1 / 11

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

NKU CSC 685 Advanced Topics in Applied Logic 2: Engineering with Modal Logics. p 56. w 2. w 1. p 23 , p 56. p 23 , p 90. w 3. true in a possible world. true in all possible worlds. Kripke Semantics: Set of possible worlds in which propositions are true/false,

mandy
Download Presentation

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

An Image/Link below is provided (as is) to download presentation Download Policy: Content on the Website is provided to you AS IS for your information and personal use and may not be sold / licensed / shared on other websites without getting consent from its author. Content is provided to you AS IS for your information and personal use only. Download presentation by click this link. While downloading, if for some reason you are not able to download a presentation, the publisher may have deleted the file from their server. During download, if you can't get a presentation, the file might be deleted by the publisher.

E N D

Presentation Transcript


  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

More Related