A central distinction for Carnap is that between definite and indefinite notions. A definite notion is one that is recursive, such as “is a formula” and “is a proof of φ”. An indefinite notion is one that is non-recursive, such as “is an ω-consequence of PA” and “is true in Vω+ω”. This leads to a distinction between (i) the method of derivation (or d-method), which investigates the semi-definite (recursively enumerable) metamathematical notions, such as demonstrable, derivable, refutable, resoluble, and irresoluble, and (ii) the method of consequence (or c-method), which investigates the (typically) non-recursively enumerable metamathematical notions such as consequence, analytic, contradictory, determinate, and synthetic.
A language for Carnap is what we would today call a formal axiom system. The rules of the formal system are definite (recursive) and Carnap is fully aware that a given language cannot include its own c-notions. The logical syntax of a language is what we would today call metatheory. It is here that one formalizes the c-notions for the (object) language. From among the various c-notions Carnap singles out one as central, namely, the notion of (direct) consequence; from this c-notion all of the other c-notions can be defined in routine fashion.
We now turn to Carnap’s account of his fundamental notions, most notably, the analytic/synthetic distinction and the division of primitive terms into ‘logico-mathematical’ and ‘descriptive’. Carnap actually has two approaches. The first approach occurs in his discussion of specific languages – Languages I and II. Here he starts with a division of primitive terms into ‘logico-mathematical’ and ‘descriptive’ and upon this basis defines the c-notions, in particular the notions of being analytic and synthetic. The second approach occurs in the discussion of general syntax. Here Carnap reverses procedure: he starts with a specific c-notion – namely, the notion of direct consequence – and he uses it to define the other c-notions and draw the division of primitive terms into ‘logico-mathematical’ and ‘descriptive’.
In the first approach Carnap introduces two languages – Language I and Language II. The background languages (in the modern sense) of Language I and Language II are quite general – they include expressions that we would call ‘descriptive’. Carnap starts with a demarcation of primitive terms into ‘logico-mathematical’ and ‘descriptive’. The expressions he classifies as ‘logico-mathematical’ are exactly those included in the modern versions of these systems; the remaining expressions are classified as ‘descriptive’. Language I is a version of Primitive Recursive Arithmetic and Language II is a version of finite type theory built over Peano Arithmetic. The d-notions for these languages are the standard proof-theoretic ones.
For Language I Carnap starts with a consequence relation based on two rules – (i) the rule that allows one to infer φ if T φ (where T is some fixed ∑10-complete formal system) and (ii) the ω-rule. It is then easily seen that one has a complete theory for the logico-mathematical fragment, that is, for any logico-mathematical sentence φ, either φ or ¬φ is a consequence of the null set. The other c-notions are then defined in the standard fashion. For example, a sentence is analytic if it is a consequence of the null set; contradictory if its negation is analytic; and so on.
For Language II Carnap starts by defining analyticity. His definition is a notational variant of the Tarskian truth definition with one important difference – namely, it involves an asymmetric treatment of the logico-mathematical and descriptive expressions. For the logico-mathematical expressions his definition really just is a notational variant of the Tarskian truth definition. But descriptive expressions must pass a more stringent test to count as analytic – they must be such that if one replaces all descriptive expressions in them by variables of the appropriate type, then the resulting logico-mathematical expression is analytic, that is, true. In other words, to count as analytic a descriptive expression must be a substitution-instance of a general logico-mathematical truth. With this definition in place the other c-notions are defined in the standard fashion.
The content of a sentence is defined to be the set of its non-analytic consequences. It then follows immediately from the definitions that logico-mathematical sentences (of both Language I and Language II) are analytic or contradictory and (assuming consistency) that analytic sentences are without content.
In the second approach, for a given language, Carnap starts with an arbitrary notion of direct consequence and from this notion he defines the other c-notions in the standard fashion. More importantly, in addition to defining the other c-notion, Carnap also uses the primitive notion of direct consequence (along with the derived c-notions) to effect the classification of terms into ‘logico-mathematical’ and ‘descriptive’. The guiding idea is that “the formally expressible distinguishing peculiarity of logical symbols and expressions [consists] in the fact that each sentence constructed solely from them is determinate”. Howsoever the guiding idea is implemented the actual division between “logico-mathematical” and “descriptive” expressions that one obtains as output is sensitive to the scope of the direct consequence relation with which one starts.
With this basic division in place, Carnap can now draw various derivative divisions, most notably, the division between analytic and synthetic statements: Suppose φ is a consequence of Γ. Then φ is said to be an L-consequence of Γ if either (i) φ and the sentences in Γ are logico-mathematical, or (ii) letting φ’ and Γ’ be the result of unpacking all descriptive symbols, then for every result φ” and Γ” of replacing every (primitive) descriptive symbol by an expression of the same genus, maintaining equal expressions for equal symbols, we have that φ” is a consequence of Γ”. Otherwise φ is a P-consequence of Γ. This division of the notion of consequence into L-consequence and P-consequence induces a division of the notion of demonstrable into L-demonstrable and P-demonstrable and the notion of valid into L-valid and P-valid and likewise for all of the other d-notions and c-notions. The terms ‘analytic’, ‘contradictory’, and ‘synthetic’ are used for ‘L-valid’, ‘L-contravalid’, and ‘L-indeterminate’.
It follows immediately from the definitions that logico-mathematical sentences are analytic or contradictory and that analytic sentences are without content. The trouble with the first approach is that the definitions of analyticity that Carnap gives for Languages I and II are highly sensitive to the original classification of terms into ‘logico-mathematical’ and ‘descriptive’. And the trouble with the second approach is that the division between ‘logico-mathematical’ and ‘descriptive’ expressions (and hence division between ‘analytic’ and ‘synthetic’ truths) is sensitive to the scope of the direct consequence relation with which one starts. This threatens to undermine Carnap’s thesis that logico-mathematical truths are analytic and hence without content.
In the first approach, the original division of terms into ‘logico-mathematical’ and ‘descriptive’ is made by stipulation and if one alters this division one thereby alters the derivative division between analytic and synthetic sentences. For example, consider the case of Language II. If one calls only the primitive terms of first-order logic ‘logico-mathematical’ and then extends the language by adding the machinery of arithmetic and set theory, then, upon running the definition of ‘analytic’, one will have the result that true statements of first-order logic are without content while (the distinctive) statements of arithmetic and set theory have content. For another example, if one takes the language of arithmetic, calls the primitive terms ‘logico-mathematical’ and then extends the language by adding the machinery of finite type theory, calling the basic terms ‘descriptive’, then, upon running the definition of ‘analytic’, the result will be that statements of first-order arithmetic are analytic or contradictory while (the distinctive) statements of second- and higher-order arithmetic are synthetic and hence have content. In general, by altering the input, one alters the output, and Carnap adjusts the input to achieve his desired output.
In the second approach, there are no constraints on the scope of the direct consequence relation with which one starts and if one alters it one thereby alters the derivative division between ‘logico-mathematical’ and ‘descriptive’ expressions. Logical symbols and expressions have the feature that sentences composed solely of them are determinate. The trouble is that the resulting division of terms into ‘logico-mathematical’ and ‘descriptive’ will be highly sensitive to the scope of the direct consequence relation with which one starts. For example, let S be first-order PA and for the direct consequence relation take “provable in PA”. Under this assignment Fermat’s Last Theorem will be deemed descriptive, synthetic, and to have non-trivial content. For an example at the other extreme, let S be an extension of PA that contains a physical theory and let the notion of direct consequence be given by a Tarskian truth definition for the language. Since in the metalanguage one can prove that every sentence is true or false, every sentence will be either analytic (and so have null content) or contradictory (and so have total content). To overcome such counter-examples and get the classification that Carnap desires one must ensure that the consequence relation is (i) complete for the sublanguage consisting of expressions that one wants to come out as ‘logico-mathematical’ and (ii) not complete for the sublanguage consisting of expressions that one wants to come out as ‘descriptive’. Once again, by altering the input, one alters the output.
Carnap merely provides us with a flexible piece of technical machinery involving free parameters that can be adjusted to yield a variety of outcomes concerning the classifications of analytic/synthetic, contentful/non-contentful, and logico-mathematical/descriptive. In his own case, he has adjusted the parameters in such a way that the output is a formal articulation of his logicist view of mathematics that the truths of mathematics are analytic and without content. And one can adjust them differently to articulate a number of other views, for example, the view that the truths of first-order logic are without content while the truths of arithmetic and set theory have content. The point, however, is that we have been given no reason for fixing the parameters one way rather than another. The distinctions are thus not principled distinctions. It is trivial to prove that mathematics is trivial if one trivializes the claim.
Carnap is perfectly aware that to define c-notions like analyticity one must ascend to a stronger metalanguage. However, there is a distinction that he appears to overlook, namely, the distinction between (i) having a stronger system S that can define ‘analytic in S’ and (ii) having a stronger system S that can, in addition, evaluate a given statement of the form ‘φ is analytic in S’. It is an elementary fact that two systems S1 and S2 can employ the same definition (from an intensional point of view) of ‘analytic in S’ (using either the definition given for Language I or Language II) but differ on their evaluation of ‘φ is analytic in S’ (that is, differ on the extension of ‘analytic in S’). Thus, to determine whether ‘φ is analytic in S’ holds one needs to access much more than the “syntactic design” of φ – in addition to ascending to an essentially richer metalanguage one must move to a sufficiently strong system to evaluate ‘φ is analytic in S’.
In fact, to answer ‘Is φ analytic in Language I?’ is just to answer φ and, in the more general setting, to answer all questions of the form ‘Is φ analytic in S?’ (for various mathematical φ and S), where here ‘analytic’ is defined as Carnap defines it for Language II, just to answer all questions of mathematics. The same, of course, applies to the c-notion of consequence. So, when in first stating the Principle of Tolerance, Carnap tells us that we can choose our system S arbitrarily and that ‘no question of justification arises at all, but only the question of the syntactical consequences to which one or other of the choices leads’, where he means the c-notion of consequence.