# Semantics

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* : tıqra_{w}(*x*). ∃*e*. τ(*e*) < t ∧ nuo_{w}(*x*)(*e*) ∧ nıe_{w}(*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.

## Models

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.