How is a priori knowledge possible? The dominant recent response on the rationalist (pro-a priori) side to Benacerraf-style worries has been the conceptualist one, featuring prominent thinkers like Peacocke, Boghossian, Jackson, perhaps Hale and C. Wright, and a host of their followers. The approach concentrates upon our grasp of relevant concepts and upon their alleged a priori connections, and attempts to account for a priori knowledge in terms of the grasp. It attempts to preserve realistically factual character or substantiality of a priori knowledge. At the same time typical recent conceptualists subscribe to for a version of naturalism, albeit a somewhat weak one, and attempt to neutralize the worries traditionally connected to causal or causal-like explanations. The idea is that the mere possession of concepts, no matter how it has been arrived at, provides the thinker with substantial (factual) knowledge about the items concepts refer to. And this is valid for all concepts, empirical, mathematical, moral, or social-conventional; there is nothing very special about the traditional “a priori concepts” like mathematical or moral ones. The conceptualist development has been quite impressive, but there are some residual worries. The story is just too good to be true.
First, the assumed uniformity and equal status of all concepts is problematic. Is the proposition “2+2 = 4” really of the same epistemic kind as “Whales are material objects”? Conceptual truths about whales seem to be rather trivial and superficial, those about numbers are probably a priori in a much deeper way. Not to mention that it has proved to be very difficult to find conceptual truths about very many empirical concepts (natural and artefactual kinds being the most obvious examples). The proposed candidates, like, for instance, complicated conceptual networks (disjunctions of conjunctions of weighted criteria) lack intuitive force; I for myself just cannot decide whether a proposed such network is believed by me a priori, or whether experience plays some role in justifying it. (Both Peacocke and Jackson and Chalmers move to meta-level and stress the richness of apparently conceptual rules of belief revision which are supposed to be a priori. But the move, although reasonable in itself, leaves the crucial, first-level issues open. And moreover, the question whether rules of revision are a priori seem to be much more difficult than the original question; there are almost no spontaneous, naïve intuitions available about such complicated matters.)

Second, there is the crucial problem of accounting for apriority. The conceptualist line is: don’t worry, all a priori truths are conceptual truths, they are explained by our having concepts, and this is not particularly mysterious. But this seems to reverse the right order of explanation: concepts are OK, but they don’t fully explain our apparent access to (alleged) a priori domains, nor fully justify our beliefs. The deeper question remains: where do these very concepts of number, of metaphysically fundamental properties or of moral qualities come from? And what justifies their use in our inquiry? They are thus themselves in need of explanation and justification, they belong to explananda (and vindicanda) not to explanantia (and vindicata). (The same question can be reiterated for higher-order conceptual items, like a priori rules of belief-revision, entitlement and the like). Neglecting explanation amounts to giving up on the one of the most exciting problems in the history of epistemology from Plato, through Descartes and Kant to Wittgenstein and Carnap. The original epistemic Platonism was not satisfactory, but at least it kept the issues alive, and focused the attention to the domain of problematic objects, mostly abstract objects, themselves.
Here is then the worry in the nutshell. By offering the most superficial, purely analytic apriority as a model, without seriously addressing the explanation issue, conceptualism might direct our attention away from real and difficult issues about serious candidates for a priori knowledge, and away from the serious a priori domains like math and morals. To use a graphic metaphor, it might turn out to place a concept-barrier, a veil of conception so to speak, between us and the domain of presumably abstract object, in the similar way in which strongly indirect theories of perception place a veil of perception between us and material objects.
So, my dear fellow bloggers, do these worries make sense? Or am I being old- fashioned and paranoid?




  1. “where do these very concepts of number, of metaphysically fundamental properties or of moral qualities come from? And what justifies their use in our inquiry?” I’m not sure that the first of these questions is of obvious epistemological relevance — though the answer to it at least in principle can help answer the second. (E.g., if they were placed in us innately by a non-deceiving God!) But the second question, I think, really gets at something important and most often neglected. Consider all the different rival analyses that have ever been put forward for KNOWLEDGE. Most of these describe perfectly coherent concepts, and they have vast extentional overlap. So what makes any one (or more) of them the concept(s) that we _should_ organize our investigations around? It is not clear how the appeal to analyticity and/or apriority is to help answer such a question.

    (The possibility of answering such questions is of course part of the promise of value-driven epistemology.)

  2. “where do these very concepts of number… come from? And what justifies their use in our inquiry?”

    I am very sympathetic to the aggregate theory of number, proposed first by John Stuart Mill, and reformulated by Glenn Kessler.

    The use of a concept is minimally justified in some context when the object or pattern which it designates may be instantiated in experience. Since a number may be instantiated, i.e., by counting your fingers, it is minimally justified. Of course, it depends on how you use the concept propositionally that makes for actual, full justification (i.e., I can’t justifiably say “Five is alive”).

    Not a very novel answer, but my sympathies are very much towards empiricism.

  3. Nenad, I am attracted to the first point you make, especially. It has long been known, I believe, that there have to be both deep and superficial analytic truths, on pain of being unable to defend the analyticity of mathematical claims. Connecting this with remarks about concepts requires a bit of work, but I’ll pass on that issue for now.

    I think the idea of there being epistemic rules (rules of belief revision) that are a priori is problematic in another respect that you don’t mention. I doubt there are such rules (I think one post here was called “Rules and Principles” and it addresses this point). If there are such rules, it makes no epistemic difference whether they are a priori, unless I am mistaken. What matters is there truth, and it is hard to see any advantage gained by their being a priori as well (except to keep the set of a priori things from being empty).

  4. Jonathan,
    thanks for the comment. Here is the beginning of an answer, concerning the question of origin of armchair beliefs.A sign that the question might be relevant is the fact the two last decades of the debate in mathematical epistemology have been fueled by Benacerraf’s dilemma, crucially featuring the notion of causal explanation (and the last two millennia of the debate have been fueled by Plato’s tantalizing attempts to explain the same matters). Their (Plato’s and Benacerraf’s) line has been that unless we have a fine explanatory story, our full, reflective justification will be defective. We need the explanation of having such beliefs, and of their reliability. Not only against mystery-mongering. Of course,even if we dont have it, we can still be weakly justified; all of us in implicitly using logic in our reasoning, in mobilizing half-implicit commonsense metaphysical categories (MATERIAL OBJECT, EVENT), and in passing factual sounding moral judgments, working mathematicians in doing their job and trusting their results, and scientists in taking mathematical framework as given and standing in no need of empirical testing. But, the full internal reflective justification will be missing. Note the contrast with the case of perception: part of our confidence in it is that we sort of understand at the level of common sense how material things can act upon us, and that, further, we have some scientific assurance that this commonsense understanding is on the right track. And the external reliability might just be a matter of good luck (a bit of change and our procedures would not track truths any more).
    Just for the record, I think that the fact that explanation is part of reflective justification, a component of wider reflective equilibrium, could undermine serious apriority of central armchair pieces of knowledge: causal explanation is a posteriori, therefore, justification might be partly a posteriori.

  5. Jonathan,
    on the second question, I am in a complete agreement. Here is a brief addition. The mainstream conceptualists take conceptual analysis ultimately to reveal factual but a priori connection between properties in the world, not just representations in people’s heads. For instance, concept KNOWLEDGE links belief and truth, not just our representations of them. But mere armchair reflection does not reveal the relevant status of the concept analyzed. In the case of knowledge D. Lewis has famously proposed a pragmatic explanation: our concept KNOWLEDGE is a shortcut, a way of ascribing some positive qualities to certain beliefs, which we use since we are incapable of doing the complicated Bayesian calculations of subjective probabilities. In that case it makes little sense to ask for what objective property the concept stands. Now, and this is the main point, WHETHER A CONCEPT IS A PRAGMATIC SHORTCUT OR AN HONEST TO GOD NATURAL KIND CONCEPT MAY NOT, AND PROBABLY IS NOT, RECOGNIZABLE A PRIORI.
    In connection with pragmatic approach there is an interesting recent exchange at the Online Phil Conference, between Justin Fisher and Jackson. Here is Jackson’s reaction to pragmatic considerations:
    “I am used to being accused of being far too partial to the a priori but here I want to side with those who insist that the nature of our concepts is a contingent a posteriori matter. (Actually, I always want to do this.)).” (Comments on Justin Fisher, ‘Pragmatic Conceptual Analysis’)

    And this is the way, I think, which an honest conceptualist should take.

  6. Jon, yes I looked at Rules and principles; I have comments there. My problem is that it all is a bit to particularistic for my taste.
    I want to leave it open that there might be both rules and principles, and question that our justification for (having, believing in, following) them is cristaly clear and pure a priori.

  7. The original post said: “Both Peacocke and Jackson and Chalmers move to meta-level and stress the richness of apparently conceptual rules of belief revision which are supposed to be a priori.”

    Could someone give me a reference to where Chalmers makes this move? Chalmers is straightforwardly grounding aprioricity in his notion of epistemic intensions (primary intensions, diagonal propositions). For him, this reveals aprioricity as a kind of “rational must” as he puts it somewhere, which is then explained in terms of his notion of a canonical description of a scenario – in his technical sense of a centred world – epistemically necessitating the proposition/statement/sentence in question. In fact, I think it is hard to see Chalmers’ notion of apriocity as anything like an epistemological notion, although he describes it as an idealised form of conclusive justification. Rather, so I think, his notion is one on which a statement is a priori iff it is analytic in that epistemic intensions are what speakers grasp in understanding the meaning of a statement.

    In general, I think it is symptomatic of the debate that it has left out the distinction between aprioricity and analyticity – even if it should turn out that there is a biconditional relation holding between them.

  8. “where do these very concepts … come from? And what justifies their use in our inquiry?”

    Great questions! I agree they are absolutely crucial to conceptualist accounts of the a priori. I have a line on how to answer them. In a nutshell, I think the best answer is one which belies the thought that conceptualism is a rationalist’s preserve. I argue for the existence of a certain epistemically important links between concepts and the world, which are mediated by sensory input. In virtue of these links, I think, our concepts can be treated as reliable maps of the structure of the world, as opposed to merely our own ways representing it. The basic idea is that through sensitivity to sensory input our concepts are sensitive to the structure of the world, and can therefore be treated as encoding information about the world’s structure which we can recover through activities like conceptual examination, conceptual analysis, attempting to conceive of various scenarios, and so on.

  9. PS Incidentally, I’m also entirely in agreement that explanation is a key notion in understanding what the relevant link between concepts and the world is. (I tend to be an explanationist about knowledge in general, and treat this as a special case.)

  10. Some of the comments here, while appreciated, are quite puzzling to me.

    I’m quite perplexed as to why “there have to be both deep and superficial analytic truths, on pain of being unable to defend the analyticity of mathematical claim”. Almost nobody believes that mathematical truths are analytic since Kant (although his argument for it wasn’t especially good). It is quite controversial to say that mathematical truths are analytic, nowadays, since the outcome of an equation can be a genuine surprise (is not self-evident). Where does this intuition come from?

    I’m not at all sure why anyone needs to take the “mathematical framework as … standing in no need of empirical testing”. (My turn to be controversial.) Mathematical truths do need to satisfy this requirement in order to count as “knowledge”, but it so happens that often, especially in very elementary equations, the tests have already been done to the complete satisfaction of the thinker, either through instantiation in experience or by definition.

    An underlying metamathematical or epistemic explanation for the validity of all this would be great, but the content of our explanations would likely differ so long as the details differ. And if we can’t even agree that mathematical truths are synthetic, there seems to be little common ground indeed.

  11. Almost nobody believes that mathematical truths are analytic since Kant…

    This claim shows an enormous lack of familiarity with the history of philosophy since Kant, I’m afraid, though it is true that it is controversial to claim that they are analytic. A grasp of the context of the remark that prompted it would have revealed that what was said was from the conceptualist point of view, assuming that analyticity and a priority are connected as Nenad was assuming, I believe.

  12. You’ve misinterpreted the point. The comment was a rhetorical means of indicating a historical turning point in the salient direction.

    It would be bad manners for me to hound your sentiments here, especially after you’ve evidently elected to drop the subject. Nevertheless, I am curious at where the intuitions lie behind this kind of conceptualism that you have presented. Hence why I wrote, “I am perplexed…”, asked a direct question, etc.

  13. Benjamin, it sure looks like the “I am perplexed…” takes its scope just over the rest of that one sentence. The rest of what you said look like flat-out assertions, and it was those assertions that Jon was looking to correct.

    Anyhow, I think that the short answer to where this is coming from could be encapsulated without much distortion as: (i) most folks think that a whole lot of math is a priori, and (ii) a whole lot of the current literature on the a priori tries to ground that category in conceptual structures or competences or possession conditions or…. So, (lots of) mathematical knowledge comes out as analytic, on a reasonable (if broad) construal of that term — we can have that knowledge primarily because of facts about our concepts. (I don’t happen to endorse the second of these trends, but it sure does seem to be a major trend.)

  14. Hi Jon (W). After the “perplexed” comment, I provided several reasons for my puzzlement. The main one — the “outcome of an equation can be a genuine surprise” comment — is an assertion, I suppose; but I think it is more or less standard, and I included it as an explanation for current opinion. Indeed, Jon K. seemed to agree on that score. In any case, no attempts at substantive (non-rhetorical) correction have been made.

    Thank you for your elucidations. What would be especially interesting is if the conceptualist theorist (in the sense you describe) were to give a rebuttal to the “genuine surprise” argument.

  15. Benjamin
    thanks for raising the issue. Here is what I meant:
    A central line of accounting for mathematics, its ontology and epistemology, has been neo-logicism. Mathematics, according to it, reduces to logic. And logic is a conceptual matter. Its core are logical constants, and they stay for fundamental logical concepts. This is how math gets analytic.
    My own worry was that math seems to be too substantial: mathematical truths seem to be more seriously a priori than truths of the kind “Every widow has lost a husband”.
    Now, there is a quick way out, tried by Giaquinto, who tries to separate conceptual from analytic, in a paper called Non-analytic conceptual truths, published in Mind.
    Unfortunately, what he does is define “analytic” in a very narrow fashion, having to do with meanings of words. He then divorces concepts from verbal meanings, and thus has conceptual distinct from analytic.
    This does not work, since “analytic” is traditionally, from Leibniz to Kant, linked to analytic of concepts and most contemporary writers take it that way.
    This is why there is a problem with math being merely a conceptual matter.

  16. Andreas,
    thanks for pointing to an over-sketchy remark of mine to the effect that “Chalmers move(s) to meta-level and stress the richness of apparently conceptual rules of belief revision which are supposed to be a priori.”
    I used the umbrella term “belief revision”, thought Chalmers does not use it. What I meant is the following:
    A central notion of Chalmers is the idea of connection between 1. and 2. extensions/intensions, which functions roughly in the following way:
    the 1-intension of “water” is given by its idealized ordinary meaning. (Chalmers uses “concept” to mean “predicate”, and what I call concept he calls intension, plus or minus small differences), like “transparent, drinkable” … stuff around here”.
    the rational speaker uses 1-intension to identify the actual water, and she can then proceed to empirically examining it. It turns out that water is H2O.,
    This then, yields, 2-intension.
    Now, 2-intension is not itself a priori, but what is a priori are the suggestions, built into the 1-intension, about how to proceed to find out the essence of actual water.

    Why is this central? Because, it is the central weapon in the arguments against physicalism. It is not so much the apriority of simple, first order characteristics built into water-concept (1-intension) that does the job. It is the apriority of indices (I subsume them under “rules”) that guide the search for 2-intension.
    For the role of a priori in the whole story please look at David J. Chalmers & Frank Jackson: Conceptual Analysis and Reductive Explanation

  17. Nenad,

    Thanks for the rundown. Believe it or not, I have some sympathy to the project of the logicist (I admire the logical positivists strongly). It’s just that the “genuine surprise” argument seems quite strong in favor of a synthetic interpretation of mathematical truths. Moreover, the aggregate theory of number, it seems to me, is quite compelling, and inclines me to a radical posteriori understanding of numbers. These seem like they would be two obstacles to the logicist if taken seriously (as I think they ought to be).

    I’m not sure your “widow” comparison was fair. It is true by definition (whatever that means), but those mathematical truths which are true by definition are also unimpressively trivial (i.e., “1=1”.) Rather, the real question is whether or not statements like “The legitimate son of a one-time married widow has lost its father” are what we’d like to attribute weighty substance to, and whether or not they’re a priori, in the same way that equations like “[(1 + 3) – 2] = 2” are purported to be.

    Earlier, the notion of epistemic rules was mentioned. Though the discussion is too sparse for me to get a solid handle on the meaning of the term in this context, it is an interesting and potentially fruitful point. Much of the terminology, as you lately agreed, uses Kant as starting point. If I’ve understood Kant correctly, concepts, the source of analytic truths, don’t have rules; and neither do intuitions (the source of the knowledge of numbers); rather, it are the schemata which have a rule-based character. This seems important to a discussion of mathematical truths, since equations are certainly rule-based to a significant extent. Do you have any thoughts on that? Is there any way we might use Kant’s apparatus against his own conclusions?

  18. I am late to this discussion, but I want to echo the original point, and add some argumentation for it as well. The implicit definition account of the a priori begins with the idea that we can intend to use a word to have whatever semantic value it must have to make a certain statement true, or a certain rule truth-preserving. However, the mere intention to use a term in such a way as to make some statements true does not explain how we actually succeed in using the term in such a way as to make those statements true. It is all too easy to intend to use terms consistently, yet fail to do so. I think that the mere fact that we do succeed (or so we think) lulls the defenders of implicit definition accounts into thinking that this fact of success has been explained, when it has not. It is obviously possible for me to introduce terms into my discourse with the intention of using them in such a way that they will have a semantic value that makes certain statements true, yet fail miserably to do so. Given that this is a genuine possibility, what explains my success in giving a term a use that actually makes the relevant statements true? I don’t think that the implicit definition accounts ever answer this question. I have put this at a completely abstract level, but I think it can be illustrated with concrete examples, like Prior’s “tonk,” and Dummett’s “Bosche.”

  19. Gordon,
    I am even later to this debate, so I seem to be the right discussant for you.
    First, I share your doubts about the relation between implicit definitions, intentions to use a term and truth. However, one might be able to quell these qualms by pointing out that any implicit definition that is e.g. free of contradictions and also doesn’t imply any contradictions will be a successful one. In other words: The hurdle may not be as high as your objection seems to imply.
    Second, I would add a concern which maybe connects a bit more to Nenad’s worries: Given that we have succesfully introduced a concept/term by implicit definition – why should the analyticities that can be generated from these concepts/terms reveal anything interesting/substantial/non-trivial about the world? Or, to put things differently: What, if anything, can we ever learn from our own stipulations?

  20. Joachim,

    Thanks for your thoughts. Avoiding contradiction might not be enough to confer a semantic value on a symbol, but even if it is, I think that avoiding contradiction is much harder than people think it is. That is because the implicit definition has to avoid not only explicit contradictions, but implicit contradictions as well, and that is harder than it looks. For example, one might have thought that you could use the term “set” in such a way as to make it true that “For every property, there is a set of objects that have that property.” But now we all know that this leads to a contradiction. And I think that examples like this can be multiplied. So I think that avoiding implicit contradictions can be much harder than it looks, and even if we do succeed in doing it, we need some explanation of how this success was achieved, because the mere intention to succeed does not explain the success. On your second point, I’m inclined to agree, though it would be nice to have more of an account of the distinction between the superficial and the substantive a priori.

  21. This is in response to Benjamin’s brief argument that the fact that mathematical discoveries can be surprising shows mathematical truths to be synthetic rather than analytic. Here’s an explanation of how analytic truths can be surprising. Suppose a certain set of claims, S, are analytic and, through a long and complicated proof, can be shown to entail another claim, P. If S is analytic, then so is P. P might be surprising because, due to the difficulty of the proof of P from S, it is far from clear at the outset that P is true. So, there can be analytic truths that are surprising, and so the fact that it is surprising that a certain claim is true is no great reason to believe that it is synthetic.

    Also, I think that this explanation of how an analytic truth can be surprising goes some way toward answering Joachim’s second concern about how we can learn things from our stipulations. Truths that are analytic by stipulation can have unexpected entailments, and it may take time to learn these, and when we do we will often be surpised. These entailments may, further, be interesting and non-trivial. Can they be substantial, though? I’m not sure, but, with Gordon, I would like to have a clearer idea of what we mean by ‘substantial.’

  22. But Josh, you’re presupposing an agreement about some chain of reasoning being analytic in the first place, which is something that a Kantian would reject. I suppose, though, this shows that the “surprise” argument isn’t very good in the first place, since it can be used one way or the other. It’s only as good as it shows that it is not only concepts which are being used.

    For instance, when Kant writes: “That 5 should be added to 7, I have indeed already thought in the concept of a sum = 7+5, but not that this sum is equivalent to the number 12”, this is what I mean by a surprise. But also, it seems to demonstrate a point about the limits of what is meant by a “concept”.

    This takes us back to the discussion of conceptualism. But in order for such an outlook to be able to handle mathematical truths, it seems to require changing the meaning of “concept” to include (for example) rule-based behavior. Assuming I have properly understood the issues there, the real question seems to be, What philosophical or semantic facts would motivate us to change the meaning of “concept” in this way?

  23. To be clear: I should have said there, not “some chain of reasoning”, but “some chain of mathematical reasoning” (though by context of conversation, this should be evident).

  24. Josh,
    I partly agree with your point that even stipulations can have interesting, non-trivial, even surprising consequences. For example, it is relatively easy to stipulate the rules of chess, but some of the smartest people on this planet spend their whole lives to explore the fascinating possibilities that are set up by these rules. However, chess isn’t about anything but itself – it is a purely self-concealed abstract universe. So if abstract entities like chess count as ordinary denizens of our everyday world, then indeed we can learn something substantial about the world as a consequence of our stipulations. But if what we mean by ‘substantial’ is ‘facts about things in space and time’, then probably nothing substantial can be learned from stipulations, implicit definitions and the like. Do you think that this is, at least prima facie, a satisfying clarification of ‘substantial’?

  25. Did someone mention arriving late? 😉

    I think Josh’s analysis of the phenomenon of surprise is right. This harks back to the hoary distinction between intuition and demonstration. A complex analytic claim may be necessary and knowable a priori without being obvious or self-evident, provided that the chain of reasoning from self-evident (intuitive) premises, step by logically necessary step, is sufficiently long that one can understand the conclusion without at the same time recognizing that it must be true. Where the cutoffs fall along the quasi-continuum of analytic claims of increasing complexity will typically be a matter that varies not only from individual to individual but over time for the same individual (e.g., for me depending on whether or not I’ve had my morning caffeine).

    Speaking of which, it’s time for some Earl Grey …

  26. Hi, Tim, just a quick question here. I liked Josh’s account as well, though I would think we need the initial premises to be analytic and not merely self-evident. We’ll need to assume, of course, that the logical rules employed are themselves analytic, and there may be some who balk at that as well. But if we start from analytic truths and apply logical devices that are themselves analytic, then given our cognitive limitations, we shouldn’t be surprised if we end up with analytic, but surprising, results.

  27. Jon,

    Exactly. Although Locke is working without the contemporary definition of analytic truth, his account of concepts lends itself nicely to interpretation along those lines.

    On the question of the status of laws of logic and of epistemic principles in general: Lydia and I have some discussion of this in our forthcoming book Internalism and Epistemology, slated for publication by Routledge in October. In our view, legitimate epistemic principles are analytic (which we’re comfortable identifying as conceptual) and a priori.

  28. Dear colleagues
    thnaks for comments, and sorry for not replying immediately; I was at a summer school. Just a brief note about what “substantial” could mean. Distinguish three broad ways for an a priori claim to be substantial. First, epistemically, in terms of novelty and fruitfulness of beliefs it expresses. Second, semantically: once their meaning is fixed, a priori claims are made true by facts independent of them, and of human mind in general (factual characther). Finally, metaphysically: some of them concern rich and metaphysically fundamental items, such as, for instance, sets and numbers, or humanly particularly interesting items such as our minds (call this sub-species “anthropologically substantial”, if you want a special name for it). Epistemic substance can probably be accounted for most easily in terms of the limitations of our cognitive grasp, which make logically trivial propositions difficult and exciting for us. However, semantic and metaphysical substance and richness are very difficult to account for.

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