Abstract
The logical analysis of epistemic paradoxes—including, for example, the Moore and Gödel-Buridan paradoxes—has traditionally been performed assuming the whole range of corresponding modal logic principles: \(\lbrace \mathsf{D, T, 4, 5}\rbrace \). In this paper, it is discovered precisely which of those principles (including also the law of excluded middle, LEM) are responsible for the paradoxical behavior of the Moore, Gödel-Buridan, Dual Moore, and Commissive Moore sentences. Further, by reproducing these paradoxes intuitionistically we reject a conjecture that these paradoxes are caused by the LEM. An exploration of the Gödel-Buridan sentence prompts the inquiry into a system, Doxastic Arithmetic, \(\mathsf{DA}\), designed to represent the arithmetical beliefs of an agent who accepts all specific arithmetical proofs and yet believes in the consistency of their own beliefs. For these reasons, \(\mathsf{DA}\) may be regarded as an epistemic way to circumvent limitations of Gödel’s Second Incompleteness Theorem.
V. A. Peluce—The author would like to thank Sergei Artemov for discussion and guidance throughout the development of this project.
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Notes
- 1.
This notion is closely related to Hintikka’s similar concept of doxastic indefensibility, [10], p. 71. Utterance, being of central importance to Hintikka’s concept, does not enjoy the same status in our notion, however. Hintikka writes, “[W]e shall call the set \(\lbrace p_{1}, p_{2}, \ldots p_{k} \rbrace \) doxastically indefensible for the person referred to by this term to utter if and only if the sentence \( \mathcal {B}_{a}(p_{1}\ \& \ p_{2}\ \& \ldots \& \ p_{k})\) is indefensible simpliciter.” A natural question is: what Hintikka meant by a formula’s indefensibility simpliciter? Given [10], p. 32, a natural answer is inconsistency.
- 2.
This sort of result is offered in [10].
- 3.
In its classical form, this sort of observation is not a new one, see for instance, [11], p. 2. It is not clear, however, that this has been demonstrated in iKD4.
- 4.
Burge writes that “In Sophism 13 of Chapter VIII Buridan supposes the following proposition is written on the wall: Socrates knows the proposition written on the wall to be doubted by him. Socrates reads it, thinks it through and is unsure (doubts) whether or not it is true. Further, he knows that he doubts it. Buridan asks whether or not the proposition is true.” [5], p. 22.
- 5.
- 6.
It would be more precise to use notation \(G_A\) here since G actually depends on A for the reasons mentioned. Since we said we would set those considerations aside for now, we stick to a shorter notation.
- 7.
The author thanks Sergei Artemov for this proof.
- 8.
Necessitation is not always unproblematic, especially in the epistemic context. See, for instance, Artemov’s [2]. It is admissible in this case, however, because \(\lnot \square A\) follows using only iKD4 and \(\square G\).
- 9.
Caie [7], p. 36, discusses the Moore sentence as relates to G. There are some things having to do with that discussion, though, that are worth clarifying. He writes, “We’ve considered two classes of sentences, the Moore-paradoxical and the Burge-Buridan sentences. It’s worth noting, however, that the latter class is really a subclass of the former. In general, a Moore-paradoxical sentence is one that has the following form \(\phi \wedge \lnot \mathcal {B} \phi \). A Burge-Buridan sentence, on the other hand, has the form \(\lnot \mathcal {B}T(\beta )\), where \(\beta \) refers to that very sentence. On the surface, of course, this does not seem to have the form of a Moore-paradoxical sentence. However, given the plausible assumption that \(T(\beta )\) and \(\lnot \mathcal {B}T(\beta )\) are logically equivalent, then we get that \(\lnot \mathcal {B}T(\beta )\) is, in fact, equivalent to \(T(\beta ) \wedge \lnot \mathcal {B}T(\beta )\). Thus a Burge-Buridan sentence, while not having the overt form of a Moore-paradoxical sentence, is equivalent to a Moore-paradoxical sentence.” Though G does in fact share features with M, Caie’s claim is too strong. For as we have shown, G is equivalent not to the Moore sentence M but to the disjunction of the Moore and Dual Moore, DM.
- 10.
For an overview of its discussion in the literature, see Green and Williams [12].
- 11.
- 12.
See [3] for some discussion.
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Peluce, V.A. (2018). From Epistemic Paradox to Doxastic Arithmetic. In: Artemov, S., Nerode, A. (eds) Logical Foundations of Computer Science. LFCS 2018. Lecture Notes in Computer Science(), vol 10703. Springer, Cham. https://doi.org/10.1007/978-3-319-72056-2_17
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