What is the rigorous definition of a symbol?

Question

In mathematical logic, there is a lot of talk about syntax. For example, a logic textbook may have the symbols $\forall$ and $\vee$, which represent the universal quantifier and disjunction, respectively. But, what is the formal and rigorous definition of those symbols? I don't think that, for instance, the universal quantifier is simply the shape of the letter $A$ inverted. True, that is how it is written in a math book, but that is merely how it is written, not what the universal quantifier actually is. So, my question is, what is the formal and rigorous set-theoretic definition of a symbol? Or, does it not even matter, and the symbols can be any distinct entities?

Answer 1

I like what Wilfred Hodges writes about this on page 1 of his book Model Theory. Here's the full quote:

Nevertheless there is something that the reader may find unsettling. Model theorists are forever talking about symbols, names and labels. A group theorist will happily write the same abelian group multiplicatively or additively, whichever is more convenient for the matter in hand. Not so the model theorist: for him or her the group with '$\cdot$' is one structure and the group with '$+$' is a different structure. Change the name and you change the structure.

This must look like pedantry. Model theory is an offshoot of mathematical logic, and I can't deny that some distinguished logicians have been pedantic about symbols. Nevertheless there are several good reasons why model theorists take the view that they do. For the moment let me mention two. In the first place, we shall often want to compare two structures and study the homomorphisms from one to the other. What is a homomorphism? In the particular case of groups, a homomorphism from group $G$ to group $H$ is a map that carries multiplication in $G$ to multiplication in $H$. There is an obvious way to generalise this notion to arbitrary structures: a homomorphism from structure $A$ to structure $B$ is a map which carries each operation of $A$ to the operation with the same name in $B$.

Secondly, we shall often set out to build a structure with certain properties. One of the maxims of model theory is this: name the elements of your structure first, then decide how they should behave. If the names are well chosen, they will serve both as a scaffolding for the construction, and as raw materials.

Aha - says the group theorist - I see you aren't really talking about written symbols at all. For the purposes you have described, you only need to have formal labels for some parts of your structures. It should be quite irrelevant what kinds of thing your labels are; you might even want to have uncountably many of them.

Quite right. In fact we shall follow the lead of A. I. Mal'tsev and put no restrictions at all on what can serve as a name. For example any ordinal can be a name, and any mathematical object can serve as a name of itself. The items called 'symbols' in this book need not be written down. They need not even be dreamed.

Answer 2

In Paul Tomassi's Logic, a symbol is defined as a "mark" that has been defined (pg. $116$). In other words, a symbol is a mark, such as $\to$ or $\forall$ or $\exists$, that has been assigned meaning.

For instance, in first-order logic, $\forall$ is a symbol because it is a mark with a unambiguous meaning and purpose. In natural language, it can translate to "for every" or "for all," but formally speaking it is used to quantify the extent to which a subject-predicate sentence applies to the elements in a given domain. Specifically, it specifies that a subject-predicate sentence applies to all elements in a given domain. It is this carefully defined meaning that makes $\forall$ a symbol and more than just a "mark," or some arbitrary lines on a sheet of paper that touch each other.