NKU CSC 685 Advanced Topics in Applied Logic 2: Engineering with Modal Logics

1. NKU CSC 685 Advanced Topics in Applied Logic 2: Engineering with Modal Logics

2. p56 w2 w1 p23, p56 p23, p90 w3 true in a possible world true in all possible worlds Kripke Semantics: Set of possible worlds in which propositions are true/false, connected by a reachability relation. Propositional Modal Logic p p is true p p is possibly true  p p is necessarily true Examples:  it is raining all bachelors are unmarried Duality p  p p  p w |= p p is true in a world reachable from w w |= p p is true in all worlds reachable from w |= p p is true in all worlds NKU CSC 685 Kirby

3. Some possible axioms for epistemic logic: Propositional Epistemic Logic Kp means: agent knows p Kp means: ? Kp K(pq)  Kq Kp  KKp  Kp  KKp This looks so much like the logic of necessity-- let's give a uniform treatment... Is this correct: Kp, pq | Kq ?

4. epistemic logic Often written as  . Why is this ok? temporal logic LOGIC ENGINEERING necessarily forever believes knows "K" ( )   "T"   "4"   "5"   

5. Natural Deduction Rules for Modal Logic: = Propositional Logic Natural Deduction Rules, plus: { ... } : "within an arbitrary world" i { ...  } |  e  | { ...  ... } ___________________ T  |  K4  |  K5  |  K "general modality" KT45 "knowledge/ necessity" Note: No rules for ! It is just an abbreviation:  =  

6. Example: |KT45p  p [ 1 p asm 1 p abbrev { 2 p e 1 [ 3 p asm 4 p K5, 3 5  e 2,4 ] 6 p PBC 3-5 7 p T 6 } 8 p i 2-7 ] qed p  pi 1-9 Try also: Exercise 5.4.2.c

7. Kripke frame w2 w1 w3 abstract away abstract away Kripkemodels formulas w2 q r w1 p q satisfaction w3 r p p  (q  p) w2 q w1 w3 r (rp)q p q LOGIC ENGINEERING formula scheme 

8. x Rxx Kripke frame w2 reflexive w1 w3 abstract away abstract away Kripkemodels formulas w2 q r w1 p p satisfaction w3 r p (q  r)  (q  r) w2 q w1 w3 r r  r p q LOGIC ENGINEERING formula scheme  T rule

9. xyz Rxy & Ryz  Rxz transitive abstract away formulas p  p satisfaction (q  r)  (q  r) r  r LOGIC ENGINEERING formula scheme Kripke frame w2 w1  w3 K4 rule abstract away Kripkemodels w2 q r w1 w3 r p w2 q w1 w3 r p q

10. abstract away formulas p  p satisfaction (q  r)  (q  r) r  r LOGIC ENGINEERING xyz Rxy & Rxz  Ryz formula scheme Kripke frame euclidean w2 w1  w3 K5 rule abstract away Kripkemodels w2 q r w1 w3 r p w2 q w1 w3 r p q

11. epistemic logic let's prove this.. temporal logic LOGIC ENGINEERING necessarily forever believes knows syntax semantics (Kripke world accessibility) "K" ( )   "T"   reflexive: x Rxx "4"   transitive: xyz Rxy & Ryz  Rxz "5"    euclidean: xyz Rxy & Rxz  Ryz