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Basic question here but I cannot find the definition:

Given a modal logic and a set of propositions $P$, a model $M=(W,R,V)$ where $W$ are possible worlds, $R$ an accesibility relation and $V$ a valuation for every $p\in P$

What is de definition of $(M,w)\models \varphi$?

And whats the name of it? Validity?

Thanks

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  • $\begingroup$ Have you looked at en.wikipedia.org/wiki/Modal_logic#Semantics ? $\endgroup$ – mrp Aug 22 '15 at 20:15
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    $\begingroup$ We say that $\varphi$ is true at world $w$ in model $M$ (in symbols: $(M,w) \vDash \varphi$); we say that $\varphi$ is true in model $M$ (in symbols: $M \vDash \varphi$) when $(M,w) \vDash \varphi$ for all $w \in W$. We say that $\varphi$ is valid ($\vDash \varphi$) when $M \vDash \varphi$ for every model $M$. See e.g : Edward Zalta, Basic Concepts in Modal Logic. $\endgroup$ – Mauro ALLEGRANZA Aug 22 '15 at 20:27
  • $\begingroup$ mauro thanks a lot $\endgroup$ – Applied mathematician Aug 23 '15 at 14:41
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  • We write $(M,w) \vDash \varphi$ if $\varphi$ is true at world $w$ in model $M$.

  • We say that $\varphi$ is true in model $M$ (in symbols: $M \vDash \varphi$) when $(M,w) \vDash \varphi$ for all $w \in W$.*

  • We say that $\varphi$ is valid ($\vDash \varphi$) when $M \vDash \varphi$ for every model $M$.

(*) This terminology is a bit unfortunate since it means that most of the time, neither $\varphi$ nor $\neg\varphi$ will be true in $M$.

Chellas, Brian F., Modal logic. An introduction, Cambridge etc.: Cambridge University Press. XII, 295 p. hbk: £ 17.50; pbk: £ 6.50 (1980). ZBL0431.03009.

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