Open main menu
Home
Random
Recent changes
Special pages
Community portal
Preferences
About Wikipedia
Disclaimers
Incubator escapee wiki
Search
User menu
Talk
Dark mode
Contributions
Create account
Log in
Editing
Common knowledge (logic)
(section)
Warning:
You are not logged in. Your IP address will be publicly visible if you make any edits. If you
log in
or
create an account
, your edits will be attributed to your username, along with other benefits.
Anti-spam check. Do
not
fill this in!
=== Modal logic (syntactic characterization) === Common knowledge can be given a logical definition in [[Multimodal logic|multi-modal logic]] systems in which the modal operators are interpreted [[epistemic logic|epistemically]]. At the propositional level, such systems are extensions of [[propositional logic]]. The extension consists of the introduction of a group ''G'' of ''agents'', and of ''n'' modal operators ''K<sub>i</sub>'' (with ''i'' = 1, ..., ''n'') with the intended meaning that "agent ''i'' knows." Thus ''K<sub>i</sub> <math>\varphi</math>'' (where <math>\varphi</math> is a formula of the logical calculus) is read "agent ''i'' knows <math>\varphi</math>." We can define an operator ''E<sub>G</sub>'' with the intended meaning of "everyone in group ''G'' knows" by defining it with the axiom : <math>E_G \varphi \Leftrightarrow \bigwedge_{i \in G} K_i \varphi,</math> By abbreviating the expression <math>E_GE_G^{n-1} \varphi</math> with <math>E_G^n \varphi</math> and defining <math>E_G^0 \varphi = \varphi</math>, common knowledge could then be defined with the axiom : <math>C \varphi \Leftrightarrow \bigwedge_{i = 0}^\infty E^i \varphi</math> There is, however, a complication. The languages of epistemic logic are usually ''finitary'', whereas the [[axiom]] above defines common knowledge as an infinite conjunction of formulas, hence not a [[well-formed formula]] of the language. To overcome this difficulty, a ''fixed-point'' definition of common knowledge can be given. Intuitively, common knowledge is thought of as the fixed point of the "equation" <math>C_G \varphi=[\varphi\wedge E_G (C_G \varphi)]=E_G^{\aleph_0} \varphi</math>. Here, <math>\aleph_0</math> is the [[Aleph-naught]]. In this way, it is possible to find a formula <math>\psi</math> implying <math>E_G (\varphi \wedge C_G \varphi)</math> from which, in the limit, we can infer common knowledge of <math>\varphi</math>. From this definition it can be seen that if <math>E_G \varphi</math> is common knowledge, then <math>\varphi</math> is also common knowledge (<math>C_G E_G \varphi \Rightarrow C_G \varphi</math>). This ''syntactic'' characterization is given semantic content through so-called ''[[Kripke structure (model checking)|Kripke structures]]''. A Kripke structure is given by a set of states (or possible worlds) ''S'', ''n'' ''accessibility relations'' <math>R_1,\dots,R_n</math>, defined on <math>S \times S</math>, intuitively representing what states agent ''i'' considers possible from any given state, and a valuation function <math>\pi</math> assigning a [[truth value]], in each state, to each primitive proposition in the language. The [[Kripke semantics]] for the knowledge operator is given by stipulating that <math>K_i \varphi</math> is true at state ''s'' iff <math>\varphi</math> is true at ''all'' states ''t'' such that <math>(s,t) \in R_i</math>. The semantics for the common knowledge operator, then, is given by taking, for each group of agents ''G'', the [[reflexive closure|reflexive]] (modal axiom '''T''') and [[transitive closure]] (modal axiom '''4''') of the <math>R_i</math>, for all agents ''i'' in ''G'', call such a relation <math>R_G</math>, and stipulating that <math>C_G \varphi</math> is true at state ''s'' iff <math>\varphi</math> is true at ''all'' states ''t'' such that <math>(s,t) \in R_G</math>.
Edit summary
(Briefly describe your changes)
By publishing changes, you agree to the
Terms of Use
, and you irrevocably agree to release your contribution under the
CC BY-SA 4.0 License
and the
GFDL
. You agree that a hyperlink or URL is sufficient attribution under the Creative Commons license.
Cancel
Editing help
(opens in new window)