Carnap, Quine, Quantification and Ontology

Carnap, Quine, Quantification and Ontology May 25, 2016 This is a prepublication version of: Carnap, Quine, quantification and ontology in A. Torza (Ed.) Quantifiers, Quantifiers, and Quantifiers:Themes in Logic, Metaphysics, and Language, vol. 373 of Synthese Li- brary. Springer. 2015. 1 Abstract At the time of The Logical Syntax of Language (Syntax), Quine was, in his own words, a disciple of Carnap’s who read this work page by page as it issued from Ina Carnap’s typewriter. The present paper will show that there were serious problems with how Syntax dealt with ontological claims. These problems were especially pronounced when Carnap attempted to deal with higher order quantification. Carnap, at the time, viewed all talk of reference as being part of the misleading material mode of speech, and as such dismissed, rather than addressed, ontological problems. Central to the analysis in the present paper is the concept of an explication, which was seen by both Carnap and Quine as being of great philosophical importance. It will be shown that the concept of explication played a significant role in how each formulated their mature position on ontology. Both these final positions on ontology can also be seen as evolving in reaction to Carnap’s flawed handling of ontological matters at the time of Syntax. Carnap, influenced by Tarski’s work on semantics, comes to believe that the concept of reference can be given an acceptable explication, and that by doing so we can see reference to abstract objects as unobjectionable. Quine, on the other hand, strongly rejected the instrumentalism of Syntax, and sought to give an explication of ontological questions that was language independent. This paper closes with a discussion of each’s understanding of the other’s position. 1 Introduction The purpose of the present paper is to provide a clear understanding of the dispute between Carnap and Quine on matters of ontology.1 I trace the dispute back to an unstable position, on existence assumptions in logic, which Carnap held at the time of The Logical Syntax of Language (Syntax hereafter). This problematic position is especially apparent in Carnap’s understanding of higher order quantification. One of the things Syntax explicitly sought to achieve was to show that philosophical claims tend to be, properly understood, claims about the features of some language. Quine, in reaction to this, from very early on, attempted to formulate ontological questions in a way that was both very clear, and, at the same time, language transcendent. I will show that Quine took himself to be giving an explication of such terms as ontology. His answer, of course, is that we are ontologically committed to all the ineliminable entities in the domain of quantification of our best scientific theories. I show that Carnap’s position in ‘Empiricism, Semantics and Ontology’ (ESO hereafter) is also tied to the notion of explication. For Carnap, to answer questions about the refer1 The present paper, although independent, is something of a sequel to my paper “On the Quinean-Analyticity of Mathematical Proposition” (Lavers (2012)). In my earlier paper I look at how the positions of Carnap and Quine on analyticity are related to their views on explication. In that paper I had to purposefully ignore their debate on ontology in order to focus on their views on analyticity. This paper is meant to do the opposite — analyticity will be considered only where it is necessary to consider it in order to understand their respective positions on ontology. 2 ence of terms for abstract objects requires a two stage explication — one at the level of the object language and one at the level of metalanguage. The case of set theory (or type theory) is of particular importance when it comes to discussing ontology. As Quine maintains, once you have set theory, all abstract objects may then be defined as sets. Carnap thought we have a considerable amount of freedom in explicating the notion of logical truth, and he himself included type theory in this category. Quine views all set theories and type theories as non-logical, because all such theories make arbitrary stipulations about which collections exist. Since he sees higher order logic as quantifying over sets, Quine thought only first-order logic ought to count as logic. Quine’s reasons for opposing higher order quantification are closely related to problems Carnap was facing in Syntax. Carnap, there, chose to accept (non-substitutional) higher order quantification and chose to take an instrumentalist stance towards the logical portion of the language. Quine sees Carnap’s instrumentalism as illegitimate, and rejects higher order quantification because he believes the ontological price is too high. Ultimately, Quine fails to understand the significant changes in Carnap’s views on ontology after Carnap was exposed to Tarskian semantics. Carnap saw semantic meta-languages as making possible the explication of notions such as reference. Carnap took such an explication to show how, without engaging in any metaphysics, it is possible to understand reference to abstract objects. Quine took Carnap’s mature position on ontology to be a minor reformulation of his position from Syntax. While Quine did not understand the role of explication in Carnap’s mature position on ontology, Carnap likewise, failed to understand that Quine was attempting to provide an explication that preserves the core meaning of the term ‘ontology’. Carnap saw his dispute with Quine as being purely terminological, but he was clearly wrong in this. In doing so, Carnap really fails to understand Quine’s goal. That said, a worry that Carnap expressed, although it may not apply to Quine, seems clearly to apply to many that have been influenced by Quine’s views on ontology. In section 2, I discuss the manner in which ontological assumptions are handled in Syntax. I show that they are handled in a way that is clearly circular, and that this problem was not discovered because of certain views Carnap held which prevented him from seeing any questions about abstract ontology as even being well formulated. Section 3 contains a brief discussion of Carnap and Quine on the notion of an explication. Section 4 will discuss Quine’s formulation of a language independent way of addressing ontological questions. Section 5 contains a discussion of Carnap’s position in ESO. Section 6 discusses their positions on the status of set theory as logic. The final two sections (7 and 8) are devoted to each’s understanding of the other’s position on ontology. 2 Carnap, Syntax and the formal mode of speech Carnap claims, in his ‘Intellectual Autobiography’ ((Carnap, 1963b, p. 53)), he and other members of the Vienna Circle had come to reject the Wittgensteinean 3 view that we can say nothing of the logical structure of language. Carnap was influenced by the metamathematical work of Hilbert, Tarski and Gödel, but sought to generalize meta-linguistic investigations beyond those of metamathematics. Ultimately, Carnap hoped his science of logical syntax would make clear which philosophical problems were really questions about the features of a certain language. As a book introducing a new and general method of linguistic analysis, Syntax begins by addressing the question of what logical syntax is. §1 opens with the following lines: By the logical syntax of a language, we mean the formal theory of the linguistic forms of that language—the systematic statement of the formal rules which govern it together with the development of the consequences that follow from these rules. A theory, a rule, a definition, or the like is to be called formal when no reference is made in it either to the meaning [Bedeutung] of the symbols (for example, the words) or to the sense [Sinn] of the expressions (e.g. the sentences), but simply on the kind and serial order of the symbols from which the expression is constructed. (Carnap (1934/1937)) Notice that ‘formal’ here is not used as we would define it today but is defined as being equivalent with ‘not concerning the sense or reference of either individual words or complete sentences’. In fact, throughout Syntax ‘formal’ is identified with being unconcerned with meaning. Carnap, at the time, views talk of meaning as part of the material (inhaltlich) mode of speech and responsible for much philosophical confusion.2 Sentences that mention the meaning of terms are to be translated into the formal mode of speech, which involves eliminating the concept of meaning. At several places in Syntax, Carnap recognizes that, concerning the languages and definitions he is outlining, he makes significant existential assumptions. However, his various strategies for dealing with these ontological worries constitute an unsatisfactory patchwork. In this section, I would like to explore the link between Carnap’s dismissing all questions concerning the meaning of terms, and his various, and not very convincing, attempts to deal with ontological questions. When Carnap defines ‘analyticity’ for Language II in §34d, he does so on the basis of of what he called valuations.3 The set of valuations of a given type is what we would now call the domain of quantification for that type of variable. If x is a variable of type 0, then class of valuations for it are the accented expressions (0, 0′ , 0′′ . . . ). A valuation for a standard first level predicate variable will be any arbitrary collection of zero level expressions. A valuation for a variable that stands for a second level predicate is an arbitrary collection of 2 In a footnote to §56 of Word & Object, Quine writes ‘It was indeed I, if I may reminisce, who in 1934 proposed ‘material mode’ to him as a translation of his German.’ 3 Sections with letters affixed to the numbers were prepared for the original German edition but not included for lack of space. 4 valuations of first level predicate variables, an so on for every type in the language.4 In this way for each type of variable there is a, usually uncountable, intended domain associated with it. Since a valuation for a numerical variables is an accented expression, that is, an actual string of symbols, it can unproblematically be called syntactic. However, for all higher types of variables, a valuation will be a class of valuations of lower type. Carnap was aware that this at least hinted at a platonistic interpretation of higher order quantification: Thus the definition must not be limited to the syntactic properties which are definable in S, but must refer to all syntactic properties whatsoever. But do we not by this means arrive at a Platonic absolutism of ideas, which is non-denumerable and therefore can never be exhausted by definitions, is something that subsists in itself, independent of all construction and definition? ((Carnap, 1934/1937, §34d)) Carnap, of course, denies that the view he is defending is platonistic. The reason he gives is that we can define the set of valuations for some language S, in a stronger syntax language S2 . Of course for this to work properly S2 must be be interpreted in a standard way and so only pushes the problem back a step. This was pointed out to Carnap, much later, in Beth (1963) to which Carnap agreed (Carnap (1963a)).5 At the time of Syntax, however, Carnap simply points to the fact that analyticity for S is defined in some distinct language as all that is required to avoid any platonistic commitments. I take it few philosophers today would view this ‘but the definition can be given in another language’ point to successfully eliminate the worry that too strong existence assumptions are being made. This is especially true since existential assumption at least as strong have to be made concerning the domain of quantification for S2 . We have just seen that Carnap’s definition of ‘analyticity’ for Language II involves quantification over uncountable totalities. Despite interpreting his quantifiers as ranging over such uncountable collections, Carnap believed his languages to be quite innocent of ontological problems. §38a is devoted exclusively to addressing the problem of existence assumptions in logic. If logic is to be independent of empirical knowledge, then it must assume nothing concerning the existence of objects. For this reason Wittgenstein rejected the Axiom of Infinity, which asserts the existence of an infinite number of objects. And, for kindred reasons, Russell himself did not include this axiom amongst the primitive sentences of his logic. ((Carnap, 1934/1937, §38a, original italics)) He begins the section by exploring how one would construct a logical system that makes no existence assumptions. He then notes that his Language I and 4 For simplicity I am avoiding discussing relations and functions. the 1963 publication date, most of the material for the Schilpp volume on Carnap was written in the mid-fifties. This is still, of course, much later than Syntax. 5 Despite 5 II are not such systems, but appeals to the distinction between co-ordinate languages and name languages to dismiss ontological worries. Name languages pick out elements of their domain by name, whereas co-ordinate languages pick out elements of the domain in a systematic way by using numbers. I have argued elsewhere (Lavers (2004) and Lavers (forthcoming)) that this does nothing to address the ontological problems that he seems to be worried about, and will not go into significant detail about this here. I will only point out that the domain of quantification could be identical between a name language and a coordinate language.6 As such it is unclear what this distinction can do to ease the concerns of those who have serious worries about ontology. From our perspective, it may seem that the Carnap of Syntax, when faced with ontological worries, would simply restate the principle of tolerance: It is not our business to set up prohibitions, but to arrive at conventions. . . In logic there are no morals. Everyone is at liberty to build up his own logic, i.e. his own language, as he wishes. All that is required of him is that, if he wishes to discuss it, he must state his methods clearly, and give syntactical rules instead of philosophical arguments. ((Carnap, 1934/1937, §17)) It can’t both be true that logic must assume nothing concerning the existence of objects, and that in logic there are no morals. That is, if we are free to outline any language we wish, then we ought to be able to outline languages with all kinds of statements of the form ‘(∃ x ) Px’ among its theorems. Carnap, however, never makes this straightforward move to defuse ontological concerns. The closest he comes is at the end of §38a where he is considering the ontological implications of accepting the axiom of choice. However, as we see, he does not merely restate the principle of tolerance as a justification for existence assumptions in logic, but asserts that the mathematical portion of the language is a mere tool for the purpose of making correct descriptive claims: The Sl [logical sentences] (and with them all sentences of mathematics) are, from the point of view of material interpretation, expedients for the purpose of operating with Sd [Descriptive sentences]. Thus, in laying down an Sl as a primitive sentence, only usefulness for this purpose is to be taken into consideration. ((Carnap, 1934/1937, §38a)) Here Carnap’s answer is that, sure, the axiom of choice makes certain existential assumptions, but we are unconcerned with these since we need only care about the material interpretation of descriptive sentences. But adopting this instrumentalist view concerning existence assumptions of the logicomathematical portion of the language is to dismiss rather than address (or dissolve) ontological concerns. 6 Since what characterizes a coordinate language is that elements of the domain are picked out in some systematic way, there is no reason why we would be limited to countable domains. We might pick out the objects in some domain systematically using real numbers or ordinals for instance. 6 When Carnap pays some attention to the ontological commitments of his languages (or definitions), as we just saw, he employs three different strategies. In the definition of analyticity for Language II, faced with quantification over uncountable totalities, Carnap simply states that the definition can be given in a distinct language. It is hard to see anything very satisfying in this response. Later, in §38a, he appeals to the distinction between name languages and coordinate languages to address the ontological commitments of his language systems. Here again, his response is unsuccessful, this distinction does nothing to address the worry, since the domain of entities may be identical between a name language and a co-ordinate language. Finally we saw that he dismisses ontological concerns because we need not worry about the interpretation of the logical portion of the language. We can treat it as a mere instrument for deriving descriptive claims. The reason we can ignore the interpretation of the logical vocabulary is that all sentences involving only logical expressions are either analytic or contradictory. Indeterminateness comes in only with the descriptive vocabulary. Once the descriptive vocabulary is given a material interpretation, then every sentence will be either true or false. So once we have interpreted the descriptive vocabulary, no further interpretation is required (see §62). But now we see that his instrumentalist stance towards the logical vocabulary depends on his definition of ‘analytic’, and we have seen that there is very little provided in terms of an argument that this definition does not involve serious ontological assumptions (at least in the case of Language II). Recall that his argument here was merely that the definition of ‘analytic in S’ could be given in some distinct language S2 . Presumably, Carnap would take an instrumentalist stance towards the logical vocabulary of S2 , but then we have clearly come full circle. Taken all together then, we can see that in Syntax Carnap has done nothing to address those who are concerned about ontological assumptions in logic. Carnap himself points out that there may be some cause for concern about the ontological assumptions regarding his Languages I and II. His response to these concerns, however, is, as we have just seen, to sweep them under the rug. We can now see this as quite inevitable. The reason he could not deal more satisfactorily with ontological questions is that certain views he held at the time of Syntax, and all of which he abandons shortly afterward, prevented him from viewing any questions about existence assumptions as even well formulated.7 In the remainder of this section I would like to examine these views that were abandoned shortly after Syntax. Let us begin then with Carnap’s view that we must use what he calls syntax languages to explore the logical features of an object language. An object language may have any vocabulary that one wishes, but Carnap held that syn7 Carnap (1942) §39 discusses which theses of Syntax need to be altered in light of of developments in semantics. His general outlook here is that, on the whole, the various theses in Syntax, including discussions of the material mode of speech and of quasi-syntactic sentences “remain valid” but ought to be “supplemented by the corresponding semantical discussions”. This is an unstable position, given the material mode of speech is predicated on the elimination of the notion of reference, and the notion of quasi-syntactic depends on the obsolete notion of a syntax language. 7 tax languages are languages whose sole function is to talk about certain object languages. As such, he thought, their descriptive vocabulary, if they have any at all, should be limited to what is needed to discuss what symbols appear at which places. Carnap realized that he could define truth for logical languages. “If S1 is a logical language, then, with respect to S1 , ‘true’ corresponds to ‘analytic’.” ((Carnap, 1934/1937, §63)) He did not see how it was possible to define truth for descriptive languages, because such a definition would have to be given in a syntax language that contains little or no descriptive vocabulary. Truth then, along with meaning, are relegated to the material mode of speech. We must now look at Carnap’s views concerning the misleading nature of what he calls the material mode of speech. The material mode of speech is characterize as involving talk of meaning or by the use of universal words. To obtain a proper understanding of a sentence, if it is not a straightforwardly empirical claim, we need to translate it into the formal mode of speech. Remember ‘formal’ is taken to mean not concerned with meaning. If we have a sentence involving the concept of meaning, say, to use Carnap’s own example, “Yesterday’s lecture was about Babylon”, we need to translate it into one that does not involve the concept of meaning. In this case, we can translate it as ‘The word ‘Babylon’ or a synonymous expression was used in the previous lecture’. In translating into the formal mode of speech we are also supposed to eliminate universal words. A universal word is a word for a property that holds of all the entities of a certain type (that is universally true for a certain type of variable). Assuming numbers make up a logical type, the statement “five is a number” involves a universal word. It should be translated as “ ‘five’ is a number word”. Here Carnap would call the statement “five is a number” a pseudo object sentence. A pseudo object sentence is defined as a quasi-syntactic sentence of the material mode of speech — where a sentence is quasi-syntactic if it is equivalent to a statement expressible in a syntax language. Carnap describes translatability into the formal mode as the touchstone of meaningfulness for all philosophical sentences. In this section we are concerned with how Carnap addresses (or avoids addressing) ontological questions, especially concerning logical objects. But if the proper understanding of a question is obtained only once our question is formulated in the formal mode, then we see we cannot ask questions about the existence assumptions in logic at all. We certainly can’t ask if “ ‘five’ refers to a number”. This would involve both a universal word and the concept of reference. When translated into the formal mode it would become “ ‘five’ is a numerical expression”. Any hint of ontological assumptions is removed. To consider just one more example, let us look at Carnap’s own translation of the claim that arithmetic involves numbers and numerical properties etc.: 10a. The sentences of arithmetic state (or express) certain properties of numbers and certain relations between numbers. 10b. The statements of arithmetic are composed of numerical expressions and one- or many-termed numerical predicate in such 8 and such a way. ((Carnap, 1934/1937, §75)) Clearly Carnap saw it as an advantage of his system that it did away with ontological questions. My goal in this section was to show how various particularities about Syntax prevented Carnap from being in a position to give a satisfactory answer to questions about the existence assumptions in logic. The pieces are now almost all in place to make this connection. The views discussed in the previous few paragraphs were all abandoned by Carnap in his semantic phase. These include the limitation to syntax languages and the need for translatability into the formal mode of speech. Once Carnap accepts semantic metalanguages, including a full translation of the object language, he realizes that he can’t dismiss certain questions for being quasi-semantic, since all statements would be quasi-semantic. “Jane is over five feet tall” could be translated as “A true sentence results from substituting ‘Jane’ for ‘x’ in the predicate ‘x is over five feet tall’ ”. I would like to point out now that already in Syntax, Carnap realized that for logical languages the property of being quasi-syntactic is trivial. In fact he explicitly says as much concerning logical languages: “in this case, the concept ‘quasi-syntactical’ becomes trivial.” ((Carnap, 1934/1937, §63)) The reason for this, we can now see, is that syntactic metalanguages could contain a full translation of a logical object language. Recall that the restriction on syntactic metalanguages is that they contain no descriptive vocabulary (beyond that needed to say which symbols appear where). They can include all the logical vocabulary one would want. Carnap dismisses such claims as “five is a number” as being a quasi-syntactic sentences of the material mode of speech, and Carnap takes it that properly understood this becomes a question about the features of a language. But this is in exactly the same sense in which, once semantic metalanguages are accepted, any assertion may be seen as making a claim about a language. We saw above that Carnap employs three strategies when dealing with ontological assumptions in logic. The distinction between name and co-ordinate languages is nothing but a red herring. The other two strategies were seen to each support the other (in a clearly circular way). He defends his definition of analyticity by claiming that the ranges for the valuations of various types could be defined in a distinct metalanguage. He then goes on to maintain an instrumentalist reading of the logical sentences of a language. The justification of this is presumably that, if true, they are analytically true. But then each of these last two strategies works only if the other does. Carnap does not seem to be aware of the problems with these various strategies. The reason for this would appear to be that Carnap views all ontological questions about the logical portion of the language as really questions about the features of certain languages. However, logical sentences are quasi-syntactic for exactly the same reason that all sentences become quasi-semantic once the move to semantic metalanguages is made. That is, in a trivial sense that does not succeed in showing that they are really questions about language. The concepts of ‘truth’ and ‘meaning’ were considered to be part of the misleading material mode of speech. We have just seen that Carnap thought 9 translation into the formal mode of speech, which lacks these concepts, was required before we could properly understand what was being claimed in sentences involving these concepts. As is now well known, what Carnap called ‘syntactic’ at the time of Syntax includes much of what we would now call semantics.8 For instance he defines the relation of consequence, analyticity, and synonymy. But there is reason to think that even for the concepts of ‘truth’ and ‘reference’, which are so strongly associated with the material mode of speech, Carnap did not see proper definition as an impossibility: The material mode of speech is not itself erroneous it only readily leads itself to wrong use. But if suitable definitions and rules are laid down and systematically applied, no obscurities or contradictions arise. Since, however, the word-language is too irregular and too complicated to be actually comprehended in a system of rules, one must guard against the dangers of the material mode of speech as it is ordinarily used in the word-language by keeping in mind the peculiar character of its sentences. ((Carnap, 1934/1937, §81, my italics)) Although Carnap abandoned many of the specific theses of Syntax, the above quote is important because it represents a view that remains constant in Carnap’s philosophy. In later years he might express very much the same thought with reference to his concept of explication. He might say: the concepts of ‘truth’, ‘reference’ and even ‘existence’ are, in ordinary language, imprecise to the point of inviting fruitless philosophical disagreements; explications of these notions, on the other hand, may be very fruitful and important. Let us, now, then, turn to the subject of explications. 3 Carnap and Quine on explication Quine ((Quine, 1970/1976, p. 41)) describes himself as “very much a disciple of Carnap’s for six years’. Early in this period (which extends roughly from 1933 to 39) Quine “attended [Carnap’s] lectures and read his Logische Syntax page by page as it issued from Ina Carnap’s typewriter.” ((Quine, 1970/1976, p. 41))9 By 1951 Quine describes Carnap’s influence over him by saying “Though no one has influenced my philosophical thought more than Carnap, an issue has persisted between us for years over the questions of ontology and analyticity.” (Quine (1951/1976)) The remainder of this paper will concern principally their disagreement in the fifties on the subject of ontology. In several works leading up to the early fifties (Carnap (1945), Carnap (1947/1956) and Carnap (1950)) Carnap develops his account of an explication. The concept of explication became a central pillar of Carnap’s thought, but Quine also saw the notion of a Carnapian explication as very important. Very many of Quine’s works, 8 See Creath (1990) for an early argument to this effect. would, then, have included the sections of Syntax prepared for the original German edition but not included for lack of space. 9 This 10 including many of the most important ones, contain a discussion of explication (for example, Word & Object (Quine (1960)), ‘Two Dogmas . . . ’ (Quine (1951/1963)), ‘Epistemology Naturalized’ (Quine (1969a)), and The Web of Belief (Quine & Ullian (1970/1978)) all contain at least some discussion of explication). I have discussed in detail the relationship between Carnap and Quine’s account of explications in Lavers (2012). Here I wish only to outline their views and then demonstrate the relationship between their views on explication and their views on ontology. Carnap’s account of explication begins by rejecting a certain more traditional view of the goal of analysis. On the traditional view the goal of an analysis is to come up with a clear definition of a concept that is identical to the concept under analysis. However, if identity is required, the definition can be no more clear than the notion being analyzed and therefore analysis cannot in principle yield anything fruitful. Once this condition of identity is dropped, we see that in giving an analysis we are introducing a new notion (Carnap calls this the explicatum) in place of the already understood notion (the explicandum). Beginning with the observation that the explicandum and explicatum cannot, on pain of making no progress whatsoever, be required to be identical, Carnap goes on to impose the weakest possible condition on the relationship that must hold between them. The condition is merely that the explicatum is similar enough to the explicandum that it could usefully be used as a replacement. In his Logical Foundations of Probability, Carnap outlines four desiderata of an explication: 1. The explicatum is to be similar to the explicandum in such a way that, in most cases in which the explicandum has been so far used, the explicatum can be used; however, close similarity is not required and considerable differences are permitted. 2. The characterization of the explicatum, that is, the rules of its use (for instance, in the form of a definition), is to be given in an exact form, so as to introduce the explicatum into a well-connected system of scientific concepts. 3. The explicatum is to be a fruitful concept, that is, useful for the formulation of many universal statements (empirical laws in the case of a nonlogical concept, logical theorems in the case of a logical concept). 4. The explicatum should be as simple as possible; this means as simple as the more important requirements (1), (2), and (3) permit. ((Carnap, 1950, §3, original italics)) The important thing to note about these is that it is only the first desideratum that mentions the explicandum, and only the loosest relation is required between the explicandum and explicatum. Quine wholeheartedly agrees with Carnap that we cannot require the concept arrived at after an analysis to be identical with the notion we had prior to an analysis. Carnap spoke of explication as replacing an existing concept with 11 a new one. Quine, in what amounts to the same thing, speaks of eliminating the old troublesome concept in favour of a clear counterpart. A similar view can be taken of every case of explication: explication is elimination. We have, to begin with, an expression or form of expression that is somehow troublesome. It behaves partly like a term but not enough so, or is vague in ways that bother us, or it puts kinks in a theory or encourages one or another confusion. But also it serves certain purposes that are not to be abandoned. Then we find a way of accomplishing those same purposes through other channels, using other less troublesome forms of expression. The old perplexities are resolved. ((Quine, 1960, §53, original italics)) Notice, however, and this is very important, in Quine’s account of explication we are preserving certain features of the explicandum. The above quote is from §53 of Word and Object. This section is given the title ‘The ordered pair as a philosophical paradigm’. Quine’s point is that the various definitions of the ordered pair disagree on many points, and are in fact mutually inconsistent, but what they disagree on can be labeled ‘don’t cares’. More importantly, what they agree on, and what is core to their meaning, can be summed up in the following condition: hx, yi = hw, zi only if x=w and y=z Concerning a proposed set theoretic definition of the ordered pair, Quine states: This construction is paradigmatic of what we are most typically up to when in a philosophical spirit we offer an “analysis” or “explication” of some hitherto inadequately formulated “idea” or expression. We do not claim synonymy. We do not claim to make clear and explicit what users of the language had in mind all along. We do not expose hidden meanings, as the words ‘analysis’ and ‘explication’ would suggest; we supply lacks. We fix on the particular functions of the unclear expression that make it worth troubling about, and then devise a substitute, clear and couched in terms of our liking, that fills those functions beyond those conditions of partial agreement, dictated by our interests and purposes, any traits of the explicans come under the head of “don’t cares”. ((Quine, 1960, §53, my italics)) This is a more traditional account of explication than Carnap’s. For Quine explications begin by identifying what it is about the explicandum that we wish to preserve. Only then do we provide a replacement that preserves these features. That this was not a feature of Carnap’s conception can easily be seen in Carnap’s paper ‘Quine on Analyticity’.10 Carnap wrote this paper in response to Quine’s ‘Two Dogmas . . . ’, but it was unpublished until its inclusion 10 Carnap does talk of explication as a two stage process. We begin by clarification of the explicandum, and then we provide the explicatum. But in the second stage we are in no way bound by what is identified in the first stage. 12 in Quine & Carnap (1990). Quine, in his attacks on Carnap’s definitions of ‘analyticity’, is often looking for what features of the explicandum are preserved by the explicatum. Carnap repeatedly accuses Quine of confusing properties of the explicatum with those of the explicandum. For Carnap, no particular features need to be preserved. The phase of identifying the core meaning of an expression, which then needs to be preserved, is simply absent from Carnap’s account. This difference in their accounts of explication is subtle, and subtle enough that neither of them seemed to notice that they did not share the same view. We will see below that understanding this difference in their views is an important for understanding their respective positions on questions of ontology. 4 Quine and Ontology Quine in his work with Goodman (Goodman & Quine (1947)) famously tried to defend a nominalism about abstract entities. Quine, also famously, eventually came to view the nominalist project as hopeless. In this sense Quine’s views on ontology certainly evolved. However, as to how to address questions of ontology, Quine’s views are remarkably stable. In his ‘A logistical approach to the ontological problems’ (Quine (1939/1976)), Quine wishes to distinguish between terms that genuinely name entities and syncategorematic expressions which do not.11 The key, Quine urges, is to look at what expressions may be replaced with a variable that can then be quantified over.12 “It thus appears suitable to describe names simply as those constant expressions which replace variables and are replaced by variables according to the usual laws of quantification. [. . . ] To be is to be the value of a variable” ((Quine, 1939/1976, 199)) Of course, so far, this distinction between names and syncategorematic expressions will be highly dependent on the specific features of the language with which one is dealing. However, Quine clearly wants to push further than this and arrive at something that is not purely linguistic: Shift of language ordinarily involves a shift of ontology. There is one important sense, however, in which the ontological question transcends linguistic convention: How economical an ontology can 11 Preprints of this paper were made available, and the paper was to be included in volume 9 of Erkenntnis, but the journal ceased publication before volume 9 was produced. 12 Church (1939) is an interesting review of this work of Quine’s. This work hints at the nominalist project, and Church already sees its demise. Church writes “Apparently it is hoped that an adequate formalized language may be devised in which all abstract nouns are syncategorematic, and the tenability of the nominalistic position thereby demonstrated. It would seem, however, that such a demonstration of the tenability of the nominalistic position must be at the same time a demonstration of its extreme artificiality. In the opinion of the reviewer, the effect is only to emphasize the illusory character of the question whether abstract nouns really have designata. For the matter is relative, on the present showing, not only to the choice of a particular language, but also the choice as to which particular notation or notations in the language shall be regarded as denoting the existential quantification (the syntax of the language will ordinarily not determine the latter choice uniquely).” 13 we achieve and still have a language adequate to all the purposes of science? In this form the question of the ontological presuppositions of science survives. ((Quine, 1939/1976, p. 201)) Remember, in Syntax, Carnap classified all questions about what the logical vocabulary referred to as, properly understood, questions about the features of certain languages. Quine is here searching for a way in which ontological questions are not merely questions about the features of a particular language. He believes he has arrived at a language transcendent manner to pose ontological questions. If, for the purpose of an adequate formulation of our scientific theories, we need to quantify over certain kinds of objects, then the claim that such things exist is not a mere feature of a particular language. As mentioned above, the approach to ontological questions first presented in this 1939 article did not change much throughout Quine’s career. We are ontologically committed to all those entities in the domain of quantification of our best scientific theories, where ontological economy is but one of many norms within science. So for Quine, ontological questions, even when they concern logico-mathematical entities, are on par with other questions in science. We can now ask if Quine thought of this as an explication of the term ‘ontology’. Of course 1939 predates Carnap’s earliest discussions of explication by six years. But what of Quine’s remarks about ontology after he was exposed to Carnap’s concept of an explication? There is clear evidence that Quine did consider this to be an explication of the term ‘ontology’: Now my ethics of terminology demand, on occasion, the avoidance of a word for given purposes when the word has been pre-empted by in a prior meaning; meaningless words, however, are precisely the words I feel freest to specify meanings for. But actually my adoption of the word ‘ontology’ for the purpose described is not as arbitrary as I make it sound. Though no champion of traditional metaphysics, I suspect that that the sense in which I use the word has been nuclear to its meaning all along. ((Quine, 1951/1976, pp. 203-4, my italics)) When Quine says that he suspects he has identified the sense that was nuclear to the metaphysicians’ use of the term, given his views on explication, he is stating that he suspects that he has successfully explicated the metaphysicians’ use of the term ‘ontology’.13 Remember, for Quine, giving an explication consists in identifying the core (or nuclear) meaning of an expression — the part of it’s traditional meaning that is clear and useful — and then giving a precise definition that preserves this feature. So in 1951 there seems to be clear evidence that Quine suspects himself to have successfully explicated the term ‘ontology’. In his 1966 paper ‘Existence and Quantification’, Quine begins by discussing the case of singular existence claims such as ‘Socrates exists’. A traditional logical analysis of language might insist that such claims are meaningless because 13 Of course, Quine does not think it worthwhile to go through a detailed study of how metaphysicians have used the term to show that this is in fact the case. 14 it is impossible to assert of an object that it exists. Quine argues that we should regard, ‘(∃ x)(x = Socrates)’ as an explication of what we mean to express when we claim that Socrates exists. He then turns his attention to statement of the form ‘Ps exists’ where P is a predicate. Here we are asking about the role of the existential quantifier in statements of the form (∃ x)Px. Quine holds that there is no unified answer that could serve as an explication of all such cases: We found an explication of “a exists” as “(∃ x)(x = a)”; but explication in turn of the existential quantifier itself, “there is,” “there are,” explication of general existence is a forlorn cause. Further understanding we may still seek even here, but not in the form of explication. We may still ask what counts as evidence for existential quantification. (Quine (1966/1969)) This may seem to conflict with the claim above that Quine saw himself as successfully explicating the term ‘ontology’. In fact, however, there is no conflict at all. The 1966 view is perfectly consistent with his 1951 view that he takes himself to have identified the core meaning of the term ‘ontology’. What our theories say exists can be given a unified explication. It is this that his explication of ontology in terms of our domain of quantification achieves. But which existence claims we should accept is not a matter to be decided by explication. To this question, of course, Quine appeals to his naturalism and holism. 5 Carnap, explication and ESO The discussion above of Carnap’s position in Syntax ended with his claim that there is nothing in principle wrong with the concepts particular to the material mode of speech, so long as they are given clear definition. It is only their use in ordinary language that is so unclear as to lead to philosophical confusion. We saw above that Carnap, at the time of Syntax, could define truth for logical languages. In such cases ‘true’ and ‘analytic’ coincide. But given his selfimposed restriction to syntactic meta-languages, he could not define truth for descriptive languages. In his ‘Intellectual Autobiography’ Carnap recounts the meeting where Tarski first told him of his definition of truth. Carnap says that he assumed Tarski meant logical truth, but was surprised to hear that Tarski meant our ordinary notion of truth, including truth as it applies to contingent factual claims. Carnap immediately challenges Tarski to give the truth conditions for a simple claim like “this table is black”. Of course, Tarski replies “The sentence ‘This table is black’ is true if and only if this table is black.” Carnap continues: In his treatise Tarski developed a general method for constructing exact definitions of truth for deductive language systems, that is, for stating rules which determine for every sentence of such a system a necessary and sufficient condition for its truth. In order to formulate these rules, it is necessary to use a metalanguage which 15 contains the sentences of the object language or translations of them and which, therefore, may contain descriptive constants, e.g., the word “black” in the example mentioned. In this respect, the semantical metalanguages go beyond the limits of syntactical metalanguages. This new metalanguage evoked my strongest interest. I recognized that it provided for the first time the means for precisely explicating many concepts used in our philosophical discussions. (Carnap, 1963b, p. 60-61, my italics) There are a couple things to notice about this quote. First, Carnap clearly identifies the liberalization from syntactic metalanguages to semantic metalanguages as making possible the definition of truth. Secondly, and more important for our purposes, Carnap speaks of explicating many further notions used in philosophical discussions. Carnap clearly sees Tarski’s definition of truth as an explication. In fact, besides Frege’s definition of number, it is Carnap’s most used example of a successful explication.14 But Carnap, thinks that a definition of truth is only one important notion that the liberalization to semantic metalanguages permits. Carnap quickly realized that semantic languages permit the definition of ‘reference’ (or ‘designates’). We simply require of a ‘designate’ predicate that all statements of the form “ ‘a’ designates a ” be provable.15 Of course, talk of reference was the hallmark of the material mode of speech, but now we see Carnap realized even this concept is capable of clear explication. Carnap’s 1939 ‘Foundations of Logic and Mathematics’ shows how quickly Carnap abandoned his Syntax thesis that we ought say nothing concerning the meaning of symbols . In §14 of this work, Carnap defines zero, the successor function, and the property of being a finite cardinal number in the manner that “Frege has shown”.16 In §17 Carnap introduces the Peano axioms with ‘b’, ‘′ ’ and ‘N’ as primitives. He then goes on to say: The customary interpretation of the Peano system may first be formulated this way: ‘b’ designates the cardinal number 0; if ‘. . . ’ designates the cardinal number n then ‘. . .′ ’ designates the next one, i.e., n + 1; ‘N’ designates the class of finite cardinal numbers. Hence on this interpretation the system concerns the progression of finite cardinal numbers ordered according to magnitude. ((Carnap, 1939/1955, 182)) 14 Concerning Frege’s explication of number Carnap writes “Before Frege, nobody was able to give an exact account of the meanings of [arithmetical] words in non-arithmetical terms. By Frege’s explication of the numerical words, which I regard as one of the greatest philosophical achievements of the last century, the logical connection between these words and logical particles like “there is”, “not”, “or”, and “the same as” became completely clear for the first time. Therefore we have to say that in spite of practical skill in usage, people in general, and even mathematicians before Frege, were not completely clear about the meaning of numerical words.” ((Carnap, 1963c, p. 935, my italics)) 15 If the metalanguage does not contain the object language, but contains a translation of the object language, this condition must be adjusted accordingly. 16 There are differences, however, between Carnap and Frege’s definition. Carnap defines the numbers as classes of the second level. It is also worth noting that Carnap is now aware that the definitions depend on a standard interpretation of the higher level quantifiers. 16 In 1934, Carnap dismisses all questions about the reference of terms, but now in 1939 he is happy to talk of the terms of Peano arithmetic designating finite cardinal numbers. The goal of the present section is to talk about the role of explication in ESO, but ESO has not even yet been mentioned. Although it might not seem like it, what has been discussed already is essential for introducing a discussion of ESO. What has been said so far might strike some, however, as having little to do with what transpires in ESO. For instance, the concepts of external questions of linguistic frameworks have not at all been mentioned (until just now). And in fact, I will discuss these concepts as little as possible. It is true, much of the discussion of ESO concerns these concepts. In turn, much of the secondary literature complains that these concepts are ill defined. The concepts of external questions and linguistic frameworks, were used by Carnap as a way of illustrating his mature position on matters of ontology, but they are not necessary for understanding that position. Neither were they used by Carnap outside of ESO, except when discussing the position of that paper. What is central to his mature philosophical position on ontology is his notion of explication and his view that semantic metalanguages can be used to give explications of notions (such as reference) that he previously dismissed. The term explication does not appear even once in ESO, but that does not mean it does not play a very important role in the paper. In fact, I would say that the paper, properly understood, is all about explications. ESO is five years after the first explicit discussion of explication, and from the same year as Carnap (1950) which contains Carnap’s most detailed discussion of explication. Also, 1950 is just prior to when all of the material for the Schilpp volume was prepared, and here it is clear that the notion of explication is central to how he approaches most philosophical problems. By leaving out the concept of explication from one’s understanding of ESO, and focusing on the concepts of external questions and linguistic frameworks, it is hard to see this work fitting in naturally with Carnap’s other writings. I want to claim, in fact, that ESO is concerned with explications from start to finish. For instance when Carnap considers, in ESO, how we introduce the system of numbers, he talks of defining the individual numbers, the general property of being a finite number, etc. Clearly what Carnap has in mind here is a Frege-type definition of number. And, as was mentioned above, Carnap views Frege’s definition of number as an exemplar of explication. Carnap is explicit about this in many places. So by talk of introducing the framework of arithmetic, Carnap clearly has in mind giving an explication of our arithmetical vocabulary — that is, providing a particular systematic treatment of number. When Carnap talks of the system of propositions, he has in mind an account of propositions similar to that given in Meaning & Necessity. He even, in a footnote to the section of ESO dealing with propositions, tries to clarify a point about this previous discussion of propositions from Meaning & Necessity, where Carnap clearly thinks of himself as giving an explication of the concept of proposition. “The greatest difficulty in the task of explicating the concept of proposition is involved in the case of the false proposition.” ((Carnap, 1947/1956, p. 29)) The same could 17 be said of all of the various linguistic frameworks that Carnap discusses. What he has in mind in each case is a formalized language that serves as an explication of a certain range of vocabulary (whether vocabulary concerning things, numbers, propositions, properties, etc.). Most importantly, explication again comes into play with what Carnap himself identifies as the main task of the paper. Carnap says the following in the introductory section: Recently the problem of abstract entities has arisen again in connection with semantics, the theory of meaning and truth. Some semanticists say that certain expressions designate certain entities, and among these designated entities they include not only concrete material things but also abstract entities, e.g., properties designated by predicates and propositions designated by sentences. Others object strongly to this procedure as violating the basic principles of empiricism and leading back to a metaphysical ontology of the Platonic kind. It is the purpose of this article to clarify this controversial issue. ((Carnap, 1947/1956, p. 206, my italics)) Remember, in Syntax, giving an interpretation of a language involves providing a material interpretation of only the descriptive vocabulary. But here he identifies as the central goal of the paper to defend the use of abstract objects as the referents of terms in a semantic theory. This goal, however, is postponed until the final section of the paper (apart form the conclusion). Here the argument is presented with such incredible brevity that it is not surprising that most commentators on ESO do not address it at all. The argument does not even take up the entire section but is contained in only a few lines. The majority of the section contains a tangentially related discussions of Ryle and British empiricists. But let us now look at the argument —filling in the required reasoning. Carnap begins by considering a semantic claim where an abstract object stands as the referent of a term: (a) ‘five’ designates a number. Before we can discuss (a) we need the ‘framework of numbers’ in which both the individual numbers and the general concept of number are defined. Of course, Carnap has in mind here a Fregean definition of number, which Carnap sees as an explication of our arithmetical vocabulary. In such a system it will hold that: (b) five is a number. But this Frege-style explication of our arithmetical vocabulary is not expressive enough for us to yet formulate (a). We need a semantic metalanguage for our language of arithmetic. So we introduce a metalanguage that contains a full translation of the object language. This metalanguage will include explications of our semantic vocabulary as they apply to statements of the object language: 18 Further, to make the statement (a) possible, L [a meta-language for the language of arithmetic] must contain an expression like “designates” or “is a name of” for the semantic relation of designation. If suitable rules are laid down, the following is likewise analytic: (c) ‘five’ designates five. ((Carnap, 1947/1956, p. 217)) Carnap then points out that from (b) — which results from a explication of our arithmetical vocabulary — and (c) — which results from an explication of semantic expressions as they relate to the object language — (a) is a trivial consequence.17 Carnap then goes on to maintain that the same argument applies no matter what we start with as our object language. “Thus the question of the admissibility of entities of a certain type or of abstract entities in general as designata is reduced to the question of the acceptability of the linguistic framework for those entities.” ((Carnap, 1947/1956, p. 217)) Everyone agrees that we can set up logical systems where we can give a Frege-style definition of number.18 Some philosophers might, however, wish to regard such a system as nothing but an empty formalism. Sure, they may say, we could define numerical vocabulary in that way, but we should not see these terms as referring to anything. Carnap’s point is that we can introduce ‘refer’ (or ‘designate’) in the precise sense explicated in the semantic metalanguage, and it will be a theorem of such a formalized language that numerical terms refer. The view that we ought to see such terms as without reference is now seen as unmotivated. No longer is “ ‘five’ refers to a number” hopelessly unclear metaphysics, it is now a theorem of a well defined formalized language.19 In this way, Carnap hopes to help empirically minded philosophers “to overcome their nominalistic scruples.” ((Carnap, 1947/1956, p. 206)) I said I would discuss the concepts of linguistic frameworks and external questions as little as possible. I have already mentioned that what Carnap has in mind, when he talks of linguistic framework, is an explication of a certain range of vocabulary. For example the framework of numbers is an explication of our arithmetical vocabulary and the framework of propositions would consist of an explication of the concept ‘proposition’. Let me now close with a brief remark about what an external question is. We just saw that, relative to an explication of number, numbers exist. And relative, to an explication of the notion of reference for an arithmetical object language, numerical terms refer. One might say 17 Quine, as is well known, makes a lot out of Carnap’s use of ‘analytic’ in the above quote. But Carnap could have equally used the term ‘provable’ here instead of ‘analytic’. 18 Of course, one might say say that because of the need for a standard interpretation of higher order logic, one cannot be sure to have completely unambiguously defined the numbers. But whatever one’s views on higher order quantification, one cannot deny that, at least with impressive clarity, we can define such a system. 19 According to an explication of our arithmetical vocabulary, and an explication of our semantic vocabulary as it applies to our system of arithmetic, numbers exist and numerical terms refer. Carnap does not take this position to amount to platonism. Platonism would involve asserting that numbers exist and numerical terms refer, in an unexplicated sense of ‘exist’ and ‘refers’ (technically, in giving an explication of our arithmetical vocabulary we do not explicate existence, but show the connection between logical notions like existential quantification and our arithmetical vocabulary — see footnote 14). 19 at this point, yes, relative to this newly introduced sense of ‘refer’, numerical terms refer, but is this the correct sense of refer — is there actually an object for which these terms stand? Since Carnap has offered an explication of the term ‘refers’ as it relates to the object language, he would say there is no question of whether the account of reference is correct. An external question then is one that asks, of ‘exists’ or ‘refers’ in some reconstructed system, if they agree with reference and existence in the unreconstructed sense — a sense Carnap saw, in Syntax and right through his semantic period, as being sufficiently unclear as to invite philosophical confusion.20 6 Carnap, Quine, and set theory For Carnap there are no deep mysteries in the philosophy of mathematics. Today, questions about the existence of numbers, or of whether numerical terms refer, are seen by many philosophers as quite mysterious. For Carnap, to answer such questions involves no mystery, but simply a two-stage explication. We begin by giving an explication of our arithmetical vocabulary — in a type theoretic background, for instance. We then explicate our semantic vocabulary relative to this object language. Once this is done ‘Numbers exist’ and ‘Numerical terms refer’ become theorems of the appropriate formalized language. Of course, as mentioned, these explications have to take place in a background theory — be it type theory (with an axiom of infinity) or set theory, or something else.21 What can we say about the status of this background theory? Carnap saw Frege, Hilbert, and Russell and Whitehead, for instance, as all involved in the project of of explicating the notion of logical truth. Carnap himself tended to prefer type theoretic languages, and explicitly states that the notion of L-truth he defines relative to these languages is meant as an explication of the notion of logical truth. So the status of the background theory is that it is itself an explication of our concept of logical truth. The question of whether all of type theory is really part of logic, is a question about the correctness of such an explication, and given Carnap’s conception of an explication, it is not a legitimate question. Notice how much turns on the explication of ‘logic’. Carnap’s account of explication requires only similarity between the explicatum and the explicandum. Type theory, complete with higher order quantification, is certainly similar to what has traditionally been called logic, and so Carnap intends to count it as such. Carnap, therefore, views the project of logicism a having already been successfully carried out by Frege. All that was needed was to import Frege’s work into a consistent background theory. Quine, of course, does not count set 20 Howard Stein briefly makes a similar point about external questions being questions concerning the correctness of an explication (see Stein (1992) p. 280). 21 Carnap discusses the axiom of infinity in §37e of Carnap (1958). Here he says that it can either be taken as a primitive sentence — an axiom, or taken as a rule in the meta-language that makes the assertion of the existence of infinitely many objects L-true. It is clear from here (and from Carnap (1963b) pp 47-48) that Carnap never had a definitive position on the axiom of infinity, but thought that under the proper interpretation it should count as analytic. 20 theory (or type theory) as part of logic. We will turn shortly to the question of why Quine did not see set theory as part of logic. First however, I should say something now about Carnap’s preference for type theoretic languages. In a letter to Quine, Carnap explains his preference for (often many-sorted) typetheoretic languages: I feel somewhat uneasy when entities like Socrates, kindness, & 7 are grouped together as “objects”. Frege did so, and it was his undoing. You can, of course, avoid contradictions by suitable restrictions. But the question is whether the contradictions are not symptoms for a fundamental unsoundness. ((Quine & Carnap, 1990, 1947-4-13)) Interestingly, Quine responds to this this very point in saying: I agree that the logical antinomies are symptoms of a fundamental unsoundness somewhere, but I suspect that this unsoundness lies in platonism itself—i.e., in the admission of abstract values of bindable variables. The contradictions which issue from platonism can indeed be staved off by various artificial devices, and in my view the theory of types is merely one such artificial device. ((Quine & Carnap, 1990, 1947-5-1)) Carnap sees us as skirting inconsistency by grouping too many intuitively distinct kinds of objects into one all encompassing domain. We will see later, that Quine took Carnap’s preference for such languages to be based on his desire to preserve his prejudice against universal words. Quine held this position for many years, even through the 1960s, but this suspicion on Quine’s part is without merit. In Meaning & Necessity Carnap explicitly rejects a prejudice against universal words as unwarranted. Quine’s response, just quoted, to Carnap’s preference for type theories leads nicely into our discussion of why Quine rejected any kind of higher order quantification as part of logic. In the quote above, Quine expresses worries about quantification over abstract entities, and also expresses his belief that type theory is merely an “artificial device”. Quine’s rejection of second-order logic as logic is tied to his views on set theory.22 Quine, in many places in his writing, expresses the same argument against set theory (or type theory). The argument is that we have one intuitive notion of set and that is the notion of set introduced by naı̈ve comprehension. The paradoxes show this notion of set to be inconsistent, and all further developments of set theories or type theories are simply ad hoc devices designed to avoid paradox. That is to say, various set theories and type theory are not an explication of our intuitive notion of set, since they do not preserve the defining feature of our intuitive notion of set (naı̈ve comprehension). Consider for example: But we cannot simply withhold each antinomy-producing membership condition and assume classes corresponding to the rest. The 22 For further discussion of Quine’s views on set theory and higher order logic see Shapiro (1991). 21 trouble is that there are membership conditions corresponding to each of which, by itself, we can innocuously assume a class, and yet these together yield a contradiction. We are driven to seeking optimum consistent combinations of existence assumptions, and consequently there is a great variety of proposals for the foundations of general set theory. Each proposal is unnatural, because the natural scheme is the unrestricted one that the antinomies discredit; and each has advantages, in power and simplicity or in attractive consequences in special directions, that its rivals lack. ((Quine, 1976, p. 16)) In §55 of Word & Object Quine begins by saying that if we have sets, then we have all we could ever need, because any other abstract object could be explicated in set theory. He then goes on to give the same argument that there’s only one natural comprehension principle and many ad hoc ones. But, so far, these are arguments against a set theory (or type theory, since he sees this too as an ad hoc means of avoiding paradox) in general, and not an argument as to why they do not count as logic. In ‘Carnap and Logical Truth’ Quine sketches how the argument concerning the ad hoc nature of set theory can be extended to an argument that set theory is not part of logic: I will not here review the important contrast between logic and set theory, except for the following one. Every truth of elementary logic is obvious (whatever this really means), or can be made so by a series of individually obvious steps. Set theory, in its present state anyway, is otherwise. [. . . N]o consistent set theory is both adequate to the purposes envisioned for set theory and capable of substantiation by steps of obvious reasoning from obviously true principles. What we do is develop one or another set theory by obvious reasoning, or elementary logic, from unobvious first principles which are set down, whether for good or the time being, by something very like convention. ((Quine, 1963, p. 388)) So here we get one answer as to why set theory might not count as logic. Quine takes it as a feature of our intuitive notion of logic that it must involve reasoning by obvious steps from obvious (in some sense) first principles, and then shows that, whatever we mean by obvious, set theory fails this test. Of course, Quine is not putting forward, as a serious theory, that logic proceeds from obvious steps from obvious first principles. His main aim is to show that Carnap’s ‘linguistic doctrine’ of logical truth is no more an explanation of how we know logical truths than the view that logic is obvious. For this reason Quine does not go into detail about what he means by ‘obvious’. But despite the not fully worked out nature of the account, this argument does give us insight into why Quine thought set theory was not logic. Set theory is not logic because it proceeds from non-obvious (arbitrarily stipulated) conventions. But given that these reasons for not including set theory as logic are based on a sketch of a criterion, which Carnap points out (Carnap (1963d)), as it stands, does not 22 even rule out ‘I have five fingers on my hand’ as a logical truth, it can hardly be seen as a definitive argument. There is another argument, in his Philosophy of Logic, for why set theory (and higher order logic) are not properly parts of logic. Here Quine defines logical truth as a truth such that sentences with the same grammatical structure is also true. That is to say a true sentence is a logical truth if truth is preserved over any substitution on its atomic components. Quine shows, for first order languages, assuming the language is expressive enough, this definition coincides with other definitions of logical truth such as being true in all models. He then argues that because set theoretic truths and truths of higher order logic cannot be captured substitutionally, they ought not be considered logical truths. Higher order quantifiers must be seen as either quantifying over attributes (intensions) or over sets (extensions). Quine clearly sees ontological economy as a norm for logic. Logic should make minimal ontological demands even at the level of metatheory. It is for this reason that he proposes to capture logical truth substitutionally instead of talking about models. It is also for this reason that he rejects the ‘staggering existential assumptions’ of set theory and higher order quantification. In this work, Quine is dealing with the same issues that Carnap faced in Syntax. There Carnap thought logic should make minimal existence assumptions, and had originally wanted to define higher order logic substitutionally. Gödel, however, showed him that it would not work. Carnap’s answer was to accept higher order quantification as quantification over uncountably many arbitrary sets, but to, at the same time, take an instrumentalist stance toward these existence claims. We saw that there are serious problems with the way ontological claims are dealt with in Syntax, and we also saw how Carnap’s position on these matters changed in response to the development of semantics. Quine, as we will see in the next section, continued to see Carnap as holding a version of the Syntax position on existential assumption in logic. It is for this reason that Quine sees Carnap as helping himself to existence assumptions without being willing to pay the ontological price. Quine’s substitutional understanding of logical truth did not become standard, but his view that set theory (or type theory) is not part of logic did become standard (and largely due to his influence). Up until the fifties most systems of logic did assume sets, extensions or other similar notions. So an explication of logical truth that includes such a notion is not a break from historical precedent. This is not to argue for a return to the view that set theory is logic, but merely to demonstrate that, at the time, it would not have seemed as unnatural as it does today to claim that logic includes set theory or type theory. 7 Quine’s understanding of Carnap on ontology If we were to describe Carnap’s Syntax period views on questions of ontology in just two principles, one would be the necessity for translation into the formal mode of speech, the other would be the need to take only an instrumentalist 23 stance towards the logical vocabulary. The translation into the formal mode of speech involved the elimination of universal words. The instrumental stance toward the logical portion of the language was supported by Carnap’s position that all logical sentences are either analytic or contradictory and so not in need of material interpretation. When Quine discusses Carnap’s mature views on ontology he sees them as a mere minor reformulation of his earlier views. Consider, for example his discussion in ‘Ontological Relativity’. “In his later writing this doctrine of universal words takes the form of a distinction between internal and external questions, in which people come to grips with the relative merits of theories.” ((Quine, 1969b, p. 52)) Quine goes on to attack the earlier view by saying that universal words are identified by their meaning (‘number’ is a universal word, but the extensionally equivalent predicate ‘less than seven or greater than five’ is thought to be unproblematic). Given his views on meaning, Quine doubts that such a distinction can be made. He then, without discussing the matter in more detail, proclaims that the ‘internal’ / ‘external’ distinction fares no better. In his ‘Carnap’s views on ontology’, Quine also makes it clear that he sees Carnap’s internal/external question distinction as reformulation of the Syntax position on universal words. But now I want to examine the dichotomy which, as we see, underlies Carnap’s distinction between external and internal, and which I am phrasing as the distinction between category questions and subclass questions. It is evident that the question whether there are numbers will be a category question only with respect to languages which appropriate a separate style of variables for the exclusive purpose of referring to numbers. ((Quine, 1951/1976, 207-208)) To rephrase ‘external questions’ as ‘category questions’ is to assume that what is wrong external questions is their use of universal words words. This quote is from 1951 and the one considered just before was from ‘Ontological Relativity’ which was originally presented in 1968. So Quine believed for at least eighteen years that the position of ESO was a fairly minor modification of the Syntax position on existential assumptions in logic. It is important, however, to note something else about this last quote. We also see here reference to Carnap’s preference for type theoretic languages through the talk of separate styles of variables. Quine continues: [Carnap] is thinking of languages which contain fundamentally segregated styles of variables before any definitional abbreviations; and he is thinking of styles of variables that are sealed off from one another so utterly that it is commonly ungrammatical to use a variable of one style where a variable of another style would be grammatical. A language which exploits this sort of basic compartmentalization of variables is that of Russell’s theory of types. However, I think many of us overstress the theory of types to the neglect of its coeval alternative, Zermelo’s set theory and its descendants. 24 Now, it is true that Carnap did prefer type-theoretic languages. But this attitude of Carnap’s, that distinct kinds of things should be assigned distinct logical types has, of course, nothing to do with his former views concerning universal words. In fact, in response to Quine’s comments on an early draft of Meaning & Necessity, Carnap adds in the published version: It is important to emphasize the point just made that, once you admit certain variables, you are bound to admit the corresponding universal concept. It seems some philosophers (not Quine) overlook this fact; they do not hesitate to admit into the language of science variables of the customary kinds, like sentence variables (‘p’, ‘q’, etc.), numerical variables, perhaps also predicate variables of at least level one, and other kinds; at the same time, however, they feel strong misgivings against words like ‘proposition’, ‘number’, ‘property’ (or ‘class’), ‘function’, etc. because they suspect in these words the dangers of an absolutist metaphysics.23 ((Carnap, 1947/1956, p. 44)) Here, Carnap is clearly agreeing with Quine that we can formulate, for any given type, a universal predicate for that type. That is, Carnap is here stating that his previous position with regards to universal words is untenable — for any type, there is a definable universal predicate for that type and thus no reason to have any prejudice against terms like ‘number’, ‘property’ etc. We have seen that Quine interprets the mature Carnap as trying to maintain some version of his Syntax position against universal words. We began this section by saying that the position in Syntax on ontology had two main components. First is the necessity of translation into the formal mode of speech — including the elimination of universal words. The second is the instumentalist stance towards the logical portion of the language. This, as we saw, was supported by Carnap’s view that because sentences of the logical portion of the language are analytic (or contradictory) no interpretation of this portion of the language is required. When Quine relates the rejection of the concept of analyticity to considerations of ontology, he takes this to block a certain move on Carnap’s part. Quine takes it that Carnap wants to divide existential claims into two groups which Quine calls emprirical and ontological existence claims, in order to then ignore the ontological existence claims on the ground that they are analytic. Consider: The contrast that [Carnap] wants between those ontological statements and empirical existence statements such as ‘there are black swans’ is clinched by the distinction between analytic and synthetic. ((Quine, 1951/1976, p. 210)) or again: 23 That Quine, more than twenty years after the publication of Meaning & Necessity, still took Carnap to be defending a version of his thesis that philosophical confusion results from the use of universal words, is reason to suspect Quine never reread the published version to see how Carnap responded to his comments on the early draft. 25 Carnap [. . . ] has recognized that he is able to preserve a double standard for ontological questions and scientific hypotheses only by assuming an absolute distinction between the analytic and the synthetic; and I need not say again that this is a distinction which I reject. The issue over there being classes seems more a question of convenient conceptual scheme; the issue over there being centaurs, or brick houses on Elm Street, seems more a question of fact. But I have been urging that this difference is only one of degree[.] ((Quine, 1951/1963, pp. 45-46)) Quine understands Carnap as needing the analytic/synthetic distinction in order to make a division in types of existence claims so that he may ultimately dismiss questions about abstract ontology. Again, then, Quine is taking Carnap’s mature position on ontology to be essentially the same as the position in Syntax. In Syntax Carnap has a clear double standard towards existence claims. He recognizes that he is making existential assumptions in the logical portion of the language, but as we saw, employs several strategies to dismiss these assumptions rather than address them. On the other hand the descriptive portion stands in need of a material interpretation. By the time of ESO, Carnap does not need a way to avoid dealing with existential assumptions concerning abstract objects. Given an explication of, for instance, our arithmetical vocabulary and given an explication of our semantic notions relative to that systematic account of number, the statement that numbers exist and that numerical terms refer become theorems of the appropriate formalized languages. It is true Carnap takes claims about abstract objects to be analytic. Of course, Carnap and Quine had very different views on the epistemology of mathematics and the empirical sciences, and analyticity played an important epistemlogical role for Carnap. But the concept of analyticity was not meant to support taking a dismissive stance towards all analytic existence claims. That was a view Carnap held at the time of Syntax, but it was abandoned shortly after. Carnap maintained that to use, for instance, the language of set theory is a practical decision of language choice. Quine interprets this to mean that talk of sets is a mere manner of speaking. Of course Quine did not think that Carnap was entitled to this position if it could not be shown that quantification over sets was eliminable from our best scientific theories. But Carnap did not think talk of sets was a mere manner of speaking. To do so would be to hold that we prove that many sets exist while working in some system of set theory, and also hold that sets do not exist according to the ordinary notion of existence in natural language. But Carnap takes no position on whether sets exist in the ordinary sense of existence, because he takes this notion to be unclear. There is nothing mere about the existence of sets for Carnap. Quine’s arguments, even in the 1960s, against Carnap’s views on ontology are all, in reality, directed toward the position of Syntax. Quine, it seems, never recognized the (double) role of explication in Carnap’s mature views on ontology. This is too bad, since Quine thought of explication as a very useful 26 philosophical/scientific activity. As it stands, Quine thought there was something clearly illegitimate about Carnap’s position on ontology. This is due to his reading the position of Syntax into Carnap’s later works. I am not claiming that had Quine understood the role of explication in Carnap’s later views he would have agreed with them, but I am trying to provide a better understanding of where their true differences lie. 8 Carnap’s understanding of Quine on ontology We have just seen that Quine seemed not to have realized the role played by explication in Carnap’s mature views on ontology. It can also be said that Carnap did not realize the role played by explication in Quine’s views on ontology. Carnap often suggested that his differences with what Quine says about ontology are purely terminological. Carnap accepts Quine’s position that to be is to be the value of a quantified variable, but dislikes the way Quine relates this position to traditional ontological debates over nominalism and realism: I, like many other empiricists, regard the alleged questions and answers occurring in the traditional realism-nominalism controversy, concerning the ontological reality of universals or any other kind of entity, as psuedo-questions and pseudo-statements devoid of any cognitive meaning. I agree, of course, with Quine that the problem of “Nominalism” as he interprets it is a meaningful problem it is the question of whether all of natural science can be expressed in a “nominalistic” language, that is, one containing only individual variables whose values are concrete objects, not classes, properties, and the like. However, I am doubtful whether it is advisable to transfer to this new problem in logic or semantics the label ‘nominalism’ which stems from an old metaphysical problem. ((Carnap, 1947/1956, p. 43)) However, it is not simply the case that Quine is giving new acceptable meanings to terms like ‘nominalism’ or ‘ontology’ from metaphysics. We saw as early as 1939, Quine is seeking a language transcendent way of asking about the existence of an entity. In 1951 he writes, speaking of the word ‘ontology’, “I suspect that that the sense in which I use the word has been nuclear to its meaning all along.” ((Quine, 1951/1976, pp. 204)) Given Quine takes an explication to involve identifying a core use that is to be preserved, this is a clear statement that Quine thought of himself as having explicated the term ‘ontology’. Of course, this talk of identifying the core meaning of a term is absent from Carnap’s account of explication. It is no surprise then Carnap does not understand that Quine is offering what he takes to be an explication of the term ‘ontology’. From the time of Syntax Carnap warns of “the dangers of the material mode of speech as it is ordinarily used in the word-language.” (Carnap, 1934/1937, 27 §81) That is, Carnap takes questions about the existence of objects or the reference of terms, as posed in ordinary language, to be so unclear as to invite philosophical confusion. This position is preserved in his later views. We cannot answer questions of existence and reference before explicating a certain range of vocabulary, and then explicating various semantic notions as they apply to the explication of that vocabulary. It is a basic feature of explications that they are not correct or incorrect. Since the notion of correctness does not apply, there is no further, sufficiently clear question that needs to be addressed according to Carnap. Quine’s goal was to rehabilitate the very question Carnap always dismissed as a psuedo-question. The difference then, between Carnap and Quine, is clearly not merely terminological. Furthermore, to understand how Quine intended to rehabilitate this general question of existence, we need to look again at the difference in their accounts of explication. For Carnap, once we have given our two stage (object language and metalanguage) explication and come to accept “ ‘five’ refers to a number”, there is of course no question of whether this is correct in some further sense. Explications are not to be evaluated in terms of correctness, but in terms of usefulness. On Quine’s view, we begin an explication by identifying the core meaning of a term — it is then a requirement of an explication that it preserve this core meaning. Everything besides this core meaning falls under the heading ‘don’t cares’. The explicandum and exclicatum, of course, are not required to be identical, but they do, for Quine, need to agree on the core meaning. When Quine says he suspects that he has identified the nuclear meaning of the term ‘ontology’, this amount to his claiming that he has identified what any explication of ‘ontology’ ought to preserve. Any explication of what we take to exist must view us as committed to all those entities we ineliminably quantify over in our best scientific theories. This is not just one explication among many, as it would be on Carnap’s account, but a general requirement on any explication of our ontological commitments. Carnap failed to fully understand Quine’s position and took their differences to be terminological. Quine was amazingly ingenious in his attempts to rehabilitate the general question of existence that Carnap dismissed. Quine saw something important preserved by his use of the terms ‘ontology’ and ‘nominalism’. Quine was not trying to identify exactly what metaphysicians meant by these terms, but does think he has identified a core meaning that is useful and preserved by his use of the terms. Consider Quine’s formulation of the problem of nominalism. Can we reformulate all of science in a language that does not involve quantification over abstract entities? Carnap, as we saw, agrees that this is a meaningful question, but sees any connection to the old problem of nominalism as undesirable. Quine is unhappy with language specific answer to existence claims — language A quantifies over abstract objects, but language B does not — and seeks a language independent way of posing ontological questions. The reformulation of the question of nominalism is a case in point. By asking if there is any nominalistic language suitable for the purposes of science, Quine has severed the ties between this problem of nominalisnm and any specific language. 28 Despite all of Quine’s ingenuity in trying to rehabilitate a language transcendent way to address ontological questions and despite Carnap mistakenly taking their differences to be mainly terminological, a certain worry we saw Carnap expressing above is entirely justified. Carnap was worried that by using the existing term ‘nominalism’ for the program Quine describes, many might view an answer to Quine’s question of nominalism as an answer to the traditional question of nominalism. That is, we are likely to draw a stonger conclusion than we are really entitled to. Consider the sentence ‘nominalism is false if quantification over abstract objects is not in principle eliminable from our best (ideal) scientific theories’. We don’t learn something else about the world when we learn that the notion of set is ineliminable from our best scientific theories. Given that Quine has eliminated the old notion of nominalism in favour of a clear counterpart, the sentence we were considering is equivalent to ‘Quantification over abstract objects is not in principle eliminable from our best (ideal) scientific theories if quantification over abstract objects is not in principle eliminable from our best (ideal) scientific theories.’ When Carnap says, as just quoted, “I am doubtful whether it is advisable to transfer to this new problem in logic or semantics the label ‘nominalism’ . . . ”, he is expressing the worry that one might view an answer to Quine’s clearly expressed problem as an answer to the old unclear problem — even if, as Quine thinks, there is something preserved between the two, they are not identical. We must forget about all features of the old notion that are not part of Quine’s explication — after all, remember, explication is elimination. I am not claiming that Quine is under any illusions about this, but certainly many people influenced by Quine take it that we would learn something else about the world if we were to learn that real numbers, for instance, are ineliminable from our best scientific theories. 9 Conclusions One of the goals of Syntax was to show which philosophical questions were really questions about the features of a certain language. All questions about the logical portion of the language were labeled quasi-syntactic, and so all questions about the abstract ontology assumed by the language of science are illposed. By the time of ESO, Carnap thought that an explication of ‘reference’ could be given. It could be shown that, relative to this explication, there was no motivation for the nominalistic scruples held by many empiricists. Carnap did not attempt to show that talk of numbers, sets or propositions was a mere manner of speaking. His goal was to show how we can speak in very clear terms about abstract objects as the referents of terms. Quine understood Carnap as continuing to hold a position on ontology similar to the one at the time of Syntax. Quine wanted to reformulate ontological questions so as to be independent of any particular language. Carnap was worried that some might view an answer to Quine’s reformulation of, for instance, the question of nominalism as an answer to the metaphysical question of nominalism. Quine, held that expli29 cation is elimination, and so was himself unlikely to fall prey to what Carnap was worried about. Carnap accepted that Quine had formulated a problem that is independent of the features of any specific language, but thought that making the connection to the traditional problem of nominalism might lead some to think that something more had been established. 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