2.1 Carnap and Quine on Explication

Quine describes himself as a disciple of Carnap’s for six years, from 1932, when Quine met him in Prague, while Carnap was composing Syntax, until 1938. But even as late as Quine (Reference Quine1947), where Quine is principally concerned with attacking quantified modal logic, he shows that he sees it as unproblematic that Carnap uses of a strong notion of analyticity to define mathematical truth. There Quine says that analytic claims, in a non-modal system, can be defined as statements “[d]educible by the logic of truth-functions and quantification from true statements containing only logical signs” (Quine, Reference Quine1947, p. 43). Importantly, Quine points out that, as per Gödel, the class of true statements containing only logical signs is not recursively enumerable. Notice that the appeal to Gödel’s first incompleteness theorem in this context only makes sense if Quine thinks that all of arithmetic is both analytic and composed of only logical vocabulary. In 1947 then, in this sense at least, Quine has not moved far from a Carnapian position on logico-mathematical truth.Footnote ⁹

Carnap first presents his account of explication in Carnap (Reference Carnap1945), but his most detailed account is contained in Carnap (Reference Carnap1950b). Reading Langford (Reference Langford and Schilpp1942/1968) shortly after its publication, the presentation, there, of the paradox of analysis had an effect on how Carnap saw the process of philosophical analysis.Footnote ¹⁰ The paradox of analysis is the claim that all analyses must be either incorrect or uninformative. For instance, if one were to say “to be A is to be B” then this is either incorrect or uninformative. For if A and B have the same meaning the claim is uninformative, but if they have different meanings, then the claim is incorrect. Inspired by the paradox of analysis, Carnap believes the idea of analysis as arriving at something identical to the explicandum is to be rejected. If the goal is not to arrive at the same thing, then, after an analysis, we must be introducing a new concept (i.e. something different) – which he calls the explicatum. Carnap proposes four conditions on successful explication. The conditions are those of similarity, simplicity, precision, and fruitfulness. The only one of these conditions which has anything to do with the explicandum, and is not solely concerned with the explicatum, is the first one: the explicatum has to be similar to the explicandum.Footnote ¹¹ How similar must it be? Carnap’s answer is that the explicatum needs to be sufficiently similar that it could be used in the place of the explicandum. The exact statement of the requirement of similarity is “[t]he explicatum is to be similar to the explicandum in such a way that, in most cases in which the explicandum has so far been used, the explicatum can be used; however, close similarity is not required, and considerable differences are permitted” (Carnap, Reference Carnap1950b, p. 7). So Carnap’s condition of similarity imposes the weakest possible condition (given that the explicatum is meant to replace the explicandum) on the link between the explicandum and explicatum. It is for this reason that Carnap thought Quine’s challenge from “Two Dogmas $\dots$” was so easy to answer. The challenge was to define a concept of analyticity that goes beyond logical truth to include statements like “all bachelors are unmarried.” Carnap proposed that this could be done via meaning postulates. A logical truth follows from logic alone, but an analytic truth follows from the meaning postulates together with the laws of logic. So long as the meaning postulates are chosen in a reasonable manner, this will lead to a concept of analyticity that is extensionally similar to what is ordinarily considered analytic.

In an article I published on this subject (Lavers, Reference Lavers2012), I argued that Carnap and Quine’s differing conception of explication explained their differences on analyticity. The difference I point to is that when Quine discusses explication, he speaks of identifying and preserving certain features of the already existing expression. Features outside of these “favored contexts” are labeled “don’t-cares.”Footnote ¹² This preservation of the use of expressions in these “favored contexts” is completely absent from Carnap’s view on explication.Footnote ¹³

For Quine explication is a two-step process. This, we saw earlier, is also true of Carnap. But unlike Carnap, for Quine, in the first stage, we must identify some central aspect of the meaning of the explicandum that we wish to preserve. For Quine, this central feature is relative to our purposes and interests, but this does not diminish the importance of this first stage.Footnote ¹⁴ Those aspects of the use of the explicandum which are not part of this are labeled “don’t-cares.” The first stage for Quine is principally concerned with distinguishing as clearly as possible between the “favored contexts” and the “don’t-cares.” We then, in the second phase, come up with a clearly defined substitute which while agreeing with the explicandum on the “favored contexts” may differ in any possible way over the “don’t-cares.”

Quine is explicit about how he views explication in a number of places, but most discussions of explication are very brief. He usually begins by citing his agreement with Carnap’s views on explication, and then Quine lays out his own view of the matter. The agreement he expresses with Carnap’s view is tied to them both rejecting the idea that the definiendum and the definiens ought to be synonymous. In an explication our goal is to introduce a newly defined notion. But beyond this agreement their views on the matter are quite different. The most detailed account is contained in Word & Object, but even the very brief discussions show that he is very consistent on the subject. There is roughly one page of “Two Dogmas $\dots$” which deals with explication. Here Quine asks whether it is possible to give a Carnapian explication of synonymy or analyticity. Then he lays out the following as his account of how explication is supposed to work:

Here we see Quine, in his discussion of explication is putting forward a view exactly in line with the interpretation given earlier. The definiens and definiendum need not be everywhere synonymous, but must agree in the “favored contexts.” He continues:

Again we see that there are favored contexts in which the ordinary notion and any newly defined ones are meant to agree. Outside of these contexts, there can be differences between two acceptable explications.

Quine’s most detailed account of explication, as mentioned, is given in Word & Object (§53), it is exactly in line with the view already briefly discussed in “Two Dogmas $\dots$.” This section is given the title “The ordered pair as a philosophical paradigm.” The reason given for the claim made in the title is that we can identify exactly what any analysis of ordered pair ought to preserve. What any account of the ordered pair must preserve is that $⟨x,y⟩=⟨w,z⟩$ only if $x=w$ and $y=z$. Quine sees the various definitions of the ordered pair as exemplars of philosophical analysis, because here it is so easy to identify the features of the ordinary notion that we wish to preserve. The various definitions are mutually inconsistent; if $⟨x,y⟩=\left\{\right\{x\},\{x,y\left\}\right\}$ then it is not, in general, identical with $\left\{\right\{x\},\{y,\varnothing \left\}\right\}$. That is, the Kuratowski and Wiener definitions of the ordered pairs are competing accounts, but they are both successful analyses. Because each preserves the central feature of ordered pairs. Where they disagree is over what Quine terms “don’t-cares.” It is true that in this same section he says that this example is atypical of explication. “Our example is atypical in just one respect: the demands of partial agreement are preternaturally succinct and explicit” (Quine, Reference Quine1960, §53) But notice the one respect in which this is atypical, is that in this case the “don’t-cares” can be separated from what we wish to preserve with an incredibly simple to state principle. Other than this, this is a very typical explication.

Quine is well known to have said (again in Word & Object §53) that explication is elimination. This has led some to think that Quine saw explication as ontological elimination, but while this is sometimes true, explication does not always involve ontological elimination. Notice the conditional nature of this quote: “If there was a question of objects, and the partial parallelism which we are now picturing obtains, the corresponding objects of the new scheme will tend to be looked upon as the old mysterious objects minus the mystery” (Quine, Reference Quine1960, §53). But importantly, Quine mentions Turing’s analysis of a computable function as a paradigm of explication, and here, there is no ontological elimination, but rather a conceptual clarification. So while explication can lead to ontological elimination, it is always the elimination of an old defective term in favor of a more clearly defined one.

Let us return now to showing that Quine sees explication as involving the preservation of some feature of the use of the antecedent expression. Consider now this discussion of explication from Word & Object:

After presenting his account of explication, Quine says, “philosophical analysis, explication, has not always been seen in this way” (Quine, Reference Quine1960, p. 259). In a footnote he acknowledges Carnap and cites Meaning and Necessity as having seen explication correctly. We have seen however, of course, that there is an important difference in their accounts. Quine discusses the idea of explication in many places and he is very consistent in what he says about the subject. In an explication, we begin by determining which aspect of the existing use of a term we wish to preserve, then we propose a substitute that agrees with the existing term in these contexts. In the next section we will look at how their accounts of explication figured in the debate over analyticity.Footnote ¹⁶

2.2 Explication and Analyticity

In the previous section we saw the differences between Carnap and Quine on explication. Both see it as a two step process, where the second step involves the introduction of a new notion which is meant to replace the existing notion. But the first stage is quite different. For Carnap all that we need to do at the first stage is understand the term well enough that we could roughly predict when a speaker of the language would apply it. This then serves as a guide when we introduce the replacing notion, but does not significantly constrain the second stage. For Quine on the other hand, it is not enough to merely come up with a similar, precisely defined concept. At the first stage, on Quine’s view, we need to identify what it is, about the use of the original expression, that we intend to preserve. We are then constrained by this at the second stage. Our newly introduced term must agree with the existing one in these “favored contexts.”

In “Two Dogmas $\dots$,” Quine poses the problem of explicating a notion of analyticity wider than logical truth. Such an explication should count not only “all bachelors are bachelors” as analytic, but also “all bachelors are unmarried.” Quine concludes, at the time, that no explication could be given. Carnap, on the other hand, believes meeting this challenge is a very straightforward task. After all, we understand the concept of analytic well enough that we could correctly predict when a competent user of the term would apply it or not. We are then free to define as we wish an expression which is similar to this notion, but defined in a manner that is more to our liking. Carnap chooses to do this by using meaning postulates (or elsewhere semantic rules). Logically true statements follow from the logical rules alone, while the wider term analytic applies to anything that follows from the logical rules together with the meaning postulates. This achieves extensional similarity, but not by identifying and preserving some aspect of the ordinary notion. So while it satisfies Carnap’s demands of an explication, it does not satisfy Quine’s.

I suggested in Lavers (Reference Lavers2012), that Carnap and Quine are completely talking past one another on the subject of analyticity. This is because, while each is addressing the question of whether the concept of analyticity can be successfully explicated, they each mean different things by successfully explicated, as we just saw. Carnap’s explication in terms of meaning postulates is perfectly successful according to Carnap’s criteria for a successful explication – the extension is similar after all. But, as following from meaning postulates is not a property of the explicandum, and as Quine requires identifying and preserving properties of the explicandum, according to Quine’s account of successful explication it is a clear failure:

Interpreted as identifying a feature of our existing conception of analyticity which is preserved by the explication, Quine sees Carnap’s explication as comically bad.Footnote ¹⁷

Carnap wrote a response to Quine’s “Two Dogmas $\dots$” shortly after it came out. This was never published during Carnap’s lifetime, but was included in Quine and Carnap (Reference Quine and Carnap1990). Here it is clear that Carnap sees Quine’s demand, that there be a feature of the use of the ordinary notion that is preserved, as confused. Carnap in this response repeatedly accuses Quine of confusing properties of the explicatum with properties of the explicandum:

Carnap is clearly saying that his explication of analyticity satisfies his own account of what constitutes a successful explication. Quine is looking for what is preserved between the explicandum and explicatum. That is, Quine is looking for a central property of the explicandum which is also a property of the explicatum. Carnap sees Quine a confusing properties of the explicandum with properties of the explicatum. While Carnap believes that the explicandum and explicatum should have considerable extensional overlap, they need not share any characteristic property in common. But Quine is not confused here, he is looking for what aspect of the meaning of our ordinary notion is preserved by Carnap’s explication. Quine is applying his own standards of successful explication. It is just that neither is aware of the differences between their accounts of explication.

2.3 Analyticity

Given the analysis of the dispute between Carnap and Quine on analyticity presented in this section, an affirmative answer to Quine’s question of whether analyticity can be successfully explicated is strongly suggested. And this even relative to Quine’s account of successful explication. Quine believes that for many concepts, it is clear, relative to our purpose and interests, what any explication of that concept ought to preserve. Now, there is a traditional connection between analytic truth and analysis. Analytic truths are, traditionally, truths discovered in the process of an analysis. This suggests that we take the features identified in the first stage of a Quinean explication (and what follows from them) as our analytic truths.Footnote ¹⁹ To use what Quine sees as the clearest example, it is an analytic truth that $⟨x,y⟩=⟨w,z⟩$ only if $x=w$ and $y=z$. It is not an analytic truth, however, that $⟨x,y⟩=\left\{\right\{x\},\{x,y\left\}\right\}$. Notice that what is identified as a “favored contexts” in the analysis of one of our concepts could then be used to define the set of analytic truths in the manner that Carnap intended to do with meaning postulates.Footnote ²⁰ The logical truths are consequences of the set of basic logical truths, while the analytic truths consequences of this set plus the set of “favored contexts” identified in the process of carrying out our various explications. Notice that Quine cannot reject this definition as he did Carnap’s own appeal to meaning postulates. Remember, he dismissed Carnap’s appeal to meaning postulates as ad hoc and artificial. But the distinction between “favored contexts” and “don’t-cares” is central to Quine’s understanding of a successful explication, and this in turn is, as Quine stresses in “Two Dogmas $\dots$,” an important “activity to which philosophers are given, and scientists also in their more philosophical moments.”

In the previous paragraph, I took logic to be given and I defined analytic as what follows logically from the favored contexts that we identify. But we would like to have logic itself be analytic in some non-trivial sense. To do this we just need that in the case of logical notions, what is identified as a favored context will include not just propositions but also rules of inference. I also want to be somewhat liberal here and allow for even infinitary rules of inference so that something like an $\omega$-rule could be analytic. That is, I do not wish to limit the identified rule of inference to formal rules of inference, but wish, following Carnap and the early Quine, to allow for analyticity to be defined so as to include a non-recursively enumerable set of analytic truths.Footnote ²¹ We then take as analytic the closure of the favored contexts under the identified rules of inference. One could accept a weaker notion of consequence and still accept the definition of analyticity provided here, but as I and others have argued in the works just cited, this will be insufficient for capturing our full understanding of mathematical truth. It is therefore straightforward to extend this account to one which allows logical truths to be analytic via an explication of logical concepts.Footnote ²²

One of the advantages of this definition of analytic is that it distinguishes between what is a conceptual truth from what is a purely stipulative truth. “ $⟨x,y⟩=⟨w,z⟩$ only if $x=w$ and $y=z$” is something that we identify as an important conceptual truth about the notion of an ordered pair, while “ $⟨x,y⟩=\left\{\right\{x\},\{x,y\left\}\right\}$” is something that is true (by stipulation) on one explication and false on others. This in turn can be used to address the main point of Benacerraf (Reference Benacerraf1965). There Benacerraf argues against any particular set-theoretic explication of number, on the basis that such an explication will confer on number properties that are foreign to them. He imagines two children of die-hard logicists named Ernie and Johnny. As a result of what must have been an oversight, Ernie uses John von Neumann’s definition of the ordinals and Johnny uses Ernst Zermelo’s. Ernie and Johnny then get into a disagreement over whether 3 is a member of 17. Ernie says it is, Johnny says it isn’t. Here we see a disagreement about what is purely stipulative on one explication and false on another. In other words, this disagreement concerns only “don’t-cares.” The mathematically interesting properties of number are exactly the analytic properties of number as just explicated.Footnote ²³

This definition does not, of course, eliminate all the vagueness of the term analytic. Especially when considering sentences of natural language it may be difficult to determine if they are analytic according to this definition. In one of his earliest expressions of serious doubt concerning the concept of analyticity, from a letter to Carnap in November of 1943, Quine writes, “Honestly I don’t know whether ‘Scott is human’ is for me analytic or synthetic, nor whether ‘human being’ means the same as ‘rational animal’ ” (Quine & Carnap, Reference Quine and Carnap1990, p. 367).Footnote ²⁴ What is important though is, first, this analysis satisfies Quine’s account of successful explication. And secondly, in mathematics, it tends to be completely straightforward to distinguish between the important properties to be preserved by an explication and the “don’t-cares.” For instance, Quine thinks it is completely clear what any satisfactory explication of number ought to satisfy. The same is obviously true of other mathematical concepts such as groups, rings or fields.

Quine famously wanted a behavioristic explication of analyticity, and the account of analyticity under consideration is clearly not given in behavioristic terms. But there are two things to be said of this. First behaviorism has significantly fallen out of favor and this demand is hardly one we see as appropriate for contemporary philosophical definitions. Second, not all explications which Quine views as successful meet this standard. Tarski’s definition of truth, which Quine so often refers to as a successful explication, is not given in behavioristic terms. One might point out that it could be grounded behavioristically by tying it to what a speaker would assent to. But presumably Quine would have to hold that analytic, as here defined, must be in principle, explicable behavioristically. Since Quine thinks we can often distinguish between features of the use of a term which ought to be preserved by an explication and “don’t-cares,” then presumably Quine would hold that a field linguist observing us could come to understand this distinction as well.

As we saw, Quine does eventually accept several definitions of analytic (for a discussion of these see Creath Reference Creath, Friedman and Creath2007), but he is quick to point out that from these one may get statements of ordinary language, such as “there are seven days a week,” counting as analytic, one does not get much more than this. In particular, Quine points out, that contra Carnap, it does not turn out that mathematics is analytic. “In short, I recognize analyticity in its obvious and useful but epistemically insignificant applications. The needs that Carnap felt for this notion in connection with mathematical truth are better met through holism” (Quine, Reference Quine2008, p. 397). This “epistemic insignificance” has to do with the specific definition of analytic that Quine was considering, a view different from the one suggested in this section. But I wish to defend the view that a definition of analytic based on Quine’s distinction between favored contexts and don’t-cares is perfectly suited to give an account of the analyticity of mathematics. Waismann in the late 1940s and early 1950s wrote a series of articles critical of the concept of analyticity.Footnote ²⁵ Here he argued that analyticity is both vague and open-textured. He also, in this period, argued against reductionism. All that said, he maintained that in mathematics, everything is perfectly clear.

Waismann defended the view that mathematics is tautological. I will not follow Waismann on this point, but I certainly endorse the spirit of Waismann’s observation that in mathematics what we mean is extremely clear, and much more so than in ordinary language. One ought to expect, then, contra the later Quine, that clear cases of analyticity or truth in virtue of meaning will be more easily found in mathematics than in ordinary language. In the next couple of sections I would like to consider what Quine says about the explication of the concept of natural numbers and sets, as he is quite explicit about these cases.

2.4 Arithmetic and Explication

We have now seen then how Quine takes explication to function and what he takes as the characteristics of a successful explication. Let us now turn our attention to what Quine says about explication and the philosophy of mathematics in particular. Quine is explicit in many places that it is possible to give a successful analysis of arithmetic. Quine defends Frege’s definition of number against a family of objections. In analyzing numbers we identify laws which hold of any series any of whose members has only finitely many predecessors. Any sequence which satisfies these laws then would constitute a successful analysis of number – differing only concerning “don’t-cares.”

Numbers can be explicated in various ways as sets. All of the various explications satisfy the arithmetical laws, so any difference between the explications, or between the explication and the ordinary notion point only to inessential features. In §54 of Word & Object Quine is clear that arithmetic lends itself to being explicated because we can so easily and clearly separate what must be preserved:

from “don’t-cares:”

We find a similar point being made, if we look at “Ontological Relativity,” from 1968:

Here what we wish to preserve are the laws of arithmetic. Any account from which we can derive the laws of arithmetic will be a successful explication, and differences between them (such as whether 3 is a member of 17) fall under the “don’t-cares.” In the case of arithmetic then, Quine does not think there is ambiguity about what is a conceptual truth of arithmetic, which ought to be preserved by any explication and a purely stipulative truth which is relative to a particular explication. Of course, as touched on earlier, although there is no problem for identifying the conceptual truths that any account of arithmetic must preserve, providing such an account must be done in set theory or type theory, and this is where Quine sees a limit to what can be done by explication. It is true, in this essay, Quine does say:

This quote seems to suggest a close connection between the cases of arithmetic and set theory. However, as we will see in the next section, Quine clearly does not take the phrase “the laws of set theory” to be univocal, as he clearly does in the case of “the laws of number.” Let us now, then, turn to the subject of set theory.

2.5 Set Theory and Explication

We saw in the last section that Quine believed we could, in the case of arithmetic clearly distinguish the favored contexts from the don’t-cares. As we will see in this section this is definitely not his view on set-theory. The case of set theory is interesting since so much of mathematics is expressible in set theory. Also, exploring the difference between Quine’s view and Boolos is useful as I will be defending a position on the analyticity of mathematics which sides with Boolos, over Quine, on conception of the status of the axioms of set theory. In this section I will show what Quine’s position is, show how Boolos objects, and then briefly give my reasons for siding with Boolos.

Quine saw the definition of various systems of set-theory as pragmatically very useful. If we have classes, then it is relatively unproblematic to explicate all the mathematical objects we wish:

The question, then, that clearly suggests itself is whether we can successfully explicate the concept of set. Sean Morris, in his recent book on Quine and set theory (Morris, Reference Morris2018), mentions one problem for this explication.Footnote ²⁶ For Quine, explication is elimination, and there is no category of entities that suggest themselves as candidates for sets to be explicated in terms of. As we saw earlier though, explication for Quine does not always involve ontological elimination. But apart from the problem of our inability to reduce sets to another kind of entity, Morris writes: “[s]till, I think explication is the right way to describe Quine’s approach to set theory” (p. 124). However, Quine is clear that we cannot give an explication of set theory. The reason is that we cannot identify, from a central feature of the intuitive notion, a comprehension principle any which account ought to preserve.

We saw that Quine took Carnap’s explications of analyticity to fail because he could not find in Carnap’s explication a feature of the intuitive notion that the explication was meant to preserve. Concerning set theory Quine repeatedly states that, if we look to the intuitive notion of set, the only comprehension principle we can identify is naive comprehension. In the face of the paradoxes, Quine is quite clear that with respect to set theory, intuition is bankrupt, and as a result we must content ourselves with arbitrary and ad hoc restrictions on naive comprehension:

An explication begins by our identifying the central conceptual truths which we wish to preserve, and then we must come up with a clear replacement, which preserves the central features. If we cannot identify what it is about the original notion that we take to be worthy of explication and merely come up with a replacement, then the newly introduced replacement is purely ad hoc and not at all an explication.

In his book, Morris takes issue with Boolos’s criticism of Quine’s New Foundations theory of sets. In particular, Morris takes issue with this claim:

Morris takes issue with the “alone” at the start of this quote. It is clear that the point of Boolos’s paper is not so much to defend the claim that all alternatives to ZF (including Quine’s New Foundations, NF) lack independent motivation. In fact, this claim is supported by little else than a reference to Russell who said that no logician unaware of the paradoxes would ever have come up with Quine’s system of NF. The main idea defended in the paper is that ZF is well motivated. I do not here want to address the main issue between Morris and Boolos over whether any independent motivation can be given for NF.Footnote ²⁷ What I want to address next the issue between Boolos and Quine over whether ZFC is arbitrary and ad hoc and divorced from intuition, or founded on a natural and intuitive conception of set.

The point I want to make here is that the situation that Boolos describes in the beginning of his paper is very much an accurate description of Quine’s view on set theory. Here is the opening of Boolos’s paper where he lays out the position that he seeks to overturn.

The view here described only as what “one might come to believe” is not attributed to Quine. That said, I want now to show that this very much is Quine’s view on the subject.

Let us begin with what Quine says about set theory in his 1939 piece “A logistical approach to the ontological problem.” Here we see Quine clearly advancing the view that all responses to the paradoxes are merely ad hoc stipulations unsupported by intuition:

In his Mathematical Logic of 1940, Quine makes much the same point:

In his “Logic and the Reification of Universals,” of 1947, Quine writes, “Classical mathematics has roughly the above theory as its foundation, subject, however, to one or another arbitrary restriction, of such kind as to restore consistency without disturbing Cantor’s result” (Quine, Reference Quine1948a, pp. 121–122, my italics). In “On what there is” of 1948, there is a very short discussion of set theory and Quine makes exactly the same point. “These contradictions had to be obviated by unintuitive, ad hoc devices; our mathematical myth-making became deliberate and evident to all” (Quine, Reference Quine1948b, p. 18).

The discussion of sets from “Ways of paradox” is probably the best illustration of this attitude on Quine’s part.

In Word & Object, Quine is quite explicit in holding exactly the kind of position that Boolos is attempting to refute. Remember, Boolos is arguing against the view that any consistent set theory is some kind of arbitrary contrivance made simply to avoid paradox. When he speaks of ways of coming up with consistent set theories Quine says:

So we see that Quine is incredibly consistent on this point. There is one natural comprehension principle for introducing sets and that is the naive comprehension principle. But given the contradictions we need to accept an unnatural, ad hoc, and unmotivated alternative. Quine clearly sees the case of set theory as bleak in a way that, for instance, the case of ordered pair is not. Sure there are mutually inconsistent explications of the ordered pair, but we can identify beforehand what is a non-arbitrary, intuitive/conceptual truth, about this concept which must be preserved by any explication. This is precisely what we can’t do in the case of sets. If we come up with some set theory which avoids paradox, it has no intuitive motivation and is justified on “bleakly pragmatic” grounds. There is no way in this case to separate the principles which are conceptual truths about our concept of set.

I do not want to rehash Boolos’s arguments against Quine’s position. Boolos (Reference Boolos1971), himself, does an excellent job of arguing that the axioms of Z (ZFC without replacement or choice) are true relative to the iterative conception. The iterative conception of set is the view that we start with a collections of individuals (if any) at the 0th stage and then we iterate the powerset operation through the ordinals. Any set which exists, first appears at some stage in this cumulative hierarchy. The cumulative hierarchy is then a structure relative to which the axioms of Z are true. Further axioms like replacement and choice are justified by their consequences. They entail intuitively correct statements and not obvious falsehoods. This is sometimes (e.g. Russell and Whitehead in the Principia) referred to as inductive evidence for the truth of these axioms.

The title of this Element is Mathematics Is (Mostly) Analytic. The reason for the qualifying word was just touched on in the previous paragraph. Some mathematical axioms are arrived a by an analysis of some existing conception. According to Boolos, this is true of Z but not the axioms of Choice and replacement. These latter axioms are arrived at inductively. There are undoubtedly many more examples of such inductively justified axioms in many sub-fields of mathematics. Such axioms as well as statements that require them for their proof do not count as analytic as I have defined it here (they are not arrived at by analysis or explication). That said, a great deal of mathematics does count as analytic in the sense defined in this Element. In particular we have seen that this applies to all of arithmetic and, as per Boolos, most of the axioms of ZF set theory.Footnote ²⁸ Additionally even if the axiom of choice is not analytic according to the present account, a statement like “If AC, then Zorn’s Lemma” would be.

2.6 Quine and Analyticity: Conclusions

We saw that Quine rejected Carnap’s attempted explication of analyticity in terms of meaning postulates because it was purely, in his mind, an ad hoc replacement for the ordinary concept, and not, by Quine’s standards, a proper explication. Quine, unlike Carnap, saw it as important, in the course of providing an explication of some concept, to identify favored contexts. That is, we must begin by identifying what it is about the concept which we wish to preserve. Carnap’s meaning postulates were never intended to do this. We also saw that Quine’s distinction between what we called favored contexts and don’t-cares is exactly what is needed to identify, in a motivated and natural way, what Carnap called meaning postulates. When doing so, however, we do not arrive at a concept of analyticity exactly like Carnap’s, but we get an account of analytic truth that separates conceptual truths from what is stipulated regarding don’t-cares.

Concerning Quine’s views on the analyticity and mathematical truth, we saw that Quine continued to hold that all of mathematics was analytic until right before “Two Dogmas $\dots$” when he rejected the concept altogether. When he did allow some statements to count as analytic, he reverses his position. Some basic logic and statements like “there are seven days a week” count as analytic, but not much else. In particular, not very much of mathematics will count as analytic. That said, it was shown that in the definition of analyticity considered here, all of arithmetic counts as analytic, according to Quine’s own pronouncements. Also if we reject Quine’s views that all set theories with restricted comprehension are purely ad hoc and unmotivated, then most of the axioms of set theory can be seen as analytic as well.

3.1 Waismann on Frege’s “Mistake”

Between 1949 and 1953, Friedrich Waismann wrote a series of papers on the subject of analyticity. Overall he is quite critical of various attempts to define analyticity. He begins by considering suggestions on how to define analyticity. One suggestion he associates with Arthur Pap, who described analytic statements as statements whose truth follows from the meaning of the terms involved. Waismann expresses his sheer bafflement at this proposal:

Since Waismann mentions Frege here, I will address Waismann on Frege rather than Pap. The reason for this is that, in this case, the difference between the literal interpretation of Frege’s words and what Frege actually meant can be clearly shown. With respect to the literal interpretation of Frege’s words, Waismann is at a loss to see how any reasonable person could assert such a thing. “If [$\dots$ someone] tells me that an equation follows from the meaning of its terms, or that an analytic statement is one whose truth follows from the meaning of its terms, I am absolutely at a loss to make head or tail of it” (Waismann, Reference Waismann and Harré1949/1968, p. 125). Despite Waismann claiming that Frege gave “not the slightest hint” of what he meant, if we dig just a little into what Frege meant here, we see that a straightforward interpretation suggests itself clearly.

Let us look at the passage that Waismann takes issue with. In this part of Grundgesetze, Frege is responding to Thomae who holds a formalist view of mathematics. In particular, that mathematicians lay down rules for symbol manipulation, but the assertions expressed should be seen as analogous to configurations on a chess board, and not really making an actual claim about any objects and their properties. Frege discusses this position in some detail and concludes by making the following claim:

So on the face of it, Frege is clearly putting forward the kind of view Waismann finds baffling. In fact, Frege’s pronouncement here may be even more baffling, as he is speaking of the rules following from the Bedeutungen and not the Sinne of the expressions. Most who think of truth in virtue of meaning are thinking of truth following from the sense not the reference of expressions. However, if we look a little closer, we see that what Frege is doing is essentially standard practice for him. I will claim that an ounce of charity would avoid this problem, but I will get to that shortly, after a brief discussion of what is contained in these sections. Frege is trying to put forward a position which avoids the problems he sees with both a more classical conception of the real numbers and a modern formalistic one. The classical position defines real numbers as ratios of spatial magnitudes, and has the problem of not being able to explain the generality of the application of real numbers. For instance, Frege points out, masses and light intensities can take on any real value. Why should the laws of relations of spatial magnitudes apply to what is not spatial? On the other hand, the formalist position of Thomae, sees the laws of real numbers as not making any claim (true or false) about anything, and so has trouble explaining the applicability of the laws of real number at all. Frege’s strategy is to define the identity of magnitude-ratios (of any kind) in terms of an identity between value-ranges. Frege explains the situation as follows:

Frege takes himself to have avoided the problems with the classical definition which is too closely tied to spatial magnitudes. At the same time, according to his definition, laws of real numbers do state facts concerning objects, and so his definition avoids the problem he sees with a formalist account. It is this that he wants to stress in the quote that Waismann is critical of. That is, Frege wants to insist that the laws are true of a range of objects (magnitudes). Frege is not making a significant philosophical error here, but merely speaking slightly imprecisely, and in a manner, as mentioned, that an ounce of charity ought to clear up. He does not really mean to say that certain propositions (or rules) follow from certain objects. What he obviously intends here is that the laws of real number (the rules he is speaking of), on his interpretation, concern objects, and they follow from the constructive definitions he used to introduce the terms for such objects. I take it to be completely clear that this is what Frege intended, and that he did not intend to assert that a certain entailment relation holds between certain objects and propositions. As I mentioned earlier and as we will see with a couple of examples subsequently, this was quite standard practice for Frege. With this clarification of what Frege intended, it is then Waismann who is making the mistake of not giving a sufficiently charitable interpretation of Frege’s words here, rather than Frege making a baffling philosophical error.

When Frege speaks of certain rules following from the meanings of symbols, he quite obviously means following from the definition, which fixes the meaning of the term. I said above that this was fairly standard practice for Frege and we can see Frege making similar moves from very early on. Take, for example, this quote from Begriffsschrift, where Frege is discussing a proposition which serves to define a newly introduced symbolism:

Notice, in this quote, Frege speaks of the meaning of a sign being fixed by a definition and that this renders the proposition analytic. Although this proposition does not introduce any new objects, we see a close connection, in Frege’s mind between the meaning of a symbol and the proposition which introduces it. What I am suggesting here is that this close connection between the meaning of a symbol and the definition which introduces it explains his slight misstep in talking about something following from the meanings when he intends to speak of something following from a definition.

Frege believed that in the construction of a system, of arithmetic for instance, even if we use our existing terms (e.g. individual numbers), the meaning of the terms are to be completely determined by the definitions which introduce them.

Clearly, Frege sees a close connection between the meaning of the terms and the propositions which introduce them. So when he speaks of something following from the meanings [bedautungen] of the terms, this is not a major confusion on Frege’s part, but simply shorthand for follows from the propositions which determine the meaning of the term.

Waismann is baffled by Frege’s talk of something following from the meaning of a term. But if we see such talk as being shorthand for following from principles which characterizes the meaning of a term, then we see that there is no reason for any puzzlement. In his Waismann (Reference Waismann and Harré1956/1968), Waismann sees the goal of philosophy as developing various important insights. I think it is somewhat ironic that Waismann’s mistake of attributing an incoherent position to Frege, leads to an important insight on the question of analyticity. I take it that, in general, those who attack as incoherent the idea of truth in virtue on meaning, are making essentially the same mistake as Waismann. Any talk of somethings being true in virtue of meaning, following from the meaning of a term or being made true by the meaning of a sign, is always to be charitably interpreted as following from principles which characterize the meaning of a term. I take it that no one, if pressed for clarification, would hold that entailment relations hold between meanings and propositions.Footnote ³⁰ In the next sections, I wish to discuss Boghossian and his distinction between what he calls the metaphysical and epistemic conceptions of analyticity. Boghossian finds the metaphysical conception mistaken and baffling, but wishes to defend an epistemic interpretation. With this clarification of the situation between Frege and Waismann in place, we can now turn to the principal target of this section: Boghossian’s criticism of truth in virtue of meaning.

3.2 Boghossian and Metaphysical Analyticity

In his very well-known paper “Analyticity Reconsidered” (Boghossian, Reference Boghossian1996), Boghossian distinguishes between what he calls metaphysical and epistemic accounts of analyticity. My plan here is not to straightforwardly defend one of these positions over the others. What I want to do is rather argue that separated from some confusions, the metaphysical view of truth in virtue of meaning is a view of analyticity much like the one discussed in the previous section. The epistemic conception of analyticity, on the other hand, essentially begins with this same conception of analyticity, and then, I will argue, superimposes an epistemic dimension on top. This epistemic dimension would allow us to show, in principle, if the details could be worked out, that we can have a priori knowledge of analytic truths. But it is important to note that the way analytic truths are singled out does not depend on this epistemic superstructure.Footnote ³¹

Let us begin with Boghossian on the distinction that is now our focus. He introduces it by saying that on his preferred interpretation of analyticity:

He then continues:

Concerning the metaphysical conception, Boghossian states that a defender will have to maintain that “our meaning p by S makes it the case that p.” He then goes on to claim “this line is itself fraught with difficulty. For how can we make sense of the idea that something is made true by our meaning something by a sentence?” (Boghossian, Reference Boghossian1996, pp. 364–365). This is the main argument against the metaphysical view. The main argument against what he calls the metaphysical view is that it is ultimately incoherent. There are two additional arguments that I will deal with shortly, but the principal objection is this one. Let us look at this now. How is the truth of some claim supposed to follow from our meaning what we do by a claim? Of course, this is exactly the objection which Waismann raised against Pap and Frege. And the response is the same too: anyone who seriously discusses something following from the meaning of a term, does not mean to posit novel entailment relations (or some new relation like entailment) between meanings and propositions, but, more charitably, means that a certain proposition follows from propositions which characterize the meaning of a term. Likewise, propositions which characterize the meaning of a term make certain claims true (that is, they entail them). These propositions which characterize the meaning of a term (or terms), would presumably fall under the favored contexts which were discussed in the last section. As such, the metaphysical view, minus the confusion pointed out, amounts to the same position that was discussed in the last section.Footnote ³² I would not, however, call this a metaphysical conception of analyticity, but rather a semantic one.

But as just stated, beyond this Waismannian bafflement there are a couple of other arguments against the metaphysical conception of analyticity. One of the further problem, that Boghossian sees with the metaphysical view, is that it is tied to the analytic conception of necessity. The analytic conception of necessity is most closely associated with Carnap, and Carnap held, for instance in The Logical Syntax of Language, that “Necessarily-P,” should be understood to be the same as “ $\left(P\right)$ is analytic.” After Syntax, Carnap became more tolerant of modal logic, but still held that there is a strong connection between necessity and logical truth. From our perspective, we can see Carnap as only ever wishing to interpret the modal operators as asserting logical necessity. We can also see that there are a range of other possible interpretations for these operators. But even if there are historical connections between the view that Boghossian is attacking and the analytic understanding of necessity, this connection is completely accidental. There does not seem to be anything preventing one from holding something resembling what Boghossian calls the metaphysical view of analyticity while rejecting the analytic theory of necessity. The only thing preventing such a view is that the metaphysical view is not a clearly defined position but a confused position associated with the view that entailment relations exist between meanings and propositions.

The other main criticism Boghossian offers for the metaphysical view is that it is too closely tied to conventionalism. Once again, it is true, at least in the case of Carnap, that there is some connection to conventionalism. Carnap, when discussing the principle of tolerance in The Logical Syntax of Language, talks of establishing conventions instead of setting up prohibitions. However, Carnap’s ties to conventionalism are a lot weaker than much of the commentary on Carnap’s work would suggest. Carnap saw certain conventionalist views, which he calls “radical conventionalism,” as leading to a coherence theory of truth. When Carnap discusses conventionalism at all it is typically to distance his position (or those of allies like Poincaré or Neurath) from such radical views. Also, when Quine charges him of holding a conventionalist position with regard to logical truth, Carnap writes:

Of course the position of conventionalism about the meaning of individual words is universally accepted. So, at least by the time of writing his responses for the Schilpp volume, Carnap held only an extremely weak form of conventionalism. Interestingly, notice that, in this quote, Carnap is guilty of misspeaking in the manner of Frege in the last section. When he speaks of “logical relation holding between the given meanings,” he means to refer to logical relations which hold given the propositions which characterize the meaning of the terms. Additionally, of course, besides the question of how much of a convetionalist Carnap was, there is the question of whether anyone defending a metaphysical conception of the analytic needs to also hold some unacceptably strong version of conventionalism. I do not see why this would be the case.

Before turning to Boghossian on the epistemic conception of analyticity, there is one more point that I wish to address. This point is a rather a historical point, but an important one. In fairness though, Boghossian writes, “I should emphasize right at the outset, however, that I am not a historian and my interest here is not historical” (Boghossian, Reference Boghossian1996, p. 361). That said, much of the article is a discussion of historical views. But there is one mistake on Boghossian’s part that, I believe, merits correction, as it will come up again in a later section. In explaining how his own view does not amount to conventionalism, Boghossian writes, “I anticipate the complaint that the entailment between Implicit Definition and Conventionalism is blocked only through the tacit use of a distinction between a sentence and the proposition it expresses, a distinction that neither Carnap nor Quine would have approved” (Boghossian, Reference Boghossian1996, p. 380). While it is true that that Quine as a “confirmed extensionalist” (see Quine, Reference Quine2008), would never have accepted an account of propositions as individuated by their intensions, this is exactly what Carnap offers in Meaning & Necessity. Here Carnap even describes propositions as “objective, nonmental, extra-linguistic entities” (Carnap, Reference Carnap1947/1956, p. 25). It is, I believe too easy, and all too common to portray Carnap as somewhat dogmatically rejecting anything that seems too metaphysical. In truth however, Carnap is happy to accept such things as propositions once they can be given a sufficiently clear definition.

3.3 Boghossian and Epistemic Analyticity

In the previous section we saw that Boghossian rejects the metaphysical concept of analyticity. In part, this was because he takes the view to be incoherent, and in part because of associations with other views that have since fallen into disrepute. We saw the connections with other views in disrepute was completely contingent. More importantly, we also saw that we can eliminate the incoherence by interpreting following form the meaning of terms or made true by the meaning of a term as following form principles which characterize the meaning of terms. This leave us with a non-epistemic relation, and, in fact, the relation which I am calling analytic is of this type. If C is a useful concept and P is a proposition which characterizes the meaning of C, then P would be part of the favored context for this concept. That is, any explication of the concept would have to satisfy this principle. I would, as mentioned, not refer to this position as a metaphysical position, but rather as a semantic one. The main purpose of Boghossian’s paper, however, is, not to attack the metaphysical view, but to defend the epistemic conception of analyticity. It is to this that we now turn.

As we saw, Boghossian states, with respect to the epistemic concept of analyticity that “a statement is ‘true by virtue of its meaning’ provided that grasp of its meaning alone suffices for justified belief in its truth” (Boghossian, Reference Boghossian1996, p. 363). Obviously, this falls far short of being a clear and informative definition on its own (and, of course, is not put forward as such). Boghossian, therefore, considers ways of augmenting this suggestion in order to arrive at a more informative idea of the analytic. He first considers the notion of what is called Frege-analytic. A proposition is Frege-analytic if it can be transformed into a logical truth by substituting synonyms for synonyms. Concerning Frege-anlyticity, Boghossian writes:

Notice that the main reason this is an incomplete account of a statement’s epistemic analyticity is that Frege-analyticity is not an epistemic relation. If, for a moment, we suppose that synonymy is an objective relation between terms and entailment is an objective relation between propositions, then statements could be Frege-analytic without anyone ever knowing them. Boghossian himself repeatedly talks of working within a background of realism about meanings, so these assumptions seem appropriate here. We see that in filling in the details of his account of epistemic analyticity, Boghossian begins with a relation that is not epistemic, and then argues that we can complete it by assuming epistemic principles, which make all Frege-analytic truths knowable. “The question whether facts about the sameness and difference of meaning are a priori cannot be discussed independently of the question what meaning is, and that is not an issue that I want to prejudge in the present context” (Boghossian, Reference Boghossian1996, p. 367). This is a major issue to set aside in arguing for an epistemic account of analyticity as it is what makes the account epistemic.

Now, as a matter of historical fact, Quine, in his last days of trying to arrive at a definition of synonymy (in order to define analyticity), found himself forced (see Verhaegh, Reference Verhaegh2018) to assume that we must be able to know if two terms are synonymous, if they are so. It was actually the problems involved in trying to make such an account workable that likely lead him to reject analyticity and synonymy altogether. In 1943, in a letter to Carnap, Quine writes that he has no idea whether human and rational animal are synonymous.Footnote ³³ So, even if Quine, later, thought it important to assume we can know two terms are synonymous when they are so, this assumption was on shaky ground from the start. On the assumption that facts about synonymy are knowable, Quine would not have thought it was a problem to explain the analyticity of “all bachelors are unmarried.” Quine, right up until his ultimate rejection of the concept of analyticity (and even after), maintained that were we to have an acceptable definition of synonymy, we could in turn define analyticity. So the assumption Boghossian makes here, we see, is exactly a principle that Quine himself thought we must accept if we are to provide a definition of synonymy (and ultimately analyticity). It is also a principle that is likely not true, or even if fairly widely discussed, there has never been a definitive argument in favor of this principle and lots of reasons to reject it. In 1949, Quine describes this principle, in a letter to Goodman (see Verhaegh, Reference Verhaegh2018, pp. 178–180) as a Tarskian condition of material adequacy on any explication of the term meaning. As Quine saw things, at the time, we face a trichotomy, either we can (1) successfully explicate meanings (and define their identity conditions via a definition of synonymy – no entity without identity, after all), or (2) we do away with talk of meanings, synonymy and analyticity, or (3) we continue to misleadingly talk of meanings even though we can’t successfully define their identity conditions. At the time, Quine was still holding out hope for the first alternative. Also at the time, Quine described Nelson Goodman as brave, but foolhardy for taking the second option.Footnote ³⁴

Boghossian says that Fregean-analyticity is incomplete as it does not explain how analytic truths are knowable a priori. It is also incomplete in the sense that it cannot explain the analyticity of such truths as “Everything colored is extended.” Another problem is that logic is assumed by the notion of Frege-analyticity (logical truths are trivially Frege-analytic). In order to account for the possibility of an informative account of the a priority of logic, Boghossian appeals to the idea of implicit definition. He calls what can be shown to be true by appeal to implicit definitions Carnap-analytic. Appealing to implicit definitions could also explain the analyticity of sentences like “Everything colored is extended” which fail to be Frege-analytic. One of his main lines of argument here is that one can accept implicit definitions as an explication of analyticity without committing oneself to conventionalism or non-factualism. I will discuss the question of the non-factuality and conventionality of analytic truths in the next section, but right now, let me say a few things about Boghossian’s account of epistemic analyticity now expanded to include what is both Frege-analytic and what is Carnap-analytic.

As we saw, in the case of Frege-analytic, the notion of synonymy is doing most of the work, and synonymy is not an overtly epistemic notion. What transforms this into an overtly epistemic account of analyticity is the principle that the synonymy of two expressions is knowable a priori. But notice, that while this principle plays the role of making the account overtly epistemic, it is not needed to characterize which truths are analytic. It is for this reason that I say the epistemic dimension of epistemic analyticity is a superstructure that sits atop a non-epistemic notion. At least this is the case for Frege-analyticity, but exactly the same thing can be said once the account is extended to include Carnap-analyticity. There is nothing particularly epistemic about Carnapian-analyticity. Boghossian would need some story about how the implicit definitions are a priori knowable in order to make this into an epistemic conception of the analytic. Perhaps the case can be made that what is Frege-analytic or what is Carnap-analytic can be known a priori. In the case of Frege analyticity, Boghossian explicitly sets aside the question of whether any synonymy relation is knowable a priori. In the case of what is true in virtue of implicit definitions (or Carnap-analytic), Boghossian writes:

Here we see Boghossian again mostly avoiding the question, but I think this quote is quite telling in another respect. If we assume the premises can be known (perhaps even a priori), then the role of the epistemic agent in this epistemic account of analyticity is merely to perform a certain logical inference (usually just modus ponens) to the truth of the analytic claim. But this then means that the role of the epistemic agent in the account could be replaced by the notion of consequence. It is the assumption that we can know the premises which is doing the real epistemic work – and this is something Boghossian does not argue for. Again, beyond what is outright assumed about our knowledge of meaning facts, there does not seem to be anything essentially epistemic about the epistemic account of analyticity.

As Boghossian presents things, what is primary is the definition which holds that a claim is analytic if knowledge of its meaning is sufficient for knowledge of its truth. But this on its own is not a useful definition without giving any idea of how this is to work. It is telling that when giving a more detailed account of how this is possible, he appeals to principles which characterize the meaning of an expression (either by stating synonymy relations or via implicit definitions), argues that this is plausibly knowable a priori, and then shows how an agent aware of all this could come to know an analytic truth. But what this suggests is that what is primary is the non-epistemic relation between a set of propositions (or rules of inference) which characterize the meaning of an expression and an analytic claim.

In fact, one gets the impression from Boghossian that there are many radically different ways one might try characterize the notion of analyticity (even if not all are as successful). One might try to define it as a metaphysical notion, as an epistemic notion, or perhaps, as something else entirely. However, what I think comes out of all this is that all reasonable attempts to define analyticity agree that what is analytic follows from principles which are characteristic of the meaning of an expression. One might even say, any explication of analyticity will have to preserve this feature.Footnote ³⁵ There are not then, radically different ways to characterize analyticity. There is not a metaphysical and an epistemic analyticity, and no one need explore if there is an aesthetic, ethical or mereological conception of analyticity. There is really but one conception of the analytic and it is a semantic concept.

Boghossian thinks it was an important mistake of early twentieth-century philosophers to prejudge that all necessary truths were a priori. Boghossian is at least in danger of assuming too close of a connection between the a priori and the analytic.Footnote ³⁶ Boghossian suggests that any account of analyticity that is not “overtly epistemic” is in danger of being incoherent. That said, the epistemic component of his epistemic conception of analyticity is grafted onto a notion that is not epistemic and perfectly coherent. While he suggests associating the analytic and necessary truths too closely is a mistake, I would maintain that we need not assume anything from the outset of a link between the analytic and the a priori. In the next section I would like to look at the relations between analytic truths and other notions including stipulations of conventions, necessary truths, and regular truth. I will also explore what can said about analytic truths which make ontological demands.

4.1 Analyticity, Stipulation and Content

The early twentieth-century view, associated most strongly with Carnap, was that analytic truths are true by stipulation and that they are void of content. Quine criticizes Carnap for maintaining there were different kinds of truths. Quine famously asserts in “Two Dogmas $\dots$” that Carnap can only preserve his double standard on ontology by appealing to the analytic/synthetic distinction.Footnote ³⁷ It is true Carnap maintained that analytic claims are void of content. For instance, in The Logical Syntax of Language, Carnap defines the content of a proposition as the set of its non-analytic consequences. If we accept such a definition, then analytic truths, including all of mathematics, are void of content. What I think this shows is that the question of whether mathematical truths, if analytic, are void of content, comes down to what we choose to mean by content. I will not settle question of what we ought to mean by content here. I will discuss Gödel’s criticism of Carnap on exactly this point. We will see that both Gödel and Carnap thought of mathematics as following from conceptual truths. What I take this to show is that the view defended here is compatible with both Carnap’s and Gödel’s positions. While I am quite sympathetic to Gödel’s arguments here, without a definition of content, there is no way to take a side on this issue. The important point is that the claim, then, that mathematics is analytic (in the sense of this Element) is not equivalent to the claim that it is tautological, empty, or devoid of meaning.

Before turning to Gödel’s criticism of Carnap, I would like to address the question of whether the view I am defending amounts to holding that mathematics is true by convention or stipulation. First of all, it can be straightforwardly pointed out that the way analytic has been defined here, it is not identical to what might be stipulated. Recall Quine’s example of the ordered pair:

$⟨x,y⟩=⟨w,z⟩\text{only if}x=w\text{and}y=z$(1)

is identified as what must be preserved by any analysis of ordered pair. In my terminology this and what follows from it are called analytic.Footnote ³⁸ We might stipulate that the ordered pairs are to be defined as per Kuratowski, but this stipulation is stronger than what I have identified as analytic of ordered pair. So, what is analytic and what is stipulated are at least not identical. Furthermore, we can ask if (1) is a matter of convention or stipulation. Here we have to be a little more careful. Of course, the meaning we associate with the particular marks “ordered pair” or “ $<,>$” is a matter of convention. But it does not seem right to say that it is purely a matter of stipulation that an ordered pair must satisfy (1). For, if one were to stipulate a definition of “ordered pairs” which does not satisfy (1), that may be a proposed definition of “ordered pairs,” but it would fail to introduce ordered pairs. If this is right, then that ordered pairs satisfy (1), is not a stipulative truth but a conceptual one. So what I am calling analytic is neither identical with, or even a subclass of purely stipulative truths.

Let us now turn to Gödel’s criticism of Carnap on the question of whether mathematical sentences have content. Gödel, after having already made contributions to the Schilpp volumes on Russell and Einstein, agreed to contribute an article for the Library of Living Philosophers volume on Carnap. Gödel wanted to show that Carnap’s positivistic “syntactical” interpretation of mathematics failed to do justice to how mathematics is actually used in the sciences. He produced six drafts of the work, over six years, but in the end none were to his satisfaction and he withdrew from the project. The third draft, the most complete, argues for two main points. The first concerns Carnap’s overly liberal interpretation of what is to count as “syntactic.” The second major point Gödel wished to establish, which is more relevant to our concern here, was that Carnap’s definition of “content,” begs the question in favor of a formalistic interpretation of mathematics:

Concerning how Carnap uses the term “content,” Gödel writes:

Gödel takes it that one of the central tenets of the syntactic interpretation of mathematics is that, in mathematics and its application in the sciences, one needs never appeal to our intuitive understanding of the standard models of our theories. To assume that we can know and speak meaningfully about the standard model is, in Gödel’s view to abandon the syntactic approach. He argues that without appeal to the intuitive model we have no reason to see the mathematical theories we use in the sciences as consistent. We could still treat them as stipulations, but then by their not leading to false empirical consequences, we obtain inductive evidence of their consistency. Gödel stresses that when we test a theoretical claim we are always assuming a great deal of mathematics. It is then only ever both mathematics and physical theory that get tested. “If mathematical intuition is accepted at face value, the existence of a content is evidently admitted. If it is rejected, mathematical axioms become open to disproof and for this reason have content” (Gödel, Reference Gödel, Feferman, D. Jr., Goldfarb, Parsons and Solovay1995a, p. 348). Given this parallelism between theoretical and mathematical claims, Gödel argues it is question begging to declare that theoretical propositions have content, but mathematical claims are contentless.

We see in this criticism an opposition between Gödel, who rejects the claim that mathematics has no content, and Carnap who has defined content in a manner such that analytic claims, by definition, are contentless. So it seems the question of whether mathematics is empty or contentless, will turn on what we mean by empty or content. That said, both Carnap and Gödel hold quite similar views about the role of meaning in determining mathematical truth. That is, they both hold that the meaning of mathematical expressions is determined by fixing the meaning of the individual words, and the rest is a consequence of this. Recall the passage from Carnap’s reply to Quine in the Schilpp volume (quoted earlier):

Compare this to a remark from Gödel’s unpublished draft IV of his “Is mathematics syntax of language?:”

So both Carnap and Gödel hold that mathematical truths are analytic in the sense of following from the propositions which fix the meanings of the individual terms. Gödel, however, wanted to stress that mathematics ultimately rests on conceptual truths:

This is very much in line with the view of analyticity proposed in this Element. Gödel is here stressing that mathematical truths are conceptual truths, as opposed to being merely stipulated by definition. One could define the content of a claim as content concerning the world of sensory experience, or one could define content more broadly so that mathematical claims have content. Without such a definition, there is no way to decide the matter between Gödel and Carnap. The focus of this Element is on the analyticity of mathematics, and I will not be putting forward a definition of content. The goal of the present section is to argue that whether mathematical claims have content or not is not decided by the present account of analyticity. Nothing about the claim that a sentence is analytic, in the sense defended in this Element, implies that such sentences are empty or tautological. All this said, Gödel clearly wants to hold that conceptual truths are not only contentful but true in a robustly realist sense. He says in “Is Mathematics the Syntax of Language” that the fundamental question of philosophy is how we can begin by reasoning about concepts and end up with objective knowledge of a range of objects. We will, in the next section, turn our discussion to the question of the sense in which conceptual truths are true, and in the following section we will deal with the ontological implications of analytic propositions.

4.2 True in Virtue of Meaning and True

I have argued that a truth of arithmetic, for instance, follows from what is analytic of our concept of number. I have also argued that the bafflement, which Waismann expresses faced with “Frege’s mistake,” is essentially the same incoherence which Boghossian sees in the notion of something “made true by our meaning something by a sentence.” Talk of truth following from meaning, or of something “made true” in virtue of what we mean, has always, reasonably interpreted, I contend, meant following from principles which characterize the meaning of a term. But so far this does not add up to truth in virtue of meaning. If what is analytic of the concept of number characterizes the meaning of this concept, what reason do we have for thinking these express truths? If the basic principles which characterize the meaning of arithmetical expressions are not themselves true, then what follows from them has no right to be described as true in virtue of meaning. Boghossian, for instance, says that the only answer we can give to what makes anything true is something like the world. It follows from what I have said so far that anything we view as a truth of arithmetic is analytic, nothing I have said so far implies that they express truths. Nominalists like Hartry Field (Field, Reference Field1980) have argued that such claims are false. Fictionalists, like Mary Leng (Leng, Reference Leng2010), believe that all mathematical statements on their straightforward interpretation are false, even if they are “true according to the story of mathematics.” Such philosophers may be happy to admit that statements of arithmetic are analytic in the sense defined here, but would clearly reject any claim that these claims are true in virtue of meaning. That is, nominalists and fictionalists may admit that any analysis of the concept of number would have to satisfy the laws of arithmetic.Footnote ³⁹ They may also not object to many statements being logical consequences of what we take to be true of our concept of number. What they object to is our calling what is true, on any analysis of our concept of number, true.

Are we not back then to saying that ultimately, it is the world that makes any claim true? This is certainly as Boghossian sees the matter. I believe we can defensibly say that the truths of arithmetic are true in virtue of meaning. The account of analyticity I have been putting forward is based on Quine’s conception of an explication. But now we are asking about the truth of these analytic claims. To deal with this problem, I will begin by outlining a relativized notion of truth. Note that the mere acceptance of this relativized version of truth does not commit one to the thesis of relativism. That is, one can talk of truth in this relativized sense without rejecting a univocal (absolute) conception of truth. I want use this relativized notion of truth to analyze the problem of certain things which are conceptual truths regarding the concepts involved, but which are generally considered false. Such cases seem to be the primary motivation for claiming that not all conceptual truths are true. But I will argue that whether or not we accept some univocal conception of truth, these do not pose an insurmountable problem for the current view.

So far, in the previous sections, we have seen that analytic claims are conceptual truths of sorts, that are not purely stipulated or conventional. The account of analyticity presented is built on top of Quine’s distinction between what must be preserved by any explication (“favored contexts”) and the “don’t-cares.” This distinction Quine takes to be important, and in many cases completely clear. But the distinction is relative to our purposes and interests. So we end up with a somewhat relative notion of analytic that is ultimately founded on something pragmatic rather than something absolute. I believe it is this pragmatism that can answer one of the problems that faces any account of “true in virtue of meaning;” namely, is everything that is true in virtue of meaning true? One obviously faces a lot of bad company objections here. For instance, is it not true of the logical notion of set held by Frege and Russell around the turn of the twentieth century, that every expressible property determines a set? This would seem to be a case of something that is true in virtue of meaning but not true. Many might also point to such examples as conceptual truths of phlogiston theory (e.g. “combustion is dephlogistication”). Others may ask about the status of the conceptual truths in logical system which includes a tonk operator. Presumably there are many examples of things that are true in virtue of meaning in a certain context, that we, from our perspective, think of as simply false. Such examples would seem to be a problem for the type of view considered in this Element.

I do not think such problems are a substantial hurdle for the view under consideration. As the account of analyticity defended here is based on a Quinean distinction, I think it is somewhat fitting that my answer to this difficulty will be appealing to pragmatism (and, perhaps, holism). No matter what, we are going to employ a range of concepts in our understanding of the world. Any such range of concepts used by a group at a particular time, I will call their conceptual scheme. Given a conceptual scheme S, we can define the notions of $analyti{c}_S$, $backgroun{d}_S$ and $tru{e}_S$. $Analyti{c}_S$ includes everything that is analytic in the sense under consideration for that group at that time, we will assume that some set of inference rules count as analytic too, and so, this includes a notion of consequence. $Backgroun{d}_S$ would include everything $analyti{c}_S$, but also all sorts of other background assumptions: things that are straightforwardly stipulated, as well as, perhaps, certain theoretical assumptions. $Tru{e}_S$ would include $backgroun{d}_S$, as well as all sort of claims accepted on the basis of experience. So, in particular, anything $analyti{c}_S$ is $tru{e}_S$. That is, those who have $backgroun{d}_S$ take every proposition in $backgroun{d}_S$ to be true. To take a simple example, anyone who shares our conceptual scheme, takes it as true that Wednesday comes immediately after Tuesday. So anything $analyti{c}_S$ is “true in virtue of meaning” in at least the (relativized) sense of $tru{e}_S$.

Now it could happen that a certain set of background assumptions $backgroun{d}_S$ lead those who hold these assumptions into an incoherent position. On the basis of $backgroun{d}_S$ and other claims that are $tru{e}_S$, it follows that some claim P is true, but at the same time, given the background we have reasons to hold that P is false.Footnote ⁴⁰ When a group finds themselves in any incoherent position, they will have to make changes somewhere in their set of assumptions $backgroun{d}_S$. There is, of course, no reason that conceptual truths are immune from revision, but to abandon a conceptual truth which characterizes a concept is to abandon that concept.

Let us assume that at some time before, a group with conceptual scheme B (for before) took some statement P to characterize the meaning of a certain expression. To take something of a more concrete example, let us say that those with conceptual scheme B are those that toward the end of the nineteenth century held the logical notion of set. As has been often pointed out (Grattan-Guinness, Reference Grattan-Guinness2011, chapter 3), Cantor, for instance, did not share these assumptions, but let us consider our community small enough such that all members accepted the principle of naive comprehension P as analytic of their conception of set. This group saw sets as defined by a rule which separates everything there is into those things which satisfy the rule and those things that don’t, with there being a set corresponding to each side of such a division. So, P expresses something which is $analyti{c}_B$ and thus $tru{e}_B$. But of course this puts members of this group in an incoherent situation. Once a set-theoretic paradox is discovered, given the logic they accepted at the time, anything becomes provable. From our perspective now we accept a conceptual scheme N (for now), which is incompatible with P. We say that P is false, but this is to say that P is $fals{e}_N$. Now, likely some, at this point, will object that P is not just false in some relative sense, but false simpliciter as it leads to a contradiction.Footnote ⁴¹ In opposition to this it can be pointed out that, for instance, moving to rough set theory, or accepting a dialethic logic could allow us to preserve naive comprehension while avoiding an incoherent position. That is, even if we modify our conceptual scheme by abandoning P, this does not imply that P is false in some absolute sense. A change has to be made somewhere to the background once those who share that background are in an incoherent position, but no particular change is forced on this group. One can insert the Neurathean sailor metaphor at this point. We can discover inconsistencies in our conceptual scheme, we can alter our conceptual scheme, but there is completely neutral perspective (no dry dock) from which we can evaluate the truth of any principle which characterizes a concept.

So far we have a picture according to which, at any time, we have a set of background assumptions, and included in this background are certain conceptual or analytic truths. Relative to each of these background assumptions there is a concept of truth. Nothing so far has been assumed about whether or not there is a non-relativized concept of truth. It certainly makes sense to compare various backgrounds to a degree. While we always have some freedom in how we alter our system of background beliefs, I do not wish to suggest that all ways of altering our background are on par. We can say that one set of background assumptions leads to incoherence in a way another set does not. We may also say that parts of our former background, including some (former) conceptual truths, are no longer tenable, given our changing state of knowledge. Perhaps, one might argue, there is an ideally coherent and complete set of background assumptions which would allow us to best describe the world. Truth simpliciter would then be truth relative to this ideal background. I do not want to take any position on the existence (or uniqueness) of such a background, which could be used to make sense of truth in an absolute sense. I am though, more inclined to look at a background as a set of tools used to try to come up with a more or less complete and coherent description of the world. We accept certain conceptual truths for their utility and we abandon the concepts relative to which they are conceptual truths when they lead to some kind of incoherence, or otherwise no longer seem useful.

Even if one is inclined to accept that there is a univocal conception of truth simpliciter, of which I am at least somewhat dubious, then all this means is that something’s being a conceptual truth of our current conceptual system is only a fallible indicator of truth. We would just have to accept that certain things, which are conceptual truths in our current conceptual scheme may in fact be false (in the absolute sense). In the last section we looked at the question of whether the analyticity of mathematics implied that mathematics was devoid of content. We saw that this depended on what one means by “content.” Carnap gave a definition of “content” as the non-analytic consequences of a claim. Gödel saw this as question-begging against the Platonist. That said, both held the view that mathematics was ultimately various conceptual/analytic truths and the consequences of these. So we saw that with the same conception of analyticity, there is a spectrum of positions one might take on the question of whether mathematics has content. Likewise on the question of relativism, we can once again see Carnap and Gödel representing opposing ends of a spectrum. Carnap stresses our freedom of choice in choosing a conceptual scheme and holds that analytic truths (in his sense – containing both analytic truths in my sense together with stipulations) are selected or discarded on pragmatic grounds. Gödel, representing the other end of the spectrum, believes that although our conceptual knowledge is fallible, mathematics, which is based on conceptual knowledge, still concerns an objective world of truth:

We saw previously that both the claim that mathematics is in some sense without content, and the claim that it does have content are compatible with the claim that mathematics is (mostly) analytic in the sense discussed in this Element. This comes down to how we define content. Likewise, both the views that mathematics is true only relative to a conceptual scheme, or that it is true in a more robust, realist sense is compatible with the account of analyticity now under discussion.

If relativism is true, that is there is no univocal conception of truth, then we can only talk about the usefulness and coherence of a conceptual scheme. We can say that one conceptual scheme is open to a certain incoherence, or that it is less useful than another. But there is no perspective here to judge the absolute truth or falsity of things which are true on a particular conceptual scheme. If there is a univocal conception of truth, then this only forces us to be fallibilists about our conceptual truths. One might ask, at this point, for the justification of the claim that something is a conceptual truth of our current conceptual scheme is an indicator of truth at all. Why might not most or all of our conceptual truths be false if there is a univocal concept of truth? Here I would just say that if the univocal conception of truth shares little in the way of conceptual truths with our own conception of truth, then it might be unreasonable to judge what is true in our background according to its agreement with that background presupposed by the univocal view. Also, we have done quite a bit of work in formulating useful concepts and eliminating incoherence. If a concept is useful, why should it not be included in an ideal background? This seems especially the case in mathematics. We may still have some incoherence to purge from our conceptual scheme, but it seems there are a lot of useful concepts in there too. Others may be more pessimistic than this, but I think this strong pessimism would require strong argument.

It should be noted that, even if mathematics is relative to a certain conceptual scheme, this does not imply that it is therefore mind or language dependent. We saw earlier that Carnap after defining propositions, states that propositions are objective, non-linguistic, extra-mental entities. The reason he says this is that we do not figure in the truth conditions for claims involving propositions. This means that, for instance, even on a Carnapian view, which would be a form of a relativist position, we can still say that mathematical claims express (objective) propositions, and that since we do not figure in the truth conditions for mathematical sentences, what the propositions assert does not depend on us. Carnap might have claimed that this does not amount to Platonism, as Platonism employs a metaphysical conception of reality. However, it should be pointed out that, as we saw, even according to Carnap’s relativist position, mathematical propositions state truths that are independent of us (as we do not figure in their truth conditions). It is not clear what more a realist could ask for.

4.3 Analyticity and Ontology

In his 1939 paper “A Logistical Approach to the Ontological Problem” Quine formulates a way of asking ontological questions, which are not relative to the language system one is working in. If ontological questions are always relative to a language system, then all we can say is that a certain language system includes talk of real numbers, for example, and another system does not. Quine there famously identified the quantifiers as identifying the ontological commitments of a language. Quine then argues that we could ask, for any type of entity, whether we would need to quantify over that type of thing in a complete scientific description of the world. Is it possible, for instance to give a scientific description of the world that does not quantify over real numbers? This is now a question that is not tied any one language system. “How economical an ontology can we achieve and still have a language adequate to all purposes of science (Quine, Reference Quine1939/1976, p. 68)?” At the time, Quine hopes to demonstrate the nominalist thesis that quantification over abstract objects is not needed for a scientific description of the world. This formulation of ontological questions, as well as the project of nominalism, was clear enough to satisfy Carnap. Of course, Carnap famously objects to the use of the term “ontology,” for its ties to medieval metaphysics. That said, this is merely a terminological dispute, and Quine’s approach to ontology and his definition of the nominalist program proved very influential.

This approach to ontology, although not universally accepted, is now standard. For all kinds of ontological questions, we can ask whether a complete scientific description of the world (regimented in first-order logic) would need to quantify over the entities that interest us. As I have argued elsewhere (Lavers, Reference Lavers and Torza2015), I see two main problems with this Quinean approach to ontology. The first main problem is that this approach to ontology leads to a plethora of unpromising research programs. For every type of entity, given that we have no idea what the best version of our scientific theories (regimented in first-order logic) will look like, both the positive and negative answer to Quine’s ontological question for that type of entity are defensible. Those in support of a certain kind of entity can write a few papers pointing to the usefulness of talking of some kind of entity, and pointing out how the opponent has not yet showed that talk of this kind of entity is dispensable. The philosopher opposing talk of a certain kind of entity can point to the ontological economy of rejecting the entity in question, can make the case that not all that much overall simplicity is gained by commitment to the type of entity, and can give a sketch of how the elimination of talk of such entities might be carried out. Never is there a complete knock out blow from either side. Olympic sized swimming pools could be filled with papers which amount to little more than speculation over whether a future ideal science would need to talk of a certain kind of thing.

The second problem is a misunderstanding of Quine’s explication of ontology. That this was intended as an explication is clear from Quine’s remark: “I suspect that the sense in which I use this crusty old word has been nuclear to its usage all along” (Quine, Reference Quine1951/1976, p. 127). How economical our ontology can be while still being adequate for science is meant to replace the question of what are the ultimate components of reality. Too many contemporary philosophers, working within the framework of a Quinean approach to ontology, think that via indispensability arguments together with a holistic attitude toward science, we get an answer to the traditional ontological questions. If for instance, sets or real numbers are indispensable for science, then these entities must be part of the ultimate constituents of metaphysical reality. We get an answer to the metaphysical question via science.

Whatever the value of much of the contemporary debate about ontology, although Quinean inspired, it is not Quine’s view. In “Carnap’s views on ontology” Quine begins emphasizing that he is “no champion of metaphysics.” Carnap did not like Quine’s use of the term ontology, but, as we saw, Quine takes himself to have explicated the term. He has therefore replaced traditional ontological questions with a clear counterpart. For Quine, the usefulness of mathematics in the physical sciences is not evidence for some metaphysical claim about the ultimate existence of mathematical objects. Ultimately Quine has a deep-seated pragmatic attitude. Quine does not think we can appeal to his holism to say that the entities presupposed by our best scientific theories probably correctly reflect the ultimate constituents of the world. Quine has replaced the traditional ontological question of concerning the ultimate constituents of reality with the ontological question as he defined it. But too much discussion accepts the broad framework of the Quinean position, but then tends to think that somehow we can take an answer to Quine’s ontological question as evidence for the metaphysical question.

On these points, I am inclined to follow Quine’s more pragmatic attitude, and deny that anything, of a metaphysical nature, follows from pragmatic questions about ontology. That the world is best described by a system which includes, let us assume, the real numbers, tells us no more than that our best description of the world includes real numbers within the scope of our quantifiers. We choose a conceptual scheme which suits our needs. If we run into incoherence we change our conceptual scheme. Obviously this position is in the end quite Carnapian (in the sense of Carnap Reference Carnap1950a).Footnote ⁴² If we are to be pragmatists about conceptual schemes, we can ask about the pragmatic value of such a framework or we could possibly expose an incoherence, but the question of the correctness of a conceptual scheme is rejected.

One might, following Quine’s lead, object that while I talk of conceptual schemes as useful tools, I have not shown that to use a certain concept is a mere façon de parlé. Quine thought that this could be done only if we show how we could do without a concept in a complete scientific description of the world. Carnap thought analytic claims make no claims about the world. As I am not committed to holding that analytic truths are empty or tautological, I need not show that conceptual truths tell us nothing about the world. But, again, what the use of real numbers in our best scientific theories tells us about the world is that it is best described in a scientific language which includes the laws of analysis among its conceptual truths. I see no reason to conclude more than this.

5.1 Vagueness and Analyticity

It is certainly true that, for many concepts, it will be very difficult to identify what is analytic of them. As Bertrand Russell said, “Everything is vague to a degree you do not realize till you have tried to make it precise (Russell, Reference Russell1918/1956, p. 180)[.]” Of course there are terms in language like tall and bald, for which we are often unsure of the exact range of applicability. If analyticity is truth in virtue of meaning, and meaning is often not clear enough to determine clearly the applicability conditions for a term, then is analyticity as I have been considering it hopelessly vague? In fact, most concepts we use in everyday life are not only vague but have what Waismann called open texture (see Waismann, Reference Waismann and Harré1953/1968). Open texture concerns the applicability of the concept to possible rather than actual applications. Take for example the concept of a blanket. Blankets tend to be wide and fairly flat. That said, they can also be quite thick, but how thick? We could start with a knitted (square) blanket and then gradually increase the thickness until we are finally left with something that is clearly not a blanket but a woolen cube. At what point did the blanket become too thick for it to be a blanket? This is just one possible way a term like blanket is vague. Since we don’t encounter borderline cases like these, we don’t think of the concept as vague in this way. Most terms of everyday language are open-textured like this; they are vague in ways we have not even considered. Even if they are not typically seen as vague, they are vague with respect to a host of possible but not actual applications.

If the meanings of our everyday concepts are to such a degree vague and open-textured, and analyticity is based on meaning, is analyticity itself then not hopelessly vague? It may be a very difficult matter to say what is analytic of our concept blanket. There are presumably all kinds of conceptual truths about any concept. In fact, we might say that even in the earlier example showing the open texture of blanket, we expose a constraint on an explication of blankets: blankets are never cubes. Even if this is not the most interesting analytic truth, this example shows that while it is not all that easy to identify what is analytic of a concept, it is not impossible. In terms of other everyday terms, it is true, for example, of our conception of weekdays that there are seven days in a week or that Tuesday comes right after Monday. Therefore, much concerning what we ordinarily take to be analytic of ordinary concepts counts as analytic according to the current view.

Of course, the goal of this Element is to defend the view that mathematics is analytic. In the case of mathematics, we can say, with mathematical precision what is true of our mathematical concepts. As we saw, Quine says that all of arithmetic is analytic in the sense I am discussing here (although he does not call it that). Or to take another example, it is certainly not a vague matter whether every group contains an identity element. What must be preserved on any explication of the mathematical concept of a group? Well, the axioms of group theory, of course.

I don’t really mean to deny that there is any vagueness in what is analytic in mathematics. Some believe that by reflecting on our conception of set, we may identify new axioms which settle the continuum hypothesis. Others might argue for a possible bifurcation of set theory on the basis of our conception of set being somewhat vague. This is not an issue I intend to settle here. To take another example, Quine, we saw, held that any explication of arithmetic must preserve arithmetical laws. But Benacerraf, for instance (Benacerraf, Reference Benacerraf1965) thinks what I have called the favored contexts, should also include links to our judgments of cardinality and that the less than relation be decidable. Again I will not take a position on these issues. Rather, I just want to point out that the existence of some debates on these questions is not evidence that the idea of what is analytic of our concept of number is hopelessly vague. In fact this points to the opposite being true. Both Quine and Benacerraf believe it is very clear what any definition of number must satisfy, it is just that Benacerraf takes this to include somewhat more than Quine.

We have seen in this section that everyday language is vague to a much greater extent than mathematics. We have seen as well that there are nonetheless at least some clear cases of analytic truths involving everyday vocabulary. But this will likely suggest to many another problem with analyticity as defined here. After presenting his account of stimulus-analytic in Quine (Reference Quine1960), Quine says that this will not play the role traditionally assigned to analyticity as it will include sentences like “there have been black dogs.” What is preventing such a sentence from counting as analytic according to the current account? Well, if we look to our best attempts to define dogs (or canis familiaris) we see that this is defined by reference to these animals occupying a certain position on the phylogenetic tree. For something to be a dog it must have descended from the (now extinct) grey wolf.Footnote ⁴³ So if we look at how we characterize the most important concept in “there have been black dogs,” we see that this statement does not follow the propositions which characterize the meaning of the concept.Footnote ⁴⁴ But could not some of our concepts be such that the principles we take to characterize them entail some ordinary empirical facts? Of course I cannot rule this out. Perhaps this would show, if discovered, that they would better be characterize in some other way. In any case, the mere possibility of such cases does not rule out the present definition as an analysis of analyticity.

The problems discussed in the previous paragraph concern applying the concept of analyticity discussed here to our concepts outside of mathematics. But, again, the main goal of the present work is to show that this concept applies to most of mathematics.Footnote ⁴⁵ There are certain mathematical axioms that are accepted on the basis of what they entail, rather than that they are recognized themselves as being true of our intuitive concepts. Boolos (Reference Boolos1971) argues that all of the axioms of Z are true of our iterative conception of set, but the axiom of replacement is not recognized as being true of this concept. It is accepted because it allows us to prove reasonable things and does not allow us to prove unreasonable things. Axioms arrived at inductively like this cannot be said to be analytic in the sense defined in this Element. Most of mathematics is derived from conceptual truths and so analytic in the sense discussed here. If some axioms are chosen for the simplicity they introduce, or because they lead to appealing consequences, then the truths that depend on these axioms are not analytic as here discussed.

5.2 Analyticity and Axiomatics

Some might say that the picture of mathematical truths as analytic discussed in the last section is really not much more than a restatement of the axiomatic method in mathematics. What counts as analytic of the concept of a group, for example, as we saw, is exactly the axioms of group theory and what follows from these. Of course as the axiomatic view is the dominant view of mathematics today, one would not want one’s account of analyticity to be in conflict with the axiomatic method. Further, on the interpretation that what I have said amounts to no more than a restatement of the axiomatic method in mathematics, then few if any should disagree with anything I have said. After all, as said, this is the dominant view of mathematics. However, while we identify some axioms by conceptual analysis and then we see what follows from this via axiomatics, not all axioms are arrived at in this way. We saw this in the previous section. Some axioms are accepted on more inductive grounds. These further axioms, and things that are only demonstrable on the basis of them, are not analytic in my sense. So it is not quite true that the view of mathematics I am putting forward here gives us exactly the same picture as the axiomatic view of mathematics.

Still others might contrast the picture of mathematics as following from conceptual analyses to the picture we get from the axiomatic method.Footnote ⁴⁶ In axiomatics we do not care about the intended meaning of the axioms at all, but merely treat them as arbitrarily reinterpretable. Hilbert is famous for saying that concerning the axioms of geometry that for “points, lines and planes” we must always be able to say “tables, chairs and beer mugs.” The point being that mathematics abstracts away from the particular meanings of terms and looks only at the logical relations between the terms as stated in the axioms.Footnote ⁴⁷

But does this abstract character of the axiomatic method imply that conceptual analysis or explication is less important in mathematics? I do not think this follows at all. Take the Peano axioms for instance. Sure when we treat them axiomatically, we can see them as arbitrarily reinterpretable, but the axioms themselves were arrived at via a conceptual analysis. Here we can say much the same thing as Frege did concerning the conception of arithmetic an aggregative mechanical thought. Frege emphasizes that it is only possible to operate with figures mechanically once real thought has taken place:

Just as we can follow algorithms blindly once they have been developed and proven to work, we can treat a structure completely abstractly once the work of conceptual analysis, to find the axioms which characterize the structure, has been carried out.

5.3 Conclusions

I have put forward an account of analyticity which is immune to the problems which Quine saw with Carnap’s attempts to explicate analyticity in terms of meaning postulates. Recall that Quine thought Carnap’s attempt in terms of meaning postulates was arbitrary and ad hoc in failing to identify what it was about the intuitive concept of analyticity that was to be preserved. This was not part of Carnap’s conception of an explication, but it was part of Quine’s conception. We used Quine’s conception of an explication itself to define a notion of analyticity. The result is an interesting notion of analyticity, which, unlike well-known early twentieth-century accounts, distinguishes between conceptual truths which are analytic and what is merely stipulated and is not analytic.Footnote ⁴⁸

It was argued that talk of something being made true in virtue of meaning should always be interpreted as following from principles which characterize the meaning of an expression. What we take to characterize the meaning of an expression is what is to be preserved in a Quinean explication. As there is nothing particularly epistemic about this concept of analyticity, I looked at Boghossian’s arguments in favor of epistemic analyticity and against metaphysical analyticity. I argued that metaphysical analyticity, or something like it, which I would call semantic analyticity, is defensible once the confusion mentioned at the start of this paragraph is cleared up. I then argued that epistemic analyticity is just this same concept, but with certain tentatively held epistemic principles which are used to turn this into an epistemic notion.

In Section 4, I looked at the relationship between this definition of analytic and other notions. For the most part, I argued that the analyticity of certain propositions does not commit one to various philosophical positions associated with earlier views of analyticity. We need not, and should not see analyticity as meaning stipulated truth. A claim’s being analytic does not imply that it is empty or contentless. I then argued that whether or not one accepts some form of relativism, with a little pragmatism, there is reason to believe that what is analytic is (fallibly) true. I also explained why I don’t see the ontological implications of analytic claims as worrying.

Finally, I argued that while there is a lot of vagueness in ordinary language, which makes it very difficult to determine exactly what is analytic of a concept, this does not extend to mathematics. For mathematical concepts it is usually quite straightforward to identify what must be preserved by any explication of them. I then showed how the view of analyticity fits with the axiomatic method in mathematics. Here I argued that the axiomatic method is useful once the hard work of explication has been carried out.