More on the Psychology of Closure Affirmation

I’ve been thinking more about closure, and what leads epistemologists to accept it in a theory of knowledge or justification. A standard way in which closure is defended is by appeal to something like intuitions. You consider all the examples you can think of, and note how closure is preserved. You also make some point about extending knowledge through competent deduction. All of this culminates in a predilection to explain away apparent counterexamples to closure, such as Dretske’s zebra/mule case.

Each of these steps can be questioned, especially when we consider how various deductive principles for first-order logic have to be abandoned when thinking about the logic of subjunctives. For a broad variety of subjunctives, hypothetical syllogism is just fine; but it only takes a good counterexample or two to undermine the rule. And notice that if you really want to preserve such principles for subjunctives, it is not that hard to see how to go contextual about them to preserve them (UPDATE: probably the best example here would be that of strengthening the antecedent, which is the example Heller uses). Moreover, extending knowledge through competent deduction doesn’t require a closure principle. It only requires that the method is justification-preserving or highly reliable, in a way that doesn’t normally introduce gettierization.

So the psychology of closure affirmation still puzzles me.

I think I understand why evidentialists affirm closure. Evidence for a claim is information that gives one a reason to think the claim is true, and there is no better reason to think a claim is true than one that logically guarantees that it is true. All other evidence should be envious of this exalted status.

I think we can say something of the same sort for non-modal versions of reliabilism. After all, if we want to use methods and procedures that get us reliably to the truth, what better method than one guaranteed to preserve truth?

The puzzling case, though, is modal epistemologists. If you think of knowledge in terms of possible worlds and closeness relations between them, then you think of knowledge in much the same way that fans of Lewis/Stalnaker semantics think of subjunctives. In such a case, you’ll have a high degree of motivation to deny closure, since the class of worlds relative to which the truth value of a knowledge claim is assessed can shift between premises and conclusion of a deductive argument.

This is the conclusion that some modal epistemologists embrace, but not all. It’s the remainder that I don’t quite understand. For example, consider Sosa. Sosa proposes safety rather than sensitivity as a condition for knowledge, and a large part of the motivation is because (he thought) safety preserves closure. Turns out he was wrong, which he now realizes; but instead of abandoning closure, he adjusts the safety view even further to try to come up with a principle that will preserve closure.

But why this attachment to closure? I would have expected an attachment to closure to derive from one’s inclinations in the theory of knowledge, but that can’t explain the attraction of modal epistemologists to closure. The case for closure, if it goes as in the first paragraph above, is far from conclusive. If I add in an attraction for certain kinds of epistemology, I can see the attraction. But not for modal epistemologists. Anyone have ideas here?


More on the Psychology of Closure Affirmation — 16 Comments

  1. Jon,

    It’s fairly hard to think about this clearly without having a more definite idea of what you mean by “a closure principle.” You say that “extending knowledge through competent deduction does not require a closure principle.” But if you spell out a principle explaining how we can extend knowledge through competent deduction, why won’t that just be a closure principle?


  2. You raise an interesting question. I’d respond with pragmatism; the mind’s requirement for closure affirmation relates to what William James referred to as the “cash value” of a belief. I think the mind is always in search of some tangible value for its beliefs (i.e., if I believe this couch is across the room, I should be able to, in fact, sit on it).

    I’m new to this site and after three years of law school, I need to reconnect to my philosophy training. Glad to be here.

  3. Rich,

    The principle that explains how we might extend knowledge through competent deduction might be a principle to the effect that nearly always, coming to believe by competent deduction extends our knowledge. Such a view wouldn’t endorse any closure principle, since such a principle would need to be an exceptionless generalization.

  4. Tanisha, welcome to the blog! My post was unnecessarily cryptic about the notion of closure, so I’ll try to clarify. Here, closure is a mathematical notion, and the notion in question is deductive closure. Roughly, the idea is this: a set of propositions is deductively closed if it contains all the deductive consequences of anything in the set. Applied in epistemology, the simplest closure principle would then be: if S knows p and q is a deductive consequence of p, then S knows q. This principle is obviously false, and it’s falsity puts fans of closure on the journey of finding a better principle. The best one I know of goes like this: Is S knows p and comes to believe q by competently deducing it from p, while retaining knowledge of p and finding no ultimately undefeated defeaters for q, then S knows q.

  5. Jon,

    You say that there might be a principle to the effect that “nearly always, coming to believe by competent deduction extends knowledge.” I assume that there is a reason knowledge is not extended in those rare cases when it is not. And so there is a more refined universal principle that is true. This principle will build in conditions to the effect that these factors are not present. And that will be a closure principle.

    One other thing. I admit that there may not be a neat way to state all the exceptions. We may have a hard time spelling out the correct principle in all its detail. Are you thinking that we should “affirm closure” only if we can formulate a correct closure principle? Would you say the same about, say, induction? Along the same lines, why isn’t your affirmation of the “nearly always” principle to which you referred an affirmation of closure?


  6. Rich, I’ll start with your last question. Closure is a mathematical concept, where the closure of an object O is defined as the smallest set that both includes X as a subset and possesses some given property. An object is closed iff it is equal to its closure. So a deductive closure principle will talk about the logical consequences of a set of propositions, in our case the known ones (or justified ones), and the the set will be closed in this respect iff … So, no “nearly always” principle will not count as a closure principle; it’s just part of the concept of closure that the set in question is closed only under exceptionless circumstances.

    Presumably, we could list all the exceptions one by one and that would get us an exceptionless principle. But, when we ask about the truth or falsity of a closure principle, we are asking about principles that pass the usual tests for philosophical adequacy, including not being ad hoc, etc. There’s no guarantee of that, even if there is a guarantee that some kind of exceptionless principle can be constructed.

  7. Jon,

    If you understand closure in that way, then I agree that knowledge is not closed. I think virtually all contributions to the literature that I know of agree about that. That is, everyone agrees that a person need not know everything that is a logical consquence of something he knows. So if there is an ongoing debate about “closure”, it is not about that.

    What I think has moved lots of people, perhaps including Ernie Sosa, is that some theories seem to be committed to egregious failures of closures. For example, some theories have seemed to imply that you could know that you saw a red barn, while not knowing that you saw a barn. My guess is that at least some of the people who have described themselves as defending “closure” were acknowledging that such implications are bad ones. I don’t see anything mysterious about that attitude.


  8. Rich, let me try again. The point was not to say what the closure property is, but rather to explain why a closure principle has to be exceptionless. The property with respect to which closure holds might be just the deductive consequence property, or it might be what is believed on the basis of competent deduction, or whatever. It doesn’t matter what the property is; the point was only that closure is by definition a property of an object that admits no exceptions.

  9. A clarification on mathematical closure: The closure of an open set A is obtained by adding to A all the limit points of A. For example, if A is the open interval (0,1), the closure of A would be obtained by adding the limit points 0 and 1.

    Now, one may also specify a function f on A to A such that f(A) is the closed set of A. One may then talk about closure of a set with respect to this function. The logical consequence operator may be viewed this way: if f satisfies set inclusion, idempotence and is monotone, then f is Tarski’s classical consequence relation, Cn.

    If Cn is the function underlying the notion of closure you’re working with (and you’re wedded to JTB), then it is doubtful that Cn is an appropriate model for epistemic closure: justification propagation is not monotone. If this is what is meant by denying closure, then, indeed, we all should do so.

    This said, the problem seems to be to account for how and when we successfully extend knowledge by otherwise sound modes of reasoning.

    To this end, a final comment. I’m not sure that one should have prima facie doubts about modal accounts. Presumably, the modal operators–if they are to mean anything–will be defined by an underlying frame. It is true that modal logics give rise to a parametric notion of logical consequence–namely, the consequence relation must refer to the class of structures specified by the frame you’ve adopted. But, there is nothing dodgy about this. It is just the nature of modal logics. Whether the expressive capacity of modal languages is sufficient to capture epistemic closure, however, is another thing. Still, this issue is to be settled by examining particular proposals and whether the pre-theoretic idea of ”nearly always” is captured by the semantics. The point is, I don’t see anything intrinsically weird about modal logics to cause a bind for so-called modal epistemologists. Nor is it true that no ‘nearly always’ principle will count as a closure principle–so long as the exceptions are captured in the semantics of your modal logic. (This isn’t to rope Rich into modal logic, of course. :^)

  10. Greg, here’s an approximation (because from memory) of the best closure principle available: If S knows p, comes to believe q by competently deducing it from p, retains knowledge of p throughout the process and no ultimately undefeated (internal) defeaters for q are learned in the process, then S knows that q. If this principle is true, then there is a property (the “competently deduced without loss of knowledge and defeater introduction” property) such that the set of propositions which one knows is closed with respect to this property. That is, the closure of the set of known propositions with respect to this property is equal to the set itself (assuming, of course, the truth of the principle in question).

    “Modal epistemology,” in my terminology, covers relevant alternatives theories as well as counterfactual theories that employ safety or sensitivity for distinguishing knowledge from true belief. The modalists identify, in one way or another, what one knows with true belief across some range of possible worlds. The range of worlds is claimed to vary depending on the proposition in question, and may also vary by context (if one is a modal contextualist).

  11. Hi Jon,

    So, there are two ideas in play here. One is the mathematical notion(s) of closure. This notion concerns sets or, perhaps, functions defined on sets. The other is an epistemic principle, which, at minimum, describes changes in and relations on psychological states through time. There is also reference to abilities–reasoning, memory access–that are broadly psychological but may not reduce cleanly to being described by transitions between states. Finally, defeasibility may be understood psychologically or ‘objectively’.

    At any rate, I’m inclined to think that these two notions of closure have very little to do with one another. You can build mathematical models to capture some of the dynamics involved in the latter case–the epistemic case–but that is a different kettle of fish. I’m not sure it makes sense to talk about an epistemic closure principle, then, at least in the way this discussion started. (This is not to deny that we can be competent deducers!)

  12. Yes, Greg, there is some question the intrusion of time and psychology in closure principles, but notice that this one (it’s John Hawthorne’s precisification of Williamson’s idea) is synchronic in one sense. I’m not quite sure why you think there are two senses of closure here. We can understand a mathematical notion of closure as a function from objects to an object that includes the first object as a subset and has some given property. So we can have, I take it, transitive closure of a binary relation in set theory; algebraic closure in algebra, and topological closure of a set as well. So we ought to be able to have “cdwlokadi” closure of a set of known propositions, too, without having to say that we’re using a different idea of closure. Of course, it’s not a mathematical notion any more, but the mathematical notion and this notion have a common definition.

    But you know the formal stuff so much better than I, I think I must be missing something?

  13. The difference between logical and psychological relations often comes up in discussions of closure as it has here. Isn’t one way of *representing* the psychological limitations of non-ideal agents (such as humans) just to say that for a given non-ideal agent the set of known propositions K is not equivalent to the closure of K under deduction, ~(K = C{D}(K). Indeed, perhaps it’s not even the closure of K under what I’ll call the Williamson Relation (being competently deduced, etc), C{W}(K). So a sufficient condition for S being a non-ideal cognitive agent is that, relativized to S, ~(K = C{W}(K)).

    You could even rate the quality of cognitive agents (along one dimension) by what closure principles were true of them. For example I take it that the first inequality is true of all of us humans, so it’s not much of a strike against one. In fact, I suspect the second inequality is true for most of us, but not perhaps for Williamson himself! So an agent for whom that inequality did not hold would be less non-ideal, a greater approximation of ideal cognitive excellence.

    This seems helpful to me and it could also be used to express notions of epistemic responsibility. Though I think rules of inference are regulative ideals, I’d say that the set R of propositions S is responsible for believing (where I’m responsible for believing p iff not to believe p is epistemically irresponsible (blame incurring)) is a proper subset of the closure C{I}(K) of K under the appropriate inference relations. I’d love to be able to define that subset by abstraction, but that has proved hard to do.

    I hope Jon sees these suggestions as forays into value-driven epistemology, for I hold that to be an important part of the future of epistemology.

  14. Trent, nice idea here. I think the search for a closure principle is supposed to be one that is true for any cognitive agent whatsoever, which is what makes the Williamson relation (as amended by Hawthorne) among the most interesting proposals. So, you start with the class of things known to some S, and assume the relation in question holds with respect to q (and that this relation has something to do with deductive consequences of the things known). Then the question is whether it follows that q is known by S.

    So I’m now on the 14th comment in this thread, but I haven’t quite heard a good story about why people are convinced that there is some such relation, Williamsonian or not, that will allow us to derive that q is known by S. I told one such story: there’s evidence from cases, from extension of knowledge by deduction, and the rest of the story is an attraction to a particular theoretical perspective (the ones that are obvious to me are evidentialism and reliabilism). I still wonder, though, if there’s another account available, since the one just cited won’t help modal epistemologists (e.g., relevant alternatives theorists and counterfactual (safety/sensitivity) theorists).

  15. I’ve been thinking more about closure, and what leads epistemologists to accept it in a theory of knowledge or justification.

    I guess for the purposes of this discussion, I count as a pro-closure epistemologist, as does Rich, who has already weighed in above. I don’t know how typical I am, but I can at least say what leads me to be in the relevant way pro-closure…

    A standard way in which closure is defended is by appeal to something like intuitions. You consider all the examples you can think of, and note how closure is preserved.

    No, that’s not it at all. Like many, I first encountered the issue of “closure” together with the (putative) counter-examples to it. These seemed to be at least arguably failures of the principle. So there’s no way I would have come to be pro-closure through the consideration of lots of examaples & noticing that closure always seems to be upheld.

    You also make some point about extending knowledge through competent deduction.

    That’s it! But then it’s not just the added extra you seem to be making it out to be. It’s just thinking in general terms: “Well, if you know one thing, and you know it entails something else, so you deduce that something else from the first thing, and you thereby come to believe the second thing, and come to believe the second thing firmly enough to satisfy the attitude requirement for knowledge (no mere tentatively held belief), then won’t you know the second thing?” Well, maybe not invariably. We may not have built enough into the conditions yet to absolutely guarantee the answer must be yes. [ For myself, I worry about the “just barely” problem I’ve described at: ] But we definitely seem — at least to me — to be closing in on some true principle, even if we don’t have it perfectly formulated yet. And I take it that in discussions like this backing “closure” just means that you think there is some true principle very nearby (though this true principle won’t be well-named a “closure” principle).

    The puzzling case, though, is modal epistemologists.

    I guess I’m a bit puzzled by your puzzlement.

    If you think of knowledge in terms of possible worlds and closeness relations between them…

    OK. I guess I count. The “toy” theory of knowledge I use in my relevant work (like “Solving the Skeptical Problem”) is that S knows that P iff S has a true belief that P, and S’s belief as to whether or not P matches the fact of the matter in the sufficiently close possible worlds (where how close is sufficiently close is a matter that varies with context: Where high standards rule, the “sphere of epistemically relevant worlds” is relatively large, but where low standards are in force, the sphere is smaller).

    …then you think of knowledge in much the same way that fans of Lewis/Stalnaker semantics think of subjunctives.

    Well, in somewhat the same way.

    In such a case, youâ??ll have a high degree of motivation to deny closure, since the class of worlds relative to which the truth value of a knowledge claim is assessed can shift between premises and conclusion of a deductive argument.

    Maybe what you have in mind is specifically non-contextualist modalists? For me, such a shift in the class of worlds amounts to chaging the knowledge relation that is being invoked, and I advocate “closure” only relative to any given knowledge relation. Of course it can happen that you know PREMISE relative to lower standards, and, even though you’ve followed all the conditions in whatever the correct closure principle is, you can fail to know CONCLUSION relative to higher standards.

    On the other hand, an invariantist modalist, at least just as such, can deny that the class of worlds can shift between premises & conclusion.

    I guess the only modalist who’s being pressured here into denying closure is an invariantist modalist who’s got some commitment to the possibility of such a shift in relevant worlds. (Like, for instance, one who uses subjunctive conditionals in certain ways in their analysis of knowledge.)

  16. Keith, this is very helpful. On the part at the end about modal epistemologists, I think there are two kinds to distinguish. One kind has settled the contextual/invariant question independently of closure. The other kind, the kind I was thinking of in the post, is one who hasn’t settled that question, but approaches the question of closure, thinking like a modal epistemologist. For such an epistemologist, I would think that one would view the argument from ignorance the same way we view hypothetical syllogism for subjunctives. That is, in both cases, we might say, “Gosh, I guess I would have thought that argument would work, but once I see how the relevant worlds shift from premise to premise, I can see why it doesn’t. So, nope, I don’t accept HS for subjunctives and nope, I don’t accept closure for knowledge.”

    Such an epistemologist might have grounds down the road to go back one of these inclinations, and one such reason might be other grounds for endorsing contextualism. I see that part, so if one’s contextualism is independent of trying to save closure in this way, then the inclination derived from one’s modal epistemological views might be swamped by the contextualism (together with whatever one’s reasons for inclining toward closure).

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