Let D(S) read as, "I disagree that S." It is possible to iterate this, so that DD(S), "I disagree that I disagree that S." Then we can go on to DDD(S), and so on. (For a peer-reviewed text that goes over its own theory of a disagreement operator, see Haret and Woltran, "Belief Revision Operators with Varying Attitudes Towards Initial Beliefs.")

At first, I was thinking that we would start losing track of the significance of iterated disagreement, like once we got to, say, D227(S) (for that many iterates), we would have trouble understanding what exactly it is that we are disagreeing with. Plainly enough, DD(S) already is not plausibly implied by just D(S), so we have a lack of a parallel with "standard" epistemic logic (where k(S) yields kk(S)) here. However, the disagreement operator can essentially figure in doxastic logic, which partly underlies/underscores epistemic logic (esp. when an epistemic logic carries a belief operator); and so its cognitive limitations would seem pertinent to the question of iterating the belief operator, and then even the knowledge operator itself.

On the other hand, I'm intuitively reading, "I disagree that I disagree that I disagree that S," as canceling out the denial of S-disagreement. Then D4(S) cancels that cancellation out, and so on. So maybe we can just flip back and forth between a pro-disagreement(!) moment in the unfolding expression, and an anti-disagreement one, rather than diffuse our cognition of our capacity for disagreement into the doxastic-modal ether.

Would the back-and-forth phenomenon occur in the iteration of the disagreement operator, or would the sense of disagreement break down eventually instead, yielding ramifications for the iteration of belief and knowledge operations in turn?

  • No, not again! Some people, it seems, want to live forever. Mar 11 at 11:50
  • @AgentSmith ??? I can usually unpack things but I'm feeling kinda dense this morning haha. Mar 11 at 11:55
  • Visited yer profile @Kristian Berry. 😊 Mar 11 at 11:58
  • @AgentSmith although I still think the idea of a justification function is useful, I'm not as worried as I was back then, about finding some sort of "selective" response to CH/forcing. In the outline from my notes, for a compiled analysis of the notes, I'm thinking the section on CH could do as an appendix, even. There are some themes that can be illustrated with reference to the powerset question, but I don't think I need to detain the audience with that subplot in the direct middle of the main storyline. Mar 11 at 12:03
  • 1
    Lim D^n(S) = -1/12 S
    – Frank
    Mar 11 at 15:29

1 Answer 1


Digital computers have call stacks, so that for any D(s), one can invoke recursively D(s), at least theoretically, for as many calls as one has computing resources. Factorials for instance can be computed for very large numbers. The recursion in such systems can be meaningfully grounded in denotational semantics which is the idea that deep recursion has mathematical meaning.

You want to apply recursion to human semantics, and I think meaning is assigned in the same way, by grounding high-order predication as an abstraction of operation. For instance:

  1. 'I disagree that s.' D(s) is a proposition. Meaningful.
  2. 'I disagree that I disagree that s'. D(D(s)) is a propositional attitude. Meaningful.
  3. 'I disagree that (2)'. D(D(D(s))) is still meaningful. One routinely disagrees with propositional attitudes all of the time. Meaningful.
  4. 'I disagree that (3)'. This is a higher-order propositional attitude. Meaningful, but highly abstract.
  5. 'I disagree that (4)'. I think it ends here, at least in terms of intuition. What does it mean to disagree with a higher-order proposition? Unless there is some way to abstract, and therefore make the claim 5, meaningful, you are in the territory of abstract nonsense at least in a broad reading.

Like the factorial function, there is no theoretical limit in the recursion, but there is a practical one for the human brain. That is, natural language semantics essentially ends around 5 invocations. How do we know? There's no word-thing definition for 5. Four can be understood as a 'disagreement with a propositional attitude that disagrees about a propositional attitude of disagreement'. It's abstract, and would be understood by logicians who are used to that sort of abstraction, but extending it, without finding a way to appealing to a meaningful analysis, synonym, or so on would quite literally leave it without meaningful semantic grounding in natural language other than as abstract nonsense. So:

  1. 'I disagree that (5).' Has meaning only in that it can be shown 5 has meaning and 6 itself isn't meaningless (since the syntax has been shown to be meaningful in 1.

But can one meaningfully paraphrase 6? I don't think so, and it would be incumbent upon anyone who disagrees to provides a meaningful, natural language construction that passes a usage panel. In fact, I think a usage panel of non-experts might even declare 5 or 4 meaningless. It's only with the education and experience regarding higher-order logics that an expert might affix a meaning of nested abstraction, a concept that is not available to everyone.

Does it break down? An emphatic yes. And where does the abstraction break? I think it wanes around 4 or 5 and is gone by the 6th invocation of predication. Much like the factorial function, the semantic grounding at high orders lies in seeing the nested predication as an abstract process of recursion, and lacks any other meaningful association.

  • Does the 4-or-5 barrier have anything to do with Russell's relatively early idea about the hierarchy of types flattening out at a 4th/5th level? From what I read, Russell quickly enough decided that the hierarchy would go on and on instead, but I've always wondered why he would have thought there was a finite limit at exactly one of those numbers of levels. Although I found a non-superstitious, but apparently trivial, reason for why 4-fold structures kept showing up at a crucial level of my own theory, I still wonder if there is a nontrivial reason for 4-or-5 to be a barrier number, there. Mar 11 at 18:14
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    @KristianBerry If you believe there to be a 4-or-5 barrier of sorts in regards to the comprehensibility of nested structures of formalism, then I'd simply argue that it's a psycholinguistic limit. Instead of some deep transcendental relationship, the human brain simply has a limit on its ability to parse syntax so any attempt to have meaningful formalisms bottoms at around the same place.
    – J D
    Mar 12 at 0:39
  • So maybe Russell experienced that neurological(?) limit, accidentally conflated it with a type-theoretic limit in abstracto, somehow realized his error (in some form, maybe not attributed to neurological barriers, but perhaps close to such attribution? IDK, really), and then moved on to his more expansive theory, without realizing the physical significance of his earlier issue? I realize that's a speculative psychological-historical assessment of a widely-read philosophy writer/teacher, so I don't want to press the matter too much, but is it remotely plausible? Mar 12 at 1:11
  • @KristianBerry I would say that Russell wasn't around late enough to understand the neurobiological nature of language. If you're familiar with Pinker's somewhat dated work How the Mind Works, you'll see that there is neurological architecture that shapes language production and use. That architecture isn't dissimilar from computer architecture, except more complicated by far. In CS, there's the notion of Chomskian hierarchy of grammars. I think you'd be very interested in finite automata as a formalism...
    – J D
    Mar 12 at 3:00
  • Finite automata is the requisite mathematical formalisms for understanding the equivalence between state machines and artificial languages. What I'm saying is that if you believe in the Curry-Howard correspondence, the nature of logical systems and the limits in their semantic grounding are best understood as mathematical formalisms of computation, and that Russell would have run into the limits of language as he was exploring logic. All logic is inherently linguistic. Philosophers were the first to formalize information processing.
    – J D
    Mar 12 at 3:04

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