| There are two kinds of components in a calculus. First there is a set of symbols and second a basic set of expressions. These components relate to each other through another set of expressions — the conventions or rules that tell how to interpret the symbols and the primitive expressions. The details about how to proceed in assigning the concepts of the discipline to the various symbols and expressions depend on the needs of the discipline being formalized. Indeed, much of the power and utility of a calculus is in the flexibility that it offers in this assignment. |
Strings of symbolsIn a calculus an expression is a finite string of symbols — a linear sequence having a beginning and an end. The symbols that string together to form an expression are often simply marks made on paper in a certain way. Symbols need not be limited to being visually distinct marks on paper. In inventing symbols for a calculus the analyst constructs the forms so that each one is distinguishable from the others by virtue of the distinct impression it makes on the physical senses. But in general symbols may be much more abstract, as long as each is kept distinct from any other. Symbols come in different instances of the same symbol (tokens) and in different varieties or classes (types). |
The meaning of the symbolsIt is the context of the calculus that defines its symbols. To see what this means compare a calculus to any of the many board games that rely on pieces that are visually and tactually distinct. In the game of chess the various pieces are the symbols of a calculus. The knight, for example, has a distinctive shape, which is usually like a horse’s head. The player recognizes this shape and attributes a specific value to the piece. The fact that a “black knight” may actually look like a red, or gray, or gold horse, is irrelevant. Its meaning in the context of the game is the same in any case. The symbol derives its specific meaning by virtue of its context in the calculus. |
Symbols of writingThe naive student of language knows that it is also composed of symbols. In English one of these symbols is the class of marks called the letter e. These marks are distinguishable from the class called the letter b, for example. In terms of the definition of a symbol as part of a calculus it makes no difference whether the language user writes letters in longhand, prints them, types them, or stores them in some binary form in a computer. The actual form that the letter realizes might be quite different depending on the language, the style, the conventions, i.e., the calculus of its context. The important point is that the different forms of e belong to the same equivalent class and the forms of b to another such equivalent class. |
Symbols of speechThe sounds of a particular language, the impressions on the auditory system, are symbols in this sense as much as are their written representations. An n-sound may be palatal in one context and velar in another. Yet the English speaker usually perceives either instance as a different form of the same sound. On the level of words it is just as obvious that the language user is dealing with symbols. The word the rhymes with he in one context (in isolation or before vowel sounds) and with huh in another (otherwise). In either context the English speaker perceives the two very different sounds as alternative pronunciations of the same word. This kind of behavior is what tells us that it is possible to consider the as a symbol in some calculus. |
Symbols are classesIt may not at first be obvious that a symbol cannot possess meaning independent of its calculus. A symbol acquires meaning only when the user sees it in context as part of a system. The same symbol “–” signifies subtraction in arithmetic, but when printed between words it is a hyphen. In attributing meaning to a symbol, the user is assigning the symbol to a particular class based on the calculus that the class belongs to. The context refers the symbol to a system of classes — it presupposes the existence of certain other classes. What the investigator calls the meaning of a symbol is a result of the various interrelationships that it participates in with the various parts of the system. These interrelationships are explicated by means of expressions. This is the one and only means by which a calculus can impart meaning to its symbols. The positivist position is that this is also the one and only means by which a symbol may acquire meaning. |
Rules of formation and transformationOne thing that the rules of a calculus do is state the conditions under which an expression belongs to a certain class of expressions. They also state the conditions under which one or more expressions may transform into certain other expressions. In other words, the rules tell the logician when to consider a certain expression to be an alternate form of the other(s). The first kind of rule, the one that categorizes expressions, is a rule of formation. The second kind of rule, the one that derives an expression from one or more other expressions, is a rule of transformation. These two kinds of rules are the only two elements essential to defining a calculus. A calculus is a system of symbols and expressions, with rules of formation and rules of transformation relating the two. |