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Geometric conventionalism and carnap's principle of tolerance
Geometric Conventionalism and Carnap’s Principle of Tolerance David DeVidi* and Graham Solomon-f Abstract-We discuss in this paper the question of the scope of the principle of tolerance about languages promoted in Carnap’s The Logical Syntax of Language and the nature of the analogy between it and the rudimentary conventionalism purportedly exhibited in the work of Poincare and Hilbert. We take it more or less for granted that Poincare and Hilbert do argue for conventionalism. We begin by sketching Coffa’s historical account, which suggests that tolerance be interpreted as a conventionalism that allows us complete freedom to select whatever language we wish-an interpretation that generalizes the conventionalism promoted by Poincare and Hilbert which allows us complete freedom to select whatever axiom system we wish for geometry. We argue that such an interpretation saddles Carnap with a theory of meaning that has unhappy consequences, a theory we believe he did not hold. We suggest that the principle of linguistic tolerance in fact has a more limited scope; but within that scope the analogy between tolerance and geometric conventionalism is quite tight.
IT IS WIDELY accepted that the conventionalism that emerged from late nineteenth-century debates about the viability of non-Euclidean geometries plays some important role in laying the groundwork for Carnap’s principle of tolerance. Albert0 Coffa, for one, says that principle was ‘developed as a response to a certain conflict between the classical (Aristotelian) conception of science and the appearance of non-Euclidean geometries’;’ the intellectual roots of Carnap’s linguistic tolerance are to be found in the conventionalism debated Michael
in the context of geometry by Russell, Friedman, for another, has remarked
Poincare, Frege and Hilbert. that it is with ‘an exuberant
*Department of Philosophy, University of Waterloo, Waterloo, Ontario, Canada. TDepartment of Philosophy, Wilfrid Laurier University, Waterloo, Ontario , Canada. Received
5 December
1993; in revisedform
7 June 1994.
‘Albert0 Coffa, ‘From Geometry to Tolerance: Sources of Conventionalism in NineteenthCentury Geometry’, in R. Colodny (ed.), From Quarks to Quasars (Pittsburgh: University of Pittsburgh Press, 1986), p. 4. We shall refer to it as Coffu. The analogy between geometrical conventionalism and linguistic tolerance is explored also in Coffa’s The Semantic Tradition from Kant to Curnap (Cambridge: Cambridge University Press, 1991) about which we will comment in note 8. Pergamon
Stud Hist. Phil. Sci.. Vol. 25, No. 5, pp. 773-183, 1994 Copyright @ 1995 Elsevier Science Ltd Printed in Great Britain. All rights reserved 0039-3681/94 $7.00+00.00 173
Studies in History and Philosophy of Science
714
sense
of
liberation
logic itself’.2 Coffa and Friedman
that
Carnap
are pointing
extends
Poincare’s
at least to an analogy
conventionalism between
to
the roles
played by statements of syntax and the axioms of geometry for Poincare and Hilbert. One might well attempt to argue that Carnap was explicitly adapting to logic the conventionalist response to problems arising out of the development of non-Euclidean geometries. After all by the late 1920s and early 1930s there were a variety of formally well-articulated and scientifically applicable alternatives in logic, just as there had been in geometry at the turn of the century. And just as with geometry, there was no generally accepted formal reason for choice among the alternative systems of logic. On the face of it, it would be reasonable to agree that if Poincare and Hilbert had never held their views Carnap would not have come to hold the view that he did. We are going to discuss here the question of the scope of the principle of tolerance about languages promoted in The Logical Syntax of Language’ and the nature of the analogy between it and the rudimentary conventionalism purportedly exhibited in the work of Poincare and Hilbert. We will take it more or less for granted that Poincare and Hilbert do argue for conventionalism. We will begin by sketching Coffa’s historical account (in Cofla), which suggests that tolerance be interpreted as a conventionalism that allows us complete freedom to select whatever language we wish-an interpretation that generalizes the conventionalism promoted by Poincare and Hilbert which allows us complete freedom to select whatever axiom system we wish for geometry. We shall argue that such an interpretation saddles Carnap with a theory of meaning that has unhappy consequences, a theory we believe he did not hold. We will suggest that the principle of linguistic tolerance in fact has a more limited scope; but within that scope the analogy between tolerance and geometric conventionalism is quite tight. As Coffa tells the story, the debates about the nature of geometry between Russell
and Poincare,
and between
Frege and Hilbert,
while different
in detail,
share the same source of disagreement. According to Coffa, all four of the protagonists share a commitment to ‘semantic atomism’, the view that ‘the grammatical units of a sentence S must have meaning before they join their partners in S, if S is to be at all capable of expressing a proposition or in any way conveying information [about the world]’ (Coffa, p. 21). Russell and Frege, however, drew rather different conclusions from this doctrine than did Poincare and Hilbert. ‘Michael Friedman, ‘The Re-evaluation of Logical Positivism’, Journal of Philosophy 88 (1991), 517. Friedman had previously remarked in his Foundations of Space-Time Theories (Princeton: Princeton University Press, 1983), p. 288, that Carnap held a ‘generalized conventionalist position in The Logical Syntax of Language’. 3Rudolf Carnap, The Logical Syntax of Language (London: Routledge and Kegan Paul, 1937). We shall refer to it as Logical Syntax.
Carnap’s Principle of Tolerance
175
Russell and Frege, according to Coffa, took it as obvious that the axioms of geometry express propositions about the world. The thesis of semantic atomism implies, then, that if these axioms express propositions, the primitive terms in these axioms (‘point’, ‘straight line’, etc.) must somehow have acquired meanings (p. 22, pp. 38-39). Now it is not an easy task to explain just how these terms have acquired meanings, and Russell and Frege said some things on the matter that are almost embarrassing to read (pp. 24-26) but they were convinced that an explanation could be given. Also, Coffa argues, if one holds both semantic atomism and the view that the axioms of geometry express propositions, then it is impossible to claim consistently that more than one geometry is ultimately acceptable. If there is a true set of axioms for geometry then any non-equivalent set must be false. This is the basis, Coffa says, for Frege’s insistence that all geometries but one should be ‘counted among the pseudosciences, to the study of which we will attach some slight importance, but only as historical curiosities’ (quoted from Frege in coffu, p. 40). Poincare and Hilbert, according to Coffa, likewise recognized that their commitment to semantic atomism entails that the primitive terms must somehow have acquired meanings if the axioms of geometry express claims about the world. But unlike Russell and Frege, Poincare and Hilbert had learned the hard lesson of the developments in nineteenth-century geometry, namely that ‘there is nothing we can say except what the axioms say, that would in any way circumscribe what the axioms mean .. . there is no particular meaning that geometry attaches to its primitives prior to its construction’ (p. 22): axioms ‘implicitly define’ the primitives. Poincare and Hilbert were not about to join Frege in rejecting all but one of the geometries. And so they drew the conclusion that the axioms of geometry do not express propositions. Coffa sees in that conclusion ‘acknowledgement of a distinction that was destined to play a dominating role in Catnap’s and Wittgenstein’s syntacticist philosophies: the distinction between sentences that genuinely convey information and sentences that, in spite of their misleading syntactic appearances are, in fact, content-free and that function as part of a system through which meaning is assigned to certain words’ (p. 23). That distinction, according to Coffa, can be found in Wittgenstein’s writings circa 1930. We aren’t going to take on the task of assessing Coffa’s interpretation of Wittgenstein, but we do want to draw attention to various features of his interpretation that are useful for understanding his account of Carnap. As is well known, there is in the Tractatus a somewhat mysterious distinction between statements which are sinnlos and those which are unsinnig, between statements which lack sense and those which are nonsense. Both types are such that they cannot be established as true or false by either a priori or empirical means. The difference between the two types lies in the fact that the statements
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Studies in History and Philosophy of Science
which lack sense but are not nonsense are somehow ‘essential to all knowledge’, and this category includes logic. By 1930 the sinnlos but not unsinnig statements were called ‘rules of grammar’, and the category had grown to include, in addition worthiest
to logic, those sentences of attention’
‘that epistemology
has traditionally
treated
as
(Cofsa, p. 43).
It is the rules of grammar which Coffa claims play a role with respect to language similar to that played by axioms in the conventionalist account of geometry. Like axioms, the rules of grammar are neither true nor false. They nevertheless have the ‘capacity to “define” or constitute concepts’. We must endorse some set of rules if we are to operate at all with concepts (p. 44). Finally, almost completing the analogy, sometimes ‘Wittgenstein, argues that these rules are in the nature of conventions, since it is entirely up to us to endorse one or another set of apparently conflicting rules’ (p. 45). However, Coffa argues, Wittgenstein does not maintain this conventionalism in anything like a consistent way (pp. 4748). How does Carnap’s syntacticism compare? Coffa takes the sentences Carnap calls ‘pseudo-object sentences’ or ‘sentences in the material mode of speech’ or ‘quasi-syntactical sentences in the material mode’ to be sentences that may appear to but do not convey any information about the world, like Wittgenstein’s rules of grammar. However, Carnap’s material mode sentences, unlike Wittgensteins’s rules ‘can be seen as perfectly meaningful assertions concerning the features of a natural or artificial language’ (p. 53); that is, they can be either true or false statements of ‘syntax’, or they can be meaningful proposals for the constitution of a new language form. Here, then, Carnap departs from Wittgenstein: material mode sentences are not sinnlos: properly interpreted, they are perfectly capable of expressing propositions-but only about the form of language, not about the world. And, argues Coffa, just as the claim that the axioms of geometry do not convey information about the world can be used to justify the view that there can be more than one acceptable system of geometry, and so too can the claim that there are ‘sentences in the material mode’ be used to justify the view that there can be more than one acceptable system of logic. There are various points at which Coffa’s story can be disputed. One might well argue that he has conflated the views of Poincare and Hilbert,4 and ignored subtle differences between rules and conventions in Wittgenstein. But, that the principle of tolerance is conventionalist to some extent is not to be doubted. Indeed, this is taken for granted in expositions of Logical Syntax aimed at nonspecialists, such as that by J. E. McGuire who remarks that Carnap 4While Coffa focusses on implicit definition in his discussion of the views of Poincart and Hilbert on geometry, a detailed historical study would have to take into account matters which he ignores, such as Poincart’s rejection of the ‘formalism’ that motivated Hilbert. However, since we are concerned with the nature of the analogy between implicit definitions and statements of syntax, we will not go into such historical details here.
Carnap’s Principle of Tolerance
‘invokes
the principle
conventional that
of tolerance,
in character
is deemed
777
here the notion
and allow the construction
preferable’.5
Our concern
that linguistic
structures
are
of any form of language
is with the possible
limits
of the
conventionalist interpretation of the principle of tolerance: just how much freedom does linguistic tolerance allow? In particular, does it allow us freedom to choose the very language with which we communicate? A key point of our discussion has been made previously by Wilfrid Hodges6 point out and elaborated by William Demopoulos. 7 Hodges and Demopoulos that Hilbert uses the primitive terms of geometry in his mathematical practice very much in accordance with our present understanding of the notion of non-logical constant, however much his explicit statements about this practice may seem confused. The distinction between variables and non-logical constants, as Hodges and Demopoulos both emphasize, is often overlooked or insufficiently understood. Non-logical constants are expressions that have determinate meaning but no fixed reference. Such constants have reference only when a structure for the language of the theory in which they occur is specified. Once we have selected a structure for the language of a given theory, the reference of the non-logical constants of the theory is fixed by the conventions governing the pairing of constants with distinguished elements, relations, and operations of the structure. So non-logical constants in formal languages play a role analogous to that of indexicals in natural language. Words such as ‘yesterday’ and ‘today’ are hardly meaningless, in spite of their lacking fixed reference prior to the specification of an occasion of utterance. Similarly, a non-logical constant of a theory is not meaningless, in spite of its lacking a fixed reference prior to the selection of a structure for the language of the theory. Variables are different. While they do admit of a conventionally stipulated reference, this reference is not ‘automatically’ fixed, given a selection of a structure for the language of the theory. Even after the structure has been specified, variables require a separate stipulation to fix their reference. Now, as we have seen, Coffa attributes to PoincarC and Hilbert the view that the axioms of geometry ‘implicitly define’ the primitive terms. Such a view is not implausible if we regard the notion of ‘implicitly defined primitive term’ as essentially our current notion of ‘non-logical constant’, In order to fix the reference of a primitive term in an axiomatized theory, we do not add anything to the meaning of the term. Instead, we select one of a class of structures for the language of the theory-we interpret the axioms within the framework of an ‘5. E. McGuire,
‘Scientific Change:
Perspectives
and Proposals’,
Chapter
4 in Merrilee H. Salmon
et al., Introductionto the Philosophyof Science (Englewood Cliffs: Prentice Hall, 1992), p. 141. ‘Wilfrid Hodges, ‘Truth in a Structure’, Proceedingsof the AristotelianSociety 86 (1985/1986), 135-151. ‘William Demopoulos, ‘Frege, Hilbert, and the Conceptual Structure of Model Theory’, History and Philosophyof Logic (forthcoming).
778 interpreted
Studies in History and Philosophy of Science metalanguage-and
the reference
automatically. So in this sense the axioms primitive terms.
of the primitive
terms
is fixed
say all there is to say about
the
It might be tempting to object to our claims here on the grounds that an implicit definition is true by stipulation, whereas the axioms governing the deductive behaviour of non-logical constants need not be true. But this temptation ought to be resisted, for such an objection would be misconceived in our view. The content of the stipulation that a set of implicit definitions be true amounts to the insistence that we should only consider interpretations of the primitive terms which are consistent with those implicit definitions being true. In other words, of the class of structures of the appropriate similarity type, we should only be concerned with the subclass which are models of the axioms-and so, of course, with cases in which the axioms are true. It must be granted that axioms which implicitly define the primitive terms of geometry are not necessarily true. Hilbert, for example, was well aware that different sets of axioms give different implicit definitions of the geometric terms. He was at some pains to construct, for instance, a non-Archimedean geometry precisely to show that Archimedes’s axiom is not always true, in spite of its indisputable status as part of the implicit definition of the geometric primitives in the other (Archimedean) geometries he considered. If we decide to adopt those Archimedean axioms for some purpose, that is, to stipulate that those definitions must be true, then that is just analogous, as we noted above, to restricting attention to the subclass of the class of algebras of the relevant type which are models. The tempting objection goes wrong by drawing the wrong analogy. Of course, the current understanding of the axiomatic method owes a great deal to Alfred Tarski, much of whose relevant work was done after the publication of Logical Syntax. Indeed, Hodges traces the first fully explicit statement of this current understanding to a 1957 paper by Tarski and Robert Vaught.
Moreover,
given
Carnap’s
insistence
that
‘the logic of science
. . . is
nothing else than the syntax of the language of science’ (Logical Syntax, p. 7), and the fact that the present understanding of the axiomatic method depends so heavily on considerations nowadays called ‘semantical’, it might seem likely that Carnap was confused about these issues. At least it might seem likely that we ought not to be casual in our attribution of our current understanding of the axiomatic method to Carnap in Logical Syntax. We want to show, however, that in Carnap’s explicit discussion of the axiomatic method (Logical Syntax 4 71) his position, despite its syntactic formulation, is not that far removed from our post-Tarskian one. According to Carnap, an axiom system is constructed by taking an existing language and adding to it a system of sentences (the axioms) which consist of (usually logical) symbols already in the language together with a stock of new
779
Carnap’s Principle of Tolerance
symbols:
the primitive symbols of the axiom
mines the meaning
of the primitive
271). ‘The primitive of substitution since the axioms
symbols
for variables determine
symbols
system.
are not [variables]’ do not apply
An axiom
‘by a sort of implicit
system deterdefinition’
(p.
(p. 273) and so the usual rules
to them (pp. 272-273).
However,
the meaning
of the primitive symbols ‘to a certain extent . . . only in relation to one another’, it is often possible to interpret the primitive symbols in several different ways. These interpretations are by means of ‘correlative definitions’, which ‘determine to which . . . concepts [of some other language] the axiomatic primitive symbols are to be equivalent in meaning’ (p. 78). We prove that an axiom system is ‘fulfilled for a certain interpretation, or at least that its fulfilment is not excluded’ (p. 271) by providing a translation of the language of the axiom system into another language (‘usually a language of science which has a potential use’ (p. 273)) where a translation is a map taking the expressions of the first language into expressions of a second which preserves the consequence relation of the first (cf. !.$61). This clearly is not all that far removed from our current understanding of what it is to provide an interpretation of an axiomatic theory. In particular, Carnap is quite explicit about the nature and role of the primitive symbols, as is evident in his insistence, mentioned above, that they are not variables, and his statement that it is normally enough merely to state the correlates of the primitive symbols of the theory, since the translation of the logical constants can be ‘assumed to be established and well-known’ (p. 273). Module the syntactic character of the formulation, Carnap’s primitive symbols are essentially the non-logical constants of current model theory. We don’t mean to suggest that Carnap was completely clear on these matters. However, we think it is important to distinguish evidence that Carnap did not have something recognizably akin to the current view on these matters from evidence that the terminology now in current use had not yet evolved. So, for instance, we do not think it particularly significant that Carnap in describing one of his three methods for putting an axiomatic system into the form of a calculus, the method which uses the notation of his Language II (p. 272) says that each of the primitive symbols of the axioms system is to be represented by a variable, and that these as a group are called the primitive variables. In our view, Carnap’s terminology reflects the limitations of his Language II more than it does his thinking that the primitive variables were merely variables. He rather cryptically indicates that he recognizes a difference between the two when he says that ‘In the material mode of speech: the primitive variables do not express universality, but indeterminateness’. To see what he means by this we need to look briefly at how the primitive variables are to be treated. There are two main points to be noted. First, from the syntactical point of view, ‘In the deductions . . . there is no substitution for the free primitive
780
variables’
Studies in History and Philosophy
(p. 272). Since we cannot
substitute
arbitrary
values
of Science
for Carnap’s
‘free primitive variables’, they lack the characteristic property of what are currently called ‘free variables’. Secondly (and quasi-semantically), Carnap assumes
that we are adding
these formal
axioms
to a formal
language
which
already includes a substantial supply of provable formulae. He defines a real model for the axioms to be a substitution of some other expressions (of the appropriate types) of the language for the primitive variables such that the resulting formulae are provable in the language. This is clearly a syntactical approximation of the modern notion of mapping the non-logical constants into a suitable structure. In a real model, of course, nothing similar is required for the variables properly so-called, that is, for the variables that are not ‘free primitive variables’. We now have enough background to discuss the scope of the analogy between geometric conventionalism and the principle of linguistic tolerance. Carnap famously expressed his principle in a well-known passage of Logical Syntax: ‘In logic there are no morals. Everyone is at liberty to build up his own logic; i.e. his own form of language, as he wishes’ (p. 52). This can be read as the strong thesis that we are free to endorse whatever set of syntactical rules pleases us, and that these syntactical rules constitute or define the primitive terms of the language we speak, as the axioms of geometry constitute or define the primitive terms of geometry. The principle of tolerance on this view entails a general theory of meaning applied to any language whatever: the primitive terms of any given language are constituted or defined by some set of syntactical rules.* But this interpretation of tolerance puts Carnap in a most unhappy position. We have seen that, while the axioms of geometry fix the meaning of the primitive terms of geometry, they are incapable of fixing the reference of such terms. Reference of these terms, as Demopoulos has emphasized, is fixed only relative to the assumption that the reference of the terms of the language in which our background model theory is expressed is fixed. If the points of our geometrical theory are to be sets of some sort, then we are presupposing that the ‘While he does not use our terminology, Demopoulos, in the article cited previously, effectively suggests that this reading of tolerance can be found in Co&. He presumably has in mind Coffa’s remarks like ‘[according to tolerance] we have the same freedom to endorse alternative mathematical or metaphysical systems [i.e., languages] as we have in our choice of geometries’ (p. 54). However, in The Semantic Tradition from Kant to Carnap, Coffa seems to argue that while this reading of tolerance is consistent with what Carnap explicitly says about his principle, Carnap’s practice, which embodies what Coffa calls ‘second level semantic factualism’, implicitly commits him to a different interpretation of a sort closely related to what we will suggest shortly. In our view, though, Carnap was well aware that tolerance did not grant complete freedom. His remarks on axiomatic method and interpretation of languages go hand in hand with his explicit elucidation of tolerance. We have suggested in ‘Tolerance and Metalanguages in Carnap’s Logical Syntax of Language’, Synthese (forthcoming), that the principle of tolerance as it emerged in Logical Syntax was the product in part of explicit reflection on the notion of interpretation required to make sense of the various metalogical investigations of the 1920s and 1930s.
Carnap’s Principle of Tolerance reference
781
of the term ‘set’ in our background
be beer mugs, then we are presupposing
theory is fixed. If the points
are to
that we know to what the term ‘beer
mug’ refers. It follows that if Carnap is proposing either have to admit that the expressions
a general theory of meaning of the background language,
he will whose
meanings are fixed by some axiom analogues, simply have no fixed reference, or else accept that he is on the road to a regress. Fortunately, there is evidence in Logical Syntax that Carnap had no commitment to such a general theory of meaning, so that the principle of tolerance ought to be interpreted more weakly than we have so far suggested. Carnap explicitly claims in Logical Syntax (0 62) that if we know only the syntax of a given language then, while we will be able to answer questions about its syntax, ‘it is not possible to use it as a language of communication, because the interpretation of the language is lacking’ (p. 227). There are, he says, two methods by which we can learn to use a language for communication. First, there is ‘the purely practical method which is employed in the case of quite small children and at the Berlitz school of languages’. Secondly, there is the ‘method of explicit statements’, which is essentially the method outlined above for constructing an interpretation of an axiom system. He says that in Logical Syntax ‘by the interpretation of a language we shall always mean . . . the method of explicit statements’. This should not be surprising because in this sense an interpretation is a mapping (which Carnap calls a translation) of one language into a second, with this mapping described in a third; any and all of these languages can but need not coincide. Thus interpretations of this sort ‘can be formally represented; [so] the construction and examination of interpretations belong to formal syntax’ (p. 227). However, Carnap immediately and explicitly insists that this is only part of the story. For when we construct
an interpretation
same position as when we attempt natural language. He writes that
of this sort in formal to give a syntactical
syntax we are in the
description
of a specific
the construction of a calculus must take place entirely within the domain of formal syntax, although the decision as to whether the calculus fulfils the given condition is not a logical but an historical and empirical one, which lies outside the domain of pure syntax. The same thing holds, analogously, for the relation between two languages designated as translation or interpretation. The ordinary requirement of a translation from the French into the German language is that it be in accordance with sense or meaning-which means simply that it must be in agreement with the historically known habits of speech of French-speaking and German-speaking people. [p. 2281 The message here is obvious, even if unstated. The interpretation which must be supplied in order to use a mere calculus as a language of communication can be supplied in two ways. But the ‘method of explicit statements’ will not transform a calculus into a language of communication unless it maps the
782
Studies in History and Philosophy of Science
translated calculus into a language we can use as a language of communication. That is, in order to avoid a regress, if we are going to transform a calculus into a language of communication we are eventually going to have to translate into a language which already has the extra component which cannot be acquired syntactically (i.e. by the method of explicit statements).9 This language will usually be one we have learned by the ‘purely practical’ method, though Carnap seems to be leaving open the possibility of learning by sufficiently intensive empirical study. Our claim is that, while there is an analogy between axioms of geometry understood as conventions and Carnap’s ‘sentences in the material mode’, this analogy should not be pushed to the conclusion that we have complete freedom in our selection of languages. As we read Carnap, we have complete freedom to construct whatever language we choose. Nonetheless, if we wish to use the newly constructed language as a language of communication, we will need to provide an interpretation in some antecedently understood language. Of course, we may decide to investigate this antecedently understood language and so attempt to recover its rules by constructing a calculus. However, the question of whether the constructed calculus captures the syntax of the language under consideration is an empirical one, and, qua calculus, it lacks interpretation. All this was unstated in Logical Syntax, but, we believe, was stated explicitly by Carnap in reply to a criticism made by Evert Beth. We don’t want to go into the details of Beth’s criticism, but the core of the relevant problem is that there might be more than one interpretation of a metalanguage in which a given object language is described. Carnap’s reply was that Since the metalanguage ML [for describing the syntax of some given language L] serves as a means of communication . I have always presupposed that a fixed interpretation of ML, which is shared by all participants, is given. This interpretation is usually not formulated explicitly; but since ML uses English words, it is assumed that these words are understood in their ordinary senses. The necessity of this presupposition of a common interpreted metalanguage seems to me obvious.1° Warren Goldfarb and Thomas Ricketts argue that Carnap’s reply to Beth is unsatisfactory: ‘Clearly, if the metalanguage is a rich one, and if our understanding of it cannot be exhaustively explicated in terms of rules, deductive procedures in axiomatic systems, or the like, then Carnap’s “presupposition” is an admission that much can never be made explicit, but must simply be tacitly
‘Indeed, right at the start of Logicul Syntax Carnap makes it clear that he does not believe that the syntactical study of language can give us the whole story. There are features of language that must be studied by other methods. ‘For instance, its words have meaning; this is the object of investigation for semasiology’ (p. 5). “Rudolf Carnap, ‘Reply to Beth’, in P. A. Schilpp (ed.), The Philosophy of Rudolf’ Carnup (La Salle: Open Court, 1963) p. 929.
Carnap’s
Principle of Tolerance
relied upon.
This fits poorly
783
with Carnap’s
proclaimed
standards
of exactitude
and rigor.“’ One
can
imagine
Carnap
responding
to this by saying
that
in fact the
presupposition fits perfectly well with his proclaimed standards of rigour. He might begin a response by pointing out that one must be careful not to suppose that the method of explicit statements supplies a theory of meaning even for constructed languages in any but a derivative sense. Meaning for a constructed language comes, ultimately, from the interpretation of that language in a language that already has meaning. Goldfarb and Ricketts, when they talk of Carnap’s ‘proclaimed standards of rigour’, have in mind the standards appropriate to syntax. But, as we have tried to show, Carnap thinks that it is empirical investigation that is appropriate to semantics, and so it will be the standards of rigour appropriate to empirical science which apply to semantics. So it is not the case that the things that must be tacitly relied upon in discussions of syntax are, for Carnap, not subject to rigorous investigation at all. Carnap’s presupposition is not ‘an admission that much can never be made explicit, but must simply be tacitly relied upon’. Granted, ‘much can never be made explicit, but must simply be tacitly relied upon’ from the point of view of strictly syntactical investigations. But, as we see it, Carnap was well aware that syntax, including the method of explicit statements, could not tell the whole story about language. The part that it could not give and must leave tacit was, for Carnap, open to empirical investigation, and so, except from the point of view of syntax, need not be left tacit at all. Acknowledgements-Our thanks to William Demopolous, who is always generous with helpful comments, and to the referees for this journal. David DeVidi also thanks the Social Sciences and Humanities Research Council of Canada for support.
“Warren Goldfarb and Thomas Ricketts, ‘Carnap and the Philosophy of Mathematics’, in D. Bell and W. Vossenkuhl (eds), Wissenschuf und Subjectivitiit (Berlin: Academic Verlag, 1992), p. 72. The point with which we take issue is a secondary one in an article that we otherwise find very persuasive.
IT IS WIDELY accepted that the conventionalism that emerged from late nineteenth-century debates about the viability of non-Euclidean geometries plays some important role in laying the groundwork for Carnap’s principle of tolerance. Albert0 Coffa, for one, says that principle was ‘developed as a response to a certain conflict between the classical (Aristotelian) conception of science and the appearance of non-Euclidean geometries’;’ the intellectual roots of Carnap’s linguistic tolerance are to be found in the conventionalism debated Michael
in the context of geometry by Russell, Friedman, for another, has remarked
Poincare, Frege and Hilbert. that it is with ‘an exuberant
*Department of Philosophy, University of Waterloo, Waterloo, Ontario, Canada. TDepartment of Philosophy, Wilfrid Laurier University, Waterloo, Ontario , Canada. Received
5 December
1993; in revisedform
7 June 1994.
‘Albert0 Coffa, ‘From Geometry to Tolerance: Sources of Conventionalism in NineteenthCentury Geometry’, in R. Colodny (ed.), From Quarks to Quasars (Pittsburgh: University of Pittsburgh Press, 1986), p. 4. We shall refer to it as Coffu. The analogy between geometrical conventionalism and linguistic tolerance is explored also in Coffa’s The Semantic Tradition from Kant to Curnap (Cambridge: Cambridge University Press, 1991) about which we will comment in note 8. Pergamon
Stud Hist. Phil. Sci.. Vol. 25, No. 5, pp. 773-183, 1994 Copyright @ 1995 Elsevier Science Ltd Printed in Great Britain. All rights reserved 0039-3681/94 $7.00+00.00 173
Studies in History and Philosophy of Science
714
sense
of
liberation
logic itself’.2 Coffa and Friedman
that
Carnap
are pointing
extends
Poincare’s
at least to an analogy
conventionalism between
to
the roles
played by statements of syntax and the axioms of geometry for Poincare and Hilbert. One might well attempt to argue that Carnap was explicitly adapting to logic the conventionalist response to problems arising out of the development of non-Euclidean geometries. After all by the late 1920s and early 1930s there were a variety of formally well-articulated and scientifically applicable alternatives in logic, just as there had been in geometry at the turn of the century. And just as with geometry, there was no generally accepted formal reason for choice among the alternative systems of logic. On the face of it, it would be reasonable to agree that if Poincare and Hilbert had never held their views Carnap would not have come to hold the view that he did. We are going to discuss here the question of the scope of the principle of tolerance about languages promoted in The Logical Syntax of Language’ and the nature of the analogy between it and the rudimentary conventionalism purportedly exhibited in the work of Poincare and Hilbert. We will take it more or less for granted that Poincare and Hilbert do argue for conventionalism. We will begin by sketching Coffa’s historical account (in Cofla), which suggests that tolerance be interpreted as a conventionalism that allows us complete freedom to select whatever language we wish-an interpretation that generalizes the conventionalism promoted by Poincare and Hilbert which allows us complete freedom to select whatever axiom system we wish for geometry. We shall argue that such an interpretation saddles Carnap with a theory of meaning that has unhappy consequences, a theory we believe he did not hold. We will suggest that the principle of linguistic tolerance in fact has a more limited scope; but within that scope the analogy between tolerance and geometric conventionalism is quite tight. As Coffa tells the story, the debates about the nature of geometry between Russell
and Poincare,
and between
Frege and Hilbert,
while different
in detail,
share the same source of disagreement. According to Coffa, all four of the protagonists share a commitment to ‘semantic atomism’, the view that ‘the grammatical units of a sentence S must have meaning before they join their partners in S, if S is to be at all capable of expressing a proposition or in any way conveying information [about the world]’ (Coffa, p. 21). Russell and Frege, however, drew rather different conclusions from this doctrine than did Poincare and Hilbert. ‘Michael Friedman, ‘The Re-evaluation of Logical Positivism’, Journal of Philosophy 88 (1991), 517. Friedman had previously remarked in his Foundations of Space-Time Theories (Princeton: Princeton University Press, 1983), p. 288, that Carnap held a ‘generalized conventionalist position in The Logical Syntax of Language’. 3Rudolf Carnap, The Logical Syntax of Language (London: Routledge and Kegan Paul, 1937). We shall refer to it as Logical Syntax.
Carnap’s Principle of Tolerance
175
Russell and Frege, according to Coffa, took it as obvious that the axioms of geometry express propositions about the world. The thesis of semantic atomism implies, then, that if these axioms express propositions, the primitive terms in these axioms (‘point’, ‘straight line’, etc.) must somehow have acquired meanings (p. 22, pp. 38-39). Now it is not an easy task to explain just how these terms have acquired meanings, and Russell and Frege said some things on the matter that are almost embarrassing to read (pp. 24-26) but they were convinced that an explanation could be given. Also, Coffa argues, if one holds both semantic atomism and the view that the axioms of geometry express propositions, then it is impossible to claim consistently that more than one geometry is ultimately acceptable. If there is a true set of axioms for geometry then any non-equivalent set must be false. This is the basis, Coffa says, for Frege’s insistence that all geometries but one should be ‘counted among the pseudosciences, to the study of which we will attach some slight importance, but only as historical curiosities’ (quoted from Frege in coffu, p. 40). Poincare and Hilbert, according to Coffa, likewise recognized that their commitment to semantic atomism entails that the primitive terms must somehow have acquired meanings if the axioms of geometry express claims about the world. But unlike Russell and Frege, Poincare and Hilbert had learned the hard lesson of the developments in nineteenth-century geometry, namely that ‘there is nothing we can say except what the axioms say, that would in any way circumscribe what the axioms mean .. . there is no particular meaning that geometry attaches to its primitives prior to its construction’ (p. 22): axioms ‘implicitly define’ the primitives. Poincare and Hilbert were not about to join Frege in rejecting all but one of the geometries. And so they drew the conclusion that the axioms of geometry do not express propositions. Coffa sees in that conclusion ‘acknowledgement of a distinction that was destined to play a dominating role in Catnap’s and Wittgenstein’s syntacticist philosophies: the distinction between sentences that genuinely convey information and sentences that, in spite of their misleading syntactic appearances are, in fact, content-free and that function as part of a system through which meaning is assigned to certain words’ (p. 23). That distinction, according to Coffa, can be found in Wittgenstein’s writings circa 1930. We aren’t going to take on the task of assessing Coffa’s interpretation of Wittgenstein, but we do want to draw attention to various features of his interpretation that are useful for understanding his account of Carnap. As is well known, there is in the Tractatus a somewhat mysterious distinction between statements which are sinnlos and those which are unsinnig, between statements which lack sense and those which are nonsense. Both types are such that they cannot be established as true or false by either a priori or empirical means. The difference between the two types lies in the fact that the statements
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which lack sense but are not nonsense are somehow ‘essential to all knowledge’, and this category includes logic. By 1930 the sinnlos but not unsinnig statements were called ‘rules of grammar’, and the category had grown to include, in addition worthiest
to logic, those sentences of attention’
‘that epistemology
has traditionally
treated
as
(Cofsa, p. 43).
It is the rules of grammar which Coffa claims play a role with respect to language similar to that played by axioms in the conventionalist account of geometry. Like axioms, the rules of grammar are neither true nor false. They nevertheless have the ‘capacity to “define” or constitute concepts’. We must endorse some set of rules if we are to operate at all with concepts (p. 44). Finally, almost completing the analogy, sometimes ‘Wittgenstein, argues that these rules are in the nature of conventions, since it is entirely up to us to endorse one or another set of apparently conflicting rules’ (p. 45). However, Coffa argues, Wittgenstein does not maintain this conventionalism in anything like a consistent way (pp. 4748). How does Carnap’s syntacticism compare? Coffa takes the sentences Carnap calls ‘pseudo-object sentences’ or ‘sentences in the material mode of speech’ or ‘quasi-syntactical sentences in the material mode’ to be sentences that may appear to but do not convey any information about the world, like Wittgenstein’s rules of grammar. However, Carnap’s material mode sentences, unlike Wittgensteins’s rules ‘can be seen as perfectly meaningful assertions concerning the features of a natural or artificial language’ (p. 53); that is, they can be either true or false statements of ‘syntax’, or they can be meaningful proposals for the constitution of a new language form. Here, then, Carnap departs from Wittgenstein: material mode sentences are not sinnlos: properly interpreted, they are perfectly capable of expressing propositions-but only about the form of language, not about the world. And, argues Coffa, just as the claim that the axioms of geometry do not convey information about the world can be used to justify the view that there can be more than one acceptable system of geometry, and so too can the claim that there are ‘sentences in the material mode’ be used to justify the view that there can be more than one acceptable system of logic. There are various points at which Coffa’s story can be disputed. One might well argue that he has conflated the views of Poincare and Hilbert,4 and ignored subtle differences between rules and conventions in Wittgenstein. But, that the principle of tolerance is conventionalist to some extent is not to be doubted. Indeed, this is taken for granted in expositions of Logical Syntax aimed at nonspecialists, such as that by J. E. McGuire who remarks that Carnap 4While Coffa focusses on implicit definition in his discussion of the views of Poincart and Hilbert on geometry, a detailed historical study would have to take into account matters which he ignores, such as Poincart’s rejection of the ‘formalism’ that motivated Hilbert. However, since we are concerned with the nature of the analogy between implicit definitions and statements of syntax, we will not go into such historical details here.
Carnap’s Principle of Tolerance
‘invokes
the principle
conventional that
of tolerance,
in character
is deemed
777
here the notion
and allow the construction
preferable’.5
Our concern
that linguistic
structures
are
of any form of language
is with the possible
limits
of the
conventionalist interpretation of the principle of tolerance: just how much freedom does linguistic tolerance allow? In particular, does it allow us freedom to choose the very language with which we communicate? A key point of our discussion has been made previously by Wilfrid Hodges6 point out and elaborated by William Demopoulos. 7 Hodges and Demopoulos that Hilbert uses the primitive terms of geometry in his mathematical practice very much in accordance with our present understanding of the notion of non-logical constant, however much his explicit statements about this practice may seem confused. The distinction between variables and non-logical constants, as Hodges and Demopoulos both emphasize, is often overlooked or insufficiently understood. Non-logical constants are expressions that have determinate meaning but no fixed reference. Such constants have reference only when a structure for the language of the theory in which they occur is specified. Once we have selected a structure for the language of a given theory, the reference of the non-logical constants of the theory is fixed by the conventions governing the pairing of constants with distinguished elements, relations, and operations of the structure. So non-logical constants in formal languages play a role analogous to that of indexicals in natural language. Words such as ‘yesterday’ and ‘today’ are hardly meaningless, in spite of their lacking fixed reference prior to the specification of an occasion of utterance. Similarly, a non-logical constant of a theory is not meaningless, in spite of its lacking a fixed reference prior to the selection of a structure for the language of the theory. Variables are different. While they do admit of a conventionally stipulated reference, this reference is not ‘automatically’ fixed, given a selection of a structure for the language of the theory. Even after the structure has been specified, variables require a separate stipulation to fix their reference. Now, as we have seen, Coffa attributes to PoincarC and Hilbert the view that the axioms of geometry ‘implicitly define’ the primitive terms. Such a view is not implausible if we regard the notion of ‘implicitly defined primitive term’ as essentially our current notion of ‘non-logical constant’, In order to fix the reference of a primitive term in an axiomatized theory, we do not add anything to the meaning of the term. Instead, we select one of a class of structures for the language of the theory-we interpret the axioms within the framework of an ‘5. E. McGuire,
‘Scientific Change:
Perspectives
and Proposals’,
Chapter
4 in Merrilee H. Salmon
et al., Introductionto the Philosophyof Science (Englewood Cliffs: Prentice Hall, 1992), p. 141. ‘Wilfrid Hodges, ‘Truth in a Structure’, Proceedingsof the AristotelianSociety 86 (1985/1986), 135-151. ‘William Demopoulos, ‘Frege, Hilbert, and the Conceptual Structure of Model Theory’, History and Philosophyof Logic (forthcoming).
778 interpreted
Studies in History and Philosophy of Science metalanguage-and
the reference
automatically. So in this sense the axioms primitive terms.
of the primitive
terms
is fixed
say all there is to say about
the
It might be tempting to object to our claims here on the grounds that an implicit definition is true by stipulation, whereas the axioms governing the deductive behaviour of non-logical constants need not be true. But this temptation ought to be resisted, for such an objection would be misconceived in our view. The content of the stipulation that a set of implicit definitions be true amounts to the insistence that we should only consider interpretations of the primitive terms which are consistent with those implicit definitions being true. In other words, of the class of structures of the appropriate similarity type, we should only be concerned with the subclass which are models of the axioms-and so, of course, with cases in which the axioms are true. It must be granted that axioms which implicitly define the primitive terms of geometry are not necessarily true. Hilbert, for example, was well aware that different sets of axioms give different implicit definitions of the geometric terms. He was at some pains to construct, for instance, a non-Archimedean geometry precisely to show that Archimedes’s axiom is not always true, in spite of its indisputable status as part of the implicit definition of the geometric primitives in the other (Archimedean) geometries he considered. If we decide to adopt those Archimedean axioms for some purpose, that is, to stipulate that those definitions must be true, then that is just analogous, as we noted above, to restricting attention to the subclass of the class of algebras of the relevant type which are models. The tempting objection goes wrong by drawing the wrong analogy. Of course, the current understanding of the axiomatic method owes a great deal to Alfred Tarski, much of whose relevant work was done after the publication of Logical Syntax. Indeed, Hodges traces the first fully explicit statement of this current understanding to a 1957 paper by Tarski and Robert Vaught.
Moreover,
given
Carnap’s
insistence
that
‘the logic of science
. . . is
nothing else than the syntax of the language of science’ (Logical Syntax, p. 7), and the fact that the present understanding of the axiomatic method depends so heavily on considerations nowadays called ‘semantical’, it might seem likely that Carnap was confused about these issues. At least it might seem likely that we ought not to be casual in our attribution of our current understanding of the axiomatic method to Carnap in Logical Syntax. We want to show, however, that in Carnap’s explicit discussion of the axiomatic method (Logical Syntax 4 71) his position, despite its syntactic formulation, is not that far removed from our post-Tarskian one. According to Carnap, an axiom system is constructed by taking an existing language and adding to it a system of sentences (the axioms) which consist of (usually logical) symbols already in the language together with a stock of new
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Carnap’s Principle of Tolerance
symbols:
the primitive symbols of the axiom
mines the meaning
of the primitive
271). ‘The primitive of substitution since the axioms
symbols
for variables determine
symbols
system.
are not [variables]’ do not apply
An axiom
‘by a sort of implicit
system deterdefinition’
(p.
(p. 273) and so the usual rules
to them (pp. 272-273).
However,
the meaning
of the primitive symbols ‘to a certain extent . . . only in relation to one another’, it is often possible to interpret the primitive symbols in several different ways. These interpretations are by means of ‘correlative definitions’, which ‘determine to which . . . concepts [of some other language] the axiomatic primitive symbols are to be equivalent in meaning’ (p. 78). We prove that an axiom system is ‘fulfilled for a certain interpretation, or at least that its fulfilment is not excluded’ (p. 271) by providing a translation of the language of the axiom system into another language (‘usually a language of science which has a potential use’ (p. 273)) where a translation is a map taking the expressions of the first language into expressions of a second which preserves the consequence relation of the first (cf. !.$61). This clearly is not all that far removed from our current understanding of what it is to provide an interpretation of an axiomatic theory. In particular, Carnap is quite explicit about the nature and role of the primitive symbols, as is evident in his insistence, mentioned above, that they are not variables, and his statement that it is normally enough merely to state the correlates of the primitive symbols of the theory, since the translation of the logical constants can be ‘assumed to be established and well-known’ (p. 273). Module the syntactic character of the formulation, Carnap’s primitive symbols are essentially the non-logical constants of current model theory. We don’t mean to suggest that Carnap was completely clear on these matters. However, we think it is important to distinguish evidence that Carnap did not have something recognizably akin to the current view on these matters from evidence that the terminology now in current use had not yet evolved. So, for instance, we do not think it particularly significant that Carnap in describing one of his three methods for putting an axiomatic system into the form of a calculus, the method which uses the notation of his Language II (p. 272) says that each of the primitive symbols of the axioms system is to be represented by a variable, and that these as a group are called the primitive variables. In our view, Carnap’s terminology reflects the limitations of his Language II more than it does his thinking that the primitive variables were merely variables. He rather cryptically indicates that he recognizes a difference between the two when he says that ‘In the material mode of speech: the primitive variables do not express universality, but indeterminateness’. To see what he means by this we need to look briefly at how the primitive variables are to be treated. There are two main points to be noted. First, from the syntactical point of view, ‘In the deductions . . . there is no substitution for the free primitive
780
variables’
Studies in History and Philosophy
(p. 272). Since we cannot
substitute
arbitrary
values
of Science
for Carnap’s
‘free primitive variables’, they lack the characteristic property of what are currently called ‘free variables’. Secondly (and quasi-semantically), Carnap assumes
that we are adding
these formal
axioms
to a formal
language
which
already includes a substantial supply of provable formulae. He defines a real model for the axioms to be a substitution of some other expressions (of the appropriate types) of the language for the primitive variables such that the resulting formulae are provable in the language. This is clearly a syntactical approximation of the modern notion of mapping the non-logical constants into a suitable structure. In a real model, of course, nothing similar is required for the variables properly so-called, that is, for the variables that are not ‘free primitive variables’. We now have enough background to discuss the scope of the analogy between geometric conventionalism and the principle of linguistic tolerance. Carnap famously expressed his principle in a well-known passage of Logical Syntax: ‘In logic there are no morals. Everyone is at liberty to build up his own logic; i.e. his own form of language, as he wishes’ (p. 52). This can be read as the strong thesis that we are free to endorse whatever set of syntactical rules pleases us, and that these syntactical rules constitute or define the primitive terms of the language we speak, as the axioms of geometry constitute or define the primitive terms of geometry. The principle of tolerance on this view entails a general theory of meaning applied to any language whatever: the primitive terms of any given language are constituted or defined by some set of syntactical rules.* But this interpretation of tolerance puts Carnap in a most unhappy position. We have seen that, while the axioms of geometry fix the meaning of the primitive terms of geometry, they are incapable of fixing the reference of such terms. Reference of these terms, as Demopoulos has emphasized, is fixed only relative to the assumption that the reference of the terms of the language in which our background model theory is expressed is fixed. If the points of our geometrical theory are to be sets of some sort, then we are presupposing that the ‘While he does not use our terminology, Demopoulos, in the article cited previously, effectively suggests that this reading of tolerance can be found in Co&. He presumably has in mind Coffa’s remarks like ‘[according to tolerance] we have the same freedom to endorse alternative mathematical or metaphysical systems [i.e., languages] as we have in our choice of geometries’ (p. 54). However, in The Semantic Tradition from Kant to Carnap, Coffa seems to argue that while this reading of tolerance is consistent with what Carnap explicitly says about his principle, Carnap’s practice, which embodies what Coffa calls ‘second level semantic factualism’, implicitly commits him to a different interpretation of a sort closely related to what we will suggest shortly. In our view, though, Carnap was well aware that tolerance did not grant complete freedom. His remarks on axiomatic method and interpretation of languages go hand in hand with his explicit elucidation of tolerance. We have suggested in ‘Tolerance and Metalanguages in Carnap’s Logical Syntax of Language’, Synthese (forthcoming), that the principle of tolerance as it emerged in Logical Syntax was the product in part of explicit reflection on the notion of interpretation required to make sense of the various metalogical investigations of the 1920s and 1930s.
Carnap’s Principle of Tolerance reference
781
of the term ‘set’ in our background
be beer mugs, then we are presupposing
theory is fixed. If the points
are to
that we know to what the term ‘beer
mug’ refers. It follows that if Carnap is proposing either have to admit that the expressions
a general theory of meaning of the background language,
he will whose
meanings are fixed by some axiom analogues, simply have no fixed reference, or else accept that he is on the road to a regress. Fortunately, there is evidence in Logical Syntax that Carnap had no commitment to such a general theory of meaning, so that the principle of tolerance ought to be interpreted more weakly than we have so far suggested. Carnap explicitly claims in Logical Syntax (0 62) that if we know only the syntax of a given language then, while we will be able to answer questions about its syntax, ‘it is not possible to use it as a language of communication, because the interpretation of the language is lacking’ (p. 227). There are, he says, two methods by which we can learn to use a language for communication. First, there is ‘the purely practical method which is employed in the case of quite small children and at the Berlitz school of languages’. Secondly, there is the ‘method of explicit statements’, which is essentially the method outlined above for constructing an interpretation of an axiom system. He says that in Logical Syntax ‘by the interpretation of a language we shall always mean . . . the method of explicit statements’. This should not be surprising because in this sense an interpretation is a mapping (which Carnap calls a translation) of one language into a second, with this mapping described in a third; any and all of these languages can but need not coincide. Thus interpretations of this sort ‘can be formally represented; [so] the construction and examination of interpretations belong to formal syntax’ (p. 227). However, Carnap immediately and explicitly insists that this is only part of the story. For when we construct
an interpretation
same position as when we attempt natural language. He writes that
of this sort in formal to give a syntactical
syntax we are in the
description
of a specific
the construction of a calculus must take place entirely within the domain of formal syntax, although the decision as to whether the calculus fulfils the given condition is not a logical but an historical and empirical one, which lies outside the domain of pure syntax. The same thing holds, analogously, for the relation between two languages designated as translation or interpretation. The ordinary requirement of a translation from the French into the German language is that it be in accordance with sense or meaning-which means simply that it must be in agreement with the historically known habits of speech of French-speaking and German-speaking people. [p. 2281 The message here is obvious, even if unstated. The interpretation which must be supplied in order to use a mere calculus as a language of communication can be supplied in two ways. But the ‘method of explicit statements’ will not transform a calculus into a language of communication unless it maps the
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Studies in History and Philosophy of Science
translated calculus into a language we can use as a language of communication. That is, in order to avoid a regress, if we are going to transform a calculus into a language of communication we are eventually going to have to translate into a language which already has the extra component which cannot be acquired syntactically (i.e. by the method of explicit statements).9 This language will usually be one we have learned by the ‘purely practical’ method, though Carnap seems to be leaving open the possibility of learning by sufficiently intensive empirical study. Our claim is that, while there is an analogy between axioms of geometry understood as conventions and Carnap’s ‘sentences in the material mode’, this analogy should not be pushed to the conclusion that we have complete freedom in our selection of languages. As we read Carnap, we have complete freedom to construct whatever language we choose. Nonetheless, if we wish to use the newly constructed language as a language of communication, we will need to provide an interpretation in some antecedently understood language. Of course, we may decide to investigate this antecedently understood language and so attempt to recover its rules by constructing a calculus. However, the question of whether the constructed calculus captures the syntax of the language under consideration is an empirical one, and, qua calculus, it lacks interpretation. All this was unstated in Logical Syntax, but, we believe, was stated explicitly by Carnap in reply to a criticism made by Evert Beth. We don’t want to go into the details of Beth’s criticism, but the core of the relevant problem is that there might be more than one interpretation of a metalanguage in which a given object language is described. Carnap’s reply was that Since the metalanguage ML [for describing the syntax of some given language L] serves as a means of communication . I have always presupposed that a fixed interpretation of ML, which is shared by all participants, is given. This interpretation is usually not formulated explicitly; but since ML uses English words, it is assumed that these words are understood in their ordinary senses. The necessity of this presupposition of a common interpreted metalanguage seems to me obvious.1° Warren Goldfarb and Thomas Ricketts argue that Carnap’s reply to Beth is unsatisfactory: ‘Clearly, if the metalanguage is a rich one, and if our understanding of it cannot be exhaustively explicated in terms of rules, deductive procedures in axiomatic systems, or the like, then Carnap’s “presupposition” is an admission that much can never be made explicit, but must simply be tacitly
‘Indeed, right at the start of Logicul Syntax Carnap makes it clear that he does not believe that the syntactical study of language can give us the whole story. There are features of language that must be studied by other methods. ‘For instance, its words have meaning; this is the object of investigation for semasiology’ (p. 5). “Rudolf Carnap, ‘Reply to Beth’, in P. A. Schilpp (ed.), The Philosophy of Rudolf’ Carnup (La Salle: Open Court, 1963) p. 929.
Carnap’s
Principle of Tolerance
relied upon.
This fits poorly
783
with Carnap’s
proclaimed
standards
of exactitude
and rigor.“’ One
can
imagine
Carnap
responding
to this by saying
that
in fact the
presupposition fits perfectly well with his proclaimed standards of rigour. He might begin a response by pointing out that one must be careful not to suppose that the method of explicit statements supplies a theory of meaning even for constructed languages in any but a derivative sense. Meaning for a constructed language comes, ultimately, from the interpretation of that language in a language that already has meaning. Goldfarb and Ricketts, when they talk of Carnap’s ‘proclaimed standards of rigour’, have in mind the standards appropriate to syntax. But, as we have tried to show, Carnap thinks that it is empirical investigation that is appropriate to semantics, and so it will be the standards of rigour appropriate to empirical science which apply to semantics. So it is not the case that the things that must be tacitly relied upon in discussions of syntax are, for Carnap, not subject to rigorous investigation at all. Carnap’s presupposition is not ‘an admission that much can never be made explicit, but must simply be tacitly relied upon’. Granted, ‘much can never be made explicit, but must simply be tacitly relied upon’ from the point of view of strictly syntactical investigations. But, as we see it, Carnap was well aware that syntax, including the method of explicit statements, could not tell the whole story about language. The part that it could not give and must leave tacit was, for Carnap, open to empirical investigation, and so, except from the point of view of syntax, need not be left tacit at all. Acknowledgements-Our thanks to William Demopolous, who is always generous with helpful comments, and to the referees for this journal. David DeVidi also thanks the Social Sciences and Humanities Research Council of Canada for support.
“Warren Goldfarb and Thomas Ricketts, ‘Carnap and the Philosophy of Mathematics’, in D. Bell and W. Vossenkuhl (eds), Wissenschuf und Subjectivitiit (Berlin: Academic Verlag, 1992), p. 72. The point with which we take issue is a secondary one in an article that we otherwise find very persuasive.