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.
There is a partial isomorphism between first-order NAL and higher-order NAL.
Overall, NAL has four basic copulas that are directly recognized by the inference rules. They all represent a certain exchangeability ("can be used as") relation between terms, and the syllogistic rules correspond to the transitivity of the copulas involved.
The higher-order copulas are not defined by truth tables, 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.
In IL-5, a consistency requirement is added on the experience of NARS.
There are three types of negation in IL-NAL: meta-level (CWA), between terms (difference), and on statement (negation, opposite statement).
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 by dominating negative evidence of a substatement. 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.
PL treats inference as purely truth-functional, while IL as syllogistic, depending on the transitivity of the copulas. 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.
In IL and NAL, statements with the same truth-value are not necessarily equivalent.
In all three systems, the Deduction Theorem holds. In NAL it takes the truth-value into account.
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.