Abstract
We propose that, for the purpose of studying theoretical properties of the knowledge of an agent with Artificial General Intelligence (that is, the knowledge of an AGI), a pragmatic way to define such an agent’s knowledge (restricted to the language of Epistemic Arithmetic, or EA) is as follows. We declare an AGI to know an EA-statement \(\phi \) if and only if that AGI would include \(\phi \) in the resulting enumeration if that AGI were commanded: “Enumerate all the EA-sentences which you know.” This definition is non-circular because an AGI, being capable of practical English communication, is capable of understanding the everyday English word “know” independently of how any philosopher formally defines knowledge; we elaborate further on the non-circularity of this circular-looking definition. This elegantly solves the problem that different AGIs may have different internal knowledge definitions and yet we want to study knowledge of AGIs in general, without having to study different AGIs separately just because they have separate internal knowledge definitions. Finally, we suggest how this definition of AGI knowledge can be used as a bridge which could allow the AGI research community to import certain abstract results about mechanical knowing agents from mathematical logic.
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Notes
- 1.
Perhaps the best example being in the Theaetetus [18].
- 2.
By a sentence, we mean a formula with no free variables. Thus, \(x^2>0\) is not a sentence, but \(\forall x (x^2>0)\) is.
- 3.
Often phrased more like “cannot know its own code”, with knowledge-of-own-truthfulness taken for granted.
- 4.
We should note that, with the AGI research field being so young, there is little consensus even on basic things. Some researchers would consider some things to be AGI which have no communication ability (applying the term to entities who have certain adaptation abilities or pattern-matching abilities, for example, even if those entities have no means of communicating), however, we believe that to be a minority opinion.
- 5.
We assume the AGI explicitly follows commands (that it is “under explicit control”, to use Yampolskiy’s terminology [24]).
- 6.
This is reminiscent of Williamson’s contextualism [23].
- 7.
This is reminiscent of Elton’s proposal that instead of trying to interpret an AI’s outputs by focusing on specific low-level details of a neural network, we should instead let the AI explain itself [12].
- 8.
Which is incorrect—see [16].
- 9.
This is reminiscent of a recent argument [17] that humans maintain superiority over the AIs they create, as, for example, today’s latest and greatest chess-playing AI is better at tactically playing individual games of chess, but is incapable of designing its own replacement (tomorrow’s latest and greatest chess-playing AI), which will instead be designed by humans (making humans still better at chess in a higher-level sense).
- 10.
We assume Peano arithmetic is true.
- 11.
It can be shown that \(W_e\) is definable in the language of Peano arithmetic, therefore we can use expressions like “\(x\in W_e\)” in EA-formulas as shorthand.
- 12.
A universal closure of a formula \(\phi \) is a sentence \(\forall x_1\cdots \forall x_k\phi \), and the universal closure of a schema of formulas is the schema of universal closures of those formulas.
- 13.
We have pointed out elsewhere [3] that (i) Reinhardt implicitly assumes that the knower knows its own truthfulness; and (ii) it is possible for a knowing machine to know its own code if it is allowed to be ignorant of its own truthfulness, despite still being truthful. See [1] and [2] for some additional discussion.
- 14.
Or rather, its own code and its own truthfulness—we have pointed out [3] that Reinhardt implicitly assumes the knower knows its own truthfulness.
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Acknowledgments
We gratefully acknowledge Alessandro Aldini, Phil Maguire, Brendon Miller-Boldt, Philippe Moser, and the reviewers for comments and feedback.
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Alexander, S.A. (2021). Short-Circuiting the Definition of Mathematical Knowledge for an Artificial General Intelligence. In: Cleophas, L., Massink, M. (eds) Software Engineering and Formal Methods. SEFM 2020 Collocated Workshops. SEFM 2020. Lecture Notes in Computer Science(), vol 12524. Springer, Cham. https://doi.org/10.1007/978-3-030-67220-1_16
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