 We can define the usual boolean operations by means of the conditional operator, as in the following examples:   \[\begin{align} \tag{2} \texttt{and} &amp;= {\lambda x:\texttt{bool}, y:\texttt{bool}{\, . \,}{\texttt{if }x\texttt{ then }y\texttt{ else }\texttt{ff}}}.                                     \label{etbivalente} \\ \tag{3} \texttt{or} &amp;= {\lambda x:\texttt{bool}, y:\texttt{bool}{\, . \,}{\texttt{if }x\texttt{ then }\texttt{tt}\texttt{ else }y}}                                 \label{velbivalente}\end{align}\]   with the usual truth tables. However, we have now to take into account the possibility of divergence of the evaluation process, for example in a term like \(\texttt{or}(\Omega,\texttt{tt})\), therefore we extend the usual truth tables by adding a new boolean value, representing absence of information, \(\bot\) (read as “undefined”) to \(\texttt{tt}\) and \(\texttt{ff}\), as the value of the term \(\Omega\). Here, the first argument to be evaluated is the one on the left, and if the evaluation of this diverges then the whole evaluation process diverges. Consider now an operator por whose interpretation is given by the table   \[\tag{4}\label{por}  \begin{array}{r|lll}  \texttt{por}&amp; \textit{tt}&amp; \textit{ff}&amp;\bot \\\hline \textit{tt}&amp; \textit{tt}&amp; \textit{tt}&amp;\textit{tt}\\  \textit{ff}&amp; \textit{tt}&amp; \textit{ff}&amp;\bot\\  \bot &amp; \textit{tt}&amp; \bot &amp;\bot  \end{array}\]   In this case \(\texttt{por}(\texttt{tt},\Omega) = \texttt{por}(\Omega,\texttt{tt}) = \texttt{tt}\): this is the parallel-or which plays a central role in the full abstraction problem for PCF. It will turn out that is is not definable by any PCF term, precisely because of its parallel nature. In order to carry out a semantical analysis of PCF, we need a theory of data types with partial elements and of functions over them that support an abstract form of recursive definition through fixed point equations: this is what is achieved in Scott’s theory of domains, the original mathematical foundation for denotational semantics of programming languages as conceived by Strachey (1966, 1967).