§ 2. Language

If logic is analysis of language, we must begin our inquiry by a consideration of language. Language consists of signs. We should not forget, however, that signs are physical things: ink marks on paper, chalk marks on a board, sound waves produced in a human throat. What makes them signs is the intermediary position they occupy between an object and a sign user, i.e., a person. The person, in the presence of a sign, takes account of an object; the sign therefore appears as the substitute for the object with respect to the sign user. The three-place relation holding between the sign, the object, and the person has therefore been called by Charles Morris1 the relation of mediated-taking-account-of. To simplify the terminology we shall speak of the sign relation, or relation of denotation.

The notion of sign is wider than that of language. Not all signs represent a language. Smoke may appear to us a sign of fire; but smoke does not represent a linguistic utterance. Signs of this kind, which acquire their sign function through a causal connection between the object and the sign, are called indexical signs. In other cases something will be the sign of an object because of a similarity in appearance, such as a photograph; such signs are named iconic signs. In the third case, that of language, the coordination of the sign to the object is purely conventional; we speak here of conventional signs, or symbols.2 The use of the symbols is determined by a set of rules which we call the rules of language.

For practical purposes, linguistic signs must be reproducible since we use different individual signs for the same logical functions. The individual sign is called a token. Thus in the two sentences 'Los Angeles is a city' and 'Los Angeles is situated in California' we have the same word 'Los Angeles', but appearing in two different tokens; and now in making the explanation a third token of this word has been used. Different tokens of the same word have the same meaning, or are equisignificant. In part, equisignificance is given by geometrical similarity of the tokens, as in our example; but we have also equisignificance between printed and handwritten tokens, and between these and spoken tokens. The coordination of these different kinds of tokens to one another is of course a matter of convention. To have a convenient term we shall call these different kinds of tokens similar to each other, using the word 'similar' in a somewhat wider sense. The class of similar tokens is called a symbol. Saying 'the same symbol occurs in different places' means 'tokens of the same symbol-class appear in different places'.3

The most important unit among signs is the proposition.4 It is usually composed of several words. The grammatical definition of a word as a group of letters separated from others by an interval is not satisfactory: what is one word in one language may be expressed by several words in another, whereas propositions are always translated into propositions. The German language is noted for its long compound nouns, such as 'Eisenbahnknotenpunkt' which means 'railroad junction' in English. The rule stating that the words 'rail' and 'road' are written in one word, whereas 'junction' is written separately, is merely conventional. Likewise the distinction between words and suffixes, which usually are considered parts of words, is without logical significance. The Turkish language, which uses many suffixes, sometimes expresses a whole sentence in one word. Thus the Turkish word 'alabilecegim' is a sentence and means 'I shall be able to buy'.5 We should not forget that in all languages the division into words more or less disappears as soon as we speak, since in talking we make no intervals between most words; in the French language this habit leads to the liaison of spoken words, i.e., the merging of words into one by pronouncing otherwise silent end consonants. We shall later inquire whether a logically satisfactory definition of words, in the sense of sentential parts, can be given.

What makes a proposition the fundamental unit is the fact that only a whole proposition can be true or false -- that, as we say, it has a truth-value. An isolated word like 'table' is not true or false. Only if a word stands for a sentence, as an abbreviation, can we speak of its truth or falsehood, as, for instance, if a child points at a table and says 'table', when the complete statement would be 'this is a table'. Likewise the property of having a meaning is originally restricted to whole sentences. If we want to communicate meanings to other persons we speak in sentences; a word does not communicate anything unless it stands for a sentence.

If none the less we sometimes speak of the meaning of words, the usage is to be interpreted in the following way. The same word may occur in different sentences, and we say that we understand the meaning of a word if we know how to use it in sentences of different failings. It seems advisable to distinguish the two terms, if necessary, by speaking of sentence-meaning and word-meaning. It is clear from the given definition that sentence-meaning is logically prior to word-meaning, i.e., that the expression 'word-meaning' is defined in terms of the expression 'sentence-meaning'.

The properties of having a truth-value and a meaning are confined to signs and do not apply to physical things which have no sign function. In the often vague usage of conversational language this rule is not always followed; we speak of a true friendship, or of true facts; we say that the refusal of an ultimatum by a government means war, etc. Sometimes this usage may be excused by the existence of indexical sign relations, as in the last example, or in such expressions as 'smoke means fire', 'smoke is a true sign of fire'. In other instances we may have iconic sign relations, as when we speak of a true portrait. It appears advisable, however, to restrict the predicates meaning and truth to linguistic signs, or symbols, since a complete interpretation of these terms can be given only within a system of rules constituting a language. Incidentally, it is easy to replace these ambiguous words by more suitable terms; thus we may speak of a genuine friendship, of smoke as indicating fire, etc.

Not all combinations of meaningful words are meaningful. The word sequence 'The emperor of and is' is a meaningless set of signs. In this particular example the lack of meaning is indicated by the violation of grammatical rules. However, the observance of grammatical rules is not a guarantee of meaning. The word sequence 'Caesar is a prime number' is also meaningless, although this combination of signs satisfies the rules of grammar. The reason for this further restriction as to meaning will be explained later (§ 40). It is important to realize that a meaningless set of signs does not become meaningful if we take the negation of it. Thus the set of signs 'the emperor of and is not' is as meaningless as the set of signs that does not contain the word 'not'. Neither is the set of signs 'Caesar is not a prime number' meaningful.

We shall use the term 'proposition' in such a way that it refers only to meaningful sets of symbols. We thereby exclude from the domain of logic certain combinations of words which otherwise would lead to serious difficulties, the so-called antinomies. The introduction of rules restricting meaning beyond the restrictions of grammatical rules is one of the most important advances made by modern logic; the credit for having recognized the necessity of such rules, formulated in his so-called theory of types (cf. § 40), belongs to Bertrand Russell.

Although the theory of types formulates some necessary requirements of meaning, it leaves open the question of sufficient requirements. In other words, it leaves unanswered the general question, 'when is a set of signs meaningful?' The answer to this question constitutes an important and much-discussed chapter of modern epistemology. We cannot enter upon this inquiry here and shall merely report that the answer has been given through the verifiability theory meaning, which, in its simplest form, is expressed by the two principles:

  1. A proposition is meaningful only if it is verifiable as true or false.
  2. Two propositions have the same meaning if they obtain the same verification, as true or false, for all possible observations.
A discussion of these two principles must be left to other expositions of the subject. Let us mention only that the second represents the modern form of a principle that has played a leading role in the history of philosophy. It emerged in the philosophy nominalism and was formulated by William of Occam as the rule -- known under the name of Occam's razor -- that entities should not be multiplied except if necessary. It was brought to the fore by G. W. Leibniz as the principle of the identity of indiscernibles, and it has acquired a particular significance through its application in Einstein's theory of relativity. It constitutes the nucleus of the theories of meaning developed in pragmatism and logical positivism, conceptions which have been united in the philosophical movement that carries the name of logical empiricism.

Analysis in our day has shown that the two principles stated above require some correction, and that the terms 'true' and 'false' should be replaced by the continuous scale oi probability.6 We shall not refer to this modified conception of verification but shall continue to consider propositions as two-valued, i.e., as being true or false. Two-valued logic, is the mother of all other logics; further, it can always be carried through, in the sense of an approximation, even if refined analysis demands a probability logic.

The preceding remarks will make it clear why we shall base the system of logic on propositions as fundamental units. Propositions may be called the atoms of language, and, just as a piece of matter will always consist of a whole number of atoms, a meaningful speech or article will always consist of a whole number of propositions. This analogy can be carried further. Atoms combine into molecules; in a similar way atomic propositions combine into molecular propositions. On the other hand, the fact that atoms constitute the fundamental units of matter does not preclude them from being themselves composed of subunits, from having an inner structure accessible to investigation. Similarly, the inner structure of propositions can be investigated. These considerations led modern logicians to a general division of logic into two parts. The first part, the calculus of propositions, deals with the operations combining propositions as wholes; the second part, the calculus of functions, treats the inner structure of propositions, relating this analysis to the results of the first part.

We cannot explain here the notion of function, which has given its name to the second part, and must refer the reader to § 17 for an explanation of the term. But we may add that the notion of function possesses a parallel in the notion of class and that thus another interpretation of the second part can be constructed, which is usually added to the given classification as a third part under the name of calculus of classes.

We shall present the three parts in the order given.