Alternatives to pure quantifier logic 
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*Are there some alternatives for pure quantifier logic? 
Pure quantifier logic is axioms and rules of inference added to proposition logic to result first order logic.

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*Are there other axioms that do not use notions "for all" or "there exists", but can be applied to extend propositional logic, and can capture many things (at least quantification over $\Bbb R^n$) captured by pure quantifier logic.


*Is it possible to use limits to define quantifiers (that would allow at least quantification over $\Bbb R^n$)?
I know that limits are defined in terms of quantifiers, but can they be redefined somehow?
 A: In general, it is possible to make a foundational program that is based on a logic different (modal logics, or other more exotic systems) from classical first-order logic. 
But the classical framework has been judged the most convenient (or least inconvenient) for most people, and was designed to be the most natural way to express mathematical statements (such as the definition of limit). 
It doesn't mean that it has been completely taken for granted, so people have investigated the rules of logic in the foundational questions, with constructive mathematics refusing the law-excluded middle. More to the point, combinatory logic has been designed precisely to remove variable quentification in mathematical logic, and thus investigate the precise power of quantifiers.
Now even for someone who would refuse all previous justifications, there are strong results like soundness, completeness and compactness that show the uniqueness of FOL (see Lindström theorem).
For your precise example, in category theory, you can define notions in terms of diagrams, and have arrows to replace quantification over $\epsilon, \delta$. Other notions like being injective are similarly characterized by diagrams/universal properties. Category theory and logic serve different purposes and inform each other quite well (see the Curry-Howard-Lambek correspondence, and categorical logic for instance).
