Exercise 4, Appendix 0

Let $A$ be a term or a relation in a theory $\mathscr{T}$. Consider the sequence of assemblies defined as follows. First of all, write $A$. If $A$ consists of a single letter, stop there. If not, write next the assembly or assemblies antecedent to $A$ (if $A$ begins with a $\tau$, choose one of the antecedent assemblies arbitrarily). Then write the assembly or assemblies antecedent to those of the preceding assemblies which do not consist of a single letter; and so on.

(a) If we reverse the order of this sequence of assemblies, we get a formative construction.

(b) Let $B$ be a balanced segment of $A$ such that no sign of $B$ is linked in $A$ to a sign outside $B$. Show that $B$ is either a term or a relation (use (a) and Exercise 1).

(c) Replace $B$ in $A$ by a term (resp. a relation) if $B$ is a term (resp. a relation). Show that the assembly thus obtained is a term if $A$ is a term, a relation if $A$ is a relation.