NAL will be introduced in multiple layers. At each layer, the grammar rules, semantic definitions, and inference rules of the logic will be further extended, to increase the expressive power of the language and the inferential power of the logic.
For each NAL-n, an idealized version of it is defined first, which is a binary axiomatic logic, called IL-n (Inheritance Logic, the n-th layer). IL itself does not acknowledge AIKR, but specifies the boundary cases of NAL.
| IL-9 | → | NAL-9 | |
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| IL-8 | → | NAL-8 | |
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| IL-7 | → | NAL-7 | |
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| IL-6 | → | NAL-6 | |
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| IL-5 | → | NAL-5 | |
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| IL-4 | → | NAL-4 | |
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| IL-3 | → | NAL-3 | |
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| IL-2 | → | NAL-2 | |
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| IL-1 | → | NAL-1 |
The logics in this IL-NAL family all belong to the "term logic" tradition, which uses subject-predicate sentences and syllogistic rules (while the predicate logic tradition use predicate-argument sentences and truth-functional rules).
In an inheritance statement, a subject term and a predicate term are linked together by a copula called "inheritance". Inheritance is defined by being reflexive and transitive, and interpreted as the generalization-specialization relation.
The language of IL-1 contains inheritance statements as sentences.
A non-empty and finite set of statements in IL-1, K, can be treated as the experience for a system using the logic.
The single inference rule of IL-1 corresponds to the transitivity of inheritance. The transitive closure of K is the system's knowledge, K*.
Given K, a statement is true if it is either in K* or is a tautology, otherwise it is false.
IL-1 accepts Closed-World Assumption.
Given K, the extension of a term includes its known specializations; its intension includes its known generalizations. The two together form the meaning of the term.
The above definitions of truth-value and meaning form an Experience-Grounded Semantics (EGS).
The system implementing IL-1 can answer simple questions by searching its knowledge.
The key point is to define evidence for an inheritance statement. Example: Raven Paradox.
Theorem: an inheritance statement is equivalent to the inclusion of the extension of the subject in the extension of the predicate, as well as to the inclusion of the intension of the predicate in the intension of the subject.
Therefore, an inheritance statement summarizes many pairs of inheritance statements, each of which provides a piece of positive evidence.
NAL uses an idealized experience in IL to define its semantic notions, while the actual experience of the system contains Narsese sentences.
Each judgment in the system has an evidential base in the system's experience. Now we can re-visit Raven Paradox to see where the problem is.
NAL-1 also achieves a unification in the representation of uncertainty: randomness comes from extension; fuzziness comes from intension; ignorance comes from the future.