From syntax to semantics

So far, we have been working with languages as a bunch of symbols of their alphabets put one after the other in various ways. This allowed us to specify, for a given language, which formulas belong to the language, and which formulas do not. However, something we have not done yet is specify the meaning of these formulas. Clearly, we use languages to convey ideas about various topics. We do this through well-formed formulas of the language at hand that have a specific meaning. This is true of formal langugages as it is true for natural ones like English.

Some of the languages above already came with some previously understood meaning. For example, for the formulas of $\mathcal{L}_{AE}$, the language of arithmetic expressions, we already know what they mean, at least intuitively, through our knowledge of mathematics. Similarly, for $\mathcal{L}_f$, we already knew what those formulas meant given our knowledge of how a computer works, and more specifically, how their file system is usually structured.

On the other hand, for langugages like $\mathcal{L}_0$, that you may not know, we specified which formulas belong to the langugage, and which formulas do not, but so far, we have not given them any meaning, which would tell us what ideas we can convey with these formulas. In the next part of the book (after a brief detour), this is precisely what we will be doing.

Technically, what we have been doing so far is specifying the syntax of our languages. Syntax is the way expressions are formulated in a language. In the next part, we will be dealing with the semantics of some languages, which is specifying the various ways in which these expressions have, or can be given, meaning.

However, before we begin talking about meaning, we have to learn a bit about set theory. In modern mathematics, set theory is a foundational discipline, since many different mathematical structures can be formulated in it. Indeed, our semantics will be formulated in set theory. Thus, we will go through some fundamental aspects of set theory, those we will need to continue on our journey through formal logic.