Rudolf Carnap’s The Logical Syntax of Language is a work of technical philosophy first written in German in the early 1930s and published in an expanded English edition, translated by Amethe Smeaton, in the late 1930s. It belongs to the tradition of logical positivism that grew out of the Vienna Circle, and it sets out to do something radical: to replace philosophy, insofar as philosophy claims to be scientific, with the logical analysis of the language of science. The book is not a treatise on how language happens to work but a construction manual for artificial languages, together with a general theory of what any language’s syntax can be.
Carnap’s central claim is that the formal rules of a language—its rules of formation and its rules of transformation—can be chosen freely, and that once chosen they determine the meaning of the language’s logical symbols. There is no prior standard of “correct” logic against which a language must be measured. He calls this the Principle of Tolerance, and states it bluntly: it is not our business to set up prohibitions, but to arrive at conventions; in logic, there are no morals. The book’s long middle is the working-out of that idea in concrete formal detail.
From a Definite Language to the Ocean of Possibilities
The English Edition and Its Additions
The volume appears in Ogden’s International Library of Psychology, Philosophy and Scientific Method, with Carnap identified as Professor of Philosophy at Chicago and Amethe Smeaton, Countess von Zeppelin, as translator. In his preface Carnap explains that the English text restores material the German manuscript, sent to press in December 1933, had lost for want of space: sections 16a, 34e–i, 38a–c, 60a–d and 71a–e, twenty-two sections in all. Parts of this material had already appeared in German journals, on a validity criterion for classical mathematics and on the antinomies and incompleteness. The old section 60 gives way to the shortened 60a–d; the bibliography is revised, corrections are listed section by section, and two theorems are dropped. Carnap thanks Tarski, McKinsey and Quine for suggestions, and Quine, Graham, Helmer and Nagel, along with the translator, for help with the English rendering of his terminology.
Philosophy Replaced by the Logic of Science
The foreword states the book’s purpose: a systematic exposition of logical syntax. Logistics has taught the precise handling of symbols, yet the surrounding prose remains murky. Around the Vienna Circle and kindred groups in Poland, France, England, America and even Germany, the conviction spreads that metaphysics is not science and that the scientific portion of philosophy is logical analysis. Philosophy is to be replaced by the logic of science — the logical analysis of the concepts and sentences of the sciences — and that is the logical syntax of the language of science. Carnap constructs two languages: Language I, simple and narrow in conceptual field, and Language II, rich enough for classical mathematics and physics. The investigation concerns synthetic empirical sentences, the core of science; mathematico-logical sentences are analytic, without real content, mere formal auxiliaries. Language I serves to show that a language’s syntax can be formulated within itself, free of the feared antinomies, and a general syntax valid for any language follows.
The Principle of Tolerance
The range of possible language-forms, Carnap insists, is far wider than modern logic had assumed. Departures from Russell’s classical form — unlimited existential sentences, rejection of excluded middle, many-valued calculi, probability logic, intensional sentences — have been few because deviation was thought to need justification as “correct.” Carnap claims complete liberty: construction rules and transformation rules may be chosen arbitrarily. Meaning, on the old approach, was fixed first and correct sentences sought afterwards, but meaning expressed in words is inexact; the clearer direction is to choose postulates and rules freely, and the choice determines the meaning of the fundamental logical symbols. The conflict over foundations thus dissolves, leaving only syntactical consequences, non-contradiction among them. This is the Principle of Tolerance, and before us, he writes, lies the boundless ocean of unlimited possibilities. He credits Frege, Hilbert’s metamathematics, the Polish logicians Ajdukiewicz, Lesniewski, Lukasiewicz and Tarski, and Gödel’s arithmetization; he thanks conversations with Tarski and Gödel, acknowledges Wittgenstein on syntax and the logic of science (with divergences noted), and reports Schlick’s word that Wittgenstein now accepts free choice of linguistic rules.
What Logical Syntax Is
The introduction defines logical syntax as the formal theory of linguistic forms: the systematic statement of formal rules together with the development of their consequences. Formal here means concerned with designs, not meaning.
Predicates, Functors, and Co-ordinate Languages
Carnap contrasts name-languages, which designate objects by proper names, with co-ordinate-languages, which use systematic positional symbols — house numbers, Ostwald’s colour letters, latitude and longitude, space-time co-ordinates — the more advanced and methodologically advantageous method. Language I adopts co-ordinate-languages as its object-languages, building series from ‘O’, ‘O”, ‘O”’ (abbreviated 1, 2) and higher dimensions from ordered couples and n-ads. Predicates express properties and relations, one- or many-termed, and divide into descriptive and logical; following Hilbert the term covers multi-term cases. Properties may themselves be arithmetized by number-symbols, allowing universal laws, and functors express them.
Gothic Symbols and the Five Kinds of Sign
Symbols alike in shape but differently placed are different yet equal, so rules must fix syntactical equality: ‘z’ and ‘3’ are unequal here, round and square brackets equal, ‘=’ and ‘≡’ equal, unlike Russell. Pure syntax deals only with designs. Five kinds of symbol occur in Language I — individual symbols, variables, constant numerals, predicates, functors — and the Gothic notation, with autonymous symbols and indices distinguishing descriptive from logical constants, guards against confusing symbols with what they symbolize.
The Limits of Language I and the Need for an Indefinite Syntax
Carnap opens this stretch by pressing on a technical difficulty in Language I. Certain definitions require unlimited operators, which Language I simply does not possess. The concepts “derivable” and “demonstrable” are indefinite in character, and Language I can capture only their definite versions—derivation or proof consisting of at most n symbols. To define the indefinite concepts properly, one must ascend to an indefinite syntax-language such as Language II. Yet some indefinite concepts, though undefinable in I, still allow their universal sentences to be formulated there: a negated unlimited existential operator can be replaced by a universal one, and unlimited universality can be expressed in I through a free variable. Thus “BewSatz(r,a)” says that every r fails to be a proof of a, meaning a is not demonstrable; “AblSatz[r,b,reihe(a)]” says b is not derivable from a.
Descriptive Syntax and the Single Primitive “zei”
The pure syntax of Language I turns out to be part of arithmetic, but descriptive syntax uses descriptive symbols and reaches beyond arithmetic—for instance, stating that at a particular place an expression of such-and-such form occurs. Rather than introducing a host of undefined primitive symbols, Carnap takes the single undefined functor “zei” as his only addition, coordinating it with term-number 243. The construction of descriptive syntax then follows any descriptive axiomatic system: first the syntax of the language is established, and for some systems the language must contain arithmetic. With “zei” no axioms are needed, since functor rules guarantee one value per place and existential sentences follow from arithmetic. A regressive definition introduces “ausdr” (expression), the most important term of descriptive syntax, and with “zei” and “ausdr” plus the pure-syntax symbols, descriptive-syntax sentences of Language I can be formulated within Language I itself.
Axiomatic, Physical, and Arithmetical Syntax Compared
Carnap distinguishes axiomatic syntax from physical syntax, the latter standing to the former as physical geometry to axiomatic geometry; physical geometry arises through correlative definitions that fix which physical concepts the axiomatic primitives match in meaning. A schematic survey compares three kinds of geometry and syntax: arithmetical (pure), descriptive (axiomatic), and physical. The axiomatic primitives differ—”Point,” “straight line,” “between” versus ‘Var,’ ‘Nu,’ ‘Präd,’ ‘Gl’ versus ‘zei’ alone—and the axioms range from Hilbert’s to numerous trivial ones to none at all. Analytic sentences use definitions but not axioms; synthetic sentences are the axioms and what is proved from them. The question of an axiomatic system’s validity, once correlative definitions are supplied, becomes the question of the validity of laws translated into the language of science—for geometry, Euclidean versus non-Euclidean; for syntax, the critical matter of existential axioms, particularly axioms of infinity.
Comparisons with Russell, Hilbert, Gödel, and Tarski
The text then compares its own system with others. Russell states PSII 16 as a primitive sentence rather than a schema, requiring a substitution rule he never formulates but tacitly applies; Hilbert likewise states PSII 16 as primitive and adds the necessary rule. Gödel avoids the existential operator altogether, so PSII 18 becomes unnecessary, and Tarski erects no primitive sentences for the calculus of functions but only rules of inference. Arithmetic follows Peano’s method, with ‘0’ and a successor symbol primitive, though Peano’s undefined ‘number’ is dropped because all expressions of the lowest type are numerical. An Axiom of Infinity proves unnecessary in Language II—the corresponding sentence is demonstrable—while Zermelo’s Axiom of Selection corresponds to PSII 21 and PSII 22 is an Axiom of Extensionality. Neither an Axiom of Reducibility nor an Axiom of Comprehension is needed, since only the simple classification of types is carried out and every predicate can be defined by a sentence with free variables.
Criteria of Validity
A chief task of the logical foundation of mathematics is a formal criterion of validity stating necessary and sufficient conditions for truth in the classical-mathematical sense. Carnap distinguishes three kinds. A definite criterion, decidable in finitely many steps, would give a method for solving every mathematical problem—one could “calculate” the truth of Fermat’s Theorem. Weyl doubted such a touchstone would ever be found, and Gödel’s findings make the search for a definite criterion for the whole system seem hopeless. An indefinite criterion based on definite rules—the method of derivation—remains the practical alternative.
Autonymous Designation and the Warsaw School
Carnap opens with a cautionary example drawn from a distinguished logician’s treatise: a sentence about a formula “a” resulting from replacing a variable “x” by a “combination of symbols p.” Try as one might, no consistent reading works—enclose the letters in inverted commas and the phrases about “combinations of symbols” and “x occurring in a” collapse; leave them bare and the symbols turn out to inhabit the object-language’s own axioms. The tangle illustrates his real target: the confusion that arises when it is left unclear whether an expression belongs to the object-language or is a syntactical designation of it, and whether it designates itself. Frege’s demand that sign and thing signified be kept apart, Carnap notes, is strictly honoured only by the Warsaw school—Lukasiewicz, Leśniewski, Tarski and their pupils—who deploy special syntactical symbols. The method has great advantages, though it is not essential for correctness; Carnap himself adopts a word-language with Gothic symbols, following preliminary steps taken by Hilbert and Church. Full symbolization, as in Leśniewski and Gödel, buys exactness at the cost of facility. In Part II, Language I serves as both object-language and syntax-language.
Definite and Indefinite Terms
A defined symbol of Language II is definite when no unrestricted operator appears anywhere in its chain of definitions, indefinite otherwise. Definite predicates yield decidable sentences for definite arguments; indefinite ones generally do not. “Prime number” can be rendered either way—by an indefinite ‘Prim₁’ or a synonymous definite ‘Prim₂’—and the two are demonstrably equal. For the indefinite ‘Bewbll’ no definite synonym is known, and there is reason to think none exists, since finding one would hand mathematics a general decision method. Poincaré, Brouwer, Wittgenstein and Kaufmann reject indefinite terms as meaningless, holding that meaning consists solely in a method of determination. Carnap answers that we know perfectly well what form an answer would take—a proof ending in ‘P₁(5)’—and whether a given series is such a proof is itself a definite question. To make significance depend on whether an answer has historically been found would tie logic to the accidents of discovery. Finding a number with a property and finding a proof of a universal sentence are, he argues, operations of essentially the same kind.
The Vicious Circle and the Principle of Tolerance
Impredicative terms, banned by Russell’s vicious-circle principle, are defined through a totality to which they themselves belong. The objection runs that deciding ‘P₃(5)’ requires deciding ‘M(P₃,5)’, which requires ‘P₃(5)’—an obvious circle. Carnap calls this beside the point: a universal sentence is established by proof of that sentence, not by verifying every substitution instance, which for infinite cases is impossible anyway. He works through a concrete demonstration, resolving ‘M(P₄,5)’ first and then deriving ‘P₃(5)’ for the low numbers and ‘P₃(8)’ from seven onwards. Since sentences with unrestricted operators can be demonstrated, decision is always possible in principle. The proper question is not admissibility—”no morals in logic”—but which language-form to construct and with what consequences. Excluding unrestricted operators, as Language I does, excludes indefinite and impredicative symbols; admitting universal predicate-variables makes impredicative definitions syntactically legitimate. The material objections so far advanced are not sound. Carnap’s attitude throughout conforms to the principle of tolerance: exclude indefinite terms or admit them, by convention. The syntactical vocabulary itself—’derivable’, ‘demonstrable’, ‘analytic’, ‘consequence’—is generally indefinite and demands an indefinite syntax-language, though ‘sentence’ is usually defined definitely, and von Neumann insists such definiteness is indispensable.
Logical and Descriptive Expressions
A class of expressions is fixed by four conditions—non-emptiness, determinacy of every sentence built from it, minimal size of members, maximal comprehensiveness—and an expression counts as logical if it can be subdivided into members of that class, descriptive otherwise. The criterion settles borderline cases: the metrical tensor ‘g_μν’ is a logical symbol in a language assuming homogeneous space, where its values follow from general law, but descriptive in Einsteinian non-homogeneous space, where empirical data on matter distribution are required. Every logical sentence is determinate; every indeterminate sentence is descriptive. Carnap then defines sub-languages, conservative sub-languages and proper sub-languages, and identifies the logical sub-language of a language as the conservative sub-language left after all descriptive sentences are eliminated. A table records when a syntactical property carries over between a language and its sub- or super-language.
Quasi-Syntactical Sentences and Their Correlates
Carnap opens this stretch of the general syntax by defining what it means for a sentence to be quasi-syntactical. The machinery is precise: given a sentence whose form attributes a property to an expression that is itself quasi-syntactical, one constructs a correlated syntactical sentence in which each such expression is replaced by a designation of its syntactic kind. His worked example pairs a sentence about a property F having a further property with a sentence about a class x standing in a relation to a y. Any sentence containing a quasi-syntactical expression counts as quasi-syntactical, and compound cases are built up in the same way. The distinction matters because it connects with the difference between syntactical concepts and truth. Were “true” a syntactical term, every sentence would be quasi-syntactical relative to every partial expression, since each sentence is equipollent to the claim that its parts are such that the sentence is true. In a purely logical language, where truth coincides with analyticity, the notion would collapse into triviality; only relative to a descriptive language does the distinction do real work. Claims of quasi-syntacticality, Carnap adds, mean that a sentence stays quasi-syntactical even when the language is expanded to become descriptive—a point he will later press against the logic of modalities.
Autonymous and Material Modes of Speech
Two interpretations of such sentences are then separated. In the first, the expression designates itself—it is autonymous—and the predicate is equivalent in meaning to its syntactical correlate; “Five is a number-word” is the model. In the second, the predicate attributes an object-property to the object designated, and the sentence belongs to the material mode of speech, translatable into a sentence about the correlated syntactical property of the designation. “Five is a number” belongs here, correlated with “‘Five’ is a number-word.” Carnap gives formal definitions: an expression is autonymous in a sentence when the sentence is equipollent to its syntactical correlate for every isogenous expression; otherwise the sentence is material. A translation replacing quasi-syntactical sentences by their syntactical correlates and leaving the rest untouched moves discourse from the material into the formal mode. Since the distinction concerns interpretation, it cannot be drawn for an isolated calculus—yet interpretation can itself be formally represented and so brought within syntax. Where an author supplies only vague word-language explanations, the classification cannot be exact; but deciding that a sentence is quasi-syntactical rather than genuinely syntactical can be as exact as the construction of the language itself.
Implication Contrasted with Consequence
The discussion turns to extensionality for partial sentences, then to the sharp separation of implication from the consequence-relation. Materially put: consequence holds between sentences; implication does not. Whether Russell’s view that implication relates propositions is wrong depends on what “proposition” means—if it names whatever a sentence designates, implication relates such designata, but consequence does not. Formally, the horseshoe belongs to the object-language while “consequence” is a predicate of the syntax-language. A theorem links two sentences, but the second cannot be inferred from the first—only from the syntactical sentence asserting the first is valid or analytic. Most symbolic languages, once their inference rules are extended, are logical languages with no indeterminate sentences, so the inference does go through there; this explains why implication-sentences get misread as statements about consequence. Carnap regrets the long neglect of indeterminate sentences. Intensional implication symbols, strict implication among them, stand to the horseshoe and to consequence exactly as “Limp” stands to “Imp” and consequence, with differences among intensional implications mirroring different definitions of consequence. Russell’s choice of “implication” for the truth-functional junction he calls very unfortunate, since “to imply” means to contain or involve; the nomenclature bred confusion, perhaps leading those who knew the difference to think the symbol ought to express consequence and to count it a failure that it does not. Carnap keeps the word in a sense wholly divorced from its original meaning, as a mere label for a kind of sentential junction.
Modal Logic as Syntax
Every prior system of modal logic, Carnap observes, has used the quasi-syntactical method—not by conscious choice but as something felt natural. All intensional sentences in existing modal systems are quasi-syntactical under either interpretation, and each system can be read so that some partial expression is autonymous. Any intensional modal system, even one admitting synthetic sentences as arguments, can be translated into an extensional syntactical language, each intensional sentence becoming its correlated syntactical sentence. Syntax, in short, already contains the whole of modal logic; no special intensional logic is needed. Choosing between methods is a matter of expedience, and Carnap states the properties of both without deciding.
The Material Mode and Its Translation
Carnap opens the final stretch by asking whether the material mode of speech—the idiom in which philosophers say things like “the thing-in-itself” or “the transcendental”—can be admitted at all. His answer is a qualified yes. A sentence in the material mode need not always be rewritten in the formal mode, but it must always be translatable into it. That translatability becomes the touchstone for every philosophical sentence and for every sentence outside the empirical sciences. Where ordinary usage or an author’s definitions leave the translation undetermined, the sentence is ambiguous and obscure; where no indication is given at all, it falls outside science and cannot even be discussed. He illustrates the danger with specimens of what he calls a “mythology of the inexpressible” and a “mythology of higher things”—”What can be shown cannot be said,” “The sense of the world must lie outside the world”—and offers deliberately deflationary translations that turn them into syntactical remarks about designations, or expose them as empty (“…is not a sentence”). Pseudo-object-sentences of the logic of science really speak about designations of objects, and all of them can be translated into syntactical sentences. The logic of science is the syntax of the language of science.
The Physical Language and the Testing of Laws
Turning to physics, Carnap treats its logical analysis as the syntax of the physical language, with epistemological problems falling partly to empirical psychology and partly to syntax. He sets out the machinery: point-expressions built from four real-number expressions, domain-expressions, descriptive functors for coefficients of state; sentences split into concrete sentences and laws. Transformation rules may rest on L-rules alone or on L-rules with P-rules, and the P-primitive sentences include primitive laws, deterministic or probabilistic. Syntax fixes the form of protocol-sentences but never decides which ones are laid down—truth and falsity are not syntactical terms, and that choice belongs to the observing physicist. Testing deduces protocol-form sentences and compares them with those actually stated; when contradiction arises, P-rules, the protocol-sentence, or L-rules may be altered, and no established rules govern the change. Laws are hypotheses: selected rather than inferred, never strictly refuted nor completely verified, only gradually confirmed, and generally testable only as part of the whole system of physics—a point he credits to Duhem and Poincaré. A primitive sentence without sufficient transformation rules is untestable and useless; the neo-vitalists’ “entelechy” is rejected not for being undefined but because no empirically testable laws are laid down for it. Construction proceeds by conventions that are influenced by practical and methodological considerations and testable by protocol-sentences, yet never univocally determined by empirical material. He notes two theses he does not depend on—physicalism and the unity of science—crediting Neurath with their early radical adoption, and contrasts the earlier Vienna Circle demand for complete verifiability, which left no place for laws, with the simpler construction that treats laws as proper sentences; by the principle of tolerance both are admissible.
Foundations of the Sciences
The foundations of physics—time and space, causality, verification, measurement, the macro/micro relation, genidentity—are syntactical problems. Biology raises the reducibility of concepts and of laws, the latter being the scientific core of vitalism. Psychology poses analogous questions; physicalism answers the first affirmatively and leaves the second open.
The Logic of Science Is Syntax
Carnap closes by stripping philosophy of its pretensions: whoever investigates the logic of science works in the same field as the scientific specialist, only with his attention turned to formal, syntactical connections. The thesis that the logic of science is syntax does not mean it can be pursued apart from empirical science. Investigating a system already given is pure mathematics, but the language of science is not given in syntactically established form; the investigator must study the language actually used in the special sciences and frame rules on that basis. Any proposed formulation is in principle a free convention, yet it can be useful only if it regards the available empirical findings—the choice between deterministic and probabilistic laws, or between Euclidean and non-Euclidean geometry, is not forced by the material but is made in consideration of it. All philosophical work is barren without close co-operation with the special sciences. The non-metaphysical philosophy of recent decades, he suggests, has at bottom been syntactical without knowing it; that character must now be recognized in theory and practised systematically. Only then can traditional philosophy give way to a strict scientific discipline. The step from the morass of subjectivist problems onto the firm ground of exact syntactical problems must be taken, and the book offers only a first working-tool of syntactical terms, whose use and improvement demand the co-operation of many minds. There the argument ends, followed by bibliography, indexes, and the publisher’s catalogue.
What the Formal Machinery Is For
The apparatus of the book can look forbidding—Gothic symbols, numbered definitions, schemata of primitive sentences—but each piece answers a philosophical worry. The distinction between derivability and consequence, for instance, exists because a sentence can follow from an infinite class of premises without following from any finite subset; derivability is a finite chain, consequence is wider, and analyticity is wider still than demonstrability. Gödel’s results had shown that not every analytic sentence is demonstrable, and Carnap builds that gap directly into his definitions rather than treating it as a scandal. Similarly, the arithmetization of syntax—assigning numbers to symbols and series-numbers to expressions, following the method Gödel applied in metamathematics—is not a technical flourish. It lets the syntax of a language be formulated inside arithmetic, so that statements about proofs and demonstrability become ordinary arithmetical statements. Without it, a syntax could say that no written proof of a sentence exists, but not that no proof is possible.
The treatment of “analytic” and “contradictory” carries the philosophical weight. Analytic sentences are those that follow from the null class; contradictory ones are those from which everything follows. Carnap uses these to define content: the content of a sentence is the class of its non-analytic consequences, so that analytic sentences have null content and contradictory ones total content. This is how the book makes good on the positivist claim that mathematics and logic are formal auxiliaries without real content, while the synthetic sentences of empirical science carry the substance.
The Antinomies and the Limits of Self-Knowledge
The most striking results in the book concern what a language can say about itself. Carnap shows that the syntax of Language I can be formulated within Language I, without the epistemological and linguistic antinomies that had been feared. But the success has a precise boundary. If a language is consistent and rich enough to contain arithmetic, then “analytic in S” cannot be defined in S. The liar paradox, restated with “non-demonstrable” or “refutable” in place of “false,” fails to generate contradiction—precisely because not all analytic sentences are demonstrable and not all contradictory sentences refutable. But replace those terms with “analytic” and “contradictory,” which completely classify logical sentences, and contradiction returns. Grelling’s heterological paradox and Richard’s paradox can be reconstructed on the same basis. The upshot is a theorem: a syntax containing “analytic in S₁” must be formulated in a richer language S₂. This is why the non-contradictoriness of a language cannot be proved within that language, and Carnap credits Gödel’s work for making the situation clear.
A Grammar of Everything
Part IV widens the lens from particular languages to syntax as such. Here Carnap defines the logical expressions of a language by formal conditions, distinguishing them from descriptive ones; the example of the metrical tensor is instructive, since the same symbol counts as logical in a language that assumes a homogeneous space and descriptive in one that assumes Einsteinian space-time. He separates L-rules, which rest on a logico-mathematical basis, from P-rules, which include laws of nature, and insists that how many P-rules to adopt is a matter of convention and expedience, not of philosophical principle. The concept of range, taken from Wittgenstein’s Tractatus, defines the domain of possibilities a sentence leaves open. The quasi-syntactical sentence—one that looks like a statement about objects but is really a disguised statement about symbols—becomes the tool for diagnosing confusion, and Carnap will later use it against sentences of modal logic.
Why the Book Still Sets Terms
The book’s legacy lies less in any single formal system than in the attitude it codified. By treating alternative logics—many-valued calculi, the logic of probability, intuitionist restrictions—as legitimate constructions rather than deviations to be justified, Carnap cleared space for the proliferation of non-classical logics that followed. His insistence that questions about foundations dissolve once you see them as questions about which language to construct, and his willingness to let a thousand languages bloom, gave later philosophy of science and philosophical logic a vocabulary it still uses. The formal details have been revised and superseded, but the Principle of Tolerance remains the book’s enduring provocation: the choice of a language is a decision, and the only test is what follows from it.
