David Lewis, Convention (1941-2001)

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Lewis introduced common knowledge into philosophy to explain how conventions work, and in doing so left a foundational tension unresolved. He gave both a formal definition (the infinite iteration of nested beliefs) and a real-world mechanism (finite shared situations that generate all levels all at once)—but never fully connected them. That gap is what every subsequent figure in this debate is responding to.

Lewis’ 1969 book Convention: A Philosophical Study tries to answer: what is a convention, and what holds them in place? Consider driving on the right side of the road. There’s nothing necessary about right rather than left—what matters is that everyone coordinates on the same side. Lewis’ answer: a convention is a behavioral pattern where everyone prefers to follow it on the condition that everyone else does, and where this whole structure is common knowledge among the participants. This is how common knowledge entered philosophy, and Lewis left enough of its foundation unresolved that forty years of subsequent work have been trying to sort it out.

But Lewis was actually doing two distinct things simultaneously, and he never fully connected them:

  • First, Lewis defined a formal definition of common knowledge as the infinite iteration of nested beliefs—everyone knows, everyone knows that everyone knows, etc. This is his answer to “what does common knowledge consist in?” It describes the logical structure.
  • Second, a real-world mechanism. Lewis also said you don’t generate common knowledge by grinding through infinite levels one by one. In actual situations, something finite—a shared environment, a public signal, something mutually visible—generates all the infinite levels simultaneously. When a plate drops loudly in a quiet restaurant and two people both see it, the whole infinite structure arises at once from that single shared moment. This is his answer to “how does common knowledge actually arise in the world?”

The tension: the formal definition says common knowledge is the infinite iteration. The mechanism says the infinite iteration results from something finite. Those are different kinds of claims: one describes the structure, the other describes what produces it. Lewis needed both but never fully worked out how they fit together. He left that connection loose and under-theorized.

Every major figure in this debate is responding to or refining Lewis.

Heal, Common Knowledge (1978)

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Heal diagnosed the problem Lewis left behind: the formal definition had displaced the real phenomenon it was supposed to capture. Her intervention is to insist that the explanation lives in the phenomenon, not the formal iterated structure—and to do the positive work of characterizing what situations of openness actually require. Analysis E is her answer: common knowledge consists not in the infinite iteration itself but in a finite, graspable awareness of being in a situation where reasoning is mutually replicable. The infinite iteration is real, but it enters the account only as the object of a finite thought—you grasp that the class of available inferences is shared and inexhaustible, not by iterating through its members. Whether this fully settles the connection between her conditions and the formal apparatus is debatable, but it’s a worked-out proposal, not a gesture.

After Lewis, most people in the field ran with this formal infinite iterated definition, becoming the standard technical apparatus. Lewis’s gesture toward the finite situational mechanism—the thing that actually produces common knowledge in the real world—got treated as mere background illustration.

Heal: common knowledge = open = public

Heal treats “open,” “public,” and “common knowledge” as synonyms for the same phenomenon, with “common knowledge” as her working term. Heal never considers that these might come apart, never arguing for this but instead presupposing it

Heal’s diagnosis: the infinite iteration might correctly describe the logical structure of common knowledge, but it doesn’t explain anything. Saying “common knowledge is the infinite iteration” is like saying “fire is rapid oxidation.” Technically true, but it doesn’t tell you how fire starts or what produces it. The explanation—the thing doing the real philosophical work—lives in Lewis’ undertheorized finite situational grounding. Her question: what features does a real-world situation require in order to produce openness?

The paper works through five candidate analyses (A through E), rejecting the first four and proposing the fifth. Heal is the first to clearly articulate the gap between the informal phenomenon and the formal analysis, and to insist that the explanation lives in the phenomenon, not in the formal structure.

Info

The “formal” tail was wagging the “philosophical” dog: infinite iteration, not the informal phenomenon of genuinely shared knowledge, had become the object of analysis.

I. The Problem (§I)

Heal opens with two dining table cases that frame the entire paper:

  • The quarreling diners (paradigm case of openness): You and I are dining together. At the next table, a loud quarrel breaks out. You catch my eye and grimace. The quarrel is “completely open, public”—common knowledge.
  • The dropped potato (paradigm case of what is not open): Same dinner. I messily drop a potato. I hope you haven’t noticed. You have, but you pretend you haven’t. We both know about the potato, but the situation is “not similarly open”—there are false beliefs, concealment, and uncertainty about each other’s knowledge.

The task: spell out explicitly what is known in the first case that isn’t known in the second. What is the structure of common knowledge?

Heal sets three criteria any adequate account must satisfy:

  1. Finite mind constraint. Must be consistent with the fact that human minds are finite and limited in informational complexity.
  2. Justificatory role. The propositional content must actually justify confident cooperative action—not just be theoretically available, but graspable and usable as a reason.
  3. Identity of content. The propositional content of common knowledge as a dispositional state, as an action-justifying state, and as an occurrent conscious state must be identical.

She introduces a kidnap coordination game that serves as a test case throughout. Twelve people are kidnapped; two (A and B) are placed before panels of colored buttons. On each round, if and only if they press the same color, one comrade is released. They can’t see each other or communicate. At the start, they can only choose randomly. But if they’re placed in an open space where each can see the cell doors and each other, and they happen to both choose blue and see a door open—it becomes common knowledge that blue was successful, and each should confidently choose blue again. Any adequate account must explain why this confidence is justified.

II. Analysis A: The Infinite Iteration (§II)

Analysis A: common knowledge between A and B that p iff A knows that p, B knows that p, A knows that B knows that p, B knows that A knows that p… and so on to infinity (“Series S”).

Heal considers two defenses.

The first defense says the higher-level knowledge is merely dispositional—(I know that you know)^999 that p in the same weak sense in which I “know” truths of advanced mathematics. Heal rejects this immediately: it abandons desideratum (3), the identity of content across occurrent, dispositional, and justificatory roles. If the higher levels are merely dispositional, they can’t explain what it’s like to have common knowledge or why it justifies action.

The second defense is bolder: the propositions are grasped not individually but as a class, via the generating relation—like understanding the truth operator (if p is true, then “it is true that p” is true, and so on). You grasp the pattern and thereby “know” all members of the series.

Heal takes this more seriously but shows it collapses back into the first defense. The analogy with truth fails at a critical point: in the truth case, each member entails the next. But in Series S, earlier members do not entail later ones. That A knows that q is one state of affairs; that A knows that B knows that q introduces “novel and logically independent elements”—this holds whatever proposition q is. So any guarantee that all members of the series are true must come from outside S itself. And such a proof can only establish the higher members in a weak dispositional sense—what Heal calls a “double dose of dispositionality” (the higher-level knowledge is dispositional, and the proof that grounds it is itself merely dispositional). So the bold defense turns out to be a subtler version of the first, inheriting the same problem: it can’t connect to occurrent common knowledge or justification for action. Analysis A cannot be the complete account—it needs a basis.

III. Analyses B and C: Finite Segments (§III)

Analysis B: a short initial segment of Series S gives necessary and sufficient conditions.

Heal tests this with a series of escalating kidnap situations (her own version of what Lederman later calls the escalating cases):

  • Situation 1: A and B can’t see each other or the cell doors. Neither knows p. Random choice only.
  • Situation 2: Each sees the success, but thinks the other is in Situation 1’s ignorance. Each thinks the other will choose randomly, so each has no reason to choose blue.
  • Situation 3: Each has levels 1–2 of Series S, but thinks the other is in Situation 2. A thinks B thinks A doesn’t know p, so A thinks B has no reason to choose blue, so A has no reason either.
  • Situation 4: Same structure, one level up. If Situation 3’s knowledge didn’t justify confident choice, then thinking the other is in Situation 3 can’t justify it either.

Conclusion: no finite initial segment of Series S, however long, provides conditions sufficient for common knowledge. The argument is recursive—at each level, thinking the other person is stuck at the previous level destroys justification.

Analysis C tries to fix B by adding a negative existential clause: neither A nor B believes anything inconsistent with the known levels. Heal rejects this because the absence of false beliefs might just reflect stupidity—unreflective people could qualify where more perceptive people with the same evidence wouldn’t. The condition needs to say not merely that A and B aren’t mistaken, but that they rationally could not be so mistaken.

IV. Analysis D: Lewis’ Account (§IV)

Analysis D (which Heal attributes to Lewis, Convention pp. 52ff):

Common knowledge between A and B that p iff: (i) they share reasoning standards (ii) a certain set-up exists in which: (a) both have good evidence the set-up exists, (b) it’s good evidence that p, (c) it’s good evidence that both know the set-up exists

The crucial element is (ii)(c): the set-up is self-evidencing. It involves the participants as elements (open eyes, direction of gaze, etc.) and because of how these elements are patterned, the set-up of its nature can be known to exist by both participants. This generates an infinite chain of potential reasonings (Series S′): you can work out that p; I can work out that you can work out that p; and so on.

Heal immediately notes that D as stated doesn’t even require that A and B realize that p—only that they have good evidence the set-up exists. It needs amendment to demand that they actually recognize the set-up and draw the relevant conclusions. But even with this amendment, D faces two problems.

First—the cream bun mirror case. You and I are in a room with a large mirror. I’m facing it; you’re behind me, also facing it. I bite a cream bun. All conditions of amended D are satisfied—the physical set-up of us, the mirror, and the bun meets (ii)(a)–(c). So D says it should be common knowledge. But: suppose my habits of thought are still dominated by situations where mirrors are not present and I think (foolishly) that you can’t see what I’m doing. I have the evidence staring me in the face but fail to draw the inference. This is possible, and it is not a situation of common knowledge. The fact that we both have reasons for not making these mistakes doesn’t make it impossible to make them. The switch to common knowledge happens when you look surprised and indignant (we’ve taken a vow to forswear buns) and my face in the mirror assumes a guilty expression. Something has changed—but the conditions of D were already met, so D can’t explain the transition.

Second—the structural problem. Even amended D only gives participants potential knowledge (Series S′)—an infinite chain of inferences they could draw, not ones they actually hold. But the arguments against Analysis B showed that no finite segment of actual knowledge suffices for justification. The only difference between amended D and B is that D’s participants have reasons for the higher levels rather than actual higher-level knowledge. Since potential knowledge doesn’t justify action, D inherits B’s insufficiency.

V. Analysis E: Heal’s Proposal (§V)

Heal’s diagnosis of what went wrong: each prior analysis tries to rule out the worry that the other person misconstrues your state of mind. Analysis A does it by demanding infinite knowledge (unrealizable). C does it with a negative existential (too weak). D does it by writing in available reasons (too weak—you can have reasons and fail to use them). What’s needed is something of intermediate strength.

Her key idea: replication of reasoning. Each person knows that the other can and will replicate his reasoning, and thus knows that his mind is, in certain respects, transparent to the other. “It is awareness of this which constitutes the sense of openness so characteristic of common knowledge.”

Analysis E: common knowledge between A and B that p iff:

  1. Shared reasoning standards: A and B know that, given the same evidence, they will come to the same conclusions.
  2. A shared set-up: A and B know that (a) a certain set-up exists, (b) it is good evidence that p, (c) it is good evidence that both know the set-up exists.
  3. Replication of reasoning (Heal’s term): on the basis of the set-up and shared standards, each knows that (a) p, (b) what he can infer, the other can infer also, (c) each thus has available the same class of beliefs, (d) hence for any belief in this class, it is not possible that some shared purpose makes it important for one person to have it and the other to know he has it, while there is uncertainty about whether this obtains.

Condition (iii)(d) does the real justificatory work and is what directly answers the escalating kidnap cases. The problem in Situations 2–5 was always the same: A worried that B misconstrued A’s state of mind, so A couldn’t trust that B would choose blue. Condition (iii)(d) says that this worry cannot arise: for any belief in the shared class, if a cooperative purpose makes it important that one person hold it and the other know they hold it, then that state of affairs actually obtains—no uncertainty gap.

The critical conceptual move: infinite potential knowledge enters the account only as itself the object of a finite and actual thought (p. 129). You don’t grasp each level individually. You grasp, in one finite thought, that you and the other person have access to the same class of inferences—and that this class is inexhaustible. The infinity is real but it’s apprehended as a class, not member by member. This is what makes Heal’s account psychologically realistic and distinguishes it from Analysis A: real people don’t iterate through infinite nested beliefs. They recognize situations. When all three conditions hold, the infinite iteration falls out automatically—the way a fully lit room guarantees that everything in it is visible without you having to check each object individually.

The dining table examples revisited (my interpretive reconstruction, not Heal's own elaboration)

  • The quarreling diners illustrate all three conditions at once: the quarrel is the set-up (publicly available, involving both participants); they catch each other’s eye (both know the set-up exists and that the other knows); and the grimace works because they share reasoning standards. The grimace is not communicating any particular conclusion—not “that’s awkward” or “poor them.” It’s doing something more basic: acknowledging that the two of you are in a set-up and making sense of it the same way. Think about when the grimace wouldn’t work: if you witnessed the quarrel with someone from a culture where public arguments are the norm, they’d wonder why you’re reacting at all. Same set-up, but divergent reasoning standards. A grandmaster and a novice can watch the same chess game and see completely different things.
  • The dropped potato fails because the set-up is not open—I’m trying to conceal it, you’re pretending not to notice. The set-up exists (we’re both at the table, the potato dropped) but condition (ii)(c) fails: the set-up’s existence is not good evidence that both of us know it exists, because concealment is actively underway.

VI. Revisiting the Intermediate Cases (§VI)

Heal returns to Situations 4 and 5. Her explanation: in any actual circumstances where several levels of Series S are realized, there will almost certainly be extra premises available that provide evidence of the kind of set-up Analysis E requires. The very process of constructing elaborate deception reveals to the participants that they’ve been “repeatedly fooled in a symmetrical manner”—which itself becomes evidence for common knowledge.

Her closing paradox does real philosophical work, not just ironic observation. The universally known fact that finite human minds can’t disentangle more than a few levels of iteration is itself a shared premise—one that can serve as the basis for a set-up in the sense of Analysis E. As the situation’s complexity mounts, each person suspects neither of them can control the ramifications. This mutual recognition of limitation constitutes evidence of symmetry: both are in the same epistemic position, both know they’re in the same epistemic position, and both know the other knows this. That is precisely the kind of self-evidencing set-up that grounds common knowledge under Analysis E. The recognition that “we’re both muddled in the same way” provides, ipso facto, a basis for common knowledge. This is Heal’s explanation for why Situations 4 and 5 feel like they’re approaching common knowledge even though they formally lack it—they leak evidence of the kind Analysis E requires. “Thus, paradoxically, the very limitations of our faculties provide grounds for certain kinds of mutual understanding.”

The Cross-Disciplinary Adoption (1976–2015)

In the decades between Heal and Lederman, the formal iterative definition—the very apparatus Heal had warned was displacing the real phenomenon—was picked up by economics, game theory, and computer science, where it accumulated enormous technical authority. Aumann’s 1976 partition model gave economists a clean, tractable formalism: common knowledge of an event falls out of the meet of agents’ information partitions, no philosophical hand-wringing required. A generation of game theorists built on this through the 1980s—Milgrom, Rubinstein, Monderer & Samet—treating common knowledge as standard equipment in equilibrium analysis. Computer scientists Halpern & Moses (1990) then showed that common knowledge is a necessary condition for guaranteed coordinated action in distributed systems, giving the concept a rigorous engineering application. The Fagin, Halpern, Moses & Vardi textbook Reasoning About Knowledge (1995) codified it all into a canonical reference spanning logic, CS, and game theory. The critical point is what this accumulation did philosophically: the sheer volume of technically impressive work done using the formal definition created the impression that the definition was well-understood and well-grounded, when what had actually been established was its technical fertility within abstract models—models that deliberately abstract away the messy particulars of actual human cognition that Heal was trying to describe. The authority was real in its own domain; the problem is that it got imported back into philosophy as if it settled questions the technical work was never designed to address.

Lederman, Common Knowledge (2017)

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Lederman’s paper is a philosophical audit, a clearing of the ground, not a constructive theory. By 2017, the Default Position—that informal public information and formal, iterated common knowledge coincide—had been the assumed benchmark for forty years. Lederman subjects it to sustained scrutiny for the first time and finds both the arguments for it and the arguments against it weaker than the field had assumed. He doesn’t establish that the Default Position is false. He establishes something more deflationary: that nobody had actually proven it true.

Lederman coins and audits the Default Position: that public information and common knowledge are coextensive—some agents have public information that p if and only if they have common knowledge that p. This had become so embedded in the literature that it was rarely argued for explicitly. Lederman’s project is to take that assumption seriously and ask: is there actually any good argument for or against it?

The Default Position: more than mere theoretical interest

Common knowledge assumptions are load-bearing in game theory, Stalnaker’s theory of common ground in conversation, Gricean communication, Lewis’s account of convention, and theories of joint action. If the connection between public information and common knowledge is poorly established, a lot of downstream work inherits that instability.

The Argument For the Default Position (§2)

The best argument for the Default Position doesn’t start by asserting that common knowledge is needed. It starts by showing, case after case, that anything less than common knowledge isn’t enough. The strategy is eliminative: rule out every finite level of mutual knowledge as insufficient for public information, and if you accept that only epistemic levels matter, common knowledge — the full infinite iteration — is the only thing left standing.

The argument comes through a series of escalating cases, which Lederman presents using Greco (2015) while noting that early versions appear in Heal (1978) and Clark & Marshall (1981). The cases construct a contrast between genuine openness and increasingly elaborate epistemic setups that fall short of it:

  • Public announcement (All levels manifest): A professor tells her class publicly that she grew up in Maine. The fact is open in Heal’s sense—the students have public information that the professor is from Maine, and would coordinate on writing “Maine” in a payoff game attached to the example.
  • Private information (Level 1 only): The professor whispers the same fact to each student individually, telling each that she’s not telling anyone else. All students know the professor is from Maine, but the fact is not open.
    • The simplest proposal—that public information just requires everyone to know p—fails here, because everyone does know p and yet the openness is absent.
  • More private information (Levels 1 and 2 only): The professor whispers to each student: “I’m privately telling everybody in the class that I grew up in Maine—but you’re the only one I’m telling that I’m telling everyone.” Now every student knows that every other student knows p (because each was told “I’m telling everyone”).
    • So the second-level proposal—everyone knows that everyone knows p—is satisfied. But the information still doesn’t feel open. Each student believes they’re the only one who was told about the broadcast, so nobody knows that others know that everyone knows p. The third level fails, and public information is still absent. The second-level proposal fails too.

Why the gap regenerates (and why it’s tempting to think it doesn’t)

The natural reaction at this point is: “But everyone is in the same position — they all got the same whisper. So once I know that everyone knows, I should be able to see that everyone else knows that too, and the whole thing collapses into full openness.” This is intuitive but wrong, and seeing exactly where it breaks is the key to understanding why infinite iteration is needed.

Track the levels concretely using the “more private information” case. Remember, each student is told: “I’m telling everybody I’m from Maine — but you’re the only one I’m telling that I’m telling everyone.” E(p) = “everyone knows p”

  • Level 1 — E(p): Everyone knows p. Does Sarah? Yes — the professor told her directly. (And the same is true of every other student.) ✓
    • Formally: K_Sarah(p) ✓, and likewise for all agents.
  • Level 2 — E(E(p)): Everyone knows that everyone knows p. Does Sarah know that everyone knows? Yes — the professor told her “I’m telling everyone.” ✓
    • Formally: K_Sarah(E(p)) ✓, and likewise for all agents.
  • Level 3 — E(E(E(p))): Everyone knows that everyone knows that everyone knows p. Does Sarah know that everyone has level 2? No — she was told she’s the only one who got the “I’m telling everyone” information. So from her perspective, the other students might only know “the professor is from Maine” and nothing more. She thinks they could be stuck at level 1. She can’t verify that they have level 2, so level 3 fails. ✗
    • Formally: K_Sarah(E(E(p))) ✗ — Sarah lacks access to other agents’ epistemic states, so she cannot verify E(E(p)).

Notice: everyone actually has level 2. But nobody knows that everyone has level 2, because each person was told they’re the only one who got the “I’m telling everyone” information. The gap is between what’s true and what’s known to be true.

The tempting inductive step — “everyone is in the same position as me, so everyone knows what I know” — is precisely the move that fails. Everyone is in the same position. But nobody knows that everyone is in the same position, because the whisper structure hides this. Getting from “X is the case” to “everyone knows X is the case” always requires one more epistemic level. That’s the gap, and it never closes on its own.

You can try to fix this by enriching the whisper. To illustrate: the professor pulls aside three students (you, Sarah, Jake) individually and says: “I’m from Maine. I’m telling everyone. And I’m telling everyone that I’m telling everyone. But you’re the only one I’m telling that I’m telling everyone that I’m telling everyone.”

Here’s what each student thinks happened.

You:

  • Level 1: “I’m from Maine” → You know p — the professor is from Maine.
  • Level 2: “I’m telling everyone” → Every student is being told level 1.
  • Level 3: “I’m telling everyone that I’m telling everyone” → Every student is being told level 2 — everyone will know that everyone knows the professor is from Maine.
  • But: “You’re the only one I’m telling level 3.” That is, Sarah and Jake were told “I’m from Maine, and I’m telling everyone.” They didn’t get the extra sentence about everyone being told that everyone is being told.

Sarah and Jake:

  • Same enriched whisper. They have levels 1, 2, and 3.
  • But they think the others were only told “I’m from Maine, and I’m telling everyone.” They think everyone else is stuck at level 2.

Now check the levels:

  • Level 1 — E(p): Does everyone know p? Yes, all three. ✓
  • Level 2 — E(E(p)): Does everyone know that everyone knows? Yes, all three. ✓
  • Level 3 — E(E(E(p))): Does everyone know that everyone knows that everyone knows? Yes, all three. ✓
  • Level 4 — E(E(E(E(p)))): Does everyone know that everyone has level 3? Do you know that Sarah has level 3? No — you were told you’re the only one who got the “I’m telling everyone that I’m telling everyone” part. You think Sarah only has level 2. ✗

Everyone actually has level 3. But nobody knows that everyone has level 3. The “but you’re the only one” clause always creates a ceiling one level above what it grants. The failure point has moved from level 3 to level 4, but it hasn’t disappeared. Each upgrade pushes the problem up by exactly one rung. You never catch up.

Formal statement

Let Kᵢ(p) = agent i knows that p. Let E(p) = everyone knows p, i.e. K₁(p) ∧ K₂(p) ∧ … ∧ Kₙ(p). Let Eⁿ(p) = n iterations of the everyone-knows operator (so E²(p) = E(E(p)), etc.). Then common knowledge is the infinite conjunction: C(p) = E(p) ∧ E²(p) ∧ E³(p) ∧ …

Each whisper case constructs a model where Eⁿ(p) holds but Eⁿ⁺¹(p) fails: private information satisfies E¹(p) but not E²(p); more private information satisfies E²(p) but not E³(p); and so on. The tempting inductive move — that E(p) entails E(E(p)) — amounts to claiming the E operator is idempotent. It isn’t. From the fact that everyone knows p, it does not follow that everyone knows that everyone knows p. That inference requires access to other agents’ epistemic states, which the whisper scenarios deliberately withhold.

The public announcement works because it makes all agents’ epistemic access converge: the one-step and two-step and n-step reachable worlds are the same. So Eⁿ(p) holds for all n, which just is C(p).

The recipe generalizes. For each level you propose as sufficient for public information, Lederman can construct a professor speech that satisfies exactly that level while preserving the felt absence of openness — each new scenario adds one more layer of “I told everyone, but only you know that,” pushing the failure point up by one. Private information fails at level 2. More private information fails at level 3. An “even more private information” scenario would fail at level 4. And so on without limit. The negative argument is that no fixed level — however deep — is sufficient for public information.

This is what makes the public announcement structurally different. When the professor speaks out loud to the whole room, she doesn’t just make the same content available to everyone — she makes the process of distribution itself visible to everyone, simultaneously. Everyone sees everyone hearing it, sees everyone seeing everyone hearing it, and so on. There is nothing hidden about the delivery mechanism. That’s what closes all the levels at once, and it’s why no sequence of private whispers — no matter how informationally rich — can replicate what a single public utterance achieves.

Everything above is the negative half of the argument: no finite level suffices. But ruling out every finite level doesn’t automatically tell you that the infinite iteration is what’s needed — that further step requires an additional premise: that public information is analyzable solely in terms of these nested layers of knowledge, with no other features of the situation mattering. Only given that premise does ruling out every finite level imply that the full infinite iteration — where everyone knows, and everyone knows that everyone knows, and everyone knows that everyone knows that everyone knows, continuing without end — is required. That infinite iteration is what Lederman calls common knowledge.

Lederman’s critical assessment: He identifies two underexamined vulnerabilities the literature has consistently overlooked.

First, the mk4 response (named for “mutual knowledge level 4” — the idea that some finite level like mk4 might actually suffice). The escalating cases work by constructing professor speeches that are increasingly complex and difficult to understand. A proponent of a finite-level analysis can argue that readers judge students to lack public information not because the cases are genuine counterexamples, but because the students don’t actually achieve the relevant mutual knowledge level—they can’t parse the nested statements or verify that others have parsed them. Kinderman et al. (1998) and Stiller & Dunbar (2007) found people essentially at chance processing statements involving six nested attitudes. The cases might be revealing the limits of human ability to achieve high-order mutual knowledge, not that high-order mutual knowledge is insufficient for public information. This response “has considerable appeal” (Lederman) but has been almost entirely ignored.

Second, the additional premise is rejectable. Even granting the negative argument, the move to the positive conclusion requires assuming that public information is analyzable purely in epistemic-iterative terms. But the felt absence of openness in the private-information cases might derive from other features of the situation—that they’re confusing, unusual, or unfamiliar. A full development of this alternative would need a systematic account of what those features are, but the general strategy is viable and has received almost no discussion.

The upshot: the escalating cases successfully demonstrate that no particular finite level of mutual knowledge is sufficient for public information. That negative result is robust. But the positive conclusion — that the full infinite iteration is therefore required — rests on assumptions that are more fragile than the literature has acknowledged. The argument for the Default Position is real, but it’s not the slam dunk it’s been treated as.

The Argument Against the Default Position (§3)

The most common objection runs in the opposite direction: common knowledge is too demanding to be realistic, because finite minds cannot hold an infinite collection of beliefs. Heal (1978) and Clark & Marshall (1981) raised early versions of this worry; Clark (1996) put it directly: “CG-iterated obviously cannot represent people’s mental states because it requires an infinitely large mental capacity.”

Lederman thinks this objection is not very powerful. People routinely hold beliefs about infinite collections without obvious cognitive impossibility. He believes, for every natural number n > 0, that no one has ever seen Santa Claus exactly n times—an infinite set of claims that clearly doesn’t impose an “unrealistic computational load.” The mere infinitary character of common knowledge isn’t a compelling objection to its psychological possibility.

Info

What is an interesting question—the one the impossibility debate should have been asking—is how common knowledge is cognitively represented. What’s the “basic belief” that grounds the infinite collection? Answering this means offering a theory of cognitive representation, not just noting that the set is infinite.

The shared environment analysis (associated with Barwise 1988, with antecedents in Lewis, Harman, Heal, and Clark & Marshall) is a candidate answer: agents have public information that p iff there is some environment E such that E is the case, E entails p, and E entails that everyone knows E. The self-referential structure of E—it entails that everyone knows it—generates the infinite iterations without anyone explicitly entertaining them. This is the formal counterpart of Heal’s point that you recognize a situation and the infinite structure falls out automatically.

Lederman also presents Stalnaker’s inversion (Stalnaker 2009): common belief might actually be easier to represent than finite levels of mutual belief. If belief is thought of negatively—in terms of the space of live possibilities one allows for, rather than sentences in a belief box—then what would be computationally demanding is believing six iterations but not the seventh, which would require representing a mind-bogglingly subtle distinction between nested possibility structures. The full infinite iteration, by contrast, is simpler. Lederman finds Stalnaker’s notion of representation “somewhat obscure” but flags it as worth taking seriously.

The field’s response to this debate mostly produced defenses of ideal common knowledge (Lederman’s term)—the weaker thesis that ideal reasoners who have public information would have common knowledge—rather than the full Default Position. Almost every author who has discussed public information at length accepts ideal common knowledge. Almost nobody actually defends the Default Position as stated.

The Electronic Mail Game (§4)

Rubinstein’s 1989 electronic mail game is a formal example where the relationship between public information, common knowledge, and coordination can be examined precisely. Lederman presents it as genuinely ambiguous between readings that support and undermine the Default Position—and argues it’s been treated as more decisive than it is.

The setup: Row and Column are uncertain which of two games they’re playing. G_A has a safe action A that guarantees 0 regardless; G_B rewards mutual B with 1 but punishes unilateral B with -2. The game is selected by fair coin flip. Row observes the outcome; if G_B is selected, an automated email chain begins between them, but each transmission has a positive probability of failure. Each player learns only how many messages their computer sent.

Rubinstein’s theorem: The unique rationalizable strategy—given common certainty of rationality—is to always play A, regardless of how many messages have been sent. Even after many rounds, even when each player is certain the game is G_B and certain the other is certain, rational agents cannot coordinate on B.

  • The standard reading: Common knowledge (or common certainty) is essential for coordination. Any finite level of mutual certainty is insufficient; the game shows what’s at stake when it’s absent.
  • Lederman’s reinterpretation: The result is derived under assumptions that include common certainty of rationality. But the prediction—that rational agents really cannot coordinate on B even after many rounds—is highly counterintuitive. If a model’s assumptions generate counterintuitive predictions, the right response might be to reject the assumptions, not accept the prediction. On this reading, the e-mail game is an argument against the game-theoretic rationality assumptions, not for the necessity of common knowledge in coordination.
  • The common p-belief alternative (Monderer & Samet 1989): For sufficiently high p < 1, agents can have common p-belief about the game setup and rationality while still rationally coordinating after finite rounds. This suggests the game might be teaching us that common p-belief is the relevant doxastic state for coordination—not common certainty or common knowledge specifically.

What the Field Actually Needs

Lederman ends with an honest assessment of the uncertain state of play. The Default Position remains the benchmark in the literature, but almost no one actually defends it as stated. The arguments for and against it are weaker than assumed. There has been almost no sustained scrutiny of whether common knowledge or public information plays the roles they’ve been claimed to play.

He highlights the common ground of conversation (Stalnaker 2002, 2014) as the clearest instance of unfinished work. It’s standardly assumed in linguistics and philosophy of language that the body of shared information among conversational participants is determined by what they commonly know. But almost no arguments have been offered for this assumption, and it’s unclear whether the linguistic data actually require it or whether something weaker would suffice.

Significantly, Lederman’s paper doesn’t leave the Default Position refuted—it leaves it undefended. The orthodoxy that everyone was treating as a foundation turns out to be a promissory note that nobody had cashed. That’s both a problem and an invitation: the most foundational concept in formal epistemology, social ontology, game theory, and philosophy of language has been resting on a connection that nobody has adequately established.

Is the Default Position even the right question?

There’s a deeper worry than whether the Default Position is true or false: the framing itself may be misconceived. The Default Position asks whether public information and common knowledge are coextensive — whether they pick out the same set of cases. But they aren’t operating at the same level of description. Common knowledge is an epistemic property: it describes what’s stacked up inside agents’ heads (everyone knows, everyone knows that everyone knows, etc.). Public information is a property of the communicative event itself: specifically, whether the event is self-disclosing — whether it makes its own occurrence, audience, and scope visible to all participants as part of its very structure. One produces the other. The Default Position collapses them into extensional equivalents.

The concept of self-disclosure makes the distinction precise. A public announcement is self-disclosing: when the professor speaks out loud to the room, you don’t just learn p — you learn that this is an event in which everyone is learning p. The event carries information about its own scope. A whisper is self-concealing: it transmits content while hiding its own distribution. No matter how much you pack into the whisper — “I’m telling everyone, and I’m telling everyone that I’m telling everyone” — you’re encoding information about scope as content, passing it through a channel that still conceals its own structure. The message can never fully substitute for the medium, because there’s always one more level: does the recipient know that the channel carrying this enriched message was also available to everyone else? The answer is always no, because it wasn’t. This is what Barwise’s shared environment analysis captures formally: the environment E entails that everyone knows E — the situation entails knowledge of itself.

The whisper cases reveal this. The standard reading is: they lack public information because they lack level n+1. But the causation arguably runs the other way. They lack public information because the communicative event is self-concealing — each student receives through a conduit only they can observe, regardless of what flows through it. The missing epistemic level is a symptom, not the cause. You don’t look at the whisper case and think “this would feel open if only Sarah had one more level of knowledge.” You think “this would feel open if the professor had just said it out loud.” The fix is always to change the medium — from self-concealing to self-disclosing — never to enrich the content by adding more nested layers to the whisper.

An analogy: think about why a handshake works. Two people reach out, clasp hands, and acknowledge each other. Why can’t you replicate a handshake by mail? You could send a letter saying “I acknowledge you,” they send one back saying “I acknowledge your acknowledgment,” and so on forever — building an infinite chain of mutual acknowledgment. But it would never be a handshake. A handshake isn’t a chain of acknowledgments. It’s a single simultaneous act where the mutuality is built into the physical structure of the event — you cannot shake hands without both parties participating at once. The simultaneity and visibility are features of the act itself, not features that emerge from stacking enough back-and-forth. The public announcement is the handshake. The whisper cases are the letters. And the Default Position is asking how many letters you need before they become a handshake.

Heal, Barwise, and Lederman’s “additional premise” vulnerability all gesture toward this. But nobody has quite assembled the explicit claim: that the Default Position is a category error because it treats a property of communicative events (self-disclosure) and a property of agents’ epistemic states (common knowledge) as candidates for extensional equivalence, when they stand in a generative relationship instead. Common knowledge is the shadow. The self-disclosing event is the object casting it.


Public Information vs Common Knowledge

Public information is an informal, pre-theoretical concept introduced by Heal designed to capture the intuitive openness of certain facts. Heal’s paradigm example is of two diners observing a conspicuous quarrel at an adjacent table. This event is completely public and transparent to both diners.

Common Knowledge is a strict, formal definition. It requires an infinite, iterative hierarchy of epistemic states. For a group to have common knowledge of a proposition , everyone must know everyone must know that everyone knows everyone must know that everyone knows… ad infinitum.

Public information is explicitly drawn from real-world scenarios. The realization of formal common knowledge, however, is highly contested. Skeptics argue that because common knowledge requires an infinite mental capacity, finite humans cannot possibly achieve it. Lederman notes that this objection is not necessarily fatal; humans routinely hold infinite sets of beliefs, such as the belief that Santa Claus has not been seen exactly times for any positive integer .

TermCategoryOperational DefinitionTheoretical Stakes
Public InformationTarget PhenomenonThe informal, pre-theoretical experience of absolute epistemic openness.This is the actual reality of coordination that requires a mechanistic explanation.
Common KnowledgeExplanatory MechanismA formal mathematical model requiring an infinite iteration of nested propositional attitudes.The orthodox assumption is that this mechanism produces public information. It is highly contested because it demands infinite computational labor from finite agents.
Mutual KnowledgeExplanatory MechanismA finite iteration of nested propositional attitudes, capped at a specific level .Proves that simply stacking discrete internal representations cannot organically produce true openness, as it remains vulnerable to higher-order deception.
Shared EnvironmentExplanatory MechanismA model where the physical scene entails both the relevant facts and the agents’ mutual perception of them.Bypasses the cognitive fatigue of infinite internal iteration by offloading the epistemic burden onto the physical dynamics of an embodied, coupled system.

Glossary

The informal phenomenon:

Public information: (Coined by Heal 1978) The informal phenomenon of openness—facts that are publicly known and shared in a situation. (e.g., Two diners observing a conspicuous argument at an adjacent table.) This is the phenomenon that the formal apparatus of common knowledge is supposed to capture. Whether the two actually coincide is the central question of the Default Position.

Shared situation: (Heal 1978, as part of Analysis E) People are together in a context that makes certain facts mutually available — not information passed privately through a network, but something simply there for everyone at once. Heal’s paradigm case: a potato falls off someone’s plate at dinner. Both diners saw it; neither needs to ask.

Mutual transparency: (Heal 1978, as part of Analysis E) Each person is aware not just of the situation itself, but that everyone else is in the same situation with the same access to it.

Shared reasoning standards: (Heal 1978, as part of Analysis E) The background assumption that people in the situation are reasoning in the same way from the same starting point. Without this, a shared situation isn’t enough: a chess grandmaster and a novice can watch the same game and see completely different things.

The formal apparatus (builds bottom-up):

Kᵢ(p): (Hintikka 1962) Agent i knows that p. The basic building block of epistemic logic. Everything else is built from this.

E(p) / the E operator: (Standard in epistemic logic since Hintikka 1962) “Everyone knows that p” — shorthand for K₁(p) ∧ K₂(p) ∧ … ∧ Kₙ(p). Applying E repeatedly produces the levels: E(p) is level 1, E(E(p)) is level 2, E(E(E(p))) is level 3. Crucially, E is not idempotent: E(p) does not entail E(E(p)). That is, from the fact that everyone knows p, it does not follow that everyone knows that everyone knows p — because knowing something yourself doesn’t guarantee you know what’s going on in other people’s heads. Each additional application of E demands access to other agents’ epistemic states, which is a genuinely new requirement that isn’t contained in the previous level. This is the formal core of why the whisper cases generate a gap at every level: each scenario satisfies Eⁿ(p) but fails at Eⁿ⁺¹(p), and no amount of finite stacking closes the gap.

Idempotent: (General mathematical term) An operation is idempotent if applying it twice gives the same result as applying it once. The E operator is not idempotent: E(p) ≠ E(E(p)). This is why the levels never collapse—each application of E adds a genuinely new epistemic requirement.

Common knowledge (formal): (Lewis 1969, formalized further by Aumann 1976) The infinite conjunction C(p) = E(p) ∧ E(E(p)) ∧ E(E(E(p))) ∧ … — everyone knows, everyone knows that everyone knows, and so on without end. No finite number of levels suffices; common knowledge is what you get when you need all of them.

The debate (does the phenomenon = the formalism?):

Default Position: (Coined by Lederman 2017) The thesis that public information and common knowledge are coextensive — a group has public information that p iff they have common knowledge that p. Widely assumed across philosophy, linguistics, and game theory, but Lederman’s central finding is that it’s unrefuted but also unestablished.

Ideal common knowledge: (Lederman’s term, 2017) The weaker thesis that ideal reasoners who have public information would have common knowledge. Almost everyone in the literature accepts this; almost nobody defends the stronger claim (the Default Position) about real human cognition.

mk4 response: (Discussed by Lederman 2017, drawing on Kinderman et al. 1998 and Stiller & Dunbar 2007) Named for “mutual knowledge level 4.” The objection that the escalating whisper cases fail not because finite mutual knowledge is genuinely insufficient, but because people can’t parse the deeply nested statements well enough to achieve the relevant level.

Candidate solutions and alternatives:

Shared environment analysis: (Barwise 1988, with roots in Lewis, Harman, Heal, and Clark & Marshall) Common knowledge arises when there is some situation E that entails p and entails that everyone knows E. The self-referential structure generates the full infinite hierarchy automatically — the formal counterpart of Heal’s insight that you recognize a situation and the levels fall out without iterating.

Common p-belief: (Monderer & Samet 1989) A weaker alternative to common knowledge requiring only that everyone believes p with probability ≥ p (rather than certainty), everyone believes that with probability ≥ p, and so on. Relevant because it offers an escape from Rubinstein’s email game — agents can rationally coordinate after finitely many rounds, unlike under full common knowledge.