Epistemic Logic I. An introduction to the course Yanjing Wang Department of Philosophy, Peking University Sept. 14th 2015
Standard epistemic logic and its dynamics Beyond knowing that: a new research program Plan of the course
Importance of reasoning about knowledge (in everyday life) We use knowledge, belief and probability to organize certainty and uncertainty, and to turn uncertainty into certainty. Knowledge is power: act properly to achieve goals; Knowledge is time: to make decisions more efficiently; Knowledge is money: can be traded; Knowledge is responsibility: to prove someone is guilty; Knowledge is you: to identify oneself; Knowledge is an immune system: to protect you; Knowledge is common ground: to communicate; Knowledge satisfies our curiosity Reasoning about knowledge: know the unknown from the known (and new info). To help us to get more power, more money, yet also more responsibility.
Standard Epistemic Logic Propositional modal logics that reason about knowledge (and belief) [von Wright 1951, Hintikka 1962]. Language: agent i knows that φ : φ ::= p φ (φ φ) K i φ Model: possibilities with (equivalence) relations. Semantics: you know that φ iff φ is true in all epistemic alternatives (of the current world). i p p K i p i p i
S5 system (strongest epistemic logic) Axioms System S5 TAUT all the instances of tautologies MP Rules DISTK K i (p q) (K i p K i q) NECK T K i p p SUB 4 K i p K i K i p 5 K i p K i K i p 4 and 5 axioms in Confucius teaching: φ, φ ψ ψ φ K i φ φ φ[p/ψ] S5 is sound and strongly complete for modal logic over S5 frames.
Core ideas Semantics: knowledge as elimination of uncertainty Syntax: (normal) modal logics ([S4, S5]) (semantic) vs. (syntactic) Powerful when combined with other modalities.
Examples of extensions: knowledge and action/time Handling knowledge and actions: Epistemic Temporal Logic (ETL) and Dynamic Epistemic Logic (DEL): 2-dimensional modal logics. language model semantics ETL time+k temporal+epistemic Kripke-like DEL K+events epistemic Kripke+dynamic Kp [e] Kp Kp [!p]kp p p p p e e!p p p p Other extensions and variants: Epistemic ATL, Epistemic STIT, justification logics, evidence logic etc. with applications in TCS, AI, Game theory and so on.
Standard epistemic logic and its dynamics Beyond knowing that: a new research program Plan of the course Examples of extensions: knowledge and action Some notable axiom schemas ( e and ˆK are duals of [e] and K): PR: e ˆKφ ˆK e φ or K[e]φ [e]kφ NL: ˆK e φ e ˆKφ or [e]kφ K[e]φ DEL-NM: ˆK e φ [e]ˆkφ or e Kφ K[e]φ DEL-INV: (p [e]p) ( p [e] p) Uniform sub. fails DEL-PRE: e pre(e) DEL-DET: e φ [e]φ (usually derivable in DEL-like logics) We learn by checking the executability of actions: p p e=!p p
Puzzles
Cheryl s birthday puzzle Cheryl doesn t want to share her birthday directly. So she gives her friends Albert and Bernard a list of 10 possible dates: May 15, May 16, May 19, June 17, June 18, July 14, July 16, August 14, August 15 or August 17. Cheryl tells Albert and Bernard separately the month and the day of her birthday, respectively. Then the following dialogue happens between Albert and Bernard: Albert: I don t know when Cheryl s birthday is, but I know that you don t know it, either. Bernard: At first I did not know it, but now I know it. Albert: Then, now I know it, too. What is Cheryl s birthday?
Muddy Children puzzle Out of n > 1 children, k 1 got mud on their faces while playing. They can see whether other kids are dirty, but there is no mirror for them to discover whether they are dirty themselves. Then father walks in and states: At least one of you is dirty! Then he requests If you know you are dirty, step forward now. If nobody steps forward, he repeats his request: If you now know you are dirty, step forward now. After exactly k requests to step forward, the k dirty children suddenly do so (assuming they are perfect reasoners).
Some issues Logical omniscience Agents=rigid indexes Locality w.r.t. indistinguishable alternatives Ignorance considering too many possibilities Learning elimination Model building or hacking (pai nao dai)?
Beyond knowing that : motivation Knowledge is not only expressed in terms of knowing that : I know whether the claim is true. I know what your password is. I know how to go to Barcelona. I know why he was late. I know who proved this theorem. I know where he was born.... power: know-how; authentication: know-what; science: know-why and so on...
Beyond knowing that : motivation Linguistically: know takes embedded questions but believe does not: factive verbs; neg raising; ways of questioning; ambiguity; semantics of questions Philosophically: reducible to knowledge-that? Logically: how to reason about knowing X? Computationally: efficient knowledge representation, and automated reasoning about knowing X
Beyond knowing that: research agenda In fact, knowing who was discussed by Hinttikka (1962) in terms of first-order modal logic: xk(yanjing = x). Knowing the answer of the embedded question. Our agenda: Take a know-x construction as a single modality, e.g., pack xk(yanjing = x) into Kwho Yanjing. Give an intuitive semantics according to some linguistic theory. Axiomatize the logics with (combinations of) those operators. Dynamify those logic with knowledge updates. Automate the inferences based on decidability. Come back to philosophy and linguistics with new insights. Stay at the appropriate abstraction level for your purpose is important!
Beyond knowing that: (technical) difficulties not normal: Kw(p q) Kw p Kw q Khowφ Khowψ Khow(φ ψ) φ Kwhyφ not strictly weaker: Kwφ Kw φ combinations of quantifiers and modalities: x φ(x); the axioms depend on the special schema of φ essentially; weak language vs. rich model: hard to axiomatize; guarded fragments of FO/SO-modal language: decidability? new use of Kripke models; relevant to different logics.
Plan of the lectures Standard epistemic logic of knowing that Dynamify the standard theory Logics of knowing whether Logics of knowing what Logics of knowing how Logics of knowing why Connections with other logics Research topics for your final papers How to write and present a paper Final presentations
Main technical contents Alternative axiomatizations of Dynamic Epistemic Logic [Wang & Cao Synthese13] [Wang & Aucher IJCAI13] Knowing whether (non-contingency): axiomatizations and completeness proofs for its logic over various frame classes [Fan, Wang & van Ditmarsch: AiML14, RSL15]; Knowing what: axiomatization for conditionally knowing what logic over FO epistemic models [Wang & Fan: IJCAI13, AiML14] [Xiong14][Ding 15] Knowing how: philosophical discussion [Lau & Wang]; alternative non-possible-world semantics [Wang: ICLA15]; a logic [Wang LORI15] Knowing why: axiomatization [Xu & Wang]
Questions? Whether? What? How? Why? y.wang@pku.edu.cn