Response: Explication as Foundations

As indicated at the outset of this chapter, it is worth noting Awodey & Carus’ (2004) very astute observation that G¨odel’s particular circularity objection is unsound. Re- call that G¨odel’s complaint was that Carnap must prove that a candidate syntactical rule is consistent in order that it can do the epistemic work of replacing G¨odel’s own appeal to mathematical intuition. But since such a proof requires mathematical notions stronger than the rule, Carnap is stuck using stronger mathematical prin- ciples than those he was attempting to show syntactical, and his account collapses. Awodey & Carus note that Carnap need not prove that a candidate syntactical rule be consistent; it need only be the case that the rule actually is consistent. G¨odel’s call for a proof of consistency goes too far, since all of our syntactical rules may very well be consistent even if this cannot be proven. The situation is analogous to our acceptance of the consistency of Peano Arithmetic. Thanks to the incomplete- ness theorems, almost any proof of the consistency of arithmetic will inevitably be uninformative. But this technical situation alone is not usually taken to put into jeopardy our confidence in the consistency of the theory.

However, there are actually two concerns with this reply. First off, this response

37_{Again from Carnap’s autobiography: “From [Frege’s] analysis I gained the conviction that}

knowledge in mathematics is analytic in the general sense that it has essentially the same nature as knowledge in logic. [. . . ] Furthermore the following conception, which derives essentially from Frege, seemed to me of paramount importance: It is the task of logic and of mathematics within the total system of knowledge to supply the forms of concepts, statements, and inferences, forms which are then applicable everywhere, hence also to non-logical knowledge. It follows from these considerations that the nature of logic and mathematics can be clearly understood only if close attention is given to their application in non-logical fields, especially in empirical science. Although the greater part of my work belongs to the fields of pure logic and the logical foundations of mathematics, nevertheless great weight is given in my thinking to the application of logic to non- logical knowledge. This point of view is an important factor in the motivation for some of my philosophical positions, for example, for the choice of forms of languages, for my emphasis on the fundamental distinction between logical and non-logical knowledge.” (Carnap, 1963, pp. 12–13)

does not respect the Principle of Tolerance, in the sense that it does not acknowledge that we may want to investigate the consequences of choosing a framework with inconsistent syntactical rules. This being the case, G¨odel would argue that we have moved out of the realm of mathematics altogether, since our supposed “L-Rules” would imply ostensibly factual sentences. To the letter of Carnap’s program however, this is not the case, since by definition every sentence of the language would be analytic, and so the framework would have no factual sentences. Certainly this is an odd framework, and it is certainly unworkable as a total language of science (since it would be wildly inadequate for explicating our empirical theories), but it should not be ruled out from the beginning, especially as the explication of some pure logico-mathematical theories (such as paraconsistent logics).

More importantly, this response is predicated upon the peculiarities of G¨odel’s criticism to the first thesis of the syntactic viewpoint. The more general concern that we have abstracted from the circularity objections is still to be addressed. The fundamental worry is that Carnap’s account of mathematics as a purely formal set of syntactical rules appeals to informal mathematical concepts in its justification or explanation, thus undermining the thesis. But we have seen that Carnap’s under- standing of logico-mathematical sentences as formal auxiliaries is not a foundational thesis in the traditional sense. Rather, it is the result of a methodological analysis of science and the role that logic and mathematics seem to play in our total sys- tem of knowledge. In other words, Carnap’s mathematical conventionalism is itself an explication of the distinction that he recognizes between the formal and factual aspects of science. In order to support this explication as successful, Carnap must show that his conventionalist account of logic and mathematics recovers our under- standing of the role that mathematics plays in the total language of science. Carnap takes this to be an account of the applicability of logico-mathematical sentences, and certain characteristics that equate to those sentences being utilized as formal aux- iliaries. He then argues that his treatment of logic and mathematics recovers these characteristics, and so is successful in this regard.

Why is this not just an appeal to verificationism? Recall in §2.1.2 above that G¨odel’s second criticism involves calling into question the Logical Empiricist notion of ‘content’. G¨odel objects that the Logical Empiricists rule out from the beginning any definition of the concept that is not limited to ‘empirical content’, thus begging the question.38 _{I suggested that while traditional Logical Empiricists such as Hahn} 38_{Note that given the response of Awodey & Carus, this objection is also unsound because it}

and Schlick do indeed assume verificationism, that Carnap in fact shows that math- ematics involves nothing that ought to be construed as content. Carnap does this by transforming foundational questions about the nature of logic and mathemat- ics into problems of explication. The question then is not to explain the nature of logico-mathematical truth in some metaphysical sense, or our epistemic access to it. Carnap presumes the veracity of logico-mathematical concepts and results, because these are matters of science, not philosophy. Rather, Carnap sees the philosophical problem as one of developing a more rigorous language that recovers the role that logic and mathematics play in our system of knowledge.

So Carnap does not just presume that logic and mathematics is analytic, but provides an informative analysis of logico-mathematical truth and related concepts which suggests that treating logic and mathematics as purely formal recovers its es- sential characteristics. This analysis does indeed presume the concepts and methods of mathematics itself, but this is part of the nature of providing an explication. The concept ‘Warmer’ does not disappear once we have offered a rigorization of it with the concept ‘Temperature’. Moreover, in order to demonstrate that our explicatum

is successful, we must rely at least in part upon our confidence in the explicandum. It is important to recognize that Carnap’s methodological characterization of the sciences does quite a bit of work in his program, and it is a point upon which his sug- gestion that philosophy be replaced by the Logic of Science (as the primarily formal explication of the concepts of science) depends. But insofar as Carnap’s program is a recommendation to adopt a particular set of methods, it is resistant to foundational criticisms. And insofar as his mathematical conventionalism is an explication of a concept of logico-mathematical truth predicated upon this methodological charac- terization of the role of logic and mathematics in the sciences, he can virtuously appeal to the concepts and methods of mathematics to argue that his explication is successful. As we will see in the second interlude, any further metaphysical or epistemic questions Carnap regards as pseudo-questions.

In the next chapter we will revisit G¨odel’s criticisms as a means to understand the Deflationary reading in greater detail. As noted, the Deflationary response to these objections is quite different than the response developed herein. Examining this response will give us an opportunity to better understand the methodological role of the Principle of Tolerance in Carnap’s program.

idea of G¨odel’s first objection generalizes, the kernal of his second objection remains a difficulty outside of the context of G¨odel’s particular presentation.

Chapter 3

‘Empirical Fact’, Tolerance, and

Conventionalism: A Reply to

Goldfarb and Ricketts

It is impossible to argue against what professes to be insight, so long as it does not argue in its own favour.

Bertrand Russell([1914]1952)

In this chapter we will examine the Deflationary interpretation in more detail, and I will argue that it is in several places either mistaken or uncharitable as an inter- pretation of Carnap’s meta-philosophy. As mentioned in our first chapter, Goldfarb and Ricketts’ reading seems partially motivated precisely because they believe it to furnish Carnap with a response to G¨odel’s criticisms—almost every paper in their series devotes a substantial amount of space to the treatment of this issue.1 _{In any}

case, G¨odel’s understanding of Carnap’s program affords an illuminating contrast for our own understanding of the Deflationary interpretation, since Goldfarb and Ricketts argue that he fundamentally misunderstands Carnap’s aims and the radical nature of his project.

With regard to the other circularity objections, although Goldfarb and Ricketts do not treat them in a systematic way, the various concerns the authors do address over the course of their many papers advocating their reading strongly suggests how

1_{But see especially Goldfarb & Ricketts (1992), Ricketts (1994; 2007; 2009), and Goldfarb}

a deflationary Carnap would respond to these objections. One of these concerns is the intuitionist complaint noted in both our first and second chapters, that Car- nap’s meta-philosophy seems to beg the question against an intuitionist because the meta-logical investigation of various frameworks will almost necessarily presuppose mathematical notions stronger than any the intuitionist considers legitimate. We will see that the Deflationary interpretation and the interpretation we have devel- oped thus far must approach this complaint in very different ways.

We will begin with a brief review of the main tenets of the Deflationary read- ing. As noted, Goldfarb and Ricketts’ interpretation will become clearer through the discussion of their treatment of the circularity objections. Their responses lean heavily upon the Principle of Tolerance to deny that Carnap recognized any notions which “cut-across” distinct linguistic frameworks, notions such as ‘empirical fact’ and ‘mathematical truth’. In§2 we will examine the Deflationary response to G¨odel in detail and see that, contra the Deflationary reading, Carnap’s program does in- deed have a rather robust notion of ‘empirical fact’ that acts in exactly the way Goldfarb and Ricketts argue it must not. §3 argues that logocentrism is a rather bold interpretive thesis in the context of Carnap’s program. This will lead us to §4, which discusses the Deflationary response to the other circularity objections. We will see that this response threatens to rob Carnap’s program of much of its potential philosophical interest, fundamentally limiting its extensibility and denying Carnap the ability to offer any reason for its adoption. As suggested in our first chapter, Carnap’s program becomes philosophically inert on the Deflationary reading. We will also observe that the Deflationary response to G¨odel does not completely ad- dress his concerns. Finally, §5 will examine the role of the Principle of Tolerance in Carnap’s meta-philosophy in more detail. We will conclude that the principle is licensed by Carnap’s methodological analysis of the sciences rather than acting as the foundation of his program.

3.1

The Deflationary Interpretation

Recall from our first chapter that the Deflationary reading suggests that Carnap’s program is grounded upon an insight into the primacy of our language as a precon- dition for inquiry and justification. Consider:

Carnap’s central notion is that of a language, or, in his later terminology, a linguistic framework. A linguistic framework provides the logical rela-

tions of consequence and contradiction among propositions. The fixing of these logical relations is a precondition for rational inquiry and discourse. There are many alternative frameworks, many different logics of inference and inquiry. There can be no question of justifying one over another, since justification is an intra-framework notion; justification can proceed only given the logical relations that a framework provides. (Goldfarb, 1995b, pp. 326–327. My emphasis.)2

Now, I take no issue with the understanding of a linguistic framework as being a model—a rational reconstruction—of our methods of inquiry and discourse. As we have seen, this is presumably the point of Carnap’s reconstructive program. My concern is the further interpretive claim that outside the auspices of some partic- ular linguistic framework Carnap does not recognize any set of intuitive epistemic notions to which our formal reconstructions can be compared. According to the Deflationary reading, for Carnap concepts like ‘justification’, ‘inference’, ‘empirical fact’, or ‘mathematical truth’ are constituted by the choice of syntactical rules for one’s framework, and so there is no notion of scientific inquiry or practice standing outside of this rigid formal structure.

We labeled this interpretive ascription made by the Deflationary reading Carnap’s

logocentrism, which I will repeat here:

Logocentrism: Our choice of linguistic framework includes the choice of all epistemic standards for justification, truth (viz., analyticity), and correctness, and therefore all such notions must be language-relative.

The consequence is that since each distinct framework will generate its own distinct set of epistemic notions, there can be no extra-framework notions which “cut-across” distinct linguistic frameworks, and so there can be no rational grounds for the cate- gorical assertion or rejection of a linguistic framework, or for the ordering of a set of proposed frameworks according to some rational criteria. Any rational argument for the rejection of a framework would by definition ensue already from the confines of one’s own particular framework, but this need-not impinge upon the very disparate notions of rationality and justification that may apply in some other language.

It follows on this reading that any choice between frameworks must be entirely a matter of our values and practical concerns—after all, the choice of a language is not

2_{In our discussion from the first chapter we quoted Goldfarb (1996, p. 225) making this same}

something that can be “correct” or “incorrect”—and so the Principle of Tolerance has primacy of place in the Deflationary reading of Carnap’s meta-philosophy. I will also repeat this principle here:

[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 form of 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. (Logical Syntax, pp. 51–52. Original emphasis.)

Tolerance suggests a logical pluralism: Our choice of logical language is not a matter of truth but instead a matter of convention. It also recommends that we state our methods clearly—in the form of syntactical rules—thus suggesting the formalized methods that Carnap favoured.

None of this is controversial with regard to Carnap interpretation. We have seen Carnap enact this process inLogical Syntax through the reconstruction of intuition- istic and classical mathematics. He does indeed suggest that our decision to utilize intuitionistic or classical reasoning in further scientific investigation is ultimately a matter of choice. The reconstructions of these positions in the form of linguistic frameworks LI and LII help to inform that choice by highlighting the formal proper- ties of each language. However, the logocentric interpretation of Carnap’s program takes this one step further by suggesting that this process is really all there is to his meta-philosophy. Because all of our epistemic notions are intra-framework, Carnap cannot provide rational criteria by which to determine that some particular frame- work acts as a more accurate reconstruction than some other, or give a rational justification for the adoption of some particular framework over another. Indeed, Carnap cannot argue for any kind of philosophical position at all. Conventional- ism and empiricism are just particular frameworks that we might choose adopt on the Deflationary reading, since Carnap’s program is “not based on any substantial theoretical commitments of its own.” (Goldfarb, 1997, p. 61). Likewise, the method- ological reforms encapsulated in the Principle of Tolerance are a suggestion, but not one for which we can provide rational arguments. “In a sense, it is not said, but only shown.” (Ibid.)

As we will see below, on the Deflationary reading Carnap’s program cannot be the explication of the concepts of science, since there can be no particular set which constitutes the concepts of science. Rather, Carnap’s program amounts to the con- struction of independent, almost free-floating, linguistic frameworks that practicing scientists may choose to utilize or to ignore. Philosophical debates are still treated as a matter of choice between linguistic proposals, but because of the logocentric character of our concepts this procedure is extended to all inquiry and investiga- tion. Since rationality and justification are intra-framework notions, the comparison of distinct frameworks, or the idea that distinct frameworks may be more or less adequate with regard to some informal scientific concept, fall completely under the methodological purview of Tolerance, that is, they are entirely matters of convention. So rather than trying to show that an explication is successful and offering reasons for the adoption of one explicatum over another, on the Deflationary interpretation such choices are a matter of our preferences and values, as in choosing between a set of Kuhnian paradigms.3

Furthermore, since a deflationary Carnap eschews the very idea that argument is appropriate in philosophy, he gives up the ability to recommend his position to any objector who does not immediately accept the Principle of Tolerance. The position also results in a troubling tension within Carnap’s program, as Tolerance’s call for clarity must ultimately remain unfulfilled due to circularity-style reasoning. But as we will see below, this tension is a consequence of the Deflationary interpretation.