Audio note: this article contains 49 uses of latex notation, so the narration may be difficult to follow. There's a link to the original text in the episode description.

Löb's Theorem:

If <span>_vdash Box x to x_</span>, then <span>_vdash x_</span>.
Or, as one formula: <span>_Box (Box x to x) to Box x_</span>

Payor's Lemma:

If <span>_vdash Box (Box x to x) to x_</span>, then <span>_vdash x_</span>.
Or, as one formula: <span>_Box big(Box (Box x to x) to xbig) to Box x_</span>.

In the following discussion, I'll say "reality" to mean <span>_x_</span>, "belief" to mean <span>_Box x_</span>, "reliability" to mean <span>_Box x to x_</span> (ie, belief is reliable when belief implies reality), and "trust" to mean <span>_Box (Box x to x)_</span> (belief-in-reliability).

Löb says that if you have trust, you have belief.

Payor says that if you can prove that trust implies reality, then you have belief.

So, both results give conditions for belief. Indeed, both results give conditions equivalent to belief, since in both cases the inference can also be reversed:

Bidirectional Löb: <span>_vdash Box x to x iff vdash x_</span>
Bidirectional Payor: <span>_vdash Box (Box x to x) to x iff vdash x_</span>

Furthermore, both results relate reliability with [...]

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First published:
November 8th, 2025

Source:
https://www.lesswrong.com/posts/EQekyqyxzzHd8MHMX/comparing-payor-and-loeb

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Narrated by TYPE III AUDIO.

“Comparing Payor & Löb” by abramdemski

LessWrong (30+ Karma)