td> 170 <td>0</td> 171 </tr> 172 173 <tr> 174 <td>1</td> 175 <td>1</td> 176 <td>1</td> 177 <td>1</td> 178 <td>1</td> 179 <td>1</td> 180 <td>1</td> 181 <td>1</td> 182 <td>1</td> 183 </tr> 184 </tbody> 185 </table> 186 187 You see that the right three columns all have the same values, so semantically the formulas are the same. 188 189 </details> 190