Logical primitives (MI)

Particles of family MI (all particles starting with m-) act as predicate words, and some of them are defined directly in logical notation with the goal to express logical primitives that the grammar itself don’t cover.

\[ \begin{align} \text{ma}(c,e) &= \text{is-atom}(e) \\ \text{mai}(c,e) &= \top \\ \text{mao}(c,E,A) &= \forall x_0 \dots \forall x_n. \ A(c,x_0,\dots,x_n) \Rightarrow E(c,x_0,\dots,x_n) \\ \text{mui}(c) &= \text{unknown} \\ \text{mue}(c,e) &= c = e \\ \text{mua}(c,e,A) &= A(e) \\ \end{align} \]

ma (intransitive) wraps the primitive concept of its argument being an atom.

mai (intransitive) puts no constraint on its argument, and is true for any possible $(e). However to be used $(e) must exist (at some point an existential variable must be created), and for some definitions expressing this existence and nothing more is useful.

mao (intransitive) allows the speaker to express the concept that $(E) is a subset of $(A), in the sense that any list of arguments that satisfy $(E) also satisfy $(A). It is useful as it doesn’t require the language itself to support a variable number of arguments. Instead both $(E) and $(A) are just considered predicates of possibly unknown arity and only the implication is relevant. $(E) and $(A) can be of different arities, in which arity mismatch resolution can be used to give them identical arity. This word can be useful for some definitions, or to express that a predicate $(E) represents multiple combinations of values that make $(A) true, that answers a question represented by $(A).

mui (intransitive) always returns the unknown truth value.

mue (intransitive) and mua (transitive) allows the speaker to interact with the usually hidden context argument. mue accepts a 1-ary predicate that is true when provided the context as an explicit argument, while mua takes a proposition $(A) and an atom $(e) such that $(A) is true when $(e) is provided as the hidden context argument. These predicate words are what allows the context argument to be really useful: the grammar automatically forwards the context parameter for the speaker, which can then be used by predicates.

Other MI particles are not logical wrappers, they are simply root words included in MI for convenience and will be explained in a later chapter.