A symbol is nothing more than a mark made in a certain way so that it can be distringuished from another mark. In English (a natural language) there are letters of the alphabet used as symbols marks falling into twenty-six distinctive classes. In our symbolic language we will use these letters and combinations of them as convenient for our symbols. Other clearly distinguishable marks will also be used.
The rules of a calculus also involve expressions. These are nothing more than strings or series of symbols. The rules tell us which sequences of symbols are legitimate in describing the various structures of the discipline. In a natural language we may have rules that tell us which words are pronouncable, which sentences grammatical. In our symbolic language the rules characterize a particular structural description as well-formed.
The real distinction between a calculus and anothe type of symbolic language is the dichotomazation between two types of rules of a calculus: the rules of formation and the rules of transformation. The rules of formation state that if an expression has certain kinds of symbols in a certain order, then it is a certain kind of expression. For example, such a rule would state that a word is a consonant plus a vowel plus a consonant. Or it might say that a sum is a number plus a plus sign plus a number.
More involved expressions must often be described by a rule of transformation. These rules state that if an expression has certain kinds of symbols in a certain order, then it may be derived from a certain other expression. In a certain sense a genealogist derives his compilation from several sources. If we can describe a source as an expression of the symbolic language using rules of formation, then the derivatrion of the compilation would be by means of rules of transformation. The derived expression consists of elements of the source though some may be reordered, lost, or modified in a regular fashion.