 By employing the idea of generalized truth value functions, one can obtain a hierarchy of valuation systems starting with a certain set-theoretic representation of the valuation system for classical logic. The representation in question is built on a single initial value which serves then as the designated value of the resulting valuation system. More specifically, consider the singleton \(\{\varnothing \}\) taken as the basic set subject to a further generalization procedure. Set-theoretically the basic set can serve as the universal set (the complement of the empty set) for the valuation system \(\mathbf{V}^{\varnothing}_{cl}\) introduced below. At the first stage \(\varnothing\) comes out with no specific intuitive interpretation, it is only important to take it as some distinct unit. Consider then the power-set of \(\{\varnothing \}\) consisting of exactly two elements: \(\{\{\varnothing \}, \varnothing \}\). Now, these elements can be interpreted as Frege’s the True and the False, and thus it is possible to construct a valuation system for classical logic, \(\mathbf{V}^{\varnothing}_{cl} = \langle \{\{\varnothing \}, \varnothing \}, \{\{\varnothing \}\}, \{f_{\wedge}, f_{\vee}, f_{\rightarrow}, f_{\sim}\}\rangle\), where the functions \(f_{\wedge}, f_{\vee}, f_{\rightarrow}, f_{\sim}\) are defined as follows (for  \[ \begin{align} X, Y \in \{\{\varnothing \}, \varnothing \}:\quad &amp; f_{\wedge}(X, Y) = X\cap Y; \\ &amp; f_{\vee}(X, Y) = X\cup Y; \\ &amp; f_{\rightarrow}(X, Y) = X^{c}\cup Y; \\ &amp; f_{\sim}(X) = X^{c}. \end{align} \]   It is not difficult to see that for any assignment \(a\) relative to \(\mathbf{V}^{\varnothing}_{cl}\), and for any formulas \(A\) and \(B\), the following holds: 