NARS Tutorial

NAL-1: The Core Logic

1. An inheritance logic

A term is an internal identifier in NARS, and its simplest form is a word in a given alphabet.

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*.

[The logics in the IL-NAL family all belong to the "term logic" tradition, which uses subject-predicate sentences and syllogistic rules, while the predicate logic tradition uses predicate-argument sentences and truth-functional rules.]

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). [EGS is very different from Model-theoretic semantics, while similar to Proof-theoretic semantics.]

The system implementing IL-1 can answer simple questions by searching its knowledge.

2. Evidence in NAL-1

NAL-1 is obtained by extending IL-1 according to AIKR. The key point is to define evidence for an inheritance statement.

In an axiomatic system, the truth-value of a statement is determined by its relation with the axioms, and indicates its relation with state of affairs in a model; in an non-axiomatic system, the truth-value of a statement is determined by its relation with evidence, and indicates its relation with the experience of the system.

Evidence is (input or derived) information that has impact on the truth-value of a statement in an inconclusive manner, while proof decides the truth-value a statement in a conclusive manner. Evidence can be positive or negative, or a mixture of them. Evidence comes to the system one piece at a time.

A quantitative representation is necessary for an adaptive system, since the amount of evidence matters when a selection is made among competitive conclusions. The advantage of a numerical measurement of evidence is its generality, not its accuracy. Furthermore, an interpretation of the measurement is required, which usually defines the measurement in an idealized situation.

A IL-1 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 evidence.

In ideal situations (ignoring fuzziness, inaccuracy, etc.), amount of evidence can be represented by a pair of non-negative integers. It can be generalized into a pair of non-negative real numbers w+ and w-. We use w for "all available evidence", which is the sum of w+ and w-.

3. The semantics of NAL-1

When truth is evaluated according to experience, "truth-value" measures evidential support, and shows degree of belief of the system. Though in principle, the information is already carried by the amount of evidence, very often a relative and bounded measurement is preferred.

A natural indicator of truth is the frequency (proportion) of positive evidence in all evidence, that is, f = w+ / w. The limit f, if exists, is the probability for the statement. However, from the value of f alone, whether its limit exists cannot be determined, not to mention where it is.

In an open system, all frequency values may be changed by new evidence, and this is a major type of uncertainty — ignorance about the future frequency value. [Related approaches include higher-order probability, probability interval, imprecise probability, Dempster-Shafer theory, etc, though none of them fully satisfies the needs of a non-axiomatic system.]

While frequency compares positive and negative evidence, a second measurement, confidence, can compare past and future evidence, in the same manner. Here the key idea is to only consider to a constant horizon in the future, that is, c = w / (w + k). A high confidence value means the statement is supported by more evidence, so less sensitive to new evidence. It doesn't mean that the statement is "closer to the reality", or the frequency is "closer to the true probability".

The ⟨frequency, confidence⟩ pair can be used as the truth-value of a statement in a non-axiomatic system. It is fully defined on available evidence, without any assumption about future evidence. Also, it captures the uncertainty caused by negative evidence and future evidence.

The frequency f value will be in the interval [lower, upper] in the near future specified by the horizon k.

The three representations: {w+, w}, ⟨f, c⟩, and [l, u] can be transformed into each other. They all represent the system's degree of belief on the statement, or its evidential support.

For the normal statements in an non-axiomatic system, the amount of supporting evidence is finite. There are two limit cases that are discussed in meta-language only: null (zero) evidence and full (infinite, or no future) evidence.

NAL uses an idealized experience in IL to define its semantic notions, while the actual experience of the system contains Narsese sentences. At the input/output interface, the truth-value of a statement can also be represented imprecisely. If there are N verbal labels, the [0, 1] interval can be divided into N equal-width subintervals. Or, default values can be used, so that the users can omit the numbers.

There are reasons to use high-accuracy representations inside the system, while allow low-accuracy representations outside the system.

[Many traditional problems are solved by this treatment of evidence and truth: [NAL-1 also achieves a unification in the representation of uncertainty: randomness comes from extension; fuzziness comes from intension; ignorance comes from the future.]

[It is not easy to revise predicate logic for this purpose, while to do it in a term logic is easy and natural, given the subject-copula-predicate structure.]

4. Revision and choice

Each judgment in the system has an evidential base in the system's experience.

The revision rule combines distinct evidential bases for the same statements. The amount of (positive or negative) evidence is the same of those of the premises.

[Compared to Dempster's rule of combination: there are different interpretations of "evidence combination". In NARS, it is a form of average, while in D-S theory, it is a form of conjunction.]

[NAL tolerates inconsistency in knowledge, though it is different from the existing paraconsistent logic and belief revision.]

The choice rule chooses among competing answers to a question.

For an evaluative question, the statement with a higher confidence value is preferred; for a selective question, the statement with a higher expectation value is preferred.

Given two competing answers, one with a confirmation record of 19 out of 20, and the other by all n cases, when we should prefer the latter when n gets larger? It depends on the value of k.

5. Forward inference

A syllogistic rule requires the two premises to share one term, and produces a conclusion between the other two terms.

For the inheritance copula, the two premises have four possible combinations, and only one of them corresponds to a valid rule in IL-1. In NAL-1, all can be valid when associated with a proper truth-value function.

Truth-value functions are determined according to the semantics, by treating the involved measurements as extended Boolean variable.

A variant of syllogistic rule is a rule for immediate inference, which only takes one premise.

An inference rule of NAL can be either "strong" or "weak", depending on whether it converges to an inference rule in IL. [This distinction is similar to the traditional distinction between "deductive" and "inductive" inference, or between "explicative" and "ampliative" inference.]

6. Backward inference

A question and a judgment can be used as premises to derive another question, if and only if the answer of the derived question and the judgment can be used as premises to derive an answer to the original question.

The syllogistic rules of NAL-1 are reversible, in the sense that to exchange a conclusion and a premise leads to another rule.

Because of the reversibility of the rules, the rule tables for forward and backward inference are the same, except truth-values.

The conclusion of any rule can be used as a premise of any other rule. To avoid circular inference, the two premises must have distinct evidential bases.