Remark

Intuitively, terms are assemblies which represent objects, and relations are assemblies which represent assertions which can be made about these objects. Condition (a) means that the letters represent objects. Condition (b) means that if $B$ is an assertion, then $\neg B$, called the negation of $B$, is an assertion (which is read : not $B$). Condition (c) means that if $B$ and $C$ are assertions, $\vee BC$, which is called the disjunction of $B$ and $C$, is an assertion (which is read : either $B$ or $C$); thus $\neg\to BC$ is an assertion (in words : “either not $B$, or $C$”, or “$B$ implies $C$”). Condition (d) means that if $B$ is an assertion and $x$ a letter, then $\tau_x(B)$ is an object. Let us consider the assertion $B$ as expressing a property of the object $X$; then, if there exists an object which has the property in question, $\tau_x(B)$ represents a distinguished object which has this property; if not, $\tau_x(B)$ represents an object about which nothing can be said. Finally, condition (e) means that if $A_1,A_2,\ldots,A_n$ are objects, and if $s$ is a relational (resp. substantific) sign of weight $n$, then $sA_1A_2\ldots A_n$ is an assertion about the objects $A_1,\ldots,A_n$ (resp. an object depending on $A_1,\ldots,A_n$).