Proving transitivity of $\vDash$ and and $\vdash$ I was given this excercise and find myself unable to tackle it (beginner here):
Prove that $\models$ is transitive, i.e. prove that if $\varphi\models\psi$ and $\psi\models\chi$, then $\varphi\models\chi$.
Similarily, prove that $\vdash$ is transitive, i.e. prove that if $\varphi\vdash\psi$ and $\psi\vdash\chi$, then $\varphi\vdash\chi$.
Thank you for any suggestions on how to approach this!
 A: The definition of "$\vDash$" (the relation of entailment or logical consequence) is :

$\varphi \vDash \psi$ iff for every interpretation $\mathcal M$, if $\varphi$ is true in $\mathcal M$, then also $\psi$ is true in $\mathcal M$.

Thus, if we have :

$\varphi \vDash \psi$ and $\psi \vDash \chi$,

clearly in every interpretation $\mathcal M$ in which $\varphi$ is true, having that also $\psi$ is, and because in every interpretation in which $\psi$ is true also $\chi$ is, we may conlcude that in every interpretation $\mathcal M$ in which $\varphi$ is true, also $\chi$ is true, i.e. :



$\varphi \vDash \chi$.



Regarding the derivation relation : $\varphi \vdash \psi$, again by definition :

$\psi$ is derivable form $\varphi$ iff there is a finite sequence of formulae $\psi_1, ..., \psi_n$ such that : $\psi_n$ is $\psi$ and each $\psi_i$, $1 \le i \le n$ can be :
(i) $\varphi$, or
(ii) a logical axiom, or
(iii) derived by previous formulae in the sequence through an inference rule, i.e. assuming that modus ponens is the only inference rule, we have $1 \le j,k < i$ such that $\psi_k$ is $\psi_j \rightarrow \psi_i$.

Having said that, starting form the derivations $\varphi \vdash \psi$ and $\psi \vdash \chi$, it is enough to "concatenate" them; the resulting sequence of formulae will satisfy the above definition and it will end with $\chi$.
Thus, it will be a derivation of $\chi$ from $\varphi$, i.e. :


$\varphi \vdash \chi$.


