So, abstraction is simply a form of generalization, which might be understood as a general and diffuse process of binding collections of things to an identifier. Linguistically, it is when graphemes map to morphemes in a succinct fashion. For instance, 'x' is a letter and can stand for a semantic expression, proposition, or even passage. From WP:
Abstraction in its main sense is a conceptual process wherein general rules and concepts are derived from the usage and classification of specific examples, literal ("real" or "concrete") signifiers, first principles, or other methods. "An abstraction" is the outcome of this process—a concept that acts as a common noun for all subordinate concepts and connects any related concepts as a group, field, or category.
That means, logic is essentially the study of the patterns of inference, and when it comes to formal systems of logic, that generally means focusing on a highly symbolized version of logic where variables, relations, and values are represented with a very bare syntax which is distinct but related to semantics in that it represents the idea economically. This has implications because the human mind is capable of keeping so much in mind at a time, a psychological concept known as chunking. Fewer symbols allows for better retention, and forms the basis for mnemonic strategies. Let's take a simple example:
Socrates is in the kitchen. If Socrates is in the kitchen, we can conclude he is in his house, and likewise, if he is not in the house, we can conclude he is not in the kitchen. If he is in the house, and we know no more, we cannot conclude with certainty he is in the kitchen. He might be in his bedroom. And if he's not in the kitchen, we cannot conclude whether or not he is or is not in the house. This is a very true and meaningful text, and guides our every day thinking, when looking for Socrates, or anyone really.
But now, it just so happens that this can be expressed as the following:
- p then q
- ~q then ~p
- q then p or ~p
- ~p the q or ~q
The first example of our text is expressed in a very concrete way and in natural language. The second example is expressed in a succinct, abstract way and is the beginning of an artificial language (though languages are usually conceived of as having more complex grammars).
The mystery is why does this work? How is it we can distill natural language to a terse, symbolic form, perform transformations on the expressions, and then arrive at a form that translates back into natural language with an often guaranteed conclusion? And even more importantly, how can we continue to build formal systems that more closely approximate human reasoning?
One way is by observing more semantic features in argumentation and adding operators. For instance, moving from sentential logic to first-order predicate logic, it becomes possible not just to represent propositions with abstractions, but predication within in the proposition. One can now consider elements of linguistic modality of natural language by creating modal logics in the broad sense. And if one observes that mathematical theorization is largely an exercise in construction of language, one again can emulate natural language more fully by dropping some of the classic laws that were once seen as Laws of Thought. Intuitionistic logic is one such program (and my personal bias). As one continues to study examples of inferences, one can see deduction, induction, and abduction, too, as abstractions. And eventually, one gets away from the toy constructions of propositional logic, and has to admit there's a fair amount of defeasible reasoning that occurs in informal logic which tends to be much more domain-specific in terms of inferences. The Toulmin model functions much more in the semantic domain than in the syntactic one.
So, it's best to think of logic as ranging over the study of very natural, informal discourse that humans use and that is related to common-sense (which has serious implications regarding semantic relationships of linguistic categories) to very artificial, formal methods that accommodate description by mathematical functions (which does it's best to ignore semantics). Perhaps one of the most challenging sorts of formal systems are those of model theory which encompass abstract algebraic techniques combined with formal logical methods resulting in ways of coming to conclusions about a collection of structures whose inputs and outputs are the truth conditions. A supreme example of intellectual challenge that results in trying to process abstraction are Kripke semantics.