(a) Every atomic formula becomes a predicate symbol. Hence, a generalization of every tautology of group 1 axioms becomes a tautology of sentential logic (moreover, in general, it becomes a specific case of the initial tautology, for example, $Px\rightarrow Py\rightarrow Px$ becomes $P\rightarrow P\rightarrow P$ ). For other axiom groups, a generalization of $\forall x\alpha\rightarrow\alpha_{t}^{x}$ becomes $\alpha^{*}\rightarrow\alpha^{*}$ , a generalization of $\forall x(\alpha\rightarrow\beta)\rightarrow(\forall x\alpha\rightarrow\forall x\beta)$ becomes $(\alpha^{*}\rightarrow\beta^{*})\rightarrow(\alpha^{*}\rightarrow\beta^{*})$ , a generalization of $\alpha\rightarrow\forall x\alpha$ becomes $\alpha^{*}\rightarrow\alpha^{*}$ , a generalization of $x=x$ becomes $\top$ , and a generalization of $x=y\rightarrow\alpha\rightarrow\alpha'$ becomes $\top\rightarrow\alpha^{*}\rightarrow\alpha^{*}$ .

(b) We can show this by induction. For a logical axiom $\phi$ we have already shown that $\phi^{*}$ is a tautology in (a). If $\beta$ is deduced from $\alpha$ and $\alpha\rightarrow\beta$ where $\alpha^{*}$ and $(\alpha\rightarrow\beta)^{*}$ are tautologies, then, since $(\alpha\rightarrow\beta)^{*}$ is $\alpha^{*}\rightarrow\beta^{*}$ , $\beta^{*}$ is a tautology as well.

(c) $(\neg\alpha)^{*}=\neg\alpha^{*}$ implies, by (b), that at least one of $\alpha$ and $\neg\alpha$ is not deducible (from the empty set).