BOOK III - Page 2

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.  The agenda made Logical Syntax obscure and contortionist.  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 principally in constructional systems and less 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 necessary and sufficient 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:

1.     the classifications of the signs in S,

2.     the classifications of the expressions in S, such as “term in S” and   “sentence in S”,

3.     the meaning of “designation in S”, and

4.     the meaning of “true in S.”

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’ is true, 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 Tarski sentence “‘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 for exclusively logical reasons in contrast to factual reasons.  This truth is called L-truth meaning 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, which is 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 in 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 new 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.  And the L-concepts are 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 ofthe 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-language 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 A-truth, which Carnap calls meaning postulates that are known to be true by virtue of the meaning relations among the descrip­tive terms in the sentence.

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 of ranges 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. 

In summary Carnap 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 and (4) rules of ranges.

Semantical Systems: Ontological vs. Linguistic Issues 

Meaning and Necessity has a more specific purpose than the earlier Introduction to Semantics.  The former is the development of a new method of semantical analysis, which Carnap calls the method of extensions and intensions, and which is based on the customary concepts of class and property respectively.  Carnap maintains that these concepts of extension and intension should be substituted for the older idea of naming of an abstract entity.  In his autobiography he notes that some philosophers, who happen to include Quine and Goodman, reject this way of speaking as the “hypostatization of entities.”  In their view it is either meaningless or at least in need of proof, to say that such entities as classes and properties actually exist.  But Carnap argues that such terms have long been used in the language of empirical science and mathematics, and that therefore very strong reasons must be offered, if such terms as “class” and “property” are to be condemned as incompat­ible with empiricism or as unscientific.  He says further­more that to label the use of such terms as “platonistic” or as “platonistic realism”, as is done by Quine and Goodman, is misleading, because these labels neglect the fundamental distinction between, say, physical laws containing real num­ber variables, and ontological theses affirming or denying the reality of universals.  Carnap dislikes the term “onto­logy”, and he maintains that the issue between nominalists and realists regarding universals is a pseudo problem, which is devoid of cognitive content.

Carnap says his method of extension and intension is a superior basis for semantical analysis than an alternative method based on the naming relation, because the latter leads to contradictions, when the names are interchanged with one another in true sentences.  He thus refers to the “antinomy of the name relation”, which is due to the fact that a predicate viewed as a name is ambiguous, since it can refer either to a class or to a property.  Some systems avoid this ambiguity by rejecting properties, and Carnap rejects this loss.  Others avoid the antinomy by having different names for properties and their correspond­ing classes, thus resulting in a higher degree of duplica­tion of expressions.  In Carnap’s method of extension and intension the expressions for properties and for their corresponding classes have the same intension and extension.  Thus both classes and properties are admitted without the inelegant duplication and without the antinomy; only one predicate is needed to speak about both a certain property and about its corresponding class.

The antinomy can be avoided by Carnap’s method of pre­scribing the principle of interchangeability for expressions with the same extension, which is distinctive of extensional contexts.  This prescription is achieved by means of the L-equivalence relation, such that extensions are defined in terms of intensions.  The extension of a given intension is defined as the one L-determinate extension that is equiva­lent to the given intension.  Extensions are thus reduced to intensions.  The result is what Carnap calls a “neutral metalanguage.”  While the metalanguage for an object lan­guage based on the name relation will contain such terms as “the class human” and “the property human”, the neutral metalanguage for an object language based on the method of extension and intension contains only the neutral expression “human.”

In “Meaning and Synonymy in Natural Language” (1955) also reprinted in the appendix to the 1956 edition of Meaning and Necessity Carnap describes how his method of extension and intension is applicable in pragmatics as well as in pure semantics.  “Pragmatic” in Carnap’s lexicon means empirical linguistics.  The purpose of this paper is to give a procedure for determining inten­sion in natural language.  This procedure is problematic, because unlike the construction of an artificial language, in which extension can be defined on the basis of inten­sions, the empirical investigation of an unknown natural language by the field linguist must begin with the identifi­cation of extensions that is not problematic.  On the basis of either spontaneous or elicited utterances of a native speaker of the unknown natural langu­age, the field linguist can ascertain whether or not the native is willing to apply a given predicate to a thing.  By such investigation the linguist determines firstly the ex­tension of the predicate, the class of things to which the native is willing to apply the predicate, secondly the extension of the contradictory class of things to which the native will not apply the predicate, and thirdly the class of things for which the native will neither affirm nor deny the applicability of the predicate.  The size of the third class indicates what Carnap calls the degree of extensional vagueness of the predicate.  Carnap admits that this determination of extension involves uncertainty and possible er­ror, either due to a failure to recognize an individual case or due to a failure to make the correct inductive infer­ence to the intended thing.  But he says that these hazards apply to all concepts in science, and they offer no reason to reject the concepts of the theory of extension.

Carnap’s thesis is that the analysis of intension for natural language is a scientific procedure, which is metho­dologically just as sound as the field linguist’s method of determining extension.  And he notes his disagreement with Quine about this thesis.  Carnap postulates the case in which two linguists agree on the extension of a native’s use of a predicate, but not on the intension.  Carnap maintains that in pragmatics the assignment of an intension is an empirical hypothesis, which like any other hypothesis can be tested by observation of linguistic behavior.  In the empir­ical investigation of the native speaker’s linguistic beha­vior, the linguist looks for what Carnap calls intensional vagueness.  Extensional and intensional vagueness are related such that a decrease in one produces a decrease in the other.  This search is directed to finding out what var­iations of a given specimen are admitted within the range of the predicate, where “range” in the context of a discussion of natural languages means those possible kinds of objects for which the predicate holds.  These are cases for which the native has never made a decision about the applicability of the predicate under investigation.  The description of these possible cases is the intensional vagueness of the predicate.  The linguist can therefore describe to the native speaker various imaginary cases, until he hits upon one that differentiates the otherwise co-extensive predi­cates.  Carnap adds that rules of intension are necessary for the language of empirical science, because without them intensional vagueness would remain, and therefore prevent mutual understanding and communication.

Carnap also elaborates his discussion to include inten­sion for a robot.  He maintains that from a logical point of view the pragmatical concept for a robot is the same as that for a human.  If the internal structure of the robot is not known, however, the same empirical method that is used to determine intension for a human speaker can be used for a robot.  In both cases the intension for a predicate for a speaker is the general condition that an object must satisfy for the speaker to apply the predicate to it.  And if the intensional structure of the robot is known, the intension of a predicate can be known even more completely.

In his “Empiricism, Semantics and Ontology” (1950) also in Meaning And Necessity (1956) Carnap deals further with the problem of classes and properties, which some philoso­phers such as Quine refer to as abstract “entities”.  Again he notes that in the language of physics it is hardly possible to avoid abstract entities, and that using a language referring to them does not imply embracing platonistic ontology.  He views such language as perfectly compatible both with empir­icism and with strictly scientific thinking.  In this paper he explains further why this compatibility is possible. Firstly he notes that there are two kinds of questions concerning the existence or reality of entities.  One kind is addressed by creating a system of new ways of speaking, which system is subject to new rules in the construction of a linguistic “framework”, i.e., a whole semantical system, for the new entities in question.  This first kind of question pertains to the existence of the entities referenced by the system as a whole, and Carnap calls these “external” questions. 

The other kind of question is appropriately called an “internal” question, since it pertains to the existence of a new kind of entity within the framework.  Internal questions can be resolved by either logi­cal or empirical scientific procedures.  The question of the reality of a kind of entity described by a theoretical term might serve as an ex­ample of an internal question.  The problem of abstract en­tities, however, is an external question, and it is this latter type of question that concerns Carnap in this paper. Carnap maintains that the introduction of a new language framework with its new linguistic forms does not imply any assertion of reality, but rather is merely a new way of speaking.  Therefore, the acceptance of a lin­guistic framework containing terms referring to abstract entities does not amount to the acceptance of platonism, because the new language framework is not a new metaphysical doctrine.  Carnap then invokes his “principle of tolerance”, which he had firstly expressed in his Logical Syntax many years earlier.  The criterion he invokes as a semanticist is not an ontological one, but rather is a pragmatical one.  The relevant criterion is whether abstract linguistic forms of variables are expedient or fruitful for the purposes for which the semantical analysis is designed, such as the clarification or construction of languages for the purpose of communication, and especially for communication in science.

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