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Language (linguistics)

The expressiveness of language comes from the ability to compose, from known, simpler signs, into new, more complex signs.

unimplemented! (center-block)

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The example sentence is composed of only 5 words, and the meaning of the sentence is derived exactly from the meaning of each individual word.

Definition

We shall model this composition with a binary operation (•) : Sign → Sign → Sign, pronounced merge.

(•) is defined with four distinct merge operations (•_S, •_L, •_M, •_P) that operate on each individual strata.

\begin{matrix} [\begin{matrix} σ_{1} \\ λ_{1} \\ μ_{1} \\ π_{1} \end{matrix}] ∙ [\begin{matrix} σ_{2} \\ λ_{2} \\ μ_{2} \\ π_{2} \end{matrix}] = [\begin{matrix} σ_{1} ∙_{S} σ_{2} \\ λ_{1} ∙_{L} λ_{2} \\ μ_{1} ∙_{M} μ_{2} \\ π_{1} ∙_{P} π_{2} \end{matrix}] \end{matrix}

Formally,

a language: is a set of signs.
a grammar: is a set of signs (called the grammar's lexicon) together with finite set of functions that operate on signs. Often, grammars for natural languages at least include (•).

A given grammar inductively generates a language. I choose to express this as you would in a language like Idris 2.This would be a better fit for Grammatical Framework.

Lexicon : Type

data MyLanguage : Type where
     FromLexicon : Lexicon -> MyLanguage
     Merge : MyLanguage -> MyLanguage -> MyLanguage

Example of merge

À propos natural languages, $∙_{P}$ (the phonological merge operation) is typically space-separated concatenation. That is, $x ∙_{P} y = x ⋄ blank ⋄ y$ , where $⋄$ denotes concatenation. Remember that $∙_{P}$ is working at the sign-level, not the grapheme/token-level; otherwise, we'dn't need the space.