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→Modal logic (syntactic characterization): fix index and bounds in infinite conjuction |
→Modal logic (syntactic characterization): an equation equates, visibly this what was meant to be equated |
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:<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>
This ''syntactic'' characterization is given semantic content through so-called ''Kripke structures''. A Kripke structure is given by (i) a set of states (or possible worlds) ''S'', (ii) ''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 (iii) a valuation function <math>\pi</math> assigning a [[truth value]], in each state, to each primitive proposition in the language. The 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 and transitive closure 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>.
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