Thoughts on the analytic/synthetic distinction

Quine was wrong. About almost everything.

Here's a paper I wrote long ago in which I tried to clarify some issues around the philosophical distinction between analytic and synthetic statements. I wasn't familiar with all the literature—e.g., at one point I referred to Jean Piaget, whose work on cognitive development had been outdated for decades—but I think I made a few points of interest anyway. One point I definitely should have made, which would have clarified some confusions, is the one I subsequently made in footnote 11, namely that 'synthetic' has two meanings that must be distinguished. Philosophically, it can mean, in effect, empirical, or it can mean the (intuitive) mental act of synthesis. I wonder how many confusions have arisen since Kant because of the conflation of these two meanings...

I'm not a philosophical empiricist, which has always struck me as an absurd thing to be (in light of modern cognitive science, and for other reasons), so I criticize Quine in the paper. Kant, not Quine, was on the right track, though the convoluted philosophy Kant laid out in The Critique of Pure Reason is of little more than historical interest given the modern brain sciences. (For his time, Kant had interesting suggestions about how the mind synthesizes experience. But cognitive science has made all his speculations hopelessly outdated and primitive. I know there are philosophers who would say I'm confusing the issue by thinking empirical science is of any relevance to so-called transcendental psychology (or transcendental idealism), but they're the confused ones. What should interest us, and what interested Kant, is how the mind constructs experience. Today, "mind" equals (certain capacities of the) "brain." The whole edifice of transcendental idealism was left in the dust by science a very long time ago.)

Anyway, by way of warning, I should also note that in the paper I make the highly contentious claim that logic is, like mathematics, actually synthetic—but in a slightly different way than math is. Neither logic nor math is empirical, so in that respect they're the same (and not synthetic). But I'd say they do both involve the mental act of "synthesis," synthesizing cognitive constructions. And yet math is not reducible to logic—as the historical failure of logicism showed—so the natures of the "syntheses" they respectively involve must differ in some way. It seems to me that mathematical reasoning is more clearly "synthetic" than logical truths are, in that, e.g., getting 5 by adding up 2 and 3 obviously involves a mental act of synthesis, of 'merging' two numbers to get a different number. (If that doesn't seem obvious to empiricists, well, so much the worse for them.)

When you start reading the philosophical literature, all these issues get pretty confusing. And I was surely confused in some way(s) or other. But at least I wasn't as wrong as Quine.

*****

The distinction between analytic propositions and synthetic propositions was first explicitly formulated by Kant. For a long time thereafter, philosophers universally believed it was not only useful but fundamental—though indeed they revised Kant’s account of it and rejected certain conclusions he had drawn, in particular that a proposition could be both synthetic and a priori. But then Quine published his paper “Two Dogmas of Empiricism”, in which he argued that the analytic-synthetic distinction is illusory, and ever since, Kant’s intuition (which is shared by most people, including non-philosophers) has been under attack. In this paper I’ll argue that Quine was wrong, that Kant’s distinction is worth keeping. I’ll also make some observations on the synthetic a priori, again defending the Kantian position.

*

Rather than reviewing the history of the concepts in question from Kant to the present day, I’ll start out by defining ‘analytic’ in what I think is the simplest and most common way: an analytic proposition is true by virtue of its meaning alone. Said differently—Fregeanly, so to speak—an analytic proposition either is a substitution-instance of a truth of logic or can be transformed into such a substitution-instance by replacing one or more of its component concepts with synonymous concepts.[1] (For example, “All bachelors are unmarried” can be turned into a logical truth by substituting ‘unmarried adult males’ for ‘bachelors’.) It could be argued that these definitions are not equivalent, but I think that ultimately they amount to the same thing. Neither, for example, explains why logical truths and logical laws themselves are true, but instead takes them as “primitive”. (The law of identity is simply self-evident; its truth does not depend on meanings or definitions.)

Quine holds that, despite intuitive appearances, there is no boundary to be drawn between analytic and synthetic statements. “All bachelors are unmarried” does not, then, differ in kind from “There are brick houses on Elm Street”; the difference is only of degree, namely of the degree to which the first statement is more important to our linguistic community’s “web of belief” than the latter. We can revise the latter without having to revise many other statements in our community’s vast and intricate web of belief, for not many statements are related to the fact that there are brick houses on Elm Street. (In other words, if it turns out there are not brick houses on Elm Street, this has very few implications with respect to the rest of our commonsensical and scientific worldview.) On the other hand, we cannot revise the statement “All bachelors are unmarried” without having to make fundamental changes in our beliefs about our understanding of language, about the reliability of empirical evidence and perhaps memory, about a variety of concepts related to ‘bachelor’, and so forth. So-called “analytic” propositions, then, are simply propositions that are relatively difficult to revise (“difficult” insofar as such a revision requires many other revisions)—and so it is simpler and easier to continue to believe them.

Essential to Quine’s theory is the argument that there are no criteria for determining whether a given proposition is analytic. Synonymy, for example, is not a good criterion, nor is definition, nor is “interchangeability salva veritate”. All such concepts are ultimately circular, in that they presuppose either the concept of analyticity itself or other problematic and related concepts (e.g., necessity).[2] What this means, says Quine, is that no philosopher has yet succeeded in making sense of the notion of analyticity—because to do so would be to give a satisfactory definition/criterion, which has not been done. As matters stand, then, it doesn’t make sense even to say there could be analytic propositions, since analyticity is an incoherent concept. Now, Quine claims that his “web of belief” theory, sketched above, can explain our intuitive conviction that there is an analytic/synthetic distinction. Therefore, it is theoretically economical simply to discard the postulated distinction, which has been shown to be senseless (at least “so far”), and to accept Quine’s own account. But because he is an empiricist—he thinks that analytic propositions, if they existed, would be the only a priori truths—an implication of his view is that a priority itself is an illusion. All epistemic justification is either directly or indirectly empirical.

Quine’s position led to his later belief in meaning-indeterminacy—and ultimately in meaning-eliminativism (in Word and Object)—because, if there is no “fact of the matter” about whether two given linguistic expressions mean the same thing (i.e., if the notion of analyticity does not make sense), then these two expressions cannot have determinate meanings. For if they did, then obviously either their meanings would be the same or they would be different. But Quine rejects the question itself, and so he rejects both possible answers, which implies that words do not have “determinate” meanings—which apparently is just to say they do not have meanings at all. This position is paradoxical, to say the least.

In fact, the counterintuitiveness of Quine’s overall view is extreme. Synonymy, insofar as the concept makes sense at all, is conceived as a posteriori, depending not on word-meanings but on facts in the world. Even logic and mathematics are a posteriori. Everything, every true proposition, is true, “in the last analysis”, solely by virtue of “being an element of a system of beliefs, some of whose members are appropriately related to experience and which as a whole satisfies certain further criteria, such as simplicity, scope, explanatory adequacy, fecundity, and conservatism”.[3] Quine is a thoroughgoing coherentist and empiricist—as well as, incidentally, a behaviorist and a nominalist. It is hard to imagine a more totally false combination of views.

Since the 1950s, a number of devastating criticisms have been made of the Quinean philosophy. I won’t rehash them all here. Instead, I’ll make a few simple observations about analyticity.

The all-important “criterion” that Quine sought unsuccessfully is in fact something he didn’t even consider: namely, intuition. Or, rather, a combination of intuition and ‘verification in practice’. The question, remember, is how we know that ‘bachelor’ is synonymous with ‘unmarried adult male’. (For we do know this, despite Quine’s perverse attempt to argue the contrary.) The answer is that our intuition, or ‘immediate understanding’, tells us so. Intuitively—immediately, non-discursively—everyone knows the two linguistic expressions have the same meaning. He experiences what Descartes would call a “clear and distinct perception” that the definition in question is true.[4] Of course, intuition in itself is not always reliable: someone can have an “intuition” that something is the case and yet be wrong. So, while intuition in this instance (involving the concept ‘bachelor’) is extremely compelling, almost as compelling as it is in the case of ‘2 + 3 = 5’, it is not quite sufficient by itself to establish conclusively that ‘bachelor’ means ‘unmarried adult male’. However, when the second criterion is added, namely that this particular use of the word will be accepted universally by people in one’s linguistic community, we are justified in saying we have established the analyticity of “A bachelor is an unmarried adult male”. Everyone would agree that (his intuition establishes that) no experience could undermine or refute the proposed definition of ‘bachelor’, for the obvious reason that the proposition in question is not about experience or the world but about meanings. To say that the difference between the analytic statement above and “There are brick houses on Elm Street” is merely one of degree is preposterous. The latter statement says nothing whatever about word-meanings; the former is wholly about meanings.

What analytic philosophers like Quine often seem to forget is that ordinary language is not an exact science. It is misguided to seek a precise criterion by which one class of propositions can be distinguished from another—as it is misguided to think that analyticity is senseless absent such a criterion. No doubt the idea of “cognitive synonymy”, which is implicit in analyticity, is elusive, perhaps necessarily so. But this fact itself is not as mysterious as philosophers have thought, nor does it invalidate the concept. The reason for the elusiveness of all such concepts—e.g., synonymy, meaning, definition, analyticity—is that linguistic meaning is a phenomenon of consciousness, and consciousness itself is remarkably elusive and difficult to analyze. What we intuitively want in an explanation of synonymy is an account of some sort of underlying fact, some quasi-mathematical equation that both proves the identity of ‘bachelor’ and ‘unmarried adult male’ and explains precisely in what sense they are “identical”;—an underlying “fact of the matter” which a particular person’s understanding more or less approximates. We want to see the identity laid out before us, as we can literally see that 2 + 3 = 5, namely by making two marks and then three marks and then adding them together. In the mathematical case, though, the identity depends only on the cognitive faculty of synthesis, of synthesizing abstract forms; not so in the case of linguistic meaning. Thus, what the mathematical equation states is really:[5]

(1 + 1) + [(1 + 1) + 1] = [{[(1 + 1) + 1] + 1} + 1]

2 + 3 = 5

The parentheses, the brackets and the plus signs signify the mental act of synthesis (of “pure synthesis”, in Kantian language), which is essentially all that the equation involves. It is but a mobilizing of the synthetic faculty in its purity, unadulterated with any empirical content.[6] An analytic proposition, on the other hand, involves “meanings”. It is tempting to ask the old question “What are meanings?”, but this question is itself, in a sense, symptomatic of the misunderstanding of language that led Quine to seek an “underlying fact” about (any given case of) analyticity/synonymy, a “criterion”, a sort of mathematical certainty. Intuitively we think of—or want to think of—meanings and concepts as something like “entities”, “abstract objects”, things to which words “correspond” somehow, a correspondence that would allow us to find a deeper fact of the matter—to get “below” linguistic expressions to their essence, and then compare these essences to see if they are “identical” (thus confirming the analyticity of the proposition in question). And then when we discover, inevitably, that there is no such criterion, no such essence or literal abstract object, we are tempted to conclude, with Quine, that words do not have meanings at all. But this too is wrong. Both platonism and nominalism are wrong. Words do have meanings—‘bachelor’ means ‘unmarried adult male’—but the latter are not what our philosophical intuition wants them to be. There is no deeper explanation of the synonymy-relation that holds between ‘bachelor’ and ‘unmarried adult male’; the “fact of the matter” is simply that we use those two linguistic expressions in the same way and intuitively they strike us as meaning the same thing, and that’s all that can said. That is why Quine could not find a criterion for analyticity: there simply is nothing “over and above” (or “underneath”) our concrete uses of words, which are context-dependent, intuition-dependent and inexact.

Indeed, the concept of ‘bachelor’ is unusual in its clarity: most words, like ‘intelligent’, ‘interesting’, ‘house’, ‘chair’, ‘plant’, ‘bad’, etc. cannot be defined by invoking a clear set of conditions that apply in all contexts. There is an element of vagueness to most words. Again, though, this does not mean it is wrong or meaningless to say that a particular sentence does a good job of articulating the meaning of, say, ‘chair’. The criteria for its doing so are, as in the case of ‘bachelor’, the intuition of the native English-speaker and/or the agreement of most (or all) other English-speakers. (If ever these two criteria should ‘conflict’, so to speak, surely the second has priority over the first. After all, in saying that two expressions mean the same thing, one is basically saying that the linguistic community uses them in the same way.) Above all, we should resist the temptation to seek a precise “standard” we can use in defining words—or in deciding whether, e.g., this object here should “really” be called a chair even though intuition hesitates, as if there is a deeper, underlying fact about what the term ‘chair’ denotes. We do not need a precise standard or criterion like that in order to make sense of analyticity and synonymy: to say that two expressions are synonymous is just to say that, generally speaking, they are used similarly in similar contexts, not that they literally share some sort of common essence or abstract object.

It is different in the case of arithmetic, which consists of synthetic a priori propositions that posit literal identity-relations. One side of an equation is literally identical with the other—but not in the same way as in a tautology, viz. a wholly uninformative equation of a thing with itself. But then how is ‘2 + 3’ identical with ‘5’? Well, evidently insofar as combining 2 and 3 is a way of arriving at 5. In this equation, 2 (like 3) is conceived as a single concept, an “abstract object” identical with itself; yet it is also a synthesis of two units. 3 is a synthesis of three units. So when you add these two concepts together, or ‘synthesize’ them (so to speak), you get five units, which is to say you get a new concept, a new unity/synthesis, the number 5. 5 represents a unity in a plurality; it is such a unity, but it is also a plurality. It is different from the other side of the equation (‘2 + 3’) insofar as it is one concept, while ‘2 + 3’ involves two concepts (plus the ‘concept’, or rather the operation, of synthesis); 5 does not exist (in this case) until 2 and 3 have been combined in intuition, which means that they are, in a sense, prior to it. On the other hand, it is identical to the other side of the equation insofar as the synthesis of 2 and 3 (which is what ‘2 + 3’ denotes) is, in fact, 5.

In short, “2 + 3 = 5” is not tautologous, at least not in the vacuous way that “An unmarried man is unmarried” is, but it is nonetheless a literal identity. Nor is it a mere definition, i.e., a phenomenon of analyticity,[7] and thus of meaning and convention; rather, it involves a sort of cognitive violence, a pushing-together of two concepts and ending up, of necessity, with one. There is no such “cognitive violence” in any definition, nor the element of absolute a priori necessity that characterizes “2 + 3 = 5”. There is merely a making-explicit of what is already and unproblematically (‘conventionally’) implicit in the definiendum—i.e., of the conditions under which the definiendum can be properly used.[8] The arithmetical proposition, then, in being neither strictly tautologous nor analytic, yet, like these two types of proposition, expressing an a priori identity-relation, has to be synthetic and a priori.[9] It cannot conceivably be revised through experience—it is necessary[10]—so it is a priori, or ‘prior to experience’; but it presupposes the cognitive faculty of synthesis—it is in fact nothing but this faculty exercising itself in abstraction from empirical conditions—which means it is “synthetic”, i.e., is not a strictly logical or tautologous truth.[11]

The view expressed here, incidentally, is the best explanation of a phenomenon that has puzzled empiricist philosophers to no end. “Why?” they have asked themselves, “is it so hard for young children to learn arithmetic? It takes them years to master this most simple of all mathematical subjects!” If empiricists were right that arithmetic deals only with definitions or conventions or tautologies or generalizations from experience or anything like that, the phenomenon in question would indeed be inexplicable. It would be a mystery why children are able to master the enormously complicated apparatus of language by the age of four or five, even as they cannot master arithmetic (this subject that supposedly deals with mere tautologies or definitions!) for several more years. The solution, of course, is that arithmetic is not quite as simple as empiricists think. It cannot be fully learned until the cognitive faculty of intuitive spatiotemporal synthesis has reached a certain level of maturity, a level that is not reached until around age 8 or 9, or even later.[12] Arithmetic involves, as it were, a cognitive smashing (or ‘mashing’) of number-concepts together to yield new ones (as well as the reverse of this process, namely the breaking-up of quantitative concepts into ‘smaller’ ones), a process that does not make logical sense and cannot be adequately analyzed through formal logic—as shown, perhaps, by the failure of logicism, Frege and Russell’s attempt to reduce mathematics to logic. The child understands, or internalizes, logic at an early age (since he understands language and can, to some degree, draw conclusions through valid arguments) but not arithmetic, evidently because two distinct modes of cognition are involved. With arithmetic he has first to grasp the concept of numerical identity—between, say, two (or three, or four...) groups of dissimilar objects, such as pencils, apples, flowers, and everything else—which is a highly abstract concept and presupposes extremely sophisticated cognitive equipment; then he has to learn to manipulate, in abstraction from empirical objects, these very abstract numerical concepts he has formed, each of which consists of a mysterious, intuitive unity-in-plurality (which unity and plurality are likewise totally abstract)—and, therefore, each of which is a pure intuitive synthesis, the “manipulation” of which consists in synthetically relating it to other such concepts in myriad ways, the results of which manipulations are, again, synthetic unities-in-pluralities (i.e., numbers). In short, the whole arithmetical process, from the original forming of number-concepts to the skillful manipulating of them, is saturated with constant cognitive syntheses, syntheses that finally become a priori in that they (1) are carried out with no reference to empirical objects and (2) are justified purely through intuition, through the intuitive necessity that characterizes them. The reason, then, for the inordinate length of the child’s arithmetic apprenticeship is that he is developing his synthetic faculty (in relation to numbers, that is), which is to say he is honing his ability to carry out mathematical operations on an a priori, intuitive basis.

However, while logical truths, and thus logical laws (such as modus ponens), apparently do not directly rely on quite the same cognitive faculty as mathematics—for logical truths are tautologies, whereas mathematical truths are not—it seems to me that even logical laws presuppose some sort of spatiotemporal synthesis, and are in fact merely ‘projections’ of such a synthesis carried out in the mind. The law, for example, that a thing is necessarily identical to itself grows out of the a priori structure that the mind imposes on experience—the structure according to which, e.g., the world is populated by objects that abide from moment to moment, remaining the same object despite the flux of appearances and their changeability. An empiricist philosopher might raise objections here, but in so doing he would be placing himself in opposition to modern cognitive science and developmental psychology. At any rate, the Kantian view I’m defending at least has the merit of being able to explain why logical laws strike everyone as self-evident: namely, because, like true arithmetical propositions, they are manifestations of synthetic a priori cognition, the cognition that structures experience for us. In assenting to the law of identity, or to the law of contradiction, I am basically “assenting” to the way I necessarily experience the world (given the structure of my mind).

I consider it misleading, therefore, to say that logical laws and logical truths are “analytic”, as philosophers tend to think. An analytic proposition is true by virtue of meanings, and thus convention. When I say a bachelor is an unmarried adult male, I am describing how English-speakers use the word ‘bachelor’. My assertion is about nothing but meanings; I am saying only that two linguistic expressions have the same meaning. If, on the other hand, I utter a tautology, such as “An unmarried adult male is unmarried”, I am, strictly speaking, saying nothing about meanings; instead, I am saying that a particular kind of thing is identical to itself. “A man who is unmarried is unmarried”: this statement is not about words, nor linguistic conventions; it says that a specific kind of thing is itself, that A = A, so to speak. It is a matter of logical form; therefore word-meanings have nothing to do with it—except, of course, in the truistic sense that the meanings of the words involved have to be understood in order for the proposition to be understood. In short, tautologies are true not because (like analytic statements) they correctly describe conventional linguistic use or articulate what is implicit in a concept, but because they are grounded in logical laws, such as the law of identity. With respect to logical laws, however, linguistic meaning is totally irrelevant. It does not even have the minimal level of relevance it has in the case of tautologies. In this regard, then, logical laws are similar to mathematical propositions—a fact that is itself suggestive. “What does it suggest?” asks the reader. It suggests the thesis I am arguing for, namely that logical laws are “synthetic a priori” (though in a slightly different way than arithmetic propositions are). Analytic statements, by contrast, are a priori but are not “synthetic”, which is effectively to say that their truth is determined not by the nature of human cognition but by the nature of the linguistic conventions relevant to them.

*

In the end, I have to agree with Quine that the analytic/synthetic distinction is simplistic—not because there is no “difference in kind” between, say, “Running involves moving your legs” (an analytic statement) and “There are trees on Elm Street”, but because each class comprises propositions that may well be, in some respects, more heterogeneous with each other than they are homogeneous. For example, it seems to me that a statement like “Running involves moving your legs” has, on one construal, more in common with tautologies than with “A bachelor is an unmarried adult male”, in that the latter is mainly a description of how two linguistic expressions are conventionally used, whereas the former states that in this activity of ‘traveling quickly by moving one’s legs rapidly’ one is moving one’s legs. In other words, the definition of bachelor describes the meaning of a word, while the statement about a person running describes a state of affairs, not a meaning. It says, essentially, that the activity of quickly moving involves movement, which is just to say that the activity is identical to itself. (‘Movement involves movement.’) On the other hand, the statement “Running involves moving your legs” can also be construed as giving the (partial) meaning of a word, namely the word ‘run’, rather than as describing an activity—in which case it has more in common with the definition of bachelor than with any tautology. So, actual linguistic practice is so complex and nuanced that any such dichotomy as “analytic vs. synthetic” is bound to be inadequate.

To give one more example: if someone says—perhaps to a person learning English—that a bachelor is an unmarried adult male, he is both giving an analysis of the concept ‘bachelor’ and saying that English-speakers use this linguistic expression in a certain way.[13] On the first interpretation, though, his utterance would commonly be called a priori, while on the second interpretation it would be called a posteriori (since its truth depends on a particular fact in the world, namely whether English-speakers do indeed use the word ‘bachelor’ in the way described). Thus, the utterance is both a priori and a posteriori—for, in saying that a word has a certain meaning, one is essentially saying that people use the word in this way[14] (which is an a posteriori statement). So even the a priori/a posteriori distinction is not quite as absolute as one would think.

Nevertheless, it is worth keeping, as is the analytic/synthetic distinction. But both should be recognized as what they are: not absolute, fixed dichotomies but merely useful philosophical tools, useful insofar as they help illuminate differences between propositions. Sometimes, adhering to them is counterproductive and can obscure understanding of the epistemological and linguistic facts (to which context is usually extremely important, more so than a simple overarching categorical dualism would seem to allow); but the fact that they highlight differences between types of cognition is a sufficient justification for retaining them. Some propositions and modes of cognition are unequivocally a priori (and synthetic); others are unequivocally a posteriori (and synthetic); still others can be interpreted as both a priori and a posteriori (as well as analytic and synthetic)—which suggests that, in these contexts, the distinctions should be ignored.

Quine was, therefore, wrong, though he made some interesting points.

[1] Taken from Laurence BonJour, Epistemology: Classic Problems and Contemporary Responses (New York: Rowman and Littlefield Publishers, Inc., 2002), p. 87. [2] See “Two Dogmas of Empiricism”, in Necessary Truth: A Book of Readings, eds. Sumner and Woods (New York: Random House, 1969). [3] Laurence BonJour, “In Defense of the a Priori”, in Contemporary Debates in Epistemology, eds. Steup and Sosa (Malden, MA: Blackwell Publishing, 2005), p. 105. [4] This Cartesian criterion for knowledge has been much maligned over the centuries, many philosophers arguing that intuition is too vague and subjective to do the epistemological work that Descartes wanted it to do. Nevertheless, no one can plausibly deny that when he assents to the common definition of ‘bachelor’, he does so just because he ‘sees’, non-discursively, that it is true. Whether his intuition justifies absolute certainty is another question; my point is that it is generally the criterion we use in judging whether two expressions are synonymous. Moreover, it does seem to be overwhelmingly reliable, especially in simple cases like the one I’m considering. [5] See Epistemology: Classic Problems and Contemporary Responses, p. 90. [6] I’ll discuss synthesis below. [7] Philosophers might take exception to the implication that a tautology is not an analytic statement, but it seems to me that their thinking has been confused on this subject. I’ll briefly discuss it below. [8] Formal logical definitions are quite different from definitions in natural language. They are not analytic, properly speaking—i.e., they do not involve linguistic meanings, or linguistic convention, or synonymy-relations between expressions that denote concepts with real content—but are instead tautologies. [9] Below I’ll argue that logical laws are also, in a sense, “synthetic a priori”, but that they are so in a slightly different way than mathematical propositions are. [See footnote 11. Logical laws obviously aren't empirical—so, like math, they aren't synthetic in that way—but they do depend on a mental act of synthesis.] [10] The necessity of analytic statements, by contrast, is, as I said earlier, not unconditional. While “intuition” may force one to assent to a particular definition of bachelor, there remains the barely conceivable possibility that one’s assent is mistaken (i.e., that it does not adequately reflect the community’s use of the word). But even this tiny possibility does not exist when one assents to “2 + 3 = 5” after having carefully considered what the sentence expresses.

Incidentally, I am aware Saul Kripke has argued that not all necessary statements are a priori—that some are a posteriori, for example “Water is H2O”. But actually, the a posteriori character of these exceptions is deceptive: the reason for their necessity is that they are disguised tautologies—tautologies whose truth has been discovered through experience. In other words, the reason that “Water is H2O” is necessarily true is that science has shown that the proper definition of water is H2O, and so what the statement says is really “H2O is H2O”, which is a tautology. So, strictly speaking, the necessity of even these Kripkean “exceptions” is a priori (since the necessity of a tautology is a priori).

[11] [Here's a point I really should have made in the paper: the fact that the word 'synthetic' has two meanings is confusing. It can mean the brain/mind's synthesizing, so to speak, of cognition, such as mathematical truths, but it also denotes, in the philosophical literature, statements that are true by virtue of facts about the world. In other words, empirical statements. Mathematics isn't synthetic in this way, but it is synthetic in the other way.] [12] See Jean Piaget’s studies, for example The Psychology of the Child. [13] Indeed, I argued earlier that to do the former is to do the latter. [14] In other words, to say a word has such-and-such a meaning is to say it is used in such-and-such a way (i.e., under such-and-such conditions).