NARS Tutorial

NAL-5 to NAL-6: Higher-order Inference

1. Higher-order statement

When a statement is taken as a term, it can be used to form "higher-order" statements, i.e., statements about statements. On the other hand, a term can name a statement. In this way, the actual difference between statements and other terms is semantic: a statement has a truth-value.

There are relations that take statements as components, such as "know" and "believe", as well as attributes of statements, such as "necessary" and "possible". In NAL, most of these relations are not built in (as in epistemic logic or modal logic), but acquired.

There is no need to further separate 2nd-order, 3rd-order, etc., nor to limit the maximal order allowed in structure.

Two higher-order copulas, implication and equivalence, are defined between statements, indicating the relation between their meanings and truth-values.

There is a partial isomorphism between first-order NAL and higher-order NAL. See Table 9.1 of the textbook.

Overall, NAL has four basic copulas that are directly recognized by the inference rules. They all represent a certain substitutability ("can be used as") relation between terms, and the syllogistic rules correspond to the transitivity of the copulas involved.

2. Derivation as implication

In NAL, implication is defined by derivation. This agrees with the Deduction Theorem in classical logic.

The higher-order copulas in IL/NAL are not defined purely by truth functions, as in propositional calculus. Here the two statements involved not only need to have truth-value relation, but also semantic relation in their contents, which is provided by the syllogistic nature of term logic.

Semantic relation among the components is also required in conjunction and disjunction.

Consequently, NAL is like a relevance logic, though it provides relevance using the intrinsic nature of term logic, rather than using multiple-world semantics.

Using Deduction Theorem, the truth-value of a statement can be taken as the truth-value of a corresponding implication statement, conditioned on the available evidence. Using this meta-level equivalence, some new inference rules can be introduced into NAL, as variants of the existing rules.

3. Negative statement

IL-5 still makes CWA, though it explicitly expresses negative statements, especially as substatements of compound statements.

In IL-5, a consistency requirement is added on the experience of NARS.

Positive and negative statements are not symmetric in NARS, either in the logic part or the control part. Negative observation comes from failed expectation. In NARS, negation is introduced when a statement has more negative than positive evidence. For a syllogistic rule, two negative premises cannot derive a conclusion.

Equivalent statements with negations in IL may have different truth-values in NAL, due to the different amounts of evidence.

4. PL, IL and NAL

IL uses connectors similar to those in Propositional Logic (PL) to build compound statements, and the connectors satisfy similar truth-conditions.

PL treats inference as purely truth-functional, while IL treats inference as substitutions. Truth-conditions in PL are definitional and primary, but derived and secondary in IL. In IL, statements with no semantic relations won't be used as components in a compound, nor as premises in an inference step. Also, in IL (and NAL) statements with the same truth-value are not necessarily equivalent.

IL is a limit case of NAL, when AIKR can be omitted. The unconditional truths (theorems) in IL are embedded in the (structural) inference rules of NAL, though not in the beliefs of the system. There is no axiom (nor theorem) in NAL. The analytic truths are only acknowledged and accepted at the meta-level.

A PL theorem becomes an IL theorem after the connective to copula replacement, if in the former there is semantic relations among the premises and conclusions. How to turn an IL theorem or rule into a NAL inference rule needs to be analyzed one by one, since different truth-value functions may be needed. By default, a true statement in IL is treated as a judgment with full positive evidence in NAL.

5. Reasoning with variable term

A variable term in NARS does not name a concept, and its name is local to a statement. Variable terms can be used to separate extensional and intensional evidence, as well as to indicate the existence of positive evidence.

[There are 8 basic forms of variable-involved statement, corresponding to the distinctions extensional-intensional, positive-negative, and having-lacking. The extensional cases are similar to the 4 figures in Aristotle's Syllogistic (A-E-I-O). The square of opposition represents immediate inference rules in IL. Also, there are rules based on the symmetry of I and E. It is possible to make IL partially isomorphic to Aristotle's Syllogistic, though this approach is not taken for the sake of simplicity.]

Variable related inference include

6. Symbolic reasoning

A variable term is a symbol that represents a constant term. It can be grounded on the latter via variable elimination. Similarly, abstract notions can be introduced or created, without grounding into empirical experience. Instead, formal models or axiomatic theories are built around these notions, with binary deductive rules. Inference within the theory is theorem proving.

When such a formal model is applied to a practical problem, model-theoretic semantics is applied to provide an interpretation to map the abstract notions into concrete concepts, so as to get derived conclusions efficiently. The notions in an formal model are "symbols" whose meaning depends on the interpretation. This is not the case for the ordinary terms in the system.

NAL can use acquired relations like define and represent to learn (or create) a language, and to relate it to its empirical concepts, respectively. NARS can emulate an arbitrary logic, by representing its truth-values and propositions as terms, and its inference rules as implication statements.

[Production systems like Soar can also implement arbitrary inference rules, though NARS is different from these systems, given its other inference rules, memory structure, and control mechanism.]

NARS uses a similar idea to handle natural language. Some preliminary results are discussed in this paper.

[The ability of formal inference is related to Piaget's notion of "formal operation".]

[The analytic vs. empirical distinction can be re-established in this way. It is related to modal logic and possible-world theory. In NARS, "analytic" is with respect to a set of assumptions made in an abstract theory, while "empirical" is with respect to the system's experience.]

[7. Traditional and classical logics re-evaluated]

IL-6 is comparable with Syllogistic and FOL.

Key differences between IL-6 and classical term logic:

Key differences between IL-6 and classical predicate logic:

IL gets ideas from set theory, propositional logic, and predicate logic.

Description logic can be seen as a combination of set theory and predicate logic, though not very deep.

NAL is built by fitting IL onto AIKR, with multi-strategy inference, both strong (deductive) and weak (non-deductive).

Many previous discussions on rationality take classical logics and probability theory as normative models with universal authority. The same happens to the psychological study of human reasoning.

Peirce initially introduced the deduction-induction-abduction trio in term logic at the inference rule level, though later use the terms informally at the inference process level, aimed at the cognitive functions. Now they are usually formalized in predicate logic, so the rules become under-specified.

After all, the differences from NAL and other logics come from AIKR.