Logical Syntax of Language

          When Carnap discovered Gestalt psychology, he reconsidered the phenomenalist constructionalism that he had undertaken in his Aufbau, and concluded that a physicalist language, a “thing language” describing things in ordinary experience, is more suitable as a basis of all scientific concepts.  At about the same time he also learned of Hilbert's metamathematics program.  The influence of Russell had led the Vienna Circle to prefer the logistic program of the foundations of mathematics to Hilbert's formalist approach.  But Carnap was attracted to the idea of a metalanguage, not just for mathematics but for a logic of all science. This was his idea of a “metalogic”, which he developed in his Logical Syntax of Language (1934).  The metalogic is the logical syntax of language viewed as a purely analytic theory of the structure of its expressions.  In his autobiography he reports that the whole theory of language structure and its possible applications in philosophy came to him like a vision during a sleepless night when he was ill in January 1931, and that on the following day he wrote down the idea in a manuscript of forty pages titled Attempt at a Metalogic, which was the first draft of his Logical Syntax.
          One of the central ideas in this book is his distinction between metalanguage and object language.  The former contains no reference to the meanings of linguistic signs occurring in the object language; it refers only to the logical structure of the expressions in the object language.  Carnap says that his chief motivation for developing this syntactical method was to formulate more precisely philosophical problems that have evaded resolution when expressed in traditional manner.  In 1934 he published "On the Character of Philosophical Problems" in the American journal Philosophy of Science, which expounded his treatment of metaphysical issues in the German edition of Logical Syntax published in the same year.  In this work he distinguishes the formal or syntactical perspective from the connotative or material perspective.  He identifies logic as a set of metalinguistic transformation rules, and the logic of the language of science, which is the object language, as one in which logical entailment is a formal transformation rule.  Thus Carnap defines the "content" of a proposition in science as a class of entailments from a synthetic proposition in the science.  Content is thus a purely formal concept, and the difference between the formal and material perspectives is merely a difference between modes of expression. Accordingly philosophical analysis consists of translating statements into the formal mode.  Meaningful state­ments in science can be translated into the formal mode of speech, but the meaningless metaphysical statements cannot be translated into the formal mode.  For this reason he maintained that differences between Positivists and realists disappear, when their respective positions are translated into the formal mode.  Similarly problems in the foundation of physics are also problems in syntax.  For example verification of physical laws is a matter concerning the syntactic deductive coherence between the general law-like propositions and singular propositions called protocol sentences, and the problem of induction is a question of how transformation rules lead from protocol sentences to laws.
          In 1937 Carnap published his English edition of Logical Syntax.  This latter edition contains additional material not in the earlier German edition, and its bibliography includes reference to Quine's "Truth by Convention" published in 1936, in which Quine rejected the idea of analytic truth.  Quine viewed the thesis of analytical truth as the Achilles heel of Carnap’s philosophy of science, its parallel postulate to be replaced with the new Pragmatist philosophy of language.  Logical Syntax is divided into five parts.  The first three set forth two artificial object languages.  Language I is designed to be acceptable to philosophers persuaded of the intuitionist philosophy of mathematics, because it includes no infinities.  Language II is adequate to all classical mathematics including what the intuitionists would not accept, and it includes Language I as a sublanguage.  The fourth part sets forth the general procedures for constructing any artificial language, and is titled "General Syntax.”  Carnap defines general syntax as a system of definitions of syntactical terms.  In general a language is any sort of calculus in the sense of a system of formation and transformation rules concerning expressions, which in turn are defined as finite, ordered series of elements called symbols.  Formation rules determine concatenations of symbolic elements to form expressions, and transformation rules determine what transformations produce valid deductions and proofs.  The interpretation of a language is the method of learning by explicit statements that are translations from an already interpreted language, and therefore can be formally represented and belongs to syntax.  A system of axioms in a calculus may firstly be given, and then interpreted in various ways by translations that establish correlations between the expressions of the language being interpreted and those already interpreted.
          The fifth and concluding part of the book pertains to philosophy and syntax, where philosophy is identified with the logic of science.  The material for the 1934 article in Philosophy of Science was taken from section A of this part.  In section B Carnap considers the logic of science as syntax, stating that the logical analysis of physics is the syntax of the physical language.  The language must have formation rules both for the protocol sentences, which express observations, and for the postulated or "P-primitive" laws, which have the form of universal sentences of implication and equivalence.  The transformation rules of the physical language consist either of only “L-rules”, which are logical rules, or of the L-rules together with “P-rules”, which are empirical rules.  A sentence in physics is tested by deducing consequences using the transformation rules, until finally sentences in the form of protocol sentences are generated.  These deduced protocol sentences are then compared with the protocol sentences that are observation reports, and the former are either confirmed or refuted by the latter.  If a sentence which is an L-consequence of certain P-primitive sentences, contradicts a sentence which has been stated as a protocol sentence, then some change must be made in the system.  But there are no established rules for the kind of change that must or must not be made, nor is it possible to set down any sort of rules as to how new primitive laws are to be established on the basis of actually stated protocol sentences.  There are no rules for induction due to the universality of laws; the laws are hypotheses in relation to protocol sentences.  Furthermore not only general laws, but also singular sentences are formulated as hypotheses, i.e. as P-primitive sentences, which are sentences about unobserved processes from which certain observed processes can be obtained.
          Carnap also treats the topic of scientific criticism, and maintains that there is no complete falsification or confirmation of any hypothesis.  When an increasing number of L-consequences of the hypothesis agree with previously acknowledged protocol sentences, then the hypothesis is increasingly confirmed, but it is never finally confirmed.  He states that it is impossible to test even a single hypothetical sentence, because the test applies not to a single hypothesis but also to a whole system of physics as a system of hypotheses.  In this context Carnap references Duhem and Poincare.  He also says that both P-rules and L-rules including those of mathematics are laid down with the reservation that they may be altered as soon as it seems expedient to do so, and that in this respect P-rules and       L-rules differ only in degree with some more difficult to renounce than others. 
          Carnap’s thesis that logical and descriptive language differs only in degree was proposed by Alfred Tarski.  Carnap explains that if every new protocol sentence introduced into a language is synthetic, then L-valid     (i.e. analytic) sentences differ from synthetic sentences, because such a new protocol sentence can be incompatible only with the P-valid synthetic sentence; it cannot be incompatible with the logical L-valid or analytic sentence.  But then he further goes on to say that in spite of the above fact, it may come about that under the inducement of new protocol sentences the language may be altered to such an extent that the L-valid or analytic sentence is no longer analytic.  He emphasizes in italics that the construction of the physical system is not effected in accordance with fixed rules, but is a product of convention.  These conventions are not arbitrary; they must be tested.  The choice of convention is influenced firstly by practical considerations such as simplicity, expediency, and fruitfulness, and secondly by their compatibility with the total system of hypotheses to which the already recognized protocol sentences belong.  Thus in spite of the subordination of hypotheses to empirical control by means of protocol sentences, hypotheses contain a conventional element, because the system of hypotheses is never "univocally" determined by empirical material however rich it may be.  Carnap never developed this thesis of the empirical underdetermination of a system of hypotheses, and the artifactual theory of language it implies, which was extensively developed by Quine in the 1950's and afterward.  Later Carnap rejected Tarski's thesis that logic and descriptive language differ only in degree, but he always maintained that definitions of L-true sentences are relative to the specific language system under construction.

Semantical Systems: Definitions and Characteristics

          Carnap's mature work in semantics is his Introduction to Semantics (1943).  When he had written his Logical Syntax he had believed that metalogic should deal only with the form of expressions of the object language, and that no reference should be made to the meanings of the signs and expressions.  In the preface to his Introduction to Semantics Carnap states that Tarski was the first to call his attention to the fact that the formal methods of syntax must be supplemented by semantical concepts, and also that these semantical concepts can be defined by means no less exact than those of syntax.  He says that his Introduction to Semantics owes more to Tarski than to any other single influence, although he also notes that he and Tarski are not in complete agreement on the distinction between syntax and semantics, and on the distinction between logical and descriptive signs.  In this new semantical perspective semantical systems were central to his philosophy for the remainder of his life.  It is a concept that is fundamental to his views in philosophy of science, his philosophy of probability, and his philosophy of information theory.
          Following the Pragmatist tradition, to which he had been introduced by Charles W. Morris in the United States, Carnap describes semiotics as the general theory of signs, which is divided into three parts based on the three factors involved in language. These factors are (1) the expression, (2) the designatum, and (3) the speaker.  The part of semiotics that deals with all three of these factors is called pragmatics.  The second part of semiotics, called semantics, abstracts from the speaker, and contains a theory of the meaning of expressions, which leads to the construction of a dictionary for translating the object language into the metalanguage.  Finally the third part of semiotics is called syntax, which abstracts from both the speaker and the designata of the signs, in order to consider only the expressions.  Carnap further distinguishes between descriptive semantics and syntactics on the one hand, and pure semantics and syntactics on the other.  The former are included in pragmatics because they are empirical, while the latter are not because they are analytic.  In pure semantics and syntactics the philosopher lays down definitions for certain concepts in the form of rules, and he studies the analytic consequences of these definitions.  Nearly all of Carnap's work is in pure semantics and pure syntactics, and the terms "semantics" and "syntactics" mean pure semantics and pure syntactics in his texts, unless otherwise noted; Carnap's interest is typically more in constructional systems than in empirical linguistics.
          A semantical system presupposes a syntactical system.  A syntactical system or calculus, denoted K, consists of rules that define syntactical concepts, such as "sentence in K" and "provable in K.”  The smallest unit of syntax in the system is called a "sign.”  Signs are combined into "expressions" according to the formation rules for the calculus.  The most important type of expression is the "sentence.”  Sentences are derivable from other sentences, i.e. are "proved", in accordance with the transformation rules of the calculus.  Transformation rules are also called the system's "logic", and for purposes of illustration Carnap typically utilizes Russell's first-order predicate calculus.  All the rules of the syntactical system are analytical rules, and are expressed in a metalanguage; the defined language system is the object language.
          Carnap defines a semantical system as a system of rules formulated in a metalanguage and referring to an object language, which rules determine a truth condition for every sentence of the language, i.e. a sufficient and necessary condition for each sentence's truth.  The semantical system supplies an interpretation of the sentences of the syntactical system or calculus, because to understand a sentence is the same as to know under what conditions it would be true.  It may be noted that truth conditions are not truth values.  The semantical rules do not determine whether or not a sentence is true; the truth value of the sentence must be determined empirically.  The truth condition need not be satisfied for the semantical rule to state it.  As a set of definitions, a semantical system denoted S must set forth certain things.  It must define:

          These definitions may be enumerations or they may be recursive definitions.  The meanings of expressions that are smaller than sentences are given by statements of designation. For example the rule for designation for predicates may include " 'H' denotes the property human.”  The meanings of sentences are given by statements of truth conditions called Tarski sentences, such as " 'The moon is round', if and only if the moon is round."  The sentence in double quotes is in the metalanguage consisting of English, and the symbol or clause in the single quotes is an expression in the object language.  The truth condition statement could also be " 'The moon is round' is true, if and only if the moon is round", since to assert that a sentence is true with the predicate "is true" is to assert the sentence.  These statements in the metalanguage are called "radical" concepts for the semantical system.
          In the Introduction to Semantics Carnap describes L-semantics, which consists of L-concepts.  In L-semantics an L-term applies whenever the term "true" applies on the basis of merely logical reasons in contrast to factual reasons.  This truth is called L-truth or logical truth.  The L-concepts are the same as those occurring in syntax, and Carnap states that logic is part of semantics even though it may also be dealt with in syntax.  Corresponding to the L-concepts in semantics, there are identical C-concepts in syntax.  The relation between syntax and semantics is such that the sentences of a calculus denoted K are interpreted by the truth conditions stated in the analytic semantical rules of the semantical system, denoted S, provided that S contains all the sentences of K.  However, not all possible interpretations of the calculus K are true interpretations.  A semantical system S is a true interpretation of K, if the C-concepts of K are in agreement with the corresponding radical concepts in S.  Furthermore not all true interpretations of the calculus K are L-true.  The semantical system S is called an L-true interpretation for the calculus K, if the C-concepts in K are in agreement with the L-concepts in S.
          Later in his Meaning and Necessity (1947) Carnap develops a definition of L-truth in terms of his concept of state description.  A state description in a semantical system denoted S, is a class of sentences in S which contains for every atomic sentence either the sentence or its negation but not both.  Such a sentence is called a state description, because it gives a complete description of a possible state of the universe of individuals with respect to all the properties and relations expressed by the predicates of the system.  It thus represents one of Leibniz's possible worlds or Wittgenstein's possible states of affairs.  To say that a sentence holds in a state description means that it would be true if the state description were true, i.e. if all the atomic sentences belong to it were true.  Thus the L-concepts are precisely those that are true in all state des­criptions, because they are true in all possible worlds, even though there is only one factually true state description.
          Carnap further elaborates on L-truth in his "Meaning Postulates" (1952) reprinted in the appendix of the 1956 edition of Meaning and Necessity.  His theory of L-truth and state descriptions initially applied to cases where the logically true statement is true only by virtue of the meanings of the logical terms in the statements, as in "Every x is either P or not P.”  But there are also cases such as "If x is a bachelor, then x is not married", which are true by virtue of the meanings of the descriptive terms.  Meaning postulates are object-langauge sentences introduced into a semantical system, that define the relations among descriptive terms in the sentence in addition to the meanings assigned by rules of designation expressed in the metalanguage.  These meaning postulates are not said to be factually true by virtue of empirical investigation, but are true by a decision of the architect of the semantical system, who uses them as semantical rules.  Carnap then introduces a modification of his concept of state description to include another kind of statement, that is the conjunction of all meaning postulates in the semantical system.  Then he says that a sentence in a given semantical system is L-true, if it is L-implied by this conjunction of meaning postulates.  This expanded notion of L-truth with meaning postulates is Carnap's explication of analyticity, by which is meant statements whose truth is known by reference to either the logical form or to the descriptive terms in the statement.  Later he refers to this expanded idea of L-truth as A-truth.
          Using his concept of state description Carnap defines the concept of ranges: the range of a sentence is the class of all state descriptions in which a sentence holds.  Rules of ranges in turn determine the range of any sentence in the semantical system S.  These rules are semantical rules that determine for every sentence in S, whether or not the sentence holds in a given state description.  By determining the ranges, these rules together with the rules of designation for the component predicates and individual variables give an interpretation for all the sentences in S.  This amounts to saying that to know the meaning of a sentence is to know in which of the possible cases it would be true.  Carnap thus describes a semantical system in terms of four types of semantical rules: (1) rules of formation for sentences, (2) rules of designation for descriptive constants, (3) rules of truth, (4) rules of ranges.