Consider the full language \(\mathcal{L}\) with the connectives \(\wedge,
\rightarrow, \perp, \leftrightarrow \vee\) and the restricted language
\(\mathcal{L}^{\prime}\) with connectives \(\wedge, \rightarrow, \perp\). Using
the appropriate derivation rules we get the derivability notions \(\vdash\) and
\(\vdash^{\prime}\). We define an obvious translation from \(\mathcal{L}\) into
\(\mathcal{L}^{\prime}\) :
\(\begin{aligned} \varphi^{+} &:=\varphi \text { for atomic } \varphi
\\\\(\varphi \square \psi)^{+} &:=\varphi^{+} \square \psi^{+} \text {for }
\square=\wedge, \rightarrow, \end{aligned}\)
\((\varphi \vee \psi)^{+}:=\neg\left(\neg \varphi^{+} \wedge \neg
\varphi^{+}\right)\), where \(\neg\) is an abbreviation,
\((\varphi \leftrightarrow \psi)^{+}:=\left(\varphi^{+} \rightarrow
\psi^{+}\right) \wedge\left(\psi^{+} \rightarrow \varphi^{+}\right)\),
$$
(\neg \varphi)^{+}:=\varphi^{+} \rightarrow \perp .
$$
Show (i) \(\vdash \varphi \leftrightarrow \varphi^{+}\),
(ii) \(\quad \vdash \varphi \Leftrightarrow \vdash^{\prime} \varphi^{+}\),
(iii) \(\varphi^{+}=\varphi\) for \(\varphi \in \mathcal{L}^{\prime}\).
(iv) Show that the full logic, is conservative over the restricted logic, i.e.
for \(\varphi \in \mathcal{L}^{\prime} \vdash \varphi \Leftrightarrow
\vdash^{\prime} \varphi\).
