Carnap on Empiricism, Semantics, and Ontology

Abstract Entities

"Carnap on Empiricism, Semantics, and Ontology," by Susan J. Fleck.  May 1993.

Are there really numbers, properties, space-time points, propositions, and even things themselves?  Rudolf Carnap's article "Empiricism, Semantics, and Ontology"  deals with the implications of accepting language which refers to abstract entities.     Empiricists have always been suspicious of abstract entities such as properties and numbers and try to stick to nominalistic language - to not have references to these entities.   For example, if the whole of mathematics were treated as a mere calculus, a formal system where no interpretation can be given, then the mathematician speaks not about numbers, functions and infinite classes, but about meaningless symbols and formulas manipulated according to given formal rules.  In physics, it is much more difficult to eliminate these dreaded entities.  The empiricist may try to declare parts of the language of physics as uninterpreted and uninterpretable - such as that which refers to real numbers as space-time coordinates or as values of physical magnitudes, to functions, limits, etc.  Within the field of semantics, the theory of meaning and truth in language forms, some declare that certain expressions designate certain entities, some of which are concrete material things, and some of which are abstract entities.  Others object, claiming that this designation violates the basic principle of empiricism and leads back to a metaphysical ontology of the platonic kind.   Carnap  sets out to show that using such language does not imply embracing  a Platonic ontology, but rather that it is compatible with empiricism and strictly scientific thinking.

Framework of Expressions

He explains how each new / different kind of entity is represented in our language by the introduction of a framework of new forms of expressions to be used according to a new set of rules.  Some names of particular entities may already occur in the language before the introduction of the new framework (e.g. 'ten' as in "I have ten fingers").  A general term, a predicate of a higher level, is introduced into the framework, so we can say of any particular entity that it belongs to this kind (e.g. 'number' as in  "Ten is a number").   New expressions and rules are introduced, such as for properties of numbers like 'odd' 'prime'; relations like 'greater than; and functions like 'plus';  leading to sentence forms like "two plus three is five".  Then, variables of the new type are introduced.  The new entities are values of these variables; the constants are substitutable for the variables.  With the help of the variables, general sentences about the new entities can be formulated (e.g. "For every n . . . " or "there is an n such that . . . ").     With these new forms in the language, you can now devise empirical or logical questions and possible answers to them.  A true answer is either factually true, using empirical methods of analysis, or analytic, using logical methods.  These types Carnap calls internal questions.  E.g. "Is there a prime number greater than a hundred?" - the answer is derived by logical analysis based on the rules for the new expressions.  The other kind of question about  entities he calls external questions which concern the existence or reality of the system of entities as a whole.  To the question "Are there numbers?", no one would consider answering 'no'.  Rather, it is a question whether numbers have a certain metaphysical characteristic called reality - an ideal reality, different from the material world.

I will briefly summarize this framework again:  A new general term (number) is introduced for the entity which incorporates all the named particulars (five, ten); new expressions and rules about the entities are developed; then with the help of variables, general sentences about the new entities are constructed.  From these new language forms, one can formulate internal empirical or logical questions and possible answers to them.  Above, we discussed the example of the system of numbers.  Because of space, I will not go into further details Carnap discusses about the systems of integers, rational numbers and real numbers (that is a little too heavy for evening philosophy anyway!).  However, let us look at a few different examples.

The World of Things

Take the world of things - the simplest kind of entities we deal with in everyday language.  We accept the thing language with its framework which enables us to ask and answer internal questions like "Is there a white piece of paper on my desk?,"  "Did King Arthur actually live?,"  "Are unicorns and centaurs real or merely imaginary?".  These are answered by empirical investigations, their results evaluated according to certain rules as confirming or disconfirming evidence for possible answers.  "To recognize something as a real thing or event means to succeed in incorporating it into the system of things at a particular space-time position so that it fits together with the other things recognized as real, according to the rules of the framework."  External questions about the reality of the thing world itself is only asked by philosophers.  Realists say 'yes' and subjective idealists say 'no' ... and the controversy goes on.   We could choose to not continue using the ordinary thing language, instead using a language of sense-data or "phenomenal" entities, or to construct another language structure entirely - or we could just refrain from speaking!  However, none of those alternatives are practical, and that is Carnap's point.  He says that the acceptance and use of thing language should not be interpreted as meaning that one believes in the reality of the thing world, but merely as an acceptance of a certain form of language; to accept rules for forming statements and for testing, accepting, or rejecting them.  "But the thesis of the reality of the thing world cannot be among these statements, because it cannot be formulated in the thing language or, it seems, in any other theoretical language."

Propositions

Within the system of propositions, the very term "proposition" is introduced, where any declarative sentence may be substituted for a variable.  Every sentence in the form  ". . . is a proposition," like "Chicago is large is a proposition," is analytic.  The following three constructs are included within this framework:  (a) For every p, either p or not-p;  (b)  There is a p such that p is not necessary and not-p is not necessary;  (c) There is a p such that p is a proposition.  The first two are internal assertions of existence.  The third may mean "There are propositions" - which can be analytic, but if meant in an external sense, it is, according to Carnap, non-cognitive.  Only a few rules for linguistic expressions of the propositional framework are needed;  further explanations are unnecessary because if correct, then they follow from the basic rules.  For example:  "Are propositions mental events  (as in Russell's theory)?"  The answer must be no, otherwise statements would have to be in the form - "If the mental state of the person in question fulfills such and such conditions, then there is a p  such that . . .".  Propositions are not mental entities.  Nor are they subjective, as there is no reference to a subject, like "There is a p which is necessary for Mr. X".

Physics

In our last example, the new entities are space-time points in the spatio-temporal coordinate system for physics.  Each point is an ordered quadruple of four real numbers, called its coordinates, of three spatial and one temporal coordinate.  The physical state of a space-time point is described either with the help of qualitative predicates, such as 'hot', or by ascribing numbers as values of a physical magnitude such as mass, temperature, etc.  The system of things only had extended objects with spatial and temporal relations between them.  To move on to this physical coordinate system is again a matter of decision of how to use language.  The choice of using real numbers instead of rational numbers or integers as coordinates is mainly due to considerations of mathematical simplicity, and is not influenced by facts of experience.  The use of three spatial coordinates is not forced on us, but is strongly suggested by common observation.  If paranormal events, such as a ball moving out of a sealed box as allegedly observed in spiritualistic sSYMBOL 233 \f "Windsor"ances, were to be proved beyond any reasonable doubt, we may choose to change our framework and use four spatial coordinates.   An internal question "Are there (really) space-time points?" - is analytic and trivial.  Again, Carnap feels that the questions of the reality of physical space and physical time are pseudo-questions.

Let us look at different ways of framing this kind of question.  The question  "Shall we introduce such and such forms into our language?"-  This is not a theoretical but a practical question, and a matter of decision.  "Are our experiences such that the use of the linguistic forms in question will be expedient and fruitful?" - this is a theoretical question of a factual, empirical nature.  But, Carnap warns, it concerns a matter of degree, and a formulation in the form "real or not?" would be inadequate.  Many philosophers think the ontological question should be raised and answered before the introduction of the new language forms;  that the language is legitimate only if it can be justified by the 'affirmative' reality of the entity.

Carnap strongly disagrees, and claims that a new way of speaking does  not need any theoretical justification because it does not imply any assertion of reality.  It only means acceptance of the new framework; i.e. of the new linguistic forms.  The acceptance of new linguistic forms cannot be judged as being either true or false, since it is not an assertion, rather it can only be judged as being more or less expedient than with prior forms.  Some nominalists label the admission of variables of abstract types as "Platonism."  Carnap calls this misleading terminology which leads to the conclusion that one who accepts the language of physics with its real number variables would be called Platonistic, even if he is a strict empiricist who rejects Platonic metaphysics.

Designation

Another strong hang-up empiricists have is with the concept of designation.  In a semantical meaning analysis certain expressions in a language are often said to designate (or name or denote or signify or refer to) certain extra-linguistic entities.  Some empiricists object to abstract entities as designata as in these examples:  "The word 'red' designates a property of things.,"  "The word 'five' designates a number.,"  "The word 'odd' designates a property of numbers."  They do not deny that these expressions are meaningful, but they reject the belief, which they regard as implicitly presupposed by those semantical statements:  ". . that to each expression, adjectives like 'red', numerals like 'five', etc., there is a particular real entity to which the expression stands in relation of designation."  They equate this with Platonic realism, and criticize it as a case of hypostatization - (what a word!)  of ascribing material existence to entities, or of treating as names expressions which are not names.  While 'Fido' designates the dog Fido, 'red' and 'five' are not names and do not designate anything.

To answer this criticism, Carnap steps us through an analysis by accepting a framework example  and clarifies abstract entities as designata:  (a) "'Five' designates a number."  This presupposes that our language L contains the forms of expressions which we have called the framework of numbers, in particular, numerical variables and the general term "number".  If  L contains these forms, the following is an analytic statement in L.  (b) "Five is a number."  To make the statement (a) possible, L must contain an expression like "designates" or "is a name of" for the semantical relation of designation.  If suitable rules for this term are laid down, the following is likewise analytic:  (c)  "'Five' designates five."  Since (a) follows from (c) and (b), (a) is likewise analytic.  You must accept these as true statements if you accept the framework of numbers.  Carnap is saying that the question of admissibility of abstract entities as designata is the same as acceptability of the linguistic framework for those entities.

The Problem of Proof

Carnap agrees that we are obliged to justify internal assertions;  to provide evidence, empirical in the case of 'there are elephants or electrons', logical proof in the case of 'there are prime numbers greater than a million'.   But this demand for theoretical justification is wrongly applied to the acceptance of a system of entities.  He cites Ernest Nagel who in 1948 asked for "evidence relevant for affirming with warrant that there are such entities as infinitesimals or propositions."  Nagel characterized the evidence required as "in the broad sense logical and dialectical."  But beyond this, there was no hint as to what might be regarded as relevant evidence.  Carnap provides the following example of this problem of proof:

"Here are three books."  The  framework  in that sentence is commonly used for us to communicate and it is easy to formulate explicit rules for the use of numerical entities.  But controversy of the external question of the ontological reality of the system of numbers continues.  One philosopher thinks numbers are real entities and that gives him the right to use the linguistic forms of the numerical framework and to make semantical statements about numbers as designata of numerals.  A nominalistic opponent says there are no numbers.  For him, the numerals may still be used as meaningful expressions, but they are not names and there are no entities designated by them.  Therefore, for him, the word "number" and numerical variables must not be used - unless there was a way to use them as merely abbreviating devices; i.e. to translate them into the nominalistic thing language.  Carnap thinks there is no possible evidence that would be regarded as relevant by both philosophers to decide this controversy.   Therefore, he regards the external question as a pseudo-question, until both parties offer a common interpretation of the question as a cognitive question; one that would indicate possible evidence regarded as relevant by both sides.

If the nominalists are right, Carnap says that they will have to offer better arguments than merely appealing to ontological insight.  He asks whether they could construct a semantical method which avoids all references to abstract entities and achieves by simpler means the same results as other methods.  He stresses that no theoretical justification is needed for our linguistic frameworks because they do not imply a belief or assertion.  The acceptance or rejection of abstract linguistic forms or of any other linguistic forms in any branch of science will be decided by their efficiency as instruments;  by the ". . ratio of the results achieved to the amount and complexity of the efforts required."  (Or, as we say in business, how much 'bang for the buck'.)  He concludes by pointing out that dogmatic prohibitions are futile and harmful;  that there are historical examples of prohibitions based on prejudices deriving from religious, mythological, metaphysical, or other irrational sources which slowed developments.  By granting freedom to use any form of expression which is useful, the work will sooner or later lead to the elimination of those forms which have no useful function.  "Let us be cautious in making assertions and critical in examining them, but tolerant in permitting linguistic forms."

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