Atomspace consistency and completeness

[Ivan V.]

Does anyone have a general opinion on Gödel's incompleteness theorem? The question that rises is: "Is Atomspace a consistent formation, or is it an incomplete formation, assuming it is an embodiment of a kind of logical system?"

I ask this question because I suspect it is possible to define different rule sets in URE, applying to different areas of knowledge contained within AGI's memory, thus overcoming limitations of Gödel's incompleteness proof of a logical system while retaining overall knowledge relative consistency.

Does OpenCog support this kind of rule divergence?

[Ben Goertzel]

Atomspace is more of a meta level framework that can be used to
implement different sorts of formal systems, including different
logics ... it does not enforce logical consistency and thus can
achieve logical completeness (given arbitrarily much resources)

In OpenCog Hyperon we are aiming to formalize this aspect more
precisely using gradual typing and paraconsistent logic, but the
current Atomspace/Atomese achieves this simply by not requiring any
particular type system or associated set of restrictions on the
nodes/links...

So basically, consistency and such exist on the level of particular
sets of node/link types with associated rules, not on the level of the
underlying infrastructure...

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--
Ben Goertzel, PhD
http://goertzel.org

“Words exist because of meaning; once you've got the meaning you can
forget the words. How can we build an AGI who will forget words so I
can have a word with him?” -- Zhuangzhi++

[Ivan V.]

Thank you for an answer.

I never thoroughly investigated it, but from what I know about it, paraconsistent logic seems like it may be a good option to consider. I never thought of it like putting it at the roots of deductive systems, but when I look at it in that light, it certainly rings some inspirative bells.

Thanks again,

ivan v.

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[Ben Goertzel]

Yeah actually OpenCog's PLN logic can be viewed as a kind of
paraconsistent logic, I have a plan to write up a paper on this when I
get time. Basically when you have evidence for two conflicting
possible truth values for a relationship, you can view this from a
paraconsistent-logic view, as the relationship being (provisionally at
least) partly true and partly false, i.e. both true and false to
different degrees

Paraconsistency of a logic maps via Curry-Howard correspondence into
gradual typing of a programming language, which is part of why we are
aiming to make Atomese 2 (a key part of Hyperon design) gradually
typed...

ben

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