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In Fitting and Mendelsohn's textbook on First order modal logic, the concept of validity is defined only for sentences (i.e, formulas without free variables). How usual is this?

Doesn't it have a strange consequence, namely that $P(x) \lor \neg P(x)$ will not be treated as valid since it contains free variables, even though it is true in all models? So there are formulas which are true in all models but not valid in their sense. What is the rationale behind that?

user65526
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  • Valid means true in every interpretation. Regarding a single interpretation, we have to ways: either (i) we say that a formula $\varphi$ is true in an interpretation $M$ (written $M \vDash \varphi$) iif its universal closure is, or ... – Mauro ALLEGRANZA May 27 '24 at 08:59
  • or (ii) we define the satisfaction relation for a formula in an interpretation under some assignment of values to the variables through a variable assignment function $s$, in which case we have $M,s \vDash \varphi$. In the second case, $\varphi$ is true in $M$ iff it is satisfied by every variable assignment. – Mauro ALLEGRANZA May 27 '24 at 09:02
  • See page 97 for the def of valuation. – Mauro ALLEGRANZA May 27 '24 at 09:09
  • In conclusion, you can simply replace Def.4.6.10 of page 99 with the following: "For a formula $\Phi$, if $M, \Gamma \Vdash_{v} \Phi$ for every valuation $v$, we say that $\Phi$ is true at $\Gamma$." – Mauro ALLEGRANZA May 27 '24 at 09:51
  • That would be nice, but that's not how they define it. Also in the second edition of the book, which I referring to. They explicitly define validity only for sentences via the notion of "true at $\Gamma$", which they reserve explicitly for sentences. So it sounds like you are proposing a modification. What I am interested in is how to understand substitution given their definition of validity. – user65526 May 28 '24 at 09:59
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    Also, in their tableaux, they do not allow tableaux to start with a formula which contains free variables. This makes sense, if only sentences can be valid (as they define it). – user65526 May 28 '24 at 10:02

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