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, corresponding to the distinctions extensional-intensional, positive-negative, 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
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.
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.
Key differences between IL-6 and classical term 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.