“common knowledge”

Has anyone working on knowledge attributions looked into the expression “common knowledge”? It’s different than a lot of uses of “knowledge” because it takes a ‘that’-clause (as compared to”knowledge of” a domain, as in: “She has knowledge of physics”). So aside from questions about what I have to know to have knowledge of a domain, there’s a question of what has to be the case for it to be common knowledge that p (or for it to be proper to say that it is common knowledge that p). I’m more interested in the “knowledge” part than in the “common” part, since there’s nothing at all defective about saying “It was common knowledge that p, but not-p.”

Suppose sentences like that can be true. Intuitively, there is a close conceptual connection between what is known and who has knowledge, i.e.:

Linking Principle: S has knowledge that p only if S knows p.

On the face of it, common knowledge is a kind of knowledge, namely, knowledge had by some salient class of (common) people, i.e.:

Definition of Common Knowledge: It is common knowledge that p (for some class of people C) iff sufficiently many of the members of C know that p.

It follows, from our supposition that “It was common knowledge that p, but not-p” can be true, that “S knows p” and “not-p” are compatible.

Is there some other account of “knowledge that” and “common knowledge” available, which isn’t committed to the Linking Principle and the Definition of Common Knowledge presented above? (It seems to me that similiar questions concern the expressions “scientific knowledge,” “folk knowledge,” etc., as well as sentences of the form: “As far as S knew, p”.)


“common knowledge” — 10 Comments

  1. There is an epistemic-temporal logic literature on common knowledge that I think would uniformly reject your proposed definition of common knowledge. The idea instead would be to treat common knowledge as a set of propositions (say) that a particular group of agents at a particular time all are presumed to accept.

    This type of approach might be outside the scope of your inquiry, for I think these conditions would be stipulated as modeling assumptions and not given an analsyis. But, if you’re interested in recent work on this topic, a good place to start would be to look at papers by Wiebe van der Hoek at Liverpool.

  2. The acceptability of your example might come from the past tense rather than from “common”. So (2) seems much worse to me than (1):
    (1) “It was common knowledge that the Earth is flat, but actually it is round.”
    (2) “It is common knowledge that the Earth is round, but actually it is flat.”
    Similarly, (3) sounds better to me than (4):
    (3) “Everybody knew that they were about to divorce. But as it turned out, they weren’t!”
    (4) “Everybody knows that they are about to divorce. But in fact, they aren’t!”
    I’ve also seen a New Yorker cartoon with the following caption: “We just discovered that everything we knew about the universe was wrong”.

    But I guess thoses uses are echoic rather than genuine factivity failures.

  3. Hi Allan,

    I think that if it’s common knowledge in group C that P, then (a) most of the members of C know P, (b) almost all of the epistemic leaders of C know P (assuming there are such leaders), and (c) most everyone in C is aware that most everyone in C knows P. That’s three necessary conditions; I don’t know how to go further. I’m not sure about (b). But I suspect that if the experts in C concerning the topics surrounding P don’t know P, then it’s not common knowledge in C that P–even if almost all the non-experts in C know P.

    I don’t think we have to add (d): most everyone in C is aware that most everyone in C is aware that most everyone in C knows P. (And we don’t need further iterations either.) But there has to be some widespread awareness in the group that people in the group know P; otherwise it doesn’t seem like common knowledge to me. Or am I just characterizing something like “reflective common knowledge”?

    I’m not sure about ‘It was common knowledge (in C) that P; but not-P’. Julien’s comments seem important on that matter.

    Looking forward to seeing you at the APA!

  4. Hello –

    This is very interesting to me. In my field (economics), common knowledge is defined in the way Bryan alludes to in (d):

    P is common knowledge among C if all the members of C know P, and all the members of C know that all the members of C know P, and all the members of C know that all the members of C know that all the members of C know P, and…

    And so on, ad infinitum. Actually, this is stronger than what Bryan suggested, replacing ‘most everyone’ with ‘all’. This is indeed a strong concept, presumed to apply only to ‘public’ events (so, if all the members of C are outside and in plain view of one another on a sunny day, it’s safe to assume that it’s common knowledge among C that it’s a sunny day).

    A good (and short) reference, for any who are interested, is Robert Aumann, 1976, ‘Agreeing to Disagree’ in the Annals of Statistics, Vol. 4. There Aumann proves the surprising claim that for those who share common prior beliefs, differences in posterior beliefs cannot be common knowledge (i.e., they cannot agree to disagree)! Much of the literature since then has, naturally, explored whether we can get more realistic results by dropping the very strong assumptions of a common prior and/or common knowledge – but Aumann’s result was extremely important for showing just how strong these assumptions really are.

    I was under the impression that this definition of common knowledge was, well, common knowledge across fields that concern themselves with knowledge and learning – apparently it isn’t! Much of my own reading on the subject has been in the AI literature (see especially Fagin et al., Reasoning about Knowledge, 1995 MIT Press, or any of Joseph Halpern’s recent work). Am I to gather from this discussion that epistemologists either haven’t defined common knowledge, or haven’t agreed on a definition?

    Thanks – this is my first post, but I’ve long enjoyed following your discussions!

  5. Hi Nathan,

    I did know of that definition of ‘common knowledge’. In fact, I think that definition of ‘common knowledge’ is common knowledge. But I think that just goes to show that the definition is too stringent to have much application. It doesn’t fit how we use ‘common knowledge’ anyway.

    In my book, it’s common knowledge among college graduates today that water is H2O and the Iraq war has gone very badly. But according to the definition you suggested, neither truth is common knowledge among that group.

  6. On “echoic” uses of ‘knowledge’ and ‘knows’:

    What do people think of this argument in favor of the view that utterances of “They had knowledge that p, but not-p” and “They knew p, but not-p” are, if proper, true:

    1. When people are employing irony, metaphor, echoic uses of a sentence (and the like) they are intentionally saying something false, to communicate a truth.
    2. To intentionally say something false, you have to know you’re saying something false. (Probably because, as a rule, you have to know that you’re F-ing to intentionally F, although the principle about saying things that are false sounds plausible even if that general principle isn’t.)
    3. But people who say “They had knowledge that p, but not-p” do not think they are saying something false. (We can tell this by asking them quesitons like “Are you serious?” or “Do you mean that literally?” or “Really?”, and getting a positive (if befuddled) answer.) So they don’t know that they’re saying something false.
    4. Therefore: prima facie, they’re not cases where people are intentionally saying something false, i.e. prima facie, they’re not cases of irony, etc.

  7. Allan,

    I think that ‘They once knew that P, but not-P’ is pretty weird. I’m not sure what “normal” people mean when they use sentences of that form. I never do. If I did, I think I might falsify your premise (3), since I would think that when I say ‘They once knew that P’ I am really saying something false or, more interestingly, something with truth conditions that aren’t what you’d expect from the words in the sentence. Maybe I’d think that I’m saying something like one of these:

    a. They once would have said with all confidence that P.
    b. They once “knew” that P.
    c. They once knew that P (but it wasn’t real knowledge).

    I’m not sure what the truth conditions are for (b) and (c). But in any case, I’m just talking about my potential uses of the sentences in question. What about the folk?

    I take it that you want to argue that often enough, when people say things of the form ‘They once knew that P, but not-P’ in contexts in which it is conversationally appropriate to do so, they are saying something true AND it has the truth conditions you’d expect the sentence to have. And the point of that thesis is the subsequent thesis that a literal use of ‘S knew P’ can be true even though P is false?

  8. Dear Allan,

    About implicatures. I think that some pragmatic phenomena are so smooth that speakers do not notice that they are saying things which are litterally false. (Have you eaten tonight? – No, I haven’t. – Is that true? Do you mean that litterally? – Yes, I swear, I haven’t eaten! – But you have eaten a peanut!)

    I think many echoic uses belong to this category. For instance, one might say:
    (a) Last year olive oil was the best thing for your health and this year it’s about as bad as alcohol.
    By which one means (b) rather than (c ):
    (b) Last year most people believed/said that … but this year they believe/say that…
    (c ) Olive oil was good for humans last year but now it is bad for them.
    I think that if you asked a non-sematicist who just uttered (a) “Is that true?” “Is that litterally true”, she will not answer “No, of course, I meant…”. And I doubt that she would clearly realize that she is not speaking litterally.

    About common knowledge. It seems to me that there are clearly two notions at stake here. There’s Widespread Knowledge: a lot of people know that p (that’s called “shared knowledge” in the epistemic literature I guess). And there’s Mututal Higher Order Knowledge: everybody knows that everybody knows that… everybody knows that p. But as a matter of fact, WK is usually also MHOK (at least to some extent) among human societies. So it might well be that the natural language phrase “common knowledge” (as in “it’s commonly known that”) has a prototypical meaning with two dimensions, with widespread mutual higher-order knowledge being at the center, and distance from it measured as a weighted sum of non-widespreadness and non-mutual-higher-orderness. Alternatively, one could try the theory that “common knowledge” actually means shared knowledge, and to explain away cases in which people would resist applying “common knowledge” to shared knowledge in which nobody knows that anybody knows.

    At any rate, that would be a interesting thing to test! Does (d) sound acceptable to you?
    (d) It was commonly known among the spies that the assassination would take place the day after, but they all ignored that the others knew.
    (I think it isn’t, so I’d go for the prototype theory!)

  9. Hey Allan,

    I think your (2) should be revised to read: 2*: To intentionally say something false, what you say must be false, and you must reasonably believe that it is false. Your (3) would then need to read, towards the end, “So they don’t reasonably believe that they’re saying something false.”

    I agree with your point that “it was common knowledge that p, but not p” is felicitous. It doesn’t sound the least bit odd or weird. It doesn’t strike me as non-literal usage, either. Common knowledge seems akin to “conventional wisdom” in this respect. It might be worth exploring that angle.

    My hunch is that, if a plurality of members of group G believe p, then it is common knowledge, in G, that p. The earlier proposals sound far too strong to me.

    Another hunch: if we can state a sufficient condition for a statement like ‘Bostonians like donuts’, there won’t be much work left for us to do to state a sufficient condition for statements like ‘Bostonians know the Big Dig is a disaster’. And this latter statement is obviously closely related to it being common knowledge in Boston that the Big Dig is a disaster. In fact, I might even go so far as to say: it is common knowledge among Bostonians that p IFF Bostonians know that p.

  10. Nathan’s definition of common knowledge, although popular in Economics (via the work of Aumann) has a clear philosophical pedigree: David Lewis first formulated the notion in question in his book Convention. Interestingly enough Lewis’s formulation is not completely identical to Aumann’s. Various articles have been devoted recently to compare both notions like Giacomo Sillari’s”

    A Logical Framework for Convention, Synthese, vol. 147(2) pp. 379-400, 2005.

    For a very nice introduction to common knowledge and philosophical applications it is a good idea to read the article in SEP, written by my ex-colleague Peter Vanderschraaf and Sillari.

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