# What the definition of validity of a formule in a possible Kripke-world in Modal Logic?

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

• Have you looked at en.wikipedia.org/wiki/Modal_logic#Semantics ? – mrp Aug 22 '15 at 20:15
• 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. – Mauro ALLEGRANZA Aug 22 '15 at 20:27
• mauro thanks a lot – Applied mathematician Aug 23 '15 at 14:41

• 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 $$M$$.