Russell discusses two kinds of symbols:
(1) Complex symbols, and (2) Incomplete symbols.
1. Complex Symbols:
A proposition according to Russell is a complex symbol. It is a complex symbol because, it has parts which are also symbols. A symbol may be defined complex when it includes within it parts which are also symbols. As a sentence is composed of words which are symbols and since the sentence itself is also a symbol, it is a complex symbol.
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It is of crucial importance to realize the distinction between a symbol and that which is symbolized by it; otherwise we will confuse the word and the thing. The proposition “Indira Gandhi is the first woman Prime Minister of India” is a complex symbol, the particles of which viz., “Indira Gandhi” and “woman Prime Minister of India” are themselves symbols. The sentence sands for a fact which also is complex. Analysis of complex facts leads us to basic elements of the universe.
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2. Incomplete symbols:
Incomplete symbol is an expression which has no meaning in isolation but, which may occur as a constituent part in, and contribute to the meaning of an expression which does have a meaning. These incomplete symbols are descriptive phrases and these descriptive phrases are proper nouns. The descriptive phrases are also of two kinds.
(A) Indefinite Descriptions:
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These are vague or ambiguous. They do not refer to any definite individual but, indifferently to any individual. The examples of indefinite descriptions are “a man,” “an author”, “a river” etc.: ‘a man’ an author’ or ‘a river’ can refer to or apply to any man, any author or any river. A river may denote Ganga, Yamuna, and Narmada, Amazon or any river in the world. Therefore, it is indefinite description.
(B) Definite Description:
The definite descriptions are the very opposite of indefinite descriptions. Whereas, the latter have no definite object of reference the former have a definite object of reference secondly, whereas, indefinite descriptions must each be applicable to at least more than one individual, the definite description must not apply to more than one.
The examples of definite description are “The Prime Minister of India.” “The author of Waverley,” “The man who shot Mahatma Gandhi,” “The father of Philosophy” etc. None of these apply to any except one individual. There is no ambiguity about what they stand for or refer to.
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The descriptive phrases, as a matter of fact, only contribute to the meaning of an expression to which they are attached. By themselves they do not mean anything. In logical propositions they can be easily eliminated. For example, instead of saying Thales, the father of philosophy, was Greek we can say without loss of meaning that Thales was Greek. “I met a man”: has been rephrased by Russell “I met an x and x was human” and Scott is the author of Waverley as “An x wrote Waverley and only x wrote it and x is Scott.” The propositions from which descriptions are eliminated are called Propositional Function by Russell.
In a Propositional Function, there is an indefinite term, the value of which is undetermined. Once its value is determined or a definite term is substituted for it, the propositional function becomes a proposition. For example, if we say “x is mortal” it is a propositional form in which “x” is indefinite. If we substitute Socrates or Plato or Russell for “x” the propositional form is transformed and becomes a proposition.