# 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 : 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.

## 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.