Toaq is a loglang, which means that from any sentence, we can unambiguously derive its meaning in logic notation. Syntax describes how this process works; semantics describes how to interpret the result.
The refgram tells you that Luı nuo sá tıqra nîe náokua translates to ∃x : tıqraw(x). ∃e. τ(e) < t ∧ nuow(x)(e) ∧ nıew(e, náokua). The reality is that this isn't "just" logic notation: it's a very specific notation that has been purpose-built for describing natural language semantics, and this article will help you understand the core concepts behind it.
To help us reason about meaning more directly, mathematicians have come up with the idea of a model: a mathematical object that tells us exactly how to interpret statements in a given formal language. In its most basic form, a model has three parts:
- A signature, which is the set of all words and symbols found in the language, along with their syntactic properties.
- A domain, which is the set of all objects, functions, relations, etc. that the language is capable of representing.
- An interpretation, which is a function defining which symbols correspond to which elements of the domain.
For example, consider the language of basic arithmetic. A model for this language might look like this:
As it turns out, Toaq is a formal language too, which means we can reason about it using models. Now, being a human language, Toaq's semantics are quite a bit more complicated than that of arithmetic, but luckily for us, models are a pretty flexible concept, and we can extend them with extra features as we need them.