Logic and Mathematica: Do logical structures exist? (In the sense explained below)

Question

Mathematica is the study of mathematical structures, where they are sets whose members are: a set, a set of functions, and a set of relations. So, this is the object of study of mathematical science, for example, geometry study a mathematical structure known as space. From the concept of structure we get the concept of signature, in the sense that a signature is a set of constants, functors (function symbols) and relators (relation symbols). But when I tried to construct a mathematical theory, this symbols are not all the symbols which I need, I need of other symbols, the so called "logical signs". These are: Variables, logical connectives, quantifiers, and sometimes descriptors. But what is a Variable?, What is a logical connective?, what is a quantifier? Now, I think that in same way we have a signature for mathematical structures, maybe we have signatures for logical structures, but what are these?

Indeed, that of the existence of logical structures is a supposition, that is the reason of my question, I hope now the context of the question be enough clear for you.

Answer 1

Welcome to MSE!

The "logical symbols" you're referring to are definitely studied in the abstract. First and foremost, this is the kind of thing people mean when they talk about "other logics".

In the usual theory of First Order Logic, people study what you can say about a structure using symbols from a signature as well as the logical symbols $\land$, $\lnot$, $\forall x$, etc. This is the most commonly studied logic, because it is extremely flexible, and satisfies extremely useful theorems (compactness and lowenheim-skolem, for instance).

However, there's no reason we have to study this logical system. For some variants, we could study

• Infintary logic, which allows infinite conjunctions/disjunctions like $\bigvee_{n \in \mathbb{N}} \varphi_n$. This lets you express countable-infinite-ness by adding countably many constant symbols $c_n$ and adding the axiom $\forall x . \bigvee_{n \in \mathbb{N}} x = c_n$. (Can you show this is not expressible in first order logic?)
• Higher order logic, which allows us to quantify not only over our model, but over subsets of our model. This lets us write well-ordered-ness, for instance, as $\forall A \subseteq X . A \neq \emptyset \to \exists x \in A . \left ( \forall y \in A . x \leq y \right )$. (Can you show that this is not expressible in first order logic?)
• We can add quantifiers like $\exists^\infty x$ or $\exists^{n \text{ mod } p} x$ which express not only that some $x$ exists, but how many there are. (Again, can you show that $\exists^\infty x $ is not expressible in first order logic?)
• etc.

There's even more "axiomatic" treatments, though! For instance, in abstract model theory, which roughly discusses a structure with a symbol "$\models$" and axioms that make $\models$ function similarly to semantic entailment in classical model theory. You can read more in the (very readable) paper by Barwise Axioms for Abstract Model Theory.

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I hope this helps ^_^