Talk:Doxastic logic

Concern from a "conceited reasoner"
i don't want to edit the main page but there are a couple of misleading things said about conceited reasoners. here are what i see as the problems. i've doublechecked myself and i really don't believe i'm wrong here:

Under "Types of reasoners"


 * "A conceited reasoner will necessarily lapse into an inaccuracy."

i don't think this is correct and think it should just be eliminated. 98.200.89.69


 * Your impression is false. The interesting truth that "A conceited reasoner (as herein defined) will necessarily lapse into an inaccuracy (as herein defined)" was proven by Smullyan. Greg Bard 03:45, 16 April 2010 (UTC)

Concern about reasoners of type 1
Under "Gödel incompleteness and doxastic undecidability"

"''A reasoner of type 1 is faced with the statement "You will never believe this sentence." The interesting thing now is that if the reasoner believes they are always accurate, then they will become inaccurate. Such a reasoner will reason: "If I believe the statement then it will be made false by that fact, which means that I will be inaccurate. This is impossible, since I'm always accurate. Therefore I can't believe the statement, it must be false.""''

this is all correct right up until the last 4 words "it must be false". the last sentence takes the form ~Bp->~p (equivalently, p->Bp). but this just isn't an inference that the conceited reasoner ever needs to make. the only inference they need to make is Bp->p. the conceited reasoner merely believes that they believe no falsehoods, not that they believe all the truths.

put another way, the conceited reasoner could have some sort of recognition of the S sentence's truth or truth-likeness or whatever but this recognition not amount to an actual belief in S. there's no inconsistency here and, in fact, this is exactly what any consistent reasoner should do. it doesn't suddenly render the conceited reasoner inconsistent. it only renders inconsistent someone who believes p->Bp.

so, in summary, please correct me if i'm wrong here but i just don't see how you get from ~Bp->~p, that is p->Bp, to its converse of Bp->p. i can't see any substitution that validates such a move and i don't see anything else in being a type 1 reasoner that requires it.

all of this is from a conceited, consistent, regular, stable, probably nonreflexive, probably nonmodest believer who is undecided on his normalcy ... ergo type 2 or 3.

by the way, i don't much like the implication that my reasoning is inferior or displays a lack of self-awareness or moral or doxastic turpitude or whatnot. me beta, you alpha and all that. 98.200.89.69


 * I'm not sure where the departure is, however, again, these interesting, if non-intuitive results have been proven rigorously by Smullyan. If you have criticisms, then perhaps others do also. If that is the case, they should be incorporated into a "criticisms" section.


 * The article isn't talking about you personally silly! These "reasoners" are types of [[conceptual model|

models]] used to better understand the logic of reasoning. There aren't any people walking around who robotically adhere to these types. However, people do, at times, reason within these patterns. There are also implications for artificial intelligence, etcetera. Perhaps some bot will feel insulted by it someday soon. Greg Bard 03:45, 16 April 2010 (UTC)

I have problems with the references to (modal axiom N)and " This is equivalent to modal system K", There is no Axiom N and the modal rule of Necessitation (RULE N) isn't valid for reasoners of  type 1, (the rule of Necessitation is about theorems, not only about tautologies). Therefore a reasoner of type 1 is much weaker than modal logic K, [][](P -> P)and []([](P->Q) -> ([]P ->[]Q)) are theorems of modal logic K but aren't always believed by a reasoner of type 1, if he did he would be a reasoner of almost type 3. — Preceding unsigned comment added by 188.28.92.100 (talk) 10:07, 1 September 2012 (UTC)

Types of reasoner
Whoever added this section of classifications deserves an Award for Pointing Out Cool Literature - very interesting paper by Smullyan which I'm happily digesting now. Thanks whoever you are. --Andy Fugard (talk) 11:21, 17 September 2009 (UTC)


 * ...and I thought contributing to WP was thankless! I am glad to have brought it to you. Be well, Greg Bard 03:45, 16 April 2010 (UTC)

Godel's sentence in Doxastic logic
The argument in the Godel's sentence section is wrong. It says that the sentence

G: "I will never believe this sentence"

is true, but will not be believed by the reasoner. As in all other presentations of Godel's theorem in a colloquial setting (without defining the notions with an underlying computation), this is incorrect, because it does not give a formal way of defining "believe" or "this sentence". Using this type of naive reasoning, the correct conclusion is that the belief system is inconsistent, because the sentence

G: "I do not believe this sentence"

can neither be believed or disbelieved. You don't solve this problem by adding additional classes like "agnostic" to "believe" "disbelieve". Because of

G: "I will neither be agnostic about this sentence nor believe this sentence"

As in all other presentations of Godel's theorem, or logic in general, in order to make sense, the ideas must be presented inside a definite system of computation. You need a computer program which produces deductions about what the reasoner believes or does not believe. Once you have the program, if the belief system includes statements about computations, then you can construct a computer program which shows that the reasoning is incomplete as follows:

Write SPITE to

1. print its code into variable R 2. deduce beliefs, looking for the belief "R does not halt" 3. if it finds this belief, halt.

This will prove incompleteness for deductive belief systems, but not in the naive way presented. The naive way is just a standard bad presentation of Godel's theorem proof. I don't know this literature, and I am not sure if it is just colloqial or if there is an underlying definition in terms of definite computations, so I can't fix the section.Likebox (talk) 21:21, 13 October 2009 (UTC)

Inaccuracy and peculiarity of conceited reasoners
This is a very helpful article summarizing nicely and precisely the extensive work of Smullyan. Thanks!

In the last sentence of the proof, no. 8, it states: "¬S [because S ≡ BS]"

I think it should read: "¬S [because ¬S ≡ BS]"

31.167.13.42 (talk) 19:35, 18 January 2018 (UTC)

Article misnamed?
The article says:
 * To demonstrate the properties of sets of beliefs, Raymond Smullyan defines the following types of reasoners:

The article seems to be about believERS and not about the logic of statements about belief, nor about belief systems. --JimWae (talk) 22:41, 24 March 2011 (UTC)


 * Are we reading the same article?!Greg Bard (talk) 23:41, 24 March 2011 (UTC)

Psychological accuracy?
It would be nice to see in the article a discussion of to what extent doxastic logic is psychologically accurate. For example, it is clear that humans in general don't possess "Type 1" rationality - their beliefs are not in practice closed over modus ponens (you can easily believe all the premises of a valid argument, yet fail to believe the conclusion; you can believe all the axioms yet fail to believe many theorems). A lot of the material here does not seem focused on trying to build a logic that can be used to realistically reason about human beliefs (which is not to say that it cannot be interesting for quite different reasons.) 60.225.114.230 (talk) 11:47, 27 April 2012 (UTC)

Personally, I don't think this is completely psychologically accurate, as we have different levels of belief: we may think something is possibly true, likely true, etc. but there are not many things we're entirely certain about. Also, if I understand correctly, a type 1 reasoner is not necessarily closed over modus ponens, but rather they will *eventually* believe every tautology (for every tautology, there is a time starting from which they will believe the tautology). But as there are infinitely many tautologies and humans don't live forever and we can only remember finitely many things, they are clearly not a type 1 reasoner. If humans did live forever and had an infinite memory, then it may be possible for such a human to be a type 1 reasoner. --65.92.20.247 (talk) 07:00, 26 May 2019 (UTC)

External links modified
Hello fellow Wikipedians,

I have just added archive links to 2 one external links on Doxastic logic. Please take a moment to review my edit. If necessary, add after the link to keep me from modifying it. Alternatively, you can add to keep me off the page altogether. I made the following changes:
 * Added archive https://web.archive.org/20060901180920/http://cs.wwc.edu/KU/Logic/Book/book/node17.html to http://cs.wwc.edu/KU/Logic/Book/book/node17.html
 * Added archive https://web.archive.org/20070505182531/http://moonbase.wwc.edu:80/~aabyan/Logic/Modal.html to http://moonbase.wwc.edu/~aabyan/Logic/Modal.html

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Conceited reasoner not necessarily inaccurate
The proof that a conceited reasoner of type 1 is inaccurate is wrong. A type 1 reasoner has a "complete knowledge of propositional logic", which does not allow self-referential sentences, yet such a sentence ("I will never believe this statement.") is used to prove the reasoner is inaccurate. An accurate conceited reasoner would not simply not draw any conclusions from such a self-referential sentence. --65.92.20.247 (talk) 06:47, 26 May 2019 (UTC)
 * The proof is essentially the same as the one in the cited reference, and I don't think it is possible to prove conceited reasoners to be inaccurate without self-referential statements. I removed the claim for this reason. Qbt937 (talk) 18:01, 5 July 2019 (UTC)

Notation and Syntax?
In this article


 * $$\mathcal{B}p$$

Is shown as indicating a belief regarding $$p$$; elsewhere very similar notation indicates the belief of agent $$p$$ and the thing about which belief is held is appended: $$\mathcal{B}_pA$$ indicates individual $$p$$'s belief that it is the case that $$A$$.

For example, the notation just described is used in a relatively recent publication - Rönnedal (2018).

Although Rönnedal innovates somewhat (introducing new operators), the notation $$\mathcal{B}_pC$$ isn't an innovation (it appears in work he cites). Furthermore, aligns well with notation for Epistemic Logic: on the Wikipedia page for Epistemic Logic, for example: "$\mathit{K}_a\varphi$ can be read as 'Agent $a$ knows that $\varphi$.'"

I realise that the $$p$$ in $$\mathcal{B}p$$ on this page is not a subscript, however it's still somewhat misleading; equally-importantly, if $$\mathcal{B}p$$ is belief about $$p$$ it says nothing about who is doing the believing.

Kratoklastes (talk) 21:31, 14 January 2020 (UTC)

Article is illogical?
This article appears to be an article on a historical view of a type of modal logic. However, I have some serious questions about the correctness of the contents. Citations and clarification, and adding context would probably fix this. I'll start with the worst problems first.

The last section, "Inconsistency of the belief in one's stability" contains a handwave. It may be possible that stability as defined by the person or persons who originated doxastic logic is inconsistent, however, B(BBp -> Bp) does not imply B(Bp -> p). In fact, I can think of a very good reason why one might hold the former belief and not the latter, so I don't know where this is coming from.

The section "increasing levels of rationality" is making the implicit assumption that the higher you get, the more rational you are. However, each level is making additional assumptions, which is reducing the likelihood of the correctness of the belief system. From the perspective of subjective evaluation of how rational an agent is, I do not see why this implication is justified.

I also think the reflexive reasoner example needs disambiguation. Currently it looks like it's saying that reflexive reasoners are imagining a proposition q which is defined as Bq -> p. A proposition which is almost definitely false, as belief cannot directly affect reality, unless the reality under examination is belief.

2001:56A:71BA:6800:516E:2307:6E53:5BE4 (talk) 17:46, 1 August 2020 (UTC)