We said that signs are physical things coordinated to other things by certain rules. The process of coordination can be repeated, and we may introduce signs referring to signs. This iteration of the coordination process is not an invention of logicians; conversational language possesses a great many terms of this kind. Thus the word 'word' refers to signs; so do the words 'sentence', 'clause', 'phrase', 'name'. We say that signs of signs constitute a language of a higher level, which we call metalanguage; the ordinary language then is called object language. From the metalanguage we proceed to meta-languages of higher levels by introducing signs denoting signs of signs.

A means of indicating the transition from a sign to a sign for a sign is offered by quotation marks. Thus 'California' is a sign that denotes California; 'California' is written with ten letters, whereas California grows oranges. The transition may be further repeated. Thus ' 'California' ' is the name of a sign, namely, of 'California', but is not the name of California. In writing quotes we have to watch that the sign combination occurring in our sentence is always one level higher than the object to which it refers. Thus ' 'California' ' is written with one pair of quotes, and 'California' is written without quotes.¹ It would be difficult to add quotes to California; we then would have to construct huge quotes and put those of the left end into the Pacific Ocean, those of the right end into Nevada.

Let us add here the remark that the function of quotes is sometimes expressed by other linguistic devices. Thus italics may assume the function of quotes. Furthermore, a formula written on a separate line is to be conceived as equivalent to being presented in quotes.

Quotes or similar linguistic devices do not represent the only means of introducing signs of signs. We may also introduce independent words as signs of signs, and the mentioned words like 'word' and 'sentence' are such signs of signs not involving quotes. The signs of signs constructed by means of quotes are of a very peculiar kind. In them the object is employed as its own sign, and the function of the quotes consists in indicating this unusual usage. We might introduce a similar usage for the names of other physical objects; thus we might, whenever we write something about sand, put some sand in the place otherwise occupied by the word 'sand'. In order to indicate that this is not an undesired sand spot on our paper, but a part of our language and the name of sand, we should have to put quotes left and right of the sand spot. Unfortunately such a practice, although perhaps suitable for sand, would often lead to serious difficulties, for instance if we wanted to use this method for denoting lions and tigers. It is for these technical reasons that the quotes method is restricted to the introduction of signs of signs.

However, there are some other restrictions to the quotes method which are of a logical nature. What we introduce by quotes are names of signs, i.e., words of the metalanguage; but we cannot introduce propositions of the metalanguage in this way. It would not help us to start from propositions of the object language; by adding quotes to a proposition we obtain, not a proposition of the metalanguage, but a word of the metalanguage. In order to construct a proposition in the metalanguage we have to use its words and to combine them in a meaningful way. Thus ' 'snow' is an English word' and ' 'snow is white' is true' are propositions of the metalanguage. Many of the words of the metalanguage will correspond to words of the object language, such as the words 'and', 'is'; each has a meaning similar to that of the corresponding word of the object language. But here the transition to the metalanguage cannot be achieved by the addition of quotes either; thus the word 'is' of the metalanguage is not obtainable by adding quotes to the word 'is' of the object language. Words which occur in different languages in similar meanings are called 'ambiguous as to level of language'. In another conception these words are regarded as identical with those of the object language; the metalanguage then is conceived as a mixture of words of the first and the second level. This conception appears preferable because such mixed sentences cannot be completely dispensed with, as is shown by a sentence like ' 'Peter' denotes Peter', where the second word 'Peter' belongs, to the object language.²

Whereas signs of signs belong to a language of a higher level, abbreviations do not. An abbreviation does not denote a sign; it stands for a sign. Thus the abbreviation 'U.S.A.' stands for 'United States of America' and belongs to the same language as that sign. An abbreviation consists merely in the introduction, by a convention, of a new kind of token. Thus the tokens of the symbol-class 'U.S.A.' are considered as equisignincant to the tokens of the symbol-class 'United States of America'. Analogously, when letters 'a' and 'b' are used for propositions, they must be regarded as abbreviations, not as names of the propositions.

With the consideration of letters standing for propositions we come to a new distinction. The letter may be an abbreviation for an individual proposition; thus 'a₁' may stand for 'snow is white'. We then call 'a₁' a propositional constant and indicate this character by a subscript. Secondly, we may introduce letters as propositional variables; we then use letters without subscripts, as 'a' and 'b'. A propositional variable is a sign which does not stand for any individual proposition but which occupies a place that can be filled by any individual proposition. We say individual propositions are special values of propositional variables. Conversational language has no particular signs for propositional variables; these signs are used only within a scientific analysis of language in order to express structural properties considered for all propositions. Thus in order to symbolize the relation of implication we write 'a ⊃ b'. (The arc sign is the sign of implication.) Here 'a' and 'b' are propositional variables whose place may be occupied by any special propositions; whether the implication holds will then depend on the special propositions. We see that a formula containing propositional variables is not true or false, in general, but will become so only after specialization. Exception is to be made for formulas that hold for all values of the propositional variables, including the formulas that form the very subject of logic (cf. § 8, § 12). The use of variables in logic thus serves the same purpose as the corresponding practice in mathematics. A numerical equation containing the variable 'x' will hold only for special numerical values of 'x'. On the other hand, a so-called identical equation, such as '(a + b)² = a² + 2ab + b²', will hold for all values of the variables.

The use of variables is connected with a specific condition. We said that for a prepositional variable we may substitute any special proposition; if the variable occurs in several places within the same context, however, such substitution is permissible only when the same proposition is substituted for the variable in all its places. We therefore say that substitutions for variables are subject to a coupling condition. Thus when we have 'a ⊃ a' we can put for the two letters 'a' any special proposition; but it must be the same for both. This coupling condition constitutes the basis for the use of variables. It is applied, for instance, when we say that the expressions 'a ⊃ b' and 'a ⊃ a' are different. Since in both expressions any proposition can be substituted for the letters, the only difference is that the second expression is connected with a coupling condition for the substitution, whereas the first is not.

Prepositional variables belong to the same language as the propositions which are their special values. Of course prepositional variables are not bound to the language of lowest level; they may be introduced in any higher language. Like propositions, the propositional variables of the higher language cannot be obtained by the addition of quotes to signs of the lower language. We may use letters of different alphabets: for instance, Latin letters for propositions and prepositional variables of the object language, Greek letters for propositions and prepositional variables of the metalanguage. Special symbols for the metalanguage are necessary, however, only when the metalanguage is formalized; we therefore shall not make much use of such symbols in this book.

When we combine the quotes with propositional variables the resulting expressions are of a peculiar nature. Substitutions made for the variable within the quotes will transform the whole expression into names of various propositions; and the expression consisting of the variable and the quotes will therefore represent a new variable. The new variable, however, is in the metalanguage when the propositional variable is in the object language. We shall call this new variable a sentence name variable because its special values are names of propositions. Thus when we say

if 'a.b' is true, 'a' is true

Quotes used in this form will be called variable-quotes. They result from ordinary quotes when a variable is put in quotes, and, in addition, substitutions are regarded as admissible for the letter variable. Sometimes substitutions are not admitted; then quotes put around a variable are ordinary quotes. Usually this distinction is expressed by a term prefixed to the expression in quotes to indicate the range of the substitutions admitted. Thus we say 'the variable 'a' '; the term 'variable' indicates here that the quotes around 'a' are variable-quotes. When we say 'the letter 'a' ', we mean ordinary quotes around 'a'. Only in exceptional cases is this notation ambiguous; then the meaning of the quotes must be understood from the context in which they are used. Thus in the sentence 'the variable 'a' in the statement 'a ⊃ b' is expressed by the first letter of the alphabet', we see from the context that we must not substitute other letters for 'a', and that therefore the quotes are ordinary quotes. For these reasons it appears unnecessary to introduce special symbols for variable-quotes. They can be dispensed with also for the reason that, when a special proposition is substituted for the variable, the resulting quotes are ordinary quotes. This is clear because a special proposition does not admit of substitutions. Furthermore, quotes used with variables will practically always be variable-quotes; and we shall therefore frequently omit a prefixed range term, the meaning of the quotes being clear from the context.³

In conversational language, variable-quotes are often evaded by means of the phrase 'such as'. We say, for instance, 'a sentence combination such as 'snow is white and water is wet' '. These quotes are the ordinary quotes. Using variable-quotes we write the same expression in the form 'a sentence combination 'a.b' '.

On the other hand, the usage of ordinary quotes may be interpreted as involving variable-quotes of a restricted nature. We said in § 2 that a symbol is defined by the use of an equisignificance relation; this relation is given by geometrical similarity, but also by such relations as consist in the correspondence of small and capital letters, or of written and spoken tokens. Now it is in the nature of the equisignificance relation that it admits the replacement of a token by another one without any coupling condition. Thus in the formula 'a ⊃ a' we can erase the first token and replace it by a similar one, leaving the second token unaltered. This is usually not admissible for a transition, for instance, to spoken tokens. We would not call it a meaningful expression if we wrote only the first 'a' of the formula and the implication sign, and then pronounced an 'a' for the second token of this letter. We allow a transition only from the written formula to the spoken formula. This restriction represents a coupling condition, and thus indicates that the transition constitutes a substitution rather than a replacement, the word 'replacement' being used for a transition to other signs that is not bound to a coupling condition. Therefore ordinary quotes should be interpreted as variable-quotes of a restricted range of substitution. The range is indicated here also by a prefixed term, such as 'the spoken letter', 'the written letter', 'the capital letter', 'the letter' (including spoken and written tokens and small and capital letters).

Let us summarize the given distinctions of levels of language by the table below.

Objects	Object language	Metalanguage
	bird thing	'bird' word, name
	the bird flies a1 situation	'the bird flies' 'a1' sentence, proposition
	a	'a' propositional variable
		' the bird flies' is true a1
		a

We consider physical objects as constituting the zero level; they may be called objects in the absolute sense, and the corresponding object language then is tne absolute object language. Sometimes other kinds of objects may be considered; for instance numbers. We then say that they are objects in a relative sense, and that the corresponding object language is a relative object language. Physical objects divide into things, such as individual human beings, tables, atoms, and situations, also called states of affairs, which constitute the denotata of sentences. Thus the sentence 'the battle­ship Bismarck was sunk' denotes a situation; the ship itself is a thing.⁴

The metalanguage is divided into three parts, corresponding to the three arguments of the sign relation. The first part, syntax, deals with relations between signs only and therefore concerns structural properties of the object language. A syntactical statement is, for instance: 'the sentence 'if water is heated it expands' is an implication'. The second part, semantics, refers to both signs and objects; in particular, it therefore includes statements concerning the truth-value of propositions, since truth is a relation between signs and objects. Of this kind is the statement: 'the sentence 'if water is heated it expands' is not always true'. The third part, pragmatics, adds a reference to persons; it therefore refers to things, signs, and persons. Of this kind is the statement: 'I consider this sentence true', or 'this sentence is a law of physics', as the latter proposition states that physicists consider the sentence true.⁵

Conversational language is a mixture of object language and metalanguage, including all three parts of the metalanguage. The occurrence of words like 'word', 'proposition', indicates the use of the metalanguage. Words like 'conclusion', 'derivable', belong to syntax; words like 'true', 'likely', 'perhaps', belong to semantics; and words like 'assertion', 'incredible', 'of course', belong to pragmatics.

The rules that we need to define a language repeat the trichotomy of the metalanguage. First we must give formation rules, which tell us under what conditions a set of signs is meaningful. Of this kind are grammatical rules; they are insufficient, however, and an ideal language would possess rules of formation showing immediately that expressions of the kind 'Caesar is a prime number' are meaningless. Second, we must give truth rules, i.e., rules that tell us what kind of truth-values a proposition can have, and how these truth-values determine the truth of compound propositions. Thus two-valued logic and probability logic differ as to their truth rules. Third, we must give derivation rules, which tell us ways of deriving new propositions from given propositions; of this kind is the rule of inference. Since the derivation of new propositions from given propositions is needed for practical purposes by persons who use the language, this third kind of rule corresponds to pragmatics.

The rules of derivation are of two kinds. The first lead from true propositions to true propositions; they are called rules of deduction. The part of logic that they establish is called deductive logic. The second sort of rules lead from true propositions to propositions that are maintained only as posits, i.e., as substitutes for true propositions where truth is not knowable and is replaced by a probability;⁶ they are the rules of induction. The part of logic that includes inductive rules is called inductive logic; it comprises both deductive and inductive derivations and deals with the theory of indirect evidence. The exposition of the present book will be restricted to deductive logic. [Ex.]