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.

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*^{-}.

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

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*}, ⟨

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:- The Confirmation Paradox does not appear here, because the Equivalence Condition is no longer held in NAL-1.
- Wason's Selection Task can be re-analyzed similarly. The common response is not a fallacy if truth-value depends on both positive and negative evidence.
- Popper's proposed asymmetry between falsification and verification can be criticized in the same way: on its assumption that a "theory" is logically a universally quantified proposition.]

[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.]

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*.

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.]

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.